ACTIVE MATHS 8
Australian Curriculum edition
Monique Miotto
Tracey MacBeth-Dunn
HomeworkProgram
This edition published in 2021 by
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Publication data
Author: Miotto, Monique and others
Title: Active Maths 8: Homework Program (Australian Curriculum edition)
ISBN: 9781420230659
Publisher: Colin McNeil and Peter Saffin
Project editor: Eve Sullivan
Editor: Liz Waud
Illustrators: Paul Lennon (cartoons). Andy Craig and Nives Porcellato (technical diagrams)
Cover designer: Dim Frangoulis
Text designer: Sunset Digital
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Typeset in Times Ten Roman 10/13pt by Sunset Digital and Nikki M Group
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The author and publisher are grateful to the following for permission to reproduce copyright material:
Extract from ‘Bottled water—the facts’, Australasian Bottled Water Institute (ABWI), 2008, http://www.bottledwater.org.au. Adapted with permission from ABWI, 18; Extract from ‘Did you know—bottled water facts’, http://www.gotap.com.au, © Do Something! 2009. Adapted with permission from Do Something!,17; The Geometer’s Sketchpad® name and images used with permission of Key Curriculum Press, 1-800-995-MATH, www.keypress.com/gsp., 44, 81, 82
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Algebra 1
1 The pronumeral in the expression 3w + 2 + 5w is
2 The constant term in the above expression is .
3 The number of terms in the expression 5xy + 7p
is
4 The coefficient of the term 5xy in the expression above is
5 The coefficient of x in the expression
p + 5w – 8x + 4 For 6–8, use the pronumerals p and q to represent two numbers.
6 Write an expression to represent the sum of the two numbers.
7 Write an expression to represent the product of the two numbers. 8 Write an expression to represent the difference between the two numbers.
[Pronumeral] [Constant] [Term] [Coefficient] [Coefficient] [Algebra words] [Algebra words] [Algebra words] [Substitution] [Substitution] [Substitution] [Substitution] [Substitution] [Substitution] [Table of values]
15 Complete the table for the rule y = 4 –2x. Express your answers as improper fractions.
[Substitution complex]
16 If a = –2, b = –3 and c = 5, evaluate 24 ab c ab + Express your answer as a mixed number.
17 If p = –11 and q = –2, evaluate 7(2p – 4q) + p 2
[Substitution complex]
For 18–30, simplify the expressions.
[Like terms]
[Simplify expression] [Simplify expression] [Simplify expression] [Simplify expression] [Simplify expression]
[Like terms]
[Like terms]
[Like terms]
[Like terms]
[Simplify expression] [Simplify expression] [Simplify expression]
Algebra 2
For 1–9, simplify each expression.
[Simplify expression]
[Like terms]
[Like terms]
[Like terms]
[Simplify expression]
[Simplify expression]
[Simplify expression]
[Simplify expression]
[Simplify expression]
For 10–15, expand the brackets using the distributive law.
[Distributive law]
[Distributive law]
[Distributive law]
[Distributive law]
[Distributive law]
[Distributive law]
For 16–20, expand the brackets and collect like terms.
For 24–28, factorise using the Highest Common Factor (HCF).
[Expand and simplify]
[Expand and simplify]
16 2(c + 4) + 4(c + 7) 17 5(p – 4) + 3(2 – p) 18 2(v + 5) + 12
[Expand and simplify]
[Expand and simplify]
8(2x – 5) – 2(x – 4) 20 x(x + 4) + x(2x – 3)
[Expand and simplify]
21 Find the common factors of 6 and 18.
[Common factors]
22 Find the common factors of 15x and 20xy
[Common factors]
[Factorise: HCF]
[Factorise: HCF]
[Factorise: HCF]
[Factorise: HCF]
[Factorise: HCF]
[Simplify expression]
[Add algebraic fractions]
23 Find the highest common factor of 4ab and 12bc
[Highest common factor]
Charlie’s garden bed
Charlie has dug a new square garden bed in the backyard. He decides to increase the length of one side of the garden bed and would like to know how much extra area that will give him for planting. Charlie has asked you to help with the calculation but has forgotten to give you the dimensions of the original square bed.
1 Let the unknown side length of the square garden bed be x m. a Sketch this garden bed in the space below showing its dimensions.
[Draw a diagram]
b What is the area of this bed in square metres?
2 Charlie asks you to add an extra metre to the length of one side of the bed. a Sketch this new garden bed in the space below showing its dimensions.
[Write a formula]
b What is the area of this bed in square metres?
c What is the extra area in square metres?
3 Charlie changes his mind and asks you to add two extra metres to the length of one side of the original bed.
a Sketch this garden bed in the space below showing its dimensions.
b What is the area of this bed in square metres?
c What is the extra area in square metres?
