Pioneer State High School

Pioneer State High School Numeracy Skills Booklet Pioneer State High School Numeracy Booklet Page 1

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 2

Number and Operations Number Facts to 10 - Strategies Count on 0, 1, 2, 3 Turn-around 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 Pioneer State High School Numeracy Booklet Page 3

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 4

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 5

Addition Algorithm 1 1 1 1 1 3 4 7 3 4 7 3 4 7 3 4 7 + 8 9 6 + 8 9 6 + 8 9 6 + 8 9 6 3 4 3 1 2 4 3 Subtraction Algorithm 6 2 7 12 6 2 7 5 12 6 2 7 5 12 6 2 7 These are the carry overs for the tens multiplication - 1 3 5-1 3 5-1 3 5-1 3 5 2 2 9 2 4 9 2 Multiplication Algorithm Single Digit Multiplier 3 2 3 2 3 2 3 5 2 3 5 2 3 5 2 3 5 6 6 6 6 0 0 1 0 1 4 1 0 Multiplication Algorithm Double Digit Multiplier 3 2 6 1 3 3 2 6 1 3 3 2 6 1 3 3 2 6 These are the carry overs for the units multiplication 2 6 2 6 2 6 2 6 6 5 6 1 9 5 6 1 9 5 6 0 Pioneer State High School Numeracy Booklet Page 6

1 1 1 1 1 3 1 3 1 3 1 3 3 2 6 3 2 6 3 2 6 3 2 6 2 6 2 6 2 6 2 6 1 9 5 6 1 9 5 6 1 9 5 6 1 9 5 6 2 0 5 2 0 6 5 2 0 6 5 2 0 8 4 7 6 Division Algorithm Single Digit 2 3) 7 1 6 2 2 5 3) 7 1 6 1 2 2 5 4 3) 7 1 6 1 2 Division Algorithm Double Digit 3 3 3 4 3 4 12) 4 1 0 4 12) 4 1 0 4 12) 4 1 0 4 12) 4 1 0 4-3 6-3 6-3 6-3 6 5 5 0 5 0 5 0-4 8-4 8 2 2 4 3 4 2 12) 4 1 0 4-3 6 5 0-4 8 2 4-2 4 0 Pioneer State High School Numeracy Booklet Page 7

100000 10000 1000 100 10 1 1/10 1/100 1/1000 1/10000 Converting Numbers to Scientific Notation Scientific Notation Number 10 5 10 4 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 3 7 0 0 3.7 10 3 3700 2 4 3 0 0 0 2.43 10 5 243 000 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 8

The following are examples of the remaining three operations on money. 1 1 1 1 1 3. 5 5 3. 5 5 3. 5 5 3. 5 5 + 8. 7 5 + 8. 7 5 + 8. 7 5 + 8. 7 5 0. 3 0 1 2. 3 0 3 2 3 2 3 2. 3 5 2. 3 5 2. 3 5 2. 3 5 6 6 6 6 0 0. 1 0 1 4. 1 0 1. 1. 3 1. 3 3 5) 5. 1 6 5 5) 5. 1 6 1 5 5) 5. 1 6 1 5 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 9

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 pm The part of the day between 12 midnight and 12 noon. 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 10

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 11

Advanced Measurement Area Calculations SUBSTITUTING FOLLOWS THE SAME RULES AS ALGEBRAIC EQUATIONS ALL AREA IS MEASURED IN SQUARE UNITS Rules Triangle Area = ½b h b = base h = vertical height Rectangle Area = w h w = width h = height Trapezium (UK) Area = ½(a+b) h h = vertical height Square Area = a 2 a = length of side Parallelogram Area = b h b = base h = vertical height Circle Area = πr 2 Circumference = 2πr r = radius 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 = 17.5cm 2 Conversions 1 cm 2 = 100 mm 2 1 m 2 = 10 000 cm 2 1 Hectare = 10 000 m 2 Pioneer State High School Numeracy Booklet Page 12

Percentage Calculations A fraction that is written out of one hundred is called a percentage. The symbol used for per cent is %. 1 For example, 100 = 1%. Say 1 per cent 20 100 = 20%. Say 20 per cent 100% means the whole. Percentages can be shown using a diagram 50, means 50% and it can be represented by the shaded part of the following grid. 100 Pioneer State High School Numeracy Booklet Page 13

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: 2 5 0.4 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 3 to 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 14

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 Write down the percentage "over" the number 100 Divide the numerator and the denominator by the same number Then simplify the fraction To calculate a percentage of a quantity Example 80 100 80 20 100 20 4 5 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. 6 Twist Hot Dog 4 Hamburger 4 Total 30 Pioneer State High School Numeracy Booklet Page 15

What is a graph? Data Presentation Graphs 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 Legend Title Border Source - Each axis must be clearly labelled. - 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. - 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. - A border must be drawn around the graph. All information must be included inside the border, including the title. - 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 16

Percentage EXAMPLE: LINE GRAPH Border Title Vertical axis clearly labelled Appropriate scale Australian households with access to computers 100 90 80 70 60 50 40 30 20 10 0 1996 1998 2000 2002 2004 2006 2008 2010 Year Source: ABS Australian Social Trends 4102.0 June 2011 Horizontal axis clearly labelled Source 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 Pioneer State High School Numeracy Booklet Page 17

Percentage EXAMPLE: COLUMN GRAPH 90 80 70 60 50 40 30 20 10 0 How Australian Children Use The Internet (2009) Educational activities Playing games Social networking Music Type of use Source: ABS Australian Social Trends 4102.0 June 2011 Equal spacing between bars All columns are equal in width 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 9 Year 10 Year 11 Year 12 Pioneer State High School Numeracy Booklet Page 18

School Numbers by Year Level Year 8 Year 9 Year 10 Year 11 Year 12 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 Pioneer State High School Numeracy Booklet Page 19

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 Solve for X: Basic Algebra X + 8 = 15 X = 15 8 X = 7 X 5 = 9 X = 9 + 5 X = 14 3X = 12 X = 12 3 X = 4 X = 5 4 X = 20 2X + 5 = 13 2X = 13 5 2X = 8 X = 8 2 X = 4 X = 3 9 X = 27 Pioneer State High School Numeracy Booklet Page 20

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 km hour Pay Rate $/h $ hour Price per $ $/kg Kilogram kg Pioneer State High School Numeracy Booklet Page 21

Scale Scales are used to represent very large or very small lengths on maps and models. Scales are often written as ratios. For example: Map/Model Length : Real Length Is the same as Is the same as 1:100 1 cm = 100 cm 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 22

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 23

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. Pioneer State High School Numeracy Booklet Page 24