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The Complete Student Workbook

In-Class Module

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Table of Contents

  1. Week 1Numbers1.1 Place Value & Number Sense
  2. Week 2Numbers1.2 Core Operations (Part 1: Addition & Subtraction)
  3. Week 3Numbers1.2 Core Operations (Part 2: Multiplication & Division)
  4. Week 4Numbers1.3 Fractions & Decimals Introduction
  5. Week 5Algebra2.1 Patterns & Simple Sequences
  6. Week 6Algebra2.2 Equality & The Balance Scale
  7. Week 7Algebra2.3 Missing Numbers (Introduction to x)
  8. Week 8Space3.1 2D & 3D Geometric Entities
  9. Week 9Space3.2 Measurement (Length, Mass, Volume)
  10. Week 10Space3.3 Perimeter & 2D Area
  11. Week 11Space3.4 Transformations & Symmetry
  12. Week 12Data4.1 Sorting & Set Classification
  13. Week 13Data4.2 Tally Charts & Pictograms
  14. Week 14Data4.3 Reading Bar Graphs
  15. Week 15Data4.4 Central Tendency (Mean, Median, Mode)
  16. Week 16Finance5.1 Currency & Denominations
  17. Week 17Finance5.2 Simple Transactions & Change
  18. Week 18Finance5.3 & 5.4 Needs, Wants & Saving Goals
  19. AppendixAnswer Key & Worked Solutions (180 items)
Week 1Pillar 1 · NumbersUnit 1.1

Place Value & Number Sense

Target mastery benchmark

Read, write, compose and decompose any whole number up to 9,999 using base-10 positional structure.

1 · Visual intuition

Every number is a tiny city. Units are people, tens are buses, hundreds are streets and thousands are whole districts. Move a digit one place to the left and it becomes ten times more powerful — the same symbol, a completely different job.

10003 x 1004 x 105 x 1Thousands1Hundreds3Tens4Units51,345 = 1000 + 300 + 40 + 5

Base-10 blocks (thousand cube, hundred flat, ten rod, unit cube) mapped onto a 4-column place value chart.

  • •Units, Tens, Hundreds and Thousands as powers of ten
  • •Positional value: the place a digit sits decides what it is worth
  • •Expanded notation and standard form
  • •Composing and decomposing whole numbers in more than one way
  • •Comparing and ordering 4-digit numbers digit by digit

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Reading the value of a digit in 4,582

  1. 1Write the number into the place value chart: 4 | 5 | 8 | 2.
  2. 2Label the columns from the right: Units, Tens, Hundreds, Thousands.
  3. 3The digit 5 sits in the Hundreds column, so its value is 5 x 100 = 500.
  4. 4Check by expanded notation: 4,582 = 4,000 + 500 + 80 + 2. The 500 term confirms the answer.

Worked Example 2 — Composing 3,000 + 400 + 20 + 9

  1. 1Match each part to its column: 3,000 -> Thousands, 400 -> Hundreds, 20 -> Tens, 9 -> Units.
  2. 2Place the leading digit of each part into its column: 3 | 4 | 2 | 9.
  3. 3Any missing column would be written as 0 — here no column is empty.
  4. 4Standard form: 3,429. Say it aloud: three thousand, four hundred and twenty-nine.

Think Like a Mathematician: a zero is never 'nothing'. It is a place-holder doing the heavy job of keeping every other digit in the right column.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    In the number 6,738, what is the value of the digit 7?

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Write 5,000 + 200 + 60 + 1 in standard form.

    Working Space & Method:
    Final Answer:
  3. Bronze3

    A base-10 diagram shows 2 thousand-cubes, 3 hundred-flats, 0 ten-rods and 6 unit-cubes. What number is modelled?

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Write 7,045 in expanded notation, showing the exact value of each non-zero digit by its place value.

    Working Space & Method:
    Final Answer:
  2. Silver5

    Order these from smallest to largest: 4,092 4,920 4,029 4,209.

    Working Space & Method:
    Final Answer:
  3. Silver6

    Using the digits 8, 0, 4 and 9 exactly once each, write the largest possible 4-digit number and the smallest possible 4-digit number.

    Working Space & Method:
    Final Answer:
  4. Silver7

    By how much does 3,478 increase when the 4 and the 7 swap places to give 3,748?

    Working Space & Method:
    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    A bank safe uses a 4-digit code. The thousands digit is 8, the hundreds digit is 0, the tens digit is half of the thousands digit, and the units digit is 1 less than the tens digit. What is the code?

    Working Space & Method:
    Final Answer:
  2. Gold9

    I am a 4-digit number. My thousands digit is 6, my hundreds digit is half of my thousands digit, my tens digit is 0, and all my digits add to 12. What number am I?

    Working Space & Method:
    Final Answer:
  3. Gold10

    Using the digits 5, 1, 7 and 2 exactly once each, build the 4-digit number closest to 5,000. Justify your answer.

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can state the value of any digit in a 4-digit number.
  • I can write a number in expanded notation more than one way.
  • I can order 4-digit numbers confidently.

My reflection

Week 2Pillar 1 · NumbersUnit 1.2

Core Operations (Part 1: Addition & Subtraction)

Target mastery benchmark

Add and subtract numbers up to 4 digits with regrouping, and choose an efficient mental strategy.

1 · Visual intuition

Regrouping is trading. Ten units trade up for one ten; one hundred trades down for ten tens. A column arithmetic page is really a currency exchange counter running at top speed.

114 7 6+ 2 5 87 3 4carries shown above4 10 125 0 2- 2 7 82 2 4borrow across the zero587884+20+6

Column arithmetic with highlighted carry and borrow indicators, plus an open jump number line.

  • •Column addition with carrying
  • •Column subtraction with borrowing, including borrowing across a zero
  • •Mental jump strategies on an open number line
  • •Checking subtraction with the inverse addition
  • •Estimating first to catch place value slips

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — 476 + 258 with carrying

  1. 1Units: 6 + 8 = 14. Write 4, carry 1 into the tens.
  2. 2Tens: 7 + 5 + 1 (carried) = 13. Write 3, carry 1 into the hundreds.
  3. 3Hundreds: 4 + 2 + 1 (carried) = 7.
  4. 4Answer 734. Estimate check: 480 + 260 = 740, close to 734.

Worked Example 2 — 502 - 278 borrowing across a zero

  1. 1Units: 2 - 8 is not possible, and the tens column is 0, so borrow from the hundreds first.
  2. 25 hundreds becomes 4 hundreds; the tens column becomes 10.
  3. 3Now borrow 1 ten for the units: tens becomes 9, units becomes 12.
  4. 4Units 12 - 8 = 4. Tens 9 - 7 = 2. Hundreds 4 - 2 = 2. Answer 224. Check: 224 + 278 = 502.

Think Like a Mathematician: estimate before you calculate. If your estimate and your answer disagree by hundreds, you have lost a carry, not a digit.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    Calculate 47 + 38.

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Calculate 62 - 27.

    Working Space & Method:
    Final Answer:
  3. Bronze3

    Use an open number line to work out 58 + 26. Start at 58, jump the tens first (+20), then jump the units (+6).

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Calculate 502 - 278.

    Working Space & Method:
    Final Answer:
  2. Silver5

    Calculate 1,405 + 2,867.

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  3. Silver6

    Calculate 3,000 - 1,246.

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  4. Silver7

    A library had 1,208 books. It received 396 new books and lent out 587. How many books remain on the shelves?

    Working Space & Method:
    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    Find all possible pairs of digits A and B that make this true: 4A7 + 2B6 = 733.

    Working Space & Method:
    Final Answer:
  2. Gold9

    In the puzzle AB - BA = 27, where A and B are digits, find one possible pair A and B.

    Working Space & Method:
    Final Answer:
  3. Gold10

    Anya thinks of a 3-digit number, adds 285, subtracts 176 and gets 1,009. What was her number? Show the inverse chain.

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can add and subtract with regrouping accurately.
  • I can borrow across a zero without losing a digit.
  • I check my answers using the inverse operation.

My reflection

Week 3Pillar 1 · NumbersUnit 1.2

Core Operations (Part 2: Multiplication & Division)

Target mastery benchmark

Model multiplication as arrays and division as sharing or grouping, including interpreting remainders.

1 · Visual intuition

Multiplication is addition that got organised. Instead of counting 7 + 7 + 7 + 7 one at a time, you build a rectangle of dots and read the whole answer in one glance.

4 rows x 7 = 2823 shared into 5 groups = 4 eachremainder 3

A 4 x 7 dot array beside a fair-sharing diagram: 23 counters shared into 5 containers, 3 left over.

  • •Multiplication as repeated addition and as an array
  • •Commutativity: 4 x 7 gives the same array as 7 x 4 rotated
  • •Division as fair sharing and as grouping
  • •Remainders and what they mean in context
  • •2-digit by 1-digit multiplication and short division

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — 4 x 7 using an array

  1. 1Draw 4 rows of 7 dots.
  2. 2Each row is 7, so the total is 7 + 7 + 7 + 7.
  3. 3Group in pairs of rows: 14 + 14 = 28.
  4. 4So 4 x 7 = 28, and rotating the array shows 7 x 4 = 28 as well.

Worked Example 2 — 23 divided by 5

  1. 1Share 23 counters one at a time into 5 containers.
  2. 2Each container receives 4 counters: 5 x 4 = 20 used.
  3. 323 - 20 = 3 counters cannot be shared equally.
  4. 4So 23 / 5 = 4 remainder 3. Check: 5 x 4 + 3 = 23.

Think Like a Mathematician: always ask what the remainder means. Sometimes you round up (buses needed), sometimes you drop it (full boxes packed).

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    Draw an array for 3 x 6 and state the product.

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Write 5 + 5 + 5 + 5 as a multiplication and solve.

    Working Space & Method:
    Final Answer:
  3. Bronze3

    Share 12 stickers equally among 4 children. How many each?

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Calculate 34 x 6.

    Working Space & Method:
    Final Answer:
  2. Silver5

    Calculate 87 / 4, giving the remainder.

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  3. Silver6

    Calculate 156 / 7.

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  4. Silver7

    A crate holds 8 bottles. How many full crates can be packed from 150 bottles, and how many bottles are left?

    Working Space & Method:
    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    A school needs to transport 137 students in buses that hold 30 students each. What is the minimum number of buses needed so that all students can fit?

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    Final Answer:
  2. Gold9

    A rectangular garden has 96 plants arranged in equal rows. If there are more than 6 rows and fewer than 12 rows, list all possible arrangements.

