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Mathematical Literacy and basic competences

in science and technology

• Lesson 1

• Mathematical Literacy

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• Definition: – Mathematical literacy is defined in the Programme for

International Student Assessment (PISA) as the capacity to identify, understand and engage in mathematics, and to make well-founded judgements about the role that mathematics plays in an individual’s current and future private life, occupational life, social life with peers and relatives, and life as a constructive, concerned and reflective citizen.

– Source Publication: Education at a Glance, OECD, Paris, 2002, Glossary

.

• A skill which is used in everyday situations. – Emphasis is on activity rather than knowledge.

An introduction

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• Definition:– It is an individual’s capability and

capacity to understand the role mathematics play in the world, and be able to apply them in ways that serve his everyday life (i.e. when shopping, traveling, cooking (PISA).

• Although, basic mathematics were covered at an early childhood, people need to be able to use them in unstructured contexts were directions are not clear and they need to decide which knowledge should be applied in each situation.

Mathematical Literacy

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• What kind of knowledge you need to solve the following problem?– If the sales tax is 6% and a €10.00 purchase is made, the

sales tax is € 10.00*6/100 =€ 0.60 • Required: knowledge of ratios and percentage

– The sum of 645 and 450• Required: Mental math (estimating and rounding numbers).

Mental math are vital as they are used on an every day basis. No time to think, no time for a calculator!

– Answer: 645 + 450= ?– 645 is close to 600 and 450 is close to 500 answer is

1100 which is close to 1095.

Mathematical Literacy

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• Addition: 15+70=85• Subtraction: 85-70=15

– If you add an amount then take it away again, you will end up at the same place and vice versa.

• A reminder about values– Have a look at the number 623.

• 6 is the hundreds digit. • 2 is the tens digit. • 3 is the units digit.

Computation methods

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• Estimate the answer: Round 314 to 300 and 82 to 100. – The total is 100+300= 400.

• Answer is close to 396

• Estimate the answer: 384-182– 384 is rounded to 400 and 182 to 200.

400-200=200• answer is close to 202

Computation methods

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• Estimate the answer: Calculate the sum of 974 and 117– 974 is rounded to 1000 and 117 to 100.

1000+100=1100 • Answer is close to 1091.

Computation methods

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• Other ways for addition:– Splitting the large numbers

into hundreds, tens and units.

• What happens in subtraction– If one of the columns has a

smaller number on top, the number on top borrows from the number to its left.

– When working out the units in this sum, as 2 is less than 4, you have to borrow 10 from the tens column. So 2 becomes 12, and in the tens column, 9 becomes 8.

Addition and Subtraction

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• Addition: – plus, add, sum, altogether,

increase, total

• Subtraction: – subtract, minus, decrease,

difference, less than, take away, fewer than, decomposition, reduce

Addition and Subtraction glossary

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Multiplication tables

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• Reverse the question – If someone asks you what 3 x 8 is

and you're not sure of your 8 times table, turn it around into 8 x 3.

• Use the facts you know well, like 10 times a number. – If you need to work out 12 x 4, start

with 10 x 4 = 40 and add 2 more 4s to give 48.

Multiplication methods

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• Doubling – Doubling is a good trick. If you

know that 4 x 4 = 16, then you can work out 8 x 4 by doubling 16, which gives 32.

• Separate and add up – If you had to work out 25 x 5 you

could use:10 x 5 = 50 plus another 10 x 5 = 50.Then 5 x 5 = 25.Added together 50 + 50 + 25 = 125

Multiplication methods

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• Tips– To find out if a number is in the 2 times table, look at the

digit at the end.• 1 357 318 is a multiple of 2 because the digit at the end is 8,

which is even.

– To find out if a number is in the 3 times table, add up the digits of the number you want to find out about. If they add up to 3, 6, or 9, then you know that it's in the 3 times table .

– All the numbers in the 4 times table are EVEN - they end with 0, 2, 4, 6 or 8 (116:look at last one digit).

Multiplication tips

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• Tips– All multiples of 5 end in a 5 or a 0.– All the numbers in the 6 times table are EVEN - they end

with 0, 2, 4, 6 or 8. Additionally, they are all a multiple of 3, they can be divided by 3.

– You can work out a 6 times sum by doubling the number and then tripling it. 5 x 6 is the same as 5 x 2 = 10, then 10 x 3 = 30.

Multiplication tips

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• Tips– There is no easy trick for finding

out if a number is in the 7 times table

– The numbers in the 8 times table are always even. That means they can be divided by 2 without remainder

– All the digits in the 9 times table add up to 9.

