TASK LIST...YEAR 7 MATHEMATICS o f W e s t ern A u s r a l i a I n c . T MA WA h e M a t h e m a t i...

YEAR 7 MATHEMATICS

o f Wes te rn Austra l ia In

c.

MA WAThe

Mathematical Association

Task 11:Task 12:Task 13:Task 14:Task 15:Task 16:Task 18:Task 19:Task 24:Task 25:Task 26:

Workshop SettingsOrderingRules for CalculatingMaking Calculations EasierRearranging NumbersEstablishing LawsRounding DecimalsSquare NumbersRatiosEquationsScoring Golf

Task 27: Task 28: Task 30: Task 32: Task 37: Task 38: Task 40: Task 101: Task 102: Task 105:

Fraction GraphicsPercentagesDiscountsMoving PointsFraction OperationsFraction ActionGraphing RelationshipsConsecutive NumbersLarge ElevensIdeal Fractions

TASK LISTTASK LIST

YEAR 7 MATHEMATICSNumber & Algebra TasksSet 2

PRODUCED BY A DEPARTMENT OF EDUCATION - MAWA PARTNERSHIP PROJECT

WRITTEN FOR THE YEAR 7 AUSTRALIAN CURRICULUM

PRODUCED BY A DEPARTMENT OF EDUCATION - MAWA PARTNERSHIP PROJECT

WRITTEN FOR THE YEAR 7 AUSTRALIAN CURRICULUM

© Department of Education, Western Australia (2015)© Department of Education, Western Australia (2015) © Department of Education, Western Australia (2015)© Department of Education, Western Australia (2015)

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MATHS7TL002 | Mathematics | Number and Algebra Activity | Workshop Settings 2 © Department of Education WA 2015

TASK 11: WORKSHOP SETTINGS

Overview

In this task, students are guided in their examination of a relationship between two variables.

The process by which relationships are examined is central to this investigation. Students

should be encouraged to look at relationships by using diagrams, tables, graphs, worded

descriptions and symbolic representation.

Students will need

No special requirements

Relevant content descriptions from the Western Australian Curriculum

Create algebraic expressions and evaluate them by substituting a given value for each

variable (ACMNA176)

Investigate, interpret and analyse graphs from authentic data (ACMNA180)

Students can demonstrate

understanding when they

o use a graph of plotted points to represent the connection between the numbers of

tables and the numbers of chairs

o determine the algebraic expressions linking the numbers of tables and the

numbers of chairs

o see the relationship between squares and square roots

o recognise different ways of determining the answer

reasoning when they

o explain why it does not make sense to join the points on the graph

problem solving when they

o independently provide several representations of the relationship between the

numbers of tables and the numbers of chairs in the final activity.


WORKSHOP SETTINGS Solutions and Notes for Teachers

1. Chris sets up the seminar room for workshops and Leah has asked that all the tables and

chairs be arranged as shown in the diagram.

A chair is shown as and a table as

(a) Give a reason to explain the fact that there are no chairs at one end of the two tables.

Maybe to leave room to see the presenter.

(b) If there were 4 tables used and the arrangement above was repeated, how many

chairs in total would be needed?

12 chairs

(c) As the number of tables increases by 2, how does the number of chairs increase?

Describe how the answer to this question can be quickly determined.

As the number of tables increases by 2, the number of chairs increases by 6.

It is easily seen in the diagram that there are 6 chairs for every 2 tables.

(d) If there were 20 tables set out using this arrangement, how many people could be

accommodated at the workshop?

There are 10 sets of 2 tables so 10 x 6 chairs = 60 chairs.

(e) Complete the table showing numbers of tables and numbers of chairs.

Number of tables 2 4 6 8 10 12 14 16 18 20

Number of chairs 6 12 18 24 30 36 42 48 54 60

http://creativecommons.org/licenses/by-nc-sa/3.0/au/











(f) Plot the points representing number of chairs and tables as in the table above.

(g) Describe the pattern seen in the location of the points.

The points lie in a straight line

(h) Does it make sense to join up the points you have plotted? Explain.

It does NOT make sense to join the points because that would imply the values

between also exist. In this case there is no situation for 3 tables.

(i) Describe (in words) the link between the number of chairs and the number of tables.

Number of chairs = 3 x number of tables

(j) Using the symbols h to represent the number of chairs and b to represent the number

of tables, write the rule linking h and b.

h = b x 3 OR h = 3b OR h = 3 x b

(k) For the rule you have developed in (j) describe the types of numbers that h and b can

represent.

For b, the numbers it represents are even, positive, integers; and for a workshop these

numbers should not be very large (less than 100).

For h, the numbers it represents are even, positive, integers, multiples of 2, 3, 6 and

composite; and should not be more than 300 for a workshop.


(l) Show how you can use your rule to determine the number of chairs needed for 100

tables.

h = b x 3 = 100 x 3 = 300

(m) How many tables would you need for 72 people?

h = b x 3 so b x 3 = 72 and b = 24

2. In another room suitable for workshops the tables have a different shape and are arranged

as shown in the diagram below.

For this arrangement, describe the link between the number of chairs and the number of

tables in the following ways:

(a) As a table of values:

Number of tables 2 4 6 8 10 12 14 16 18 20

Number of chairs 8 16 24 32 40 48 56 64 72 80

(b) As a series of plotted points:

a series of plotted points












(c) As a rule described in words:

For every two tables there are 8 people.

There 4 people per table.

(d) As a rule described in mathematical symbols:

If h represents the number of chairs and b represents the number of tables,

h = b x 4 OR h = 4b OR h = 4 x b


STUDENT COPY WORKSHOP SETTINGS

Chris sets up the seminar room for workshops and Leah has asked that all the tables and

chairs be arranged as shown in the diagram.

A chair is shown as and a table as

(a) Give a reason to explain the fact that there are no chairs at one end of the two tables.

(b) If there were 4 tables used and the arrangement above was repeated, how many

chairs in total would be needed?

(c) As the number of tables increases by 2, how does the number of chairs increase?

Describe how the answer to this question can be quickly determined.

(d) If there were 20 tables set out using this arrangement, how many people could be

accommodated at the workshop?












(e) Complete the table showing numbers of tables and numbers of chairs.

Number of tables 2 4 6 8 10 12 14 16 18 20

Number of chairs 6

(f) Plot the points representing number of chairs and tables as in the table above.

(g) Describe the pattern seen in the location of the points.

(h) Does it make sense to join up the points you have plotted? Explain.

(i) Describe (in words) the link between the number of chairs and the number of tables.


(j) Using the symbols h to represent the number of chairs and b to represent the number

of tables, write the rule linking h and b.

(k) For the rule you have developed in (j) describe the types of numbers that h and b can

represent.

(l) Show how you can use your rule to determine the number of chairs needed for 100

tables.

(m) How many tables would you need for 72 people?

2. In another room suitable for workshops the tables have a different shape and are arranged

as shown in the diagram below.

For this arrangement, describe the link between the number of chairs and the number of

tables in the following ways:

(a) A table of values:

Number of tables 2 4 6 8 10 12 14 16 18 20

Number of chairs












(b) As a series of plotted points:

(c) As a rule described in words:

(d) As a rule described in mathematical symbols:

MATHS7TL002 | Mathematics | Number and Algebra Activity | Ordering 1 © Department of Education WA 2015



TASK 12: ORDERING

Overview

For this activity students need to use their knowledge and understanding of the relative sizes

of positive and negative numbers including fractions and decimals.

Students will need

no material needed


Compare fractions using equivalence. Locate and represent positive and negative

fractions and mixed numbers on a number line (ACMNA152)

Connect fractions, decimals and percentages and carry out simple conversions

(ACMNA157)


fluency when they

o move flexibly between decimal and fractional representations of numbers as they

attempt to order numbers


o recognise equivalence of decimals and fractions

o can create comparative statements of numbers as in Activity 1 (b)-(d)

reasoning when they

o explain errors in the students‟ thinking in Activity 2


o can place a set of numbers in numerical order as in Activity 1 (h) and (j)


ORDERING Solutions and Notes for Teachers

Activity 1

1. Kate, Liz and Jon were given a set of twelve numbers to write in numerical order. The

table below shows the original numbers, Kate‟s list, Liz‟s list and Jon‟s list.

Original numbers Kate‟s list Liz‟s list Jon‟s list

1 0.75 0 -0.7 0

2 0 -0.7 -1

3

-1 -1.8

4 -0.7 -1.8 -

5 0.079 -

0 -0.7

6

0.079

-1

7 0.5678 0.5678

-1.8

8

0.75

-

9 -

1.7 0.079 0.079

10 -1.8

0.5678 0.5678

11 -1

0.75 0.75

12 1.7

1.7 1.7

(a) What is meant by „in numerical order‟? Give an example.

Order of size from smallest to greatest; e.g., 5, 7, 9, 10


(b) According to Kate‟s list,

>

Is this statement TRUE or FALSE? TRUE

Create another five “greater than” (>) statements from Kate‟s list and identify for each

one if they are TRUE or FALSE.

-

> 0 FALSE

> 0.5678 TRUE

> 0 TRUE

1.7 > 0.75 TRUE

-1 > 0 FALSE

(c) From Liz‟s list create five “less than” (<) statements that Liz would make and identify for

each one if they are TRUE or FALSE.

-0.7 < -1 FALSE

-1 < -1.8 FALSE

FALSE

<

TRUE

TRUE

(d) Jon has indicated that 1.7 > 0.5678, and this is TRUE.

Locate five more TRUE statements according to Jon‟s list.

1.7 > 0.75

>

1.7 > 0

>

Locate five FALSE statements according to Jon‟s list.

-1 > 0

-1.8 > -0.7

0.079 >

-1 >

>


(e) Kate and Jon both think that 0 is the smallest number in the list. Are they correct?

Explain.

No, zero is larger than any negative numbers.

(f) For each of the three persons, identify, describe and show by examples from the list, at

least ONE error in their thinking about the order of numbers.

Kate:

Kate thinks 0 is smaller than all negative numbers because she has put 0 as the

smallest number.

All Kate‟s negative numbers are out of order because she thinks -1.8 > -1 and has

ignored the difference that the negative sign makes.

Kate‟s decimals and fractions are correct but they have been separated into two

sections which is not correct because

is less than 0.75. She has put all the fractions

as being greater than all the decimals.

Liz:

Liz makes the same mistake as Kate with regard to the negative numbers. All Liz‟s

negative numbers are out of order because she thinks -1 > -0.7 and has ignored the

difference that the negative sign makes.

Liz‟s decimals and fractions are correct but they have been separated into two sections

which is not correct because

is greater than 0.079. She has put all the decimals as

being greater than all the fractions.

Jon

Jon makes a mistake when he puts all the fractions first, then all the negative numbers

then all the decimals. According to his list

is less than -1, which is incorrect.

Jon thinks that 0 is less than all negative numbers, which is incorrect.

Jon‟s negative numbers are in the reverse order because -1 is not less than -1.8

(g) Write the list of numbers in the correct order.

-1.8 -

-1 -0.7 0 0.079

0.5678

0.75 1.7

(h) Display the numbers in their approximate positions on the number line.

(i) Write a list of rules that provide instructions for placing numbers on a number line.

Discuss your list with another student and refine if necessary.


Zero is left of any positive numbers

To the right of zero the numbers are in the counting order with the decimals and

fractions placed according to their size.

All the negative numbers are to the left of zero.

To locate negative numbers, ignore the negative sign so the numbers start at 0 and

go left as they “increase”.

Positive fractions less than 1 go on the right between 0 and 1 while all negative

fractions are left of 0.

Decimals are placed in order of size; positive ones go right of 0 and the greater they

are the further right they will be. If they are negative, they are left of 0 and the smaller

they are, the further left they will be. Example -5 is further left than -4.

(j) Use your rules to order these ten numbers.

0.8 0 -0.81

-1.8 1.08 -

0.0008

-1.8 -0.81 -

0 0.0008

0.8

1.08

(k) Create a similar list of 12 numbers (include decimals, fractions and negative numbers)

and give them to another student to place in ascending order.

Various answers as appropriate.


Activity 2

Identify the errors made by each of these students when they placed numbers into ascending

order.

Situation 1

Numbers to order 0.99 0.9 0.9099 0.09999 0.909

Student‟s answer 0.9 0.99 0.909 0.9000 0.09999

Error in thinking The more digits there are after the decimal point, the greater the number.

Correct order 0.09999 0.9000 0.9 0.909 0.99

Situation 2

Numbers to order

Student‟s answer

Error in thinking The greater the denominator, the greater the fraction.

Correct order

Situation 3

Numbers to order -4 -7 -21 -5 -8 -9

Student‟s answer -4 -5 -7 -8 -9 -21

Error in thinking Negative numbers increase in the same order as if the negative sign is not there.

Correct order -21 -9 -8 - 7 -5 -4

Situation 4

Numbers to order 56.103 56.303 56.031 56.014 56.150 56.320

Student‟s answer 56.103 56.014 56.150 56.031 56.320 56.303

Error in thinking Zeroes after the decimal point have no value

Correct order 56.014 56.031 56.103 56.150 56.303 56.320


STUDENT COPY ORDERING

Activity 1

1. Kate, Liz and Jon were given a set of twelve numbers to write in numerical order. The

table below shows the original numbers, Kate‟s list, Liz‟s list and Jon‟s list.

Original numbers Kate‟s list Liz‟s list Jon‟s list

1 0.75 0 -0.7 0

2 0 -0.7 -1

3

-1 -1.8

4 -0.7 -1.8 -

5 0.079 -

0 -0.7

6

0.079

-1

7 0.5678 0.5678

-1.8

8

0.75

-

9 -

1.7 0.079 0.079

10 -1.8

0.5678 0.5678

11 -1

0.75 0.75

12 1.7

1.7 1.7

(a) What is meant by „in numerical order‟? Give an example.


(b) According to Kate‟s list,

>

Is this statement TRUE or FALSE?

Create another five “greater than” (>) statements from Kate‟s list and identify for each

one if they are TRUE or FALSE.

(c) From Liz‟s list create five “less than” (<) statements and identify for each one if they are

TRUE or FALSE.

(d) Jon has indicated that 1.7 > 0.5678 and this is TRUE.

Locate five more TRUE statements according to Jon‟s list.

Locate five FALSE statements according to Jon‟s list.


(e) Kate and Jon both think that 0 is the smallest number in the list. Are they correct?

Explain.

(f) For each of the three persons, identify, describe and show by examples from the list, at

least ONE error in their thinking about the order of numbers.

Kate:

Liz:

Jon:


(g) Write the list of numbers in the correct order.

(h) Display the numbers in their approximate positions on the number line.

(i) Write a list of rules that provide instructions for placing numbers on a number line.

Discuss your list with another student and refine if necessary.

(j) Use your rules to order these ten numbers.

0.8 0 -0.81

-1.8 1.08 -

0.0008

(k) Create a similar list of 12 numbers (include decimals, fractions and negative numbers)

and give them to another student to place in ascending order.


Activity 2

Identify the errors made by each of these students when they placed numbers into ascending

order.

Situation 1

Numbers to order 0.99 0.9 0.9099 0.09999 0.909

Student‟s answer 0.9 0.99 0.909 0.9000 0.09999

Error in thinking

Correct order

Situation 2

Numbers to order

Student‟s answer

Error in thinking

Correct order

Situation 3

Numbers to order -4 -7 -21 -5 -8 -9

Student‟s answer -4 -5 -7 -8 -9 -21

Error in thinking

Correct order

Situation 4

Numbers to order 56.103 56.303 56.031 56.014 56.150 56.320

Student‟s answer 56.103 56.014 56.150 56.031 56.320 56.303

Error in thinking

Correct order

MATHS7TL002 | Mathematics | Number and Algebra Activity | Rules for Calculating 2 © Department of Education WA 2015

TASK 13: RULES FOR CALCULATING

Overview

In this task students will have the opportunity to investigate and review / establish the

commutative law of arithmetic and to determine when the rule applies. Students should

attempt as many calculations as possible without using calculators. This task builds on Year

6 curriculum content and it is not necessary to have covered any other Year 7 content.

Students will need

Calculators to check answers for a few activities.


Apply the associative, commutative and distributive laws to aid mental and written

computation (ACMNA151)



o describe and summarise the commutative rule

reasoning when they

o determine that the commutative rules also apply to decimals and fractions


RULES FOR CALCULATING Solutions and Notes for Teachers

In this task, there will be a series of activities in which the known rules for calculating are

reviewed and other rules are developed. Calculators should only be used to check answers

or where indicated by the following symbol.

Activity 1

(a) Complete the following table by entering the results of the calculations given.

1 21 + 8 = 39 8 + 21 = 39

2 42 + 7 = 49 7 + 42 = 49

3 8 + 66 = 74 66 + 8 = 74

4 15 + 16 = 31 16 + 15 = 31

5 22 + 32 = 54 32 + 22 = 54

6 18 + 10 = 28 10 + 18 = 28

7 100 + 200 = 300 200 + 100 = 300

8 180 + 1000 = 1180 1000 + 180 = 1180

(b) In every row, there are two sets of numbers to add. What do you notice about these

sets of numbers?

The numbers are the same but the order is different.

(c) What do you notice about the answers in each row?

In each row the answers are the same.

(d) Write a sentence to describe the rule that is highlighted in these examples.

When two whole numbers are added, the order in which the numbers are written or

added does not influence the answer.


(e) Does your rule apply to decimals? Write down three examples to support your decision.

Yes. Examples:

0.6 + 0.4 = 1 and 0.4 + 0.6 = 1

1.3 + 1.2 = 2.5 and 1.2 + 1.3 = 2.5

10.1 + 11.5 = 21.6 and 11.5 + 10.1 = 21.6

(f) Does your rule apply to fractions? Write down three examples to support your decision.

Yes. Examples:

+ = 1 and + = 1

+ = and + =

+ = and + =

(g) Does the rule work when you add -

(i) a fraction to a whole number; e.g., 3 + ? Yes

(ii) a decimal to a whole number; e.g., 0.2 + 3? Yes

(iii) a decimal to a fraction; e.g., 0.5 + ? Yes

(iv) three numbers; e.g., 14 + 13 +1 2? Yes

(h) Write a number sentence to give another example of each type given in (g)

Examples:

(i) 1 + = + 1 =

(ii) 9.3 + 10 = 10 + 9.3 = 19.3

(iii) 0.5 + = + 0.5 = 0.75 or

(iv) 1 + 2 + 3 = 3 + 2 + 1

(i) Write a conclusion for this activity.

When two or more whole, fractional or decimal numbers are added the answer is the same

regardless of the order in which the numbers are added (or written).


Activity 2

(a) Complete the following table be entering the results of the calculations given.

1 3 x 4 = 12 4 x 3 = 12

2 5 x 10 = 50 10 x 5 = 50

3 20 x 10 = 200 10 x 20 = 200

4 6 x 11 = 66 11 x 6 = 66

5 12 x 20 = 240 20 x 12 = 240

6 8 x 7 = 56 7 x 8 = 56

7 9 x 7 = 63 7 x 9 = 63

8 150 x 10 = 1500 10 x 150 = 1500

(b) Write a sentence to describe the rule that has applied to the multiplication of these

whole numbers.

Multiplying one number by a second number gives the same answer as if you multiply

the second number by the first number.

(c) Does your rule apply to decimals? Write down five examples to support your decision.

Examples: 0.1 x 0.2 = 0.2 x 0.1 = 0.02

1.2 x 0.4 = 0.4 x 1.2 = 0.48

2.3 x 1.2 = 1.2 x 2.3 = 2.76

0.9 x 1.5 = 1.5 x 0.9 = 0.45

5.6 x 2.4 = 2.4 x 5.6 = 13.44

(d) Does your rule apply to fractions? Write down five examples to support your decision.

Examples: x = x =

x = x =

x = x =

x = x =

x = x =


Activity 3


1 20 – 8 = 12 8 – 20 = -12

2 32 – 20 = 8 20 – 32 = -8

3 4 – 50 = -46 50 – 4 = 46

4 7 – 73 = -66 73 – 7 = 66

5 81 – 90 = -9 90 – 81 = 9

6 45 – 100 = -55 100 – 45 = 55

7 68 – 12 = 56 12 – 68 = -56

8 36 – 13 = 23 13 – 36 = -23

(b) The rule for subtraction of two numbers is different from the rules for addition and

multiplication. Explain how the rule is different.

The order in which the numbers are written influences the answer.

Subtracting the second number from the first number gives a different answer from

subtracting the first number from the second number.

(c) Looking at the pattern in your answers, describe a way to find the answer when a

larger whole number is subtracted from a smaller whole number.

Take the smaller number from the larger number and use a negative sign with your

result; e.g., for 5 – 56, take 5 from 56 (51) and use a negative sign with the 51.

So 5 – 56 = -51

(d) Use your method from part (c) to determine the following answers mentally and THEN

use your calculator if necessary to check your answer.

(i) 90 – 110 = -20

(ii) 63 – 100 = -37

(iii) 45 – 90 = -45

(iv) 8 – 70 = -62


Activity 4

1. Complete the following table by entering the results of the calculations given.Express your answers as both fractions and decimals.

1 20 ÷ 8 = = 2.5 8 ÷ 20 = = 0.4

2 25 ÷ 4 = 6.25 4 ÷ 25 = 0.16

3 64 ÷ 4 = 16 4 ÷ 64 = 0.0625

4 50 ÷ 100 = = 0.5 100 ÷ 50 = 2

5 15 ÷ 60 = = 0.25 60 ÷ 15 = 4

6 6 ÷ 30 = = 0.2 30 ÷ 6 = 5

7 7 ÷ 70 = = 0.1 70 ÷ 7 = 10

8 8 ÷ 10 = 0.8 10 ÷ 8 = 1.25

2. Does order matter when dividing?

Yes

Activity 5

When a mathematical operation is applied to two numbers, does the order in which the two

numbers are written make a difference? Write a paragraph which provides an answer to this

question, explains when it is true and when it is not true, and which provides evidence for

your conclusion.

When a mathematical operation is applied to two numbers, the order in which the two

numbers are written makes a difference during subtraction and division but not for addition

and multiplication. Adding 5 to 3 is the same as adding 3 to 5 but subtracting 3 from 5 is not

the same as subtracting 5 from 3. Multiplying 7 by 6 gives 42 and this is the same as

multiplying 6 by 7. However, in division the order matters because 100 divided by 10 is 10,

but 10 divided by 100 is 0.1. This rule applies for all numbers - whole numbers, fractions and

decimals.


STUDENT COPY RULES FOR CALCULATING

In this task, there will be a series of activities in which the known rules for calculating are

reviewed and other rules are developed. Calculators should only be used to check answers

or where indicated by the following symbol.

