Pioneer State High School Numeracy Skills Booklet€¦ · Pioneer State High School Numeracy Booklet Page 1 Pioneer State High School Numeracy Skills Booklet . ... The symbol used

Pioneer State High School Numeracy Booklet Page 1

Pioneer State High School

Numeracy Skills Booklet


Contents Page

Number and Operations Page 3

Number Facts to 10 – Strategies Page 3

Even and Odd Numbers Page 4

Prime Numbers Page 5

Addition Algorithm Page 6

Subtraction Algorithm Page 6

Multiplication Algorithm – Single Digit Multiplier Page 6

Multiplication Algorithm – Double Digit Multiplier Page 6

Division Algorithm – Single Digit Page 7

Division Algorithm – Double Digit Page 7

Converting Numbers to Scientific Notation Page 8

Operating With Money Page 8

Currency Page 8

Exchange between Australian dollars and American dollars Page 9

Basic Measurement Page 10

Time Page 10

Measurement Conversions Page 11

Advanced Measurement Page 12

Area Calculations Page 12

Percentage Calculations Page 13

Changing Percentages to Fractions and Decimals Page 14

Data Collection Page 15

Data Presentation Page 16

Graphs Page 16

Data Analysis Page 19

Mean, Median, Mode and Range Page 19

Basic Algebra Page 20

Scale, Ratio and Rate Page 21

Estimation Strategies Page 23


Number and Operations

Number Facts to 10 - Strategies

Count on 0, 1, 2, 3

When adding 0 to any number you get the number you started with.

E.g. 7 + 0 = 7

When adding two single digit numbers, you should never count on any more than 3.

E.g. 7 + 2, would be 7, count on 2 which equals 9.

Students need to memorise their doubles facts.

E.g. 4 + 4 = 8

Doubles plus 1 is a strategy used for numbers that are close together.

E.g. 5 + 6

5 + 5 + 1 = 11

Doubles plus 2 can be used as a strategy.

E.g. 6 + 8

6 + 6 + 2 = 14

Make a 10 or near 10 is used when one of the numbers is either an 8 or a 9.

E.g. 9 + 4

9 + 1 + 3 = 13

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n 0

, 1, 2

, 3

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nd


Even and Odd Numbers

Even Numbers

It is any integer that can be divided exactly by 2.

The last digit will be 0, 2, 4, 6 or 8

Example: -24, 0, 6 and 38 are all even numbers

Odd Numbers

If it is not an even number, therefore not divisible by 2, it is called an odd number.

The last digit will be 1, 3, 5, 7 or 9

Example: -3, 1, 7 and 35 are all odd numbers

Adding and Subtracting

When you add (or subtract) odd or even numbers the results are always:

Operation Result Example

(red is odd, blue is even)

Even + Even Even 2 + 4 = 6

Even + Odd Odd 6 + 3 = 9

Odd + Even Odd 5 + 12 = 17

Odd + Odd Even 3 + 5 = 8


Prime Numbers

A prime number can be divided, without a remainder, only by itself and by 1. For example, 17 can

be divided only by 17 and by 1.

Some facts:

The only even prime number is 2. All other even numbers can be divided

by 2.

Zero and 1 are not considered prime numbers.

To prove whether a number is a prime number, first try dividing it by 2, and see if you get a whole

number. If you do, it can't be a prime number. If you don't get a whole number, next try dividing it

by prime numbers: 3, 5, 7, 11 (9 is divisible by 3) and so on, always dividing by a prime number

(see table below).

