Maths Program.s3 Yr5 t1

MATHS PROGRAM : STAGE THREE

YEAR FIVE

WEEKLY ROUTINE

Monday Tuesday Wednesday Thursday Friday

Whole Number 1 Terms 1-4 Number & Algebra Terms 1-4: Addition and Subtraction 1 Terms 1-4 : Multiplication & Division 1 Terms 1 & 3: Patterns and Algebra 1 Terms 2 & 4: Fractions and Decimals 1

Statistics & Probability Terms 1 & 3: Data 1 Terms 2 & 4: Chance 1

Measurement & Geometry Term 1: Length 1 / Time 1 / 2D 1 / Position 1 Term 2: Mass 1 / 3D 1 / Angles 1 Term 3: Volume and Capacity 1 / Time 1 / 2D 1 / Position 1 Term 4: Area 1 / 3D1 / Angles 1

K-6 MATHEMATICS SCOPE AND SEQUENCE

NUMBER AND ALGEBRA MEASUREMENT AND GEOMETRY STATISTICS & PROBABILITY

TERM

Whole Number

Addition & Subtraction

Multiplication & Division

Fractions & Decimals

Patterns & Algebra

Length Area Volume & Capacity

Mass Time 3D 2D Angles Position Data Chance

K 1 2 3 4

Yr 1 1 2 3 4

Yr 2 1 2 3 4

Yr 3 1 2 3 4

Yr 4 1 2 3 4

Yr 5 1 2 3 4

Yr 6 1 2 3 4

NB: Where a content strand has a level 1 & 2, the 1 refers to the lower grade within the stage, eg. Whole Number 1 in S1 is for Yr 1, Whole Number 2 is for Yr 2.

MATHEMATICS PROGRAM PROFORMA

STAGE: Year 5 ES1 S1 S2 S3

STRAND: NUMBER AND ALGEBRA

TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Whole Number 1 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations MA3-2WM › orders, reads and represents integers of any size and describes properties of whole numbers MA3-4NA

Background Information Students need to develop an understanding of place value relationships, such as 10 thousand = 100 hundreds = 1000 tens = 10 000 ones. Language Students should be able to communicate using the following language: ascending order, descending order, zero, ones, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, digit, place value, expanded notation, round to, whole number, factor, highest common factor (HCF), multiple, lowest common multiple (LCM). In some Asian languages, such as Chinese, Japanese and Korean, the natural language structures used when expressing numbers larger than 10 000 are 'tens of thousands' rather than 'thousands', and 'tens of millions' rather than 'millions'. For example, in Chinese (Mandarin), 612 000 is expressed as '61 wàn, 2 qiān', which translates as '61 tens of thousands and 2 thousands'. The abbreviation 'K' is derived from the Greek word khilios, meaning 'thousand'. It is used in many job advertisements to represent salaries (eg a salary of $70 K or $70 000). It is also used as an abbreviation for the size of computer files (eg a size of 20 K, meaning twenty thousand bytes).

Recognise, represent and order numbers to at least tens of millions • apply an understanding of place value & the role of zero to read & write numbers of any size • state the place value of digits in numbers of any size • arrange numbers of any size in ascending & descending order • record numbers of any size using expanded notation, eg 163 480 = 100 000 + 60 000 + 3000 + 400 + 80 • partition numbers of any size in non-standard forms to aid mental calculation, eg when adding 163 480 & 150 000, 163 480 could be partitioned as 150 000 + 13 480, so that 150 000 could then be doubled & added to 13 480 • use numbers of any size in real-life situations, including in money problems -interpret information from the internet, the media, the environment & other sources that use large numbers • recognise different abbreviations of numbers used in everyday contexts, eg $350 K represents $350 000 • round numbers to a specified place value, eg round 5 461 883 to the nearest million Identify and describe factors and multiples of whole numbers and use them to solve problems • determine all 'factors' of a given whole number, eg 36 has factors 1, 2, 3, 4, 6, 9, 12, 18 & 36 • determine the 'highest common factor' (HCF) of two whole numbers, eg the HCF of 16 & 24 is 8 • determine 'multiples' of a given whole number, eg multiples of 7 are 7, 14, 21, 28, … • determine the 'lowest common multiple' (LCM) of two whole numbers, eg the LCM of 21 &63 is 63 • determine whether a particular number is a factor of a given number using digital technologies - recognise that when a given number is divided by one of its factors, the result must be a whole number • solve problems using knowledge of factors & multiples, eg 'There are 48 people at a party. In how many ways can you set up the tables & chairs, so that each table seats the same number of people & there are no empty chairs?'

Learning Across The Curriculum

Cross-curriculum priorities Aboriginal &Torres Strait Islander histories & cultures Asia & Australia’s engagement with Asia Sustainability General capabilities

Critical & creative thinking Ethical understanding Information & communication technology capability Intercultural understanding Literacy Numeracy Personal & social capability Other learning across the curriculum areas Civics & citizenship Difference & diversity Work & enterprise

CONTENT WEEK TEACHING, LEARNING and ASSESSMENT

ADJUSTMENTS RESOURCES

Recognise, represent and order numbers to at least tens of millions Identify and describe factors and multiples of whole numbers and use them to solve problems

1-2

Place Value Write 15 642 on the board and ask a student to read it. Discuss the value of each digit, establishing particularly that the one represents ten thousand, but the number is read as ‘fifteen thousand’. Change the 15 to 3 and repeat. Point out that a space is left between the thousands and hundreds to make the number easier to read. Repeat with numbers such as 156 342, then 1 243 675, to establish the value of hundred thousands and millions and the way in which the numbers are read. Show a place value chart on the board with the ones (units) column labelled, e.g.

Alternatively draw an abacus diagram:

Discuss the value of the other columns, starting from the right, and label them. Draw small circles in each column, to represent a number. Write the number in figures on the board, point out the spacing, then ask a student to read the number. Change the circles to make a different five, six or seven-digit number and ask students to write it in figures, then read it aloud together. Repeat with other numbers. Read out some five, six and seven-digit numbers and ask students to write them in figures. Focus on a seven-digit number and ask: - How could I increase this number by ten thousand? - What would the new number be? Repeat with other changes to the number, e.g. 300 thousand smaller, 2 million bigger etc, asking students to record the new number each time. Ask students to write any seven-digit number. Ask them to raise their hands if their number contains: fifty thousand; nine thousand; eight hundred thousand, forty; six; three million etc.

Reduce, enlarge the numbers according to ability. Provide a concrete example of an abacus for those students who require this level of support

whiteboard and markers, abacus, paper and pencils

3-4

Whole Numbers Ask children to make the largest possible six-digit number on their calculators without pressing any key more than once. Possible questions: - Which number should everyone have made? (987 654) Why? Repeat with different criteria such as: - the smallest possible number, - the largest/smallest odd number etc. Check that children are using the calculator correctly as they do these. If necessary clarify how to key in numbers. Write on the board: 25 24 < < 630 800. Revise the meaning of the symbols, then ask students to make a number on their calculators which could go in the box. Ask a few children to read their numbers and discuss whether they are correct. Repeat with a few more outer numbers and include < symbols. Write 52 < 063 . Explain that the new symbol means greater than or equal to and ask children to

Provide assistance as required

calculators, whiteboard and markers, paper and pencils

suggest numbers that could be placed in the box. (If 52 063 is not suggested, then point out that it is one of the possible numbers and explain why.) Repeat using the < symbol. Write 73 73 > > 832 833 and ask students to show numbers on their calculators which could go in the box. - How many possible numbers are there? Discuss responses. Repeat with a few more outer numbers and include > symbols. In pairs, students make up and complete statements similar to those done with the class, recording them in their books. Allow about 10 minutes. Ask children to make a number with between four and seven digits on their calculators. Ask about five students to write their numbers on the board. Ask the class to write the numbers in ascending order in their books, then discuss strategies used. Repeat with five different numbers for the class to record in descending order.