Activity 1


1 21 + 8 = 8 + 21 =

2 42 + 7 = 7 + 42 =

3 8 + 66 = 66 + 8 =

4 15 + 16 = 16 + 15 =

5 22 + 32 = 32 + 22 =

6 18 + 10 = 10 + 18 =

7 100 + 200 = 200 + 100 =

8 180 + 1000 = 1000 + 180 =

(b) In every row, there are two sets of numbers to add. What do you notice about these

sets of numbers?

(c) What do you notice about the answers in each row?

(d) Write a sentence to describe the rule that is highlighted in these examples.


(e) Does your rule apply to decimals? Write down three examples to support your decision.

(f) Does your rule apply to fractions? Write down three examples to support your decision.

(g) Does the rule work when you add -

(i) a fraction to a whole number; e.g., 3 + ?

(ii) a decimal to a whole number; e.g., 0.2 + 3

(iii) a decimal to a fraction; e.g., 0.5 + ?

(iv) three numbers; e.g.,14 + 13 +1 2

(h) Write a number sentence to give another example of each type given in (g)

(i)

(ii)

(iii)

(iv)

(i) Write a conclusion for this activity.


Activity 2

(a) Complete the following table be entering the results of the calculations given.

1 3 x 4 = 4 x 3 =

2 5 x 10 = 10 x 5 =

3 20 x 10 = 10 x 20 =

4 6 x 11 = 11 x 6 =

5 12 x 20 = 20 x 12 =

6 8 x 7 = 7 x 8 =

7 9 x 7 = 7 x 9 =

8 150 x 10 = 10 x 150 =

(b) Write a sentence to describe the rule that has applied to the multiplication of these

whole numbers.

(c) Does your rule apply to decimals? Write down five examples to support your decision.

(d) Does your rule apply to fractions? Write down five examples to support your decision.


Activity 3


Use calculators to check your answers if necessary.

1 20 – 8 = 8 – 20 =

2 32 – 20 = 20 – 32 =

3 4 – 50 = 50 – 4 =

4 7 – 73 = 73 – 7 =

5 81 – 90 = 90 – 81 =

6 45 – 100 = 100 – 45 =

7 68 – 12 = 12 – 68 =

8 36 – 13 = 13 – 36 =

(b) The rule for subtraction of two numbers is different from the rules for addition and

multiplication. Explain how the rule is different.

(c) Looking at the pattern in your answers, describe a way to find the answer when a

greater whole number is subtracted from a smaller whole number.

(d) Use your method from part (c) to determine the following answers mentally and THEN

use your calculator if necessary to check your answer.

(i) 90 – 110 =

(ii) 63 – 100 =

(iii) 45 – 90 =

(iv) 8 – 70 =


Activity 4

Complete the following table by entering the results of the calculations given.

Express your answers as both fractions and decimals.

1 20 ÷ 8 = 8 ÷ 20 =

2 25 ÷ 4 = 4 ÷ 25 =

3 64 ÷ 4 = 4 ÷ 64 =

4 50 ÷ 100 = 100 ÷ 50 =

5 15 ÷ 60 = 60 ÷ 15 =

6 6 ÷ 30 = 30 ÷ 6 =

7 7 ÷ 70 = 70 ÷ 7 =

8 8 ÷ 10 = 10 ÷ 8 =

2. Does order matter when dividing?

Activity 5

When a mathematical operation is applied to two numbers, does the order in which the two

numbers are written make a difference? Write a paragraph which provides an answer to this

question, explains when it is true and when it is not true, and provides evidence for your

conclusion.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Making Calculations Easier 2 © Department of Education WA 2015

TASK 14: MAKING CALCULATIONS EASIER

Overview

This task involves using the associative rule for addition and multiplication. Knowing the rule

and its name is not expected. Rather, it is important that students can use the process and

can appreciate its usefulness in making calculations much easier.

Associative rule:

Adding 3 + 4 + 5 gives the same answer whether you add the 3 and the 4 first, or the 4 and

the 5 first. Thus, (3 + 4) + 5 = 3 + (4 + 5).

Multiplying 2 x 4 x 10 gives the same answer whether you multiply the 2 and the 4, and then

multiply your answer by 10, OR multiply the 4 and the 10 and then multiply this answer by 2.

Thus, (2 x 4) x 10 = 2 x (4 x 10).

Students should be familiar with commutativity (even if not the word) and

appreciate that 3 + 4 = 4 + 3, as well as 5 x 2 = 2 x 5

Students will need

Students should attempt to do all calculations without a calculator




Multiply and divide fractions and decimals using efficient written strategies and digital

technologies (ACMNA154)


fluency when they

o represent fractions and decimals in various ways

reasoning when they

o apply the associative and commutative laws to numbers


o complete Activities 6 and 7


MAKING CALCULATIONS EASIER Solutions and Notes for Teachers

The purpose of this task is to make calculations easier by rearranging numbers in ways that

can be shown to work. You may recall that when numbers are multiplied, changing the order

of the numbers does not change the answer.

Activity 1

Consider

6 x 2 = 12

60 x 2 must be 10 times the answer above because 60 = 10 times 6, and 60 = 6 x 10 x 2

600 x 2 must be 100 times 12 because 600 is a hundred times larger than 6.

1. Without using a calculator, use the patterns and the above information to complete the

table below.

6 x 2 = 12 60 x 2 = 120 600 x 2 = 1200 6000 x 2 = 12 000

7 x 2 = 14 70 x 2 = 140 700 x 2 = 1400 7000 x 2 = 14 000

8 x 2 = 16 80 x 2 = 160 800 x 2 = 1600 8000 x 2 = 16 000

9 x 2 = 18 90 x 2 = 180 900 x 2 = 1800 9000 x 2 = 18 000

10 x 2 = 20 100 x 2 = 200 1000 x 2 = 2000 10 000 x 2 = 20 000

15 x 2 = 30 150 x 2 = 300 1500 x 2 = 3000 15 000 x 2 = 30 000

18 x 2 = 36 180 x 2 = 360 1800 x 2 = 3600 18 000 x 2 = 36 000

15 000 x 2 can also be written as -

15 x 10 x 10 x 10 x 2

15 x 10 x 100 x 2

15 x 1000 x 2

2. Write 65 000 x 2 using the same pattern.

65 x 10 x 10 x 10 x 2

65 x 10 x 100 x 2

65 x 1000 x 2


Activity 2

1. This approach to multiplication can be used with numbers other than 2.

Use the method outlined above to complete the table below.

It may help to refer back to the first entry in each row.

2. Consider number sentences with 6000 x 3:

6000 x 3 = 6 x 100 x 10 x 3

6 x 100 x 10 x 3 = 6 x 10 x 100 x 3 (for this exercise considered the same)

6000 x 3 = 2 x 3000 x 3 (is another example)

3. Complete four more different number sentences for 6000 x 3.

6000 x 3 = 6 x 1000 x 3

6000 x 3 = 3 x 2000 x 3

6000 x 3 = 3 x 2 x 100 x 10 x 3

6000 x 3 = 3 x 200 x 30

4. Compare your list with that of another student. Identify similarities and differences.

Summarise your observations.

6 3 = 18 60 3 = 180 30 6 = 180 600 3 = 1800 6000 3 = 18 000

7 4 = 28 70 4 = 280 40 7 = 280 700 4 = 2800 7000 4 = 28 000

9 5 = 45 90 5 = 450 50 9 = 450 900 5 = 4500 9000 5 = 45 000

3 8 = 24 30 8 = 240 80 3 = 240 300 8 = 2400 3000 8 = 24 000

6 6 = 36 60 6 = 360 6 60 = 360 600 6 = 3600 6000 6 = 36 000

4 9 = 36 40 9 =360 90 4 =360 400 9 =3600 4000 9 = 36 000

7 8 = 56 70 8 = 560 80 7 = 560 700 8 = 5600 7000 8 = 56 000


Activity 3

1. Complete each of these tables using mental arithmetic.

(a)

0.5 x 2 = 1 0.5 x 20 = 10 0.5 x 200 = 100 0.5 x 2000 = 10 000

1.5 x 4 = 6 1.5 x 40 = 60 1.5 x 400 = 600 1.5 x 4000 = 6000

0.8 x 5 = 4 0.8 x 50 = 40 0.8 x 500 = 400 0.8 x 5000 = 4000

2.5 x 8 = 20 2.5 x 80 = 200 2.5 x 800 = 2000 2.5 x 8000 =20 000

1.4 x 5 = 7 1.4 x 50 = 70 1.4 x 500 = 700 1.4 x 5000 = 7000

2.8 x 5 = 14 2.8 x 50 = 140 2.8 x 500 = 1400 2.8 x 5000 = 14 000

14.5 x 2 = 29 14.5 x 20 = 290 14.5 x 200 = 2900 14.5 x 2000 = 29 000

(b) Hint: Think of the first number in each row in words.

For Row 1, what is 8 quarters?

For Row 8, what number is formed if you have 9 thirds?

x 8 = 2 x 80 = 20 x 800 = 200 x 8000 = 2000

x 8 = 4 x 80 = 40 x 800 = 400 x 8000 = 4000

x 6 = 2 x 60 = 20 x 600 = 200 x 6000 = 2000

x 4 = 6 x 40 = 60 x 400 = 600 x 4000 = 6000

x 8 = 10 x 80 = 100 x 800 = 1000 x 8000 = 10 000

6 x = 15 60 x = 150 600 x = 1500 6000 x = 15 000

7 x = 1 70 x = 10 700 x = 100 7000 x = 1000

9 x = 3 90 x = 30 900 x = 300 9000 x = 3000

2. Describe a step-by-step process (in words) to determine the answer to x 80 000.

Multiply 2 by 8 (gives 16)

Multiply a quarter by 8 (gives 2)

Add 16 plus 2 (gives 18)

Multiply 18 by 10 000 gives 180 000


Activity 4

The process can be used in reverse to “break down” a calculation and make it easier.

Example: 25 x 2000 = 25 x 2 x 1000 = 50 x 1000 = 50 000

1. “Break down” these calculations as shown in the example given and calculate the

answers.

x 2000 = x 2 x 1000 = 17 x 1000 = 17 000

2.1 x 6000 = 2.1 x 6 x 1000 = 12.6 x 1000 = 12.6 x 10 x 100 = 126 x 100 = 12 600

0.9 x 300 = 0.9 x 3 x 100 = 2.7 x 100 = 270

x 12 000 = x 12 x 1000 = 3 x 1000 = 3000

25 x 4000 = 25 x 4 x 1000 = 100 x 1000 = 100 000

33 x 900 = 33 x 9 x 100 = 297 x 100 = 29 700

44 x 20 000 = 44 x 2 x 10 000 = 88 x 10 000 = 880 000

1.3 x 3000 = 1.3 x 3 x 1000 = 3.9 x 1000 = 3 900

x 60 000 = x 6 x 10 000 = 15 x 10 000 = 150 000

2. Now create some to give to another student to “break down” and answer.

Various answers


Activity 5

The process can also be used when numbers are added. You may recall that when numbers

are added, changing the order of the numbers does not change the answer.

Example: 67 + 73 = 60 + 7 + 70 + 3 = 60 + 70 + 7 + 3 = 130 + 10 = 140

Break down these additions so that the answer is easily calculated.

681 + 128 = 600 + 60 + 1 + 100 + 20 + 8 = 700 + 80 + 9 = 789

+ + = + + + + + = 9 + 1 = 10

3.2 + 1.4 + 16.3 = 3 + 1 + 16 + 0.2 + 0.4 + 0.3 = 20 + 0.9 = 20.9

1010 + 7081 = 1000 + 7000 + 10 + 80 + 1 = 8091

196 + 214 = 100 + 200 + 90 + 10 + 10 = 300 + 100 + 10 = 410

+ + = + + + + + = 14 + = 14

892 + 618 = 800 + 600 + 90 + 10 + 2 + 8 = 1400 + 100 + 10 = 1510

41.7 + 55.6 + 21.4 + 80.3 = 40 + 50 + 20 + 80 + 1 + 5 + 1 + 0.7 + 0.6 + 0.4 + 0.3

= 190 + 7 + 2 = 199

+ 11 + + = + 11 + + + + + + = 28 + 2 = 30

Activity 6

In the mathematical puzzle called DocDoc the user is given a focus number.

The user has to identify the possible whole numbers which add to the focus number.

The numbers used can only be the numbers 1 to 9.

Examples: If the focus number is 7 and addition is the activity, then some possibilities are -

1 + 6, 2 + 5, 3 + 4, 1 + 2 + 4, 1 + 3 + 3, 1 + 1 + 5, 1 + 2 + 2 + 2, 1 + 1 + 2 + 3

Rearranging is not seen as a different possibility. 1 + 3 + 3 = 3 + 3 + 1 [same]

This gives 3 ways of adding 2 numbers, 3 ways of adding 3 numbers and 1 way of adding 4

numbers for a sum of 7.


Using the same rules, determine the numbers of ways that the following focus numbers can

be made with 2, 3, 4 or 5 numbers and record your answers in the table provided (only the

digits 1 to 9 can be used). Then write all the equations for each set.

Number of ways of adding

Focus number 2 numbers 3 numbers 4 numbers 5 numbers

5 2 2 1 1

8 4 5 5 2

9 4 6

16 2

17 1 13

24 0 3

27 0 1

5 = 3 + 2 = 1 + 4 = 1 + 1 + 3 = 2 + 2 + 1 = 1 + 1 + 2 + 1 = 1 + 1 + 1 + 1 + 1

8 = 1 + 7 = 6 + 2 = 5 + 3 = 4 + 4 = 1 + 2 + 5 = 1 + 3 + 4 = 1 + 1 + 6 = 2 + 2 + 4 = 3 + 3 + 2

8 = 2 + 2 + 2 + 2 = 1 + 3 + 1 + 3 = 1 + 1 + 1 + 5 = 1 + 1 + 2 + 4 = 1 + 2 + 2 + 3

8 = 1 + 1 + 1 + 1 + 4 = 1 + 1 + 1 + 2 + 3 =

9 = 1 + 8 = 2 + 7 = 3 + 6 = 4 + 5 = 2 + 3 + 4 = 1 + 2 + 6 = 1 + 3 + 5 = 1 + 1 + 7 = 1 + 4 + 4

9 = 2 + 2 + 5

16 = 8 + 8 = 7 + 9

17 = 8 + 9 = 1 + 8 + 8 = 1 + 7 + 9 = 2 + 8 + 7 = 2 + 6 + 9 = 3 + 9 + 5 = 3 + 8 + 6 = 3 + 7 + 7

17 = 4 + 6 + 7 = 4 + 5 + 8 = 4 + 4 + 9 = 5 + 9 + 3 = 5 + 6 + 6 = 5 + 7 + 5

24 = 7 + 8 + 9 = 8 + 8 + 8 = 9 + 9 + 6

27 = 9 + 9 + 9


Activity 7

In the same mathematical puzzle called DocDoc the user can be given a focus number with

a different operation.

The user has to identify the possible whole numbers which multiply to produce the

focus number.

If the focus number is greater than 100, then the number 1 cannot be used. The

numbers used can then only be the numbers 2 to 9.

Examples: If the focus number is 14 then possibilities are 2 x 7 and 1 x 2 x 7

This gives 1 way with 2 numbers and 1 way with 3 numbers.


be made with 2, 3, or 4 numbers and record your answers in the table provided (only the


Number of ways

Focus number 2 numbers 3 numbers 4 numbers

6 2 2 2

12 2 3 3

24 2 3 5

81 1 2 3

100 0 1 2

216 0 3 5

224 0 1 2

6 = 2 x 3 = 1 x 6 = 1 x 2 x 3 = 1 x 1 x 6 = 1 x 2 x 3 x 1 = 1 x 1 x 1 x 6

12 = 3 x 4 = 2 x 6 = 1 x 2 x 6 = 1 x 3 x 4 = 2 x 2 x 3

12 = 1 x 2 x 2 x 3 = 1 x 1 x 2 x 6 = 1 x 1 x 3 x 4

24 = 6 x 4 = 8 x 3 = 2 x 3 x 4 = 1 x 4 x 6 = 1 x 3 x 8

24 = 1 x 1 x 4 x 6 = 1 x 1 x 3 x 8 = 1 x 2 x 3 x 4 = 2 x 2 x 3 x 2 = 1 x 2 x 2 x 6

81 = 9 x 9 = 1 x 9 x 9 = 3 x 3 x 9 = 1 x 9 x 9 x 1 = 3 x 3 x 3 x 3 = 1 x 3 x 3 x 9

100 = 4 x 5 x 5 = 4 x 5 x 5 x 1 = 2 x 2 x 5 x 5

216 = 9 x 6 x 4 = 9 x 8 x 3 = 6 x 6 x 6

216 = 2 x 3 x 6 x 6 = 3 x 3 x 4 x 6 = 2 x 2 x 6 x 9 = 2 x 3 x 4 x 9 = 3 x 3 x 3 x 8

224 = 7 x 4 x 8 = 7 x 4 x 2 x 4 = 7 x 2 x 2 x 8


STUDENT COPY MAKING CALCULATIONS EASIER

The purpose of this task is to make calculations easier by rearranging numbers in ways that

can be shown to work. You may recall that when numbers are multiplied, changing the order

of the numbers does not change the answer.

Activity 1

Consider

6 x 2 = 12

60 x 2 must be 10 times the answer above because 60 = 10 times 6, and 60 = 6 x 10 x 2

600 x 2 must be 100 times 12 because 600 is a hundred times greater than 6.

1. Without using a calculator, use the patterns and the above information to complete the

table below.

6 x 2 = 12 60 x 2 = 600 x 2 = 6000 x 2 =

7 x 2 = 70 x 2 = 700 x 2 = 7000 x 2 =

8 x 2 = 80 x 2 = 800 x 2 = 8000 x 2 =

9 x 2 = 90 x 2 = 900 x 2 = 9000 x 2 =

10 x 2 = 100 x 2 = 1000 x 2 = 10 000 x 2 =

15 x 2 = 150 x 2 = 1500 x 2 = 15 000 x 2 =

18 x 2 = 180 x 2 = 1800 x 2 = 18 000 x 2 =

15 000 x 2 can also be written as -

15 x 10 x 10 x 10 x 2

15 x 10 x 100 x 2

15 x 1000 x 2

2. Write 65 000 x 2 using the same pattern.


Activity 2

1. This approach to multiplication can be used with numbers other than 2.

Use the method outlined above to complete the table below.

It may help to refer back to the first entry in each row.

2. Consider number sentences with 6000 x 3:

6000 x 3 = 6 x 100 x 10 x 3

6 x 100 x 10 x 3 = 6 x 10 x 100 x 3 (for this exercise considered the same)

6000 x 3 = 2 x 3000 x 3 (is another example)

3. Complete four more different number sentences for 6000 x 3.

4. Compare your list with that of another student. Identify similarities and differences.

Summarise your observations.

6 3 = 60 3 = 30 6 = 600 3 = 6000 3 =

7 4 = 70 4 = 40 7 = 700 4 = 7000 4 =

9 5 = 90 5 = 50 9 = 900 5 = 9000 5 =

3 8 = 30 8 = 80 3 = 300 8 = 3000 8 =

6 6 = 60 6 = 6 60 = 600 6 = 6000 6 =

4 9 = 40 9 = 90 4 = 400 9 = 4000 9 =

7 8 = 70 8 = 80 7 = 700 8 = 7000 8 =


Activity 3

1. Complete each of these tables using mental arithmetic.

(a)

0.5 x 2 = 1 0.5 x 20 = 0.5 x 200 = 0.5 x 2000 =

1.5 x 4 = 1.5 x 40 = 1.5 x 400 = 1.5 x 4000 =

0.8 x 5 = 0.8 x 50 = 0.8 x 500 = 0.8 x 5000 =

2.5 x 8 = 2.5 x 80 = 2.5 x 800 = 2.5 x 8000 =

1.4 x 5 = 1.4 x 50 = 1.4 x 500 = 1.4 x 5000 =

2.8 x 5 = 2.8 x 50 = 2.8 x 500 = 2.8 x 5000 =

14.5 x 2 = 14.5 x 20 = 14.5 x 200 = 14.5 x 2000 =

(b) Hint: Think of the first number in each row in words.

For Row 1, what is 8 quarters?

For Row 8, what number is formed if you have 9 thirds?

x 8 =

x 80 =

x 800 =

x 8000 =

x 8 =

x 80 =

x 800 =

x 8000 =

x 6 =

x 60 =

x 600 =

x 6000 =

x 4 =

x 40 =

x 400 =

x 4000 =

x 8 =

x 80 =

x 800 =

x 8000 =

6 x

= 60 x

= 600 x

= 6000 x

=

7 x

= 70 x

= 700 x

= 7000 x

=

9 x

= 90 x

= 900 x

= 9000 x

=

2. Describe a step-by-step process (in words) to determine the answer to

x 80 000.


Activity 4

The process can be used in reverse to “break down” a calculation and make it easier.

Example: 25 x 2000 = 25 x 2 x 1000 = 50 x 1000 = 50 000

1. “Break down” these calculations as shown in the above example and calculate the

answers.

x 2000

2.1 x 6000

0.9 x 300

x 12 000

25 x 4000

33 x 900

44 x 20 000

1.3 x 3000

x 60 000

2. Now create some to give to another student to “break down” and answer.


Activity 5

The process can also be used when numbers are added. You may recall that when numbers

are added, changing the order of the numbers does not change the answer.

Example: 67 + 73 = 60 + 7 + 70 + 3 = 60 + 70 + 7 + 3 = 130 + 10 = 140

Break down these additions so that the answer is easily calculated.

681 + 128

+ +

3.2 + 1.4 + 16.3

1010 + 7081

196 + 214

+ +

892 + 618

41.7 + 55.6 + 21.4 + 80.3

+ 11 + +

Activity 6

In the mathematical puzzle called DocDoc the user is given a focus number.

The user has to identify the possible whole numbers which add up to the focus number.

The numbers used can only be the numbers 1 to 9.

Examples: If the focus number is 7 and addition is the activity, then the possibilities are:

1 + 6, 2 + 5, 3 + 4, 1 + 2 + 4, 1 + 3 + 3, 1 + 1 + 5, 1 + 2 + 2 + 2, 1 + 1 + 2 + 3

Rearranging is not seen as a different possibility. 1 + 3 + 3 = 3 + 3 + 1 [same]

This gives 3 ways of adding 2 numbers, 3 ways of adding 3 numbers and 1 way of adding 4

numbers for a sum of 7.



be made with 2, 3, 4 or 5 numbers and record your answers in the table provided (only the


Number of ways of adding

Focus number 2 numbers 3 numbers 4 numbers 5 numbers

5

8

9

16

17

24

27


Activity 7

In the same mathematical puzzle called DocDoc the user can be given a focus number with

a different operation.

The user has to identify the possible whole numbers which multiply to produce the

focus number.