    Working Space & Method:
    Final Answer:
  3. Gold10

    When a number is divided by 6 the answer is 14 remainder 5. What is the number? What is the same number divided by 7?

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can model multiplication with an array.
  • I can divide with remainders and explain what the remainder means.
  • I can decide whether to round a remainder up or down.

My reflection

Week 4Pillar 1 · NumbersUnit 1.3

Fractions & Decimals Introduction

Target mastery benchmark

Recognise, compare and convert simple fractions and decimals, and place them on a 0-to-1 number line.

1 · Visual intuition

A fraction and a decimal are the same amount wearing different clothes. One half, two quarters and 0.5 all stop at exactly the same point on the number line.

1 wholehalvesquarterseighths00.251/40.51/20.753/414/4

Fraction strip comparison bar above a 0-to-1 number line labelled with 1/4 = 0.25, 1/2 = 0.5 and 3/4 = 0.75.

  • •Halves, thirds, quarters and eighths as equal parts of a whole
  • •Equivalent fractions through fraction strips
  • •Tenths and hundredths as decimal place value
  • •Converting between simple fractions and decimals
  • •Placing fractions and decimals on a 0-to-1 number line

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Showing 2/8 = 1/4

  1. 1Draw a strip split into 8 equal parts and shade 2 of them.
  2. 2Draw an identical strip split into 4 equal parts and shade 1.
  3. 3The shaded lengths match exactly, so 2/8 and 1/4 are equivalent.
  4. 4Numerically: divide numerator and denominator of 2/8 by 2 to get 1/4.

Worked Example 2 — Writing 3/4 as a decimal

  1. 1Change 3/4 into hundredths: multiply top and bottom by 25 to get 75/100.
  2. 275 hundredths is written 0.75.
  3. 3Check on the number line: 0.75 sits three quarters of the way from 0 to 1.

Think Like a Mathematician: only compare fractions when the parts are the same size. Rewrite to a common denominator before deciding which is bigger.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    A circle is divided into 4 equal parts and 3 are shaded. Write the fraction shaded.

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Shade 5/8 of an eight-part strip. What fraction is unshaded?

    Working Space & Method:
    Final Answer:
  3. Bronze3

    Write one half as a fraction, then as a decimal.

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Simplify 6/8 fully.

    Working Space & Method:
    Final Answer:
  2. Silver5

    Convert 7/10 and 43/100 into decimals.

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    Final Answer:
  3. Silver6

    Which is larger, 2/5 or 0.5? Explain.

    Working Space & Method:
    Final Answer:
  4. Silver7

    Write 0.25 as a fraction in simplest form.

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    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    Place 1/3, 0.3, 2/5 and 0.35 in order from smallest to largest.

    Working Space & Method:
    Final Answer:
  2. Gold9

    A jug holds 1 litre. Maya pours out 0.4 L then 1/4 L. How much is left, as a decimal and as a fraction?

    Working Space & Method:
    Final Answer:
  3. Gold10

    Find a fraction that lies exactly halfway between 1/4 and 1/2 on the number line.

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can name equivalent fractions using strips.
  • I can convert simple fractions to decimals and back.
  • I can order fractions and decimals on a number line.

My reflection

Week 5Pillar 2 · AlgebraUnit 2.1

Patterns & Simple Sequences

Target mastery benchmark

Extend, describe and generalise repeating and arithmetic patterns, including the nth term informally.

1 · Visual intuition

Patterns are maths making a promise. Once you find the rule, you can jump ahead to the 20th term without drawing 19 pictures first.

Stage 1: 1Stage 2: 4Stage 3: 7Stage 4: 10+3+3+3

A matchstick growth pattern 1, 4, 7, 10 showing the constant difference of +3 between stages.

  • •Repeating patterns and their core unit
  • •Arithmetic sequences and the constant difference
  • •Recursive rules (+k, -k) versus position rules
  • •Growing geometric patterns built from matchsticks or blocks
  • •Predicting far-away terms without listing every step

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Continuing 1, 4, 7, 10, ...

  1. 1Find the differences: 4 - 1 = 3, 7 - 4 = 3, 10 - 7 = 3.
  2. 2The constant difference is +3, so the rule is 'add 3'.
  3. 3Next terms: 10 + 3 = 13, then 13 + 3 = 16.
  4. 4Position rule in words: multiply the position by 3, then subtract 2. Check position 4: 3 x 4 - 2 = 10.

Worked Example 2 — Finding the 20th term of 5, 9, 13, 17, ...

  1. 1Constant difference is +4 and the first term is 5.
  2. 2Position rule in words: multiply the position by 4, then add 1 (check position 1: 4 x 1 + 1 = 5).
  3. 3For position 20: 4 x 20 + 1 = 81.
  4. 4So the 20th term is 81 without listing all terms.

Think Like a Mathematician: the constant difference is the multiplier in the position rule. Find it first, then adjust the starting value.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    Write the next two terms: 2, 5, 8, 11, …

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Write the next two terms: 40, 35, 30, 25, …

    Working Space & Method:
    Final Answer:
  3. Bronze3

    A pattern repeats: red, blue, blue, red, blue, blue, … What is the 9th colour?

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Find the missing term: 7, …, 19, 25, 31.

    Working Space & Method:
    Final Answer:
  2. Silver5

    State the rule for 6, 12, 18, 24 and give the 10th term.

    Working Space & Method:
    Final Answer:
  3. Silver6

    A matchstick pattern uses 4 sticks for 1 square, 7 for 2 squares, 10 for 3 squares. How many for 8 squares?

    Working Space & Method:
    Final Answer:
  4. Silver7

    Which term of the sequence 3, 7, 11, 15, … is equal to 43?

    Working Space & Method:
    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    A block staircase uses 1, 3, 6, 10 blocks for stages 1 to 4. How many blocks at stage 7?

    Working Space & Method:
    Final Answer:
  2. Gold9

    Find the 20th term of the number sequence 1, 4, 7, 10, … and explain your rule.

    Working Space & Method:
    Final Answer:
  3. Gold10

    Two sequences start together: A = 2, 5, 8, 11, … and B = 3, 7, 11, 15, … What is the next number they share after 11?

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can find the rule of an arithmetic sequence.
  • I can predict a far-away term without listing them all.
  • I can describe a growing shape pattern in words and numbers.

My reflection

Week 6Pillar 2 · AlgebraUnit 2.2

Equality & The Balance Scale

Target mastery benchmark

Interpret the equals sign as balance and keep an equation true when both sides change.

1 · Visual intuition

The equals sign is not a button that means 'the answer is'. It is the pivot of a set of scales: whatever you do to one pan, you must do to the other.

x+ 4 kg11 kgx + 4 = 11, so x = 7

A dual-pan balance scale comparing labelled mass tokens with unknown shape weights.

  • •Equality as balance, not as an instruction
  • •Inequality symbols <, >, <=, >=
  • •Doing the same thing to both sides
  • •Inverse operations as un-doing
  • •Reasoning about unknown masses on a scale

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Is 8 + 5 = 6 + 7 true?

  1. 1Evaluate the left pan: 8 + 5 = 13.
  2. 2Evaluate the right pan: 6 + 7 = 13.
  3. 313 = 13, so the scale balances and the statement is true.
  4. 4If the right pan had been 6 + 8 = 14, the scale would tip right and we would write 13 < 14.

Worked Example 2 — A square plus 4 kg balances 11 kg

  1. 1Write the balance as square + 4 = 11.
  2. 2Remove 4 kg from BOTH pans to keep the balance.
  3. 3Left pan: square. Right pan: 11 - 4 = 7.
  4. 4So the square has mass 7 kg. Check: 7 + 4 = 11.

Think Like a Mathematician: never remove a weight from one pan only. Fairness to both sides is the entire law of algebra.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    True or false: 9 + 6 = 15 + 0.

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Insert <, > or =: 7 x 3 __ 25.

    Working Space & Method:
    Final Answer:
  3. Bronze3

    A pan holds 6 kg and balances a pan with 2 kg and a box. What is the mass of the box?

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Solve for Δ: Δ + 8 = 20.

    Working Space & Method:
    Final Answer:
  2. Silver5

    Two identical circles balance 18 kg. What is the mass of one circle?

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    Final Answer:
  3. Silver6

    Complete so the scale balances: 24 + 13 = 30 + __.

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    Final Answer:
  4. Silver7

    Solve: 3 x □ = 21, then find □ + 9.

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    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    One ■ plus two ● balance 26 kg. One ■ balances two ●. Find the mass of each.

    Working Space & Method:
    Final Answer:
  2. Gold9

    Scale A: ★ + ★ = 14. Scale B: ★ + ☾ = 19. Find ☾.

    Working Space & Method:
    Final Answer:
  3. Gold10

    Three ⬢ and 5 kg balance one ⬢ and 17 kg. Find the mass of one ⬢.

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I read the equals sign as balance.
  • I can keep an equation true by changing both sides equally.
  • I can find unknown masses by reasoning about scales.

My reflection

Week 7Pillar 2 · AlgebraUnit 2.3

Missing Numbers (Introduction to x)

Target mastery benchmark

Solve one-step equations with an unknown and translate simple word statements into equations.

1 · Visual intuition

The empty box grows up and becomes x. Same job, shorter name — a placeholder for the number you have not discovered yet.

RULEx 3 then - 4input xoutput 26inverse: + 4 then / 3

A function machine showing input, rule and output, with reverse arrows for the inverse operation.

  • •Box equations and symbolic variables
  • •Inverse operations to isolate the unknown
  • •Function machines and their reverse
  • •Checking a solution by substitution
  • •Translating English sentences into equations

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Solve x + 5 = 12

  1. 1The unknown has had 5 added to it.
  2. 2Undo by subtracting 5 from both sides: x + 5 - 5 = 12 - 5.
  3. 3x = 7.
  4. 4Substitute back to check: 7 + 5 = 12. Correct.

Worked Example 2 — Solve 4x = 36

  1. 14x means 4 lots of x.
  2. 2Undo multiplication by dividing both sides by 4.
  3. 3x = 36 / 4 = 9.
  4. 4Check: 4 x 9 = 36. Correct.

Think Like a Mathematician: to free the unknown, apply the opposite operation. Addition and subtraction undo each other; so do multiplication and division.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    Solve: □ + 9 = 21.

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Solve: 15 - □ = 9.

    Working Space & Method:
    Final Answer:
  3. Bronze3

    A function machine adds 7. If the output is 20, what was the input?

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Solve: x + 18 = 45.

    Working Space & Method:
    Final Answer:
  2. Silver5

    Solve: 6x = 54.