5

Repeated Operations Using an online calculator, press 10 x 7 = and ask students to look at the display. Now press the = key repeatedly as children watch the display. Ask: - What was happening to the digits? - What has the calculator been doing? - How do you know? Establish that it multiplied by 10 repeatedly because the digits moved one place to the left each time, the empty column being filled with zero. Repeat with 100 x 7 and discuss to revise the effect as before. Enter 7 000 000 ÷ 10 on the online calculator then repeatedly press =, asking students to watch the display as before. Stop when the display shows 7 and repeat the questions. Establish that the calculator divided by 10 repeatedly because the digits moved one place to the right each time. Continue to press = as students watch the display. Discuss that the digits continue to move to the left of the decimal point. Repeat with 7 000 000 ÷ 100 = =. - What will happen if we enter 10 x 2 = =? And 100 x 2 = = =? Ask the students to test their predictions on their calculators. - What will happen if we enter 2 x 10 = = ? and 2 x 100 = = =? Ask the students to test their predictions. Discuss their reasoning. Explain that the calculator was performing a constant function, i.e. a quick way to repeat the same operation, (N.B. on most simple calculators the multiplication constant operates on the first number entered, so 10 x 7 = = results in 70, 700 etc. 7 x 10 = = results in 70, 490 etc. The division constant acts on the second number entered). In pairs, challenge the students to predict the answers to 10 x 5 = = = and 5 x 10 = = = and then to test their predictions. Collect their responses and discuss their strategies. Ask students to explore multiplying and dividing different numbers by 10 and 100 on their calculators, including two and three-digit starting numbers. Ask them to record their results to show patterns and write the numbers alongside. - What could you do to 6 000 to get an answer of 60? - What could you do to 6 to get an answer of 0.06? Repeat for other pairs.


computers, calculators, paper and pencils

6

Factors Explore and discuss (encouraging student predictions as you go): - The factors of 2 are 1 and 2 because 2 divides by 1 and 2 - The factors of 3 are 1 and 3 because 3 divides by 1 and 3 - The factors of 4 are 1, 2 and 4 because 4 divides by 1, 2 and 4 - The factors of 5 are 1 and 5 because 5 divides by 1 and 5 - The factors of 6 are 1, 2, 3 and 6 because 6 divides by 1, 2, 3 and 6 - The factors of 7 are 1 and 7 because 7 divides by 1 and 7 - The factors of 12 are 1, 2, 3, 4, 6 and 12 because 12 divides by 1, 2, 3, 4, 6 and 12 Factors Pairs Explore and discuss (encouraging student predictions as you go): - The factors of 6 are 1, 2, 3 and 6 because 6 divides by 1, 2, 3 and 6 1 and 6 are a factor pair of 6 since 1 x 6 = 6 2 and 3 are a factor pair of 6 since 2 x 3 = 6 - The factors of 9 are 1, 3, and 9 because 9 divides by 1, 3 and 9 1 and 9 are a factor pair of 9 since 1 x 9 = 9 3 and 3 are a factor pair of 9 since 3 x 3 = 9 - The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 because 24 divides by 1, 2, 3, 4, 6, 8, 12 and 24 1 and 24 are a factor pair of 24 since 1 x 24 = 24 2 and 12 are a factor pair of 24 since 2 x 12 = 24 3 and 8 are a factor pair of 24 since 3 x 8 = 24 4 and 6 are a factor pair of 24 since 4 x 6 = 24


whiteboard and markers, paper and pencils

7

Common Factors Explore and discuss (encouraging student predictions as you go): - The factors of 4 are 1, 2, and 4 because 4 divides by 1, 2, and 4 - The factors of 6 are 1, 2, 3 and 6 because 6 divides by 1, 2, 3 and 6 The common factors of 4 and 6 are 1 and 2 since 1 and 2 are factors of both 4 and 6 - The factors of 8 are 1, 2, 4, and 8 because 8 divides by 1, 2, 4, and 8 - The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 because 24 divides by 1, 2, 3, 4, 6, 8, 12 and 24 The common factors of 8 and 24 are 1, 2, 4 and 8 since 1, 2, 4 and 8 are factors of both 8 and 24



8

Highest Common Factor [HCF] Explore and discuss (encouraging student predictions as you go): - The factors of 4 are 1, 2, and 4 because 4 divides by 1, 2, and 4 - The factors of 6 are 1, 2, 3 and 6 because 6 divides by 1, 2, 3 and 6 The common factors of 4 and 6 are 1 and 2 The highest common factor of 4 and 6 is 2 - The factors of 8 are 1, 2, 4, and 8 because 8 divides by 1, 2, 4, and 8 - The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 because 24 divides by 1, 2, 3, 4, 6, 8, 12 and 24 The common factors of 8 and 24 are 1, 2, 4 and 8 The highest common factor of 8 and 24 is 8



9

Revisit a Selection of Above Activities

10

Revision Assessment

ASSESSMENT OVERVIEW




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Addition and Subtraction 1 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations MA3-2WM › gives a valid reason for supporting one possible solution over another MA3-3WM › selects and applies appropriate strategies for addition and subtraction with counting numbers of any size MA3-5NA

Background Information In Stage 3, mental strategies need to be continually reinforced. Students may find recording (writing out) informal mental strategies to be more efficient than using formal written algorithms, particularly in the case of subtraction. Eg, 8000 − 673 is easier to calculate mentally than by using a formal algorithm. Written strategies using informal mental strategies (empty number line): The jump strategy can be used on an empty number line to count up rather than back.

The answer will therefore be 7000 + 300 + 20 + 7 = 7327. Students could share possible approaches and compare them to determine the most efficient. The difference can be shifted one unit to the left on an empty number line, so that 8000 – 673 becomes 7999 − 672, which is an easier subtraction to calculate.

Written strategies using a formal algorithm (decomposition method):

An inverse operation is an operation that reverses the effect of the original operation. Addition and subtraction are inverse operations; multiplication and division are inverse operations. Language Students should be able to communicate using the following language: plus, sum, add, addition, increase, minus, the difference between, subtract, subtraction, decrease, equals, is equal to, empty number line, strategy, digit, estimate, round to, budget. Teachers should model & use a variety of expressions for the operations of addition & subtraction, & should draw students' attention to the fact that the words used for subtraction may require the operation to be performed with the numbers in the reverse order to that in which they are stated in the question. Eg, '9 take away 3' & 'reduce 9 by 3' require the operation to be performed with the numbers in the same order as they are presented in the question (ie 9 – 3). However, 'take 9 from 3', 'subtract 9 from 3' and '9 less than 3' require the operation to be performed with the numbers in the reverse order to that in which they are stated in the question (ie 3 – 9).

