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 =

Pattern on the negative side

Farther left always means smaller.

Read from right to left: 0, −1, −2, −3, −4, −5.

The value keeps decreasing: each move left makes the number 1 smaller.

Do not look at the digit alone: although 5 is greater than 2, −5 is smaller than −2 because −5 is farther left and farther below zero.

−5 < −4 < −3 < −2 < −1 < 0

Remember: For negative numbers, the number that looks “larger” after the negative sign is actually smaller when it is farther left.

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 · Adding and subtracting signed numbers

Understand the signs before calculating.

First decide whether a change is being added or removed. Then simplify the two signs and calculate.

Read two signsOutside sign = action
Inside sign = change

Four sign meanings

Use each button to add or remove a record from the transport card history, update the balance and simplify the signs—one step at a time.

+(+2) means +2

Record a positive top-up

The card opens with a $3 balance. Press the button to add a +$2 top-up to its transaction history.

TRANSPORT CARD HISTORY

Opening balance$3

CARD BALANCE$3

The history currently shows only the opening balance.

3 + (+2) = 3 + 2 = 5
So +(+2) = +2.

+(−2) means −2

Record a negative fare

The card opens with a $5 balance. Press the button to add a −$2 fare to its transaction history.

TRANSPORT CARD HISTORY

Opening balance$5

CARD BALANCE$5

The history currently shows only the opening balance.

5 + (−2) = 5 − 2 = 3
So +(−2) = −2.

−(+2) means −2

Remove a top-up record

First recreate the card's earlier history. Then remove the +$2 top-up record and update the balance.

TRANSPORT CARD HISTORY

Opening balance$3

CARD BALANCE$3

The card begins with an opening balance of $3.

5 − (+2) = 5 − 2 = 3
So −(+2) = −2.

−(−2) means +2

Remove a fare record

First recreate the card's earlier history. Then remove the −$2 fare record and update the balance.

TRANSPORT CARD HISTORY

Opening balance$5

CARD BALANCE$5

The card begins with an opening balance of $5.

3 − (−2) = 3 + 2 = 5
So −(−2) = +2.

The four rules

Say what the two signs mean together.

Expression	Read it as	What it means
+(+2)	DO INCREASE	INCREASE → +2
+(−2)	DO DECREASE	DECREASE → −2
−(+2)	UNDO INCREASE	DECREASE → −2
−(−2)	UNDO DECREASE	INCREASE → +2

Remember: DO keeps the direction, but UNDO reverses it. Therefore, UNDO INCREASE means DECREASE, and UNDO DECREASE means INCREASE.

Sign foundation · Question 1 of 10

3 + (+2) =

Next · Calculate from a negative number

Begin on a number line, then learn to calculate.

1. Start at the first number.

2. Read the two signs as DO or UNDO, followed by INCREASE or DECREASE.

3. Describe the change: INCREASE means the value becomes greater; DECREASE means the value becomes smaller.

4. Use the number line to see the result.

For this section: Focus only on whether the value increases or decreases. Calculation shortcuts are taught later in a separate section.

Number-line demonstrationExerciseMeaning of the signsApply

Demonstration 1 · DO INCREASE

Calculate −2 + (+5)

Start: −2.

+(+5) means DO INCREASE: increase by 5.

Result: +3.

−2 + (+5) = +3

Hint: DO INCREASE means increase the starting value by 5.

Exercise 1

Increase −3 by 7.

Hint: +(+7) means DO INCREASE, so increase by 7.

−3 + (+7) =

Demonstration 2 · DO DECREASE

Calculate −2 + (−3)

Start: −2.

+(−3) means DO DECREASE: decrease by 3.

Result: −5.

−2 + (−3) = −5

Hint: DO DECREASE means decrease the starting value by 3.

Exercise 2

Decrease −1 by 4.

Hint: +(−4) means DO DECREASE, so decrease by 4.

−1 + (−4) =

Demonstration 3 · UNDO INCREASE

Calculate −2 − (+3)

Start: −2.

−(+3) means UNDO INCREASE: UNDO INCREASE means DECREASE, so decrease by 3.

Result: −5.

−2 − (+3) = −2 − 3 = −5

Hint: UNDO INCREASE means DECREASE.

Exercise 3

Undo an increase of 4.

Hint: −(+4) means UNDO INCREASE, so decrease by 4.

−1 − (+4) =

Demonstration 4 · UNDO DECREASE

Calculate −2 − (−4)

Start: −2.

−(−4) means UNDO DECREASE: UNDO DECREASE means INCREASE, so increase by 4.

Result: +2.

−2 − (−4) = −2 + 4 = +2

Hint: UNDO DECREASE means INCREASE.

Exercise 4

Undo a decrease of 4.

Hint: −(−4) means UNDO DECREASE, so increase by 4.

−1 − (−4) =

Separate section · No number line

Calculation training

Now use number skills only. Do not count spaces on a number line.

Simplify the signsCompare the number partsChoose the signCalculate

Worked calculation skills

Use the number parts and decide the sign.

When both parts are negative

−2 − 3 = −2 + (−3) = −(2 + 3) = −5

Why can we add 2 and 3? Spending $2 gives a −$2 change. Spending another $3 gives a −$3 change. Combining the two expenses means spending $2 + $3 altogether.

−$2 + (−$3) = −($2 + $3) = −$5

The negative sign applies to the whole combined amount because both transactions are decreases. This is the distributive law in reverse: −2 − 3 = −(2 + 3).

When one part is negative and one is positive

+1 + (−1) = 0

Why do we find the difference? One positive unit cancels one negative unit. Make as many zero pairs as possible; only the extra units remain.

−5 + 2: The card owes $5. A $2 top-up repays $2 of the debt, so $3 of debt remains. The result is −3.

−2 + 5: The card owes $2. A $5 top-up first repays the $2 debt, so $3 of credit remains. The result is +3.

−5 + 2 = 2 − 5 = −3

−2 + 5 = 5 − 2 = +3

The sign is not chosen randomly. The smaller part is completely cancelled, so the sign belongs to the uncancelled units from the larger part.

After simplifying	Calculate	Choose the sign
Both parts negative	Add the number parts	Put − before the combined total
One positive and one negative	Larger number part − smaller number part	Use the sign of the uncancelled units

Why? The smaller part cancels the same amount from the larger part. Only the larger part has units left, so its sign remains.

No-number-line training · Question 1 of 5

Add numbers with different signs.

Rule: Different signs: subtract the number parts and keep the sign of the larger part.

−4 + 6 =

Checkpoint 3 mastered

You can calculate signed addition and subtraction without counting spaces on a number line.

Checkpoint 4 · 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 do two top-ups. +$3 means each top-up moves right 3.

Transport card balance$0

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

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

Number lineDo two +3 moves

Start at 0. A positive change moves right.

(+2) × (+3) = +6

Try 1

Three top-ups of $1

(+3) × (+1) =

Positive × negative

Watch two journey charges of $3

+2 means do two charges. −$3 means each charge moves left 3.

Transport card balance$0

Charge 1−$3: balance becomes −$3

Charge 2−$3: balance becomes −$6

Number lineDo two −3 moves

Start at 0. A negative change moves left.

(+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.

Number lineUndo two +3 moves

First watch the top-ups. The line moves when each +3 is undone.

(−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.

Number lineUndo two −3 moves

First watch the charges. The line moves when each −3 is undone.

(−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 4 mastered

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

Checkpoint 5 · 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 5 mastered

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

Checkpoint 6 · 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 six checkpoints mastered

You connected signed calculations, repeated movement and multiplication.