If the focus number is greater than 100, then the number 1 cannot be used. The

numbers used can then only be the numbers 2 to 9.

Examples: If the focus number is 14 then the only possibility is 2 x 7.

This gives 1 way with 2 numbers and no ways with 3 numbers.


be made with 2, 3 or 4 numbers and record your answers in the table provided. [Note: Only

the digits 2 to 9 can be used if the focus number is greater than 100]

Number of ways

Focus number 2 numbers 3 numbers 4 numbers

6

12

24

81

100

216

224

MATHS7TL002 | Mathematics | Number and Algebra Activity | Rearranging Numbers 2 © Department of Education WA 2015

TASK 15: REARRANGING NUMBERS

Overview

The aim of this task is to introduce students to the recognition and use of the distributive law,

but the law will not be named nor generalised in these activities. Students should be already

familiar with the commutative and associative laws. These ideas are developed and reviewed

in Tasks 13 and 14.

Students will need

calculators








o determine the equivalence of number expressions

o recognise different ways of expressing numbers


o interpret and find evidence to support statements as in Activity 6


REARRANGING NUMBERS Solutions and Notes for Teachers

This task supports the investigation to learn how numbers can be rearranged to make

calculations easier.

Remember that for addition and multiplication operations the order does not matter if there is

only one type of operation in the number sentence. If addition and multiplication occur in the

same expression (with no brackets) then multiplication is done first.

Examples: 6 + 7 = 7 + 6 5 x 2 = 2 x 5

3 + 8 + 10 = 11 + 10 or 3 + 8 + 10 = 3 + 18 5 x 2 x 4 = 10 x 4 or 5 x 2 x 4 = 5 x 8

Activity 1

1. Consider 61 x 3. Which of the following expressions are equivalent to 61 x 3? Use your

calculator if necessary. Write TRUE or FALSE to show your thinking.

(60 + 1) x 3 TRUE 3 x (60 + 1) TRUE

60 x 3 + 1 x 3 TRUE 60 x 3 x 60 x 3 FALSE

60 x 3 – 1 x 3 FALSE 60 x 3 ÷ 3 FALSE

3 x 60 + 3 x 1 TRUE 3 x 60 x 1 FALSE



(50 + 1 ) x 4 TRUE 4 x (50 + 1) TRUE

50 x 4 + 1 x 4 TRUE 50 x 4 x 50 x 4 FALSE

50 x 4 – 1 x 4 FALSE 50 x 4 ÷ 4 FALSE

4 x 50 + 4 x 1 TRUE 4 x 50 x 1 FALSE



(100 + 1 ) x 7 TRUE 7 x (100 + 1) TRUE

100 x 7 + 1 x 7 TRUE 100 x 7 x 100 x 7 FALSE

100 x 7 – 1 x 7 FALSE 100 x 7 ÷ 7 FALSE

7 x 100 + 7 x 1 TRUE 7 x 100 x 1 FALSE


Activity 2



(100 - 1 ) x 8 TRUE 8 x (100 - 1) TRUE

100 x 8 + 1 x 8 FALSE 100 x 8 x 100 x 8 FALSE

100 x 8 – 1 x 8 TRUE 100 x 8 ÷ 8 FALSE

8 x 100 - 8 x 1 TRUE 8 x 100 x 1 FALSE



(50 - 2 ) x 2 TRUE 2 x (50 - 2) TRUE


50 x 2 – 1 x 2 TRUE 50 x 2 ÷ 2 FALSE

2 x 50 - 2 x 1 TRUE 2 x 50 x 1 FALSE



(70 - 1 ) x 9 TRUE 9 x (70 - 1) TRUE


70 x 9 – 1 x 9 TRUE 70 x 9 ÷ 9 FALSE

9 x 70 - 9 x 1 TRUE 9 x 70 x 1 FALSE

4. Consider 81 x 7. Write 4 TRUE statements and 4 FALSE statements similar to the above.

TRUE FALSE

81 x 7 = (80 + 1) x 7 81 x 7 = 80 x 1 x 7

81 x 7 = 7 x (80 + 1) 81 x 7 = 80 x 7 – 7 x 1

81 x 7 = 80 x 7 + 1 x 7 81 x 7 = 80 x 7 x 1 x 7

81 x 7 = 7 x 1 + 7 x 80 81 x 7 = 10 x 7 x 1 ÷ 7


Activity 3

Calculate the following expressions by firstly creating statements similar to the ones provided

above, thus making the calculations easier. The first two are done for you.

91 x 8 = (90 + 1) x 8 = 90 x 8 + 1 x 8 = 720 + 8 = 728

29 x 7 = (30 – 1) x 7 = 30 x 7 – 1 x 7 = 210 – 7 = 203

21 x 20 = (20 + 1) x 20 = 20 x 20 + 1 x 20 = 400 x 20 = 420

59 x 8 = (60 – 1) x 8 = 60 x 8 – 1 x 8 = 480 – 8 = 472

99 x 4 = (100 – 1) x 4 = 100 x 4 – 1 x 4 = 400 – 4 = 396

399 x 6 = (400 – 1 ) x 6 = 400 x 6 – 1 x 6 = 2400 – 6 = 2394

42 x 8 = (40 + 2) x 8 = 40 x 8 + 2 x 8 = 320 + 16 = 336

121 x 6 = (120 + 1) x 6 = 120 x 6 + 1 x 6 = 720 + 6 = 726

999 x 5 = (1000 – 1) x 5 = 1000 x 5 – 1 x 5 = 5000 – 5 = 4995

1001 x 15 = (1000 + 1) x 15 = 1000 x 15 + 1 x 15 = 15 000 + 15 = 15 015

51 x 11 = (50 + 1) x 11 = 50 x 11 + 1 x 11 = 550 + 11 = 561

599 x 7 =(600 – 1 ) x 7 = 600 x 7 – 1 x 7 = 4200 – 7 = 4193

Activity 4

Write a description of the process you have used in the previous activities and explain how

the calculations are made easier.

The high number to be multiplied is expressed as an addition or subtraction of two numbers.

The numbers are smaller and can be made close to a multiple of 10.

Then the multiplication is applied to each of the numbers in turn.

If the high number to be multiplied is expressed as an addition then the two products are

added. If the high number to be multiplied is expressed as a subtraction then you find the

difference between the two products.


Activity 5

1. If this process works for whole numbers, then should it work for fractions and decimals?

Justify your conclusion.

Yes, it should because they are also numbers which can all be expressed as the addition or

subtraction of two other numbers.

2. Use the process on these decimals and fractions. Two examples are provided.

8 x 1.99 = 8 x (2 – 0.01) = 8 x 2 – 8 x 0.01 = 16 – 0.08 = 15.92

5 x 2 = 5 x (2 + ) = 5 x 2 + 5 x =

9 x 3.01 = 9 x (3 + 0.01) = 9 x 3 + 9 x 0.01 = 27 + 0.09 = 27.09

2 x 3 = 2 x ( 3 + ) = (2 x 3 + 2 x ) = 6 + = 6

3 x 100.02 = 3 x (100 + 0.02) = 3 x 100 + 3 x 0.02 = 300 + 0.06 = 300.06

6 x 10.1 = 6 x (10 + 0.1) = 6 x 10 + 6 x 0.1 = 60 + 0.6 = 60.6

7 x 99.9 = 7 x (100 – 0.1) = 7 x 100 – 7 x 0.1 = 700 – 0.7 = 699.3

8 x 49.9 = 8 x (50 – 0.1) = 8 x 50 – 8 x 0.1 = 400 – 0.8 = 399.2

20 x 8.1 = 20 x (8 + 0.1) = 20 x 8 + 20 x 0.1 = 160 + 2 = 162

20.1 x 8 = (20 + 0.1) x 8 = 20 x 8 + 0.1 x 8 = 160 + 0.8 = 160.8

4 x 10 = (4 + ) x 10 = 4 x 10 + x 10 = 40 + 5 = 45

63.9 x 2 = (64 – 0.1) x 2 = 64 x 2 – 0.1 x 2 = 128 – 0.2 = 127.8

22.002 x 4 = (22 + 0.002) x 4 = 22 x 4 + 0.002 x 4 = 88 + 0.008 = 88.008

11 x 4 = (11 + ) x 4 = 11 x 4 + x 4 = 44 + 2 = 46

39.99 x 6 = (40 – 0.01) x 6 = 40 x 6 – 0.01 x 6 = 240 – 0.06 = 239.94

3 x 89.9 = 3 x (90 – 0.1) = 3 x 90 – 3 x 0.1 = 270 – 0.3 = 269.7

9 x 100.2 = 9 x (100 + 0.2) = 9 x 100 + 9 x 0.2 = 900 + 1.8 = 901.8

Check your expressions and answers.


Activity 6

For each of the following statements, create number sentences to decide if the statement is

(a) possibly true (b) definitely true (c) false. For each example make THREE statements

which are true or find ONE that is false.

1. Adding two numbers together and then subtracting 8 from that total gives the same

answer as subtracting 8 from each original number and then adding the results of these

two subtractions.

40 + 16 = 56 and 56 – 8 = 48

40 – 3 = 32 and 16 – 8 = 8

32 + 8 = 40 and this is not the same as 48 so the statement is FALSE

2. Adding two numbers together and then dividing that total by 2 total gives the same

answer as dividing each original number by 2 and then adding the results of these two

divisions.

10 + 8 = 18 and 18 ÷ 2 = 9 ALSO 10 ÷ 2 = 5 and 8 ÷ 2 = 4 and 5 + 4 = 9

20 + 16 = 36 and 36 ÷ 2 = 18 ALSO 20 ÷ 2 = 10 and 16 ÷ 2 = 8 and 10 + 8 = 18

12 + 14 = 26 and 26 ÷ 2 = 13 ALSO 12 ÷ 2 = 6 and 14 ÷ 2 = 7 and 6 + 7 = 13

Possibly true but we have only checked for a few numbers.

3. Subtracting a small even number from a larger even number and then dividing that total

by 2 total gives the same answer as dividing each original number by 2 and then

subtracting the smaller result from the larger result.

10 - 8 = 2 and 2 ÷ 2 = 1 ALSO 10 ÷ 2 = 5 and 8 ÷ 2 = 4 and 5 - 4 = 1

20 - 16 = 4 and 4 ÷ 2 = 2 ALSO 20 ÷ 2 = 10 and 16 ÷ 2 = 8 and 10 - 8 = 2

12 - 4 = 8 and 8 ÷ 2 = 4 ALSO 12 ÷ 2 = 6 and 4 ÷ 2 = 2 and 6 - 2 = 4

Possibly true, but we have only checked for a few numbers.


STUDENT COPY REARRANGING NUMBERS

This task supports the investigation to learn how numbers can be rearranged to make

calculations easier.

Remember that for addition and multiplication operations the order does not matter if there is

only one type of operation in the number sentence. If addition and multiplication occur in the

same expression (with no brackets) then multiplication is done first.

Examples: 6 + 7 = 7 + 6 5 x 2 = 2 x 5

3 + 8 + 10 = 11 + 10 or 3 + 8 + 10 = 3 + 18 5 x 2 x 4 = 10 x 4 or 5 x 2 x 4 = 5 x 8

Activity 1



(60 + 1 ) x 3 3 x (60 + 1)

60 x 3 + 1 x 3 60 x 3 x 60 x 3

60 x 3 – 1 x 3 60 x 3 ÷ 3

3 x 60 + 3 x 1 3 x 60 x 1



(50 + 1 ) x 4 4 x (50 + 1)

50 x 4 + 1 x 4 50 x 4 x 50 x 4

50 x 4 – 1 x 4 50 x 4 ÷ 4

4 x 50 + 4 x 1 4 x 50 x 1



(100 + 1 ) x 7 7 x (100 + 1)

100 x 7 + 1 x 7 100 x 7 x 100 x 7

100 x 7 – 1 x 7 100 x 7 ÷ 7

7 x 100 + 7 x 1 7 x 100 x 1


Activity 2



(100 - 1 ) x 8 8 x (100 - 1)

100 x 8 + 1 x 8 100 x 8 x 100 x 8

100 x 8 – 1 x 8 100 x 8 ÷ 8

8 x 100 - 8 x 1 8 x 100 x 1



(50 - 2 ) x 2 2 x (50 - 2)

50 x 2 + 1 x 2 50 x 2 x 50 x 2

50 x 2 – 1 x 2 50 x 2 ÷ 2

2 x 50 - 2 x 1 2 x 50 x 1



(70 - 1 ) x 9 9 x (70 - 1)

70 x 9 + 1 x 9 70 x 9 x 70 x 9

70 x 9 – 1 x 9 70 x 9 ÷ 9

9 x 70 - 9 x 1 9 x 70 x 1

4. Consider 81 x 7. Write 4 TRUE statements and 4 FALSE statements similar to the above.

TRUE FALSE


Activity 3

Calculate the following expressions by firstly creating statements similar to the ones provided

above, thus making the calculations easier. The first two are done for you.

91 x 8 = (90 + 1) x 8 = 90 x 8 + 1 x 8 = 720 + 8 = 728

29 x 7 = (30 – 1) x 7 = 30 x 7 – 1 x 7 = 210 – 7 = 203

21 x 20

59 x 8

99 x 4

399 x 6

42 x 8

121 x 6

999 x 5

1001 x 15

51 x 11

599 x 7

Activity 4

Write a description of the process you have used in the previous activities and explain how

the calculations are made easier.


Activity 5

1. If this process works for whole numbers, then should it work for fractions and decimals?

Justify this conclusion.

2. Use the process on these decimals and fractions. Two examples are provided.

8 x 1.99 = 8 x (2 – 0.01) = 8 x 2 – 8 x 0.01 = 16 – 0.08 = 15.92

5 x 2

= 5 x (2 +

) = 5 x 2 + 5 x

=

9 x 3.01

2 x 3

3 x 100.02

6 x 10.1

7 x 99.9

8 x 49.9

20 x 8.1

20.1 x 8

4

x 10

63.9 x 2

22.002 x 4

11

x 4

39.99 x 6

3 x 89.9

9 x 100.2

Check your expressions and answers.


Activity 6

For each of the following statements, create number sentences to decide if the statement is

(a) possibly true (b) definitely true (c) false. For each example make THREE statements

which are true or find ONE that is false.

1. Adding two numbers together and then subtracting 8 from that total gives the same

answer as subtracting 8 from each original number and then adding the results of these

two subtractions.

2. Adding two numbers together and then dividing that total by 2 total gives the same

answer as dividing each original number by 2 and then adding the results of these two

divisions.

3. Subtracting a small even number from a larger even number and then dividing that total

by 2 total gives the same answer as dividing each original number by 2 and then

subtracting the smaller result from the larger result.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Establishing Laws 2 © Department of Education WA 2015

TASK 16: ESTABLISHING LAWS

Overview

In this series of activities students are given the opportunity to identify and test the laws of

arithmetic given in algebraic notation. Students will need to have a sound knowledge and

understanding of operations with whole numbers, decimals and fractions. They also need to

appreciate the role of the variable in an algebraic expression and to understand the values

that variables can assume. Students should be able to calculate all answers without

calculators.

Some students may need

calculators



variable (ACMNA176)

Extend and apply the laws and properties of arithmetic to algebraic terms and

expressions (ACMNA177)


fluency when they

o substitute values into algebraic expression and compute the values of the

expressions


o connect the laws and properties of numbers to algebraic terms and expressions

reasoning when they

o justify their conclusions about statements which are true and false


ESTABLISHING LAWS Solutions and Notes for Teachers

Introduction

In this task, you are given activities which will help you to identify the laws governing

operations with numbers when those laws are written in algebraic terms. In these tasks, the

lower case letters represent variables which can be either positive or negative or zero as well

as being whole numbers, fractions or decimals.

Activity 1: Adding two numbers

When adding two numbers represented by a and b the rule is a + b = b + a

Using any numbers for a and b, write down examples of true statements which follow this

rule. Provide examples as described, and all examples should be different.

The first one is done for you. Answers will vary.

Example a b a + b = b + a

Add two decimals 0.4 1.6 0.4 + 1.6 = 1.6 + 0.4

Add two whole numbers 5 6 5 + 6 = 6 + 5

Add two prime numbers 7 11 7 + 11 = 11 + 7

Add two odd numbers 1 9 1 + 9 = 9 + 1

Add two multiples of 3 6 12 6 + 12 = 12 + 6

Add two fractions +

Add two negative numbers -6 -5 -6 + -5 = -5 + -6

Add two factors of 6 3 2 3 + 2 = 2 + 3

Add a whole number to a decimal 4 2.7 4 + 2.7 = 2.7 + 4

Add a fraction and a decimal 0.3 + 0.3 = 0.3 +

Add a negative and a positive 8 -3 8 + -3 = -3 + 8

Add a fraction to a whole number 11 11 + = + 11

Add two perfect squares 9 25 9 + 25 = 25 + 9

Add a prime and a composite 3 12 3 + 12 = 12 + 3

Add two 4-digit numbers 2315 1436 2315 + 1436 = 1436 + 2315


Activity 2: Subtracting two numbers

Generally a – b b – a

1. Interpret this statement and explain what it means.

If you subtract a first number from a second number, the answer is not equal to

subtracting the second number from the first. This statement is true: The order in

which the numbers are written and subtracted is important.

2. Give three examples that support this statement.

3 – 5 5 – 3

8 – 4 4 – 8

10 – 5 5 – 10

3. Give an example of a situation when the statement is false.

If a = b then a – b = b – a

e.g., 3 – 3 = 3 – 3

4. Is this statement generally true for fractions? Justify your conclusion.

Yes the statement is true for fractions unless the fractions are equal;

e.g.,

-

=

and this is not equal to

-

, which must be negative.

5. Is this statement generally true for decimals? Justify your conclusion.

Yes the statement is true for decimals unless the decimals are equal,

e.g. 0.7 – 0.3 = 0.4 and 0.3 – 0.7 is negative and less than 0.4

6. Make a true statement by writing an algebraic expression which uses the “=” sign.

a – b = – (b – a )


Activity 3: Multiplying two numbers

When multiplying two numbers represented by a and b is it true that a b = b a ?



The first one is done for you. Answers will vary.

Example a b a b = b a

Multiply two decimals 0.5 1.6 0.5 x 1.6 = 1.6 x 0.5

Multiply two whole numbers 7 8 7 x 8 = 8 x 7

Multiply two composite numbers 100 10 100 x 10 = 10 x 100

Multiply two even numbers 8 12 8 x 12 = 12 x 8

Multiply two multiples of 3 30 3 30 x 3 = 3 x 30

Multiply two fractions x = x

Multiply two negative numbers -3 -1 -3 x -1 = -1 x -3

Multiply two factors of 8 4 8 4 x 8 = 8 x 4

Multiply any number by a decimal 9 0.1 9 x 0.1 = 0.1 x 9

Multiply a fraction by an integer 6 x 6 = 6 x

Multiply any two positive numbers 35 2 35 x 2 = 2 x 35

Multiply zero by a fraction 0 0 x = x 0

Multiply a number by itself 4 4 4 x 4 = 4 x 4

Multiply a prime and a composite 12 3 12 x 3 = 3 x 12

Multiply two 1-digit numbers 7 9 7 x 9 = 9 x 7

Is the statement a b = b a true or false? Write your conclusion.

It is true. When two numbers are multiplied the order in which they are written does not affect

the product of the two numbers.


Activity 4: Dividing two numbers

Investigate the following statement a ÷ b = b ÷ a

This statement is false, since dividing a first number by a second does not give the same

answer as dividing the second number by the first.

Examples:

30 ÷ 5 5 ÷ 30

8 ÷ 4 4 ÷ 8

100 ÷ 10 10 ÷ 100

Note that you cannot divide by 0, although you can divide 0 by another number.

If a = b then a ÷ b = b ÷ a

Example: 4 ÷ 4 = 4 ÷ 4 = 1

This statement is also false for decimals and fractions (and negative numbers)

a ÷ b = = 1 ÷ and b ÷ a = = 1 ÷

Activity 5: Other Statements

Create statements of equality and provide examples for -

(a) the addition of three variables

a + b + c = a + c + b e.g., 1 + 2 + 3 = 1 + 3 + 2

a + b + c = c + a + b e.g., 2 + 3 + 4 = 4 + 2 + 3

a + b + c = c + b + a e.g., 4 + 5 + 6 = 6 + 5 + 4

a + b + c = b + a + c e.g., 2 + 1 + 4 = 1 + 2 + 4

a + b + c = b + c + a e.g., 5 + 1 + 3 = 1 + 3 + 5

(b) the multiplication of three variables

a x b x c = a x c x b e.g., 2 x 3 x 4 = 2 x 4 x 3

a x b x c = c x a x b e.g., 1 x 5 x 4 = 4 x 1 x 5

a x b x c = c x b x a e.g., 2 x 7 x 3 = 3 x 7 x 2

a x b x c = b x a x c e.g., 1 x 9 x 2 = 9 x 1 x 2

a x b x c = b x c x a e.g., 3 x 6 x 2 = 6 x 2 x 3


Activity 6:

1. Determine the answers in the table below.

7 + 10 + 1 = 18 (7 + 10) + 1 = 18 7 + (10 + 1) = 18

5 + 8 + 9 = 22 (5 + 8) + 9 = 13 + 9 = 22 5 + (8 + 9) = 5 + 17 = 22

1.1 + 2.7 + 4.8 = 8.6 (1.1 + 2.7) + 4.8 = 8.6 1.1 + (2.7 + 4.8) = 8.6

35 + 29 + 38 = 102 (35 + 29) + 38 = 102 35 + (29 + 38) = 102

9

9

9

2. Describe the pattern of the addition operation in these calculations.

If there are three numbers to add you get the same answer no matter in which order you add

the numbers.

3. Complete this statement in the same way: m + k + w = ( m + k) + w = m + ( k + w)


7 x (20 x 10) = 1400 7 x 20 x 10 = 1400 (7 x 20) x 10) = 1400

2 x (8 x 3) = 48 (2 x 8) x 3 = 48 2 x 8 x 3 = 48

0.5 x 8.6 x 4 = 17.2 (0.5 x 8.6) x 4 = 17.2 0.5 x (8.6 x 4) = 17.2

5 x 30 x 3 = 450 (5 x 30) x 3 = 450 5 x (30 x 3) = 450

(

)

5. Describe the pattern of the multiplication operation in these calculations.

When you multiply three numbers the order of multiplication does not affect the answer.

6. Complete this statement in the same manner:

m k w = (m k ) w = m ( k w )


Activity 7: Reflection

1. Use algebraic statements of equality to summarise the four main rules of operating with

numbers that have been used in this task.

a + b = b + a

a x b = b x a

m + k + w = ( m + k) + w = m + ( k + w)

m k w = (m k) w = m ( k w)

2. Explain, with the use of examples, how knowing such rules can support mental

calculations.