    Working Space & Method:
    Final Answer:
  3. Silver6

    Solve: x / 4 = 8.

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    Final Answer:
  4. Silver7

    Solve: 3x + 2 = 17.

    Working Space & Method:
    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    Write an equation and solve: 'I think of a number, multiply by 5 and get 65.'

    Working Space & Method:
    Final Answer:
  2. Gold9

    Sara has x marbles. Ben has 12 more than Sara. Together they have 46. Write an equation and find x.

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    Final Answer:
  3. Gold10

    A function machine multiplies by 3 then subtracts 4. The output is 26. Find the input.

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can solve one-step and multi-step equations such as 3x + 2 = 17 using inverse operations in reverse order.
  • I can check my solution by substituting it back.
  • I can turn a word sentence into an equation.

My reflection

Week 8Pillar 3 · SpaceUnit 3.1

2D & 3D Geometric Entities

Target mastery benchmark

Name and classify 2D polygons and 3D solids by their faces, edges and vertices.

1 · Visual intuition

A cube does not stop being a cube when you tip it over. Shapes keep their identity no matter how they are turned — that stubbornness is called invariance.

Cube 6F 12E 8VPrism 5F 9E 6VPyramid 5F 8E 5VCylinder 3 surfacesEuler: F - E + V = 2

Isometric wireframes of a cube, triangular prism, square pyramid and cylinder with labelled parts.

  • •Polygons named by side count: a polygon is a closed 2D shape with straight sides (triangle 3, quadrilateral 4, pentagon 5, hexagon 6)
  • •A prism has two identical polygon ends joined by rectangles, and is named after its end shape (e.g. pentagonal prism)
  • •Faces, edges and vertices of prisms and pyramids
  • •Curved solids: cylinder, cone, sphere
  • •Nets folding into 3D solids
  • •Euler's formula F - E + V = 2 for simple polyhedra

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Counting parts of a triangular prism

  1. 1Faces: 2 triangular ends + 3 rectangular sides = 5 faces.
  2. 2Vertices: 3 corners on each triangle = 6 vertices.
  3. 3Edges: 3 on each triangle + 3 joining edges = 9 edges.
  4. 4Euler check: F - E + V = 5 - 9 + 6 = 2. Correct.

Worked Example 2 — Matching a net to a solid

  1. 1A net with 1 square and 4 triangles attached to its sides folds up to a square pyramid.
  2. 2The square becomes the base; the triangles meet at a single apex.
  3. 3Count the result: 5 faces, 8 edges, 5 vertices.
  4. 4Euler check: 5 - 8 + 5 = 2. Correct.

Think Like a Mathematician: when counting edges, trace them with a finger and mark each one. Double counting is the number one error in solid geometry.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    How many faces, edges and vertices does a cube have?

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Name the 2D shape with 5 straight sides.

    Working Space & Method:
    Final Answer:
  3. Bronze3

    A cylinder has flat faces and curved surfaces. How many flat faces does it have, and how many curved surfaces?

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    A net has 6 identical squares in a cross shape. Which solid does it fold into?

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  2. Silver5

    How many vertices does a square pyramid have?

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  3. Silver6

    Which solid has 2 pentagonal faces and 5 rectangular faces?

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  4. Silver7

    Sketch the net of a triangular prism and label the shapes used.

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GoldMaster · 3 questions
  1. Gold8

    Verify Euler's formula for a hexagonal prism.

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  2. Gold9

    A polyhedron has 9 faces and 16 edges. How many vertices must it have?

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  3. Gold10

    Two identical cubes are glued face to face. How many faces, edges and vertices does the new solid have?

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4 · Mastery checkpoint & reflection

  • I can count faces, edges and vertices accurately.
  • I can match nets to their solids.
  • I can use Euler's formula as a check.

My reflection

Week 9Pillar 3 · SpaceUnit 3.2

Measurement (Length, Mass, Volume)

Target mastery benchmark

Select suitable metric units, read scales accurately and convert between metric steps.

1 · Visual intuition

The metric system is a staircase. Every step you climb multiplies by ten, every step you descend divides by ten — no awkward numbers to memorise.

kmx1000mx10cmx10mm0 cm10 cm20 cm30 cm12 cm = 120 mm500 mL400 mL300 mL200 mL100 mLreading: 450 mL = 0.45 L

A metric conversion staircase alongside a ruler readout and a graduated beaker.

  • •Length: mm, cm, m, km
  • •Mass: g, kg
  • •Volume and capacity: mL, L
  • •Reading rulers, scales and graduated beakers
  • •Converting between units by multiplying or dividing by 10, 100 or 1000

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Convert 3.4 m into centimetres

  1. 11 m = 100 cm, so we move down two steps of the staircase.
  2. 2Multiply by 100: 3.4 x 100 = 340.
  3. 3So 3.4 m = 340 cm.
  4. 4Sense check: centimetres are smaller, so the number should be bigger.

Worked Example 2 — Reading a beaker

  1. 1The beaker is marked every 50 mL up to 500 mL.
  2. 2The liquid sits three marks above 300 mL.
  3. 33 x 50 = 150, so the level is 300 + 150 = 450 mL.
  4. 4In litres that is 0.45 L.

Think Like a Mathematician: after every conversion ask 'should this number get bigger or smaller?'. That single question prevents most conversion errors.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    A pencil measures from 0 cm to 12 cm on a ruler. How long is it in mm?

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  2. Bronze2

    A beaker is marked every 100 mL. The water reaches the 4th mark. How much water is there?

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  3. Bronze3

    Which unit is best for the mass of a school bag: g, kg or mL?

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SilverPractice · 4 questions
  1. Silver4

    Convert 2.75 km into metres.

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  2. Silver5

    Convert 4,300 g into kilograms.

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  3. Silver6

    Convert 1.6 L into millilitres, then subtract 250 mL.

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  4. Silver7

    Add 1 m 45 cm and 2 m 80 cm. Give the answer in metres.

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GoldMaster · 3 questions
  1. Gold8

    A tank holds 12 L. It is filled using a 750 mL jug. How many full jugs are required?

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  2. Gold9

    A ribbon 3.5 m long is cut into pieces of 40 cm. How many pieces are cut, and how much ribbon is left over?

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  3. Gold10

    A parcel weighs 2 kg 350 g. Four identical parcels are posted. What is the total mass in kilograms?

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4 · Mastery checkpoint & reflection

  • I choose sensible metric units.
  • I can read scales and rulers precisely.
  • I can convert between metric units in both directions.

My reflection

Week 10Pillar 3 · SpaceUnit 3.3

Perimeter & 2D Area

Target mastery benchmark

Calculate perimeter and area of rectangles and composite rectilinear shapes.

1 · Visual intuition

Perimeter is the fence you walk around a field. Area is the grass you have to mow inside it. Same shape, two totally different questions.

AB6 cm10 cmarea = A + Bwalk the edge = perimeter

A grid-based rectilinear L-shape split into two rectangles with all dimensions labelled.

  • •Perimeter as total boundary length
  • •Area as a count of unit squares
  • •Area of a rectangle = length x width
  • •Splitting L-shapes into rectangles
  • •Same perimeter can give different areas

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Rectangle 8 cm by 5 cm

  1. 1Perimeter = 2 x (8 + 5) = 2 x 13 = 26 cm.
  2. 2Area = 8 x 5 = 40 square cm.
  3. 3Note the units: perimeter uses cm, area uses square cm.

Worked Example 2 — L-shape area by splitting

  1. 1The L-shape is 10 cm wide and 8 cm tall, with a 4 cm by 3 cm notch cut from the top right.
  2. 2Split into rectangle A (10 x 5 = 50) and rectangle B (6 x 3 = 18).
  3. 3Total area = 50 + 18 = 68 square cm.
  4. 4Alternative: whole rectangle 10 x 8 = 80 minus the notch 4 x 3 = 12, giving 68. Both agree.

Think Like a Mathematician: for composite shapes, either add rectangles together or subtract the missing piece. Do it both ways and your answers should match.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    A shape on a grid covers 14 unit squares. What is its area?

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  2. Bronze2

    Find the perimeter of a 6 cm by 3 cm rectangle.

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  3. Bronze3

    Find the area of a square with side 7 cm.

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SilverPractice · 4 questions
  1. Silver4

    A rectangle has area 48 square cm and length 8 cm. Find its width and perimeter.

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  2. Silver5

    An L-shape is made of a 10 cm x 5 cm rectangle and a 6 cm x 3 cm rectangle. Find the total area.

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  3. Silver6

    A 12 m by 9 m garden has a 3 m by 3 m pond. What area of grass remains?

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  4. Silver7

    The perimeter of a rectangle is 30 cm and its length is 9 cm. Find the width and area.

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GoldMaster · 3 questions
  1. Gold8

    A farmer has 24 m of fencing. What whole-number rectangle dimensions give the greatest area?

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  2. Gold9

    Two rectangles both have area 36 square cm. One is 4 x 9 and the other is 6 x 6. Which has the smaller perimeter?

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  3. Gold10

    A path 1 m wide runs around the outside of a 5 m by 4 m pool. Find the area of the path.

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4 · Mastery checkpoint & reflection

  • I can tell perimeter and area apart, with correct units.
  • I can find the area of composite L-shapes.
  • I can reason about maximum area for a fixed perimeter.

My reflection

Week 11Pillar 3 · SpaceUnit 3.4

Transformations & Symmetry

Target mastery benchmark

Perform and describe reflections, translations and rotations, and identify symmetry.

1 · Visual intuition

Slide, flip, turn. Three simple moves that never change the size or shape of an object — only where it sits and which way it faces.

Translation (slide)Reflection (flip)Rotation (turn)original shape (dark) · image after the move (light) · rotation: 90 degrees clockwise about the dot

A coordinate grid showing a triangle reflected across a mirror line and rotated 90 degrees clockwise.

  • •Translation (slide) described by steps right/left and up/down
  • •Reflection (flip) across a mirror line
  • •Rotation (turn) about a centre through 90, 180 or 270 degrees
  • •Lines of symmetry in 2D shapes
  • •Order of rotational symmetry

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Reflecting a point across a vertical mirror

  1. 1The mirror line is x = 4 and the point sits at (2, 3).
  2. 2Measure the horizontal distance to the mirror: 4 - 2 = 2.
  3. 3Count the same distance on the other side: 4 + 2 = 6.
  4. 4The image is at (6, 3); the vertical value never changes for a vertical mirror.

