Operations with Integers

1A Quick Recap of Integers

Integers are …, −2, −1, 0, 1, 2, … — whole numbers and their opposites. A recap is: place them on a line, add as moves, subtract as add-the-opposite.

A minus sign that only means “subtract later” without a place on the line is a recap you skipped.

How it works

1. Place a few integers on a line0 in the middle.
2. Add as a move right or leftThe recap of +.
3. Subtract as adding the oppositeThe recap of −.

2Multiplication of Integers

Multiply integers with two jobs: multiply the sizes, then choose the sign. Same signs give +; different signs give −. (−3)×4 = −12; (−3)×(−4)=12.

Multiplying sizes and forgetting the sign is a half-job.

Figure. Multiplication of integers is equal jumps on the line. Three jumps of +2 land on +6; three jumps of -2 land on -6. The count of jumps is the first factor; the size and direction of each jump is the second.

How it works

1. Multiply the absolute values3×4=12.
2. Same signs → +; different → −The sign job.
3. Write the signed product−12 or 12.

3A definition is a test you can run

Integer here means you can place it and run + or × with the sign-rules. A definition that is only “negative numbers exist” is a poster.

A temperature story without a written integer is not the recap.

How it works

1. Write an integer−4, or 7.
2. Place it or compute with itThe test.
3. If you cannot place or compute, you do not yet have itRun it.

4Name the given before the unknown

The given are the two integers and the operation. The unknown is the product or the sum. Write the signs before you multiply sizes.

Dropping both minuses because they “feel like they cancel” without checking × is a scramble.

How it works

1. Copy both signed numbersThe given.
2. Name × or +The unknown’s job.
3. Then run the sign-ruleGiven first.

5One worked case is enough at this class

One product with a sign is enough. A page of (−2)×(−3) clones does not add a new sign-rule.

Ten twins of −12 are still one idea.

How it works

1. Compute one signed productThe case.
2. A second with a changed sign as a checkOptional.
3. StopThis class.

6A miss: swapping the school name for the picture

A lift-floor story is a setting. The maths is (−2)×3 if that is the writing. Riding the lift and skipping the sign-rule is the swap-miss.

Bela’s building (or any floor story) does not replace (−)×(+)=−.

How it works

1. Keep the lift as a storyA setting.
2. Write the integer jobThe maths.
3. Run the sign-rule on the writingDo not swap.

7Check by the opposite action or the opposite test

Check a product by another grouping: (−3)×4 = −(3×4). Or multiply back if you were dividing. A check that repeats the same rush is not a check.

Saying “negatives are small” is not the opposite test.

How it works

1. Get a candidate−12.
2. Rewrite as −(size product) or redo signs slowlyThe check.
3. If they disagree, mend the sign jobHonest.

8Keep the claim at this chapter, not the next

This chapter multiplies (and recaps add). It does not finish every later integer division trick or Class 8 power-play. Keep the claim at signed × and the recap of +.

Inventing (−2)^8 rules here is the next book stealing this one.

How it works

1. Stay with signed multiply and add-movesThis chapter.
2. Get a valueThe job.
3. Leave later powers for their chapterNot the next.