It is particularly useful when there are more than two numbers to add or multiply and you

can split the numbers into smaller numbers.

Example: 546 + 600 = 500 + 46 + 600 = 500 + 600 + 46 = 1100 + 46 = 1146

Example: 25 x 6 = 5 x 5 x 6 = 5 x (5 x 6) = 5 x 30 = 150

3. Write a test of 20 mixed questions based on these rules. Prepare the answers and swap

your test for one written by another student.

Examples: Provide the missing number or variable.

h x w x n = ____ x w x n h + w + n = ____ + w + n

66 + 24 + 55 = 24 + ____ + 66 33 x 22 x 55 = ____ x 22 x 33

6.3 + 4.5 = 3.3 + ____ + 4.5 20 x 6 x 90 = 90 x 6 x 10 x ____

45 x 32 = 5 x _____ x 32 55 x 99 = 99 x 5 x ______

k x m x t = m x ______ x t k + m + t = t + m + ______

2 x h x w = h x ______ x 2 d + 3 + t = t + 3 + ______

0.5 x 444 = 0.5 x 2 x ______ 1.9 + 15 = 10 + 1.9 + ______

7 x 36 = 7 x 6 x _______ 1289 + 99 = 90 + 1289 + ______

(34 + b ) + 5 = 34 + ( _____ + 5) 66 x v x 2 = 2 x ( _____ x 66)


STUDENT COPY ESTABLISHING LAWS

Introduction

In this task, you are given activities which will help you to identify the laws governing

operations with numbers when those laws are written in algebraic terms. In these tasks, the

lower case letters represent variables which can be either positive or negative or zero as well

as being whole numbers, fractions or decimals.

Activity 1: Adding two numbers

When adding two numbers represented by a and b the rule is a + b = b + a



The first one is done for you.

Example a b a + b = b + a

Add two decimals 0.4 1.6 0.4 + 1.6 = 1.6 + 0.4

Add two whole numbers

Add two prime numbers

Add two odd numbers

Add two multiples of 3

Add two fractions

Add two negative numbers

Add two factors of 6

Add a whole number to a decimal

Add a fraction and a decimal

Add a negative and a positive

Add a fraction to a whole number

Add two perfect squares

Add a prime and a composite

Add two 4-digit numbers


Activity 2: Subtracting two numbers

Generally a – b b – a

1. Interpret this statement and explain what it means.

2. Give three examples that support this statement.

3. Give an example of a situation when the statement is false.

4. Is this statement generally true for fractions? Justify your conclusion.

5. Is this statement generally true for decimals? Justify your conclusion.

6. Make a true statement by writing an algebraic expression which uses the “=” sign.

a – b =


Activity 3: Multiplying two numbers

When multiplying two numbers represented by a and b is it true that a b = b a ?



The first one is done for you.

Example a b a b = b a

Multiply two decimals 0.5 1.6 0.5 x 1.6 = 1.6 x 0.5

Multiply two whole numbers

Multiply two composite numbers

Multiply two even numbers

Multiply two multiples of 3

Multiply two fractions

Multiply two negative numbers

Multiply two factors of 8

Multiply any number by a decimal

Multiply a fraction by an integer

Multiply any two positive numbers

Multiply zero by a fraction

Multiply a number by itself

Multiply a prime and a composite

Multiply two 1-digit numbers

Is the statement a b = b a true or false? Write your conclusion.


Activity 4: Dividing two numbers

Investigate the following statement a ÷ b = b ÷ a

Activity 5: Other Statements

Create statements of equality and provide examples for

(a) the addition of three variables

(b) the multiplication of three variables


Activity 6


7 + 10 + 1 = (7 + 10) + 1 = 7 + (10 + 1) =

5 + 8 + 9 = (5 + 8) + 9 = 5 + (8 + 9) =

1.1 + 2.7 + 4.8 = (1.1 + 2.7) + 4.8 = 1.1 + (2.7 + 4.8) =

35 + 29 + 38 = (35 + 29) + 38 = 35 + (29 + 38) =

2. Describe the pattern of the addition operation in these calculations.

3. Complete this statement in the same way: m + k + w =


7 x (20 x 10) = 7 x 20 x 10 = (7 x 20) x 10) =

2 x (8 x 3) = (2 x 8) x 3 = 2 x 8 x 3 =

0.5 x 8.6 x 4 = (0.5 x 8.6) x 4 = 0.5 x (8.6 x 4) =

5 x 30 x 3 = (5 x 30) x 3 = 5 x (30 x 3) =

5. Describe the pattern of the multiplication operation in these calculations.

Complete this statement in the same manner: m k w =


Activity 7: Reflection

1. Use algebraic statements of equality to summarise the rules of operating with numbers

that have been used in this task.

2. Explain, with the use of examples, how knowing such rules can support mental

calculations.

3. Write a test of 20 mixed questions based on these rules. Prepare the answers and swap

your test for one written by another student.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Rounding Decimals 2 © Department of Education WA 2015

TASK 18: ROUNDING DECIMALS

Overview

This task provides a basis for considering the size and nature of errors that are made when

numbers are rounded before calculations are performed. It would be useful to provide a

further opportunity for discussing the importance of precision for measuring and to recognise

that there are different levels of precision in different situations. The numbers chosen will

support minimal use of calculators.

Students will need

calculators (for Activity 3 only)


Round decimals to a specified number of decimal places (ACMNA156)



Add and subtract decimals, with and without digital technologies, and use estimation

and rounding to check the reasonableness of answers (Year 6: ACMNA128)


reasoning when they

o explain patterns relating to the size and nature of the error


o determine a process for completing Activity 5

http://www.australiancurriculum.edu.au/glossary/popup?a=M&t=Rounding

http://www.australiancurriculum.edu.au/curriculum/contentdescription/ACMNA128


ROUNDING DECIMALS Solutions and Notes for Teachers

Activity 1: Rounding review

(a) Summarise the rules for rounding decimals. Consider the following forms of rounding:

1. Rounding down to the nearest whole number.

2. Truncating a number.

3. Rounding to the nearest tenth, hundredth, thousandth.

1. If the first digit after the decimal point is 5, 6, 7, 8 or 9 then the whole number goes up by

1, otherwise it stays as it is.

2. Truncate means to chop – so the decimal part is removed.

3. To round to the nearest tenth (first decimal point):

If the second digit after the decimal point is 5, 6, 7, 8 or 9 then the first digit goes up by 1,

otherwise it stays as it is and any digits following are removed..

To round to the nearest hundredth (second decimal point):

If the third digit after the decimal point is 5, 6, 7, 8 or 9 then the second digit goes up by

1, otherwise it stays as it is and any digits following are removed. The first digit after the

decimal point is unchanged.

The pattern continues for rounding to the nearest thousandth.

(b) Write a test consisting of 10 items which test the ability to round decimals as described

above. Give your test to another student and then mark their responses.

Example of such a Test:

1. Truncate these to a whole number: 56.3 1093.89

Ans: 56 1093

2. Round these numbers to 2 decimal places: 100.001 43.5623

Ans: 100.00 43.56

3. Round these numbers to the nearest thousandth: 0.09876 1.290545

Ans: 0.098 1.291

4. The answer was 8.79 when rounded to the nearest hundredth.

What might the answer have been originally? Give 3 possibilities

Ans: 8.791 8.788 8.79000005

5. 7.55 is between 7.5 and 7.6. How many other numbers lie between 7.5 and 7.6?

An infinite number of numbers.


Activity 2: Investigating the size of “error”

(a) You are given nine values for A and nine values for B. Round these numbers to the

nearest tenth and place them in the columns AR and BR. Add the totals of the original

numbers and enter them in the A + B column. Add the totals for the rounded decimals and

enter them in the AR + BR column. The last error column is the difference between A+B and

AR+BR.

A AR B BR A + B AR + BR Error

1 0.34 0.3 0.43 0.4 0.77 0.7 0.07

2 0.11 0.1 0.54 0.5 0.65 0.6 0.05

3 0.07 0.1 0.21 0.2 0.28 0.3 0.02

4 0.89 0.9 0.59 0.6 1.48 1.5 0.02

5 0.48 0.5 0.67 0.7 1.15 1.2 0.05

6 0.66 0.7 0.75 0.8 1.41 1.5 0.09

7 0.19 0.2 0.24 0.2 0.43 0.4 0.03

8 0.45 0.5 0.33 0.3 0.78 0.8 0.02

9 0.86 0.9 0.92 0.9 1.78 1.8 0.02

Give examples of some situation where this degree of error -

(i) would be important; If the numbers represented litres of fertiliser for a lawn.

(ii) would not be significant. If these were the number of mm on a running track.

(b) As a general rule, when was the error greatest? When both decimals were rounded up?

When both were rounded down? Or when one was rounded up and the other down?

Justify your answer with evidence from the table.

The error was greatest (mean of 0.16 ÷ 3) for both numbers being rounded up but it was only

slightly higher than when both numbers were rounded down (0.14 ÷ 3). Both of these means

were much higher than when one number was rounded up and the other number was

rounded down (0.07 ÷ 3).

(c) What can you do to be more certain of your conclusion?

Need to try many more numbers and a greater variety.


Activity 3: Investigating the degree of “error” when multiplying


nearest tenth and place them in the columns AR and BR. Multiply the original numbers ()

and enter the product in the A x B column. Multiply the rounded decimals (without calculator)

and enter the product in the AR x BR column.

For the last error column calculate the difference between A x B and AR x BR.

A AR B BR A x B AR x BR Error

1 0.34 0.3 0.43 0.4 0.1462 0.12 0.0262

2 0.11 0.1 0.54 0.5 0.0594 0.05 0.0094

3 0.07 0.1 0.21 0.2 0.0147 0.02 0.0053

4 0.89 0.9 0.59 0.6 0.5251 0.54 0.0149

5 0.48 0.5 0.67 0.7 0.3216 0.35 0.0284

6 0.66 0.7 0.75 0.8 0.495 0.56 0.065

7 0.19 0.2 0.24 0.2 0.0456 0.04 0.0056

8 0.45 0.5 0.33 0.3 0.1485 0.15 0.0015

9 0.86 0.9 0.92 0.9 0.7912 0.81 0.0188




The first three examples had numbers which were both rounded down and the error ranged

from 0.0053 to 0.0262. The next three examples had numbers which were both rounded up

and the error ranged from 0.0149 to 0.065 and this is much higher. The last three numbers

had different rounding and the errors were much lower, form 0.0015 to 0.0188.

(c) However, while the size of the error is useful as shown above, it’s the degree of error that

is important. Thus we need to consider what percentage of the actual product is the error;

e.g., the No. 1 percentage error is 0.0262/0.1462 = 0.1792 = 18 hundredths = 18%.

Calculate the percentage errors in the same way for No. 2 to No. 9.

No. 2 percentage error:

0.0094/0.0594 = 0.1582 = 15.8 hundredths = 16% to the nearest per cent.

No,. 3: 26.5%; No. 4: 3% ; No. 5: 9%; No. 6: 13%; No. 7: 12%; No. 8: 1%; No. 9: 2%


Activity 4: Investigating the “error” when dividing


nearest whole number and place them in the columns AR and BR. Divide B into A and enter

the result into the A ÷ B column. Divide BR into AR and enter the results into the AR ÷ BR

column. The last column is the difference between A ÷ B and AR ÷ BR; i.e., the error. The

numbers have been determined so that the calculations can be done mentally.

A AR B BR A ÷ B AR ÷ BR Error

1 2.66 3 1.33 1 2 3 1

2 8.88 9 2.22 2 4 4.5 0.5

3 15.5 16 3.1 3 5 5.333 0.333

4 24.5 25 3.5 4 7 6.25 0.75

5 22.8 23 3.8 4 6 5.75 0.25

6 18.5 19 3.7 4 5 4.75 0.25

7 10.05 10 2.01 2 5 5 0

8 100.4 100 25.1 25 4 4 0

9 33.3 33 11.1 11 3 3 0

Compare A ÷ B with AR ÷ BR when -

(i) one number is rounded up and the other is rounded down;

A ÷ B < AR ÷ BR

(ii) both numbers are rounded up;

A ÷ B > AR ÷ BR

(iii) both numbers are rounded down.

A ÷ B = AR ÷ BR

(b) Try some other numbers and see if you can find some examples that contradict these

findings.

If A = 7.2 and B = 2.4 then 7.2 ÷ 2.4 = 3 and 7 ÷ 2 = 3.5. This contradicts (iii)


Activity 5

Jon asks his mum who is driving the car, “How much further to go?”

Mum estimates the distance to the nearest kilometre and says “25 km”.

Not long after Jon asks the same question and his mother’s estimate was 20 km.

Determine, to the nearest tenth of a kilometre both the maximum distance and the minimum

distance that the car could have travelled between the two estimates.

When Mum gave the estimate of 25 km, the value could have been from 24.5 km to 25.4 km.

Similarly when Mum gave the estimate of 20 km, the value could have been from 19.5 km to

20.4 km.

The greatest difference between the two estimates is 25.4 – 19.5 = 5.9 km

The least difference between the two estimates is 24.5 – 20.4 = 4.1 km


STUDENT COPY ROUNDING DECIMALS

Activity 1: Rounding review

(a) Summarise the rules for rounding decimals. Consider the following forms of rounding:

1. Rounding down to the nearest whole number.

2. Truncating a number.

3. Rounding to the nearest tenth, hundredth, thousandth.

(b) Write a test consisting of 10 items which test the ability to round decimals as described

above. Give your test to another student and then mark their responses.


Activity 2: Investigating the size of “error”


nearest tenth and place them in the columns AR and BR. Add the totals of the original

numbers and enter them in the A + B column. Add the totals for the rounded decimals and

enter them in the AR + BR column. The last error column is the difference between A+B and

AR+BR.

A AR B BR A + B AR + BR Error

1 0.34 0.43

2 0.11 0.54

3 0.07 0.21

4 0.89 0.59

5 0.48 0.67

6 0.66 0.75

7 0.19 0.24

8 0.45 0.33

9 0.86 0.92

Give examples of some situation where this degree of error -

(i) would be important;

(ii) would not be significant.




(c) What can you do to be more certain of your conclusion?


Activity 3: Investigating the degree of “error” when multiplying


nearest tenth and place them in the columns AR and BR. Multiply the original numbers ()

and enter the product in the A x B column. Multiply the rounded decimals (without calculator)

and enter the product in the AR x BR column.

For the last error column calculate the difference between A x B and AR x BR.

A AR B BR A x B AR x BR Error

1 0.34 0.43

2 0.11 0.54

3 0.07 0.21

4 0.89 0.59

5 0.48 0.67

6 0.66 0.75

7 0.19 0.24

8 0.45 0.33

9 0.86 0.92




(c) However, while the size of the error is useful as shown above, it’s the degree of error that

is important. Thus we need to consider what percentage of the actual product is the error;

e.g., the No. 1 percentage error is 0.0263/0.1462 = 0.1792 = 18 hundredths = 18%.

Calculate the percentage errors in the same way for No. 2 to No. 9.


Activity 4: Investigating the “error” when dividing


nearest whole number and place them in the columns AR and BR. Divide B into A and enter

the result into the A ÷ B column. Divide BR into AR and enter the results into the AR ÷ BR

column. The last column is the difference between A ÷ B and AR ÷ BR; i.e., the error. The

numbers have been determined so that the calculations can be done mentally.

A AR B BR A ÷ B AR ÷ BR Error

1 2.66 1.33

2 8.88 2.22

3 15.5 3.1

4 24.5 3.5

5 22.8 3.8

6 18.5 3.7

7 10.05 2.01

8 100.4 25.1

9 33.3 11.1

Compare A ÷ B with AR ÷ BR when -

(i) one number is rounded up and the other is rounded down;

(ii) both numbers are rounded up;

(iii) both numbers are rounded down.

(b) Try some other numbers and see if you can find some examples that contradict these

findings.


Activity 5

Jon asks his mum who is driving the car, “How much further to go?”

Mum estimates the distance to the nearest kilometre and says, “25 km”.

Not long afterwards Jon asks the same question and his mother’s estimate was 20 km.

Determine, to the nearest tenth of a kilometre both the maximum distance and the minimum

distance that the car could have travelled between the two estimates.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Square Numbers 2 © Department of Education WA 2015

TASK 19: SQUARE NUMBERS

Overview

In this task students are given opportunities to develop further understanding of squares and square roots. They are also invited to review their knowledge of the terminology of types of numbers and investigate the properties of some numbers.

Students will need • calculators

Relevant content descriptions from the Western Australian Curriculum • Investigate and use square roots of perfect square numbers (ACMNA150)• Identify and describe properties of prime, composite, square and triangular numbers

(Year 6:ACMNA122)

Students can demonstrate • fluency when they

o generate mathematical expressions to represent word descriptions• understanding when they

o use the relationships between different types of numbers to make conclusionso describe patterns between sets of numberso generate mathematical expressions to represent word descriptions

• reasoning when theyo generalise the patterns for Activities 4 and 5



http://www.australiancurriculum.edu.au/glossary/popup?a=M&t=Square

http://www.australiancurriculum.edu.au/curriculum/contentdescription/ACMNA122


SQUARE NUMBERS Solutions and Notes for Teachers

Activity 1: Review

1. Answer these questions and provide evidence/calculations to support your conclusions.

What are square numbers?

Numbers made by multiplying a number by itself; e.g., 9 is a square number as 9 = 3 x 3.

Can square numbers be both odd and even?

Yes. 9 is an odd square number and 4 is an even square number.

Can square numbers be prime numbers?

No. Prime numbers have only two different factors, 1 and the number itself.

How can your calculator be used to determine square numbers?

There is a button to square numbers. x2

What is the name given to the process of “undoing” a square number?

Finding the square root.

2. Determine these numbers:

A palindromic square number between 100 and 200.

121

Two square numbers that add to 100.

64 + 36 = 100

A 5-digit square number ending in 0.

10 000 = 100 x 100

A number which is equal to the square of itself.

1 because 1 x 1 = 1

Four square numbers which are factors of 100.

1, 4, 25, 100




Activity 2 What can be the last digit that a square number can have? Here is a list of the last digits that could exist for a square number. For each digit write Y or N according to its existence as the last digit of a square number. If the last digit can be the ending of a square number, what might have been the ending of the number used to make the square number?

Last digit of number 0 1 2 3 4 5 6 7 8 9

Can it be a square number? Y/N

Y Y N N Y Y Y N N Y

Possible last digit/s of the number?

0 1 9

2

8 5

4

6

3

7

Activity 3

Use 5 different examples to test the truth of each of these statements. 1. The square of any even number is another even number.

22 = 4 82 = 64 202 = 400 142 = 196 162 = 256 True 2. The square of any odd number is another odd number.

32 = 9 52 = 25 212 = 441 132 = 169 192 = 361 True 3. The sum of a number added to its square is always even. 20 + 202 = 420 44 + 442 = 1980 132 + 13 = 182

192 + 19 = 380 632 + 63 = 4032

True




Activity 4

Sums of Odd Numbers.

Write down your answers to the following additions.

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

1 + 3 + 5 + 7 + 9 + 11 = 36

2. What will be the sum of the next set of consecutive odd numbers in this pattern?

49

3. Write a rule to find the sum of any number of odd numbers?

The sum of a set of consecutive odd numbers starting at 1 is equal to the square of the number of odd numbers which have been added.

4. How many consecutive odd numbers were added to make a total of -

(a) 100 (b) 441 (c) 1024 (d) 5000

(a) 10 (b) 21 (c) 32 (d) does not work

5. Does your rule apply for the sums of the even numbers? Show the procedure you use todetermine your answer.

2 + 4 = 6

No, two consecutive even numbers are added but the sum is not a square number




Activity 5

One less than a square number.

1. For this activity you are asked to• square a number• subtract 1 from the square• write the above answer as different multiplications of 2 factors. Enter as many as you

can. An example is provided. [Be systematic in determining these factors.]• Complete the table below.

Number Number squared

One less than the number squared

Factor forms of one less than the number squared

N N x N N x N -1

100 10 000 9999 1 x 9999, 3 x 3333, 9 x 1111, 33 x 303, 11 x 909, 99 x 101

11 121 120 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, 10 x 12

10 100 99 1 x 99, 3 x 33, 9 x 11

9 81 80 1 x 80, 2 x 40, 4 x 20, 5 x 16, 8 x 10

8 64 63 1 x 63, 3 x 21, 7 x 9

7 49 48 1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8

6 36 35 1 x 35, 5 x 7

5 25 24 1 x 24, 2 x 12, 3 x 8, 4 x 6

4 16 15 1 x 15, 3 x 5

3 9 8 1 x 8, 2 x 4

2 4 3 1 x 3

2. Study the numbers in the final column and look for a particular pair of factors that occursin every row. Describe this pair of factors.

In every row one pair of factors is the number one more than N multiplied by the number one less than N.

3. Use this pattern to write an expression for 2202 -1.

221 x 219




Activity 6

1. Investigate the final digits of the squares of numbers ending in 5.Select any 5 numbers ending in 5, square them and record your answers. Can you see apattern in the final digit(s) of these squares? Check your findings with another student’sresults. Record your findings and suggest a reason for this pattern.

Sample of possible answers:

352 = 1225

652 = 4225

852 = 7225

1052 = 11 025

2152 = 46 225

They all end in 25. Could be related to the last digit 5 because 52 = 25

2. Investigate the product of two square numbers; e.g., 4 x 25 = 100. What can you sayabout the product of two square numbers? Provide evidence to support your conclusion.

Sample of possible answers:

32 x 22 = 9 x 4 = 36 which is a square number because 36 = 62


42 x 22 =16 x 4 = 64 which is a square number because 64 = 82



The product of two square numbers is a square number.

3. Does the addition of two square numbers produce another square number? Justify yourconclusion.

72 + 52 = 49 + 25 = 74 and 74 is not a square number, so the sum of two squares is not always a square number.

4. Determine two numbers that multiply to give one million yet neither number ends in 0 andboth numbers are whole numbers.

64 x 15 625 = 1 000 000




STUDENT COPY SQUARE NUMBERS

Activity 1: Review

1. Answer these questions and provide evidence/calculations to support your conclusions.

What are square numbers?

Can square numbers be both odd and even?

Can square numbers be prime numbers?

How can your calculator be used to determine square numbers?

What is the name given to the process of “undoing” a square number?

2. Determine these numbers:

A palindromic square number between 100 and 200.

Two square numbers that add to 100.

A 5-digit square number ending in 0.

A number which is equal to the square of itself.

Four square numbers which are factors of 100.




Activity 2

What can be the last digit that a square number can have?

Here is a list of the last digits that could exist for a square number. For each digit write Y or N according to its existence as the last digit of a square number. If the last digit can be the ending of a square number, what might have been the ending of the number used to make the square number?

