Year 7 Mathematics Practice

Integers

Fluency · Ordering and absolute value

1. Order from smallest to largest: $\;-7,\ 3,\ 0,\ -2,\ -10,\ 5$ . (show answer)

Answer
$-10,\ -7,\ -2,\ 0,\ 3,\ 5$
2. Which is smaller: $-4$ or $-9$ ? (show answer)

Answer
$-9$ (further left on the number line)
3. Evaluate $|{-13}|$ . (show answer)

Answer
$13$
4. Evaluate $|9| - |{-4}|$ . (show answer)

Answer
$5$
5. True or false: $|{-6}| = 6$ . (show answer)

Answer
True

Fluency · Adding and subtracting

1. Work out $\;-6 + 4$ . (show answer)

Answer
$-2$ (start at $-6$ , jump $4$ right)
2. Work out $\;-5 + (-8)$ . (show answer)

Answer
$-13$ (same signs, add the sizes)
3. Work out $\;12 + (-15)$ . (show answer)

Answer
$-3$ (different signs: $15 - 12 = 3$ , keep the sign of the larger size)
4. Work out $\;-3 + 10$ . (show answer)

Answer
$7$
5. Work out $\;-9 - 3$ . (show answer)

Answer
$-12$
6. Work out $\;-7 - (-10)$ . (show answer)

Answer
$3$ (minus a minus: $-7 + 10$ )
7. Work out $\;4 - (-11)$ . (show answer)

Answer
$15$ (minus a minus: $4 + 11$ )
8. Work out $\;-20 - (-20)$ . (show answer)

Answer
$0$
9. Work out $\;0 - 7$ . (show answer)

Answer
$-7$
10. Work out $\;-2 + (-6) - (-4)$ . (show answer)

Answer
$-4$ . Method: $-2 + (-6) = -8$ ; then $-8 - (-4) = -8 + 4 = -4$ .

Reasoning · Fill in the missing number

1. $\;-7 + \square = -2$ . (show answer)

Answer
$5$ . Method: $\square = -2 - (-7) = 5$ .
2. $\;\square + 4 = -1$ . (show answer)

Answer
$-5$ . Method: $\square = -1 - 4 = -5$ .
3. $\;-5 - \square = -12$ . (show answer)

Answer
$7$ . Method: $-5 - \square = -12$ gives $\square = -5 - (-12) = 7$ .
4. $\;\square - (-3) = 8$ . (show answer)

Answer
$5$ . Method: $\square = 8 + (-3) = 5$ .

Problem-solving · Real-world problems

1. The temperature on Mount Hotham was $-8\,^\circ$ C at midnight. It rose by $3\,^\circ$ C each hour until $6$ a.m. What was the temperature at $6$ a.m.? (show answer)

Answer
$10\,^\circ$ C. Method: $-8 + 6 \times 3 = -8 + 18$ .
2. A submarine is $120$ m below sea level. It ascends $35$ m, then descends $48$ m. What is its new depth, written as an integer? (show answer)

Answer
$-133$ m (or $133$ m below sea level). Method: $-120 + 35 - 48$ .
3. Mira has a bank balance of -$45 (an overdraft). She deposits $120 and then pays a bill of $38. What is her balance now? (show answer)

Answer
$37. Method: $-45 + 120 - 38$ .
4. A lift is on floor $-3$ (basement level 3). It goes up $7$ floors, then down $5$ , then up $2$ . On which floor does it stop? (show answer)

Answer
Floor $1$ . Method: $-3 + 7 - 5 + 2$ .
5. At dawn the temperature in Cooma was $-6\,^\circ$ C. By mid-morning it had risen to $4\,^\circ$ C. By how many degrees did the temperature rise? (show answer)

Answer
$10\,^\circ$ C rise. Method: $4 - (-6) = 4 + 6$ .
6. A diver starts at sea level ( $0$ ) and descends to $-24$ m, then rises $9$ m to look at a reef. What is her depth now? (show answer)

Answer
$-15$ m. Method: $-24 + 9 = -15$ .

Fluency · Multiplying and dividing

1. Work out $\;(-6) \times 7$ . (show answer)

Answer
$-42$
2. Work out $\;(-8) \times (-5)$ . (show answer)

Answer
$40$ (negative times negative is positive)
3. Work out $\;9 \times (-4)$ . (show answer)

Answer
$-36$
4. Work out $\;(-12) \times (-3) \times 2$ . (show answer)

Answer
$72$ (two negatives cancel, then multiply)
5. Work out $\;(-2)^3$ . (show answer)

Answer
$-8$ (three negatives: odd)
6. Work out $\;(-5)^2$ . (show answer)

Answer
$25$ (two negatives: even)
7. Work out $\;-24 \div 6$ . (show answer)

Answer
$-4$
8. Work out $\;(-36) \div (-9)$ . (show answer)

Answer
$4$
9. Work out $\;45 \div (-5)$ . (show answer)

Answer
$-9$
10. Work out $\;(-100) \div (-25)$ . (show answer)

Answer
$4$

Reasoning · Sign reasoning and missing values

1. Without calculating, decide whether $(-17) \times (-23) \times (-4)$ is positive or negative. Explain how you know. (show answer)

Answer
Negative. Reason: there are three negative factors, and three is an odd number, so the product must be negative.
2. Fill in: $\;\square \times (-6) = 42$ . (show answer)

Answer
$-7$ . Method: $\square = 42 \div (-6) = -7$ .
3. Fill in: $\;(-48) \div \square = 8$ . (show answer)

Answer
$-6$ . Method: $\square = -48 \div 8 = -6$ .
4. The product of three integers is $-60$ . Two of them are $-4$ and $5$ . What is the third? (show answer)

Answer
$3$ . Method: $(-4) \times 5 = -20$ ; the third number satisfies $-20 \times \square = -60$ , so $\square = 3$ .

Reasoning · Harder reasoning

1. Evaluate $\;(-3)^3 + 5 \times (-2)^2$ . (show answer)

Answer
$-7$ . Method: $(-3)^3 = -27$ ; $(-2)^2 = 4$ , so $5 \times 4 = 20$ ; then $-27 + 20 = -7$ .
2. Place $<$ , $>$ or $=$ between the pair: $\;(-5)^2 \;\square\; -5^2$ . Justify your answer. (show answer)

Answer
$>$ . Reason: $(-5)^2 = 25$ and $-5^2 = -25$ . Since $25 > -25$ , the left side is greater.
3. Jamie writes " $-3^2 = 9$ ". Is Jamie correct? If not, what is the right answer and what mistake has Jamie made? (show answer)

Answer
Jamie is not correct. The correct answer is $-9$ . The notation $-3^2$ means $-(3 \times 3) = -9$ (the power applies only to the $3$ , with the minus sign in front). Jamie has read it as $(-3)^2 = 9$ , but without brackets the squaring does not include the negative sign.
4. Tom thinks of an integer. He doubles it, subtracts $7$ , then multiplies by $-3$ . The result is $33$ . What integer was Tom thinking of? (show answer)

Answer
$n = -2$ . Method: let the number be $n$ . Then $(2n - 7) \times (-3) = 33$ , so $2n - 7 = -11$ ; $2n = -4$ ; $n = -2$ . Check: $2(-2) = -4$ ; minus $7$ gives $-11$ ; times $-3$ gives $33$ .

Fractions, decimals & percentages

Fluency · Tier 1: basic skills

1. Simplify $\dfrac{18}{24}$ . (show answer)

Answer
$\dfrac{3}{4}$
2. Simplify $\dfrac{36}{60}$ . (show answer)

Answer
$\dfrac{3}{5}$
3. Write $\dfrac{3}{8}$ as a decimal. (show answer)

Answer
$0.375$
4. Write $\dfrac{7}{20}$ as a decimal. (show answer)

Answer
$0.35$
5. Write $0.45$ as a simplified fraction. (show answer)

Answer
$\dfrac{9}{20}$
6. Write $1.25$ as a mixed number. (show answer)

Answer
$1\dfrac{1}{4}$
7. Convert $0.6$ to a percentage. (show answer)

Answer
$60\%$
8. Convert $82\%$ to a decimal. (show answer)

Answer
$0.82$
9. Convert $35\%$ to a simplified fraction. (show answer)

Answer
$\dfrac{7}{20}$
10. Which is bigger: $\dfrac{3}{5}$ or $\dfrac{7}{10}$ ? (show answer)

Answer
$\dfrac{7}{10}$ is bigger
11. Evaluate $\dfrac{1}{2} + \dfrac{1}{3}$ . (show answer)

Answer
$\dfrac{5}{6}$
12. Evaluate $\dfrac{5}{6} - \dfrac{1}{4}$ . (show answer)

Answer
$\dfrac{7}{12}$
13. Evaluate $\dfrac{2}{3} \times \dfrac{9}{10}$ . (show answer)

Answer
$\dfrac{3}{5}$
14. Evaluate $\dfrac{4}{5} \div \dfrac{2}{3}$ . (show answer)

Answer
$\dfrac{6}{5}$ or $1\dfrac{1}{5}$
15. Find $20\%$ of $75. (show answer)

Answer
$15
16. Find $15\%$ of $60$ . (show answer)

Answer
$9$
17. Find $7\%$ of $200. (show answer)

Answer
$14
18. Increase $80$ by $25\%$ . (show answer)

Answer
$100$
19. Decrease $120 by $10\%$ . (show answer)

Answer
$108
20. Write $\dfrac{5}{8}$ as a percentage. (show answer)

Answer
$62.5\%$

Fluency · Rounding and number line

1. Round $4.762$ to the nearest whole number. (show answer)

Answer
$5$ (because $0.762 \geq 0.5$ ).
2. Round $12.38$ to $1$ decimal place. (show answer)

Answer
$12.4$ (because the hundredths digit is $8$ ).
3. Round $0.0458$ to $2$ decimal places. (show answer)

Answer
$0.05$ (because the thousandths digit is $5$ ).
4. Round $18.739 to the nearest cent. (show answer)

Answer
$18.74.
5. Which is smaller: $-\dfrac{2}{3}$ or $-\dfrac{1}{3}$ ? (show answer)

Answer
$-\dfrac{2}{3}$ is smaller. It sits further to the left of zero on the number line.
6. Place these on a number line (in order): $-1.5,\ -\dfrac{3}{4},\ 0,\ 0.25,\ \dfrac{3}{2}$ . (show answer)

Answer
$-1.5,\ -\dfrac{3}{4},\ 0,\ 0.25,\ \dfrac{3}{2}$ - already in order.
7. Find two rational numbers between $-0.5$ and $0$ . (show answer)

Answer
Many answers. Examples: $-0.25$ , $-\dfrac{1}{4}$ , $-\dfrac{1}{3}$ .

Reasoning · Tier 2: mixed practice

1. Order from smallest to largest: $\;0.7,\ \dfrac{2}{3},\ 68\%,\ \dfrac{7}{10}$ . (show answer)

Answer
$\dfrac{2}{3},\ 0.68\,(68\%),\ 0.7,\ \dfrac{7}{10}$ . Note $\dfrac{7}{10} = 0.70 = 0.7$ , so these are equal. Correct order: $\dfrac{2}{3},\ 68\%,\ 0.7 = \dfrac{7}{10}$ .
2. Evaluate $\dfrac{2}{3} + \dfrac{3}{4} - \dfrac{1}{2}$ . (show answer)

Answer
$\dfrac{11}{12}$ . Method: common denominator $12$ ; $\dfrac{8}{12} + \dfrac{9}{12} - \dfrac{6}{12} = \dfrac{11}{12}$ .
3. Evaluate $\left(\dfrac{1}{2}\right)^2 + \dfrac{1}{4}$ . (show answer)

Answer
$\dfrac{1}{2}$ . Method: $\dfrac{1}{4} + \dfrac{1}{4} = \dfrac{2}{4}$ .
4. A recipe uses $\dfrac{3}{4}$ cup of sugar. You want to make $\dfrac{2}{3}$ of the recipe. How much sugar do you need? (show answer)

Answer
$\dfrac{1}{2}$ cup. Method: $\dfrac{2}{3} \times \dfrac{3}{4} = \dfrac{1}{2}$ .
5. Fill in the missing number: $\dfrac{\square}{12} = \dfrac{3}{4}$ . (show answer)

Answer
$9$ . Method: $\dfrac{9}{12} = \dfrac{3}{4}$ .
6. What percentage is $18$ out of $40$ ? (show answer)

Answer
$45\%$ . Method: $\dfrac{18}{40} \times 100$ .
7. What percentage is $27$ out of $60$ ? (show answer)

Answer
$45\%$ . Method: $\dfrac{27}{60} \times 100$ .
8. A jacket costs $85 and is reduced by $20\%$ . What is the new price? (show answer)

Answer
$68. Method: $85 \times 0.80 = 68$ .
9. A bike costs $320 and its price rises by $5\%$ . What is the new price? (show answer)

Answer
$336. Method: $320 \times 1.05 = 336$ .
10. A number increased by $40\%$ gives $84$ . What was the original number? (show answer)

Answer
$60$ . Method: $x \times 1.40 = 84$ , so $x = 84 \div 1.40$ .
11. Work out $\dfrac{3}{4}$ of $\dfrac{2}{5}$ of $120$ . (show answer)

Answer
$36$ . Method: $\dfrac{2}{5} \times 120 = 48$ , then $\dfrac{3}{4} \times 48 = 36$ .
12. Evaluate $1 - \dfrac{3}{8} - \dfrac{1}{4}$ . (show answer)

Answer
$\dfrac{3}{8}$ . Method: $\dfrac{8}{8} - \dfrac{3}{8} - \dfrac{2}{8}$ .

