Number Line Lab

Checkpoint 1 · Number line 0 to 10

Choose a move to reach the target.

Look at Milo and the target. Decide whether to add or subtract, then enter the number of spaces.

Movement ruleAdd → right
Subtract → left

ModelTryModelTry

Watch the thinking

Milo is at 3. The target is 7.

1. The target is to the right.

2. Moving right means add.

3. Count the spaces: 4.

Try 1

Complete Milo’s command.

3 = 7

Watch the thinking

Milo is at 8. The target is 5.

The target is left of Milo. Count 3 spaces left, so subtract 3.

Try 2

Complete the movement.

8 = 5

One more model

From 2 to 9

The target is 7 spaces to the right: 2 + 7 = 9.

Try 3

What command reaches 9?

2 = 9

Final model

From 10 to 4

The target is 6 spaces to the left.

Try 4

Write the command.

10 = 4

Checkpoint 1 mastered

You can choose addition or subtraction to control a position.

Checkpoint 2 · Understanding negative numbers

Negative numbers appear in daily life.

Begin with money owed and floors below ground. Then connect those ideas to a number line.

MeaningNegative = below zero

Daily storyTryDiagramTry

Story 1 · Transport card

The card starts with $2.

TRANSPORT CARDBefore the journey

Balance: $2

💸 $3

Subtract the $3 fare− $3

$2 − $3 = −$1

What does −$1 mean?
The card has used $1 more than its balance. You owe the card company $1.

Try 1

A card has $3. A journey costs $4.

What balance will the card show?

$3 − $4 = $

Story 2 · Building floors

Milo starts on G/F, which represents 0.

3/Fthree floors above ground+3

2/Ftwo floors above ground+2

1/Fone floor above ground+1

G/Fground level0Milo

LG1one floor below ground−1

LG2two floors below ground−2

LG3three floors below ground−3

G/F represents 0. Moving up makes the number greater; moving down makes it smaller.

Try 2

Start on G/F and go down one floor.

Which number do you reach?

Now use numbers only

Milo starts at 0. Move two spaces left.

On a number line: moving left makes the number smaller. Two spaces left from 0 lands on −2.

Try 3

What number does Milo land on?

0 − 2 =

One small movement

Milo is at −2. Move one space right.

1. Right means add.

2. Move only one space.

3. −2 + 1 lands on −1.

Try 4

Where does Milo land?

−2 + 1 =

Consolidation · Question 1 of 5

Milo starts at 0 and moves 3 spaces left.

0 − 3 =

Checkpoint 2 mastered

You understand negative balances, underground floors and positions left of zero.

Checkpoint 3 · Multiplying positive and negative numbers

Use top-ups, charges and undo actions.

Learn one sign combination at a time. “Undo” gives meaning to a negative first factor.

Sign ruleSame signs +
Different signs −

Daily exampleTryNext signTry

Positive × positive

Watch two top-ups of $3

+2 means two top-ups. +$3 is the amount of each top-up.

Transport card balance$0

Top-up 1+$3: balance becomes $3

Top-up 2+$3: balance becomes $6

(+2) × (+3) = +6

Try 1

Three top-ups of $1

(+3) × (+1) =

Positive × negative

Watch two journey charges of $3

+2 means two charges. −$3 is the change from each charge.

Transport card balance$0

Charge 1−$3: balance becomes −$3

Charge 2−$3: balance becomes −$6

(+2) × (−3) = −6

Try 2

Three journey charges of $1

(+3) × (−1) =

Negative × positive

Watch two $3 top-ups, then undo both

Transport card balance$0

Top-up 1+$3

Top-up 2+$3

Total change caused by undoing$0

Start at $0. First make the two positive changes that will be reversed.

(−2) × (+3) = −6

Try 3

Undo three $1 top-ups.

(−3) × (+1) =

Negative × negative

Watch two −$3 charges, then undo both

Transport card balance$0

Charge 1−$3

Charge 2−$3

Total amount restored by undoing$0

Start at $0. First make the two negative changes that will be reversed.

(−2) × (−3) = +6

Try 4

Undo three $1 charges.

(−3) × (−1) =

Small conclusion · Sign rules

The four stories make one simple pattern.

Signs	Formula	Result
Same	(+) × (+)	(+)
Different	(+) × (−)	(−)
Different	(−) × (+)	(−)
Same	(−) × (−)	(+)

Remember: Same signs → positive. Different signs → negative.