4 Charlie then asks you if there is an easy way he can work out the extra area. What would you tell him?
[Write a rule] [Generalise]
5 Imagine Charlie adds some number of extra metres, a m, to the length of one side of the original bed.
a Sketch this new garden bed in the space below showing its dimensions.
b What is the area of this bed in square metres?
c What is the extra area in square metres?
6 Charlie decides to add extra length to both sides of the original square bed. He adds 2 m to one side and 3 m to the other side.
[Apply findings to a new situation]
a Sketch this new garden bed in the space below showing its dimensions.
b What is the area of this bed in square metres?
c What is the extra area in square metres?
Farmer Sprout’s fence
Farmer Sprout wishes to create a rectangular enclosure outside his barn, using the barn wall as one side of the enclosure and fencing as the other sides. Entry to the enclosure will be through a door in the barn wall. He wants to use exactly 50 m of fencing.
1 The diagram below shows Farmer Sprout’s plan for the enclosure.
a What must the total length of the fence be?
b How many sides of the enclosure are fenced?
c Let x represent the width of the enclosure. What must the length of the enclosure be if x = 10 m?
d Is it possible to have an x value of 13.26 m? Explain. If so, what would be the length of the enclosure for this width?
e Write an expression for the length of the enclosure if the width is x m. Remember to specify the units.
Enclosure Barn 40 m 2 m x
2 Now create a table of possible dimensions for this enclosure in Microsoft Excel.
[To add borders to a table in Excel select the table cells, right-click, choose Format cells and open the Borders tab.]
• Open a new spreadsheet and set up the table using the headings as shown below. Enter widths of 1.0 m to 26.0 m in column C.
• To save time, enter your expression for the length of the enclosure in cell D5. Remember to type ‘C5’ instead of the x and use ‘*’ for multiply. Use the fill handle to copy the formula down. (Alternately, you can work out the corresponding lengths and enter them individually.)
• Highlight the values in column C and D, right-click, choose Format cells. Set the Category to Number and number of decimal places to ‘0’ on the Number tab so that measurements are whole numbers.
a If you can, print out your spreadsheet and paste it in the space below.
b Looking at Farmer Sprout’s plan in 1 and the dimensions of the enclosure in your table, would it be practical to build an enclosure with a width less than 5 m? Explain.
c Is it possible to build an enclosure with a width of 24 m? If so, is it practical?
d Comment on the lengths you obtained for widths of 25 m and 26 m. What do they mean?
3 For the remainder of this investigation you will consider enclosures with widths in the range 6.0 m to 20.0 m.
a Calculate the area of the enclosure for x = 6.0 m. Hint: You have the dimensions in your spreadsheet in 2
b Write an expression for the area of the rectangular enclosure using your answers from 1.
c By substituting x = 6.0 m into your expression in 3b, show that the area obtained is the same as in 3a.
• Make a copy of the current sheet by right-clicking the ‘Sheet 1’ tab at the bottom of the screen. Choose Move or copy and check the Create a copy box. You should now have two sheets, ‘Sheet 1’ and ‘Sheet 1(2)’. Rename ‘Sheet 1(2)’ by double-clicking on the tab. Call it ‘Area’. Rename ‘Sheet 1’ as ‘Dimensions’.
• Now highlight the rows of your spreadsheet containing widths of 1.0 m to 5.0 m, right-click and Delete. Also delete the rows containing widths of 21.0 m to 26.0 m. This leaves just dimensions for enclosures with widths of 6.0 m to 20.0 m.
• Now add an ‘Area’ heading and column to your table as shown below.
[To write m2 in Excel, expand the Font tab and check the Superscript box on the Format cells menu, under Format tab. ]
• To save time, enter your expression for the area of the enclosure in cell E5. Remember to type ‘C5’ instead of the x and use ‘*’ for multiply. Use the fill handle to copy the formula down. (Alternately, you can work out the corresponding areas and enter them individually.)
• Format the area values as numbers to 1 decimal place. Paste a copy of your spreadsheet in the space below.
d How does the area change as the value of x increases?
e Looking at the values in your table, estimate the maximum possible area for the enclosure to the nearest tenth of a square metre.
f What are the dimensions of the enclosure with the maximum possible area?
g Make a sketch of the barn and your enclosure in the space below. Mark on the dimensions of the enclosure.
What if Farmer Sprout starts with 30 m of fencing instead of 50 m? Create a spreadsheet like the one in 3d for widths between 4.0 m to 13.0 m. Estimate the maximum area that can be enclosed and the dimensions of that enclosure. Paste a copy of your spreadsheet in the space below. Try this!