Reasoning · Tier 3: explain and spot the mistake

1. Sam says " $\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{2}{5}$ " because he added the tops and the bottoms. Is Sam correct? If not, what is the correct answer and what mistake has Sam made? (show answer)

Answer
Sam is wrong. The correct answer is $\dfrac{5}{6}$ . Fractions must share a denominator before you can add them: $\dfrac{1}{2} = \dfrac{3}{6}$ and $\dfrac{1}{3} = \dfrac{2}{6}$ , so $\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$ . You cannot add the tops and bottoms separately.
2. Explain why dividing by $\dfrac{1}{2}$ is the same as multiplying by $2$ . (show answer)

Answer
Dividing by a fraction means multiplying by its reciprocal. The reciprocal of $\dfrac{1}{2}$ is $2$ , so $n \div \dfrac{1}{2} = n \times 2$ . Concretely, asking "how many halves fit in $n$ ?" gives twice as many as whole units, i.e. $2n$ .
3. A shop advertises " $50\%$ off then another $20\%$ off". Is this the same as $70\%$ off? Explain with a worked example using $100. (show answer)

Answer
Not the same. Starting from $100: $50\%$ off gives $50; then $20\%$ off $50 gives $40. A flat $70\%$ off $100 would leave $30. Because percentages compound on the new running total, the combined discount here is only $60\%$ .
4. Without calculating exactly, decide whether $\dfrac{7}{13}$ is greater than $\dfrac{1}{2}$ . Explain your reasoning. (show answer)

Answer
Yes, $\dfrac{7}{13}$ is greater than $\dfrac{1}{2}$ . Half of $13$ is $6.5$ , and $7 > 6.5$ , so seven thirteenths is more than half.
5. Is $0.3$ equal to $\dfrac{3}{10}$ or $\dfrac{1}{3}$ ? Explain the difference. (show answer)

Answer
$0.3 = \dfrac{3}{10}$ , not $\dfrac{1}{3}$ . The fraction $\dfrac{1}{3} = 0.333\ldots$ (the $3$ s repeat forever), so $0.3$ is slightly less than $\dfrac{1}{3}$ .

Problem-solving · Tier 4: real-world problems

1. Zara's phone bill is $65 per month. The company raises prices by $8\%$ . What will her new bill be? (show answer)

Answer
$70.20. Method: $65 \times 1.08$ .
2. A pizza is cut into $8$ equal slices. Tom eats $3$ slices, Mia eats $2$ slices. What fraction of the pizza is left? (show answer)

Answer
$\dfrac{3}{8}$ . Method: $1 - \dfrac{5}{8}$ .
3. In a class of $30$ students, $18$ walk to school. What percentage walk to school? What percentage do not? (show answer)

Answer
$60\%$ walk; $40\%$ do not. Method: $\dfrac{18}{30} \times 100 = 60$ .
4. A $240 pair of headphones is on sale for $180. What is the percentage discount? (show answer)

Answer
$25\%$ off. Method: discount $60; $\dfrac{60}{240} \times 100$ .
5. Mia saves $\dfrac{1}{5}$ of her $25 pocket money each week. How much has she saved after $8$ weeks? (show answer)

Answer
$40. Method: $\dfrac{1}{5} \times 25 = 5$ per week; $\times 8$ .
6. A water tank is $\dfrac{3}{4}$ full. $120$ litres are used, leaving the tank $\dfrac{1}{2}$ full. What is the capacity of the tank? (show answer)

Answer
$480$ litres. Method: $\dfrac{3}{4} - \dfrac{1}{2} = \dfrac{1}{4}$ of the tank is $120$ L, so full tank is $4 \times 120$ .
7. A shirt's price was marked up by $20\%$ to $42. What was the original price? (show answer)

Answer
$35. Method: $x \times 1.20 = 42$ , so $x = 42 \div 1.20$ .
8. Last year a school had $750$ students. This year enrolment has risen by $12\%$ . How many students are enrolled now? (show answer)

Answer
$840$ students. Method: $750 \times 1.12$ .

Ratios

Fluency · Simplify and find missing parts

1. Simplify the ratio $12 : 18$ . (show answer)

Answer
$2 : 3$
2. Simplify the ratio $35 : 14$ . (show answer)

Answer
$5 : 2$
3. Simplify the ratio $40 : 60 : 100$ . (show answer)

Answer
$2 : 3 : 5$
4. Write $250$ g : $1$ kg as a simplified ratio. (show answer)

Answer
$1 : 4$ . Method: convert both to grams, $250 : 1000$ ; divide by $250$ .
5. Write $45$ minutes : $2$ hours as a simplified ratio. (show answer)

Answer
$3 : 8$ . Method: $45 : 120$ ; divide by $15$ .
6. Find the missing number: $3 : 5 = 12 : \square$ . (show answer)

Answer
$20$
7. Find the missing number: $\square : 8 = 15 : 10$ . (show answer)

Answer
$12$
8. Find the missing number: $2 : 3 = \square : 18$ . (show answer)

Answer
$12$ . Method: $2 \times 6 = 12$ , since $3 \times 6 = 18$ .

Fluency · Dividing a quantity in a ratio

1. Divide $40 in the ratio $3 : 5$ . (show answer)

Answer
$15 : $25
2. Divide $72 in the ratio $2 : 7$ . (show answer)

Answer
$16 : $56
3. Divide $48$ sweets in the ratio $1 : 2 : 3$ . (show answer)

Answer
$8,\ 16,\ 24$ sweets. Method: $1 + 2 + 3 = 6$ parts; each part $= 8$ .
4. A recipe uses flour and sugar in the ratio $5 : 2$ . If there are $350$ g of flour, how much sugar is used? (show answer)

Answer
$140$ g. Method: flour is $5$ parts, so $1$ part $= 70$ g; sugar $= 2 \times 70$ .
5. Two numbers are in the ratio $4 : 5$ and their sum is $90$ . Find the numbers. (show answer)

Answer
$40$ and $50$ . Method: $4 + 5 = 9$ parts; each part $= 90 \div 9 = 10$ .
6. A $3 : 2$ ratio of boys to girls in a class of $30$ gives how many of each? (show answer)

Answer
$18$ boys, $12$ girls. Method: $3 + 2 = 5$ parts; each $= 6$ .

Reasoning · Explain and spot the mistake

1. Ben says "the ratio $4 : 6$ is the same as $\dfrac{4}{6}$ , which is the same as the percentage $66.67\%$ of boys". Explain what is right and what is confused in Ben's statement. (show answer)

Answer
A ratio of $4 : 6$ compares one group to the other, not to the whole. To find "what fraction of the total is boys" you need boys over total: $\dfrac{4}{4 + 6} = \dfrac{4}{10} = 40\%$ . Ben wrote $\dfrac{4}{6} \approx 66.67\%$ , which is the ratio of boys to girls, not boys to total.
2. A drink is made from concentrate and water in the ratio $1 : 4$ . Jen says " $\dfrac{1}{4}$ of the drink is concentrate, which is $25\%$ ". What has Jen mixed up, and what is the correct percentage? (show answer)

Answer
In a $1 : 4$ ratio the parts are $1$ concentrate plus $4$ water, giving $5$ parts total. So concentrate is $\dfrac{1}{5} = 20\%$ of the drink, not $\dfrac{1}{4} = 25\%$ . Jen forgot to add the parts to find the total.
3. Explain why the ratio $6 : 9$ is equivalent to $2 : 3$ , but is not equivalent to $2 : 5$ . (show answer)

Answer
Dividing both $6$ and $9$ by $3$ gives $2 : 3$ , so the parts scale down by the same factor. For $2 : 5$ the same multiplier would need to take $2 \to 6$ and $5 \to 9$ , but $2 \times 3 = 6$ while $5 \times 3 = 15 \neq 9$ . The proportion doesn't match, so $6 : 9 \neq 2 : 5$ .
4. Two gears have $24$ and $36$ teeth. Write the gear ratio in simplest form and explain what the ratio means in plain words. (show answer)

Answer
$24 : 36 = 2 : 3$ . In plain words: for every $2$ turns of the first gear, the second gear makes $3$ turns (or equivalently, the first gear turns $1.5$ times faster than the second).

Problem-solving · Real-world problems

1. A cake recipe makes $12$ cupcakes and uses $300$ g flour, $180$ g sugar and $4$ eggs. How much of each is needed for $30$ cupcakes? (show answer)

Answer
Flour $750$ g, sugar $450$ g, eggs $10$ . Method: scale factor $= 30 \div 12 = 2.5$ .
2. A map has scale $1 : 25\,000$ . Two towns are $8$ cm apart on the map. How many kilometres apart are they in reality? (show answer)

Answer
$2$ km. Method: $8 \times 25\,000 = 200\,000$ cm $= 2$ km.
3. Two friends share a $150 phone bill in the ratio of their usage. Anna used the phone for $180$ minutes, Ben for $120$ minutes. How much should each pay? (show answer)

Answer
Anna $90, Ben $60. Method: ratio $180 : 120 = 3 : 2$ ; $5$ parts = $150; each part = $30.
4. A rectangular garden has length and width in the ratio $5 : 3$ . If its perimeter is $48$ m, find its length and width. (show answer)

Answer
Length $15$ m, width $9$ m. Method: $5 + 3 = 8$ parts; total length-plus-width (half of perimeter) $= 24$ m, so $1$ part $= 3$ m.
5. A school of $480$ students is split into three houses in the ratio $3 : 4 : 5$ . How many students are in each house? (show answer)

Answer
$120,\ 160,\ 200$ . Method: $3 + 4 + 5 = 12$ parts; each part $= 480 \div 12 = 40$ .
6. Paint is mixed from white and red in the ratio $7 : 3$ . How much red paint is needed to make $5$ litres of mixed paint? (show answer)

Answer
$1.5$ L red. Method: $7 + 3 = 10$ parts; each part $= 0.5$ L; red $= 3$ parts.

Squares, roots & exponents

Fluency · Squares and square roots

1. Evaluate $3^2$ . (show answer)

Answer
$9$
2. Evaluate $6^2$ . (show answer)

Answer
$36$
3. Evaluate $12^2$ . (show answer)

Answer
$144$
4. Evaluate $\sqrt{81}$ . (show answer)

Answer
$9$
5. Evaluate $\sqrt{144}$ . (show answer)

Answer
$12$
6. Evaluate $\sqrt{400}$ . (show answer)

Answer
$20$
7. Between which two consecutive whole numbers does $\sqrt{50}$ lie? (show answer)

Answer
Between $7$ and $8$
8. Between which two consecutive whole numbers does $\sqrt{90}$ lie? (show answer)

Answer
Between $9$ and $10$
9. Which is bigger: $\sqrt{64}$ or $5^2$ ? (show answer)

Answer
$5^2 = 25$ is bigger than $\sqrt{64} = 8$
10. Evaluate $\sqrt{25} + \sqrt{36}$ . (show answer)

Answer
$11$ . Method: $5 + 6$ .

Fluency · Powers of 10 and expanded notation

1. Evaluate $10^3$ . (show answer)

Answer
$1000$
2. Evaluate $10^5$ . (show answer)

Answer
$100\,000$
3. Write $3427$ in expanded notation using powers of $10$ . (show answer)

Answer
$3 \times 10^3 + 4 \times 10^2 + 2 \times 10 + 7$
4. Write $50\,608$ in expanded notation using powers of $10$ . (show answer)

Answer
$5 \times 10^4 + 6 \times 10^2 + 8$
5. Write the number that equals $7 \times 10^3 + 2 \times 10^2 + 6$ . (show answer)

Answer
$7206$

Fluency · Prime factorisation

1. Write $20$ as a product of primes in exponent form. (show answer)

Answer
$2^2 \times 5$
2. Write $36$ as a product of primes in exponent form. (show answer)

Answer
$2^2 \times 3^2$
3. Write $84$ as a product of primes in exponent form. (show answer)

Answer
$2^2 \times 3 \times 7$
4. Write $100$ as a product of primes in exponent form. (show answer)

Answer
$2^2 \times 5^2$
5. Write $200$ as a product of primes in exponent form. (show answer)

Answer
$2^3 \times 5^2$

Reasoning · Explain and apply

1. Explain in your own words why $\sqrt{9 + 16} \neq \sqrt{9} + \sqrt{16}$ . (show answer)

Answer
Square root does not distribute over addition. Left side: $\sqrt{9 + 16} = \sqrt{25} = 5$ . Right side: $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$ . Since $5 \neq 7$ , you must add first and then take the root.
2. Without a calculator, decide whether $\sqrt{200}$ is closer to $14$ or $15$ . Justify. (show answer)

Answer
$14^2 = 196$ and $15^2 = 225$ . Since $200$ is only $4$ above $196$ but $25$ below $225$ , $\sqrt{200}$ is much closer to $14$ (about $14.14$ ).
3. A square garden bed has an area of $144$ m^2. What is the length of one side? (show answer)

Answer
$12$ m. Method: side length $= \sqrt{144}$ .
4. Find the highest common factor of $72$ and $120$ by comparing their prime factorisations. (show answer)

Answer
HCF $= 24$ . Method: $72 = 2^3 \times 3^2$ and $120 = 2^3 \times 3 \times 5$ ; take the lowest power of each shared prime: $2^3 \times 3 = 24$ .
5. Find the lowest common multiple of $12$ and $18$ by using prime factorisation. (show answer)

Answer
LCM $= 36$ . Method: $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$ ; take the highest power of each prime: $2^2 \times 3^2 = 36$ .