Last digit of number 0 1 2 3 4 5 6 7 8 9

Can it be a square number? Y/N Possible last digit/s of the number?

Activity 3

Use 5 different examples to test the truth of each of these statements.

1. The square of any even number is another even number.

2. The square of any odd number is another odd number.

3. The sum of a number added to its square is always even.




Activity 4

Sums of Odd Numbers

1. Write down your answers to the following additions.

1 + 3

1 + 3 + 5

1 + 3 + 5 + 7

1 + 3 + 5 + 7 + 9

1 + 3 + 5 + 7 + 9 + 11

2. What will be the sum of the next set of consecutive odd numbers in this pattern?

3. Write a rule to find the sum of any number of odd numbers?

4. How many consecutive odd numbers were added to make a total of -

(a) 100 (b) 441 (c) 1024 (d) 5000

5. Does your rule apply for the sums of the even numbers? Show the procedure you use todetermine your answer.




Activity 5

One less than a square number.

1. For this activity you are asked to• square a number• subtract 1 from the square• write the above answer as different multiplications of 2 factors. Enter as many as you

can. An example is provided. [Be systematic in determining these factors.]• Complete the table below.

Number Number squared

One less than the number squared

Factor forms of one less than the number squared

N N x N N x N -1

100 10 000 9999 1 x 9999, 3 x 3333, 9 x 1111, 33 x 303 11 x 909, 33 x 303, 99 x 101

11

10

9

8

7

6

5

4

3

2

2. Study the numbers in the final column and look for a particular pair of factors that occursin every row. Describe this pair of factors.

3. Use this pattern to write an expression for 2202 -1.




Activity 6 1. Investigate the final digits of the squares of numbers ending in 5. Select any 5 numbers ending in 5, square them and record your answers. Can you see a pattern in the final digit(s) of these squares? Check your findings with another student’s results. Record your findings and suggest a reason for this pattern.

2. Investigate the product of two square numbers; e.g., 4 x 25 = 100. What can you say about the product of two square numbers? Provide evidence to support your conclusion. 3. Does the addition of two square numbers produce another square number? Justify your conclusion. 4. Determine two numbers that multiply to give one million yet neither number ends in 0 and both numbers are whole numbers.



MATHS7TL002 | Mathematics | Number and Algebra Activity | Ratios 2 © Department of Education WA 2015

TASK 24: RATIOS

Overview

Students would have had the necessary exposure to ratios before Year 7 but need not be

experienced in sharing amounts according to a ratio to engage in this task.

Students will need

calculators

scissors and glue

counters – various colours


Recognise and solve problems involving simple ratios (ACMNA173)


fluency when they

o manipulate ratios with operations on both numbers


o rank the weedkiller ratios in correct order in Activity 5

reasoning when they

o identify the effect of different operations on proportions


RATIOS Solutions and Notes for Teachers

Introduction

For the supervision of swimming in open water one lifesaver with a bronze medallion is

needed for every 10 students. This can be written without reference to units as -

one lifesaver for every ten students, or 1:10

Activity 1

Using the following ratios 1:2 1:3 2:3 1:5 2:5 3:4 1:4

Working in groups

Each group is allocated one of the ratios listed above.

Each group collects a handful of counters – two colours only.

Each group makes their ratio as follows, with the first number of counters in one colour

and the second number of counters in the other colour.

o

These would be the counters for the ratio 3:5

Use colours and draw a diagram similar to the one above, to represent your ratio.

Solutions will vary according to the student’s ratio. Some solutions are provided to exemplify

the solutions required.

Now consider changing your ratio. For each change, draw a diagram showing the numbers

of counters of each colour.

1. Doubling the numbers of both colours.

2. Tripling the numbers of both colours.


















































3. Quadrupling the numbers of both colours.

4. Doubling the number of counters of the first colour only.

5. Tripling the number of counters of the second colour only.

6. Doubling the number of counters of the first colour and tripling the number of counters

of the second colour.

Changing the numbers in a ratio. The following diagram shows 3 shaded smiley faces to 1

unshaded smiley face.

The diagram below has the same objects but more of them. Are they in the same ratio? Are

there still 3 shaded smiley faces for every 1 unshaded smiley face?

This can be determined by grouping them as above.

Ungrouped Grouped
























































































































7. Use the grouping method (or otherwise) to decide if the ratios were changed when you -

(a) Doubled the numbers of both colours.

NO

(b) Tripled the numbers of both colours.

NO

(c) Quadrupled the numbers of both colours.

NO

(d) Doubled the number of counters of the first colour only.

Yes

(e) Tripled the number of counters of the second colour only.

Yes

(f) Doubled the number of counters of the first colour and tripling the number of counters


Yes


Activity 2

Solutions are not provided, as they will vary from student to student.

A. Swap your “ratio” with another student. Draw a coloured diagram of your new ratio.

B. Now consider changing your ratio. For each change, draw a diagram showing the

numbers of counters of each colour. Use the grouping method to present (draw) your

counters and determine the change to the ratio.

1. Add two more counters of each colour.

The numbers in the ratio are no longer in the same proportion; the ratio has been

changed.

2. Add three more counters of each colour.


changed.

3. Add four more counters of each colour.


changed.

4. Add two more counters of the first colour only.


changed.

5. Add three more counters of the second colour only.


changed.

6. Add two more counters of the first colour and three more counters of the second

colour.


changed.


Activity 3

A. The operations on the numbers in the ratio for Activities 1 and 2 only included

addition and multiplication. Do you think the numbers will still be in proportion (or in

the same ratio) if division or subtraction is used. Provide evidence for your conclusion.

The numbers will still be in proportion for division but not for subtraction.

Example: If there is 1 lifesaver for every 10 swimmers; i.e., 1:10 then subtracting 1

from each ratio is 0:9 and this is not the same as 1:10.

Division is just the multiplication of an inverse so it should apply.

100:10 means 100 for every 10; i.e., 10 for every 1 and this can be achieved by

dividing both the 100 and the 10 by 10, in the same way that we can simplify fractions.

B. Consider the following question to write a summary of the results of these

investigations.

When the numbers in a ratio are changed, are the numbers still in the same proportion

or ratio?

The numbers in the ratio will still be in the same proportion as long as all numbers in

the ratio are either multiplied OR divided by the SAME number. Mostly the ratio will be

changed if the numbers in the ratio have another number added or subtracted; even if

it is the same number.



Activity 4

The statement below is assumed to be true – the results of the investigation above.

A ratio does not change if both numbers in the ratio are multiplied or divided by the same

number.

Investigate if this statement applies when -

(i) the numbers in the ratio are decimals or fractions; and

(ii) the numbers used to multiply and divide the numbers in the ratio, are decimals or

fractions.

(i)

Ratio 0.1 : 0.4 0.2 : 0.5 0.5 :

Doubled 0.2 : 0.8 0.4 : 1.0 1.0 :

Tripled 0.3 : 1.2 0.6 : 1.5 1.5 :

quadrupled 0.4 : 1.6 0.8 : 2.0 2.0 :

In each column, you can divide the second number by the first and get the same result each

time so the ratio is preserved with multiplication even when there are fractions or decimals in

the ratio.

(ii)

Ratio 10 : 20 2 : 3 12 : 6 100 : 10

x 1.5 15 : 30 3 : 4.5 18 : 9 150 : 15

x 2.5 25 : 50 5 : 7.5 30 : 15 250 : 25

x 5 : 10 1 : 1.5 6 : 3 50 : 5

x 1 : 2 0.2 : 0.3 1.2 : 0.6 10 : 1

In each column, you can divide the second number by the first and get the same result each

time so the ratio is preserved with multiplication even when there are fractions or decimals in

the operating number.

Activity 5

You are given a page of ratios for weedkiller. The ratios are presented as follows.

The first number represents the number of parts of the mixture that are the weed killer and

the second number represents the number of parts of the mixture that are water.

Cut out these ratios and paste them into your workbook in order of strength of the mixture.

1:3

Water Weed

killer






11 1:3 7 3:4

10 2:5 9 3:5

15 1:8 10 4:10

1 10:1 17 2:40

6 3:2 14 4:28

4 12:4 8 2:3

9 6:10 13 1:4

17 1:20 15 3:24

12 2:7 9 6:15

16 10:90 3 4:1

19 3:100 18 3:50

5 5:3 16 1:9

2 8:1 3 20:5


STUDENT COPY RATIOS

Introduction

For the supervision of swimming in open water one lifesaver with a bronze medallion is

needed for every 10 students. This can be written without reference to units as -

one lifesaver for every ten students, or 1:10

Activity 1

Using the following ratios 1:2 1:3 2:3 1:5 2:5 3:4 1:4

Working in groups

Each group is allocated one of the ratios listed above.

Each group collects a handful of counters – two colours only.

Each group makes their ratio as follows, with first number of counters in one colour and

the second number of counters in the other colour.

o

These would be the counters for the ratio 3:5

A. Use colours and draw a diagram similar to the one above, to represent your ratio.

B Now consider changing your ratio. For each change, draw a diagram showing the

numbers of counters of each colour.

1. Doubling the numbers of both colours.

2. Tripling the numbers of both colours.










3. Quadrupling the numbers of both colours.

4. Doubling the number of counters of the first colour only.

5. Tripling the number of counters of the second colour only.

6. Doubling the number of counters of the first colour and tripling the number of counters


Changing the numbers in a ratio. The following diagram shows 3 shaded smiley faces to 1

unshaded smiley face.

The diagram below has the same objects but more of them. Are they in the same ratio? Are

there still 3 shaded smiley faces for every 1 unshaded smiley face?

This can be determined by grouping them as above.

Ungrouped Grouped






































7. Use the grouping method (or otherwise) to decide if the ratios were changed when you -

(a) Doubled the numbers of both colours.

(b) Tripled the numbers of both colours.

(c) Quadrupled the numbers of both colours.

(d) Doubled the number of counters of the first colour only.

(e) Tripled the number of counters of the second colour only.

(f) Doubled the number of counters of the first colour and tripling the number of counters



Activity 2

A. Swap your “ratio” with another student. Draw a coloured diagram of your new ratio.

B. Now consider changing your ratio. For each change, draw a diagram showing the

numbers of counters of each colour. Use the grouping method to present (draw) your

counters and determine the change to the ratio.

1. Add two more counters of each colour.

2. Add three more counters of each colour.

3. Add four more counters of each colour.

4. Add two more counters of the first colour only

5. Add three more counters of the second colour only.

6. Add two more counters of the first colour and three more counters of the second

colour.


Activity 3

A. The operations on the numbers in the ratio for Activities 1 and 2 only included

addition and multiplication. Do you think the numbers will still be in proportion (or in

the same ratio) if division or subtraction is used. Provide evidence for your conclusion.

B. Consider the following question to write a summary of the results of these

investigations.

When the numbers in a ratio are changed, are the numbers still in the same proportion

or ratio?



Activity 4

The statement below is assumed to be true – the results of the investigation above.

A ratio does not change if both numbers in the ratio are multiplied or divided by the same

number.

Investigate if this statement applies when -

(i) the numbers in the ratio are decimals or fractions, and

(ii) the numbers used to multiply and divide the numbers in the ratio, are decimals or

fractions.

Activity 5

You are given a page of ratios for weedkiller. The ratios are presented as follows.

The first number represents the number of parts of the mixture that are the weed killer and

the second number represents the number of parts of the mixture that are water.

Cut out these ratios and paste them into your workbook in order of strength of the mixture.

1:3

Water Weed

killer






1:3 3:4

2:5 3:5

1:8 4:10

10:1 2:40

3:2 4:28

12:4 2:3

6:10 1:4

1:20 3:24

2:7 6:15

10:90 4:1

3:100 3:50

5:3 1:9

8:1 20:5

MATHS7TL002 | Mathematics | Number and Algebra Activity | Equations 2 © Department of Education WA 2015

TASK 25: EQUATIONS

Overview

This focus in this task is on getting ready to interpret, write and solve equations by

establishing values for variables in number sentences. The process for solving the equations

in this task is mainly by guess and check, but more sophisticated methods can still be used.

Students will not need any extra materials


Introduce the concept of variables as a way of representing numbers using letters

(ACMNA175)


variable (ACMNA176)




fluency when they

o determine the solutions to the equations without using technology


o create number sentences in Activity 2

reasoning when they

o explain their answers in Activities 3 and 4


o complete Activity 5


EQUATIONS Solutions and Notes for Teachers

Activity 1

Each shape in these number sentences represents one of the digits from 1 to 9.

Which is which?

+ =

+ =

+ =

x =

+ =

x =

– =

x =

The digit represented by is missing from these number sentences. Which digit is

missing?

Start with . It represents 1 because 1 x 1 = 1.

Then + = means = 2

Then = 4 because + =

= 5 because + =

– = means that = 9

x = means that = 3

x = means that = 6

+ = means that = 8

= 7


Activity 2

Use a similar process to the one used in Activity 1 to design and write number sentences

with symbols for the following sets of numbers. Use as many number sentences as you need

to be able to work out the numbers. Ask another student to identify the numbers represented

by the symbols in your number sentences.

(a) The decimals 0.1, 0.2, 0.3, ..., 0.9. Give your number sentences to another student to

solve.

Answers will vary. This is a good opportunity to review the fact that 0.1 x 0.1 0.1

It is also an opportunity to get students to work with simple decimals without using

technology.

(b) The following fractions: , , , , , ,

Answers will vary. This is a good opportunity to review operations with fractions. The

fractions have been chosen so that students can do this exercise without using technology.

Activity 3

The symbols and are used to represent numbers.

Given x = 36:

(a) What numbers might and represent if only whole numbers are allowed?

1 2 3 4 6 9 12 18 36

36 18 12 9 6 4 3 2 1

(b) Give 3 examples of x = 36 when one of the numbers has a value between 0

and 1.

0.5 x 72 = 36

x 144 = 36

360 x 0.1 = 36

(c) Can both numbers be negative? Can one number be negative? Give examples or

explain your answers.

Both numbers can be negative but just one number being negative is not possible:

-9 x -4 = 36 but 9 x -4 = -36

A negative number multiplied by a positive number will give a negative number.


Activity 4

In this activity the letters a, b, k, and m represent numbers. For each equation

give 5 different examples of what the letters could represent; and

determine the number of possible values that each letter could represent and explain

the reasons for your decision.

(a) a + b = 16

a 10 -2 16 0.4 -32

b 6 18 0 15.6 16

The above are examples. The letters a and b can represent any numbers at all as long as

the relationship between them is maintained. Any number can be allocated to a, then b is

determined.

(b) k + 5 = 5 x m

k 10 0 0.5 -8 200

m 3 1 1.1 -0.6 41

The above are examples. The letters k and m can represent any numbers at all as long

as the relationship between them is maintained. Any number can be allocated to k, 5 is

added and m is determined by dividing this result by 5.

(c) k ÷ m = m ÷ k

k 10 -2 16 0.4 -32

m 10 -2 16 0.4 -32

The above are examples. The letters k and m can represent any numbers at all, except

0, as long as the relationship between them is maintained. Any number can be allocated

to k, and m must represent the same number as k.


Activity 5

You are given a set of problems to solve. Use the process described to solve these

problems. An example is provided.

Problem 1: The sum of three consecutive odd numbers is 33. What are the numbers?

Let the first number be represented by k

The next three numbers are k + 2, k + 4

The equation is k + k + 2 + k + 4 = 33

Solving: The numbers must all be about 10 because 3 x 10 = 30

Try 11 + 13 + 15 = 39 (too large)

Try 9 + 11 + 13 = 33 correct The numbers are 9, 11, 13

Problem 2: A number plus its square adds to 650. What is the number?

Let the number be represented by k

Then the square is k2

The equation is k + k2 = 650

The number must be larger than 20 because 20 x 20 + 20 = 420

The number must be less than 30 because 30 x 30 + 30 = 930

The number could end in 5 because 650 is a multiple of 5

Try 25

25 x 25 + 25 = 650 correct The number is 25

Example:

Problem: Four consecutive even numbers add up to 60. What are they?


The next three numbers are k + 2, k + 4, k + 6

The equation is k + k + 2 + k + 4 + k + 6 = 60

Solving: The numbers must all be less than 20 because 4 x 20 = 80

Try 10 + 12 + 14 + 16 = 52 (bit short)

Try 12 + 14 + 16 + 18 = 60 correct



Problem 3: The sum of the ages of Granddad’s children is 24. Nickie is the eldest and is

three years older than his sister Lottie. The youngest, Bertie, is 10 years younger than Nickie

and three years younger than Gracie. How old are the children?

Let Nick’s age be represented by m

Then Lottie is m – 3

Bertie is m – 10

Gracie is m – 7

The equation is m + (m – 3) + (m – 10) + (m – 7) = 24.

Nick must be at least 10 (if Bertie is a very small baby), so try 10 for Nick:

10 + 7 + 0 + 3 = 20 (not enough)

Try 11:

11 + 8 + 1 + 4 = 24 correct

Nickie’s age is 11

Lottie is 8

Bertie is 1

Gracie is 4.

Problem 4: Two punnets of strawberries and one punnet of cherry tomatoes together cost $11

and the strawberries were 25c per punnet more expensive than the cherry tomatoes. What

did each punnet cost?

Let the cost of a punnet of strawberries by m.

Then the cost of a punnet of cherry tomatoes is m – $0.25

The equation is m + m + m – $0.25 = $11

The cost must be between $3 and $4 and end in 5c or 10c.

Try $3.50:

$3.50 + $3.50 + $3.25 = $10.25 (too little).

Try $3.75:

$3.75 + $3.75 + $3.50 = $11 correct

The cost of a punnet of strawberries is $3.75.

The cost of a punnet of cherry tomatoes is $3.50.


STUDENT COPY EQUATIONS

Activity 1

Each shape in these number sentences represents one of the digits from 1 to 9.

Which is which?

+ =

+ =

+ =

x =

+ =

x =

– =

x =

The digit represented by is missing from these number sentences. Which digit is

missing?


Activity 2

Use a similar process to the one used in Activity 1 to design and write number sentences

with symbols for the following sets of numbers. Use as many number sentences as you need

to be able to work out the numbers. Ask another student to identify the numbers represented

by the symbols in your number sentences.

(a) The decimals 0.1, 0.2, 0.3, ..., 0.9. Give your number sentences to another student to

solve.

(b) The following fractions: , , , , , ,


Activity 3

The symbols and are used to represent numbers.

Given x = 36:

(a) What numbers might and represent if only whole numbers are allowed?

(b) Give 3 examples of x = 36 when one of the numbers has a value between 0

and 1.

(c) Can both numbers be negative? Can one number be negative? Give examples or

explain your answers.


Activity 4

In this activity the letters a, b, k, and m represent numbers. For each equation

give 5 different examples of what the letters could represent; and

determine the number of possible values that each letter could represent and explain

the reasons for your decision.

(a) a + b = 16

(b) k + 5 = 5 x m

(c) k ÷ m = m ÷ k


Activity 5

You are given a set of problems to solve. Use the process described to solve these

problems. An example is provided.

Problem 1: The sum of three consecutive odd numbers is 33. What are the numbers?

Problem 2: A number plus its square adds to 650. What is the number?

Example:

Problem: Four consecutive even numbers add up to 60. What are they?


The next three numbers are k + 2, k + 4, k + 6

The equation is k + k + 2 + k + 4 + k + 6 = 60

Solving: The numbers must all be less than 20 because 4 x 20 = 80

Try 10 + 12 + 14 + 16 = 52 (bit short)

Try 12 + 14 + 16 + 18 = 60 correct



Problem 3: The sum of the ages of Granddad’s children is 24. Nickie is the eldest and is

three years older than his sister Lottie. The youngest, Bertie, is 10 years younger than Nickie

and three years younger than Gracie. How old are the children?

Problem 4: Two punnets of strawberries and one punnet of cherry tomatoes together cost $11

and the strawberries were 25c per punnet more expensive than the cherry tomatoes. What

did each punnet cost?

MATHS7TL002 | Mathematics | Number and Algebra Activity | Scoring Golf 2 © Department of Education WA 2015

TASK 26: SCORING GOLF

Overview

For this task students should have developed an understanding of the number line below

zero and have had some experience with operations on negative integers. In this task

students are provided with an opportunity to consolidate and demonstrate their

understanding of working with negative numbers.

Students will not need any particular equipment


Compare, order, add and subtract integers (ACMNA280)


fluency when they

o calculate accurately with integers in all activities

o represent integers in flexible ways


o can work backwards to determine operations to produce negative numbers

o interpret and exemplify statements summarising features of operations with

negative integers.


SCORING GOLF Solutions and Notes for Teachers

Activity 1

Golf courses are designed so that good golfers will usually take the same number of strokes

to hit the ball into the hole. Each hole is usually listed as being one of the following:

Par 3: requiring 3 strokes


Par 5: requiring 5 strokes and so on . . .

When the hole is played the player is said to be either;

Over par: takes more strokes than the par for that hole,

Under par: takes fewer strokes than the par for that hole, or

Par: takes the 'par' number of strokes for that hole.

Scoring for a Par 4 hole

Number of strokes to get

the ball in Description Score Name

7 3 over par 3 Triple Bogey

6 2 over par 2 Double Bogey

5 1 over par 1 Bogey

4 Par 0 Par

3 1 under par – 1 Birdie

2 2 under par – 2 Eagle

1 3 under par – 3 Albatross

1. Complete this table for a Par 3 hole



7 4 over par 4




3 Par 0 Par




2. Using positive and negative numbers, determine the Score for the following results.

(a) 5 on a par 3 (d) 4 on a par 4

2 0

(b) 6 on a par 5 (e) 2 on a par 3

1 –1

(c) 7 on a par 4 (f) 3 on a par 5

3 –2

3. What name would be given to each of the above results?


Double Bogey Par


Bogey Birdie


Triple Bogey Eagle

A Birdie is 'one under' and on a par 5 hole, the golfer would succeed in 4 strokes.

4. How many strokes would a golfer take to get an Albatross on a hole that is -

Par 3? Impossible Par 4? 1 Par 5? 2

5. Create a table, similar to Table 1 on the previous page, for a hole of par 5 on a golf

course. Show how you would score and describe the number of strokes (between 1 and 8)

that might be taken to get the ball in.



1 Four under par –4 Condor

2 Three under par –3 Albatross

3 Two under par –2 Eagle

4 One under par –1 Birdie

5 Par 0 Par

6 One over par 1 Bogey

7 Two over par 2 Double Bogey

8 Three over par 3 Triple Bogey


6. The total number of strokes to complete all 18 holes on a golf course is usually 72.

The results of four players for 18 holes are shown in the table. Complete the table to

determine the score for each hole for each player and then identify the winner.