Here is a table of all prime numbers up 600

2 3 5 7 11 13 17 19 23

29 31 37 41 43 47 53 59 61 67

71 73 79 83 89 97 101 103 107 109

113 127 131 137 139 149 151 157 163 167

173 179 181 191 193 197 199 211 223 227

229 233 239 241 251 257 263 269 271 277

281 283 293 307 311 313 317 331 337 347

349 353 359 367 373 379 383 389 397 401

409 419 421 431 433 439 443 449 457 461

463 467 479 487 491 499 503 509 521 523

541 547 557 563 569 571 577 587 593 599


Addition Algorithm

3 4 7

+ 8 9 6

1

3 4 7

+ 8 9 6

3

1 1

3 4 7

+ 8 9 6

4 3

1 1

3 4 7

+ 8 9 6

1 2 4 3

Subtraction Algorithm

6 2 7

- 1 3 5

2

12

6 2 7

- 1 3 5

2

5 12

6 2 7

- 1 3 5

9 2

5 12

6 2 7

- 1 3 5

4 9 2

Multiplication Algorithm – Single Digit Multiplier

2 3 5

× 6

0

3

2 3 5

× 6

0

2 3

2 3 5

× 6

1 0

2 3

2 3 5

× 6

1 4 1 0

Multiplication Algorithm – Double Digit Multiplier

3 2 6

× 2 6

6

1 3

3 2 6

× 2 6

5 6

1 3

3 2 6

× 2 6

1 9 5 6

1 3

3 2 6

× 2 6

1 9 5 6

0

These are the carry overs for the units multiplication

These are the carry overs for the tens multiplication


1

1 3

3 2 6

× 2 6

1 9 5 6

2 0

1

1 3

3 2 6

× 2 6

1 9 5 6

5 2 0

1

1 3

3 2 6

× 2 6

1 9 5 6

6 5 2 0

1

1 3

3 2 6

× 2 6

1 9 5 6

6 5 2 0

8 4 7 6

Division Algorithm – Single Digit

2

3) 7 16 2

2 5

3) 7 16 12

2 5 4

3) 7 16 12

Division Algorithm – Double Digit

3

12) 4 1 0 4

- 3 6

5

3 4 2

12) 4 1 0 4

- 3 6

5 0

- 4 8

2 4

- 2 4

0

3

12) 4 1 0 4

- 3 6

5 0

3 4

12) 4 1 0 4

- 3 6

5 0

- 4 8

2

3 4

12) 4 1 0 4

- 3 6

5 0

- 4 8

2 4


Converting Numbers to Scientific Notation

100000

10000

1000

100

10

1

1/1

0

1/1

00

1/1

00

0

1/1

0000

Scientific Notation

Number

105 104 103 102 101 100 10-1 10-2 10-3 10-4

3 7 0 0 3.7 × 103 3700

2 4 3 0 0 0 2.43× 105 243 000

0 4 4× 10-2 0.04

0 0 2 7 2.7× 10-3 0.0027

The power of ten is determined by the position of the first significant figure.

Operating With Money

Currency

The unit of currency is the Australian dollar which is divided into 100 cents.

The notes are: $5, $10, $20, $50, and $100. Coins: 5c 10c 20c, 50c, $1 and $2.

Example:

Sam has $5.00 in his pocket to buy bread. The bread cost $2.60. How much change must Sam receive back?

$5. 0 0

- $2. 6 0

0

4 10

$5. 0 0

- $2. 6 0

. 4 0

4 10

$5. 0 0

- $2. 6 0

$2. 4 0


The following are examples of the remaining three operations on money.

3. 5 5

+ 8. 7 5

1

3. 5 5

+ 8. 7 5

0

1 1

3. 5 5

+ 8. 7 5

. 3 0

1 1

3. 5 5

+ 8. 7 5

1 2. 3 0

2. 3 5

× 6

0

3

2. 3 5

× 6

0

2 3

2. 3 5

× 6

. 1 0

2 3

2. 3 5

× 6

1 4. 1 0

1.

5) 5. 16 5

1. 3

5) 5. 16 15

1. 3 3

5) 5. 16 15

Exchange between Australian dollars and American dollars.

Australian Dollar US Dollar

1 AUD = 0.985141 USD

How many US dollars will I get from $120 Australian dollars?

120 x 0.985141 = $118.21692

= $118.22

How many Australian dollars will I get from $150 US dollars?

150 ÷ 0.985141 = $152.26

http://www.xe.com/currency/aud-australian-dollar

http://www.xe.com/currency/usd-us-dollar


Basic Measurement Time

We can use our knowledge of basic time facts to help convert between hours, seconds and minutes.