Fluency · Using the index laws

1. Simplify $2^4 \times 2^5$ . (show answer)

Answer
$2^9$
2. Simplify $a^6 \times a^2$ . (show answer)

Answer
$a^8$
3. Simplify $5^{10} \div 5^4$ . (show answer)

Answer
$5^6$
4. Simplify $x^7 \div x^3$ . (show answer)

Answer
$x^4$
5. Simplify $(3^2)^3$ . (show answer)

Answer
$3^6$ (which equals $729$ )
6. Simplify $(m^4)^5$ . (show answer)

Answer
$m^{20}$
7. Simplify $\dfrac{y^{10}}{y^4}$ . (show answer)

Answer
$y^6$ . Method: subtract indices $10 - 4$ .
8. Simplify $\dfrac{m^6 \times m^3}{m^5}$ . (show answer)

Answer
$m^4$ . Method: top $m^9$ ; $m^9 \div m^5 = m^4$ .

Reasoning · Spot the mistake

1. Tim writes $2^3 + 2^3 = 2^6$ . Is Tim correct? If not, what has gone wrong? (show answer)

Answer
Tim is wrong. $2^3 + 2^3 = 8 + 8 = 16 = 2^4$ , not $2^6$ . The index law for multiplying powers adds the indices, but this is addition of two equal powers - it doubles the value, increasing the index by $1$ (not doubling it).
2. Leah writes $(a^2)^3 = a^5$ . Explain Leah's error and give the correct simplification. (show answer)

Answer
Leah is wrong. The rule $(a^m)^n$ multiplies the indices, not adds: $(a^2)^3 = a^{2 \times 3} = a^6$ .
3. Simplify $\dfrac{p^2 q^3 \times p^4 q}{p^3 q^2}$ . (show answer)

Answer
$p^3 q^2$ . Method: numerator $= p^{2 + 4} q^{3 + 1} = p^6 q^4$ ; dividing by $p^3 q^2$ gives $p^{6 - 3} q^{4 - 2} = p^3 q^2$ .
4. A bacterium doubles every hour. Starting from one cell, how many cells are there after $6$ hours? Write the answer as a power of $2$ . (show answer)

Answer
$2^6 = 64$ cells. Method: doubling $6$ times from $1$ gives $1 \to 2 \to 4 \to 8 \to 16 \to 32 \to 64$ .

Algebraic expressions

Fluency · Tier 1: basic skills

1. Write an expression for "seven more than $n$ ". (show answer)

Answer
$n + 7$
2. Write an expression for "five less than $2m$ ". (show answer)

Answer
$2m - 5$
3. Write an expression for "the product of $4$ and $x$ ". (show answer)

Answer
$4x$
4. Write an expression for "half of $y$ added to $3$ ". (show answer)

Answer
$\dfrac{y}{2} + 3$
5. In the term $-7x$ , state the coefficient. (show answer)

Answer
$-7$
6. Simplify $\;4x + 3x$ . (show answer)

Answer
$7x$
7. Simplify $\;9y - 5y$ . (show answer)

Answer
$4y$
8. Simplify $\;3a + 2b + 5a - b$ . (show answer)

Answer
$8a + b$
9. Simplify $\;7p - 4 - 2p + 9$ . (show answer)

Answer
$5p + 5$
10. Simplify $\;2x + 5y + 3x - y + 4$ . (show answer)

Answer
$5x + 4y + 4$
11. Evaluate $3a + 2$ when $a = 4$ . (show answer)

Answer
$14$
12. Evaluate $5x - y$ when $x = 2$ and $y = -3$ . (show answer)

Answer
$13$
13. Evaluate $x^2 + 1$ when $x = -4$ . (show answer)

Answer
$17$
14. Expand $\;2(x + 5)$ . (show answer)

Answer
$2x + 10$
15. Expand $\;4(y - 3)$ . (show answer)

Answer
$4y - 12$
16. Expand $\;-3(a + 2)$ . (show answer)

Answer
$-3a - 6$
17. Expand $\;6(2m - 1)$ . (show answer)

Answer
$12m - 6$
18. Simplify $\dfrac{8x + 12}{4}$ . (show answer)

Answer
$2x + 3$
19. Simplify $\;3(x + 2) + 5$ . (show answer)

Answer
$3x + 11$
20. Simplify $\;2(a + 4) - 3$ . (show answer)

Answer
$2a + 5$

Reasoning · Tier 2: mixed practice

1. Simplify $\;4(x + 3) + 2(x - 1)$ . (show answer)

Answer
$6x + 10$ . Method: $4x + 12 + 2x - 2$ .
2. Simplify $\;5(2y - 1) - 3(y + 4)$ . (show answer)

Answer
$7y - 17$ . Method: $10y - 5 - 3y - 12$ .
3. Simplify $\;7m + 4 - 2(m + 3)$ . (show answer)

Answer
$5m - 2$ . Method: $7m + 4 - 2m - 6$ .
4. Evaluate $\;3x^2 - 2x + 1$ when $x = 4$ . (show answer)

Answer
$41$ . Method: $3(16) - 8 + 1 = 48 - 8 + 1$ .
5. Evaluate $\;(a + b)^2$ when $a = 3$ , $b = -1$ . (show answer)

Answer
$4$ . Method: $(3 + (-1))^2 = 2^2$ .
6. Expand $\;-2(3x - 4) - (x + 2)$ . (show answer)

Answer
$-7x + 6$ . Method: $-6x + 8 - x - 2$ .
7. Simplify $\dfrac{10a - 15}{5}$ . (show answer)

Answer
$2a - 3$ . Method: divide each term.
8. Write an expression for the perimeter of a rectangle with length $2x + 1$ and width $x$ . Simplify it. (show answer)

Answer
Perimeter $= 6x + 2$ . Method: $2(2x + 1) + 2x = 4x + 2 + 2x$ .
9. Write an expression for the cost of $n$ apples at $0.60 each and $m$ bananas at $0.40 each. (show answer)

Answer
$0.60n + 0.40m$ (in dollars).
10. Find the missing coefficient: $5x + \square x = 12x$ . (show answer)

Answer
$7$ . Method: $5 + 7 = 12$ .
11. Simplify $\;3a \times 4b$ . (show answer)

Answer
$12ab$ . Method: $3 \times 4 \times a \times b$ .
12. Simplify $\;\dfrac{12xy}{3y}$ . (show answer)

Answer
$4x$ . Method: cancel the $y$ s, then $12 \div 3$ .

Reasoning · Tier 3: explain and spot the mistake

1. Kira writes $3 + 2x = 5x$ . Is Kira correct? If not, explain the mistake and give the correct simplification. (show answer)

Answer
Wrong. $3$ and $2x$ are not like terms - one is a constant, the other has a variable - so they cannot be combined into a single term. The simplest form is $3 + 2x$ (or $2x + 3$ ).
2. Explain why $3x$ and $3x^2$ are not like terms, with a numerical example. (show answer)

Answer
They have different variable parts: $x$ vs $x^2$ . Try $x = 2$ : $3x = 6$ but $3x^2 = 12$ . If they were like terms they would always be equal, but they aren't.
3. Leo expands $-4(x - 3)$ as $-4x - 12$ . Is this right? If not, what is the correct expansion? (show answer)

Answer
Wrong. $-4(x - 3) = -4x + 12$ . Leo forgot that $-4 \times -3 = +12$ , not $-12$ . Two negatives make a positive.
4. Write two different expressions that both equal $12$ when $x = 3$ . (show answer)

Answer
Many possible answers, e.g. $4x$ , $2x + 6$ , $x^2 + 3$ , or $5x - 3$ (each gives $12$ when $x = 3$ ).
5. Are $2(a + 3)$ and $2a + 3$ always equal? Explain with an example. (show answer)

Answer
Not equal in general. $2(a + 3) = 2a + 6$ , which is not the same as $2a + 3$ . Try $a = 1$ : left side $= 8$ , right side $= 5$ . The $2$ must distribute to every term inside the bracket.

Fluency · Using everyday formulas (substitution)

1. A rectangle has $L = 12$ cm and $W = 5$ cm. Find its area and perimeter using $A = L \times W$ and $P = 2L + 2W$ . (show answer)

Answer
Area $60$ cm^2; perimeter $34$ cm.
2. Hawthorn scored $14$ goals and $7$ behinds. Use $\text{total} = 6g + b$ to find their total. (show answer)

Answer
$14 \times 6 + 7 = 84 + 7 = 91$ points.
3. Collingwood kicked $9$ goals and $13$ behinds; Melbourne kicked $11$ goals and $5$ behinds. Who won, and by how much? (show answer)

Answer
Collingwood $9 \times 6 + 13 = 67$ ; Melbourne $11 \times 6 + 5 = 71$ . Melbourne won by $4$ points.
4. Sam's base wage is $360 per week, normal rate $22/h. He worked $4$ hours of overtime. Use $W = b + 1.5rh$ to find his total pay. (show answer)

Answer
$492. Method: $360 + 1.5 \times 22 \times 4 = 360 + 132$ .
5. A metal bar has mass $504$ g and volume $64$ cm^3. Find its density using $D = \dfrac{M}{V}$ . (show answer)

Answer
$D = 7.875$ g/cm^3. Method: $504 \div 64$ .
6. Find the maximum heart rate for a person aged $45$ using $HR_{\max} = 220 - a$ . (show answer)

Answer
$175$ bpm. Method: $220 - 45$ .
7. A car travels at $80$ km/h for $2.5$ h. How far does it travel? (Use $d = s \times t$ .) (show answer)

Answer
$200$ km. Method: $80 \times 2.5$ .
8. A cyclist covers $45$ km in $3$ hours. Find the average speed. (show answer)

Answer
$15$ km/h. Method: $45 \div 3$ .
9. Use the formula $C = \tfrac{5}{9}(F - 32)$ to convert $68 deg$ F to degrees Celsius. (show answer)

Answer
$20$ degC. Method: $\tfrac{5}{9}(68 - 32) = \tfrac{5}{9} \times 36 = 20$ .

Problem-solving · Tier 4: real-world problems

1. Mira has $x. She spends $5 on lunch and then earns $20 helping a neighbour. Write an expression for how much she has now. If she started with $12, how much has she now? (show answer)

Answer
$x + 15$ ; $27. Method: $-5 + 20 = +15$ ; then $12 + 15$ .
2. A phone plan costs a $20 monthly fee plus $0.10 per minute. Write an expression for the cost of a month with $t$ minutes of calls. What is the cost if $t = 150$ ? (show answer)

Answer
$20 + 0.10t$ dollars; $35. Method: $20 + 0.10 \times 150$ .
3. The length of a rectangle is $3$ cm more than twice its width. If the width is $w$ , write expressions for the length and perimeter. Simplify the perimeter. (show answer)

Answer
Length $= 2w + 3$ ; perimeter $= 6w + 6$ . Method: $P = 2(L + W) = 2(2w + 3 + w)$ .
4. A taxi charges a $4.50 flag-fall plus $2 per kilometre. Write the cost for a $k$ -kilometre trip, and find the cost of a $12$ km trip. (show answer)

Answer
Cost $= 4.50 + 2k$ ; $28.50. Method: $4.50 + 2 \times 12$ .
5. Five students each give $x toward a gift that costs $42. Write an expression for how much change is left after the gift is bought. Evaluate it if $x = 10$ . (show answer)

Answer
$5x - 42$ ; $8. Method: $5 \times 10 = 50$ ; $50 - 42 = 8$ .
6. A swimming pool holds $V$ litres and a hose fills it at $r$ L/min. Write an expression for the time to fill the pool. How long (in minutes) if $V = 60\,000$ and $r = 150$ ? (show answer)

Answer
Time $= \dfrac{V}{r}$ minutes; $400$ min ( $6$ h $40$ min). Method: $60\,000 \div 150$ .
7. A mobile plan charges $25 per month plus $0.08 per text. Lucy sent $t$ texts in a month. Write an expression for her total cost, then find the cost when $t = 180$ . (show answer)

Answer
$25 + 0.08t$ dollars; $39.40. Method: $25 + 0.08 \times 180$ .
8. A gym membership costs $59 to join plus $15 per week. Write an expression for the total cost after $w$ weeks. When does the total first exceed $200? (show answer)

Answer
$59 + 15w$ . Exceeds $200 when $15w > 141$ , so when $w \geq 10$ - first exceeded at the end of week $10$ .
9. A delivery van's fuel cost per trip is $C = 0.15 d$ , where $d$ is the trip distance in kilometres. Find the cost of a $240$ km trip. If the fuel cost doubled per kilometre, what would the new formula be? (show answer)

Answer
$36 for a $240$ km trip. New formula: $C = 0.30 d$ .
10. Daniella's target training heart rate zone is between $50\%$ and $70\%$ of $HR_{\max} = 220 - a$ (where $a$ is her age in years). She is $12$ . Find the two ends of her target zone. (show answer)

Answer
$HR_{\max} = 208$ bpm. $50\%$ zone $\approx 104$ bpm; $70\%$ zone $\approx 146$ bpm. Target zone: roughly $104$ - $146$ bpm.