Use efficient mental and written strategies and apply appropriate digital technologies to solve problems • use the term 'sum' to describe the result of adding two or more numbers, eg 'The sum of 7 and 5 is 12' • add three or more numbers with different numbers of digits, with and without the use of digital technologies, eg 42 000 + 5123 + 246 • select and apply efficient mental, written and calculator strategies to solve addition and subtraction word problems, including problems involving money - interpret the words 'increase' and 'decrease' in addition and subtraction word problems, eg 'If a computer costs $1599 and its price is then decreased by $250, how much do I pay?' • record the strategy used to solve addition and subtraction word problems - use empty number lines to record mental strategies -use selected words to describe each step of the solution process • check solutions to problems, including by using the inverse operation Use estimation and rounding to check the reasonableness of answers to calculations • round numbers appropriately when obtaining estimates to numerical calculations • use estimation to check the reasonableness of answers to addition and subtraction calculations, eg 1438 + 129 is about 1440 + 130 Create simple financial plans • use knowledge of addition and subtraction facts to create a financial plan, such as a budget, eg organise a class celebration on a budget of $60 for all expenses -record numerical data in a simple spreadsheet -give reasons for selecting, prioritising and deleting items when creating a budget






Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

Use estimation and rounding to check the reasonableness of answers to calculations

Create simple financial plans

1

Rounding Revise the rules for rounding numbers such as 68, 23, 214, 675, 2998 etc. and ask children to show the nearest multiple of 10, 100 or 1000 using their whiteboards. Confirm some answers using number lines, e.g.

- How much did we adjust our number by to reach the nearest multiple of (10, 100 or 1000)? Explain that rounding to the nearest 10, 100 or 1000 can be used as a strategy for addition and subtraction. Write 93 – 69 on the board. Ask students: - What multiple of 10 is nearest to 69? - What is 93 subtract 70? - Have we subtracted more or less than 69? - How should we adjust the answer to make it correct? Emphasise that the extra 1 we subtracted must be added to 23 for the answer to 93 – 69. Record the process as:

93 – 69 = (93 – 70) + 1 = 23 + 1 = 24

and demonstrate on a number line, e.g.

Repeat with calculations such as: 368 + 51, 286 – 97, 5250 – 1998, 458 + 199 etc. Each time emphasise the adjustment to be made and have students record the process on the board. Provide examples for students to practise the strategy. Go over some of the practice examples, asking students to explain how they rounded and adjusted the numbers involved. Write a variety of addition and subtraction calculations on the board, e.g. 73 + 26, 182 – 95, 6003 – 5994, 56 – 29, 73 + 200, 583 – 71 etc. - For which of these would you use the rounding and adjusting strategy? For each suggestion ask how the calculation would be performed, focussing on the rounding and adjusting. Ask children to suggest how they would tackle the other calculations.

Provide assistance and extension as required


2

Strategies Write 40 + 90 + 60 + 50 on the board. - How would you find the total? Discuss different methods such as looking for pairs with a sum of 100, and starting with largest number first. Repeat with other sets of two-digit numbers. Include numbers that involve doubling and near doubling, e.g. 60 + 70 + 80 + 20 + 30 + 80 + 70; 20 + 80 + 10; 50 + 60. Record the strategies in a list on the board. Introduce three two-digit numbers, e.g. write 28 + 35 + 12 on the board and ask students to



suggest strategies for finding the total, such as looking for unit pairs that make 10, and starting with the largest number. Write on the board the digits: 1 + 2 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9. Ask children to add them up. Recap the strategy of finding pairs to 10. Identify that there are 5 pairs that sum to 10. Establish that: 1 + 2 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 is equivalent to 10 x 5. Write on the board: 4 + 4 + 3 + 5. - What is multiplication is this equivalent to? Agree it is 4 x 4. Discuss the method the children used to arrive at this. Write on the board: 18 + 20 + 22. - How can we represent this as a multiplication? Establish that the calculation is equivalent to 20 x 3. Discuss the idea of balancing the numbers. Repeat for sets such as 48 + 49 + 50 + 51 + 52 and 26 + 28 + 30 + 32 + 34, and discuss the strategy. Provide addition questions for children to discuss and answer in pairs. Ask them to decide on an appropriate strategy for each, using the list on the board for reference, then find the total. Ask them to record their method so that they will remember how they worked out the answer.

3

Magic Tools Show the table below. Ask the students to find the total of given rows and then given columns.

13 18 11

12 14 16

17 10 15

- What do you notice about the totals? Establish the totals are the same. - Can you find any other patterns in the square? Allow time for students to discuss in pairs, then take feedback, drawing out and listing points such as the following on the board: - All rows, columns and diagonals have the same total (42). - The sum of each pair of numbers on opposite sides of the centre is twice the middle number. - The odd numbers are at the corners. - The sum of the corner numbers is equal to the sum of the numbers in the middle of each side. Remind students that this is a 3 by 3 magic square. - Suppose we subtract 7 from each number, will it still be a magic square? Discuss children’s views and get them to confirm it is still a magic square. - What is the ‘magic total’ for this square? How could you have predicted this? Collect answers and work through other cases where a number is added to or subtracted from the numbers in the square. List some of the ‘magic totals’ for the different squares. - What do you notice about these totals? Establish they are multiples of 3. Refer back to the table and remind students that the total is the middle number multiplied by 3. Show the table below. Ask students to find the sum of the first row, then the first column. Confirm these each total 46. - Is 46 a multiple of 4? Agree it is not. Ask students to sum the other rows and columns to confirm they all total 46.



17 10 15 4 14 5 16 11 8 19 6 13 7 12 9 18

- Can you find any other sets of four numbers that total 46? Ask students to work in pairs to find sets of four numbers.

4

Inverse Operations Write 468 + 573 on the board and ask students to suggest how they might work out the total. Discuss suggestions and demonstrate the informal method of adding the most or the least significant digits first, i.e. 468 468 + 573 + 573 900 11 130 130 11 900 1041 1041 Repeat with 4676 + 768. Emphasise the importance of lining up the digits correctly according to their place value. Set similar questions for students to practise the method on. Check answers and go over any the students found difficult, asking students to prompt each stage of the calculations. - How can we check that the answers are correct? Discuss suggestions and remind students addition and subtraction are inverse operations. Demonstrate an informal counting-up method for 1041 – 573. Record as:

Set subtractions questions for students to practise the method. Check answers and correct any misunderstandings. Give students two three-digit numbers. Ask students to find the difference and check the answer using addition. Set students questions that include three-digit and four-digit numbers.



