Operations with Integers · Lesson 6 of 6
Chapter Summary and Practice
“Review signed movement, token reasoning, multiplication, division, and expressions together.”
• Connect movement and zero pairs to signed arithmetic. • Explain multiplication and division signs instead of memorising them. • Use properties to evaluate longer integer expressions. • Model real situations and check results by inverse operations.
One connected picture of integer operations
A sign can encode direction or an opposite quantity: right versus left, gain versus loss, above versus below. Adding signed changes gives a final position; subtracting an integer adds its inverse. Multiplication repeats a signed change, while division asks for the missing factor that would produce a known total. These ideas work together in questions with an initial position and several changes.
| Idea | What to do | Quick check |
|---|---|---|
| Addition and subtraction | Add signed changes; replace subtraction by addition of the inverse. | Use a number line or zero pairs. |
| Multiplication | Multiply magnitudes; determine sign from factors. | Token placement/removal or a times-table pattern. |
| Division | Ask which factor gives the dividend. | Multiply divisor by quotient. |
| Several factors | Regroup and count negative nonzero factors. | Any zero factor makes zero. |
| Distribution | Multiply every term in the grouped sum. | Compare with evaluating brackets first. |
Problem
A lift starts at +15 m and descends 3 m per minute for 45 minutes.
- 1.Represent the repeated change as 45 × (−3) = −135 m.
- 2.Add the start: +15 + (−135) = −120 m.
- 3.The result is 120 m below the reference level.
Problem
Calculate (32 × (−18)) ÷ (−36).
- 1.First product: 32 × (−18) = −576.
- 2.Divide: −576 ÷ (−36) is positive; 36 × 16 = 576.
- 3.The expression equals +16.
Problem
If 47 − 56 + 14 − 8 + 2 − 8 + 5 = −4, find its term-by-term opposite.
- 1.Negate each original term to obtain −47 + 56 −14 + 8 −2 + 8 −5.
- 2.Negating a whole sum negates its total.
- 3.The new value is +4.
Apply, explain, and test
Check the reason for a sign before multiplying: how many factors are negative, and is any factor zero? Use parentheses to make a negative factor clear. For a longer question, write the story as an expression, calculate in stages, then interpret the sign in context. A pattern machine, a magic grid, or the “pibs” activity asks for more than arithmetic: describe a rule and test whether it always works.
The closing Terhüchü board is an optional cultural strategy-game extension played with two differently coloured sets of nine coins. Players alternate moving one coin along a line to a neighbouring vacant intersection, or jumping over an adjacent opposing coin to an empty intersection beyond it and capturing that coin. Multiple captures may follow in one move, including a change of direction. In the outer triangular corners, a coin may pass over an empty intersection. A player wins by capturing all opposing coins or blocking every legal opposing move; if a draw is unavoidable, the player with more coins wins. This spatial activity is separate from the integer sign rules.
Five factors of −1 multiply to −1. The source’s displayed five-factor line has a sign error; use the consistent even/odd pattern.
Quiz
A movement of −7 from +5 ends at
What is 7 − (−18)?
What is (−8) × (−5)?
What is (−56) ÷ (−2)?
Which equals (−4) × (2 + (−3))?
A product with four negative factors and no zero factors is
Practice Problems
- Find integers with sum −7 and ordered difference −13; verify both conditions.
- A coin moves +5, −7, +4 and −9 from 0. Find and interpret its final position.
- Explain (−4) × (−2) using zero pairs, then calculate (−9) × (−10).
- A temperature starts at 8°C and falls 5°C each hour. Write an expression and find it after four hours.
- A firm gains ₹8 per white bag and loses ₹5 per grey bag. Find the net from 3000 white and 5000 grey bags.
- Calculate (25 × (−12)) ÷ (45 × (−27)), checking the sign before simplifying.
- Find three consecutive integers whose product is −6, then another three whose product is 120.
- Construct a three-input pattern machine using at least one subtraction and describe how another student could infer its rule.
- Compare the signs of five versus six factors, all equal to −1, and explain the printed-pattern correction.
- In a 50-question challenge, each right response gives +5 and each wrong response −2. What are the largest and smallest possible totals if every question is answered?
Key Takeaways
• Signed movement and zero pairs explain addition and subtraction. • Multiplication sign rules follow from repeated placement, removal and patterns. • Division reverses multiplication and requires a nonzero divisor. • Associativity, commutativity and distribution make long expressions manageable. • An initial value must be added to a repeated signed change. • Reasoning puzzles and applications should be checked against the operation’s meaning.
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Expressions, Structure, and Number Puzzles
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