Operations with Integers · Lesson 2 of 6
Multiplication with Signed Tokens
“Build every integer multiplication sign case from adding or removing tokens.”
• Model a positive multiplier as repeated placement of tokens. • Model a negative multiplier as repeated removal using zero pairs. • Explain why two negative factors give a positive product. • Separate product sign from product magnitude.
A bag that starts empty
Multiplication of whole numbers can mean placing the same group into a bag repeatedly: four placements of two positive tokens give eight positives. To extend this idea, make each operation start from an empty bag. A positive multiplier tells us to place groups; a negative multiplier tells us to remove groups. Each group itself may contain positive or negative tokens.
Problem
Interpret 4 × (−2).
- 1.Place two negative tokens into the empty bag four times.
- 2.There are eight negative tokens.
- 3.The product is −8.
Remove a group from an empty bag
For (−4) × 2, remove two positive tokens four times. There are none to remove at first, so place two zero pairs before each removal. Each time, the positives leave and two negatives remain. After four repetitions, eight negatives are left. This agrees with taking the inverse of 4 × 2.
Problem
Interpret (−4) × 2.
- 1.At each of four rounds, add two zero pairs and remove two positive tokens.
- 2.Each round leaves two negatives, for eight negatives in total.
- 3.Thus (−4) × 2 = −8.
Now remove two *negative* tokens four times. Each round begins with two zero pairs; removing the two negative tokens leaves two positives. Four rounds leave eight positives. This explains why (−4) × (−2) = +8 rather than simply asking students to remember a sign table.
Problem
Interpret (−4) × (−2).
- 1.Add two zero pairs and remove two negative tokens per round.
- 2.Two positives remain per round, repeated four times.
- 3.The bag ends with eight positives, so the product is +8.
| Multiplier | Multiplicand | Operation | Product |
|---|---|---|---|
| +4 | +2 | Add two positives four times | +8 |
| +4 | −2 | Add two negatives four times | −8 |
| −4 | +2 | Remove two positives four times | −8 |
| −4 | −2 | Remove two negatives four times | +8 |
Different token collections can represent the same number because zero pairs may be added. Four copies of any valid representation of −2 must produce −8. This observation connects the token model to a well-defined number operation, independent of how many harmless zero pairs were drawn.
The magnitude of 4 × (−2) is 4 × 2 = 8. Determine the sign from the meanings of the two factors, then attach it to the magnitude.
Quiz
What is 4 × (−2)?
What is (−4) × (−2)?
A negative multiplier means we model multiplication by
Removing negative tokens from zero pairs leaves
What is (−7) × 3?
Practice Problems
- Draw or describe tokens for 3 × (−2).
- Use zero pairs to explain (−5) × (−2).
- Explain in words why (−4) × (−1) is +4.
- Find 4 × (−6) and 9 × (−7), stating the magnitude and sign separately.
- If 123 × 456 = 56088, find (−123) × 456 and (−123) × (−456) without repeating the full multiplication.
- Represent −2 with two different token sets containing zero pairs, then multiply each set by 4.
Key Takeaways
• Start each token multiplication with an empty bag. • A positive multiplier places groups; a negative multiplier removes them. • Zero pairs make a removal possible without changing the bag’s value. • Removing negatives leaves positives, explaining negative times negative. • Find the magnitude from the positive magnitudes, then determine the sign.