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Pioneer State High School Numeracy Booklet Page 1
Pioneer State High School
Numeracy Skills Booklet
Pioneer State High School Numeracy Booklet Page 2
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
Pioneer State High School Numeracy Booklet Page 3
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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, 1, 2
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nd
Pioneer State High School Numeracy Booklet Page 4
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
Pioneer State High School Numeracy Booklet Page 5
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
Pioneer State High School Numeracy Booklet Page 6
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
Pioneer State High School Numeracy Booklet Page 7
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
Pioneer State High School Numeracy Booklet Page 8
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
Pioneer State High School Numeracy Booklet Page 9
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
Pioneer State High School Numeracy Booklet Page 10
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.
Pioneer State High School Numeracy Booklet Page 11
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
Pioneer State High School Numeracy Booklet Page 12
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
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Conversions 1 cm2 = 100 mm2
1 m2 = 10 000 cm2
1 Hectare = 10 000 m2
Pioneer State High School Numeracy Booklet Page 13
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.
Pioneer State High School Numeracy Booklet Page 14
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%
Pioneer State High School Numeracy Booklet Page 15
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
Pioneer State High School Numeracy Booklet Page 16
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.
Pioneer State High School Numeracy Booklet Page 17
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
Pioneer State High School Numeracy Booklet Page 18
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
Pioneer State High School Numeracy Booklet Page 19
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
Pioneer State High School Numeracy Booklet Page 20
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
Pioneer State High School Numeracy Booklet Page 21
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
$
Pioneer State High School Numeracy Booklet Page 22
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
Pioneer State High School Numeracy Booklet Page 23
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.
Pioneer State High School Numeracy Booklet Page 24
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.