Hole Par

Paul Cath Tom Jill

strokes score strokes score strokes score strokes score

1 3 3 0 2 –1 1 –2 2 –1

2 4 5 1 4 0 3 –1 5 1

3 3 3 0 2 –1 1 –2 3 0

4 4 6 2 5 1 3 –1 4 0

5 3 5 2 6 3 7 4 8 5

6 3 4 1 3 0 6 3 3 0

7 4 4 0 2 –2 5 1 4 0

8 3 7 4 3 0 4 1 5 2

9 5 6 1 4 –1 5 0 4 –1

10 4 5 1 5 1 6 2 5 1

11 4 1 –3 3 –1 7 3 5 1

12 5 3 -2 3 –2 2 –3 4 –1

13 5 2 -3 3 –2 3 –2 4 –1

14 4 3 -1 4 0 4 0 5 1

15 4 3 -1 4 0 4 0 5 1

16 4 4 0 5 1 2 –2 5 1

17 5 5 0 4 –1 5 0 4 –1

18 4 7 3 5 1 4 0 4 0

Final scores: 5 –4 1 8

The winner is Cath.


Activity 2

2

c) Which player wins?

Investigation Negative Numbers

Use your calculator (if necessary) to investigate the truth of each of the given

statements.

If you think that the statement is false, write one example to support your conclusion.

If you think that the statement is true, write at least three examples to support your

conclusion.

Statements

1. Adding two negative numbers always results in a negative number.

True

–4 + –5 = –9

–6 + –9 = –15

–3 + –7 = –10

2. Doubling a negative number will produce another negative number.

True

2 x –9 = –18

10 x –2 = –20

4.5 x –2 = –9

3. Squaring a negative number will produce another negative number.

False

–9 x –9 = 81

4. Dividing a negative number by a counting number (1, 2, 3, . . . always gives a

quotient that is negative.

True (counting numbers are positive).

–40 ÷ 8 = –5

–100 ÷ 10 = –10

–45 ÷ 5 = –9

5. The product of a negative number and a whole number is always positive.

False (whole numbers, except 0, are positive).

–4 x 8 = – 32

–7 x 0 = 0




Investigation (cont’d) Negative Numbers

6. Negative numbers are always whole numbers.

False

Decimals can be negative; e.g., –0.5.

7. Adding a negative number to a positive number gives a negative answer.

False

– 4 + 5 = 1

8. Subtracting a positive number from a negative number always gives a negative

answer.

True

-4 – 9 = -13

-6 – 20 = -14

-10 – 1 = -11

9. Subtracting any positive number from a smaller positive number always gives a

negative answer.

True

9 – 12 = -3

10 – 100 = -90

1 – 2 = -1

10. The sum of any positive number and a whole number is another whole

number.

False

1 + 0.5 = 1.5 0.5 is a positive number, but 1.5 is not a whole number.

11. Adding a negative fraction to a positive decimal can result in 0.

True

- 1

2 + 0.5 = 0

0.1 + - 1

10 = 0

- 3

4+ 0.75 = 0



Activity 3

The card below is part of a game of Neggo where the number is crossed off if it is the result

of the operation that is called out. Your task is to devise three different operations for each

number on the card. The first one is shown as an example.

–30 –2 –42 –16 –24

–10 –8 –9 –18 –60

–12 –36 –1 –20 –100

Examples are given below, but there are many possibilities.

Operation 1 Operation 2 Operation 3

–30 6 x –5 10 – 40 –10 – 20

–2 1 – 3 2 x –1 4 – 6

–42 – 6 x 7 –7 x 6 2 – 44

–16 – 2 x 8 16 x –1 3 – 19

–24 – 6 x 4 3 x –8 2 x –12

–10 2 x – 5 –2 – 8 –3 – 7

–8 8 x –1 1 x –8 – 3 – 5

–9 –3 – 6 –3 x 3 9 x –1

–18 – 36 ÷ 2 –9 x 2 6 – 24

–60 –20 x 3 10 – 70 –3 x 20

–12 24 ÷ –2 3 x –4 6 x –2

–36 9 x – 4 4 x –9 6 x –6

–1 9 ÷ – 9 –4 ÷ 4 6 – 7

–20 –10 x 2 –4 x 5 0.5 x –40

–100 –10 x 10 – 100 ÷ 1 1000 ÷ –10


STUDENT COPY SCORING GOLF

Activity 1

Golf courses are designed so that good golfers will usually take the same number of strokes

to hit the ball into the hole. Each hole is usually listed as being one of the following:



Par 5: requiring 5 strokes and so on . . .

When the hole is played the player is said to be either;

Over par: takes more strokes than the par for that hole,

Under par: takes fewer strokes than the par for that hole, or

Par: takes the 'par' number of strokes for that hole.

Scoring for a Par 4 hole






4 Par 0 Par



1 3 under par – 3 Albatross

1. Complete this table for a Par 3 hole



7 4

6

5

4

3 Par

2 Birdie

1


2. Using positive and negative numbers, determine the Score for the following results.




3. What name would be given to each of the above results?




A Birdie is 'one under' and on a par 5 hole, the golfer would succeed in 4 strokes.

4. How many strokes would a golfer take to get an Albatross on a hole that is;

Par 3? Par 4? Par 5?

5. Create a table, similar to Table 1 on the previous page, for a hole of par 5 on a golf

course. Show how you would score and describe the number of strokes (between 1 and 8)

that might be taken to get the ball in.




6. The total number of strokes to complete all 18 holes on a golf course is usually 72.

The results of four players for 18 holes are shown in the table. Complete the table to

determine the score for each hole for each player and then identify the winner.

Hole Par

Paul Cath Tom Jill

strokes score strokes score strokes score strokes score

1 3 3 2 1 2

2 4 5 4 3 5

3 3 3 2 1 3

4 4 6 5 3 4

5 3 5 6 7 8

6 3 4 3 6 3

7 4 4 2 5 4

8 3 7 3 4 5

9 5 6 4 5 4

10 4 5 5 6 5

11 4 1 3 7 5

12 5 3 3 2 4

13 5 2 3 3 4

14 4 3 4 4 5

15 4 3 4 4 5

16 4 4 5 2 5

17 5 5 4 5 4

18 4 7 5 4 4

Final scores:


Activity 2


Investigation Negative Numbers

Use your calculator (if necessary) to investigate the truth of each of the given

statements.

If you think that the statement is false, write one example to support your conclusion.

If you think that the statement is true, write at least three examples to support your

conclusion.

Statements

1. Adding two negative numbers always results in a negative number.

2. Doubling a negative number will produce another negative number.

3. Squaring a negative number will produce another negative number.

4. Dividing a negative number by a counting number (1, 2, 3, …) always gives a

quotient that is negative.

5. The product of a negative number and a whole number is always positive.



c) Which player wins?Investigation (cont’d) Negative Numbers

6. Negative numbers are always whole numbers.

7. Adding a negative number to a positive number gives a negative answer.

8. Subtracting a positive number from a negative number always gives a

negative answer.

9. Subtracting any positive number from a smaller positive number always gives a

negative answer.

10. The sum of any positive number and a whole number is another whole

number.

11. Adding a negative fraction to a positive decimal can result in 0.



Activity 3

The card below is part of a game of Neggo where the number is crossed off if it is the result

of the operation that is called out. Your task is to devise three different operations for each

number on the card. The first one is shown as an example.

–30 –2 –42 –16 –24

–10 –8 –9 –18 –60

–12 –36 –1 –20 –100

Operation 1 Operation 2 Operation 3

–30 6 x –5 10 – 40 –10 – 20

–2

–42

–16

–24

–10

–8

–9

–18

–60

–12

–36

–1

–20

–100

MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Graphics 1 © Department of Education WA 2015



TASK 27: FRACTION GRAPHICS

Overview

This task requires students to demonstrate an understanding of fractions and operations on

fractions in the creation of a poster highlighting one particular fraction.

Students will need

calculators

access to the internet

printed media; e.g., newspapers




Express one quantity as a fraction of another with and without the use of digital


Connect fractions, decimals and percentages and carry out simple conversions

(ACMNA157)



o represent fractions in various ways

o create accurate statements of operations on their allocated fraction


FRACTION GRAPHICS Solutions and Notes for Teachers

Create an information poster for the fraction you have been allocated. For your fraction you

will need to show -

(i) Other number forms for your fraction, including three other equivalent fractions.

(ii) Models showing your fraction as a proportion of the model. These models should

include number lines, continuous area models; e.g., circles, rectangles as well as

discrete models (sets of numbers). Try to show a few examples of each type of model.

(iii) Write questions which require the use of operations with your fraction.

(iv) Write questions for which your fraction is the answer.

(v) Applications of your fraction to situations outside the classroom; e.g., media, news, the

environment etc.

Examine the fraction graphic for on the next page.

Students could work on this task in pairs or small groups.

Any fraction could be considered, such as follows:

The fractions that could be considered are endless in number and could be selected by the

students themselves.

(i) Equivalent number forms that students should consider are percentages, decimals and

equivalent fractions.

(ii) Students should be encouraged to think beyond the traditional representations of

fractions (circles, rectangles, squares) and examine a variety of uncommon

representations.

(iii) Operations should include addition, subtraction, multiplication, division and squaring.

Students may also consider negating and inverting the fraction.

(iv) Students should be encouraged to write questions in a context.

(v) Students could look on the internet or in printed material to determine some applications

of their fraction.


is one half

0.5 is one half

50% is one half

4=

50

00 =

33

66=

3

6 =

Half the smiley faces are blue

(discrete model)

Area models of one half

Number line

Operations

x 4 = 2 3.5 + = 4

5 ÷ = 10 of 8.8 = 4.4

+ + = 3 - = 0

x = 4

+ = 1

÷ 2 = 4

5 + 2 x = 6 Questions

1. Roughly what proportion of people

are males?

2. What will a shirt marked at $22 cost

if it is reduced to “half-price”?

3. After round 20 in 2015, the Dockers

had 64 points but 6 teams had less

than half that number. What was the

greatest number of points any of the 6

teams could have had?

4. The half-price of a polar fleece

jacket is $26. What would it normally

cost?

5. Where on the number line is - ?

6. If 10 oranges are cut into halves,

how many halves would there be?

Did you know that -

Dividing by a half is the same as

multiplying by 2.

The chance of a baby being a boy

or a girl is .

The inverse or reciprocal of is 2.

The opposite of is – .

If you toss a normal die, the

chance of the number coming up

being a prime number is .

























STUDENT COPY FRACTION GRAPHICS

Create an information poster for the fraction you have been allocated. For your fraction you

will need to show -

(i) Other number forms for your fraction, including three other equivalent fractions.

(ii) Models showing your fraction as a proportion of the model. These models should

include number lines, continuous area models; e.g., circles, rectangles as well as

discrete models (sets of numbers). Try to show a few examples of each type of model.

(iii) Write questions which require the use of operations with your fraction.

(iv) Write questions for which your fraction is the answer.

(v) Applications of your fraction to situations outside the classroom; e.g., media, news, the

environment etc.

Examine the fraction graphic for on the next page.


FRACTION GRAPHIC

is one half

0.5 is one half

50% is one half

4=

50

00 =

33

66=

3

6 =

Half the smiley faces are blue

(discrete model)

Area models of one half

Number line

Operations

x 4 = 2 3.5 + = 4

5 ÷ = 10 of 8.8 = 4.4

+ + = 3 - = 0

x = 4

+ = 1

÷ 2 = 4

5 + 2 x = 6 Questions

1. Roughly what proportion of people

are males?

2. What will a shirt marked at $22

cost if it is reduced to “half-price”?

3. After round 20 in 2015, the

Dockers had 64 points but 6 teams

had less than half that number. What

was the greatest number of points any

of the 6 teams could have had?

4. The half-price of a polar fleece

jacket is $26. What would it normally

cost?

5. Where on the number line is - ?

6. If 10 oranges are cut into halves,

how many halves would there be?

Did you know that -

Dividing by a half is the same as

multiplying by 2.

The chance of a baby being a boy

or a girl is .

The inverse or reciprocal of is 2.

The opposite of is – .

If you toss a normal dice, the

chance of the number coming up

being a prime number is .
























MATHS7TL002 | Mathematics | Number and Algebra Activity | Percentages 2 © Department of Education WA 2015

TASK 28: PERCENTAGES

Overview

This task consists of a series of activities designed to encourage thinking about percentages.

Students will need

calculators



Find percentages of quantities and express one quantity as a percentage of another,

with and without digital technologies. (ACMNA158)


fluency when they

o decide which operations to use in the routine questions and execute these

operations efficiently; e.g., 2(a) and 2(b)


o use 100 as the base for a percentage

reasoning when they

o explain the processes by which they obtained the solutions to questions involving

percentages


o determine the processes necessary to answer the questions asked in context –

particularly 2(f) and 2(g)


PERCENTAGES Solutions and Notes for Teachers

Activity 1

1. Here is some of the information printed on two packets of cereal.

Cereal Quantity per 100 g

Carbohydrate Fat Protein

A 69.5 g 1.7 g 8.6 g

B 64.7 g 1.7 g 7.8 g

(a) Complete the table to show the percentages of nutrients in each packet.

Cereal Percentage of nutrients in packets of cereal

Carbohydrate Fat Protein Sugar Fibre

A 69.5% 1.7% 8.6% 28.7% 10.2%

B 69.5% 1.7% 7.8% 24.7% 16.1%

(b) Explain how you determined your answer to part (a).

The figures are give as grams per 100 grams. The units are the same so the

proportions are amounts per 100 and as this is what percentages are, then the amount

in grams per 100 g is the same as the percentage. Per cent = Hundredths.

(c) Percentages are given for sugar and fibre and these are both types of carbohydrates.

Cereals A and B come from different size packets.

Cereal A is in a 460 g packet and Cereal B is in a 775 g packet.

What percentage of fibre would you expect in an 800 g packet of Cereal A?

10.2%

What percentage of sugar would you expect in a 400 g packet of Cereal B?

24.7%

Explain how you determined your answers.

They are the same as the percentages given.

If the product does not vary, then the percentage of its components does not change

as the amount of the product changes.


2. More data about Cereals A and B.

Cereal Quantity per 100 g Percentage of daily requirements per

average serve (40 g)

Sodium* Riboflavin Niacin Energy Sugar Iron

A 30 mg 1.06 mg 6.2 mg 7% 13% 25%

B 125 mg 1.05 mg 6.2 mg 7% 11% 20%

*Sodium: Not pure sodium, but compounds containing sodium.

(a) Is the percentage of niacin the same in both cereals? Explain.

Yes. There is 6.2 mg per 100 gm in each product. The same amount per 100 g indicates

the percentages are the same.

(b) Which % best represents the percentage of *sodium in Cereal B? Justify your choice.

125% 12.5% 1.25% 0.125% 0.0125%

125 mg per 100 g. These need to be the same units. There are 1000 mg in 1 g.

125 mg = 125 ÷ 1000 g = 0.125 g

0.125 g per 100 g means 0.125%

(c) Is the percentage of *sodium in Cereal A approximately four times the percentage of

*sodium in Cereal B? Explain.

Yes. Comparing 30 mg per 100 g to 125 g per 100 g, they are both out of 100 g. The

*sodium is measured in the same units in each case, so the percentages can be

compared. 125 is about 4 times 30 so the statement is true.

(d) To obtain all your daily iron requirements, how many grams would you need to eat of -

Cereal A? Cereal B?

160 g 200 g

(e) According to information on the packet, 12 mg of *sodium is equivalent to about 0.1 g

salt. Determine the % of salt in Cereal B.

125 mg salt per 100 g cereal is about 10 x 0.1 g salt.

This is about 1 g of salt per 100 g cereal.

So the percentage of salt is about 1%

(f) How many packets of Cereal B would you need to guarantee you have 1 kg sugar?

Cereal B has 24.7% of sugar; so 25% is a good estimate.

Each packet has 25% of sugar so 25% of 775 g = 193.75 g

There are 1000 g in 1 kg

1000 ÷ 193.75 = 5.16 packets

You would need 6 packets of cereal.


(g) On the packet of Cereal B, a statement is given

More than 20% of your daily fibre needs.

Assuming a standard serve (40 g) of Cereal B has 21% of the recommended daily

intake, determine the number of grams of fibre recommended each day.

40 g of cereal contains 16.1% fibre and this is 21% of the recommended daily intake

Each serve has 16.1% of 40 g = 6.44 g

6.44 g is 21% of the daily need

100 ÷ 21 x 6.44 = 30.7 g of fibre

Activity 2

Search the internet for “nutritional information” about the foods you eat for breakfast or lunch.

Locate three of four different foods and summarise the percentages of carbohydrates,

proteins and fats in each of these foods. Determine the percentages of daily-recommended

intakes in a standard serve for energy, sugar and iron. How do these foods compare with the

cereals presented on the previous pages?

Answers will vary

Activity 3

Write a dot-point summary of features of percentages that you think are important for

students to know, understand and be able to use.

Answers will vary


STUDENT COPY PERCENTAGES

Activity 1

1. Here is some of the information printed on two packets of cereal.

Cereal Quantity per 100 g

Carbohydrate Fat Protein

A 69.5 g 1.7 g 8.6 g

B 64.7 g 1.7 g 7.8 g

(a) Complete the table to show the percentages of nutrients in each packet.

Cereal Percentage of nutrients in packets of cereal

Carbohydrate Fat Protein Sugar Fibre

A 28.7% 10.2%

B 24.7% 16.1%

(b) Explain how you determined your answer to part (a).

(c) Percentages are given for sugar and fibre and these are both types of carbohydrates.

Cereals A and B come from different size packets.

Cereal A is in a 460 g packet and Cereal B is in a 775 g packet.

What percentage of fibre would you expect in an 800 g packet of Cereal A?

What percentage of sugar would you expect in a 400 g packet of Cereal B?

Explain how you determined your answers.


2. More data about Cereals A and B.

Cereal Quantity per 100 g Percentage of daily requirements per

average serve (40 g)

Sodium* Riboflavin Niacin Energy Sugar Iron

A 30 mg 1.06 mg 6.2 mg 7% 13% 25%

B 125 mg 1.05 mg 6.2 mg 7% 11% 20%

*Sodium: Not pure sodium, but compounds containing sodium.

(a) Is the percentage of niacin the same in both cereals? Explain.

(b) Which % best represents the percentage of *sodium in Cereal B? Justify your choice.

125% 12.5% 1.25% 0.125% 0.0125%

(c) Is the percentage of *sodium in Cereal A approximately four times the percentage of

*sodium in Cereal B? Explain.

(d) To obtain all your daily iron requirements, how many grams would you need to eat of -

Cereal A? Cereal B?


(e) According to information on the packet, 12 mg of *sodium is equivalent to about 0.1 g

salt. Determine the % of salt in Cereal B.

(f) How many packets of Cereal B would you need to guarantee you have 1 kg sugar?

(g) On the packet of Cereal B, a statement is given

More than 20% of your daily fibre needs

Assuming a standard serve (40 g) of Cereal B has 21% of the recommended daily

intake, determine the number of grams of fibre recommended each day.


Activity 2

Search the internet for “nutritional information” about the foods you eat for breakfast or lunch.

Locate three of four different foods and summarise the percentages of carbohydrates,

proteins and fats in each of these foods. Determine the percentages of daily-recommended

intakes in a standard serve for energy, sugar and iron. How do these foods compare with the

cereals presented on the previous pages?

Activity 3

Write a dot-point summary of features of percentages that you think are important for

students to know, understand and be able to use.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Discounts 2 © Department of Education WA 2015

TASK 30: DISCOUNTS

Overview

In this task students investigate discounts. They consider discounting by a fixed amount and

by a percentage. Students are directed to consider applying known facts to ease the

calculation of discounts rather than use an algorithm that may be difficult to remember.

Students will need

calculators


Find percentages of quantities and express one quantity as a percentage of another,

with and without digital technologies.(ACMNA158)


fluency when they

o determine further percentages given 10%


o distinguish between fixed change and proportional change

o explain their solution process

o recognise equivalences of percentages and fractions

reasoning when they

o explain which is the best discount offer in Activity 3


DISCOUNTS Solutions and Notes for Teachers

Review activity

1. Changing by a fixed amount. ‘Discount’ means to ‘not count’ or to ‘count off’.

The table below shows the original prices and the discounted prices for various goods.

Item Original

price Discounted

price Value of discount

Bluetooth speaker $144 $118 $26

Big red umbrella $329 $269 $60

Diamond ring (1.25, grade f) $13 210 $11 500 $1710

2 litres milk $2.19 $1.99 20c

Dual cab $22 800 $21 700 $1100

1 ball knitting wool $3.99 $2.99 $1

16 GB iPad mini $298 $258 $40

Vintage Lancia $45 609 $45 069 $540

2. Changing by a fixed percentage. Determine the amount by which these items are

discounted. Show how you determined your answer.

Item Original price

% of discount

Amount of discount

lawnmower $200 10% $20

pool pump $550 10% $55

steel-framed gazebo $150 10% $15

esky $80 20% $16

pyjamas $40 20% $8

sunglasses $250 20% $50


Activity 1

Determine the amount of the discount when the percentage of the discount varies.

Use your results to complete the table below.

Item Original price

10 % discount

20% discount

30% discount

cot set $100 $10 $20 $30

pillow set $30 $3 $6 $9

sewing machine $120 $12 $24 $36

bed sheets $80 $8 $16 $24

bed quilt $45 $4.50 $9 $13.50

doona $95 $9.50 $19 $28.50

1. How did you calculate 10% of each item?

Divide the original price by 10.


Double the 10% discount or divide the original price by 5.

3. Can you calculate 20% by multiplying your answer for 10% by 2? Justify your conclusion.

Yes. 10% of an amount is one tenth and 20% of the same amount is one fifth.

One fifth = 2 x one tenth

4. What will the sunglasses cost if the discount is 100%?

Nothing


5. For these items you are given the value of the 10% discount. Determine the value of the

other discounts and complete the table.

Item Original

price

Discounts

10% 1% 5% 70%

plastic chair $40 $4 40c $2 $28

park bench $150 $15 $1.50 $7.50 $105

barbeque $420 $42 $4.20 $21 $294

ladder $100 $10 $1 $5 $70

rake $4 40c 4c 20c $2.80

Explain how, given 10%, you can determine the other percentages.

To determine 1% of the original amount you can divide the 10% discount by 10.

To determine 5% of the original amount you can -

(a) halve the 10% discount OR

(b) multiply the 1% discount by 5

To determine the 70% discount you can -

(a) multiply the 5% discount by 14 OR

(b) multiply the 1% discount by 70 OR

(c) multiply the 10% discount by 7.


6. For these items you are given the value of the 30% discount. State the values of the

other discounts and determine the original price.

Item Original

price

Discounts

30% 10% 5% 1%

wheelbarrow $120 $36 $12 $6 $1.20

floodlight $80 $24 $8 $4 80c

tool cabinet $150 $45 $15 $7.50 $1.50

smoke alarm $10 $3 $1 50c 10c

shed $7000 $2100 $700 $350 $70

Explain how, given 30%, you can determine 10% and the original price.