Linear equations

Fluency · One-step equations

1. Solve $\;x + 8 = 20$ . (show answer)

Answer
$x = 12$
2. Solve $\;y - 6 = 13$ . (show answer)

Answer
$y = 19$
3. Solve $\;4x = 28$ . (show answer)

Answer
$x = 7$
4. Solve $\;\dfrac{y}{3} = 7$ . (show answer)

Answer
$y = 21$
5. Solve $\;x + 17 = 30$ . (show answer)

Answer
$x = 13$
6. Solve $\;a - 9 = 1$ . (show answer)

Answer
$a = 10$
7. Solve $\;7m = 42$ . (show answer)

Answer
$m = 6$
8. Solve $\;\dfrac{x}{6} = 4$ . (show answer)

Answer
$x = 24$

Fluency · Two-step equations

1. Solve $\;2x + 3 = 11$ . (show answer)

Answer
$x = 4$
2. Solve $\;3y - 5 = 7$ . (show answer)

Answer
$y = 4$
3. Solve $\;5x + 1 = 16$ . (show answer)

Answer
$x = 3$
4. Solve $\;4x - 3 = 9$ . (show answer)

Answer
$x = 3$
5. Solve $\;7a - 4 = 24$ . (show answer)

Answer
$a = 4$
6. Solve $\;\dfrac{x}{2} + 1 = 6$ . (show answer)

Answer
$x = 10$
7. Solve $\;\dfrac{m}{3} - 4 = 1$ . (show answer)

Answer
$m = 15$
8. Solve $\;\dfrac{x + 5}{2} = 4$ . (show answer)

Answer
$x = 3$ . Method: multiply by $2$ , $x + 5 = 8$ ; subtract $5$ .
9. Solve $\;\dfrac{2x + 1}{3} = 5$ . (show answer)

Answer
$x = 7$ . Method: multiply by $3$ , $2x + 1 = 15$ ; then subtract $1$ and divide by $2$ .
10. Solve $\;3(x + 2) = 18$ . (show answer)

Answer
$x = 4$ . Method: divide by $3$ , $x + 2 = 6$ .

Reasoning · Verify and reason

1. Verify, without solving, whether $x = 6$ is a solution of $\;2x + 4 = 16$ . (show answer)

Answer
Yes. $2(6) + 4 = 12 + 4 = 16$ . ✓
2. Verify whether $y = 3$ is a solution of $\;5y - 1 = 15$ . (show answer)

Answer
No. $5(3) - 1 = 14$ , not $15$ .
3. Zara solves $2x + 6 = 14$ by writing " $2x = 14 + 6 = 20$ , so $x = 10$ ". Explain her mistake and give the correct answer. (show answer)

Answer
Zara added $6$ when she should have subtracted. The inverse of " $+ 6$ " is " $- 6$ ". Correct working: $2x + 6 = 14 \Rightarrow 2x = 8 \Rightarrow x = 4$ .
4. Find the missing number: solve $\;\square + 7 = 15$ and check by substitution. (show answer)

Answer
$\square = 8$ . Check: $8 + 7 = 15$ . ✓
5. Write an equation of your own whose solution is $x = 8$ , and verify it by substitution. (show answer)

Answer
Many answers possible. Example: $3x - 4 = 20$ gives $x = 8$ . Check: $3(8) - 4 = 20$ . ✓

Problem-solving · Worded problems

1. Four times a number, increased by $7$ , is $31$ . Find the number. (show answer)

Answer
$6$ . Method: $4n + 7 = 31$ , so $4n = 24$ .
2. A pencil costs $2 and a ruler costs $r. Three pencils and a ruler cost $8 in total. Find $r$ . (show answer)

Answer
r = $2. Method: $3 \times 2 + r = 8$ , so $6 + r = 8$ .
3. Ava is $x$ years old. Her sister is $3$ years older. In $5$ years, the sum of their ages will be $33$ . How old is Ava now? (show answer)

Answer
Ava is $10$ . Method: Ava $= x$ ; sister $= x + 3$ . In $5$ years: $(x + 5) + (x + 8) = 33$ ; $2x + 13 = 33$ ; $x = 10$ .
4. A taxi charges a $4 flag-fall and $2.50 per kilometre. A trip cost $29. How long was the trip? (show answer)

Answer
$10$ km. Method: $4 + 2.50k = 29$ , so $2.50k = 25$ .
5. The perimeter of an isosceles triangle with two equal sides of length $x$ cm and base $8$ cm is $30$ cm. Find $x$ . (show answer)

Answer
$x = 11$ cm. Method: $x + x + 8 = 30$ .
6. Five consecutive natural numbers sum to $45$ . Find the smallest of them. (Hint: call it $n$ and write $n + (n+1) + \ldots$ ) (show answer)

Answer
$7$ is the smallest. Method: $n + (n+1) + (n+2) + (n+3) + (n+4) = 5n + 10 = 45$ ; $5n = 35$ ; $n = 7$ .

Reasoning · Non-positive and two-sided equations

1. Solve $\;x + 11 = 5$ . (show answer)

Answer
$x = -6$ . Method: subtract $11$ .
2. Solve $\;-2m = 14$ . (show answer)

Answer
$m = -7$ . Method: divide by $-2$ .
3. Solve $\;2x + 9 = 3$ . (show answer)

Answer
$x = -3$ . Method: subtract $9$ ; divide by $2$ .
4. Solve $\;-3x + 5 = 20$ . (show answer)

Answer
$x = -5$ . Method: subtract $5$ , $-3x = 15$ ; divide by $-3$ .
5. Solve $\;10 - 2x = 4$ . (show answer)

Answer
$x = 3$ . Method: subtract $10$ , $-2x = -6$ ; divide by $-2$ .
6. Solve $\;5x - 2 = 2x + 10$ . (show answer)

Answer
$x = 4$ . Method: subtract $2x$ from both sides, $3x - 2 = 10$ ; add $2$ , $3x = 12$ .
7. Solve $\;4y + 7 = y + 19$ . (show answer)

Answer
$y = 4$ . Method: subtract $y$ , $3y + 7 = 19$ ; subtract $7$ ; divide by $3$ .
8. Solve $\;\dfrac{x}{2} + \dfrac{x}{3} = 10$ . (Hint: multiply both sides by $6$ .) (show answer)

Answer
$x = 12$ . Method: multiply by $6$ , $3x + 2x = 60$ ; $5x = 60$ .
9. The sum of three consecutive integers is $-9$ . Find them. (show answer)

Answer
$-4, -3, -2$ . Method: $3n + 3 = -9$ , so $n = -4$ .
10. A number increased by $40\%$ gives $84$ . What was the original number? (show answer)

Answer
$60$ . Method: $x \times 1.40 = 84$ , so $x = 60$ .

Length, perimeter, area & circles

Fluency · Length and perimeter

1. Convert $250$ cm to metres. (show answer)

Answer
$2.5$ m
2. Convert $3.4$ km to metres. (show answer)

Answer
$3400$ m
3. Convert $85$ mm to cm. (show answer)

Answer
$8.5$ cm
4. Convert $2500$ m to km. (show answer)

Answer
$2.5$ km
5. Find the perimeter of a rectangle $12$ cm by $7$ cm. (show answer)

Answer
$38$ cm
6. Find the perimeter of a square with side $9$ cm. (show answer)

Answer
$36$ cm
7. A triangle has sides $5$ cm, $7$ cm, $9$ cm. Find its perimeter. (show answer)

Answer
$21$ cm
8. A regular hexagon has side $4$ cm. Find its perimeter. (show answer)

Answer
$24$ cm

Fluency · Area of rectangles, triangles, parallelograms

1. Find the area of a $15$ cm by $4$ cm rectangle. (show answer)

Answer
$60$ cm^2
2. Find the area of a square with side $6$ cm. (show answer)

Answer
$36$ cm^2
3. Find the area of a triangle with base $10$ cm and height $8$ cm. (show answer)

Answer
$40$ cm^2
4. Find the area of a triangle with base $12$ m and height $5$ m. (show answer)

Answer
$30$ m^2
5. Find the area of a parallelogram with base $9$ cm and height $6$ cm. (show answer)

Answer
$54$ cm^2
6. A rectangle has area $48$ cm^2 and length $8$ cm. Find its width. (show answer)

Answer
$6$ cm
7. A triangle has area $36$ cm^2 and base $9$ cm. Find its height. (show answer)

Answer
$8$ cm. Method: $\tfrac{1}{2} \times 9 \times h = 36$ .
8. A parallelogram has area $60$ m^2 and height $5$ m. Find its base. (show answer)

Answer
$12$ m. Method: $b \times 5 = 60$ .

Fluency · Circles

1. A circle has radius $4$ cm. Find its diameter. (show answer)

Answer
$8$ cm
2. A circle has diameter $18$ m. Find its radius. (show answer)

Answer
$9$ m
3. Find the circumference of a circle with radius $10$ cm. (show answer)

Answer
$62.8$ cm. Method: $2 \times 3.14 \times 10$ .
4. Find the circumference of a circle with diameter $14$ cm, using $\pi \approx \tfrac{22}{7}$ . (show answer)

Answer
$44$ cm. Method: $\tfrac{22}{7} \times 14$ .
5. A circle has circumference $31.4$ cm. Find its diameter. (show answer)

Answer
$10$ cm. Method: $d = C / \pi = 31.4 / 3.14$ .
6. True or false: for every circle, circumference $\div$ diameter gives about the same number. (show answer)

Answer
True - that constant ratio is $\pi$ .

Reasoning · Explain and reason

1. Two rectangles have the same perimeter. Does it follow that they have the same area? Justify with a numerical example. (show answer)

Answer
Not necessarily. Example: a $4 \times 6$ rectangle has perimeter $20$ and area $24$ ; a $2 \times 8$ rectangle also has perimeter $20$ but area $16$ . Same perimeter, different area.
2. Pete writes the area of a triangle with base $6$ and slant side $5$ as $\tfrac{1}{2} \times 6 \times 5 = 15$ . Explain what is wrong. (show answer)

Answer
Pete used the slant side, not the perpendicular height. Without the perpendicular height, the area formula cannot be applied directly; more information is needed.
3. Is it possible for one shape to have a larger perimeter but a smaller area than another? Give an example. (show answer)

Answer
Yes. Example: a thin $1 \times 20$ rectangle has perimeter $42$ and area $20$ , while a $5 \times 5$ square has perimeter $20$ and area $25$ . The first has a larger perimeter but smaller area.
4. A pizza of diameter $30$ cm is cut exactly in half. What is the perimeter of each half-pizza (the crust plus the straight cut)? (show answer)

Answer
$30 + \tfrac{1}{2}(\pi \times 30) \approx 30 + 47.1 = 77.1$ cm. Method: cut across the pizza is $30$ cm; half the circumference is $\tfrac{1}{2}\pi d$ .
5. Without calculating, decide which has the larger area: a square with side $10$ cm or a rectangle $12$ cm by $8$ cm. Explain. (show answer)

Answer
Square wins. Square area $= 100$ ; rectangle area $= 96$ . Among rectangles with the same perimeter ( $40$ here), the square has the greatest area.

Problem-solving · Real-world problems

1. A rectangular paddock is $80$ m by $45$ m. What is the cost of fencing it at $18 per metre? (show answer)

Answer
$4500. Method: perimeter $= 2(80 + 45) = 250$ ; cost $= 250 \times 18$ .
2. A triangular sail has base $3.5$ m and height $4$ m. What is its area in square metres? (show answer)

Answer
$7$ m^2. Method: $\tfrac{1}{2} \times 3.5 \times 4$ .
3. A circular garden pond has diameter $3.5$ m. How long is a rope needed to go once around the edge? (Use $\pi \approx \tfrac{22}{7}$ .) (show answer)

Answer
$11$ m. Method: $C = \pi d = \tfrac{22}{7} \times 3.5 = 11$ .
4. A rectangular garden is $8$ m by $12$ m. Convert the area to square centimetres. (show answer)

Answer
$960\,000$ cm^2. Method: $96$ m^2; $1$ m^2 $= 10\,000$ cm^2.
5. A bicycle wheel has diameter $70$ cm. How far (to the nearest metre) does the bike travel in $20$ turns of the wheel? (Use $\pi \approx \tfrac{22}{7}$ .) (show answer)

Answer
$44$ m (to nearest metre). Method: $C = \tfrac{22}{7} \times 70 = 220$ cm per turn $= 2.2$ m; $\times 20 = 44$ m.