By knowing these facts: 1 minute = 60 seconds 1 hour = 60 minutes 1 day = 24 hours 1 week = 7 days 1 fortnight = 14 days 1 year = 52 weeks 10 years = 1 decade 100 years = 10 decade = 1 century

We can convert times such as: 3 minutes = 180 seconds (3 × 60) 240 seconds = 4 minutes (240÷60) 1 ½ hours = 90 minutes (60 + 30) 360 minutes = 6 hours (360÷60) 1 week = 7 days = 168 hours (7 × 24) 216 hours = 9 days (216÷24) 2 years = 104 weeks (2 x 52)

We use am and pm with digital time.

am The part of the day between 12 midnight and 12 noon. pm The part of the day between 12 noon and 12 midnight.

Time can be measured using 12 hour time, using am/pm, or 24 hour time.


Measurement Conversions

E.g. Convert 3 km into cm 3km x 1000 = 3000m 3000m x 100 = 300 000cm There are 300 000cm in 3 km E.g. 400 000mm is the same as how many m 400 000mm ÷ 10 = 40 000 cm 40 000cm ÷ 100 = 400m 400m is the same distance as 400 000mm

Conversions

1 cm = 10 mm

1 m = 100 cm

1 km = 1000 m


Advanced Measurement

Area Calculations SUBSTITUTING FOLLOWS THE SAME RULES AS ALGEBRAIC EQUATIONS

ALL AREA IS MEASURED IN SQUARE UNITS

Rules

E.g. Find the area of the following Triangle

Base = 5cm = b Vertical Height =7cm = h Area of Triangle = ½ b x h = ½ x 5cm x 7cm = 17.5cm2

Triangle Area = ½b × h

b = base h = vertical height

Square Area = a2

a = length of side

Rectangle Area = w × h

w = width h = height

Parallelogram Area = b × h

b = base h = vertical height

Trapezium (UK)

Area = ½(a+b) × h h = vertical height

Circle Area = πr2

Circumference = 2πr r = radius

[Type a quote from the document or the summary of an interesting point. You can position the text box anywhere in the document. Use the Drawing Tools tab to change the formatting of the pull quote text box.]

Conversions 1 cm2 = 100 mm2

1 m2 = 10 000 cm2

1 Hectare = 10 000 m2

http://www.mathsisfun.com/triangle.html

http://www.mathsisfun.com/quadrilaterals.html




http://www.mathsisfun.com/geometry/circle.html


Percentage Calculations

A fraction that is written out of one hundred is called a percentage. The symbol used for per cent is %.

For example, 100

1

= 1%. Say “1 per cent”

100

20

= 20%. Say “20 per cent” 100% means the whole. Percentages can be shown using a diagram

100

50, means 50% and it can be represented by the shaded part of the following grid.


Changing Percentages to Fractions and Decimals From Percentage to Decimal

To convert from percent to decimal: divide by 100, and remove the "%" sign.

The easiest way to divide by 100 is to move the decimal point 2 places to the left.

From Percentage To Decimal

move the decimal point 2 places to the left, and remove the "%" sign.

From Decimal to Percentage

To convert from decimal to percentage: multiply by 100, and add a "%" sign.

The easiest way to multiply by 100 is to move the decimal point 2 places to the right. So:

From Decimal To Percentage

move the decimal point 2 places to the right, and add the "%" sign.

From Fraction to Decimal

The easiest way to convert a fraction to a decimal is to divide the top number by the bottom number (divide the numerator by the denominator in mathematical language)

Example: Convert 5

2 to a decimal

Divide 2 by 5, 2 ÷ 5 = 0.4

Answer: 4.05

2

From Fraction to Percentage

The easiest way to convert a fraction to a percentage is to divide the top number by the bottom number, then multiply the result by 100, and add the "%" sign.

Example: Convert 8

3to a percentage

First divide 3 by 8: 3 ÷ 8 = 0.375,

Then multiply by 100: 0.375 x 100 = 37.5%


From Percentage to Fraction

To convert a percentage to a fraction, first convert to a decimal (divide by 100), then use the steps for converting decimal to fractions (like above).

Example: To convert 80% to a fraction

Steps Example

Write down the percentage "over" the number 100 100

80

Divide the numerator and the denominator by the same number 20100

2080

Then simplify the fraction 5

4

To calculate a percentage of a quantity Example: 25% of $120 = 0.25 x $120 (Change 25% to decimal by dividing by 100) = $30

Data Collection

Data is tabulated using a frequency distribution table. For example:

The whole class stood near the line at the tuckshop to record the types of hot food that the students were buying.