9 Revisit a Selection of Above Activities

10

Revision Assessment

ASSESSMENT OVERVIEW




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Multiplication and Division 1 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations MA3-2WM › gives a valid reason for supporting one possible solution over another MA3-3WM › selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation MA3-6NA

Background Information Students could extend their recall of number facts beyond the multiplication facts to 10 × 10 by memorising multiples of numbers such as 11, 12, 15, 20 and 25. They could also utilise mental strategies, eg '14 × 6 is 10 sixes plus 4 sixes'. In Stage 3, mental strategies need to be continually reinforced. Students may find recording (writing out) informal mental strategies to be more efficient than using formal written algorithms, particularly in the case of multiplication. An inverse operation is an operation that reverses the effect of the original operation. Addition and subtraction are inverse operations; multiplication and division are inverse operations. The area model for two-digit by two-digit multiplication in Stage 3 is a precursor to the use of the area model for the expansion of binomial products in Stage 5. Language Students should be able to communicate using the following language: multiply, multiplied by, product, multiplication, multiplication facts, area, thousands, hundreds, tens, ones, double, multiple, factor, divide, divided by, quotient, division, halve, remainder, fraction, decimal, equals, strategy, digit, estimate, round to

Solve problems involving multiplication of large numbers by 1 or 2 digit numbers using efficient mental & written strategies & appropriate digital technologies • use mental & written strategies to multiply 3 & 4 digit numbers by 1 digit numbers, including: − multiplying the 1000s, then the 100s, then the 10s and then the 1s, eg

− using an area model, eg 684 × 5

− using the formal algorithm, eg 432 × 5

• use mental & written strategies to multiply 2 & 3 digit numbers by 2 digit numbers, including: − using an area model for 2 digit by 2 digit multiplication, eg 25 × 26

− factorising the numbers, eg 12 × 25 = 3 × 4 × 25 = 3 × 100 = 300 − using extended form (long multiplication) of the formal algorithm,

• use digital technologies to multiply numbers of up to 4 digits - check answers to mental calculations using digital technologies • apply appropriate mental 7 written strategies, 7 digital technologies, to solve multiplication word problems - use the appropriate operation when solving problems in real-life situations - use inverse operations to justify solutions • record the strategy used to solve multiplication word problems - use selected words to describe each step of the solution process Solve problems involving division by a 1 digit number, including those that result in a remainder • use the term 'quotient' to describe the result of a division




calculation, eg 'The quotient when 30 is divided by 6 is 5' • recognise 7 use different notations to indicate division, eg 25 ÷ 4,

, • record remainders as fractions 7 decimals, eg or 6.25 • use mental 7 written strategies to divide a number with 3 or more digits by a 1 digit divisor where there is no remainder, including: − dividing the 100s, then the 10s, and then the 1s, eg 3248 ÷ 4

− using the formal algorithm, eg 258 ÷ 6

• use mental & written strategies to divide a number with 3 or more digits by a 1 digit divisor where there is a remainder, including: − dividing the 10s and then the 1s, eg 243 ÷ 4

− using the formal algorithm, eg 587 ÷ 6

- explain why the remainder in a division calculation is always less than the number divided by (the divisor) • show the connection between division & multiplication, including where there is a remainder, eg 25 ÷ 4 = 6 remainder 1, so 25 = 4 × 6 + 1 • use digital technologies to divide whole numbers by 1 & 2 digit divisors - check answers to mental calculations using digital technologies • apply appropriate mental & written strategies, & digital technologies, to solve division word problems - recognise when division is required to solve word problems - use inverse operations to justify solutions to problems • use & interpret remainders in solutions to division problems, eg recognise when it is appropriate to round up an answer, such as 'How many 5-seater cars are required to take 47 people to the beach?' • record the strategy used to solve division word problems - use selected words to describe each step of the solution process Use estimation & rounding to check the reasonableness of answers to calculations • round numbers appropriately when obtaining estimates to numerical calculations • use estimation to check the reasonableness of answers to multiplication & division calculations, eg '32 × 253 will be about, but more than, 30 × 250'



Solve problems involving multiplication of large numbers by 1 or 2 digit numbers using efficient mental & written strategies & appropriate digital technologies Solve problems involving division by a 1 digit number, including those that result in a remainder Use estimation & rounding to check the reasonableness of answers to calculations

5

Comparing Mental and Written Strategies Students estimate, then multiply three- and four-digit numbers by one-digit numbers, to compare mental and written strategies when solving problems eg ‘There are 334 students in a school. If each student watches 3 hours of television per day, how many hours of television is this?’ Students share their strategies and determine which is the most efficient. Possible questions include: - how did your estimation help? - which operation did you use? - can you describe your strategy? - is your strategy efficient? Why? - did your answer make sense in the original situation? - how can you check whether your answer is correct? Students write their own problems using large numbers. They check answers on a calculator.


calculators, whiteboards and marker, paper and pencils

6

Factor Game Part A In pairs, students are provided with a pack of playing cards with tens and picture cards removed. The Aces remain and count as 1 and the Jokers remain and count as 0. The students flip a card each and place them together to make a one- or two-digit number. Students use a calculator to find all of the factors of the number created. They record the number and the factors in two groups: composite numbers and prime numbers. Part B In pairs, students select 5 composite numbers and 5 prime numbers. They use counters to make arrays for their numbers. Possible questions include: - why does a prime number, when modelled as an array, have only one row?

Extension: Students record and discuss square and triangular numbers and look for patterns eg numbers with 3 factors are squares of prime numbers.

packs of playing cards, paper and pencils

7

Multiples of 10 Part A Students are asked to multiply some two-digit numbers by ten and discuss their findings. They are asked to determine mental strategies for doing this. Students then try multiplying the same two-digit numbers by 20, 30,……100. They are asked to determine mental strategies for doing this. Part B Students are asked to divide some two-digit numbers by ten and discuss their findings. They are asked to form a rule for doing this. Students then try dividing the same two-digit numbers by 20, 30, ….100. They are asked to determine mental strategies for doing this. Possible questions include: - does your strategy apply to all two-digit numbers? - does your strategy apply to multiplying/dividing by 20, 30,….100?


paper and pencils

Dividing by Ten The teacher poses the scenario: ‘On the way to school 4 children found a $50 note. They handed it


paper and pencils

8 in to the principal. They will get a share of the $50 if no one claims it after a week.’ Possible questions include: - how much would each child get? - how much would each child get if $5 was found? - how much would each child get if 50c was found? - which operation would you use to check if your answer is correct? Students discuss the solutions and make generalisations about placement of the decimal point when dividing by ten. They investigate similar problems to test their ideas.

9

Revision

10

Assessment

ASSESSMENT OVERVIEW




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Patterns and Algebra 1 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations MA3-2WM › gives a valid reason for supporting one possible solution over another MA3-3WM › analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane MA3-8NA

Background Information Students should be given opportunities to discover and create patterns and to describe, in their own words, relationships contained in those patterns. This sub strand involves algebra without using letters to represent unknown values. When calculating unknown values, students need to be encouraged to work backwards and to describe the processes using inverse operations, rather than using trial-and-error methods. The inclusion of number sentences that do not have whole-number solutions will aid this process. To represent equality of mathematical expressions, the terms 'is the same as' and 'is equal to' should be used. Use of the word 'equals' may suggest that the right-hand side of an equation contains 'the answer', rather than a value equivalent to that on the left. Language Students should be able to communicate using the following language: pattern, increase, decrease, missing number, number sentence, number line. In Stage 3, students should be encouraged to use their own words to describe number patterns. Patterns can usually be described in more than one way and it is important for students to hear how other students describe the same pattern. Students' descriptions of number patterns can then become more sophisticated as they experience a variety of ways of describing the same pattern. The teacher could begin to model the use of more appropriate mathematical language to encourage this development.

Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction • identify, continue and create simple number patterns involving addition and subtraction • describe patterns using the terms 'increase' and 'decrease', eg for the pattern 48, 41, 34, 27, …, 'The terms decrease by seven' • create, with materials or digital technologies, a variety of patterns using whole numbers, fractions or decimals, eg

… or 2.2, 2.0, 1.8, 1.6, … • use a number line or other diagram to create patterns involving fractions or decimals Use equivalent number sentences involving multiplication and division to find unknown quantities • complete number sentences that involve more than one operation by calculating missing numbers, eg , 5 x = 4 x 10, 5 x = 30 - 10 - describe strategies for completing simple number sentences and justify solutions • identify and use inverse operations to assist with the solution of number sentences, eg 125 ÷ 5 = becomes x 5 = 125 - describe how inverse operations can be used to solve a number sentence • complete number sentences involving multiplication and division, including those involving simple fractions or decimals, eg 7 x = 7.7 - check solutions to number sentences by substituting the solution into the original question • write number sentences to match word problems that require finding a missing number, eg 'I am thinking of a number that when I double it and add 5, the answer is 13. What is the number?'






Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

Use equivalent number sentences involving multiplication and division to find unknown quantities

4

Identifying and completing number patterns With practice, a variety of common number patterns can be identified. This requires careful observation. Identifying patterns involves being able to look at features such as differences (the amounts between numbers) and rates of change (how quickly the numbers are seen to increase or decrease). Often, a pattern will start and you will be required to continue the series. You can do this by first identifying the pattern and then making use of the last number to extend the sequence. Examples: 2, 5, 8, 11, 14, 17, ..., ... addition (+ 3) 100, 96, 92, 88, 84, ..., ... subtraction (- 4) 2, 4, 8, 16, 32, 64, ..., ... multiplication (doubling) 160, 80, 40, 20, 10, ..., ... division (halving) 81, 64, 49, 36, 25, ..., ... decreasing square numbers Sometimes the missing numbers may be located within the number sequence. You can use the surrounding numbers as a guide. Examples: 14, 26, ..., 50, ..., 74, 86 addition (+ 12) 93, 82, 71, ..., ..., 38, 27 subtraction (- 11) 1, ..., ..., 125, 625, 3 125 multiplication (x 5) 1 000, ..., 10, 1, ..., 0.01 division (÷ 10)


paper and pencils

5

Number patterns within tables Sometimes you may be provided with a table or grid which has a series of numbers and a rule to follow. Provided the series of numbers given is already an uninterrupted sequence, the answers which complete the grid should form a pattern. Example:

The answers to this grid (90, 140, 190, 240, 290 and 340) maintain the pattern which has already been established.


paper and pencils, rulers

6

Identifying the rule and completing the pattern You might be given a sequence of paired numbers and asked to identify a rule for the pattern. You may also then be asked to complete the sequence. Example:


paper and pencils, rulers

You must work out what number operation has been used to create the pattern and then use the rule to calculate the missing values in the pattern.

7

Patterns and problem solving We can also make use of patterns for solving problems. Being able to identify patterns can save a great deal of time in working out a solution to a problem. 1. Where will the postman call next? Have a large number line at the front of the class for reference. Introduce the idea of a postman calling at houses to drop off letters. Have your postman call at regular intervals e.g. doors numbered 3, 6, 9, 12. Ask the students if they can predict where the postman will call next. Questions to ask. - Can you explain to your partner how you knew it was that house? - Can you explain the pattern using a number line? - Can you tell me a house number over 20 that you know he would not call at? How did you know that? - What house number will he deliver his seventh / seventieth letter to? How could you check this? 2. Make it bigger Present students with a multilink model of roughly 6 blocks. E.g. an “h” or “s” shape. Ask the children if they can work out how to make the model exactly twice as big. Questions to ask - Can you estimate how many cubes you will need - Can you explain to your partner how you decided on your estimate? - Which part of the model was the hardest part to estimate? Why? - Which part was the easiest? Why - How are you going to check your answer? - How many cubes would you need if you were going to make it 2, 3, 100 times bigger? How do you know? How could you check? Students may want to record their predictions or findings on squared paper. Etc

extension: include prime numbers or sequences such as +1 +2 +3 +4 Students may find it helpful to draw a picture to help

paper and pencils

9 Revision

10 Assessment

ASSESSMENT OVERVIEW



STRAND: MEASUREMENT AND GEOMETRY

TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Length 1 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › gives a valid reason for supporting one possible solution over another MA3-3WM › selects and uses the appropriate unit and device to measure lengths and distances, calculates perimeters, and converts between units of length MA3-9MG

Background Information When students are able to measure efficiently and effectively using formal units, they should be encouraged to apply their knowledge and skills in a variety of contexts. Following this, they should be encouraged to generalise their method for calculating the perimeters of squares, rectangles and triangles. When recording measurements, a space should be left between the number and the abbreviated unit, eg 3 cm, not 3cm. Language Students should be able to communicate using the following language: length, distance, kilometre, metre, centimetre, millimetre, measure, measuring device, ruler, tape measure, trundle wheel, estimate, perimeter, dimensions, width. 'Perimeter' is derived from the Greek words that mean to measure around the outside: peri, meaning 'around', and metron, meaning 'measure'.

Choose appropriate units of measurement for length • recognise the need for a formal unit longer than the metre for measuring distance • recognise that there are 1000 metres in one kilometre, ie 1000 metres = 1 kilometre - describe one metre as one thousandth of a kilometre • measure a kilometre & a half-kilometre • record distances using the abbreviation for kilometres (km) • select & use the appropriate unit & measuring device to measure lengths & distances - describe how length or distance was estimated & measured - question & explain why two students may obtain different measures for the same length, distance or perimeter • estimate lengths & distances using an appropriate unit & check by measuring • record lengths & distances using combinations of millimetres, centimetres, metres & kilometres, eg 1 km 200 m

Calculate the perimeters of rectangles using familiar metric units • use the term dimensions to describe the lengths & widths of rectangles • measure & calculate the perimeter of a large rectangular section of the school, eg a playground, netball courts • calculate perimeters of common 2D shapes, including squares, rectangles, triangles & regular polygons with more than 4 sides (ie regular polygons other than equilateral triangles & squares) - recognise that rectangles with the same perimeter may have different dimensions - explain that the perimeters of 2D shapes can be found by finding the sum of the side lengths - explain the relationship between the lengths of the sides and the perimeters for regular polygons (including equilateral triangles & squares) • record calculations used to find the perimeters of 2D shapes






Choose appropriate units of measurement for length

Calculate the perimeters of rectangles using familiar metric units

1

How Many Centimetres in a Metre? Students make a metre strip using 1 cm grid paper. In groups, students randomly cut their metre into 3 pieces and place all the group’s strips into a bag. Students take turns to select and measure one strip. Students estimate and calculate what length strip they would need to add to their selected length to make exactly 1 metre. They are asked to explain how they know it will be 1 metre. Calculations for each strip are recorded in a table. Variation: Students take two strips that together are less than 1 metre, measure them and add the lengths together. They estimate and calculate how long a third strip would need to be to make exactly 1 metre. Students also record the measurements using decimal notation.