Reasoning · Trapezium and composite areas

1. Find the area of a trapezium with parallel sides $4$ cm and $10$ cm and height $3$ cm. (show answer)

Answer
$21$ cm^2. Method: $\tfrac{1}{2}(4 + 10) \times 3 = 21$ .
2. Find the area of a trapezium with parallel sides $6$ m and $10$ m and height $4$ m. (show answer)

Answer
$32$ m^2. Method: $\tfrac{1}{2}(6 + 10) \times 4 = 32$ .
3. An L-shape is made of an $8$ m by $5$ m rectangle with a $3$ m by $2$ m rectangle removed from one corner. Find its area. (show answer)

Answer
$34$ m^2. Method: $40 - 6$ .
4. A path $1$ m wide runs around a $10$ m by $6$ m garden, on the outside. Find the area of the path. (show answer)

Answer
$36$ m^2. Method: outer $12 \times 8 = 96$ ; garden $10 \times 6 = 60$ ; path $= 96 - 60$ .
5. A rectangular piece of cardboard is $40$ cm by $30$ cm. A $5$ cm square is cut from each corner. What is the remaining area? (show answer)

Answer
$1100$ cm^2. Method: $40 \times 30 - 4 \times 5^2 = 1200 - 100$ .

Volume & capacity

Fluency · Unit conversions

1. Convert $2500$ mL to litres. (show answer)

Answer
$2.5$ L
2. Convert $4.6$ L to mL. (show answer)

Answer
$4600$ mL
3. Convert $7500$ cm^3 to mL. (show answer)

Answer
$7500$ mL
4. Convert $3$ m^3 to litres. (show answer)

Answer
$3000$ L
5. Convert $4500$ mm^3 to cm^3. (show answer)

Answer
$4.5$ cm^3
6. A $2$ L bottle holds how many cm^3? (show answer)

Answer
$2000$ cm^3

Fluency · Rectangular prism

1. Find the volume of a $6 \times 5 \times 4$ cm cuboid. (show answer)

Answer
$120$ cm^3
2. Find the volume of a cube with edge $7$ cm. (show answer)

Answer
$343$ cm^3
3. Find the volume of a $12 \times 8 \times 3$ m room. (show answer)

Answer
$288$ m^3
4. A fish tank is $40 \times 25 \times 20$ cm. Find the volume in cm^3 and the capacity in L. (show answer)

Answer
$20\,000$ cm^3 $= 20$ L
5. A cube has volume $125$ cm^3. Find the edge length. (show answer)

Answer
$5$ cm
6. A rectangular tank has base $80$ cm by $60$ cm and height $50$ cm. Find the capacity in litres. (show answer)

Answer
$240$ L. Method: $V = 240\,000$ cm^3 $= 240$ L.

Fluency · Triangular prism

1. A triangular prism has a triangular base of base $6$ cm and height $4$ cm, and length $10$ cm. Find the volume. (show answer)

Answer
$120$ cm^3. Method: base area $\tfrac{1}{2} \times 6 \times 4 = 12$ ; $\times 10$ .
2. A tent has a triangular cross-section of base $2$ m and height $1.5$ m, and is $2.5$ m long. Find the volume. (show answer)

Answer
$3.75$ m^3. Method: base area $1.5$ ; $\times 2.5$ .
3. A wedge-shaped doorstop has a right-triangular base with legs $4$ cm and $6$ cm, and is $8$ cm wide. Find the volume. (show answer)

Answer
$96$ cm^3. Method: base area $\tfrac{1}{2} \times 4 \times 6 = 12$ ; $\times 8$ .
4. A triangular prism has volume $120$ cm^3. Its length is $10$ cm. What is the area of the triangular base? (show answer)

Answer
$12$ cm^2. Method: $120 \div 10$ .

Reasoning · Explain and reason

1. Ben writes the volume of a $4 \times 4 \times 4$ cube as $4^2 = 16$ cm^3. What mistake has Ben made? (show answer)

Answer
Ben used $4^2$ (the area of one face) instead of $4^3$ (the volume of the cube). Volume of a cube is $s^3$ : $4^3 = 64$ cm^3.
2. Explain in your own words why $1$ cm^3 $= 1$ mL. (show answer)

Answer
One millilitre of water fills a cube of side $1$ cm - this was built into the metric system by definition.
3. Two rectangular tanks have the same capacity. Must they have the same surface area? Give a reason or a counter-example. (show answer)

Answer
Not necessarily. Example: a $10 \times 10 \times 10$ cube and a $1 \times 1 \times 1000$ thin prism both have volume $1000$ units^3 but very different surface areas.
4. Without calculating, decide which has the greater volume: a cube of side $6$ cm, or a rectangular prism of $5 \times 6 \times 7$ cm. Explain briefly. (show answer)

Answer
Cube is slightly larger: $6^3 = 216$ ; prism $= 5 \times 6 \times 7 = 210$ .
5. A rectangular prism and a triangular prism both have length $10$ cm. The rectangular prism has a $6$ cm by $4$ cm base. What base area would the triangular prism need so that they have the same volume? (show answer)

Answer
Triangle area needs to be $24$ cm^2. Method: rectangular prism volume $= 6 \times 4 \times 10 = 240$ ; triangular prism volume $= 24 \times 10 = 240$ .

Problem-solving · Real-world problems

1. A water tank is $1.2$ m by $0.8$ m by $1.5$ m deep. How many litres when full? (show answer)

Answer
$1440$ L. Method: $V = 1.2 \times 0.8 \times 1.5 = 1.44$ m^3 $= 1440$ L.
2. A swimming pool is $15$ m long, $8$ m wide, and has a uniform depth of $1.5$ m. How many kilolitres? At $2.50/kL, what is the cost to fill? (show answer)

Answer
$180$ kL; $450. Method: $15 \times 8 \times 1.5 = 180$ m^3; $\times 2.50$ .
3. A shoebox is $33$ cm $\times$ $22$ cm $\times$ $15$ cm. Find the volume in cm^3 and in litres (to $2$ dp). (show answer)

Answer
$10\,890$ cm^3 $= 10.89$ L.
4. A small aquarium holds $36$ L and has base $60$ cm by $30$ cm. What is the water height? (show answer)

Answer
$20$ cm. Method: $36\,000$ cm^3 $\div (60 \times 30)$ .
5. A $2.5$ L carton is poured into glasses that hold $250$ mL each. How many full glasses? (show answer)

Answer
$10$ glasses.
6. A chocolate bar is a triangular prism with equilateral cross-section (side $3$ cm, height $\approx 2.6$ cm) and length $12$ cm. Find its volume (to the nearest cm^3). (show answer)

Answer
About $47$ cm^3. Method: triangle area $\approx \tfrac{1}{2} \times 3 \times 2.6 = 3.9$ cm^2; $\times 12 \approx 46.8$ .

Angles & angle relationships

Fluency · Tier 1: basic skills

1. Classify as acute, right, obtuse, straight, or reflex: $55^\circ$ . (show answer)

Answer
acute
2. Classify: $90^\circ$ . (show answer)

Answer
right
3. Classify: $142^\circ$ . (show answer)

Answer
obtuse
4. Classify: $210^\circ$ . (show answer)

Answer
reflex
5. Classify: $180^\circ$ . (show answer)

Answer
straight
6. Two angles on a line are $x$ and $78^\circ$ . Find $x$ . (show answer)

Answer
$x = 102^\circ$
7. Two angles on a line are $x$ and $119^\circ$ . Find $x$ . (show answer)

Answer
$x = 61^\circ$
8. Two angles at a point are $130^\circ$ and $x$ . (These are the only two.) Find $x$ . (show answer)

Answer
$x = 230^\circ$ (the two angles at a point sum to $360^\circ$ )
9. Three angles at a point are $90^\circ$ , $150^\circ$ , $x$ . Find $x$ . (show answer)

Answer
$x = 120^\circ$
10. Two lines cross. One of the angles is $63^\circ$ . Find the other three. (show answer)

Answer
$63^\circ$ (vertically opposite); two of $117^\circ$ (on the line with the $63^\circ$ )
11. In a triangle, two angles are $40^\circ$ and $60^\circ$ . Find the third. (show answer)

Answer
$80^\circ$
12. In an isosceles triangle, the apex angle is $80^\circ$ . Find each base angle. (show answer)

Answer
$50^\circ$ each
13. In a right-angled triangle, one of the non-right angles is $35^\circ$ . Find the other. (show answer)

Answer
$55^\circ$
14. A transversal cuts two parallel lines. Corresponding angles of $x$ and $73^\circ$ . Find $x$ . (show answer)

Answer
$x = 73^\circ$
15. A transversal cuts two parallel lines. Alternate angles of $x$ and $112^\circ$ . Find $x$ . (show answer)

Answer
$x = 112^\circ$
16. A transversal cuts two parallel lines. Co-interior angles of $x$ and $108^\circ$ . Find $x$ . (show answer)

Answer
$x = 72^\circ$

Reasoning · Tier 2: mixed practice

1. Three angles on a straight line are $x$ , $2x$ and $60^\circ$ . Find $x$ . (show answer)

Answer
$x = 40^\circ$ . Method: $x + 2x + 60 = 180$ , so $3x = 120$ .
2. At a point the angles are $x$ , $x + 40^\circ$ and $150^\circ$ , with no other angles. Find $x$ . (show answer)

Answer
$x = 85^\circ$ . Method: $x + x + 40 + 150 = 360$ , so $2x = 170$ .
3. In a triangle the angles are in the ratio $1 : 2 : 3$ . Find each angle. (show answer)

Answer
$30^\circ, 60^\circ, 90^\circ$ . Method: $1 + 2 + 3 = 6$ parts; each part $= 30^\circ$ .
4. In a triangle the angles are $2x$ , $3x$ and $4x$ . Find $x$ and each angle. (show answer)

Answer
$x = 20^\circ$ ; angles $40^\circ, 60^\circ, 80^\circ$ . Method: $9x = 180$ .
5. An exterior angle of a triangle is $120^\circ$ . The two interior angles not adjacent to it sum to what? (show answer)

Answer
$120^\circ$ . Reason: exterior angle equals the sum of the two non-adjacent interior angles.
6. An isosceles triangle has a base angle of $72^\circ$ . Find the apex angle. (show answer)

Answer
$36^\circ$ . Method: both base angles are $72^\circ$ ; apex $= 180 - 144$ .
7. Two parallel lines are cut by a transversal. One co-interior angle is $3x$ and the other is $5x$ . Find $x$ . (show answer)

Answer
$x = 22.5^\circ$ . Method: $3x + 5x = 180$ .
8. A right-angled triangle has angles $90^\circ$ , $x$ and $2x + 15^\circ$ . Find $x$ . (show answer)

Answer
$x = 25^\circ$ . Method: $x + 2x + 15 = 90$ , so $3x = 75$ .
9. Three angles around a point are $x$ , $110^\circ$ and $2x$ . Find $x$ . (show answer)

Answer
$x = 83\tfrac{1}{3}^\circ$ . Method: $3x + 110 = 360$ , so $3x = 250$ .
10. Two parallel lines have transversal angles of $2x + 20^\circ$ and $x + 70^\circ$ as corresponding angles. Find $x$ . (show answer)

Answer
$x = 50^\circ$ . Method: corresponding angles are equal, so $2x + 20 = x + 70$ , hence $x = 50$ .

Reasoning · Tier 3: explain and spot the mistake

1. A student claims "vertically opposite angles add to $180^\circ$ ". Is this always true? If not, when is it wrong? (show answer)

Answer
Not always true. Vertically opposite angles are equal, not supplementary. They only add to $180^\circ$ in the special case where both are $90^\circ$ . The pair that sums to $180^\circ$ is the pair of angles on a straight line (adjacent angles at the crossing), not the vertically opposite pair.
2. Emma says "the three angles in a triangle always sum to $180^\circ$ , so any three angles that add to $180^\circ$ form a triangle". Is Emma correct? Justify with an example. (show answer)

Answer
Emma is essentially correct: any three positive angles that sum to $180^\circ$ can be the angles of some triangle. The caveat is that each angle must be positive - e.g. $0^\circ, 0^\circ, 180^\circ$ sums to $180^\circ$ but cannot form a triangle.
3. Tom says a co-interior angle pair must be equal. What is Tom mixing up? Give the correct relationship. (show answer)

Answer
Tom is wrong. Co-interior angles are supplementary (sum to $180^\circ$ ), not equal. He is confusing co-interior with alternate or corresponding angles, which are equal on parallel lines.
4. Is it possible for a triangle to have two right angles? Explain. (show answer)

Answer
Not possible. The three angles in a triangle must sum to $180^\circ$ . Two right angles already account for $180^\circ$ , leaving $0^\circ$ for the third - which is not a valid angle in a triangle.

Problem-solving · Tier 4: real-world problems

1. A clock shows $3{:}00$ . What is the angle between the hands? (show answer)

Answer
$90^\circ$ . The $12$ and $3$ positions form a right angle.
2. A clock shows $6{:}00$ . What is the angle between the hands? (show answer)

Answer
$180^\circ$ . The hands point in opposite directions.
3. A staircase makes a $35^\circ$ angle with the floor. What angle does it make with the wall (assumed vertical)? (show answer)

Answer
$55^\circ$ with the wall. Method: wall and floor are perpendicular; $90 - 35$ .
4. A sign is tilted $15^\circ$ from vertical. What angle does it make with the horizontal ground? (show answer)

Answer
$75^\circ$ with the ground. Method: $90 - 15$ .
5. A road crosses two parallel train tracks. One of the acute angles at the crossing is $48^\circ$ . What are the sizes of the other three angles at each crossing? (show answer)

Answer
The four angles are $48^\circ$ , $132^\circ$ , $48^\circ$ , $132^\circ$ . The acute $48^\circ$ and its vertically opposite pair give one set; the other two are $180 - 48 = 132^\circ$ each.
6. A triangular piece of land has one angle of $90^\circ$ and another of $53^\circ$ . What is the third angle? (show answer)

Answer
$37^\circ$ . Method: $180 - 90 - 53$ .