Before the proof · A zero pair

Why does 1 + (−1) = 0?

Use one transport card. A +$1 top-up and a −$1 journey charge cancel each other.

Transport card balance$0

Top up $1+$1: balance becomes $1

Pay a $1 fare−$1: balance returns to $0

Start with a $0 balance.

1 + (−1) = 0

Zero pair: +1 and −1 are opposites, so together they make 0.

Now prove −1 × −1

The missing product must balance the equation.

(−1) × 0 = 0

(−1) × [1 + (−1)] = 0

(−1) × 1 + (−1) × (−1) = 0

−1 + (−1) × (−1) = 0

(−1) × (−1) = 1

Try 5 · Sign only

What sign should the product have?

(−4) × (−2)

Consolidation · Question 1 of 5

Multiply two positive numbers.

Two steps: Decide whether the signs are the same or different. Then multiply the number parts.

(+2) × (+3) =

Checkpoint 3 mastered

You can explain all four sign combinations using daily actions and algebra.

Checkpoint 4 · Difference between numbers

See the distance first. Then calculate without a number line.

The difference is how far apart two numbers are. It is always zero or positive.

Difference ruleLarger − smaller

SeeCountRewrite signsCalculate

Visual example 1

Difference between 2 and 6

See: the highlighted section joins 2 and 6.

Count: 4 spaces.

Calculate: 6 − 2 = 4.

Try 1 · Use the number line

Difference between 1 and 5

5 − 1 =

Visual example 2 · Across zero

Difference between −3 and 2

−3 to 0: 3 spaces.

0 to 2: 2 spaces.

Total: 3 + 2 = 5 spaces.

Try 2 · Use the number line

Difference between −2 and 3

The numbers arespaces apart.

Move from pictures to calculation

What does −(−2) mean?

1. The outside − means “take the opposite”.

2. The opposite of −2 is +2.

3. Therefore, −(−2) = +2.

−(−2) = (−1) × (−2) = +2

Shortcut: two negative signs together become a positive sign.

Try 3 · Rewrite the signs

What does −(−3) become?

A useful rearrangement

Why is −2 + 3 the same as 3 − 2?

1. Change the order: −2 + 3 = 3 + (−2)

2. Adding a negative: + (−2) means subtract 2.

3. Calculate: 3 − 2 = 1.

−2 + 3 = 3 + (−2) = 3 − 2 = 1

Remember: When adding, you may change the order. Put the positive number first, then change “+ a negative” into subtraction.

Calculation only · Across zero

Difference between −2 and 4

1. Larger − smaller: 4 − (−2)

2. Rewrite: −(−2) becomes +2

3. Calculate: 4 + 2 = 6

4 − (−2) = 4 + 2 = 6

Try 4 · No number line

Difference between −3 and 2

2 − (−3) =

Calculation only · Two negatives

Difference between −5 and −2

1. Larger number: −2 is farther right than −5.

2. Larger − smaller: −2 − (−5)

3. Rewrite and calculate: −2 + 5 = 3

−2 − (−5) = −2 + 5 = 3

Try 5 · No number line

Difference between −4 and −1

−1 − (−4) =

Consolidation · Question 1 of 10

Put the positive number first, then subtract.

Rewrite: −2 + 3 = 3 − 2

−2 + 3 =

Checkpoint 4 mastered

You can see a difference, simplify its signs and calculate it.

Checkpoint 5 · Compound repeated movement

Use two lines of repeat blocks to reach the target.

Each mission gives two movements. Put one movement on each line and choose its repeat number.

Coding ideaLine 1 movement
+ Line 2 movement

Model

Start at −1. Reach 6.

Line 1: repeat +2 twice: −1 → 1 → 3

Line 2: repeat +3 once: 3 → 6

Total change: 2 × (+2) + 1 × (+3) = +7

Result: −1 + 7 = 6

Read the two-line code

The program runs from the top line to the bottom line.

Example program

Line 1 repeat 2 times

move by +2

Line 2 repeat 1 time

move by +3

Both lines matter. Line 1 adds 4. Line 2 adds 3. Together they add 7.

Mission 1 of 5

Move Milo from −1 to 6.

Target: 6

Available movements: +2 and +3. Use one movement on each line.

Your two-line program

Line 1 repeat times

Line 2 repeat times

Milo is waiting at −1.

All five checkpoints mastered

You connected repeated movement, multiplication and signed numbers.