Adjust the length required to measure if students are having difficulty with task

grid paper, scissors, 30cm and 1m rulers, paper and pencils

2

Investigating Perimeters Students use geo-boards to investigate perimeters of shapes. They use shapes that have square corners. Students construct shapes that have perimeters of 4 units, 6 units, 8 units, etc. They record the shapes on dot or square paper. Students try to make different shapes that have the same perimeters. Students are asked if it is possible to make shapes which have a perimeter of 3 units, 5 units, 7 units, etc. They use the geo-board to make a shape which has: - the smallest perimeter - the largest perimeter

Use online geo-boards for students who are unable to manipulate concrete materials

geo-boards, dot or square paper

3

Perimeter Match In pairs, students are given a length (eg 16 cm) and are required to construct a two-dimensional shape on a card with this perimeter. The teacher collects, shuffles and re-allocates cards to each pair. Students estimate and then measure the perimeter of their allocated shape. They then find their partner and compare and contrast their shapes.

Peer tutoring strategies for pairing

lengths of string, shape cards, 30 cm rulers, paper and pencils

4

Millimetres Students make a table of things that have a dimension of 10 mm, 5 mm and 1 mm eg the width of a toothpick, the thickness of ten sheets of paper.

Support and extension as required

paper, pencils, 30cm rulers, assorted objects

5

Perimeters Students estimate and then measure, to the nearest centimetre, the perimeters of small items such as book covers, art paper, leaves. Students record the results and discuss.


variety of items, 30cm rulers, paper and pencils

6

How Far is a Kilometre? Students discuss how kilometres are used as a unit to measure distance, and the relationship between metres and kilometres. Students discuss the distance represented by 1 kilometre, in terms of distance to local landmarks or walking routes in the school grounds, and the possible time taken to walk 1 kilometre. Students discuss how to measure 1 kilometre in the school grounds, possibly by measuring 100 metres and multiplying by 10. Students estimate, then measure to see how long it takes them to walk 1 kilometre, for example by walking 100 metres 10 times. Variations: students estimate, then measure, how many steps they would take when walking 1 kilometre, or time taken by different age groups of students, or time taken to ride a bicycle or skateboard for 1 kilometre.


trundle wheels, tape measures, watches or stop watches, pencils and paper

7

Desks Over the Horizon Students estimate, then calculate how many desks aligned end to end would fit into a line 1 kilometre long. Students record measurements and calculations. Variation: students calculate how many times their body length would need to be repeated to measure 1 kilometre or how many times the length of a pair of students would need to be repeated.


desks, measuring tapes, 30cm and 1m rulers, calculators, pencils and paper

8

How Long? Students work in small groups to answer: How long is the wool in a ball of wool? Students may need to discuss a range of strategies before commencing to measure. Students express the measurement in kilometres, and in metres.

Peer tutoring strategy for group formations

balls of wool, measuring devices, paper and pencils

9

Introduce Scale Students investigate how the representation of an object is reduced, when the object is drawn to scale. Small groups of students photocopy an object such as a pencil. The pencil is copied again, reduced to 50% of the original size (1:2). The pencil is copied a third time, reduced to 25% of its original size (1:4). Students discuss the lengths of the pencil in the second and third copies, compared with the original length. Students measure the length of an object (watch, pencil case, strip of paper) and predict the length when the object is drawn to a scale of 1:4. Check by cutting a strip of paper the predicted length, folding and using this to measure the object. Whole class discusses why and how maps are drawn to scale, and the units of measure which are commonly cited on a scale.


different objects, 30cm and 1m rulers, photocopier, strips of paper, pencils

10

Finding the Detail Whole class discuss how to use a scale to represent kilometres or metres on a street map. Students are given a map of the local area, showing the location of the school. Students use scale and a drawing compass to mark the area within 500 metres of the school in all directions. Students list the street names or landmarks within this area.

Questioning techniques local maps, grid paper, 30cm rulers, pencils, paper

ASSESSMENT OVERVIEW




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Time 1 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › uses 24-hour time and am and pm notation in real-life situations, and constructs timelines MA3-13MG

Background Information Australia is divided into three time zones. In non-daylight saving periods, time in Queensland, New South Wales, Victoria and Tasmania is Eastern Standard Time (EST), time in South Australia and the Northern Territory is half an hour behind EST, and time in Western Australia is two hours behind EST. Typically, 24-hour time is recorded without the use of the colon (:), eg 3:45 pm is written as 1545 or 1545 h and read as 'fifteen forty-five hours'. Language Students should be able to communicate using the following language: 12-hour time, 24-hour time, time zone, daylight saving, local time, hour, minute, second, am (notation), pm (notation).

Compare 12- and 24-hour time systems and convert between them • tell the time accurately using 24-hour time, eg '2330 is the same as 11:30 pm' - describe circumstances in which 24-hour time is used, eg transport, armed forces, digital technologies • convert between 24-hour time and time given using am or pm notation • compare the local times in various time zones in Australia, including during daylight saving Determine and compare the duration of events • select an appropriate unit to measure a particular period of time • use a stopwatch to measure and compare the duration of events • order a series of events according to the time taken to complete each one • use start and finish times to calculate the elapsed time of events, eg the time taken to travel from home to school






Compare 12- and 24-hour time systems and convert between them

Determine and compare the duration of events

1

Timetables Students access timetables on the Internet or the teacher provides students with a variety of timetables eg bus, plane, train, ferry, theme parks, movies. Students describe any visible patterns eg ‘Buses leave every 15 minutes on weekday mornings.’ Students calculate the duration of different journeys or events using start and finish times. They develop an itinerary for a given time-frame eg 4 hours. Students plan their ‘ultimate’ 24-hour itinerary. Students record their itinerary in 12-hour time using am and pm notation, and 24-hour time. Students discuss which timetables use 24-hour time and why it is important.


timetables, computers, paper and pencils

2

Stopwatches Students read digital stopwatch displays showing time from left to right in minutes, seconds and hundredths of a second.

2:34:26

Students use stopwatches to time various events and order them according to the time taken. Students discuss cases where accurate timing is important eg athletics, swimming, television advertisements.

Extension: Students research the world records of different sports. They then record and order them.

stop watches, paper and pencils

3

Reading a Timeline The teacher displays a timeline related to real life or a literary text. Students write what they can interpret from the timeline.

Olympic Timeline 1896 The first modern Olympic Games held in Athens, Greece. 1900 Women first compete in the Games, in tennis and golf. 1904 1908 1912 1916 Games cancelled due to the First World War. 1920 1924 1928 1932 1936 1940 Games cancelled because of the Second World War 1944 Games cancelled because of the Second World War. 1948 1956 Olympic Games held in Melbourne 1960 1964 1968


timeline, paper and pencil

1972 Munich Olympics marred by terrorist attack 1976 Montreal hosts the games. 1980 The United States, Canada and 50 other countries boycott the Moscow Games following the invasion of Afghanistan by the Soviet Union. 1984 The Soviet Union boycotts the Olympics in Los Angeles. 1988 1992 South Africa permitted to the games for the first time after a 30-year ban. 12 separate teams represent the countries formerly part of the USSR. 1996 2000 Olympic Games held in Sydney.