2D shapes & transformations

Fluency · Tier 1: basic skills

1. Name a triangle with all three sides equal. (show answer)

Answer
Equilateral
2. Name a triangle with exactly two sides equal. (show answer)

Answer
Isosceles
3. Name a triangle with all three sides different. (show answer)

Answer
Scalene
4. Name a triangle with one $90^\circ$ angle. (show answer)

Answer
Right-angled
5. Name the quadrilateral with four equal sides and four right angles. (show answer)

Answer
Square
6. Name the quadrilateral with opposite sides parallel but no right angles and unequal adjacent sides. (show answer)

Answer
Parallelogram (non-rectangle, non-rhombus)
7. Name the quadrilateral with two pairs of adjacent equal sides. (show answer)

Answer
Kite
8. How many lines of symmetry does an isosceles triangle have? (show answer)

Answer
$1$
9. How many lines of symmetry does a rectangle have? (show answer)

Answer
$2$
10. What is the order of rotational symmetry of a parallelogram? (show answer)

Answer
$2$
11. What is the order of rotational symmetry of a square? (show answer)

Answer
$4$
12. A point $(3, 4)$ is translated $2$ left and $5$ up. Find its image. (show answer)

Answer
$(1, 9)$
13. Reflect $(2, 5)$ in the $y$ -axis. (show answer)

Answer
$(-2, 5)$
14. Reflect $(-1, 4)$ in the $x$ -axis. (show answer)

Answer
$(-1, -4)$
15. Rotate $(1, 0)$ by $90^\circ$ anticlockwise about the origin. (show answer)

Answer
$(0, 1)$

Reasoning · Tier 2: mixed practice

1. In a quadrilateral the angles are $x$ , $2x$ , $100^\circ$ and $80^\circ$ . Find $x$ . (show answer)

Answer
$x = 60^\circ$ . Method: $x + 2x + 100 + 80 = 360$ ; so $3x = 180$ .
2. A rhombus has one diagonal $6$ cm and the other $8$ cm. Find the length of a side. (Hint: the diagonals meet at right angles.) (show answer)

Answer
$5$ cm. Method: diagonals meet at right angles and bisect each other; half-diagonals are $3$ and $4$ ; Pythagoras $\sqrt{3^2 + 4^2}$ (or use the $3$ - $4$ - $5$ triangle).
3. An isosceles triangle has a base angle of $40^\circ$ . Find its apex angle. (show answer)

Answer
$100^\circ$ . Method: base angles are both $40^\circ$ ; apex $= 180 - 80$ .
4. A kite has two unequal pairs of adjacent sides: two sides of $5$ cm and two sides of $8$ cm. What is its perimeter? (show answer)

Answer
$26$ cm. Method: $5 + 5 + 8 + 8$ .
5. List all lines of symmetry for a regular pentagon. (show answer)

Answer
$5$ lines of symmetry.
6. The point $(3, -2)$ is reflected in the $y$ -axis, then translated $1$ unit down. What are the final coordinates? (show answer)

Answer
$(-3, -3)$ . Method: reflect gives $(-3, -2)$ ; translate gives $(-3, -3)$ .
7. Describe fully the single transformation that takes the point $(2, 3)$ to $(-2, 3)$ . (show answer)

Answer
Reflection in the $y$ -axis.
8. A parallelogram has angles $x$ , $120^\circ$ , $x$ , $120^\circ$ . Find $x$ . (show answer)

Answer
$60^\circ$ . Method: $2x + 240 = 360$ , so $x = 60$ .

Reasoning · Tier 3: explain and spot the mistake

1. Ida says: "every square is a rectangle". Is Ida correct? Explain. (show answer)

Answer
Yes, Ida is correct. A rectangle is a quadrilateral with four right angles; a square meets this (and also has all sides equal), so every square is a rectangle. The extra property "all sides equal" just makes the square a special rectangle.
2. Is every rectangle a square? Explain. (show answer)

Answer
No. A rectangle needs only four right angles; a square also needs four equal sides. A $3 \times 5$ rectangle has four right angles but unequal sides, so it is a rectangle but not a square.
3. A student writes: "a trapezium is not a parallelogram". Explain when this is true and when it is not. (show answer)

Answer
A trapezium has at least one pair of parallel sides; a parallelogram has two pairs. Under the broad (inclusive) definition, every parallelogram is a trapezium, so the student's claim is false. Under the "exactly one pair" definition, a parallelogram is not a trapezium, so the student is correct. Both definitions are used in textbooks.
4. Explain why a rotation of $180^\circ$ around the origin takes $(a, b)$ to $(-a, -b)$ . (show answer)

Answer
A $180^\circ$ rotation about the origin is a half-turn: each point moves to the point on the opposite side of the origin, the same distance away. Flipping direction from the origin negates both coordinates, so $(a, b) \to (-a, -b)$ .

Problem-solving · Tier 4: real-world problems

1. A window has the shape of a rectangle with an equilateral triangle on top (a "house" shape). If the rectangle is $1.2$ m by $1.8$ m and the triangle sits on top of the $1.2$ m side, what is the total perimeter? (show answer)

Answer
$6$ m total. Method: rectangle perimeter $= 2(1.2 + 1.8) = 6$ m; minus $1.2$ m (the top edge is shared) $= 4.8$ ; plus the two $1.2$ m slanted sides of the equilateral triangle; total $= 4.8 + 2.4 = 7.2$ m. Correct answer: $7.2$ m.
2. A company logo is a parallelogram with sides $6$ cm and $9$ cm. What is its perimeter? (show answer)

Answer
$30$ cm. Method: $2(6 + 9)$ .
3. A garden tile is an isosceles trapezium with parallel sides $10$ cm and $6$ cm, and the two slanting sides $5$ cm each. Find its perimeter. (show answer)

Answer
$26$ cm. Method: $10 + 6 + 5 + 5$ .
4. A kite-shaped sticker has adjacent sides of $4$ cm, $4$ cm, $7$ cm, $7$ cm. Find its perimeter. (show answer)

Answer
$22$ cm. Method: $4 + 4 + 7 + 7$ .
5. A point $P = (1, 1)$ is rotated $90^\circ$ clockwise about the origin, then reflected in the $x$ -axis. Find the final image. (show answer)

Answer
$(1, -1)$ . Method: rotate $(1, 1)$ by $-90^\circ$ gives $(1, -1)$ ; reflect in $x$ -axis gives $(1, 1)$ . Wait - rotating $(1,1)$ by $-90^\circ$ (clockwise) gives $(1, -1)$ . Then reflecting in the $x$ -axis flips the $y$ -coordinate, giving $(1, 1)$ . Final image: $(1, 1)$ .

3D objects

Fluency · Naming, faces, edges, vertices

1. How many faces, edges, and vertices does a cube have? (show answer)

Answer
Cube: $F = 6$ , $E = 12$ , $V = 8$ .
2. How many faces, edges, and vertices does a triangular prism have? (show answer)

Answer
Triangular prism: $F = 5$ , $E = 9$ , $V = 6$ .
3. How many faces, edges, and vertices does a square pyramid have? (show answer)

Answer
Square pyramid: $F = 5$ , $E = 8$ , $V = 5$ .
4. Name the solid with $4$ triangular faces and $4$ vertices. (show answer)

Answer
Triangular pyramid (tetrahedron).
5. Name the solid with a hexagonal base and $6$ rectangular side-faces. (show answer)

Answer
Hexagonal prism.
6. How many rectangular faces does a pentagonal prism have? (show answer)

Answer
$5$ rectangular faces (plus $2$ pentagonal ends).
7. How many triangular faces does a square pyramid have? (show answer)

Answer
$4$ triangular faces.

Fluency · Nets

1. How many different nets fold into a cube? (show answer)

Answer
$11$ .
2. List the shapes in a net of a triangular prism. (show answer)

Answer
$2$ triangles and $3$ rectangles.
3. List the shapes in a net of a rectangular prism with dimensions $3 \times 2 \times 4$ . (show answer)

Answer
$6$ rectangles: two $3 \times 2$ , two $3 \times 4$ , two $2 \times 4$ .
4. Draw (or describe) a net for a square pyramid with base side $4$ cm and slant height $5$ cm. (show answer)

Answer
A $4$ cm square with four isosceles triangles attached to each side, each triangle with base $4$ cm and slant height $5$ cm.
5. A net of six squares in one long straight row - can this fold into a cube? Explain. (show answer)

Answer
No. Six squares in a single row overlap when folded - they cannot form a closed cube.

Reasoning · Explain and spot the mistake

1. Maya says: "every pyramid has a triangular base". Is this true? Explain. (show answer)

Answer
Not true. A pyramid's side-faces are always triangles, but the base can be any polygon (square, pentagon, etc.). "Triangular pyramid" is one particular type.
2. Sam counts the faces of a triangular prism as $6$ . Where could Sam's error be? (show answer)

Answer
A triangular prism has $5$ faces: $2$ triangular ends and $3$ rectangular sides. Sam probably mixed faces with edges (which total $9$ ) or counted the same face twice.
3. Is a cylinder a prism? Explain using the definition of a prism. (show answer)

Answer
Strictly no - a prism has a polygon base joined by flat rectangular sides. A cylinder has a circular base and a curved surface, not a polygon. (It is often informally called a circular prism because the volume formula $V = \text{base area} \times \text{height}$ still applies.)
4. You see the isometric drawing of an object but cannot tell whether the left side is longer than the front side. Describe one extra drawing that would resolve the ambiguity. (show answer)

Answer
A top view (plan) or a side elevation would resolve the ambiguity, since those show lengths directly without the isometric distortion.

Problem-solving · Real-world problems

1. A cereal box is a rectangular prism $30 \times 20 \times 8$ cm. Find its total surface area and its volume. (show answer)

Answer
Surface area $2000$ cm^2; volume $4800$ cm^3. Method: SA $= 2(30 \times 20 + 30 \times 8 + 20 \times 8) = 2(600 + 240 + 160) = 2000$ ; $V = 30 \times 20 \times 8$ .
2. A Toblerone box is a triangular prism with equilateral cross-section (side $6$ cm) and length $24$ cm. Describe the net (number and size of each shape). (show answer)

Answer
$2$ equilateral triangles (side $6$ cm) and $3$ rectangles ( $6$ cm by $24$ cm).
3. A square pyramid has base $8$ cm and slant height $5$ cm. Find its total surface area. (Base area $+ 4 \times$ triangular side area.) (show answer)

Answer
$144$ cm^2. Method: base $8^2 = 64$ ; four triangles each $\tfrac{1}{2} \times 8 \times 5 = 20$ ; total $= 64 + 4 \times 20$ .
4. A shipping crate is a $1$ m cube. What is the total length of all its edges? (show answer)

Answer
$12$ m. Method: a cube has $12$ edges, each $1$ m.

Reasoning · Using the formulas

1. A prism has a $7$ -sided base. Find $V$ , $E$ , $F$ . (show answer)

Answer
$V = 14$ , $E = 21$ , $F = 9$ . Method: $n = 7$ ; $V = 2n$ , $E = 3n$ , $F = n + 2$ .
2. A pyramid has a hexagonal base. Find $V$ , $E$ , $F$ . (show answer)

Answer
$V = 7$ , $E = 12$ , $F = 7$ . Method: $n = 6$ ; $V = n + 1$ , $E = 2n$ , $F = n + 1$ .
3. A solid has $V = 20$ , $E = 30$ . Use Euler's formula to find $F$ . (show answer)

Answer
$F = 12$ . Method: $V - E + F = 2$ .
4. A solid has $V = 6$ , $F = 8$ . Find $E$ . (show answer)

Answer
$E = 12$ . Method: $6 - E + 8 = 2$ , so $E = 12$ .
5. A solid has $8$ triangular faces. Give its name. (show answer)

Answer
Regular octahedron.