To find 10%, given 30%, you can divide the discount amount by 3.

To determine the original price, you can multiply the 10% discount by 10.

Activity 2

A lounge dining setting is priced at $2000, and for the sale the proposed discount is 35%.

Show FOUR different ways by which you can determine the discount amount.

1. 10% of $2000 is $200.

30% is 3 x 10% so 30% of $2000 = 3 x $200 = $600

5% is half of 10% and half of $200 is $100.

35% = 30% + 5% = $600 + $100 = $700

2. Determine 1%. 1% of $2000 = $20

Determine 35%: 35% = 1% x 35 = $20 x 35 = $700

3. 10% of $2000 is $200

5% of $200 = $2000 ÷ 20 = $100

35% = 10% + 10% + 10% + 5% = $200 + $200 + $200 + $100 = $700

4. 35% = 35 hundredths = 0.35

0.35 x $2000 = $700


Activity 3

Your grandmother wants to buy a new dress. Which of the following discounts will be better?

Justify your choice.

If the dress costs exactly $100 then 30% off is equal to $30 off.

If the dress costs less than $100; e.g., $90, then 30% off is $27 and this is less than $30.

If the dress costs more than $100; e.g., $120, then 30% off is $36 and this is more than $30.

Conclusion

You get a greater value of discount with 30% if the dress is more than $100.

You get a smaller discount with $30 if the dress is more than $100.

You get the same discount with 30% and $30 off if the dress costs $100.

Activity 4

Jody wanted to buy a pair of shoes. The shop had a week-long sale and claimed that all

shoes were already marked down 10%. On the last day of the sale the owner of the shoe

store marked all shoes down another 10%. Was this the same as a 20% reduction on the

original price?

Use three different starting prices to test this theory and write a conclusion.

Price Price with first 10% off Price with second 10% off Price reduced by 20%

$100 $90 $81 $80

$60 $54 $48.60 $48

$250 $225 $202.50 $200

Conclusion: A one-off discount of 20% gives a greater discount than a 10% discount followed

by another 10% discount. Note that the second 10% is calculated on a smaller amount than

the original price.




STUDENT COPY DISCOUNTS

Review activity

1. Changing by a fixed amount. ‘Discount’ means to ‘not count’ or to ‘count off’.

The table below shows the original prices and the discounted prices for various goods.

Item Original

price Discounted

price Value of discount

Bluetooth speaker $144 $118

Big red umbrella $329 $269

Diamond ring (1.25, grade F) $13 210 $11 500

2 litres milk $2.19 $1.99

Dual cab $22 800 $21 700

1 ball knitting wool $3.99 $2.99

16 GB iPad mini $298 $258

Vintage Lancia $45 609 $45 069

2. Changing by a fixed percentage. Determine the amount by which these items are

discounted. Show how you determined your answer.

Item Original price

% of discount

Amount of discount

lawnmower $200 10%

2-bedroom apartment $550 000 10%

steel-framed gazebo $150 10%

esky $80 20%

pyjamas $40 20%

sunglasses $250 20%


Activity 1

Determine the amount of the discount when the percentage of the discount varies.

Use your results to complete the table below.

Item Original price

10 % discount

20% discount

30% discount

cot set $100

pillow set $30

sewing machine $120

bed sheets $80

bed quilt $45

doona $95



3. Can you calculate 20% by multiplying your answer for 10% by 2? Justify your conclusion.

4. What will the sunglasses cost if the discount is 100%?


5. For these items you are given the value of the 10% discount. Determine the value of the

other discounts and complete the table.

Item Original

price

Discounts

10% 1% 5% 70%

plastic chair $40 $4

park bench $150 $15

bbq $420 $42

ladder $100 $10

rake $4 40c

Explain how, given 10%, you can determine the other percentages.

6. For these items you are given the value of the 30% discount. State the values of the

other discounts and determine the original price.

Item Original

price

Discounts

30% 10% 5% 1%

wheelbarrow $36

floodlight $24

tool cabinet $45

smoke alarm $3

shed $2100

Explain how, given 30%, you can determine 10% and the original price.


Activity 2

A lounge dining setting is priced at $2000, and for the sale the proposed discount is 35%.

Show FOUR different ways by which you can determine the discount amount.

Activity 3

Your grandmother wants to buy a new dress. Which of the following discounts will be better?

Justify your choice.




Activity 4

Jody wanted to buy a pair of shoes. The shop had a week-long sale and claimed that all

shoes were already marked down 10%. On the last day of the sale the owner of the shoe

store marked all shoes down another 10%. Was this the same as a 20% reduction on the

original price?

Use three different starting prices to test this theory and write a conclusion.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Moving Points 2 © Department of Education WA 2015

TASK 32: MOVING POINTS

Overview

This task provides an opportunity for students to relate specific movements on the Cartesian

plane to particular changes to the points‟ coordinates.

Students will not need any special equipment


Given coordinates, plot points on the Cartesian plane, and find coordinates for a given

point (ACMNA178)



fluency when they

o calculate accurately with integers

o read and plot points on the Cartesian plane


o connect the relationship between changes to the coordinates and movement on

the Cartesian plane

o express changes to coordinates as algebraic expressions


MOVING POINTS Solutions and Notes for Teachers

Activity 1

1. Review plotting points by plotting the following points and labelling the points with the

letters provided.

A (2,3) B (1,0) C (7,10) D (-4,6) E (-9,0) F (-5,-5) G (-7,8) H (0,-8)

2. Plot any points J K L M N P and ask another student to identify them.

Various recordings & answers.

In the following activities you are asked to “move” a point. The point doesn‟t really move as it

is a position. The new point is positioned away from the original point. When asked to move

a point, label the new point with the same letter but add a dash; e.g., A „moves‟ to A‟.


Activity 2

1. Enter the coordinates of the points A to F into the table below. Then “move” the points as

instructed in the table and label the “moved” points.

,

Point Coordinates of given point

„Move‟ the point ... Coordinates of „moved‟ point

A (6,2) Left 4 units (2,2)

B (4,8) Left 5 units (-1,8)

C (-1,3) Left 7 units (-8,3)

D (1,-5) Right 6 units (7,-5)

E (-2,-2) Right 2 units (0,-2)

F (-9,-1) Right 3 units (-6,-1)

2. When a point is moved left or right:

(a) Does the x-value of the point change? YES

(b) Does the y-value of the point change? NO


Activity 3

1. Enter the coordinates of the points G to L into the table below. Then “move” the points as


,



G (8,2) Up 6 units (8,8)

H (5,-5) Up 3 units (5,-2)

I (-7,-3) Up 7 units (-7,4)

J (-2,8) Down 4 units (-2,4)

K (-2,-1) Down 3 units (-2,-4)

L (2,5) Down 7 units (2,-2)

2. When a point is „moved‟ up or down:

(a) Does the x-value of the point change? No

(b) Does the y-value of the point change? Yes


Activity 4

Enter the coordinates of the points M to T into the table below. Then “move” the points as




M (8,-3) Up 3 units then left 2 units (6,0)

N (2,4) Up 3 units then right 2 units (4,7)

O (-4,6) Down 5 units then left 5 units (-9,1)

P (0,0) Left 9 units then down 9 units (-9,-9)

Q (1,-5) Left 1 unit then up 10 units (0,5)

R (-4,-5) Right 4 units then down 4 units (0,-9)

S (-8,-4) Up 4 units then right 4 units (-4,0)

T (9,9) Down 9 units then right 1 unit (10,0)


Activity 5

1. Points W, X, Y and Z were moved to W‟, X‟, Y‟ and Z‟. The movements are given below. Give

the coordinates of the original points W, X, Y and Z.

(a) W‟ is at (4, 2) after W was moved 4 units left. Where was W to start with? (8, 2)

(b) X‟ is at (1, 0) after X was moved 3 units up. Where was X to start with? (1,-3)

(c) Y‟ is at (-6, 7) after Y was moved 1 unit right and 1 unit down.

Where was Y to start with? (-7,8)

(d) Z‟ is at (3, -6) after Z was moved 3 units up and 6 units left. Where was Z to start with?

(0,0)

2. State the coordinates of the new points that will be formed from the following movements.

Point to be „moved‟ Movement New coordinates

(20, 20) Right 4 units (24, 20)

(-10, 30) Left 6 units (-16, 30)

(-20, 0) Up 7 units (-20, 7)

(7, 50) Down 10 units (7, 40)

(20, 25) Left 20 units, down 25 units (0, 0)

(12, 16) Up 2 units, right 3 units (15, 18)

(100, 100) Down 50 units, right 50 units (150, 50)

(20, 50) Right 10 units, up 5 units (30, 55)

(x, y) Up 6 units (x, y + 6)

(x, y) Down 5 units (x, y - 5)

(x, y) Right 3 units (x + 3, y)

(x, y) Left 3 units (x - 3, y)

(x, y) Up m units (x , y + m)

(x, y) Right k units (x + k, y)

(x, y) Down w units then left b units (x - b, y - w)

(x, y) Left k units then up h units (x - k, y + h)


3. Create a table with points to be moved and movements listed (similar to the previous table)

and ask another student to determine the new coordinates.


Various answers

4. Write a summary of what happens to the coordinates of a point when it is “moved”

vertically or horizontally.

If a point is moved horizontally then the first number of the coordinates, the x-value,

increases by the number of units moved if the movement is to the right, and decreases if the

movement is to the left.

If a point is moved vertically then the second number of the coordinates, the y-value

increases by the number of units moved if the movement is up, and decreases if the

movement is down.


STUDENT COPY MOVING POINTS

Activity 1

1. Review plotting points by plotting the following points and labelling the points with the

letters provided.

A (2,3) B (1,0) C (7,10) D (-4,6) E (-9,0) F (-5,-5) G (-7,8) H (0,-8)

2. Plot any points J K L M N P and ask another student to identify them.

In the following activities you are asked to “move” a point. The point doesn‟t really move as it

is a position. The new point is positioned away from the original point. When asked to move

a point, label the new point with the same letter but add a dash; e.g., A „moves‟ to A‟.



Activity 2

1. Enter the coordinates of the points A to F into the table below. Then “move” the points as


,



A Left 4 units

B Left 5 units

C Left 7 units

D Right 6 units

E Right 2 units

F Right 3 units

2. When a point is moved left or right:

(a) Does the x-value of the point change?

(b) Does the y-value of the point change?



Activity 3

1. Enter the coordinates of the points G to L into the table below. Then “move” the points as


,



G Up 6 units

H Up 3 units

I Up 7 units

J Down 4 units

K Down 3 units

L Down 7 units

2. When a point is „moved‟ up or down:

(a) Does the x-value of the point change?

(b) Does the y-value of the point change?



Activity 4

Enter the coordinates of the points M to T into the table below. Then “move” the points as




M Up 3 units then left 2 units

N Up 3 units then right 2 units

O Down 5 units then left 5 units

P Left 9 units then down 9 units

Q Left 1 unit then up 10 units

R Right 4 units then down 4 units

S Up 4 units then right 4 units

T Down 9 units then right 1 unit



Activity 5

1. Points W, X, Y and Z were moved to W‟, X‟, Y‟ and Z‟. The movements are given below. Give

the coordinates of the original points W, X, Y and Z.

(a) W‟ is at (4, 2) after W was moved 4 units left. Where was W to start with?

(b) X‟ is at (1, 0) after X was moved 3 units up. Where was X to start with?

(c) Y‟ is at (-6, 7) after Y was moved 1 unit right and 1 unit down.

Where was Y to start with?

(d) Z‟ is at (3, -6) after Z was moved 3 units up and 6 units left. Where was Z to start with?

2. State the coordinates of the new points that will be formed from the following movements.


(20, 20) Right 4 units

(-10, 30) Left 6 units

(-20, 0) Up 7 units

(7, 50) Down 10 units

(20, 25) Left 20 units, down 25 units

(12, 16) Up 2 units, right 3 units

(100, 100) Down 50 units, right 50 units

(20, 50) Right 10 units, up 5 units

(x, y) Up 6 units

(x, y) Down 5 units

(x, y) Right 3 units

(x, y) Left 3 units

(x, y) Up m units

(x, y) Right k units

(x, y) Down w units then left b units

(x, y) Left k units then up h units


3. Create a table with points to be moved and movements listed (similar to the previous table)

and ask another student to determine the new coordinates.


4. Write a summary of what happens to the coordinates of a point when it is “moved”

vertically or horizontally.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Operations 2 © Department of Education WA 2015

TASK 37: FRACTION OPERATIONS

Overview

The focus of this task is on the addition and subtraction of fractions with unrelated

denominators. Students should attempt to complete these operations using mental

arithmetic.



Solve problems involving addition and subtraction of fractions, including those with

unrelated denominators (ACMNA153)




fluency when they

o represent fractions in various ways

o determine fraction addition using different models


o can develop a model to simulate fraction addition

o identify fractions to create matching number sentences

reasoning when they

o explain their use of models for fraction addition

o determine a strategy to locate a fraction halfway between two others


o determine the processes needed to answer the questions in Activity 1


FRACTION OPERATIONS Solutions and Notes for Teachers

Activity 1

1. Three family-sized pizzas were shared by five people and the table below shows the

number of pieces eaten by each person and the number left. For any particular pizza all the

pieces were the same size but each pizza was cut into a different number of pieces.

Person Alf Buzz Cody Dani Elly Leftover

Pizza 1 2 3 1 4 2 0

Pizza 2 1 2 2 1 2 0

Pizza 3 1 1 1 0 2 1

This table is reproduced below as a tool to assist with answering the questions. What fraction

of each pizza did each person eat? See table below.


Pizza 1

Pizza 2

Pizza 3

2. Buzz and Elly ate the same number or pieces, and so did Alf and Cody. Did each pair eat

the same amount of pizza? Explain your answer mathematically?

Buzz ate = + + =

Elly ate 2 2 2 4 6 8 18

12 8 6 24 24 24 24 Elly ate more than Buzz

Alf ate 2 1 1 4 3 4 11

12 8 6 24 24 24 24

Cody ate 1 2 1 2 6 4 12

12 8 6 24 24 24 24 Cody ate more than Alf

3. What fraction of the three pizzas was eaten by Dani?

Dani ate 4 1 8 3 11

12 8 24 24 24 of 1 pizza. This is equivalent to

11

72 of 3 pizzas.


Activity 2

I have five fraction tiles as shown below.

Select any of these fractions to create true statements using the number sentences below.

A. + + < 1

B. + + > 1.5

C. _ + < 0

D.

+ _ =

E. _ _ =

55

5

5

5

7

5
























Activity 3

1. Consider the following pairs of fractions.

Determine the fraction that is halfway between each member of the pairs. For each pair, use

a number line to show the positions of the three fractions.

A. and 3

B. and

C. and

D. and

2. Describe a strategy that could be used to determine the fraction halfway between any pair

of fractions.

(1) Add the two fractions and divide the result by 2,

OR

(2) Express the original fractions in an equivalent form – both with the same denominator.

If the numerators add to an even number then halve that sum and express the answer as a

fraction with the denominator.

If the numerators add to an odd number; e.g. 5, then double the denominator. Example:

1 9 2 9

5 10 10 10

2 9 11 20

11

20

and and

so double the denominator is

is


Activity 4

Use the models provided to show that

+

Explain your use of these models.

First model:

Divide the model vertically into five even columns. Colour 2 columns which is 2 out of 5.

Divide the model horizontally into 4 even rows. Colour in two rows – or the equivalent of two

rows – you cannot colour in parts that are already coloured. Count up the number of parts

coloured. Consider how many parts are coloured (13) and how many parts altogether (20).

So the answer is 13 out of 20 or

Second model:

Colour in 2 out of every 5 smiley faces. Now colour in 1 for every 4 of all of them but do not

colour in one already coloured. So the answer is 13 out of 20 or



STUDENT COPY FRACTION OPERATIONS

Activity 1

1. Three family-sized pizzas were shared by five people and the table below shows the

number of pieces eaten by each person and the number left. For any particular pizza all the

pieces were the same size but each pizza was cut into a different number of pieces.


Pizza 1 2 3 1 4 2 0

Pizza 2 1 2 2 1 2 0

Pizza 3 1 1 1 0 2 1

This table is reproduced below as a tool to assist with answering the questions. What fraction

of each pizza did each person eat?


Pizza 1

Pizza 2

Pizza 3

2. Buzz and Elly ate the same number or pieces, and so did Alf and Cody. Did each pair eat

the same amount of pizza? Explain your answer mathematically?

3. What fraction of the three pizzas was eaten by Dani?


Activity 2

I have five fraction tiles as shown below.

Select any of these fractions to create true statements using the number sentences below.

A. + + < 1

B. + + > 1.5

C. _ + < 0

D.

+ _ =

E. _ _ =

55

7
























Activity 3

1. Consider the following pairs of fractions.

Determine the fraction that is halfway between each member of the pairs. For each pair, use

a number line to show the positions of the three fractions.

A. and 3

B. and

C. and

D. and

2. Describe a strategy that could be used to determine the fraction halfway between any pair

of fractions.


Activity 4

Use the models provided to show that + Explain your use of these models.

First model:

Second model:






















MATHS7TL002 | Mathematics | Number and Algebra Activity | Fraction Action 2 © Department of Education WA 2015

TASK 38: FRACTION ACTION

Overview

This task involves the use of different activities to foster a deeper understanding of fractions

and to provide opportunities to represent fractions in various ways. Use of mental arithmetic

to add and subtract fractions is expected for all questions except the final one in Activity 4.

Students will need

calculators





Solve problems involving addition and subtraction of fractions, including those with

unrelated denominators (ACMNA153)

Express one quantity as a fraction of another with and without the use of digital


Recognise and solve problems involving simple ratios (ACMNA173)


fluency when they

o represent the given fraction in various ways in Activity 1


o develop a fraction wall showing equivalent fractions

reasoning when they

o apply their understanding of ratios


o determine processes to answer questions in Activity 4


FRACTION ACTION Solutions and Notes for Teachers

Activity 1

The table below shows multiple representations for two thirds. Create a similar table for nine

fifths. Examples shown on next page.

One third plus one third 3

2Two thirds

Four thirds take away two thirds

One third times two

Two oranges pizzas shared

equally between 3 people

3

1

3

1 3

211

3

2

3

4

2 3

1One third of

two 2 ÷ 3

3

1 2 3

11

3

2of 1

3

21

3

11 -

3

2

6

4

3

1

3

1

3

1

3

1

3

1

3

1

One minus one third 30

20

0 1 2 3


10 1

5 5

19

5

Nine fifths

Two minus one fifth

One plus four fifths

Nine oranges shared equally

between 5 people

1.8 8 1

5 5

2 3 4

5 5 5

3 3 3

5 5 5

9

5

18

10

1

9

5÷ 1

51

9

182

5

9

5of 1

33

5

20 11

5 5 Nine divided by 5

1

180%


Activity 2

1. Money has been shared out between two people, but not equally. Determine the ratio of

Ted’s share to Pat’s share as well as the fraction of the whole amount that each person

received. Complete the table provided.

Ted’s amount

Pat’s amount

Ratio Ted: Pat

Total money

Ted’s fraction

Pat’s fraction

$10 $20 1: 2 $30

$40 $10 4:1 $50

$35 $350 1:10 $385

$80 $10 8:1 $90

$75 $5 15:1 $80

$22 $44 1:2 $66

$70 $30 7:3 $100

2. Elia and Ali are sharing different amounts of money according to the ratios given.

Complete the table provided.

Amount of money to share

Ratio Elia: Ali

Elia’s amount Ali’s amount Elia’s

fraction Ali’s

fraction

$100 2:3 $40 $60

$200 1:1 $100 $100

$500 3:7 $150 $350

$350 5:2 $250 $100

$400 5:3 $250 $150

$800 9:1 $720 $80

$135 3:2 $81 $54

3. If the ratio was not known and was written as a : b, what fraction of the money would each

person get?

anda b

a b a b


Activity 3

Design and create a “fraction wall” to show the relationship between the following fractions:

Twentieths, tenths, quarters, eighths, fifths, halves.

1

Activity 4

1. A truck is one-third full of sand. When 45 kg is added it is half full.

How much more sand can be placed in the truck?

1 145

2 3

145

6

45 6 1 270

truckload truckload

truckload

truckload kg

2. A bus starts from the terminal with all seats occupied. At the first stop one third of the

passengers get off and 12 people get on. At the next stop, a half of the new total of

passengers get off and four people get on. There are now 30 passengers on the bus. How many passengers started the trip?


60

1 230 4 (12

2 3

1 2(12

2 3

2(12

3

240

3

of the original number




original number

)

26 = )

52 = )

3. The perimeter of a triangle is 52 cm. All measurements are whole numbers. The shortest

side is half the length of the longest side and the third side is two-thirds of the longest side.

Determine the length of each side.

The longest side must be a multiple of 3 and it must be more than a third of 52. It also must

be even. More than 17, 18 too close, 21 not even. Try 24.

The shortest side would be 12 and the third side must be two-thirds of 24 which is 16.

12 + 16 + 24 = 52 ... good thinking!

4. One Friday afternoon during sport, a quarter of the Year 7 students elected to play

soccer, one-third played frizzball, and that left 40 students who had chosen swimming. How

many students were in Year 7?

1 1 7

3 4 12

5

12

1

12

of Year 7 students is equivalent to 40.

of Year 7 students is equivalent to 8

So 96 students in Year 7

5. A crate half-full of apples has a mass of 130 kg. The same crate had a mass of 90 kg

when it was one-third full of apples. Determine the mass of the empty crate.

Crate + half the apples = 130 kg

Crate + a third of the apples = 90 kg

Difference of 40 kg is mass of

40 kg =

240 kg = mass of the apples

120 kg = mass of half the apples

Crate has a mass of 130 kg – 120 kg = 10 kg


6. When John Isner played Roger Federer at the US Open in 2015 some of their statistics

were as follows:

(i) Federer got 93 of his 133 first serves in and won the point 63 times. When the first

serve was not in he had a second serve (40 second serves) and he won the point 22 times.

(ii) Isner got 65 out of 111 first serves in and won the point 54 times. On the second serve

Federer won the point 22 out 40 times and Isner won the point 33 out of 46 times.

[Data obtained from the US Open tennis official website.]

Compare the statistics of both players and use mathematical arguments to show who won

the highest percentage of points on their serves.

Federer Isner

First serves in = = 0.699 = 70% = 0.586 = 59%

Points won on first serve in = 0.677 = 68% = 0.830 = 83%

Points won on second serve = 0.55 = 55% = 0.717 = 72%


STUDENT COPY FRACTION ACTION

Activity 1

The table below shows multiple representations for two thirds. Create a similar table for nine

fifths.