Worked Example 2 — Rotational symmetry of a rectangle

  1. 1Rotate the rectangle 90 degrees: it does not match itself.
  2. 2Rotate 180 degrees: it matches exactly.
  3. 3Rotate 270 degrees: no match. Rotate 360: match (the full turn always matches).
  4. 4So the order of rotational symmetry is 2.

Think Like a Mathematician: a transformation that preserves size and shape is called a congruence. Reflections, translations and rotations are all congruences.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    How many lines of symmetry does a square have?

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  2. Bronze2

    How many lines of symmetry does the capital letter H have?

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  3. Bronze3

    Name the transformation: a shape slides 4 squares right without turning.

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SilverPractice · 4 questions
  1. Silver4

    Reflect the point (3, 5) across the vertical line x = 6. Give the image.

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  2. Silver5

    Translate the point (2, 7) by 5 units to the right and 3 units down.

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  3. Silver6

    Rotate the point (4, 0) by 90 degrees clockwise about the origin.

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  4. Silver7

    An equilateral triangle is rotated about its centre. State its order of rotational symmetry.

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GoldMaster · 3 questions
  1. Gold8

    A regular hexagon emblem is used as a logo. State its lines of symmetry and rotational order.

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  2. Gold9

    On squared paper, a flag is flipped over a vertical mirror line, then flipped again over a horizontal mirror line. What single move gives the same result?

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  3. Gold10

    A parallelogram (not a rectangle) has how many lines of symmetry and what rotational order?

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4 · Mastery checkpoint & reflection

  • I can describe slides, flips and turns precisely.
  • I can find lines of symmetry.
  • I can state the order of rotational symmetry.

My reflection

Week 12Pillar 4 · DataUnit 4.1

Sorting & Set Classification

Target mastery benchmark

Sort data by one or two attributes using Venn and Carroll diagrams.

1 · Visual intuition

Sorting is the first act of every data scientist. Before you can count anything, you must decide what belongs where — and what belongs in both places at once.

VENN DIAGRAM (numbers 1–12)EvenGreater than 62 46810127 911neither: 1, 3, 5CARROLL DIAGRAM (same numbers)EvenNot evenGreaterthan 6Not greaterthan 68, 10, 127, 9, 112, 4, 61, 3, 5

A two-set Venn diagram with item tokens in each region including the overlap and the outside region.

  • •Sorting rules and attributes
  • •Two-set Venn diagrams and the intersection
  • •Carroll diagrams with yes/no rows and columns
  • •Mutually exclusive groups
  • •Inclusion-exclusion: total = A + B - both

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Sorting 1 to 12 into 'Even' and 'Greater than 6'

  1. 1Even only: 2, 4, 6.
  2. 2Greater than 6 only: 7, 9, 11.
  3. 3Both (even and greater than 6): 8, 10, 12.
  4. 4Neither: 1, 3, 5. All 12 numbers are placed exactly once.

Worked Example 2 — Inclusion-exclusion

  1. 1In a class of 30, 18 play football and 14 play chess; 7 do both.
  2. 2Football only = 18 - 7 = 11. Chess only = 14 - 7 = 7.
  3. 3Total playing at least one = 11 + 7 + 7 = 25.
  4. 4Playing neither = 30 - 25 = 5 students.

Think Like a Mathematician: fill the overlap of a Venn diagram FIRST. Every other region depends on it.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    Sort 3, 8, 12, 15 into 'Odd' and 'Even'.

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  2. Bronze2

    Place 24 in a Venn diagram of 'Even' and 'Greater than 20'. Which region?

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  3. Bronze3

    Which numbers from 1 to 10 belong outside both circles of 'Even' and 'Greater than 6'?

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SilverPractice · 4 questions
  1. Silver4

    Build a Carroll diagram for numbers 1-20 using 'multiple of 3' and 'even'. List the both cell.

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  2. Silver5

    In a Venn of 'multiples of 4' and 'multiples of 6' up to 30, list the intersection.

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  3. Silver6

    20 pets: 12 are dogs, 5 are dogs that are brown, 9 are brown in total. How many are neither dogs nor brown?

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  4. Silver7

    Give a sorting rule that puts 2, 4, 16 in one set and 3, 5, 7 in the other.

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GoldMaster · 3 questions
  1. Gold8

    In a class of 35, 20 study art and 18 study music. Every student studies at least one. How many study both?

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  2. Gold9

    In a survey of 50 people, 28 like tea, 25 like coffee and 8 like neither. How many like both?

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  3. Gold10

    Explain why the intersection of 'multiples of 5' and 'odd numbers' below 30 contains exactly 3 elements.

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4 · Mastery checkpoint & reflection

  • I can sort data with a clear rule.
  • I can complete Venn and Carroll diagrams.
  • I can solve two-set problems using inclusion-exclusion.

My reflection

Week 13Pillar 4 · DataUnit 4.2

Tally Charts & Pictograms

Target mastery benchmark

Record frequencies with tallies and read or build scaled pictograms.

1 · Visual intuition

A tally is the oldest data tool in the world — five strokes, gate by gate. A pictogram is its modern cousin: one picture standing in for many.

tally = 5Group AGroup BGroup CGroup DKey: 1 icon = 5 books (half icon = 2.5 books)

A tally counter beside a scaled pictogram of books read by four reading groups, key = 5 books.

  • •Tally marks in groups of five
  • •Frequency tables
  • •Pictograms with a key (1 icon = n items)
  • •Half icons and part icons
  • •Reading totals and comparisons from a pictogram

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Reading a tally

  1. 1A tally shows two full gates and three extra strokes.
  2. 2Each gate is 5, so 2 x 5 = 10.
  3. 3Add the loose strokes: 10 + 3 = 13.
  4. 4Frequency = 13.

Worked Example 2 — Scaled pictogram with a half icon

  1. 1Key: 1 book icon = 5 books.
  2. 2Group B shows 3 full icons and 1 half icon.
  3. 3Full icons: 3 x 5 = 15 books. The half icon is half of 5, so 2.5 books.
  4. 4Total = 15 + 2.5 = 17.5 books. Choosing a key of 1 icon = 1 book would avoid the half altogether.

Think Like a Mathematician: always read the key before the picture. The same drawing can mean 5 or 50.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    A tally shows three gates and two strokes. What is the frequency?

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    Final Answer:
  2. Bronze2

    In a pictogram where 1 icon = 2 pets, a row has 6 icons. How many pets?

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  3. Bronze3

    Draw tally marks for the number 12.

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SilverPractice · 4 questions
  1. Silver4

    Key: 1 icon = 5 books. A group shows 4 icons. How many books?

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  2. Silver5

    Key: 1 icon = 10 votes. A row shows 3 icons and a half icon. How many votes?

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  3. Silver6

    Group A read 25 books and the key is 1 icon = 5 books. How many icons should be drawn?

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  4. Silver7

    Four groups read 20, 15, 35 and 10 books. How many more books did the group that read the most read than the group that read the least?

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GoldMaster · 3 questions
  1. Gold8

    A survey table records Cats 12, Dogs 18, Fish 9. Design a pictogram key that avoids awkward fractions and state the icons per row.

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  2. Gold9

    With key 1 icon = 4 items, a row shows 5 and a quarter icons. How many items?

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  3. Gold10

    A pictogram shows 7.5 icons with key 1 icon = 6 apples. Two more crates of 9 apples arrive. How many icons should the updated row show?

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4 · Mastery checkpoint & reflection

  • I can record and read tally charts.
  • I always check the key on a pictogram.
  • I can build a pictogram from a frequency table.

My reflection

Week 14Pillar 4 · DataUnit 4.3

Reading Bar Graphs

Target mastery benchmark

Read, compare and critique bar graphs including scale intervals.

1 · Visual intuition

A bar graph turns numbers into height. Your eye compares heights faster than your brain compares digits — which is exactly why a dishonest scale is so dangerous.

05101520253035MathsArtSciHistMaths25Art30Sci18Hist22same data, horizontal bars

A vertical bar chart with gridlines and a horizontal bar chart comparing the same data set.

  • •Axes, labels and titles
  • •Reading values from scaled intervals
  • •Comparing categories and finding differences
  • •Totals and aggregates from bar data
  • •Misleading scales and truncated axes

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Reading a bar between gridlines

  1. 1The axis is marked every 10 units with a faint line at every 5.
  2. 2A bar stops halfway between 30 and 40.
  3. 3Halfway is 35.
  4. 4So the category value is 35.

Worked Example 2 — Spotting a misleading graph

  1. 1A chart shows sales of 102 and 108 but the vertical axis starts at 100.
  2. 2The second bar looks three times taller than the first.
  3. 3The real difference is only 6 out of about 100 — around 6 percent.
  4. 4Redrawing with the axis starting at 0 shows two nearly identical bars.

Think Like a Mathematician: check where the vertical axis begins before you believe any comparison.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    A bar showing the number of students reaches the gridline labelled 40. How many students does the bar represent?

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  2. Bronze2

    Which category has the tallest bar if Maths = 25, Art = 30, Science = 18?

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  3. Bronze3

    The vertical axis is labelled 0, 5, 10, 15, 20, 25, … (steps of 5). A bar ends exactly on the next gridline above 20. What is its value?

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SilverPractice · 4 questions
  1. Silver4

    Maths 25, Art 30, Science 18, History 22. Find the difference between the tallest and shortest bars.

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  2. Silver5

    Using the same data, find the total across all four subjects.

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  3. Silver6

    The axis steps by 20 and a bar stops a quarter of the way between 60 and 80. What is its value?

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  4. Silver7

    Two bars show 45 and 60. Express the smaller as a fraction of the larger in simplest form.

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GoldMaster · 3 questions
  1. Gold8

    A graph starts its vertical axis at 90 and shows bars of 92 and 98. Explain why the graph is misleading.

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  2. Gold9

    Four bars total 200. Three of them are 60, 45 and 35. Find the fourth and state what fraction of the total it is.

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  3. Gold10

    A bar chart shows Red 10 and Blue 15. Every value is doubled. What are the new values, and how does the difference between the bars change?

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    Final Answer:

4 · Mastery checkpoint & reflection

  • I can read values accurately between gridlines.
  • I can calculate totals and differences from a bar graph.
  • I can spot a misleading scale.

My reflection

Week 15Pillar 4 · DataUnit 4.4

Central Tendency (Mean, Median, Mode)

Target mastery benchmark

Calculate mean, median, mode and range and choose the most suitable average.

1 · Visual intuition

If you could squash a whole data set into a single number, which number would you pick? Statisticians keep three different answers on hand — and each one tells a different truth.

before: 4, 7, 6, 3after levelling: mean = 5→

Stacks of blocks of unequal height being redistributed until every stack is level — the mean.