4

Timing Experiments Students estimate the amount of time selected events will take and then check by timing the events with a stopwatch eg - the time for a ball dropped from the top floor of a building to reach the ground - the time for a car seen in the distance to reach a chosen point. Students record the times in a table and order the events.


stop watches, paper and pencils

9 Revision

10

Assessment

ASSESSMENT OVERVIEW




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: 2D 1 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations MA3-2WM › gives a valid reason for supporting one possible solution over another MA3-3WM › manipulates, classifies and draws two-dimensional shapes, including equilateral, isosceles and scalene triangles, and describes their properties MA3-15MG

Background Information A shape has rotational symmetry if a tracing of the shape, rotated part of a full turn around its centre, matches the original shape exactly. The order of rotational symmetry refers to the number of times a figure coincides with its original position in turning through one full rotation, eg

'Scalene' is derived from the Greek word skalenos, meaning 'uneven'; our English word 'scale' is derived from the same word. 'Isosceles' is derived from the Greek words isos, meaning 'equals', and skelos, meaning 'leg'. 'Equilateral' is derived from the Latin words aequus, meaning 'equal', and latus, meaning 'side'. 'Equiangular' is derived from aequus and another Latin word, angulus, meaning 'corner'. Language Students should be able to communicate using the following language: shape, two-dimensional shape (2D shape), triangle, equilateral triangle, isosceles triangle, scalene triangle, right angled triangle, quadrilateral, parallelogram, rectangle, rhombus, square, trapezium, kite, pentagon, hexagon, octagon, regular shape, irregular shape, features, properties, side, parallel, pair of parallel sides, opposite, length, vertex (vertices), angle, right angle, line (axis) of symmetry, rotational symmetry, order of rotational symmetry, translate, reflect, rotate, enlarge. A 'feature' of a shape or object is a generally observable attribute of a shape or object. A 'property' of a shape or object is an attribute that requires mathematical knowledge to be identified.

Classify two-dimensional shapes and describe their features • manipulate, identify & name right-angled, equilateral, isosceles & scalene triangles - recognise that a triangle can be both right-angled & isosceles or right-angled & scalene • compare & describe features of the sides of equilateral, isosceles & scalene triangles • explore by measurement side & angle properties of equilateral, isosceles & scalene triangles • explore by measurement angle properties of squares, rectangles, parallelograms & rhombuses • select and classify a 2D shape from a description of its features - recognise that 2Dshapes can be classified in more than 1 way • identify & draw regular & irregular 2D shapes from descriptions of their side & angle properties - use tools such as templates, rulers, set squares & protractors to draw regular & irregular 2D shapes - explain the difference between regular & irregular shapes - use computer drawing tools to construct a shape from a description of its side & angle properties Describe translations, reflections and rotations of 2D shapes • use the terms translate, reflect & rotate to describe movement of 2D shapes - rotate a graphic or object through a specified angle about a particular point, including by using the rotate function in a computer drawing program • describe the effect when a 2D shape is translated, reflected or rotated - recognise that the properties of shapes do not change when shapes are translated, reflected or rotated Identify line and rotational symmetries • identify & quantify the total number of lines (axes) of symmetry (if any exist) of 2D shapes, including the special quads & triangles • identify shapes that have rotational symmetry & determine the order of rotational symmetry - construct designs with rotational symmetry, with/out the use of digital technologies Apply the enlargement transformation to familiar 2D shapes & explore the properties of the resulting image compared with the original • make enlargements of 2D shapes, pictures & maps, with/ out the use of digital technologies - overlay an image with a grid composed of small squares - investigate and use functions of digital technologies that allow shapes and images to be enlarged without losing the relative proportions of the image • compare representations of shapes, pictures & maps in different sizes - measure an interval on an original representation and its enlargement to determine how many times larger than the original the enlargement is






Classify two-dimensional shapes and describe their features

Describe translations, reflections and rotations of 2D shapes

Identify line and rotational symmetries Apply the enlargement transformation to familiar 2D shapes & explore the properties of the resulting image compared with the original

2

What am I? Students select a shape and write a description of its side and angle properties. Students share their descriptions with the class who attempt to identify the shape eg ‘My shape has four sides and four equal angles. The opposite sides are the same length. What am I?’ Variation: Students create flipbooks recording clues and share with a friend. Students reproduce shapes and clues using a computer software package


paper, pencils, rulers

3

Barrier Game In pairs, students are positioned back to back. One student is the ‘sketcher’, the other student is the ‘describer’. The ‘describer’ describes a given two-dimensional shape focusing on side and angle properties. The ‘sketcher’ listens to the description and sketches the two-dimensional shape described. The ‘sketcher’ names the two-dimensional shape sketched and then compares their sketch with the describer’s shape. The students swap roles and repeat the activity.

Peer tutoring strategy for pairings. 2D shape pictures to prompt drawings

paper and pencil

4

Properties of Two-Dimensional Shapes Students examine regular and irregular two-dimensional shapes and name their parts. Angle testers, set squares or protractors could be used to compare the size of angles and to identify equal angles. Rulers could be used to compare lengths of sides and to identify sides of equal length. Students are asked to identify shapes that have rotational symmetry. Students could present the information as descriptions of each shape’s side and angle properties.


angle testers, set squares, protractors, rulers, 2D shape pictures, paper and pencils

5

Circles In small groups, students draw a large circle in the playground using a range of materials eg ropes, stakes, chalk, tape measures. Students assess their circle and the strategy they used. They label parts of their circle: centre, radius, diameter, circumference, sector, semi-circle and quadrant. Students then investigate materials in the classroom they can use to draw circles eg a pair of compasses, a protractor, round containers, templates. They then draw and label circles.

Extension: to students drawing squares, equilateral triangles, regular hexagons, and regular octagons with in circles.

chalk, rope, tape measures, pencils, paper

6 Diagonals Students explore diagonals by joining two geo-strips of equal length at their centres. They then join the ends of these to other geo-strips to form a two-dimensional shape. eg Students join three or more geo-strips of different lengths at their centres and use other geo-strips to join the ends of these to make various two-dimensional shapes. Possible questions include: - what is the relationship between the number of sides and the number of diagonals? - which shapes are the strongest? - what happens when the diagonals are removed? In groups, students draw their two-dimensional shapes complete with diagonals, and record their findings. The students’ posters could be displayed.

Assistance to use geo-boards or online geo-boards

geo-boards, paper, pencils, dot or grid paper, chart paper

7

Enlarging and Reducing Students are given drawings of a variety of two-dimensional shapes on grid paper. Students enlarge or reduce the shapes onto another piece of grid paper. Possible questions include: - what features change when a two-dimensional shape is enlarged or reduced? - what features remain the same? - do properties change or remain the same? Why? Students explain the process they used to enlarge and reduce two-dimensional shapes.

Assistance to draw shapes, online drawing programs

grid paper, pencils, paper, rulers

ASSESSMENT OVERVIEW




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Position KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › locates and describes position on maps using a grid reference system MA3-17MG

Background Information In Stage 2, students were introduced to the compass directions north, east, south and west, and north-east, south-east, south-west and north-west. In Stage 3, students are expected to use these compass directions when describing routes between locations on maps. By convention when using grid-reference systems, the horizontal component of direction is named first, followed by the vertical component. This connects with plotting points on the Cartesian plane in Stage 3 Patterns and Algebra, where the horizontal coordinate is recorded first, followed by the vertical coordinate. Language Students should be able to communicate using the following language: position, location, map, plan, street directory, route, grid, grid reference, legend, key, scale, directions, compass, north, east, south, west, north-east, south-east, south-west, north-west.

