Negative Number Multiplication and Division Worksheet | Cazoom ... - Free Printable

Let's solve this step-by-step, starting with Section A, then moving through Section B and Section C, explaining the rules and patterns along the way.

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✔ Section A: Multiplication Grid

We are given a multiplication grid for numbers from -3 to 3. We multiply the row number by the column number.

#### Step 1: Fill in the grid

We'll go row by row:

| × | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|----|----|----|---|---|---|---|
| -3 | 9 | 6 | 3 | 0 | -3| -6| -9|
| -2 | 6 | 4 | 2 | 0 | -2| -4| -6|
| -1 | 3 | 2 | 1 | 0 | -1| -2| -3|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| 2 | -6 | -4 | -2 | 0 | 2 | 4 | 6 |
| 3 | -9 | -6 | -3 | 0 | 3 | 6 | 9 |

Let’s verify some entries:

- (-3) × (-3) = 9 → negative × negative = positive
- (-3) × 0 = 0
- (-3) × 1 = -3
- (-3) × 2 = -6
- (-3) × 3 = -9
- (-2) × (-3) = 6 → negative × negative = positive
- (-2) × 1 = -2 → negative × positive = negative
- (2) × (1) = 2 → already filled in
- (3) × (-1) = -3 → already filled in

✔ All values follow the pattern:
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative

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#### 🔍 What do you notice?

> Observation:
> When two numbers with the same sign are multiplied, the result is positive.
> When two numbers with different signs are multiplied, the result is negative.
> Any number multiplied by zero is zero.

So, fill in the sentences:

- Positive × Positive = Positive
- Negative × Positive = Negative
- Positive × Negative = Negative
- Negative × Negative = Positive

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✔ Section B: Complete the grids using the rules

We’ll use the rules above to fill in missing values.

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#### Grid 1:

| × | 8 | -3 | 7 | -4 |
|---|---|----|---|----|
| 5 | 40 | -15 | 35 | -20 |
| -6 | -48 | 18 | -42 | 24 |
| 4 | 32 | -12 | 28 | -16 |
| -11 | -88 | 33 | -77 | 44 |

Check:
- 5 × 8 = 40 ✔
- 5 × (-3) = -15 ✔
- 5 × 7 = 35 ✔
- 5 × (-4) = -20 ✔
- -6 × 8 = -48 ✔
- -6 × (-3) = 18 ✔
- -6 × 7 = -42 ✔
- -6 × (-4) = 24 ✔
- 4 × (-4) = -16 ✔
- -11 × 7 = -77 ✔
- -11 × (-4) = 44 ✔

All correct.

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#### Grid 2:

| × | -7 | 3 | 1 | -4 |
|---|----|---|---|----|
| -2 | 14 | -6 | -2 | 8 |
| 8 | -56 | 24 | 8 | -32 |
| -9 | 63 | -27 | -9 | 36 |
| 10 | -70 | 30 | 10 | -40 |

Check:
- -2 × (-7) = 14 ✔
- -2 × 3 = -6 ✔
- -2 × 1 = -2 ✔
- -2 × (-4) = 8 ✔
- 8 × (-7) = -56 ✔
- 8 × 3 = 24 ✔
- 8 × 1 = 8 ✔
- 8 × (-4) = -32 ✔
- -9 × (-7) = 63 ✔
- -9 × 3 = -27 ✔
- -9 × 1 = -9 ✔
- -9 × (-4) = 36 ✔
- 10 × (-7) = -70 ✔
- 10 × 3 = 30 ✔
- 10 × 1 = 10 ✔
- 10 × (-4) = -40 ✔

✔ All correct.

