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Lesson 5 of 6

Operations with Integers · Lesson 5 of 6

Expressions, Structure, and Number Puzzles

“Use grouping, distribution, and sign patterns to reason through longer expressions.”

Learning Objectives

• Regroup and reorder integer products correctly. • Predict the sign of a product with many negative factors. • Explain distribution over a signed sum with a token array. • Infer expression rules from input-output patterns.

Group factors efficiently

For 5 × (−3) × 4, multiplying the first two factors gives (−15) × 4 = −60. Multiplying the last two first gives 5 × (−12) = −60. Integer multiplication is associative: changing brackets among consecutive multiplications does not change the product. It is also commutative, so reordering factors can make the numbers easier to multiply.

Associative propertyLaTeX
This changes the grouping of integer factors; it does not permit arbitrary regrouping across addition or division.
Example — Pick easy factors first

Problem
Calculate 25 × (−6) × 12.

  1. 1.Regroup and reorder to multiply 25 × 12 = 300 first.
  2. 2.Then multiply 300 × (−6) = −1800.
  3. 3.The value is −1800 regardless of the chosen order.

Each multiplication by −1 reverses the sign. An even number of negative nonzero factors produces a positive product, and an odd number produces a negative one. Thus two or four copies of −1 make +1, while three or five copies make −1. If any factor is zero, the entire product is zero regardless of the other signs.

Example — Count negative factors

Problem
Find (−2) × (−1) × (−5) × (−3).

  1. 1.There are four negative factors, an even count, so the product is positive.
  2. 2.Multiply their magnitudes: 2 × 1 × 5 × 3 = 30.
  3. 3.The product is +30.

Distribute across a signed sum

The expression 4 × (2 + (−3)) means four copies of a group containing two positive and three negative tokens. You may combine each group first to get four copies of −1, or count all eight positives and twelve negatives separately. Both routes give −4. This is why multiplying a sum distributes over both its parts, even when one part or the multiplier is negative.

Four copies of (+2) + (−3)++−−−++−−−++−−−++−−−4 × (+2) = +84 × (−3) = −12+8 + (−12) = −4
Token array for the distributive property— Four rows show eight positives and twelve negatives; both grouping routes give −4.
Distributive propertyLaTeX
The letters represent any integers. Explain the grouped quantity before expanding it.
Example — Distribute a negative multiplier

Problem
Evaluate (−2) × (4 + (−3)) in two ways.

  1. 1.Inside the brackets: 4 + (−3) = 1, then (−2) × 1 = −2.
  2. 2.Distributed: (−2) × 4 + (−2) × (−3) = −8 + 6 = −2.
  3. 3.The two routes agree.

A pattern machine takes three integers and returns a value. Machine 1 follows first + second − third, so (−10, −12, −9) gives −13. Machine 2 gives −29 for (4, 8, −3) and 54 for (6, −11, 12): its rule is −(first × second) − third. Compare several rows before stating a rule, then test all of them. A later machine follows first − (second × third); for (4, 8, −3) it gives 28. The modified even/odd iteration and “pibs” totals are further invitations to form and test numerical conjectures.

Inputs (a, b, c)Machine 1: a + b − cMachine 2: −(a × b) − cLater machine: a − b × c
4, 8, −315−2928
6, −11, 12−1754138
−3, 9, −8143569
Example — Decode an input-output rule

Problem
A machine computes first + second − third. What does it output for −10, −12, −9?

  1. 1.Substitute in order: −10 + (−12) − (−9).
  2. 2.The first two sum to −22; subtracting −9 adds 9.
  3. 3.The output is −13.
Respect brackets and signs

In (−2) × (4 + (−3)), distribute −2 to both terms. In a longer expression, a single mistaken minus sign can reverse the result; estimate its sign before calculating.

Quiz

Quick check

Which is equivalent to 5 × (−3 × 4)?

Quick check

The product of five factors, each equal to −1, is

Quick check

What is (−2) × (4 + (−3))?

Quick check

What is (−7) × 4 × (−1)?

Quick check

The machine a + b − c outputs what for −10, −12, −9?

Practice Problems

Practice Problems
  1. Calculate (−5) × (18 + (−3)), showing a bracket-first and a distributed route.
  2. Find (−7) × 4 × (−1) and explain its sign before its magnitude.
  3. Find the sign and value of (−2) × (−1) × (−5) × (−3).
  4. Calculate −47 + 56 −14 + 8 −2 + 8 −5 from the given opposite expression 47 −56 +14 −8 +2 −8 +5 = −4, without recomputing every sum.
  5. Iterate the chapter’s rule “halve even integers; multiply odd integers by −3 and add 1” from −7 for several steps and describe what happens.
  6. Make 85 using ten +13 coins and five −9 coins; find another reachable total and justify it.
  7. Use 3, −2, 5 and −6 once each with +, − and × once each to make two expressions with very different values.
  8. Infer the later machine’s rule from (4,8,−3)→28 and (6,9,6)→−48; predict its output for (−16,−6,−9).
  9. Can 1568 pibs be made using nonnegative counts of +13 and −9 coins? Give a calculation or explain why not.

Key Takeaways

Key Takeaways

• Associativity changes grouping of a product; commutativity changes factor order. • With nonzero factors, an even number of negatives gives a positive product and an odd number gives a negative product. • Zero in a product makes the entire product zero. • Distribute an integer multiplier to every term in the brackets. • A pattern rule should predict every given input-output example.

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Integer Operations in Situations

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Chapter Summary and Practice