Chance and data 1
[Definition]
1 An even chance means there is a % chance of a favourable outcome.
[Definition]
2 If an event is impossible, that means there is a % chance of it happening.
For 9–10, refer to the following information. All 400 Year 8 students at a school voted for their year level captain. The top 3 students who received votes were Jenna 180, Harry 163, Jessica 45. The remaining 12 votes went to other students in the class.
Expressing your answers as fractions in simplest form, calculate the probability that a student chosen at random:
[Definition]
3 All probabilities must be between 0 and .
4
[Definition]
The set of possible outcomes of a probability experiment is called a
For 5–8: Express all probabilities as fractions in simplest form. ‘A die labelled with the numbers from 1 to 6 is rolled once by a person playing a board game.’
5 What is the sample space for one roll of the die?
[Sample space]
6 What is the probability of rolling a 6?
[Theoretical probability]
7 What is the probability of not rolling a 6?
9 voted for Jessica.
[Theoretical probability] [Theoretical probability]
10 voted for neither Jessica nor Harry.
[Complementary events]
11 The probability of hitting a target is 0.7. What is the probability of not hitting the target?
[Complementary events]
12 If the probability of an event is 3 8 , what is the probability of the complement?
[Theoretical probability] [Theoretical probability]
[Theoretical probability]
13 In a class of 30, a total of 5 students had red hair and 4 students had glasses. Out of these there were 2 students who had both red hair and glasses. What is the P (red hair or glasses but not both)?
8 What is the probability of rolling at least a 3?
For 14–16, write the answers in the table below. Ethan and Kane conduct an experiment by flicking their spinners at the same time. Ethan’s spinner is coloured red and green, whereas Kane’s is yellow and purple. This incomplete two-way table shows some of their results.
14 Calculate the total number of spins that result in green.
[Two-way table]
15 Calculate the number of times the outcome was red and yellow.
[Two-way table]
16 Calculate the total number of spins.
[Two-way table]
For 17–19, use the completed table above to find the relative frequencies. Give your answers as simple fractions.
17 red
[Relative frequency]
18 red and yellow
[Relative frequency]
19 green or purple
[Relative frequency]
A survey of 150 Year 8 students showed that 86 use Facebook (F) and 78 use Myspace (M). Only 20 of the students surveyed use neither Facebook nor Myspace. For 20–24, write each of your answers on the Venn diagram below. n(U) = F M
20 Find the total number of Year 8 students surveyed, n(U).
[Venn diagram]
21 Find the number of students who use neither Facebook nor Myspace.
[Venn diagram]
22 Find the number of students who use both Facebook and Myspace.
[Venn diagram]
23 Find the number of students who use Facebook only.
[Venn diagram]
24 Find the number of students who use Myspace only.
[Venn diagram]
For 25–30, use the Venn diagram to estimate the probability as a simple fraction.
25 Find the probability that a student chosen at random uses Myspace.
[Estimate probability]
26 Find Pr(F ).
[Estimate probability]
27 Find Pr(not F).
[Estimate probability]
28 Find Pr(F or M).
[Estimate probability]
29 Find Pr(F and M).
[Estimate probability]
30 Find Pr(not F or not M).
[Estimate probability]
Chance and data 2
For 1–2, state whether you would get representative data or biased data.
[Collecting data]
1 A record company wants to know the favourite music of Year 8 students in Australia. They go to the local high school and ask 50 Year 8 students what their favourite music is.
[Histogram and polygon]
3 How many students had: a 3 days absent b 6 days absent
c 5 days absent.
[Collecting data]
2 A school wished to find out what students want in their school canteen. They survey every 10th student on the roll.
For 3–5, refer to the following histogram and polygon for class 8M.
[Histogram and polygon]
4 How does the graph show you that no students had 0 days or 8 or more days absent?
5 How many students are in 8M?
[Histogram and polygon]
For 6–13, refer to the following. The number of shots a golfer has on every hole of a golf course is listed below.
4, 3, 1, 6, 5, 2, 7, 3, 4, 6, 4, 4, 5, 4, 5, 5, 24
6 Calculate the following values.
[Measures of centre and spread]
a mean (1 d.p.)
b median
c mode
d range
[Outlier]
7 Which data value would be considered an outlier?
[Histogram and polygon]
8 The golfer goes back over her score sheet and realises she has made a mistake. The 24 should have been a 2 and a 4. Which measure of centre will be most affected by this mistake? In what way?
11
[Histogram]
Draw a histogram representing the data in question 9
Frequency of golf shots per hole
For 9–13, use the corrected results from question 8.
9 Complete the frequency table.
[Frequency table]
10 Recalculate the:
[Measures of centre and spread]
a mean (1 decimal place) b median c mode
Number of golf shots per hole
12 Draw a frequency polygon on your histogram in 11
[Polygon]
13 In your own words, describe the distribution of golf shots per hole.
[Describe distribution]