Use a grid-reference system to describe locations • find locations on maps, including maps with legends, given their grid references • describe particular locations on grid-referenced maps, including maps with a legend, eg 'The post office is at E4' Describe routes using landmarks and directional language • find a location on a map that is in a given direction from a town or landmark, eg locate a town that is north-east of Broken Hill • describe the direction of one location relative to another, eg 'Darwin is north-west of Sydney' • follow a sequence of two or more directions, including compass directions, to find and identify a particular location on a map • use a given map to plan and show a route from one location to another, eg draw a possible route to the local park or use an Aboriginal land map to plan a route - use a street directory or online map to find the route to a given location • describe a route taken on a map using landmarks and directional language, including compass directions, eg 'Start at the post office, go west to the supermarket and then go south-west to the park'






Use a grid-reference system to describe locations

Describe routes using landmarks and directional language

5

Coordinates Students are given a map with grid references. The teacher models questions such as: - what town is at G3? - what feature is located at D4? - what are the coordinates of Smith Street? Students then write a variety of questions related to the map using coordinates.


map with grid references, paper and pencils

6

The Best Route Students are given a scaled map of their suburb or a section of a city and are asked to locate two points of interest. On the map, students show the shortest or best route between the two points. Students write a description of the route using grid references, compass directions and the approximate distance travelled. Variation: On a large map of the local area, all students plot their home and the route they use to get to school. They then write a description of their route.


scaled maps, rulers, paper and pencils

7

Enlarge Me/Reduce Me Students are given a simple map, with a scale, covered by a two-centimetre grid. On a separate piece of paper they draw a four-centimetre grid and copy the map. They then draw a one-centimetre grid and copy the map. Possible questions include: - did doubling/halving the size of the grid double/halve the scale? Why? - did doubling/halving the size of the grid double/halve the size of the map? Why? - how could you use this method to enlarge/reduce a smaller section of the map?


map with a scale, 2cm grid overlays, paper, pencils, rulers

8

Paper Rounds In pairs, students are given a street directory of the local area. The teacher gives them the addresses of the places where they will start and finish their paper delivery and students use coordinates to find these places. They design a route for effective delivery of the papers and calculate the distance travelled using the scale. Possible questions include: - how long is your route? - can you devise a shorter route?


street directories, paper, pencils, rulers

9 Revision

10 Assessment

ASSESSMENT OVERVIEW



STRAND: STATISTICS AND PROBABILITY

TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Data 1 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: › describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions MA3-1WM › gives a valid reason for supporting one possible solution over another MA3-3WM › uses appropriate methods to collect data and constructs, interprets and evaluates data displays, including dot plots, line graphs and two-way tables MA3-18SP

Background Information Column graphs are useful in recording categorical data, including results obtained from simple probability experiments. A scale of many-to-one correspondence in a column graph or line graph means that one unit is used to represent more than one of what is being counted or measured, eg 1 cm on the vertical axis used to represent 20 cm of body height. Line graphs should only be used where meaning can be attached to the points on the line between plotted points, eg temperature readings over time. Dot plots are an alternative to a column graph when there are only a small number of data values. Each value is recorded as a dot so that the frequencies for each of the values can be counted easily. Students need to be provided with opportunities to discuss what information can be drawn from various data displays. Advantages and disadvantages of different representations of the same data should be explicitly taught. Categorical data can be separated into distinct groups, eg colour, gender, blood type. Numerical data is expressed as numbers and obtained by counting, or by measurement of a physical attribute, eg the number of students in a class (count) or the heights of students in a class (measurement). Language Students should be able to communicate using the following language: data, survey, category, display, tabulate, table, column graph, vertical columns, horizontal bars, equal spacing, title, scale, vertical axis, horizontal axis, axes, line graph, dot plots, spreadsheet.

Pose questions & collect categorical or numerical data by observation or survey • pose & refine questions to construct a survey to obtain categorical & numerical data about a matter of interest • collect categorical & numerical data through observation or by conducting surveys Construct displays, including column graphs, dot plots & tables, appropriate for data type, with/out the use of digital technologies • tabulate collected data, including numerical data, with/out the use of digital tech such as spreadsheets • construct column & line graphs of numerical data using a scale of many-to-1 correspondence, with/out the use of digital tech - name & label horizontal & vertical axes when constructing graphs - choose an appropriate title to describe data represented in a data display - determine an approp scale of many-to-1 correspondence to represent the data in a data display - mark equal spaces on the axes when constructing graphs, & use the scale to label the markers • construct dot plots for numerical data • consider data type to determine & draw most approp display/s - discuss & justify the choice of data display used - recognise that line graphs are used to represent data that demonstrates continuous change - recognise which types of data display are most approp to represent categorical data Describe and interpret different data sets in context • interpret line graphs using the scales on the axes • describe & interpret data presented in tables, dot plots, column graphs & line graphs - determine total number of data values represented in dot plots & column graphs - identify & describe relationships that can be observed in data displays - use info presented in data displays to aid decision making






Pose questions & collect categorical or numerical data by observation or survey Construct displays, including column graphs, dot plots & tables, appropriate for data type, with/out the use of digital technologies Describe and interpret different data sets in context

2

Picture Graph Students collect data for organisation into a picture graph eg daily canteen sales of pies, drinks, ice blocks. Students decide on an appropriate scale, symbol, and key eg = 10 drinks. Possible questions include: - what key did you use? - have you given your graph a title and a key? - what is the mean for the set of data? - how did you determine the scale? - how do the scale and key enable interpretation of your graph? - can you pose three questions that can be answered using the information from your picture graph? The students could represent data in a picture graph using a computer.

Use online graphs for students with fine motor difficulties

computers, paper and pencils

3

Temperature The teacher divides the students into two groups. Students in the first group record the temperature in the playground every hour, while the students in the second group record the temperature every half hour, for a day. In groups, students draw a line graph to display their data. The first group estimates the half-hourly temperatures from their line graph and compares with the actual recordings taken by the second group. Possible questions include: - how have you labelled the axes? - how did you determine a suitable scale for the data you collected? - how did the ‘hourly’ line graph help you to predict half hourly temperature changes? - is a line graph the most suitable way to represent this data? Why? - who could use a graph like this? Why? - can you record the data another way?

Extension: determining the average temperature for the day

thermometers, paper and pencils

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Populations of Countries Students use the Internet to find the population of ten countries. They graph their findings using an appropriate scale to represent large numbers. Students are encouraged to represent the data using different types of graphs and discuss the advantages and disadvantages of each representation.


computers, paper and pencils

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Sector Graphs Students collect sector graphs from sources such as newspapers and the Internet, or the teacher provides a graph. Students discuss the relative sizes of sectors, stating absolute quantities only where half and quarter circles are involved. Students answer questions using the data in the sector graph eg Favourite Sports


computers, newspapers, paper and pencils

Possible questions include: - what sport do half the people surveyed prefer? - what sport do a quarter of the people surveyed prefer? - which two sports combined are preferred by a quarter of the people surveyed?

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Revision

10

Assessment

ASSESSMENT OVERVIEW

Maths Program.s3 Yr5 t1

Documents

Transcript of Maths Program.s3 Yr5 t1