  • •Mean as fair-share levelling — when people say 'average' they usually mean the arithmetic mean
  • •Median as the middle ranked value
  • •Mode as the most frequent value
  • •Range as a measure of spread
  • •Working backwards from a target mean

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Mean of 4, 7, 6, 3

  1. 1Add the values: 4 + 7 + 6 + 3 = 20.
  2. 2Count the values: 4 of them.
  3. 3Divide: 20 / 4 = 5.
  4. 4The mean is 5 — every stack levelled to height 5.

Worked Example 2 — Median of 8, 3, 9, 4, 7, 6

  1. 1Order the data: 3, 4, 6, 7, 8, 9.
  2. 2There are 6 values, an even count, so the median lies between the 3rd and 4th.
  3. 3Those are 6 and 7, so median = (6 + 7) / 2 = 6.5.
  4. 4Range = 9 - 3 = 6.

Think Like a Mathematician: always order the data before finding the median. Unordered data gives a meaningless middle.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    Find the mode of 3, 5, 5, 7, 9.

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    Final Answer:
  2. Bronze2

    Find the range of 12, 4, 19, 7.

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    Final Answer:
  3. Bronze3

    Find the mean of 2, 4, 6.

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    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Find the median of 8, 3, 9, 4, 7.

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    Final Answer:
  2. Silver5

    Find the median of 10, 2, 6, 8.

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  3. Silver6

    Find the mean of 15, 21, 18, 22, 19.

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  4. Silver7

    A data set is 4, 4, 6, 10. State the mean, median, mode and range.

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GoldMaster · 3 questions
  1. Gold8

    Rina scored 78, 85 and 91 on three tests. What must she score on the fourth to average 85?

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  2. Gold9

    Five numbers have a mean of 12. Four of them are 9, 14, 11 and 16. Find the fifth.

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  3. Gold10

    A data set of 5 values has mode 7, median 7, range 6 and smallest value 4. Give one possible set.

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    Final Answer:

4 · Mastery checkpoint & reflection

  • I can calculate mean, median, mode and range.
  • I can work backwards from a target mean.
  • I can choose the most suitable average for a situation.

My reflection

Week 16Pillar 5 · FinanceUnit 5.1

Currency & Denominations

Target mastery benchmark

Recognise coin and note values and build target amounts efficiently.

1 · Visual intuition

Money is a number system you can hold. Every note and coin is a pre-packed bundle of value, and paying smartly means choosing the fewest bundles that add up exactly.

€50€10€5NOTES€2€150c20c10cCOINSFewest tokens for €8.70: €5 + €2 + €1 + 50c + 20c = €8.70

Banknote and coin denomination tokens grouped into equivalent-value combinations.

  • •Coins and banknote denominations
  • •Value equivalence between combinations
  • •Minimising the number of coins used
  • •Counting mixed money totals
  • •Digital payments as the same value without physical tokens

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Total a mixed handful of euros

  1. 1You hold one €10 note, two €2 coins and three 50c coins.
  2. 2Notes: €10.00. Coins: 2 x €2.00 = €4.00.
  3. 3Cents: 3 x €0.50 = €1.50.
  4. 4Total = €10.00 + €4.00 + €1.50 = €15.50.

Worked Example 2 — Make €8.70 with the fewest coins and notes

  1. 1Euro coins go 1c, 2c, 5c, 10c, 20c, 50c, €1 and €2; notes start at €5.
  2. 2Start with the largest that fits: a €5 note (€3.70 remaining).
  3. 3Next: a €2 coin (€1.70 remaining), then a €1 coin (€0.70 remaining).
  4. 4Then 50c (€0.20 left), then 20c (nothing left).
  5. 5Fewest tokens: €5 + €2 + €1 + 50c + 20c = 5 tokens.

Think Like a Mathematician: the greedy strategy — always take the biggest denomination that still fits — gives the fewest coins for standard currency systems.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    Add: two €5 notes and three €1 coins.

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    Final Answer:
  2. Bronze2

    How much is four 20c coins?

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    Final Answer:
  3. Bronze3

    Find the total in euros: one €10 note, one €2 coin and two 50c coins.

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    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Make €3.60 using the fewest coins (available: €2, €1, 50c, 20c, 10c).

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    Final Answer:
  2. Silver5

    Make €7.35 using the fewest notes and coins (€5, €2, €1, 50c, 20c, 10c, 5c).

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    Final Answer:
  3. Silver6

    How many 20c coins equal one €5 note?

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  4. Silver7

    Ali has three €2 coins and five 50c coins. Does he have enough for an €8.00 item?

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GoldMaster · 3 questions
  1. Gold8

    Using only 50c and 20c coins, list every way to make exactly €2.00.

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    Final Answer:
  2. Gold9

    A book costs €6.40 and you pay with a €10 note. The cashier only has €1, 50c and 20c coins. What is the fewest number of coins the cashier can give as change, and which coins?

    Working Space & Method:
    Final Answer:
  3. Gold10

    A vending machine accepts only €1 coins and 50c coins and a drink costs €4.50. In how many ways can it be paid exactly?

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can total mixed coins and notes.
  • I can make an amount using the fewest tokens.
  • I can list combinations systematically.

My reflection

Week 17Pillar 5 · FinanceUnit 5.2

Simple Transactions & Change

Target mastery benchmark

Calculate totals, change and simple discounts for multi-item purchases.

1 · Visual intuition

The shopkeeper never subtracts. They count up from the price to the money you handed over — and every coin they place in your palm is a step on a number line.

VERT-X MARTBread6.45Milk3.20Apples4.99TOTAL14.64Tendered20.00CHANGE5.36

A point-of-sale receipt showing itemised prices, subtotal, tender and calculated change.

  • •Adding item prices to find a total
  • •Change as tender minus total
  • •The counting-up (shopkeeper's) method
  • •Simple discounts and offers
  • •Budget constraints

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Change from 20.00

  1. 1Items: 6.45 + 3.20 + 4.99.
  2. 2Total: 6.45 + 3.20 = 9.65; 9.65 + 4.99 = 14.64.
  3. 3Change = 20.00 - 14.64 = 5.36.
  4. 4Counting up check: 14.64 -> 14.65 (1c) -> 15.00 (35c) -> 20.00 (5.00). Total 5.36.

Worked Example 2 — A 25 percent discount

  1. 1A jacket costs 48.00 with 25 percent off.
  2. 225 percent is one quarter: 48 / 4 = 12.00 discount.
  3. 3Sale price = 48 - 12 = 36.00.
  4. 4Change from 50.00 = 14.00.

Think Like a Mathematician: counting up in friendly jumps (to the next 10 cents, then the next dollar, then the next ten) makes mental change calculation nearly error-free.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    An item costs 3.50 and you pay with 5.00. What is the change?

    Working Space & Method:
    Final Answer:
  2. Bronze2

    An item costs 7.25 and you pay with 10.00. What is the change?

    Working Space & Method:
    Final Answer:
  3. Bronze3

    Two items cost 2.40 and 1.30. What is the total?

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Basket: 4.85 + 2.60 + 6.15. Paid with 20.00. Find the change.

    Working Space & Method:
    Final Answer:
  2. Silver5

    A 30.00 jacket has 10 percent off. Find the sale price and the change from 50.00.

    Working Space & Method:
    Final Answer:
  3. Silver6

    Three notebooks cost 4.20 each. Find the total and the change from 20.00.

    Working Space & Method:
    Final Answer:
  4. Silver7

    Use the counting-up method to find the change from 50.00 for a bill of 36.85.

    Working Space & Method:
    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    You have 25.00. Pens cost 3.50 and notebooks 4.25. Buy at least one of each and spend as much as possible without exceeding the budget. What do you buy?

    Working Space & Method:
    Final Answer:
  2. Gold9

    A shop offers 'buy 2 get 1 free' on 6.00 books. What is the cost of 7 books?

    Working Space & Method:
    Final Answer:
  3. Gold10

    A bill of 84.00 is split equally among 6 friends, then each adds a 2.50 tip. What does each pay, and what is the total handed over?

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can total a basket accurately.
  • I can calculate change by counting up.
  • I can apply simple discounts and budgets.

My reflection

Week 18Pillar 5 · FinanceUnit 5.3 & 5.4

Needs, Wants & Saving Goals

Target mastery benchmark

Distinguish needs from wants and plan a time-bound savings goal.

1 · Visual intuition

Every ringgit, dollar or rupee you spend is a ringgit you cannot spend on something else. That invisible second choice is called opportunity cost — and grown-up money decisions live or die by it.

Needs50%Wants30%Savings20%

Three allocation jars labelled Needs 50 percent, Wants 30 percent and Savings 20 percent.

  • •Needs versus wants
  • •Scarcity and trade-offs
  • •Trade-off: choosing one thing means giving up another (e.g. spending on a toy instead of saving)
  • •Opportunity cost: the next-best option you give up (e.g. a cinema ticket costs you the book you could have bought)
  • •Delayed gratification
  • •SMART savings goals: specific, measurable, time-bound
  • •The 50 / 30 / 20 allocation jars

My notes

Use this space to take notes while your teacher explains the theory and worked examples.

2 · Core theory & worked examples

Worked Example 1 — Allocating 40 pocket money

  1. 1Needs jar 50 percent: 40 / 2 = 20.
  2. 2Wants jar 30 percent: 40 x 3 / 10 = 12.
  3. 3Savings jar 20 percent: 40 / 5 = 8.
  4. 4Check: 20 + 12 + 8 = 40. The whole amount is allocated.

Worked Example 2 — Planning a savings goal

  1. 1A scooter costs 96 and 8 is saved each week.
  2. 296 / 8 = 12 weeks to reach the goal.
  3. 3Milestone check at week 6: 6 x 8 = 48, exactly halfway.
  4. 4If weekly saving rises to 12, the time drops to 96 / 12 = 8 weeks.

Think Like a Mathematician: a savings goal is just a division problem wearing a life-decision costume — target divided by rate equals time.

3 · In-class guided practice (10 questions)

BronzeExplore · 3 questions
  1. Bronze1

    Classify as need or want: drinking water.

    Working Space & Method:
    Final Answer:
  2. Bronze2

    Classify as need or want: a new video game.

    Working Space & Method:
    Final Answer:
  3. Bronze3

    From this list, pick the needs: school shoes, cinema ticket, medicine, sweets.

    Working Space & Method:
    Final Answer:
SilverPractice · 4 questions
  1. Silver4

    Pocket money is 30 a week. Using the 50/30/20 guideline (50% needs, 30% wants, 20% savings), how much goes into each jar?