One third plus one third

Two thirds

Four thirds take away two thirds

One third times two

Two oranges pizzas shared

equally between 3 people

3

211

2

One third of two 2 ÷ 3

2 of 1

-

One minus one third

3

2

3

1

3

1

3

2

3

4

3

1

3

13

11

3

2

3

21

3

11

3

2

6

4

3

1

3

1

3

1

3

1

3

1

3

1

30

20

0 1 2 3















Nine fifths


Activity 2

1. Money has been shared out between two people, but not equally. Determine the ratio of

Ted’s share to Pat’s share as well as the fraction of the whole amount that each person

received. Complete the table provided.

Ted’s amount

Pat’s amount

Ratio Ted: Pat

Total money

Ted’s fraction

Pat’s fraction

$10 $20 1: 2

$40 $10

$35 $350

$80 $10

$75 $5

$22 $44

$70 $30

2. Elia and Ali are sharing different amounts of money according to the ratios given.

Complete the table provided.

Amount of money to share

Ratio Elia: Ali

Elia’s amount Ali’s amount Elia’s

fraction Ali’s

fraction

$100 2:3 $40 $60

$200 1:1

$500 3:7

$350 5:2

$400 5:3

$800 9:1

$135 3:2

3. If the ratio was not known and was written as a : b, what fraction of the money would each

person get?


Activity 3

Design and create a “fraction wall” to show the relationship between the following fractions:

Twentieths, tenths, quarters, eighths, fifths, halves.


Activity 4

1. A truck is one-third full of sand. When 45 kg is added it is half full.

How much more sand can be placed in the truck?

2. A bus starts from the terminal with all seats occupied. At the first stop one third of the

passengers get off and 12 people get on. At the next stop, a half of the new total of

passengers get off and four people get on. There are now 30 passengers on the bus. How many passengers started the trip?

3. The perimeter of a triangle is 52 cm. All measurements are whole numbers. The shortest

side is half the length of the longest side and the third side is two-thirds of the longest side.

Determine the length of each side.

4. One Friday afternoon during sport, half of the Year 7 students elected play soccer, one-

third played frizzball, and that left 40 students who had chosen swimming. How many

students in Year 7?


5. A crate half-full of apples has a mass of 130 kg. The same crate had a mass of 90 kg

when it was one-third full of apples. Determine the mass of the empty crate.

6. When John Isner played Roger Federer at the US Open in 2015 some of their statistics

were as follows:

(i) Federer got 93 of his 133 first serves in and won the point 63 times. When the first

serve was not in he had a second serve (40 second serves) and he won the point 22 times.

(ii) Isner got 65 out of 111 first serves in and won the point 54 times. On the second serve

Federer won the point 22 out 40 times and Isner won the point 33 out of 46 times.

[Data obtained from the US Open tennis official website.]

Compare the statistics of both players and use mathematical arguments to show who won

the highest percentage of points on their first and second serves.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Graphing Relationships 2 © Department of Education WA 2015

TASK 40: GRAPHING RELATIONSHIPS

Overview

For this task students may need further support if they have not previously used letters to

represent variables or if they are lacking in experience with line graphs. For all of these tasks

there is a simple linear relationship between the variables. Each situation can be described

as a rate, though the term is not used in the student activities.



Investigate, interpret and analyse graphs from authentic data (ACMNA180)

Introduce the concept of variables as a way of representing numbers using letters

(ACMNA175)


fluency when they

o complete tables by identifying and continuing patterns


o plot data points

reasoning when they

o interpret graphs of authentic data

o describe the features common to the graphs and tables in this task


o create formulae to represent relationships between variables


GRAPHING RELATIONSHIPS Solutions and Notes for Teachers

Situation 1

Susie works at the local pharmacy store where she is paid $20 per hour.

Examine the table and related graph showing Susie's pay.

Number of hours 0 1 2 3 4 5 6 7

Pay 0 $20 $40 $60 $80 $100 $120 $140

1. How much would Susie get paid for 8 hours’ work? $160

Add this point to the graph above.

2. Does it make sense to join the points? (Can you read the values in between the points?)

Use an example in your explanation.

Yes, you can read values between points. Working for 1.5 hours, Susie could get $30 pay

3. Describe the rule linking number of hours worked and amount paid in -

(i) words

Susie’s pay = number of hours x 20

(ii) symbols (Use P for pay and h for hours worked)

P = 20 x h

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6 7 8 9

Pay ($)

HOURS WORKED

Susie's Pay



Situation 2

Jon is graphing the cost of making apple cakes.

The table and graph show some of these costs.

Number of cakes

0 1 2 3 4 5 6 7

Cost ($) 0 $6 $12 $18 $24 $30 $36 $42

1. Complete both the graph and the table, given that there is a fixed cost per cake.



It does not make sense to join the points because you do not make half a cake. There is not

value for 1.5 cakes.

3. Describe the rule linking number of cakes made and the cost using -

(i) words

Cost = number of cakes x 6

(ii) symbols (Use P for cost and c for number of cakes made)

P = c x 6

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8 9 10

Cost ($)

Number of cakes

Costing cakes



Situation 3

1. There are five tables below showing the costs of different fruits. For each one -

complete the table provided;

plot the graph showing the relationship between number of kilograms and cost;

write a rule linking number of kilograms and cost; and

check that each graph is labelled and points joined if appropriate.

Pears

Number of

kilograms (d) 1 2 3 4 5 6 7 8 9

Total Cost (C) $5 $10 $15 $20 $25 $30 $45 $40 $45

Cost per kg $5 Rule C = d x 5 OR C = 5 d

Peaches

Number of

kilograms (d) 1 2 3 4 5 6 7 8

Total Cost (C) $5.50 $11 $16.50 $22 $27.50 $33 $38.50 $44

Cost per kg $5.50 Rule C = d x 5.5 OR C = 5.5 d

Grapes

Number of

kilograms (d) 1 2 3 4 5 6 7 8 9 10

Total Cost (C) $6 $12 $18 $24 $30 $36 $42 $48 $54 $60


Nectarines

Number of

kilograms (d) 1 2 3 4 5 6 7 8 9

Total Cost (C) $8 $16 $24 $32 $40 $48 $56 $64 $72


Oranges

Number of

kilograms (d) 1 2 3 4 5 6 7 8 9

Total Cost (C) $3 $6 $9 $12 $15 $18 $21 $24 $27


Note: These graphs may be presented by students as connected points.


2. Write the fruits in order of cost per kilogram.

A: Oranges B: Pears C: Peaches D: Grapes E: Nectarines

3. Write the names of the graphs in order of steepness.

A: Oranges B: Pears C: Peaches D: Grapes E: Nectarines

4. Comment on your findings.

The more the fruit costs per kilogram, the steeper the graph.


Situation 4

For the following vegetable prices

Write the rule linking mass (w) and cost (C)

Create a table of costs.

1. Carrots cost $1.50 per kilogram

Rule: Cost = number of kilogram x $1.50 C = 1.50 x w

Number of

kilograms (w) 0 1 2 3 4 5 6 7 8

Cost (C) 0 $1.50 $3 $4.50 $6 $7.50 $9 $10.50 $12

2. Tomatoes cost $3.85 per kilogram


Number of

kilograms (w) 0 1 2 3 4 5 6 7 8

Cost (C) 0 $3.85 $7.70 $11.55 $15.40 $19.25 $23.10 $26.95 $$30.80

3. Potatoes cost $2.50 per kilogram


Number of

kilograms (w) 0 1 2 3 4 5 6 7 8

Cost (C) 0 $2.50 $5 $7.50 $10 $12.50 $15 $17.50 $20

4. Onions cost $1.20 per kilogram


Number of

kilograms (w) 0 1 2 3 4 5 6 7 8

Cost (C) 0 $1.20 $2.40 $3.60 $4.80 $6.00 $7.20 $8.40 $9.60

5. Plot the costs of these vegetables (up to 10 kg) on the grid on the next page.


Reflection

Summarise the features of the tables and graphs that have been used in this task.

All of these graphs are straight lines.

The more the food costs per kilogram the steeper the line for its graph.

All graphs have the point (0,0) because it does not cost money to buy “nothing”.

It makes sense to join the points because you can read values in between the given points.

The changing mass is plotted on the horizontal axis.

The changing total cost is plotted on the vertical axis.

The tables all have the cost increasing by the same amount (as does the mass of the food

considered) which is the cost per kilogram.

The rules are all similar: Total cost = cost per kilogram x number of kilograms.


STUDENT COPY GRAPHING RELATIONSHIPS

Situation 1

Susie works at the local pharmacy store where she is paid $20 per hour.

Examine the table and related graph showing Susie's pay.

Number of hours 0 1 2 3 4 5 6 7

Pay 0 $20 $40 $60 $80 $100 $120 $140

1. How much would Susie get paid for 8 hours’ work?

Add this point to the graph above.

2. Does it make sense to join the points? Can you read the values in between the points?


3. Describe the rule linking number of hours worked and amount paid using -

(i) words

(ii) symbols (Use P for pay and h for hours worked).

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8

Pay ($)

HOURS WORKED

Susie's Pay



Situation 2

Jon is graphing the cost of making apple cakes.

The table and graph show some of these costs.

Number of cakes

0 1 2 3 4 5 6 7

Cost ($) $6 $12 $18 $24

1. Complete both the graph and the table, given that there is a fixed cost per cake.



3. Describe the rule linking number of cakes made and the cost using -

(i) words

(ii) symbols (Use P for cost and c for number of cakes made).

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8 9 10

Cost ($)

Number of cakes

Costing cakes



Situation 3

1. There are five tables below showing the costs of different fruits. For each one -

complete the table provided;

plot the graph showing the relationship between number of kilograms and cost;

write a rule linking number of kilograms and cost; and

check that each graph is labelled and points joined if appropriate.

Pears

Number of

kilograms (d) 1 2 3 4 5

Total Cost (C) $5 $10 $15 $20 $25

Cost per kg Rule

Peaches

Number of


Total Cost (C) $5.50 $11 $16.50 $22 $27.50

Cost per kg Rule

Grapes

Number of


Total Cost (C) $6 $12 $18

Cost per kg Rule

Nectarines

Number of


Total Cost (C) $8 $16 $24 $32 $40

Cost per kg Rule

Oranges

Number of


Total Cost (C) $3 $6 $9 $12 $15

Cost per kg Rule


2. Write the fruits in order of cost per kilogram.

3. Write the names of the graphs in order of steepness.

4. Comment on your findings.


Situation 4

For the following vegetable prices

Write the rule linking mass (w) and cost (C)

Create a table of costs.

1. Carrots cost $1.50 per kilogram

Rule:

2. Tomatoes cost $3.85 per kilogram

Rule:

3. Potatoes cost $2.50 per kilogram

Rule:

4. Onions cost $1.20 per kilogram

Rule:

5. Plot the costs of these vegetables (up to 10 kg) on the grid on the next page.


Reflection

Summarise the features of the tables and graphs that have been used in this task.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Consecutive Numbers 2 © Department of Education WA 2015

TASK 101: CONSECUTIVE NUMBERS

Overview

In this task, students will investigate the addition and subtraction of consecutive numbers.

The activities provide opportunities for students to make connections between consecutive

numbers and the associative law. Students are allowed to choose appropriate methods and

apply their existing strategies to seek solutions. Explaining connections or patterns will help

in the promotion of reasoning mathematically.

No special equipment required






fluency when they

o identify a way of recording the information

o use their process to record the information and calculate solutions


o identify any connections or patterns in their results

reasoning when they

o show how to identify these connections or patterns


o explain how they know they have all of the combinations


CONSECUTIVE NUMBERS Solutions and Notes for Teachers

Activity 1

1. Choose any four consecutive numbers between 1 and 9, for example, 1, 2, 3, and 4.

1, 2, 3, 4

2. You want to find all of the different ways of ADDING and SUBTRACTING all of your

chosen numbers. (NOTE: your chosen numbers must always be in numerical order)

How can you do this in a logical way? Can you think of more than one way?

Write a column of each number, with spaces.

Start with all ‘+’ signs.

Change one sign to a ‘-‘, starting from right and moving it left until all signs have been

changed.

Change two signs to a ‘-‘, starting from the right and moving left until all possible

combinations have been changed.

Lastly, all ‘-‘ signs.

3. Using one of your methods from above, record all of the different combinations.

Calculate each answer.

You may want the students to number the operations to help when explaining a rule.

a. 1 + 2 + 3 + 4 = 10

b. 1 + 2 + 3 – 4 = 2

c. 1 + 2 – 3 + 4 = 4

d. 1 – 2 + 3 + 4 = 6

e. 1 + 2 – 3 – 4 = -4

f. 1 – 2 – 3 + 4 = 0

g. 1 – 2 + 3 – 4 = -2

h. 1 – 2 – 3 – 4 = -8

4. How do you know you have all of the combinations?

8 combinations

1 x all +

1 x all –

3! = 6

1 + 1 + 6 = 8

Students may provide a different answer.


Activity 2:

1. Try choosing a different set of numbers and apply your process from above.

Ensure the students calculate the solutions in the same order as in Activity 1.

Ask students show how many combinations should there are.

a. 2 + 3 + 4 + 5 = 14

b. 2 + 3 + 4 – 5 = 4

c. 2 + 3 – 4 + 5 = 6

d. 2 – 3 + 4 + 5 = 8

e. 2 + 3 – 4 – 5 = -4

f. 2 – 3 – 4 + 5 = 0

g. 2 – 3 + 4 – 5 = -2

h. 2 – 3 – 4 – 5 = -10

2. Are there any patterns or connections between this set of solutions and the previous

set of solutions?

The answers to e, f and g are the same in both sets.

They are all even numbers.

Three negative numbers.

Four positive numbers.

Zero in each set.

Any other reasonable connection.

3. Develop a rule to help you find a set of solutions without doing the calculations.

a. The sum of the two middle numbers multiplied by 2.

b. The first number multiplied by 2.

c. The second number multiplied by 2.

d. The third number multiplied by 2.

e. -4

f. 0

g. -2

h. The negative fourth number multiplied by 2.


Activity 3: Extension

1. Investigate how changing the consecutive numbers to a descending order will

affect the results.

Answers will vary

2. Can your previous rule be applied to the process with only three consecutive

numbers?

Answers will vary

3. Can your previous rule be applied to the process with five consecutive numbers?

Answers will vary

4. Consider what might happen if you changed the sign in front of your first number

also.

Answers will vary


STUDENT COPY CONSECUTIVE NUMBERS

Activity 1

2. Choose any four consecutive numbers between 1 and 9, for example, 1, 2, 3, and 4.

3. You want to find all of the different ways of ADDING and SUBTRACTING all of your

chosen numbers. (NOTE: your chosen numbers must always be in numerical order)

How can you do this in a logical way? Can you think of more than one way?

4. Using one of your methods from above, record all of the different combinations.

Calculate each answer.

5. How do you know you have all of the combinations?


Activity 2:

1. Try choosing a different set of numbers and apply your process from above.

2. Are there any patterns or connections between this set of solutions and the previous

set of solutions?

3. Develop a rule to help you find a set of solutions without doing the calculations.



1. Investigate how changing the consecutive numbers to a descending order will

affect the results.

2. Can your previous rule be applied to the process with only three consecutive

numbers?

3. Can your previous rule be applied to the process with five consecutive numbers?

4. Consider what might happen if you changed the sign in front of your first number

also.

MATHS7TL002 | Mathematics | Number and Algebra Activity | Large Elevens 2 © Department of Education WA 2015

TASK 102: LARGE ELEVENS

Overview

In this task, students will investigate how to mentally multiply 2- and 3-digit numbers by 11.

Students will apply the number laws to develop and test rules to aid mental computation.

They are then required to deduce and justify strategies used and reached when they adapt

the known to the unknown.

Students will need

calculators

Relevant content descriptors from the Western Australian Curriculum

Apply the associative, commutative and distributive laws to aid mental and writtencomputation (ACMNA151)


fluency when they

o use a calculator to check solutions


o identify the connection between problems and solutions

reasoning when they

o develop a rule for multiplying 2-digit numbers by 11

o investigate whether their rule always works


o investigate whether they can apply their rule to 3-digit numbers

o use previous strategies to develop a rule for multiplying 3-digit numbers by 11


LARGE ELEVENS Solutions and Notes for Teachers

Activity 1

Multiplying large numbers without a calculator can be difficult but there are some tricks that

can help us!!

Consider the following 2-digit numbers multiplied by 11:

21 x 11 = 231

25 x 11 = 275

53 x 11 = 583

62 x 11 = 682

1. Using a calculator, check that the solutions above are correct.

All solutions above are correct.

2. Take the first problem above, write it out in the space below and examine it carefully.

Explain the connection between the problem and its solution.

HINT: Look closely at the digits.

21 x 11 = 231

When multiplying 21 by 11, I place the 2 as the first digit of the answer and the 1 as the last

digit of the answer. The middle digit is the sum of the 2 and 1, which is 3. So the answer is

231.

3. Is this connection between the problem and its solution the same for the other three

problems? Show your reasoning below.

25 x 11 = 275. 2 is the first digit, 5 is the last digit, 2+5=7, which is the middle digit.



4. Develop a rule for multiplying 2-digit numbers by 11.

When multiplying a 2-digit number by 11, the first digit of the number is the first digit of the

answer. The second digit of the number is the last digit of the answer. The sum of the two

digits becomes the middle digit of the answer.

5. Explain why this rule works in this instance.

The sum of the two digits is less than 10 in these examples.


6. Show how this rule does not always work for 2-digit numbers.

65 x 11 = 715

5 is the last digit but 6 is not the first digit and the sum of the two digits is not the middle digit.

7. Alter your rule from above so that it works for all 2-digit numbers.

When multiplying a 2-digit number by 11, if the sum of the two digits is less than 10, the first

digit of the number is the first digit of the answer. The second digit of the number is the last

digit of the answer. The sum of the two digits becomes the middle digit of the answer.

When multiplying a 2-digit number by 11, if the sum of the two digits is greater than 10, the

second digit of the number is the last digit of the answer. Using the sum of the two digits, the

unit value becomes the middle digit of the answer and the tens value added to the first digit

becomes the first digit of the answer.

Activity 2

Using your work from Activity 1, investigate whether the same rule works for multiplying 3-

digit numbers by 11. If it doesn’t, can you develop a different rule?

Answers will vary.

110 x 11 = 1210

121 x 11 = 1331

123 x 11 = 1353

142 x 11 = 1562

153 x 11 = 1683

1st digit of answer – 1st digit of number

2nd digit of answer – 2nd digit of number +1

3rd digit of answer – sum of the 2nd and 3rd digits of number

4th digit of answer – last digit of number

159 x 11 = 1749

185 x 11 = 2035

247 x 11 = 2717

528 x 11 = 5808

275 x 11 = 3025


4th digit of answer – last digit of number.

3rd digit of answer – unit value of the sum of the 2nd and 3rd digits of number (‘carry’ the tens).

2nd digit of answer – unit value of the sum of the 1st and 2nd digits of number plus tens value

carried over.

1st digit of answer – 1st digit of number plus tens value carried over.


STUDENT COPY LARGE ELEVENS

Activity 1

Multiplying large numbers without a calculator can be difficult, but there are some tricks that

can help us!!

Consider the following 2-digit numbers multiplied by 11:

21 x 11 = 231

25 x 11 = 275

53 x 11 = 583

62 x 11 = 682

1. Using a calculator, check that the solutions above are correct.

2. Take the first problem above, write it out in the space below and examine it carefully.

Explain the connection between the problem and its solution.

HINT: Look closely at the digits.

3. Is this connection between the problem and its solution the same for the other three

problems? Show your reasoning below.

4. Develop a rule for multiplying 2-digit numbers by 11.


5. Explain why this rule works in this instance.

6. Show how this rule does not always work for 2-digit numbers.

7. Alter your rule from above so that it works for all 2-digit numbers.

Activity 2

Using your work from Activity 1, investigate whether the same rule works for multiplying 3-

digit numbers by 11. If it doesn’t, can you develop a different rule?

MATHS7TL002 | Mathematics | Number and Algebra Activity | Ideal Fractions 2 © Department of Education WA 2015

TASK 105: IDEAL FRACTIONS

Overview

In this task, students will investigate the notion of ideal fractions. They will need to carry out

the addition and subtraction of fractions fluently in order to obtain the same solution. They

are required to identify the relationship between these fractions and build their

understanding. Through investigation they will attempt to explain how ideal fractions can be

found.

Students will need

calculators

Relevant content descriptors from the Western Australian Curriculum

Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153)

Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)


fluency when they

o calculate the solution to the addition and multiplication of fractions

o describe, using mathematical language, how the operations where performed


o identify a connection between the solutions of the operations

reasoning when they

o describe, using a rule, how to find ideal fractions


o attempt to find other examples of ideal fractions

o investigate if this relationship is true for any fraction

o look at the outcome from including subtraction and multiplication

o attempt to source other types of ideal operations


IDEAL FRACTIONS Solutions and Notes for Teachers

The fractions 5

3 and

5

2 can be described as Ideal Fractions. Below are two operations that

can be performed with these fractions.

5

3+

5

2

5

3´

5

2

Activity 1

1. Find the solution to the above operations.

Both have a solution of 25

6

2. Use mathematical language to describe how you performed each operation.

When adding fractions, the denominators must be the same. If they are not, they

must be changed. To do this you must find a common multiple of the denominators or

the least common multiple (LCM). This in turn will change the numerators.

To multiply fractions, simply multiply the numerators and multiply the denominators.

3. What is the connection between these two operations?

They both have the same solution.

4. Find 4 other examples that have the connection you have described above.

1. 7

3 and

7

4

2. 9

4 and

9

5

3. 11

5 and

11

6

4. 13

6 and

13

7

5. Is this connection true for any fractions?

No, this does not work with any fractions.

6. Investigate what happens if we take subtraction and division into account?

There is no connection with subtraction or division.


7. Is there a rule that you could use to describe how to find ideal fractions?

Improper fractions.The numerators are identical odd numbers.The denominators are two consecutive numbers that add to give the value of thenumerators.


Are there any other types of ideal operations?

Answers will vary.


STUDENT COPY IDEAL FRACTIONS

The fractions 5

3 and

5

2 can be described as Ideal Fractions. Below are two operations that

can be performed with these fractions.

5

3+

5

2

5

3´

5

2

Activity 1

1. Find the solution to the above operations.

2. Use mathematical language to describe how you performed each operation.

3. What is the connection between these two operations?

4. Find 4 other examples that have the connection you have described above.

5. Is this connection true for any fractions?


6. Investigate what happens if we take subtraction and division into account?

7. Is there a rule that you could use to describe how to find ideal fractions?


1. Are there any other types of ideal operations?

TASK LIST...YEAR 7 MATHEMATICS o f W e s t ern A u s r a l i a I n c . T MA WA h e M a t h e m a t i...

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