Dividing Integers Worksheets - Free Printable

We are given a flowchart with three columns: START → CHECKPOINT, CHECKPOINT ← CHECKPOINT, and CHECKPOINT → END.

We need to fill in all the empty circles by following the operations indicated on the arrows. Let’s solve each column step-by-step, starting from known values and working forward or backward as needed.

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## 🔹 LEFT COLUMN: START → CHECKPOINT (from -8 to +40)

Start: -8

1. × (-2)
→ -8 × (-2) = +16

2. × (+3)
→ 16 × 3 = +48

3. × (-3)
→ 48 × (-3) = -144

4. ÷ (+2)
→ -144 ÷ 2 = -72

5. × (+5)
→ -72 × 5 = -360

6. ÷ (-9) → leads to +40 (CHECKPOINT)
→ -360 ÷ (-9) = +40 ✔ (matches)

✔ So, left column filled:

- Start: -8
- After ×(-2): 16
- After ×(+3): 48
- After ×(-3): -144
- After ÷(+2): -72
- After ×(+5): -360
- After ÷(-9): +40 (CHECKPOINT)

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## 🔹 RIGHT COLUMN: CHECKPOINT → END (from -150 to 100)

Start at CHECKPOINT: -150

1. × (-5)
→ -150 × (-5) = +750

2. ÷ (-5)
→ 750 ÷ (-5) = -150

3. ÷ (-2)
→ -150 ÷ (-2) = +75

4. × (-4)
→ 75 × (-4) = -300

5. ÷ (-3) → leads to 100 (END)
→ -300 ÷ (-3) = +100 ✔

✔ Right column filled:

- Start: -150
- After ×(-5): 750
- After ÷(-5): -150
- After ÷(-2): 75
- After ×(-4): -300
- After ÷(-3): 100 (END)

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## 🔹 MIDDLE COLUMN: CHECKPOINT ← CHECKPOINT (from +40 to -150)

This column goes upward from +40 to -150, so we must reverse the operations.

Operations shown (from bottom to top):

- From +40: × (-5) ← Wait, no — the arrow points up, and the operation is written next to the arrow pointing up.

Let’s read carefully:

The arrow goes up from +40 to next circle, labeled × (-5) — meaning:
To get from the lower number to the upper number, you multiply by -5.

But since we’re going upward from +40, and the operation is written on the upward arrow, it means:

> Upper value = Lower value × (-5)

So if we’re going up, and want to find the next higher value, we multiply.

BUT — we are trying to go from +40 up to -150, so we need to compute each step upward using the given operations.

Wait — let’s list the operations in order from bottom to top:

From bottom (near +40) to top (near -150):

1. From +40: × (-5) → next value = 40 × (-5) = -200

2. Then ÷ (+2) → -200 ÷ 2 = -100

3. Then × (-4) → -100 × (-4) = +400

4. Then × (+3) → 400 × 3 = +1200

5. Then ÷ (-4) → 1200 ÷ (-4) = -300

6. Then ÷ (+2) → -300 ÷ 2 = -150 ✔ (reaches CHECKPOINT)

Perfect!

✔ Middle column filled (from bottom to top):

- Bottom: +40
- After ×(-5): -200
- After ÷(+2): -100
- After ×(-4): +400
- After ×(+3): +1200
- After ÷(-4): -300
- After ÷(+2): -150 (CHECKPOINT)

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## ✔ FINAL ANSWER: All Empty Circles Filled

Left Column (START → CHECKPOINT):

1. 16
2. 48
3. -144
4. -72
5. -360

Middle Column (CHECKPOINT ← CHECKPOINT, bottom to top):

1. -200
2. -100
3. +400
4. +1200
5. -300

Right Column (CHECKPOINT → END):

1. 750
2. -150
3. 75
4. -300
5. 100 (END)

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## 🧾 Summary of All Values:

| Position | Value |
|--------------------------|--------|
| Left Column | |
| After ×(-2) | 16 |
| After ×(+3) | 48 |
| After ×(-3) | -144 |
| After ÷(+2) | -72 |
| After ×(+5) | -360 |
| Middle Column | |
| After ×(-5) | -200 |
| After ÷(+2) | -100 |
| After ×(-4) | 400 |
| After ×(+3) | 1200 |
| After ÷(-4) | -300 |
| Right Column | |
| After ×(-5) | 750 |
| After ÷(-5) | -150 |
| After ÷(-2) | 75 |
| After ×(-4) | -300 |
| After ÷(-3) | 100 |

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✔ All calculations verified and consistent with the flowchart operations.

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