    Working Space & Method:
    Final Answer:
  2. Silver5

    A bike costs 144 and you save 12 a week. How many weeks to reach the goal?

    Working Space & Method:
    Final Answer:
  3. Silver6

    You save 5 a week for 8 weeks then 9 a week for 4 weeks. What is the total saved?

    Working Space & Method:
    Final Answer:
  4. Silver7

    A goal of 200 must be reached in 10 weeks. What is the required weekly saving?

    Working Space & Method:
    Final Answer:
GoldMaster · 3 questions
  1. Gold8

    You have 60. A concert ticket costs 45 and a pair of shoes you need costs 50. Explain the trade-off and state the opportunity cost of buying the ticket.

    Working Space & Method:
    Final Answer:
  2. Gold9

    A tablet costs 480. You save 25 a week but spend 5 a week on snacks from the same money. How long until you can buy it?

    Working Space & Method:
    Final Answer:
  3. Gold10

    Design a 12-week roadmap to save 360 with weekly milestones and one contingency if a week is missed.

    Working Space & Method:
    Final Answer:

4 · Mastery checkpoint & reflection

  • I can tell needs from wants and explain why.
  • I can split an income into needs, wants and savings.
  • I can plan a time-bound savings goal with milestones.

My reflection

Answer Key & Worked Solutions

All 180 practice questions across the 18 weekly units, with justifications.

Week 1 · 1.1 Place Value & Number Sense

  1. 1.In the number 6,738, what is the value of the digit 7?
    700 — the 7 sits in the hundreds column, so 7 x 100 = 700.
  2. 2.Write 5,000 + 200 + 60 + 1 in standard form.
    5,261
  3. 3.A base-10 diagram shows 2 thousand-cubes, 3 hundred-flats, 0 ten-rods and 6 unit-cubes. What number is modelled?
    2,306 — 2,000 + 300 + 0 + 6.
  4. 4.Write 7,045 in expanded notation, showing the exact value of each non-zero digit by its place value.
    7,045 = 7,000 + 40 + 5. The 7 is worth 7 thousands (7,000), the 4 is worth 4 tens (40) and the 5 is worth 5 units (5). The 0 holds the hundreds place.
  5. 5.Order these from smallest to largest: 4,092 4,920 4,029 4,209.
    4,029 < 4,092 < 4,209 < 4,920
  6. 6.Using the digits 8, 0, 4 and 9 exactly once each, write the largest possible 4-digit number and the smallest possible 4-digit number.
    Largest 9,840. Smallest 4,089 (0 cannot lead, so the smallest non-zero digit 4 goes first).
  7. 7.By how much does 3,478 increase when the 4 and the 7 swap places to give 3,748?
    3,748 - 3,478 = 270 increase.
  8. 8.A bank safe uses a 4-digit code. The thousands digit is 8, the hundreds digit is 0, the tens digit is half of the thousands digit, and the units digit is 1 less than the tens digit. What is the code?
    Thousands = 8, hundreds = 0, tens = 8 / 2 = 4, units = 4 - 1 = 3. The code is 8,043.
  9. 9.I am a 4-digit number. My thousands digit is 6, my hundreds digit is half of my thousands digit, my tens digit is 0, and all my digits add to 12. What number am I?
    6,303 — 6 + 3 + 0 + 3 = 12.
  10. 10.Using the digits 5, 1, 7 and 2 exactly once each, build the 4-digit number closest to 5,000. Justify your answer.
    5,127. Only numbers starting with 5 can be near 5,000; the smallest tail from {1,2,7} is 127, giving a gap of 127. Any 2,000s or 1,000s number is more than 2,000 away.

Week 2 · 1.2 Core Operations (Part 1: Addition & Subtraction)

  1. 1.Calculate 47 + 38.
    85
  2. 2.Calculate 62 - 27.
    35
  3. 3.Use an open number line to work out 58 + 26. Start at 58, jump the tens first (+20), then jump the units (+6).
    58 + 20 = 78, then 78 + 6 = 84.
  4. 4.Calculate 502 - 278.
    224 (borrow across the zero tens column).
  5. 5.Calculate 1,405 + 2,867.
    4,272
  6. 6.Calculate 3,000 - 1,246.
    1,754
  7. 7.A library had 1,208 books. It received 396 new books and lent out 587. How many books remain on the shelves?
    1,208 + 396 = 1,604; 1,604 - 587 = 1,017 books.
  8. 8.Find all possible pairs of digits A and B that make this true: 4A7 + 2B6 = 733.
    Units 7 + 6 = 13, write 3 carry 1. Hundreds 4 + 2 = 6, so the tens must carry 1 into the hundreds: A + B + 1 = 13, so A + B = 12. All pairs: (A, B) = (3, 9), (4, 8), (5, 7), (6, 6), (7, 5), (8, 4), (9, 3) — seven pairs.
  9. 9.In the puzzle AB - BA = 27, where A and B are digits, find one possible pair A and B.
    AB - BA = 9(A - B) = 27, so A - B = 3. Examples: 41 - 14 = 27, 52 - 25 = 27, 96 - 69 = 27.
  10. 10.Anya thinks of a 3-digit number, adds 285, subtracts 176 and gets 1,009. What was her number? Show the inverse chain.
    1,009 + 176 = 1,185; 1,185 - 285 = 900. Her number was 900.

Week 3 · 1.2 Core Operations (Part 2: Multiplication & Division)

  1. 1.Draw an array for 3 x 6 and state the product.
    3 rows of 6 dots = 18.
  2. 2.Write 5 + 5 + 5 + 5 as a multiplication and solve.
    4 x 5 = 20
  3. 3.Share 12 stickers equally among 4 children. How many each?
    12 / 4 = 3 stickers each.
  4. 4.Calculate 34 x 6.
    204 — (30 x 6) + (4 x 6) = 180 + 24.
  5. 5.Calculate 87 / 4, giving the remainder.
    21 remainder 3 (4 x 21 = 84, 87 - 84 = 3).
  6. 6.Calculate 156 / 7.
    22 remainder 2 (7 x 22 = 154).
  7. 7.A crate holds 8 bottles. How many full crates can be packed from 150 bottles, and how many bottles are left?
    18 full crates (8 x 18 = 144) with 6 bottles left over.
  8. 8.A school needs to transport 137 students in buses that hold 30 students each. What is the minimum number of buses needed so that all students can fit?
    137 / 30 = 4 remainder 17, so 5 buses are needed — the remaining 17 students still need a bus.
  9. 9.A rectangular garden has 96 plants arranged in equal rows. If there are more than 6 rows and fewer than 12 rows, list all possible arrangements.
    The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96. Only 8 lies strictly between 6 and 12, so the arrangement is 8 rows of 12 plants.
  10. 10.When a number is divided by 6 the answer is 14 remainder 5. What is the number? What is the same number divided by 7?
    6 x 14 + 5 = 89. Then 89 / 7 = 12 remainder 5.

Week 4 · 1.3 Fractions & Decimals Introduction

  1. 1.A circle is divided into 4 equal parts and 3 are shaded. Write the fraction shaded.
    3/4
  2. 2.Shade 5/8 of an eight-part strip. What fraction is unshaded?
    3/8 unshaded.
  3. 3.Write one half as a fraction, then as a decimal.
    1/2 = 0.5
  4. 4.Simplify 6/8 fully.
    3/4 (divide both by 2).
  5. 5.Convert 7/10 and 43/100 into decimals.
    0.7 and 0.43
  6. 6.Which is larger, 2/5 or 0.5? Explain.
    0.5 is larger. 2/5 = 4/10 = 0.4, and 0.4 < 0.5.
  7. 7.Write 0.25 as a fraction in simplest form.
    25/100 = 1/4
  8. 8.Place 1/3, 0.3, 2/5 and 0.35 in order from smallest to largest.
    0.3 < 1/3 (0.333...) < 0.35 < 2/5 (0.4).
  9. 9.A jug holds 1 litre. Maya pours out 0.4 L then 1/4 L. How much is left, as a decimal and as a fraction?
    0.4 + 0.25 = 0.65 poured; 1 - 0.65 = 0.35 L left, which is 35/100 = 7/20.
  10. 10.Find a fraction that lies exactly halfway between 1/4 and 1/2 on the number line.
    1/4 = 2/8 and 1/2 = 4/8, so halfway is 3/8 (= 0.375).

Week 5 · 2.1 Patterns & Simple Sequences

  1. 1.Write the next two terms: 2, 5, 8, 11, …
    14, 17 (rule: add 3).
  2. 2.Write the next two terms: 40, 35, 30, 25, …
    20, 15 (rule: subtract 5).
  3. 3.A pattern repeats: red, blue, blue, red, blue, blue, … What is the 9th colour?
    Blue — the core unit is 3 long, and 9 is a multiple of 3, so it is the last colour of the unit, blue.
  4. 4.Find the missing term: 7, …, 19, 25, 31.
    13 (constant difference +6).
  5. 5.State the rule for 6, 12, 18, 24 and give the 10th term.
    Rule: multiply the position by 6. 10th term = 6 x 10 = 60.
  6. 6.A matchstick pattern uses 4 sticks for 1 square, 7 for 2 squares, 10 for 3 squares. How many for 8 squares?
    Rule: multiply the number of squares by 3, then add 1. For 8 squares: 3 x 8 + 1 = 25 matchsticks.
  7. 7.Which term of the sequence 3, 7, 11, 15, … is equal to 43?
    Rule: multiply the position by 4, then subtract 1. Work backwards: 43 + 1 = 44, then 44 / 4 = 11. It is the 11th term.
  8. 8.A block staircase uses 1, 3, 6, 10 blocks for stages 1 to 4. How many blocks at stage 7?
    These are triangular numbers: each stage adds one more block than the last. Stage 5 = 15, 6 = 21, 7 = 28 blocks.
  9. 9.Find the 20th term of the number sequence 1, 4, 7, 10, … and explain your rule.
    Rule: multiply the position by 3, then subtract 2. 20th term = 3 x 20 - 2 = 58.
  10. 10.Two sequences start together: A = 2, 5, 8, 11, … and B = 3, 7, 11, 15, … What is the next number they share after 11?
    A goes up in 3s and B goes up in 4s, so shared numbers repeat every 12 (3 x 4). Common values are 1 less than a multiple of 12: 11, 23, 35. The next shared number is 23.