Chips Hot dog Chick. twist Pie Hamburger Curry Pie Chips Chips Curry Chick. twist Curry Curry Pie Hot dog Chick. twist Chips Hamburger Chick. twist Chick. twist Chips Hamburger Hot dog Pie Hot dog Hamburger Pie Chips Chips Chick. twist

Food

Tally

Frequency

Chips

7

Pie

5

Curry

4

Chick. Twist

6

Hot Dog

4

Hamburger

4

Total

30


Data Presentation

Graphs What is a graph?

It is a tool used to effectively communicate statistics.

A graph enables the easy identification of important patterns. What to consider when constructing a graph.

The type of graph selected must suit the type of data being represented.

Accuracy Is absolutely necessary. The points and lines on the graph must be precise. GRAPHING CONVENTIONS. Scale – The scale must use appropriate intervals. - The units of measurement must be clearly indicated. E.g. Metres, Tonnes.

Axis - Each axis must be clearly labelled. Legend - A legend (key) explains colour, shading, lines or symbols used on the graph

when required. For example a simple line, column and bar graph would not require a key.

Title - What the data on the graph is about. - Place at the top of the graph. - If the graph is part of an assignment, the title will be part of the figure number.

Border - A border must be drawn around the graph. All information must be included inside the border, including the title.

Source - This state where the data came from. Only needs to be included when appropriate. If the source is unknown state “Source unknown”.

OTHER

Construct a graph using a pencil and a ruler. (Mistakes can be easily fixed)

Any colour used on the graph must have meaning.

Keep colour to a minimum.

Neatness and accuracy are essential.

TYPES OF GRAPHS.

1. LINE GRAPHS. 2.

These are commonly used to show trends over time.

x axis is usually the time variable, for example year. Data is plotted sequentially.

y axis is used to plot the data such as the total, amount, percentage.

Types of line graphs: Simple line graph Multiple line graph – shows a comparison in the change of items. Cumulative or compound line graph – shows the total quantity of an item and the various

parts of the total.


EXAMPLE: LINE GRAPH

Source: ABS Australian Social Trends 4102.0 June 2011

3. COLUMN AND BAR GRAPHS

column graphs are vertical and bar graphs are horizontal.

used to show a single variable over a period of time OR

two or more variables at one point in time.

Types Simple Multiple Complex Divergence

0

10

20

30

40

50

60

70

80

90

100

1996 1998 2000 2002 2004 2006 2008 2010

Pe

rce

nta

ge

Year

Australian households with access to computers

Source

Vertical axis clearly labelled

Title

Horizontal axis clearly labelled

Appropriate scale

Border


EXAMPLE: COLUMN GRAPH

Source: ABS Australian Social Trends 4102.0 June 2011

4. PIE CHARTS

Construct a Pie Chart to display the proportion of students in each year level.

Year 8 – 192; Year 9 – 176; Year 10 – 160; Year 11 – 144; Year 12 – 128; and Total - 800.

Calculate the degrees for each sector

Year 8

Year 11

Year 9

Year 12

Year 10

0

10

20

30

40

50

60

70

80

90

Educationalactivities

Playing games Social networking Music

Pe

rce

nta

ge

Type of use

How Australian Children Use The Internet (2009)

All columns are equal in width

Equal spacing between bars


School Numbers by Year Level

Data Analysis

Mean, Median, Mode and Range.

The Mean is given by the sum of the scores divided by the number of scores.

The Median is the middle score when they are arranged from lowest to highest.

The Mode is the most frequently occurring score.

The Range is the difference between the lowest and highest scores.