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#### Grid 3:

| × | -5 | 4 | -9 | 2 |
|---|----|---|----|---|
| -2 | 10 | -8 | 18 | -4 |
| -3 | 15 | -12 | 27 | -6 |
| -7 | 35 | -28 | 63 | -14 |
| 6 | -30 | 24 | -54 | 12 |

Check:
- -2 × (-5) = 10 ✔
- -2 × 4 = -8 ✔
- -2 × (-9) = 18 ✔
- -2 × 2 = -4 ✔
- -3 × (-5) = 15 ✔
- -3 × 4 = -12 ✔
- -3 × (-9) = 27 ✔
- -3 × 2 = -6 ✔
- -7 × (-5) = 35 ✔
- -7 × 4 = -28 ✔
- -7 × (-9) = 63 ✔
- -7 × 2 = -14 ✔
- 6 × (-5) = -30 ✔
- 6 × 4 = 24 ✔
- 6 × (-9) = -54 ✔
- 6 × 2 = 12 ✔

✔ All correct.

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✔ Section C: Work backwards — find missing row/column numbers

We need to figure out what number was multiplied in each row or column based on the results.

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#### Grid 1:

| × | -8 | 6 | 7 | -4 |
|---|----|---|---|----|
| ? | -24 | 18 | 21 | -12 |
| ? | -8 | 6 | 7 | -4 |
| ? | 56 | -42 | -49 | 28 |
| ? | -72 | 54 | 63 | -36 |

Let’s find the first row multiplier:

- First row:
Let’s say the multiplier is $ x $. Then:
- $ x × (-8) = -24 $ → $ x = (-24)/(-8) = 3 $
- Check: 3 × 6 = 18 ✔, 3 × 7 = 21 ✔, 3 × (-4) = -12 ✔ → So row = 3

Second row:
- $ x × (-8) = -8 $ → $ x = (-8)/(-8) = 1 $
- Check: 1 × 6 = 6 ✔, 1 × 7 = 7 ✔, 1 × (-4) = -4 ✔ → So row = 1

Third row:
- $ x × (-8) = 56 $ → $ x = 56 / (-8) = -7 $
- Check: -7 × 6 = -42 ✔, -7 × 7 = -49 ✔, -7 × (-4) = 28 ✔ → So row = -7

Fourth row:
- $ x × (-8) = -72 $ → $ x = (-72)/(-8) = 9 $
- Check: 9 × 6 = 54 ✔, 9 × 7 = 63 ✔, 9 × (-4) = -36 ✔ → So row = 9

✔ So the rows are: 3, 1, -7, 9

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#### Grid 2:

| × | ? | ? | ? | ? |
|---|---|---|---|---|
| -3 | -9 | 24 | -6 | 36 |
| -8 | -24 | 64 | -16 | 96 |
| -2 | -6 | 16 | -4 | 24 |
| -12 | -36 | 96 | -24 | 144 |

Let’s find the column multipliers (top row).

First column: Multiply by -3, -8, -2, -12 → so we can use any row to find the top number.

Use first row:
- First column: $ x × (-3) = -9 $ → $ x = (-9)/(-3) = 3 $
- Second column: $ x × (-3) = 24 $? No → wait, that would be $ x = 24 / (-3) = -8 $ → but that doesn't match.

Wait — better: The first column has values: -9, -24, -6, -36

Let’s find the top number (column header) for each column.

Let’s call the top numbers: $ a, b, c, d $

From first row:
- $ a × (-3) = -9 $ → $ a = 3 $
- $ b × (-3) = 24 $ → $ b = -8 $
- $ c × (-3) = -6 $ → $ c = 2 $
- $ d × (-3) = 36 $ → $ d = -12 $

Now check second row:
- $ 3 × (-8) = -24 $ ✔
- $ -8 × (-8) = 64 $ ✔
- $ 2 × (-8) = -16 $ ✔
- $ -12 × (-8) = 96 $ ✔

Third row:
- $ 3 × (-2) = -6 $ ✔
- $ -8 × (-2) = 16 $ ✔
- $ 2 × (-2) = -4 $ ✔
- $ -12 × (-2) = 24 $ ✔

Fourth row:
- $ 3 × (-12) = -36 $ ✔
- $ -8 × (-12) = 96 $ ✔
- $ 2 × (-12) = -24 $ ✔
- $ -12 × (-12) = 144 $ ✔

✔ So the top row is: 3, -8, 2, -12

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#### Grid 3:

| × | ? | ? | ? | ? |
|---|---|---|---|---|
| ? | -14 | 18 | 22 | 8 |
| ? | 63 | -81 | -99 | -36 |
| ? | 21 | -27 | -33 | -12 |
| ? | -70 | 90 | 110 | 40 |

Let’s denote the row multipliers as R1, R2, R3, R4
And column multipliers as C1, C2, C3, C4

Start with Row 1: values: -14, 18, 22, 8

Try to find a common factor.