Week 6 · 2.2 Equality & The Balance Scale

  1. 1.True or false: 9 + 6 = 15 + 0.
    True — both sides equal 15.
  2. 2.Insert <, > or =: 7 x 3 __ 25.
    21 < 25, so the symbol is <.
  3. 3.A pan holds 6 kg and balances a pan with 2 kg and a box. What is the mass of the box?
    4 kg.
  4. 4.Solve for Δ: Δ + 8 = 20.
    Δ = 12.
  5. 5.Two identical circles balance 18 kg. What is the mass of one circle?
    9 kg (18 / 2).
  6. 6.Complete so the scale balances: 24 + 13 = 30 + __.
    7 — the left pan totals 37, so the right pan needs 37 - 30 = 7.
  7. 7.Solve: 3 x □ = 21, then find □ + 9.
    □ = 7, so □ + 9 = 16.
  8. 8.One ■ plus two ● balance 26 kg. One ■ balances two ●. Find the mass of each.
    Replace the ■ with 2 ●: 4 ● = 26 kg, so one ● weighs 6.5 kg and the ■ weighs 13 kg.
  9. 9.Scale A: ★ + ★ = 14. Scale B: ★ + ☾ = 19. Find ☾.
    ★ = 7, so ☾ = 19 - 7 = 12.
  10. 10.Three ⬢ and 5 kg balance one ⬢ and 17 kg. Find the mass of one ⬢.
    Remove one ⬢ and 5 kg from both sides: 2 ⬢ = 12 kg, so one ⬢ is 6 kg.

Week 7 · 2.3 Missing Numbers (Introduction to x)

  1. 1.Solve: □ + 9 = 21.
    □ = 12
  2. 2.Solve: 15 - □ = 9.
    □ = 6
  3. 3.A function machine adds 7. If the output is 20, what was the input?
    13
  4. 4.Solve: x + 18 = 45.
    x = 27
  5. 5.Solve: 6x = 54.
    x = 9
  6. 6.Solve: x / 4 = 8.
    x = 32
  7. 7.Solve: 3x + 2 = 17.
    3x = 15, so x = 5.
  8. 8.Write an equation and solve: 'I think of a number, multiply by 5 and get 65.'
    5x = 65, x = 13.
  9. 9.Sara has x marbles. Ben has 12 more than Sara. Together they have 46. Write an equation and find x.
    x + (x + 12) = 46, so 2x = 34 and x = 17. Sara 17, Ben 29.
  10. 10.A function machine multiplies by 3 then subtracts 4. The output is 26. Find the input.
    3x - 4 = 26, so 3x = 30 and x = 10.

Week 8 · 3.1 2D & 3D Geometric Entities

  1. 1.How many faces, edges and vertices does a cube have?
    6 faces, 12 edges, 8 vertices.
  2. 2.Name the 2D shape with 5 straight sides.
    Pentagon.
  3. 3.A cylinder has flat faces and curved surfaces. How many flat faces does it have, and how many curved surfaces?
    2 flat faces (the circles at each end) and 1 curved surface — 3 surfaces in total.
  4. 4.A net has 6 identical squares in a cross shape. Which solid does it fold into?
    A cube.
  5. 5.How many vertices does a square pyramid have?
    5 — four base corners plus the apex.
  6. 6.Which solid has 2 pentagonal faces and 5 rectangular faces?
    A pentagonal prism (7 faces, 15 edges, 10 vertices).
  7. 7.Sketch the net of a triangular prism and label the shapes used.
    2 triangles and 3 rectangles, with the rectangles joined edge to edge in a strip.
  8. 8.Verify Euler's formula for a hexagonal prism.
    F = 8, E = 18, V = 12. 8 - 18 + 12 = 2. Verified.
  9. 9.A polyhedron has 9 faces and 16 edges. How many vertices must it have?
    F - E + V = 2 gives 9 - 16 + V = 2, so V = 9.
  10. 10.Two identical cubes are glued face to face. How many faces, edges and vertices does the new solid have?
    The result is a cuboid: 6 faces, 12 edges, 8 vertices (the joined faces disappear inside).

Week 9 · 3.2 Measurement (Length, Mass, Volume)

  1. 1.A pencil measures from 0 cm to 12 cm on a ruler. How long is it in mm?
    120 mm.
  2. 2.A beaker is marked every 100 mL. The water reaches the 4th mark. How much water is there?
    400 mL.
  3. 3.Which unit is best for the mass of a school bag: g, kg or mL?
    kg.
  4. 4.Convert 2.75 km into metres.
    2,750 m.
  5. 5.Convert 4,300 g into kilograms.
    4.3 kg.
  6. 6.Convert 1.6 L into millilitres, then subtract 250 mL.
    1,600 mL - 250 mL = 1,350 mL.
  7. 7.Add 1 m 45 cm and 2 m 80 cm. Give the answer in metres.
    145 cm + 280 cm = 425 cm = 4.25 m.
  8. 8.A tank holds 12 L. It is filled using a 750 mL jug. How many full jugs are required?
    12 L = 12,000 mL. 12,000 / 750 = 16 jugs.
  9. 9.A ribbon 3.5 m long is cut into pieces of 40 cm. How many pieces are cut, and how much ribbon is left over?
    350 / 40 = 8 pieces with 30 cm left over.
  10. 10.A parcel weighs 2 kg 350 g. Four identical parcels are posted. What is the total mass in kilograms?
    2,350 g x 4 = 9,400 g = 9.4 kg.

Week 10 · 3.3 Perimeter & 2D Area

  1. 1.A shape on a grid covers 14 unit squares. What is its area?
    14 square units.
  2. 2.Find the perimeter of a 6 cm by 3 cm rectangle.
    2 x (6 + 3) = 18 cm.
  3. 3.Find the area of a square with side 7 cm.
    49 square cm.
  4. 4.A rectangle has area 48 square cm and length 8 cm. Find its width and perimeter.
    Width 6 cm; perimeter 2 x (8 + 6) = 28 cm.
  5. 5.An L-shape is made of a 10 cm x 5 cm rectangle and a 6 cm x 3 cm rectangle. Find the total area.
    50 + 18 = 68 square cm.
  6. 6.A 12 m by 9 m garden has a 3 m by 3 m pond. What area of grass remains?
    108 - 9 = 99 square m.
  7. 7.The perimeter of a rectangle is 30 cm and its length is 9 cm. Find the width and area.
    9 + w = 15, so w = 6 cm; area = 54 square cm.
  8. 8.A farmer has 24 m of fencing. What whole-number rectangle dimensions give the greatest area?
    Length + width = 12. Options: 1x11=11, 2x10=20, 3x9=27, 4x8=32, 5x7=35, 6x6=36. The 6 m by 6 m square gives the maximum area 36 square m.
  9. 9.Two rectangles both have area 36 square cm. One is 4 x 9 and the other is 6 x 6. Which has the smaller perimeter?
    4 x 9 gives 26 cm; 6 x 6 gives 24 cm. The square has the smaller perimeter.
  10. 10.A path 1 m wide runs around the outside of a 5 m by 4 m pool. Find the area of the path.
    Outer rectangle 7 x 6 = 42; pool 5 x 4 = 20; path = 42 - 20 = 22 square m.

Week 11 · 3.4 Transformations & Symmetry

  1. 1.How many lines of symmetry does a square have?
    4 — two diagonals and two through the midpoints of opposite sides.
  2. 2.How many lines of symmetry does the capital letter H have?
    2 (one vertical, one horizontal).
  3. 3.Name the transformation: a shape slides 4 squares right without turning.
    A translation.
  4. 4.Reflect the point (3, 5) across the vertical line x = 6. Give the image.
    (9, 5).
  5. 5.Translate the point (2, 7) by 5 units to the right and 3 units down.
    (7, 4).
  6. 6.Rotate the point (4, 0) by 90 degrees clockwise about the origin.
    (0, -4).
  7. 7.An equilateral triangle is rotated about its centre. State its order of rotational symmetry.
    Order 3 (matches at 120, 240 and 360 degrees).
  8. 8.A regular hexagon emblem is used as a logo. State its lines of symmetry and rotational order.
    6 lines of symmetry and rotational symmetry of order 6.
  9. 9.On squared paper, a flag is flipped over a vertical mirror line, then flipped again over a horizontal mirror line. What single move gives the same result?
    A half turn (rotation of 180 degrees) about the point where the two mirror lines cross.
  10. 10.A parallelogram (not a rectangle) has how many lines of symmetry and what rotational order?
    0 lines of symmetry, rotational symmetry of order 2.

Week 12 · 4.1 Sorting & Set Classification

  1. 1.Sort 3, 8, 12, 15 into 'Odd' and 'Even'.
    Odd: 3, 15. Even: 8, 12.
  2. 2.Place 24 in a Venn diagram of 'Even' and 'Greater than 20'. Which region?
    The intersection — it is both even and greater than 20.
  3. 3.Which numbers from 1 to 10 belong outside both circles of 'Even' and 'Greater than 6'?
    1, 3, 5.
  4. 4.Build a Carroll diagram for numbers 1-20 using 'multiple of 3' and 'even'. List the both cell.
    Both: 6, 12, 18.
  5. 5.In a Venn of 'multiples of 4' and 'multiples of 6' up to 30, list the intersection.
    12 and 24 (multiples of 12).
  6. 6.20 pets: 12 are dogs, 5 are dogs that are brown, 9 are brown in total. How many are neither dogs nor brown?
    Dogs only 7, brown only 4, both 5 => 16 covered, so 4 are neither.
  7. 7.Give a sorting rule that puts 2, 4, 16 in one set and 3, 5, 7 in the other.
    Even versus odd (2, 4, 16 are even).
  8. 8.In a class of 35, 20 study art and 18 study music. Every student studies at least one. How many study both?
    20 + 18 - both = 35, so both = 3.
  9. 9.In a survey of 50 people, 28 like tea, 25 like coffee and 8 like neither. How many like both?
    At least one = 42. 28 + 25 - both = 42, so both = 11.
  10. 10.Explain why the intersection of 'multiples of 5' and 'odd numbers' below 30 contains exactly 3 elements.
    Odd multiples of 5 below 30 are 5, 15, 25 — exactly 3 elements.