   18 = 1 + 8 = 9 27 = 2 + 7 = 9 36 = 3 + 6 = 9

Multiplication tips (cont.)

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Estimating:– When you divide any numbers, it is a

good idea to estimate a rough answer first. Your estimate can then be checked against your actual answer.

92 ÷ 3 is approximately90 ÷ 3 which is 30

143 ÷ 7 is approximately140 ÷ 7 which is 20

994 ÷ 5 is approximately1 000 ÷ 5 which is 200

Division

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• Divide 22972/4– 4 into 2 won't go - so carry 2

– 4 into 22 (5 x 4 = 20) - so carry 2

– 4 into 29 (7 x 4 = 28) - so carry 1

– 4 into 17 (4 x 4 = 16) - so carry 1

– 4 into 12, that will be 3 exactly

Division

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• Multiplication: – multiply, multiple, times, sets of, lots of, groups of, product,

factor, prime numbers

• Division: – divide, divisible, left over, remainder, share, groups

– Adapted from BBC: Skillwise

Multiplication and division

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gy• Algebra is all about:

– addition, division, multiplication, subtraction and formulas

• Example: volume=width x height x depth

– Inequalities are also part of algebra

Algebra

Symbol Words Example

> greater than x + 3 > 2

< less than 7x < 28

≥ greater than or equal to 5 ≥ x - 1

≤ less than or equal to 2y + 1 ≤ 7

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• To determine the percent of a number follow the steps:– (eg calculate 87% of 68)

• Multiply the number by the percent (e.g. 87 * 68 = 5916) • Divide the answer by 100 (Move decimal point two places to

the left) (e.g. 5916/100 = 59.16) • Round to the desired precision (e.g. 59.16 rounded to the

nearest whole number = 59) • ANSWER IS 59.16%

Percentages

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Calculate the 20% of 100?

Example

203040

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Calculate the 5% of 15?

Example

0.330.757.5

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• Ratios tell how one number is related to another number.

– A ratio may be written as A:B or A/B or by the phrase "A to B".

– A ratio of 1:2 says that the second number is two times as large as the first.

Ratios

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• Example: – Determine the value of B if A=6 and the ratio of A:B = 2:5

15 (6/B=2/5B=6*5/2=30/2=15)

• A more detailed explanation– Determine how many times the number A is divisible by the

corresponding portion of the ratio. (6/2=3) – Multiply this number by the portion of the ratio representing B

(3*5=15) – Therefore if the ratio of A:B is 2:5 and A=6 then B=15

• Adapted from:http://www.aaamath.com/rat62ax2.htm

Ratios

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• Percent Ratios are used in several situations:

– Commissions- A sales person receiving 10% commission on sales

• If sales is equal to 2000 then commission is 10/100* 2000=200

– Discounts• During sales an item that costs €200 has a 20% discount.

Therefore discount equals to 20/100*200=40 and therefore price now is

• 200-40=160

Percent Ratios

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• Tax- eg VAT– Vat in Cyprus is 15%.

• If price excluding VAT is 100 then price including VAT is 115

• Interest– When money is borrowed, interest is charged for the use of

that money for a certain period of time.

• Interest = Principal * Rate * Time. – If €100 was borrowed for 2 years at an interest rate of 10%,

the interest would be € 100*10/100*2 = € 20. The total amount that would be due would be € 100+ € 20= € 120.

Percent Ratios

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• Graphs

Graphs

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• Formulas

Formulas

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• Statistics– Statistics is the study of the collection, organization, and

interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.

Statistics

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• Standard Deviation– Standard deviation is a widely used measurement of

variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average (mean, or expected value). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

Statistics Example

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• A line is one of the basic terms in geometry. It extends in both directions forever

• Points are also used in geometry and are marked by a letter.

• Intersection is when lines, rays or figures meet• Examples

– Line 1 meets the square at points N and M– Line 2 intersects the circle at point P

Geometry

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• Line segment are simply named as segment AB or segment HG.

• Rays are used to show the direction of segments• Example:

Geometry

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• Angles: use this symbol < to indicate the way that points are allocated. Example: <PBW, <CBP, and <WBA

• Tip: Angles C and B have the same degrees

• Degrees

Geometry

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– In the right triangle, the hypotenuse has length 5. According to the Pythagorean Theorem 16 (4*4)+ 9 (3*3) = 25 (5*5) ή 42*32=25 (52)

– Area of a Triangle= h*b*1/2

– Area of a Circle =3.14*r2 (or π*r2)• 3.14*52

Geometry

                      

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Area of a Rectangle= A*B

Area of a Parallelogram= b *h

Area of a Trapezoid= 1/2 × h × (a + b)

Geometry

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