Look at first column: -14, 63, 21, -70

Let’s try to find the column multipliers.

Suppose C1 is the first column header.

Then:
- R1 × C1 = -14
- R2 × C1 = 63
- R3 × C1 = 21
- R4 × C1 = -70

Let’s divide equations:

R2/R1 = 63 / (-14) = -9/2 → not helpful.

But let’s look at ratios between rows.

Notice:
- R2 / R1: 63 / (-14) = -4.5 → maybe not integer.

Try another approach: assume one value.

Look at second column: 18, -81, -27, 90

Try dividing:
- R1 × C2 = 18
- R2 × C2 = -81
- R3 × C2 = -27
- R4 × C2 = 90

Let’s suppose C2 = 9 → then:
- R1 = 18 / 9 = 2
- R2 = -81 / 9 = -9
- R3 = -27 / 9 = -3
- R4 = 90 / 9 = 10

Now test with other columns.

Try Column 1:
- R1 × C1 = -14 → 2 × C1 = -14 → C1 = -7
- R2 × C1 = -9 × (-7) = 63 ✔
- R3 × C1 = -3 × (-7) = 21 ✔
- R4 × C1 = 10 × (-7) = -70 ✔

Good!

Column 3: values: 22, -99, -33, 110

Test with R1 = 2 → 2 × C3 = 22 → C3 = 11
- R2 = -9 → -9 × 11 = -99 ✔
- R3 = -3 → -3 × 11 = -33 ✔
- R4 = 10 → 10 × 11 = 110 ✔

Column 4: 8, -36, -12, 40

R1 × C4 = 8 → 2 × C4 = 8 → C4 = 4
- R2 × C4 = -9 × 4 = -36 ✔
- R3 × C4 = -3 × 4 = -12 ✔
- R4 × C4 = 10 × 4 = 40 ✔

Perfect!

So:
- Row multipliers: R1 = 2, R2 = -9, R3 = -3, R4 = 10
- Column multipliers: C1 = -7, C2 = 9, C3 = 11, C4 = 4

✔ Final answer:

| × | -7 | 9 | 11 | 4 |
|---|----|---|----|---|
| 2 | -14 | 18 | 22 | 8 |
| -9 | 63 | -81 | -99 | -36 |
| -3 | 21 | -27 | -33 | -12 |
| 10 | -70 | 90 | 110 | 40 |

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✔ Summary of Answers

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Section A: Completed Grid

| × | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|----|----|----|---|---|---|---|
| -3 | 9 | 6 | 3 | 0 | -3| -6| -9|
| -2 | 6 | 4 | 2 | 0 | -2| -4| -6|
| -1 | 3 | 2 | 1 | 0 | -1| -2| -3|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| 2 | -6 | -4 | -2 | 0 | 2 | 4 | 6 |
| 3 | -9 | -6 | -3 | 0 | 3 | 6 | 9 |

What do you notice?
> The product of two numbers with the same sign is positive; with different signs is negative. Zero times anything is zero.

Complete the sentences:
- Positive × Positive = Positive
- Negative × Positive = Negative
- Positive × Negative = Negative
- Negative × Negative = Positive

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Section B: All grids completed correctly

(As shown above)

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Section C: Missing numbers

Grid 1: Rows are: 3, 1, -7, 9

Grid 2: Top row (columns): 3, -8, 2, -12

Grid 3:
- Rows: 2, -9, -3, 10
- Columns: -7, 9, 11, 4

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📌 Final Answer Key

> Rules discovered:
> - Positive × Positive = Positive
> - Negative × Negative = Positive
> - Positive × Negative = Negative
> - Negative × Positive = Negative

This rule helps us solve all parts of the worksheet.

✔ All sections solved and verified.