Calculating the Mean, Mode and Range for the following scores

12, 13, 18, 16, 15, 14, 16, 14, 19, 16, 20

Year 8

Year 9

Year 10

Year 11

Year 12


Calculating the Median for an Odd number of scores

Middle Score

Ascending order: 12, 13, 14, 14, 15, 16, 16, 16, 18, 19, 20

Calculating the Median for an Even number of scores

Middle Score

Ascending order: 12, 13, 14, 14, 15, 16, 16, 16, 18, 19

Basic Algebra

Solve for X:

X + 8 = 15

X = 15 – 8

X = 7

X = 5 × 4

X = 20

X – 5 = 9

X = 9 + 5

X = 14

2X + 5 = 13

2X = 13 – 5

2X = 8

X = 8 ÷ 2

X = 4

3X = 12

X = 12 ÷ 3

X = 4

X = 3 × 9

X = 27


Scale, Ratio and Rate

Ratios A ratio compares two or more quantities of the same kind. For Example:

There are 3 shaded squares to 1 unshaded square. The ratio of shaded to unshaded is 3:1.

Write Say 3:1 “3 is to 1”

Ratios can be multiplied or divided by a number to give an equivalent ratio. For example: Concrete is made by mixing cement, sand, stones and water. A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6. You can multiply all values by the same amount and you will still have the same ratio.

10:20:60 is the same as 1:2:6

So if you used 10 buckets of cement, you should use 20 of sand and 60 of stones. Quantities can be divided by a ratio in the following way: For example: $40 is to be shared with two people in the ratio of 3:2. Add the number of parts in the ratio 3+2 = 5 “parts” Divide the quantity by the number of parts $40 ÷ 5 = $8 Each “part” is worth $8 First person receives 3 × $8 = $24 Second person receives 2 × $8 = $16

Rates Rates compare quantities of different kinds. Some common rates include speed, pay rates, and price per kilogram. To calculate a rate, divide the one quantity by the other. The rate tells you what to calculate. i.e.

Word Rate (symbols) Method

Speed Km/h hour

km

Pay Rate $/h hour

$

Price per Kilogram

$/kg kg

$


Scale

Scales are used to represent very large or very small lengths on maps and models. Scales are often written as ratios. Map/Model Length : Real Length For example:

1:100 Is the same as

1 cm = 100 cm

Is the same as

1cm = 1m

The scale factor is used to calculate unknown lengths.

Scale Factor

= Real Length

Map Length To find a real length Real Length = Map Length × Scale Factor To find a map length Map Length = Real Length ÷ Scale Factor


Estimation Strategies

Estimation strategies for number

There are many different approaches to numerical estimation, and good estimators use a variety of strategies.

Front-end strategy

This strategy has its strongest application in addition. The left-most digits (front-end) are the most significant in forming an initial estimate.

1.52 6.25 0.93 2.55 +

Front-end process: Add the front-end amounts: $1 + $6 + $2 = $9

Adjust the total by grouping the cents to form dollars 52c + 25c makes $1 approx.

93c is nearly $1

55c is nearly 50c

cents estimate: $2.50 overall estimate is $11.50 ($9 + $2.50).

This front-end process can be applied to multiplication.

369 x 6 300 x 6 = 1800 70 x 6 = 420 Estimate is 2220

Clustering strategy

This is best suited to groups of numbers that 'cluster' around a common value, for example

Numbers of people who came to our concert

Monday 425 Tuesday 506 Wednesday 498 Thursday 468 Friday 600

The average attendance was about 500 per night. 500x5 nights = 2500.


Rounding strategy

Numbers can be rounded to any selected place value. The choice of rounding place will produce different but reasonable results. 37 x 59: in this case it would be best to round both numbers up:

40 x 60 = 2400 51 x 22: here we would round both numbers down to 50 and 20:

50 x 20 = 1000 24 x 65: they are both close to the middle so you can try rounding one down (20) and one up (70):

20 x 70 = 1400 Rounding can be used with the four operations but is very useful in division. In division it is often better to round up: 419 ÷ 65 could be rounded to

420÷70=6.

Special numbers strategy

This strategy looks for numbers that make patterns, for example tens or hundreds. (a) 3 5 7 4 6 +

3 and 7 are ten, 6 and 4 are ten, that's 20; add the 5, and this gives a total of 25.

(b) 37 54 71 42 69+ Group the tens using a mixture of rounding and compatibility, for example 37 and 42 is about 80, 69 and 71 is 140 and 54 is approximately 50. This gives a total of 80 + 140 + 50 = 270.

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Pioneer State High School Numeracy Skills Booklet€¦ · Pioneer State High School Numeracy Booklet Page 1 Pioneer State High School Numeracy Skills Booklet . ... The symbol used