Week 13 · 4.2 Tally Charts & Pictograms

  1. 1.A tally shows three gates and two strokes. What is the frequency?
    17 (3 x 5 + 2).
  2. 2.In a pictogram where 1 icon = 2 pets, a row has 6 icons. How many pets?
    6 x 2 = 12 pets.
  3. 3.Draw tally marks for the number 12.
    Two gates and two single strokes.
  4. 4.Key: 1 icon = 5 books. A group shows 4 icons. How many books?
    20 books.
  5. 5.Key: 1 icon = 10 votes. A row shows 3 icons and a half icon. How many votes?
    35 votes.
  6. 6.Group A read 25 books and the key is 1 icon = 5 books. How many icons should be drawn?
    5 icons.
  7. 7.Four groups read 20, 15, 35 and 10 books. How many more books did the group that read the most read than the group that read the least?
    Most = 35 (Group 3), least = 10 (Group 4). 35 - 10 = 25 more books.
  8. 8.A survey table records Cats 12, Dogs 18, Fish 9. Design a pictogram key that avoids awkward fractions and state the icons per row.
    Key 1 icon = 3 pets: Cats 4 icons, Dogs 6 icons, Fish 3 icons — all whole icons.
  9. 9.With key 1 icon = 4 items, a row shows 5 and a quarter icons. How many items?
    5 x 4 + 1 = 21 items.
  10. 10.A pictogram shows 7.5 icons with key 1 icon = 6 apples. Two more crates of 9 apples arrive. How many icons should the updated row show?
    7.5 x 6 = 45 apples; plus 18 = 63 apples; 63 / 6 = 10.5 icons.

Week 14 · 4.3 Reading Bar Graphs

  1. 1.A bar showing the number of students reaches the gridline labelled 40. How many students does the bar represent?
    40 students.
  2. 2.Which category has the tallest bar if Maths = 25, Art = 30, Science = 18?
    Art with 30.
  3. 3.The vertical axis is labelled 0, 5, 10, 15, 20, 25, … (steps of 5). A bar ends exactly on the next gridline above 20. What is its value?
    25 (20 + one step of 5).
  4. 4.Maths 25, Art 30, Science 18, History 22. Find the difference between the tallest and shortest bars.
    30 - 18 = 12.
  5. 5.Using the same data, find the total across all four subjects.
    25 + 30 + 18 + 22 = 95.
  6. 6.The axis steps by 20 and a bar stops a quarter of the way between 60 and 80. What is its value?
    65.
  7. 7.Two bars show 45 and 60. Express the smaller as a fraction of the larger in simplest form.
    45/60 = 3/4.
  8. 8.A graph starts its vertical axis at 90 and shows bars of 92 and 98. Explain why the graph is misleading.
    The truncated axis exaggerates a difference of 6 into a bar that looks several times taller. Starting the axis at 0 would show nearly equal bars — a difference of about 6 percent.
  9. 9.Four bars total 200. Three of them are 60, 45 and 35. Find the fourth and state what fraction of the total it is.
    200 - 140 = 60, which is 60/200 = 3/10 of the total.
  10. 10.A bar chart shows Red 10 and Blue 15. Every value is doubled. What are the new values, and how does the difference between the bars change?
    New values: Red 20, Blue 30. The difference was 15 - 10 = 5 and is now 30 - 20 = 10 — the difference doubles too.

Week 15 · 4.4 Central Tendency (Mean, Median, Mode)

  1. 1.Find the mode of 3, 5, 5, 7, 9.
    5.
  2. 2.Find the range of 12, 4, 19, 7.
    19 - 4 = 15.
  3. 3.Find the mean of 2, 4, 6.
    12 / 3 = 4.
  4. 4.Find the median of 8, 3, 9, 4, 7.
    Ordered: 3, 4, 7, 8, 9. Median = 7.
  5. 5.Find the median of 10, 2, 6, 8.
    Ordered: 2, 6, 8, 10. Median = (6 + 8) / 2 = 7.
  6. 6.Find the mean of 15, 21, 18, 22, 19.
    95 / 5 = 19.
  7. 7.A data set is 4, 4, 6, 10. State the mean, median, mode and range.
    Mean 6, median 5, mode 4, range 6.
  8. 8.Rina scored 78, 85 and 91 on three tests. What must she score on the fourth to average 85?
    Target total = 4 x 85 = 340. Current total = 254. She needs 86.
  9. 9.Five numbers have a mean of 12. Four of them are 9, 14, 11 and 16. Find the fifth.
    Total = 60; known = 50; fifth = 10.
  10. 10.A data set of 5 values has mode 7, median 7, range 6 and smallest value 4. Give one possible set.
    Smallest 4, so largest = 10. Median 7 and mode 7 need at least two 7s: {4, 7, 7, 8, 10} works (median 7, mode 7, range 6).

Week 16 · 5.1 Currency & Denominations

  1. 1.Add: two €5 notes and three €1 coins.
    €10 + €3 = €13.00.
  2. 2.How much is four 20c coins?
    €0.80 (80c).
  3. 3.Find the total in euros: one €10 note, one €2 coin and two 50c coins.
    €10 + €2 + €1 = €13.00.
  4. 4.Make €3.60 using the fewest coins (available: €2, €1, 50c, 20c, 10c).
    €2 + €1 + 50c + 10c = 4 coins.
  5. 5.Make €7.35 using the fewest notes and coins (€5, €2, €1, 50c, 20c, 10c, 5c).
    €5 + €2 + 20c + 10c + 5c = 5 tokens.
  6. 6.How many 20c coins equal one €5 note?
    25 coins.
  7. 7.Ali has three €2 coins and five 50c coins. Does he have enough for an €8.00 item?
    €6.00 + €2.50 = €8.50, so yes, with €0.50 spare.
  8. 8.Using only 50c and 20c coins, list every way to make exactly €2.00.
    Let a = 50c, b = 20c: 50a + 20b = 200. Solutions: a = 0, b = 10; a = 2, b = 5; a = 4, b = 0. Three ways.
  9. 9.A book costs €6.40 and you pay with a €10 note. The cashier only has €1, 50c and 20c coins. What is the fewest number of coins the cashier can give as change, and which coins?
    Change = €10.00 - €6.40 = €3.60. Three €1 coins leave 60c. A 50c coin would leave 10c, which cannot be made from 20c coins, so use three 20c coins instead. Fewest coins: 3 x €1 + 3 x 20c = 6 coins.
  10. 10.A vending machine accepts only €1 coins and 50c coins and a drink costs €4.50. In how many ways can it be paid exactly?
    Let a ones and b halves: a + 0.5b = 4.5, so b must be odd: (a,b) = (4,1), (3,3), (2,5), (1,7), (0,9). Five ways.

Week 17 · 5.2 Simple Transactions & Change

  1. 1.An item costs 3.50 and you pay with 5.00. What is the change?
    1.50.
  2. 2.An item costs 7.25 and you pay with 10.00. What is the change?
    2.75.
  3. 3.Two items cost 2.40 and 1.30. What is the total?
    3.70.
  4. 4.Basket: 4.85 + 2.60 + 6.15. Paid with 20.00. Find the change.
    Total 13.60; change 6.40.
  5. 5.A 30.00 jacket has 10 percent off. Find the sale price and the change from 50.00.
    Discount 3.00, sale 27.00, change 23.00.
  6. 6.Three notebooks cost 4.20 each. Find the total and the change from 20.00.
    12.60 total; change 7.40.
  7. 7.Use the counting-up method to find the change from 50.00 for a bill of 36.85.
    36.85 -> 37.00 (0.15) -> 40.00 (3.00) -> 50.00 (10.00). Change = 13.15.
  8. 8.You have 25.00. Pens cost 3.50 and notebooks 4.25. Buy at least one of each and spend as much as possible without exceeding the budget. What do you buy?
    4 pens (14.00) + 2 notebooks (8.50) = 22.50; 3 pens (10.50) + 3 notebooks (12.75) = 23.25; 2 pens (7.00) + 4 notebooks (17.00) = 24.00; 1 pen (3.50) + 5 notebooks (21.25) = 24.75 — the best is 1 pen and 5 notebooks with 0.25 left.
  9. 9.A shop offers 'buy 2 get 1 free' on 6.00 books. What is the cost of 7 books?
    Groups of 3 cost 12.00. Two groups = 6 books for 24.00, plus 1 more book at 6.00 = 30.00.
  10. 10.A bill of 84.00 is split equally among 6 friends, then each adds a 2.50 tip. What does each pay, and what is the total handed over?
    84 / 6 = 14.00 each, plus 2.50 = 16.50 each; total 6 x 16.50 = 99.00.

Week 18 · 5.3 & 5.4 Needs, Wants & Saving Goals

  1. 1.Classify as need or want: drinking water.
    Need.
  2. 2.Classify as need or want: a new video game.
    Want.
  3. 3.From this list, pick the needs: school shoes, cinema ticket, medicine, sweets.
    School shoes and medicine.
  4. 4.Pocket money is 30 a week. Using the 50/30/20 guideline (50% needs, 30% wants, 20% savings), how much goes into each jar?
    Needs 15, Wants 9, Savings 6.
  5. 5.A bike costs 144 and you save 12 a week. How many weeks to reach the goal?
    12 weeks.
  6. 6.You save 5 a week for 8 weeks then 9 a week for 4 weeks. What is the total saved?
    40 + 36 = 76.
  7. 7.A goal of 200 must be reached in 10 weeks. What is the required weekly saving?
    20 per week.
  8. 8.You have 60. A concert ticket costs 45 and a pair of shoes you need costs 50. Explain the trade-off and state the opportunity cost of buying the ticket.
    You cannot afford both (95 > 60). Buying the 45 ticket leaves 15, not enough for the needed 50 shoes. The opportunity cost of the ticket is the shoes you need — the sound decision is to buy the shoes first.
  9. 9.A tablet costs 480. You save 25 a week but spend 5 a week on snacks from the same money. How long until you can buy it?
    Net saving = 20 per week; 480 / 20 = 24 weeks. Cutting the snacks would reduce this to 480 / 25 = 19.2, so 20 weeks.
  10. 10.Design a 12-week roadmap to save 360 with weekly milestones and one contingency if a week is missed.
    30 per week; milestones at week 3 = 90, week 6 = 180, week 9 = 270, week 12 = 360. Contingency: if one week is missed, spread the shortfall of 30 across the remaining weeks (for example an extra 10 per week over the final 3 weeks) or extend by one week.
01
NUMBERS
Place Value Structure, Multi-Digit Operations, Visual Fractions & Decimals.
02
ALGEBRA
Pattern Extraction, Dynamic Balance Equivalence, Single-Step Variables (x).
03
SPACE
2D/3D Invariant Solids, Metric Measurement, Rectilinear Area & Symmetry.
04
DATA
Venn & Carroll Sets, Scaled Pictograms, Bar Graphs, Measures of Center.
05
FINANCE
Currency Minimization, Point-of-Sale Transactions, 50/30/20 Savings Logic.