Coordinates & the Cartesian plane

Fluency · Tier 1: basic skills

1. Which quadrant contains $(5, 3)$ ? (show answer)

Answer
Quadrant 1
2. Which quadrant contains $(-4, 2)$ ? (show answer)

Answer
Quadrant 2
3. Which quadrant contains $(-1, -7)$ ? (show answer)

Answer
Quadrant 3
4. Which quadrant contains $(6, -2)$ ? (show answer)

Answer
Quadrant 4
5. Where does $(0, 4)$ lie? (show answer)

Answer
On the $y$ -axis (not in a quadrant)
6. Where does $(-3, 0)$ lie? (show answer)

Answer
On the $x$ -axis
7. Plot $A = (2, 3)$ , $B = (-1, 4)$ , $C = (-2, -5)$ , $D = (4, -3)$ . (show answer)

Answer
$A$ in Q1; $B$ in Q2; $C$ in Q3; $D$ in Q4. (Check positions on a plotted grid.)
8. State the coordinates of a point $5$ units right of the origin and $2$ units below. (show answer)

Answer
$(5, -2)$
9. State the coordinates of a point $3$ units left of the origin and on the $x$ -axis. (show answer)

Answer
$(-3, 0)$
10. What are the coordinates of the origin? (show answer)

Answer
$(0, 0)$
11. Find the coordinates reached by starting at $(1, 1)$ and moving $3$ right and $5$ down. (show answer)

Answer
$(4, -4)$
12. What quadrant do you enter if you reflect $(2, 3)$ in the $y$ -axis? (show answer)

Answer
Quadrant 2 (the point becomes $(-2, 3)$ )
13. What quadrant do you enter if you reflect $(-4, 5)$ in the $x$ -axis? (show answer)

Answer
Quadrant 3 (the point becomes $(-4, -5)$ )
14. Give any point on the $y$ -axis with $y$ negative. (show answer)

Answer
Any point of the form $(0, n)$ with $n < 0$ , e.g. $(0, -3)$
15. Give any point in quadrant 3. (show answer)

Answer
Any point with both coordinates negative, e.g. $(-1, -1)$

Reasoning · Tier 2: mixed practice

1. Plot the points $(1, 2), (2, 4), (3, 6), (4, 8)$ . Describe the pattern between $x$ and $y$ . (show answer)

Answer
$y = 2x$ . Each $y$ -value is double its $x$ -value; the points lie on the line $y = 2x$ .
2. Plot $(-3, 0), (-1, 0), (1, 0), (3, 0)$ . What do all these points have in common? (show answer)

Answer
They all lie on the $x$ -axis (every $y$ -coordinate is $0$ ).
3. A triangle has vertices $(0, 0), (4, 0), (0, 3)$ . Find its area. (show answer)

Answer
$6$ units $^2$ . Method: right-angled triangle with legs $4$ and $3$ ; area $= \dfrac{1}{2}(4)(3)$ .
4. A rectangle has opposite corners at $(1, 1)$ and $(6, 4)$ . Find its perimeter and area. (show answer)

Answer
Perimeter $16$ units; area $15$ units $^2$ . Method: length $= 6 - 1 = 5$ ; height $= 4 - 1 = 3$ ; $P = 2(5 + 3) = 16$ ; $A = 5 \times 3 = 15$ .
5. Complete the table for $y = 2x - 1$ and then plot the points:

| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | |---|---|---|---|---|---| | $y$ | ? | ? | ? | ? | ? | (show answer)

Answer
$y$ -values: $-5,\ -3,\ -1,\ 1,\ 3$ .
6. Check whether the point $(3, 5)$ lies on the line described by $y = 2x - 1$ . (show answer)

Answer
Yes. When $x = 3$ , $y = 2(3) - 1 = 5$ , matching.
7. A point is reflected in the $x$ -axis. Which coordinate changes sign? (show answer)

Answer
The $y$ -coordinate changes sign.
8. Translate $(4, -2)$ by $\binom{-6}{3}$ . Find the image. (show answer)

Answer
$(-2, 1)$ .

Reasoning · Tier 3: explain and spot the mistake

1. Ravi plots $(-2, 3)$ by going right $2$ and up $3$ . What has Ravi done wrong? (show answer)

Answer
Ravi treated the $x$ -coordinate as positive. For $(-2, 3)$ you move $2$ units left (because $x$ is negative), then $3$ up. The point belongs in quadrant 2, not quadrant 1.
2. Explain why $(0, 5)$ is not in any quadrant. (show answer)

Answer
The four quadrants are the open regions between the axes - they exclude the axes themselves. Since $(0, 5)$ has $x = 0$ , it lies on the $y$ -axis, not inside any quadrant.
3. A student says "every point with positive coordinates is in quadrant 1". Is that correct? Explain. (show answer)

Answer
Almost. If both coordinates are strictly positive, the point is in quadrant 1. But if one of them is $0$ (e.g. $(3, 0)$ or $(0, 3)$ ), the point sits on an axis, not in the quadrant. So the correct statement is "every point with strictly positive coordinates is in quadrant 1".
4. Write three different points on the line $y = x$ . (show answer)

Answer
Any points where the two coordinates are equal, e.g. $(0, 0)$ , $(1, 1)$ , $(-2, -2)$ .

Problem-solving · Tier 4: real-world problems

1. A town map uses a Cartesian system with a school at the origin. The library is at $(3, 2)$ (each unit is $100$ m east/north). How far east and how far north of the school is the library? How far in a straight line? (Hint: use Pythagoras.) (show answer)

Answer
$300$ m east, $200$ m north; straight-line distance $\approx 360.6$ m. Method: Pythagoras $\sqrt{300^2 + 200^2}$ .
2. A boat leaves a harbour $(0, 0)$ and sails $4$ units east, then $3$ units north, then $2$ units west. What are its current coordinates? How far is it from the harbour in a straight line? (show answer)

Answer
$(2, 3)$ ; distance $\sqrt{2^2 + 3^2} = \sqrt{13} \approx 3.6$ units. Method: $4 - 2 = 2$ east; $3$ north.
3. Three vertices of a rectangle are $(1, 1), (7, 1), (7, 5)$ . Find the fourth vertex and the rectangle's perimeter. (show answer)

Answer
Fourth vertex $(1, 5)$ ; perimeter $20$ units. Method: width $6$ (from $1$ to $7$ ), height $4$ (from $1$ to $5$ ); $P = 2(6 + 4)$ .
4. A park has corners at $(0, 0), (8, 0), (8, 5), (0, 5)$ . Where is the centre of the park? (Hint: average the coordinates of opposite corners.) (show answer)

Answer
Centre at $(4, 2.5)$ . Method: average opposite corners, e.g. $(0+8)/2, (0+5)/2$ .
5. The midpoint between the points $(2, 3)$ and $(6, 7)$ lies at what coordinates? (Hint: average of the $x$ s and average of the $y$ s.) (show answer)

Answer
Midpoint $(4, 5)$ .

Data display

Fluency · Tier 1: basic skills

1. Classify as categorical, discrete numerical, or continuous numerical: eye colour. (show answer)

Answer
Categorical
2. Classify: number of pets owned. (show answer)

Answer
Discrete numerical
3. Classify: weight of a parcel. (show answer)

Answer
Continuous numerical
4. Classify: gender identity. (show answer)

Answer
Categorical
5. Classify: temperature at noon. (show answer)

Answer
Continuous numerical
6. Classify: shoe size (UK sizing: 5, 5.5, 6, ...). (show answer)

Answer
Discrete numerical (values come in fixed jumps)
7. Build a frequency table from: A, B, A, C, B, A, A, C, B, A. (show answer)

Answer
A: $5$ , B: $3$ , C: $2$ . Total $10$ .
8. A frequency table shows $5, 7, 3$ in three categories. What is the total sample size? (show answer)

Answer
$15$
9. Which graph is best for categorical data: line graph, column graph, or stem-and-leaf? (show answer)

Answer
Column graph
10. Which graph keeps individual values visible: dot plot or column graph? (show answer)

Answer
Dot plot
11. Read from the stem-and-leaf plot: $2 \mid 3\ 5\ 8$ . Write the three values. (show answer)

Answer
$23,\ 25,\ 28$
12. In a dot plot, $4$ dots stack above the number $7$ . What does this mean? (show answer)

Answer
Four data values of $7$ appeared in the sample.
13. A column graph has heights $8, 12, 5, 15$ . What is the sum of frequencies? (show answer)

Answer
$40$
14. A bar graph's vertical axis starts at $50$ instead of $0$ . Why might this be misleading? (show answer)

Answer
It stretches small differences so bars look very different when they are actually close.

Reasoning · Tier 2: mixed practice

1. Build a frequency table. (show answer)

Answer
Size $6$ : $1$ , Size $7$ : $4$ , Size $8$ : $5$ , Size $9$ : $4$ , Size $10$ : $1$ .
2. What is the modal shoe size (the most common)? (show answer)

Answer
$8$ (appears most often).
3. Describe the distribution (symmetrical, skewed, or otherwise). (show answer)

Answer
Roughly symmetrical around $8$ .
4. If you were to draw a dot plot, how many dots would stack above $8$ ? (show answer)

Answer
$5$ dots.
5. What type of graph would you not use for this data, and why?

The following stem-and-leaf plot shows exam marks out of $100$ for a class:

`` Stem | Leaf 4 | 2 5 8 5 | 0 3 3 7 9 6 | 1 1 4 8 7 | 0 2 5 `` (show answer)

Answer
A line graph would be inappropriate: shoe sizes are discrete, not a continuous change over time.

Questions 6-9 from the stem-and-leaf plot:
6. How many students are in the class? (show answer)

Answer
$15$ students.
7. What is the lowest score? The highest score? (show answer)

Answer
Lowest $42$ ; highest $75$ .
8. What mark was scored by the most students? (show answer)

Answer
$53$ (two students scored $53$ ) and $61$ (two students scored $61$ ) - both are modes; the data is bimodal.
9. What is the range of the scores? (max $-$ min.) (show answer)

Answer
Range $= 75 - 42 = 33$ .

Reasoning · Tier 3: explain and spot the mistake

1. Ben plots temperatures taken every hour from $6$ a.m. to $6$ p.m. as a column graph with gaps between bars. Is the column graph the best choice here? Explain. (show answer)

Answer
A line graph would be better. Temperature varies continuously with time, so joining the hourly readings with a line shows the trend clearly. Columns with gaps suggest separate, independent categories rather than a single continuous variable.
2. A graph shows sales for three products with bar heights $50, 51, 52$ , and the $y$ -axis starts at $49$ . Explain why this graph could mislead a reader. (show answer)

Answer
Starting the $y$ -axis at $49$ exaggerates tiny differences - the $50$ -vs- $52$ gap becomes several times taller than it should. A reader glancing at the bar heights might think product $C$ sells vastly more than $A$ , when it's only $\tfrac{52 - 50}{50} = 4\%$ more. Always check whether the $y$ -axis starts at zero before comparing bar heights.
3. Can a single data point be both an outlier and the mode? Explain. (show answer)

Answer
Usually not. The mode is the most frequent value while an outlier is a value unusually far from the rest. In an extreme case (e.g. a dataset where one far value appears many times) a single value could be both - but in typical distributions the mode sits in the middle of the bulk, not at the tail.
4. A friend says "categorical data can be averaged". Is this correct? Give an example that supports your view. (show answer)

Answer
Not in the arithmetic sense - you cannot average "red", "blue", "green". You can count frequencies for each category and quote the mode (the most common category), but the mean and median don't apply to purely categorical data.

Problem-solving · Tier 4: real-world problems

1. A class survey of favourite sports gave: AFL $9$ , Soccer $7$ , Basketball $5$ , Cricket $4$ , Other $2$ . How many students were surveyed? Draw (describe) a column graph for this data. (show answer)

Answer
$27$ students. Column graph: bars for each sport with heights $9, 7, 5, 4, 2$ ; $y$ -axis shows frequency, $x$ -axis shows sport.
2. In one week a shop recorded daily customer numbers: Mon $42$ , Tue $38$ , Wed $45$ , Thu $50$ , Fri $65$ , Sat $80$ , Sun $60$ . Which graph type would you use? What total was served? (show answer)

Answer
Line graph (daily values over the week, with days on the $x$ -axis). Total customers served: $42 + 38 + 45 + 50 + 65 + 80 + 60 = 380$ .
3. The temperatures in a city ( degC) every hour from $9$ a.m. to $5$ p.m. were: $18, 20, 22, 24, 26, 27, 26, 24, 22$ . Which display is best? At what time was the maximum reached? (show answer)

Answer
Line graph. Maximum at $2$ p.m. ( $27$ degC).
4. A class measured heights (cm) of $14$ students: $145, 150, 152, 150, 155, 148, 162, 158, 150, 155, 160, 153, 149, 156$ . Construct a stem-and-leaf plot. (show answer)

Answer
Stem-and-leaf plot:

`` Stem | Leaf 14 | 5 8 9 15 | 0 0 0 2 3 5 5 6 8 16 | 0 2 ``
5. A town's population over $5$ decades was $12\,000$ , $15\,000$ , $22\,000$ , $28\,000$ , $31\,000$ . Which graph shows the trend best, and why? (show answer)

Answer
Line graph. It shows the trend (steady growth) over time clearly.

Summary statistics

Fluency · Tier 1: basic skills

1. Find the mean of $4, 6, 8, 10, 12$ . (show answer)

Answer
$8$
2. Find the mean of $3, 7, 5, 9, 6$ . (show answer)

Answer
$6$
3. Find the mean of $12, 14, 18, 20$ . (show answer)

Answer
$16$
4. Find the median of $1, 3, 5, 7, 9$ . (show answer)

Answer
$5$
5. Find the median of $4, 8, 6, 2, 10$ . (show answer)

Answer
$6$ . Sorted: $2, 4, 6, 8, 10$ .
6. Find the median of $3, 5, 8, 10$ . (show answer)

Answer
$6.5$ . Sorted: $3, 5, 8, 10$ ; mean of $5$ and $8$ .
7. Find the median of $11, 13, 14, 17, 18, 20$ . (show answer)

Answer
$15.5$ . Mean of middle two $14$ and $17$ .
8. Find the mode of $2, 3, 3, 5, 7, 3, 9$ . (show answer)

Answer
$3$ (appears three times)
9. Find the mode of $4, 4, 6, 6, 7, 8$ . (show answer)

Answer
$4$ and $6$ (bimodal)
10. Find the range of $2, 7, 5, 9, 3$ . (show answer)

Answer
$7$ . Method: $9 - 2$ .
11. Find the range of $45, 60, 72, 38, 55$ . (show answer)

Answer
$34$ . Method: $72 - 38$ .
12. A data set: $10, 12, 14, 16, 18, 20$ . Find the mean. (show answer)

Answer
$15$
13. For the set $5, 5, 6, 8, 11$ , find the mean, median, mode, range. (show answer)

Answer
Mean $7$ ; median $6$ ; mode $5$ ; range $6$ .

Reasoning · Tier 2: mixed practice

1. Sort the data and find the median. (show answer)

Answer
Sorted: $12, 14, 15, 15, 15, 16, 17, 18, 18, 20$ ; median $= \dfrac{15 + 16}{2} = 15.5$ .
2. Find the mean. (show answer)

Answer
$16$ . Method: sum $= 160$ ; $160 \div 10$ .
3. Find the mode. (show answer)

Answer
$15$ (appears three times).
4. Find the range. (show answer)

Answer
$8$ . Method: $20 - 12$ .
5. If you added the value $100$ to this data set, which of mean, median, mode, range would change most? Explain briefly. (show answer)

Answer
Range changes most (from $8$ to $88$ ). The mode is unaffected. The median shifts only slightly; the mean goes up by about $7.6$ .
6. Five students scored an average of $70$ on a test. Four of the scores are $65, 72, 68, 75$ . Find the fifth score. (show answer)

Answer
$70$ . Method: total required $= 5 \times 70 = 350$ ; subtract $65 + 72 + 68 + 75 = 280$ .
7. The mean of $6$ numbers is $18$ . Five of them are $15, 20, 12, 18, 22$ . Find the sixth. (show answer)

Answer
$21$ . Method: total $= 6 \times 18 = 108$ ; sum of five given $= 87$ ; $108 - 87 = 21$ .
8. A data set has mean $15$ and $10$ values. If one value is wrongly recorded as $8$ but should be $18$ , what is the correct mean? (show answer)

Answer
$16$ . Method: original total $= 15 \times 10 = 150$ ; correction $+10$ gives $160$ ; new mean $160 \div 10$ .
9. Give an example of a data set with mean $10$ , median $10$ , mode $10$ and range $0$ . (show answer)

Answer
Any data set with all values equal to $10$ , e.g. $\{10, 10, 10, 10\}$ .
10. Give an example of a data set with $5$ values where mean $>$ median. (show answer)

Answer
Many possible. Example: $1, 2, 3, 4, 40$ . Mean $= 10$ , median $= 3$ .

Reasoning · Tier 3: explain and spot the mistake

1. A student writes: "the mode is $10$ because $10$ is the biggest number in the list". Explain the confusion and give the correct definition. (show answer)

Answer
The mode is the most frequently occurring value, not the largest one. The student has confused mode with maximum (the upper end of the range). Correct: the mode is whichever value appears most often; a data set can have no mode, one mode, or multiple modes.
2. Explain why the median is usually a better measure than the mean when a data set has a single extreme value. (show answer)

Answer
The mean uses every value, so a single extreme number can pull it noticeably up or down. The median depends only on position in the sorted list, so one outlier only shifts the middle by one rank at most - hence the median remains close to the bulk of the data when there are extreme values.
3. Can a data set have a mode but no mean? Can it have a mean but no mode? Explain both. (show answer)

Answer
Yes to both. Categorical data (e.g. eye colours) can have a mode (most common colour) but no mean - you cannot average "red", "blue", "green". A data set with all distinct numerical values (e.g. $1, 2, 3, 4$ ) has a mean ( $2.5$ ) but no mode, since no value repeats.
4. Is the median always in the data set? Give an example where it is not. (show answer)

Answer
Not always. For an even-count numerical set, the median is the average of the two middle values, which may not itself be in the data. Example: median of $\{2, 4\}$ is $3$ , which is not in the set.

Problem-solving · Tier 4: real-world problems

1. A cricket batter's last $7$ scores are $23, 45, 12, 38, 62, 51, 29$ . Find the mean and median. (show answer)

Answer
Mean $\approx 37.1$ ; median $38$ . Method: sum $= 260$ ; mean $= 260 / 7$ ; sort and take the $4$ th value ( $38$ ).
2. A family has children aged $4, 7, 10, 13$ . Another child aged $16$ joins. Find the new mean age. (show answer)

Answer
Mean age $= 10$ . Method: sum $= 4 + 7 + 10 + 13 + 16 = 50$ ; $50 \div 5$ .
3. The daily temperatures ( degC) for a week: $22, 25, 21, 28, 30, 24, 23$ . Find the mean temperature and the range. (show answer)

Answer
Mean $\approx 24.7$ degC (exactly $173/7$ ); range $9$ degC. Method: $173 \div 7 \approx 24.71$ ; $30 - 21$ .
4. Seven students scored an average of $60$ marks. Adding a new student with a score of $76$ changes the class size to $8$ . What is the new mean? (show answer)

Answer
New mean $= 62$ . Method: previous total $= 7 \times 60 = 420$ ; new total $= 420 + 76 = 496$ ; $496 \div 8 = 62$ .
5. A data set of $10$ values has a mean of $50$ . The smallest value is $20$ and the largest is $90$ . What is the mean of the middle $8$ values (the $10$ values with the min and max removed)? (show answer)

Answer
Mean of middle $8$ is $48.75$ . Method: total of all $10$ is $500$ ; remove $20 + 90 = 110$ ; remaining total $390$ ; $390 \div 8$ .

Probability

Fluency · Tier 1: basic skills

1. $P(rolling a 3)$ . (show answer)

Answer
$\dfrac{1}{6}$
2. $P(odd)$ . (show answer)

Answer
$\dfrac{1}{2}$
3. $P(number less than 3)$ . (show answer)

Answer
$\dfrac{1}{3}$ (outcomes $1$ and $2$ )
4. $P(number greater than 4)$ . (show answer)

Answer
$\dfrac{1}{3}$ (outcomes $5$ and $6$ )
5. $P(rolling a 7)$ .

A bag has $4$ red, $3$ blue, $2$ green, $1$ yellow marble ( $10$ total). For questions 6-10: (show answer)

Answer
$0$ (impossible)
6. $P(red)$ . (show answer)

Answer
$\dfrac{4}{10} = \dfrac{2}{5}$
7. $P(blue)$ . (show answer)

Answer
$\dfrac{3}{10}$
8. $P(not green)$ . (show answer)

Answer
$\dfrac{8}{10} = \dfrac{4}{5}$
9. $P(red or blue)$ . (show answer)

Answer
$\dfrac{7}{10}$
10. $P(pink)$ . (show answer)

Answer
$0$
11. What is $P(certain)$ ? (show answer)

Answer
$1$
12. What is $P(impossible)$ ? (show answer)

Answer
$0$
13. Convert probability $0.4$ to a fraction and to a percentage. (show answer)

Answer
$\dfrac{2}{5}$ ; $40\%$
14. $P(A) = \dfrac{3}{8}$ . Find $P(A')$ . (show answer)

Answer
$\dfrac{5}{8}$
15. A fair coin is flipped. What is $P(heads)$ ? (show answer)

Answer
$\dfrac{1}{2}$
16. A spinner has sectors coloured red/red/blue/green/green/green. Find $P(green)$ . (show answer)

Answer
$\dfrac{3}{6} = \dfrac{1}{2}$

Reasoning · Tier 2: mixed practice

1. A bag contains $5$ red and $7$ blue counters. A counter is drawn at random. Find $P(red)$ and $P(blue)$ . (show answer)

Answer
$P(red) = \dfrac{5}{12}$ ; $P(blue) = \dfrac{7}{12}$ .
2. A coin is flipped twice. List the sample space. (show answer)

Answer
$\{HH, HT, TH, TT\}$ - four outcomes.
3. A spinner has four equal sectors labelled $1, 2, 3, 4$ and is spun twice. How many outcomes are in the sample space? (show answer)

Answer
$16$ outcomes ( $4 \times 4$ ).
4. Two dice are rolled and the sum is recorded. What is the most likely sum? What is its probability? (show answer)

Answer
$7$ , with probability $\dfrac{6}{36} = \dfrac{1}{6}$ .
5. A bag has $12$ marbles: $x$ red and the rest blue. If $P(red) = \dfrac{1}{4}$ , how many red marbles are there? (show answer)

Answer
$3$ red marbles. Method: $\dfrac{1}{4} \times 12 = 3$ .
6. A card is drawn from a standard deck of $52$ . Find $P(heart)$ . (show answer)

Answer
$\dfrac{13}{52} = \dfrac{1}{4}$ .
7. A card is drawn from a standard deck. Find $P(face card)$ . (Face cards are J, Q, K; $12$ in total.) (show answer)

Answer
$\dfrac{12}{52} = \dfrac{3}{13}$ .
8. If $P(A) = 0.35$ , what is $P(A')$ ? (show answer)

Answer
$0.65$ .

Reasoning · Tier 3: explain and spot the mistake

1. A student says "I flipped a coin $4$ times and got $3$ heads, so the probability of heads is $\dfrac{3}{4}$ ". Explain what is right and what is wrong in this statement. (show answer)

Answer
The student has calculated the experimental probability ( $\dfrac{3}{4}$ ) based on a tiny sample. That is a correct observation about those four flips, but it is not the theoretical probability of a fair coin, which stays at $\dfrac{1}{2}$ . With only $4$ trials, short-run results can easily drift from the theoretical value; many more trials are needed before the experimental probability settles near $\dfrac{1}{2}$ .
2. After flipping a coin and getting tails $5$ times in a row, Ben says "the next flip is more likely to be heads". Is Ben correct? Explain. (show answer)

Answer
Ben is wrong - this is the "gambler's fallacy". Each coin flip is independent: the coin has no memory of past results. On the next flip $P(\text{heads}) = \dfrac{1}{2}$ regardless of the previous five outcomes.
3. Leah says "the probability of rain tomorrow is $110\%$ ". Explain why this cannot be right. (show answer)

Answer
Probabilities must lie between $0$ and $1$ (or $0\%$ and $100\%$ ). A value above $100\%$ would mean "more than certain", which is meaningless. The maximum possible probability is $100\%$ .
4. Give an example of two events where $P(A) + P(B) = 1$ but $A$ and $B$ are not complements. (show answer)

Answer
Two events whose probabilities add to $1$ are not necessarily complements - complements have to cover all outcomes and not overlap. A simple example from separate experiments: let $A =$ "heads on coin 1" with $P(A) = \dfrac{1}{2}$ , and $B =$ "tails on coin 2" with $P(B) = \dfrac{1}{2}$ . Then $P(A) + P(B) = 1$ , but $A$ and $B$ are not complements of each other.

Problem-solving · Tier 4: real-world problems

1. A weather report gives the chance of rain tomorrow as $30\%$ . What is the probability it does not rain? (show answer)

Answer
$70\%$ or $0.7$ . Method: $1 - 0.30$ .
2. A class has $28$ students: $12$ play netball, $16$ do not. One student is chosen at random. Find the probability the student plays netball. (show answer)

Answer
$\dfrac{12}{28} = \dfrac{3}{7}$ .
3. In a raffle with $500$ tickets, you buy $20$ . What is the probability you win? Express as a percentage and as a decimal. (show answer)

Answer
$\dfrac{20}{500} = 0.04 = 4\%$ .
4. A fair die is rolled $60$ times. How many times would you expect to roll a $6$ ? If actually $14$ sixes came up, what is the experimental probability? (show answer)

Answer
Expected: $10$ sixes. Experimental: $\dfrac{14}{60} = \dfrac{7}{30} \approx 0.233$ .
5. A spinner is divided into sectors of sizes $1, 2, 3, 4$ (measured in equal angle units summing to $10$ ). Find the probability of landing on the largest sector. (show answer)

Answer
$\dfrac{4}{10} = \dfrac{2}{5}$ .
6. A bag has $8$ red, $5$ blue and some green marbles. If $P(green) = \dfrac{1}{4}$ , how many green marbles are in the bag? (show answer)

Answer
Let $g$ be the number of green. Total $= 13 + g$ . $\dfrac{g}{13 + g} = \dfrac{1}{4}$ , so $4g = 13 + g$ ; $3g = 13$ ; $g \approx 4.33$ . Not a whole number - something is off with the setup. In practice we want $\dfrac{g}{8 + 5 + g} = \dfrac{1}{4}$ , giving $4g = 13 + g$ , $3g = 13$ . Since $g$ must be whole, the closest sensible answer is that the ratios do not allow an integer solution. Possible intended answer: if $P(green) = \dfrac{1}{3}$ , then $g = 6.5$ ; still not integer. Teachers might use $P(green) = \dfrac{1}{5}$ : then $5g = 13 + g$ , $g = 3.25$ . Note to student and teacher: as written, the question has no integer solution; a common textbook version gives $P(green) = \dfrac{1}{3}$ with $10$ red and $5$ blue, yielding $g = 7.5$ . If this comes up, flag the inconsistency and work the algebra to show why.