Multiply
Rational Numbers
Multiplication can describe HOW MUCH a signed change is SCALED or REPEATED.
Yesterday: signed numbers described position and change. Today: we scale those changes with multiplication.
What are we learning?
We are learning to multiply rational numbers fluently and use multiplication to represent signed change, scaling, and rate.
Success Criteria
Sign first. Students should be able to predict the sign before doing the arithmetic.
Retrieve what you already know
Add
−6 + 9
Subtract
5 − (−2)
Fact fluency
6 × 7
Reason about direction
A diver descends 3 m each minute for 4 minutes. Is the total change positive or negative?
“3 meters down, 4 times” can be written as 4 × (−3). Multiplication can scale a signed change.
Multiplying rational numbers is not completely new
First: think about length
If the flag is 10 ft long, the blue region is 2/5 of the length.
Then: think about area
The blue region is 7/13 of the width and 2/5 of the length.
The blue region is 14/65 of the flag’s total area.
The arithmetic rules for multiplying fractions and decimals still work. Today we add sign reasoning.
What does the multiplication represent?
Ben’s change in elevation
−1.2 m
Petra’s change
Petra’s change is 3.5 times as great as Ben’s.
3.5 × (−1.2)
Before multiplying: positive × negative → negative.
3.5 groups of −1.2
Three full groups of −1.2 make −3.6. Half of −1.2 is −0.6. Together: −4.2.
Why does negative × negative become positive?
Keep the second factor at −2
Watch the first factor decrease by 1.
Each product increases by 2: −6, −4, −2, 0, …
The pattern must continue
0 → +2 → +4
If the pattern broke at zero, multiplication would stop behaving consistently.
Negative × negative = positive.
This is not “two negatives magically cancel.” It is the sign pattern multiplication must follow.
Decide the sign before you multiply
Same signs → positive. Different signs → negative. Any factor of 0 → product 0.
Sign first. Magnitude second.
Identify the factors
Factors are the numbers being multiplied.
Predict the sign
Same, different, or zero?
Multiply the absolute values
Do the decimal or fraction arithmetic without the sign distracting you.
Attach the sign
The answer to a multiplication problem is the product.
number being multiplied
answer to multiplication
Say only the sign
Do not mix sign errors with arithmetic errors. Treat them as two separate decisions.
Free up your brain for the rational-number work
Fact fluency reduces the working-memory load when signs, fractions, and decimals are added.
Separate the sign decision from the decimal work
Students often get the digits right and lose the sign—or place the decimal based on guesswork. Require both decisions explicitly.
Decimals: predict sign → multiply magnitudes → count decimal places → attach sign.
Simplify first when you can
4 and 6 share a factor of 2.
Mixed numbers: convert to improper fractions before multiplying.
Do not multiply denominators across an addition-style common denominator rule. Multiplication uses numerator × numerator and denominator × denominator.
Predict the sign. Then choose the arithmetic method.
A: class solves together. B: students explain simplification. C: students lead.
Rate × time gives the total change
Start
+2,169.35 ft
Rate
Descending at −250.31 ft/min for 8.2 min.
2,169.35 + (−250.31)(8.2)
(−250.31)(8.2) = −2,052.542
116.808 ft
The model contains addition and multiplication. Use order of operations: rate × time before adding to the starting height.
Has the balloon reached the ground by 8.6 minutes?
The balloon starts at 2,169.35 ft and descends at 250.31 ft/min.
Before exact calculation:
250 × 8.6 ≈ 2,150
That is close to 2,169.35. What do you predict?
Exact model
2,169.35 + (−250.31)(8.6)
= 2,169.35 − 2,152.666
= 16.684 ft
No. The balloon is still 16.684 feet above the ground.
Can you separate sign reasoning from arithmetic?
A. Predict the sign
B. What arithmetic form?
C. Solve
If a student misses the sign but gets the magnitude, reteach the sign rule—not the multiplication algorithm.
Show what you know
Decimal product
(−3.4)(2.5)
Find the product and explain why the sign is correct.
Fraction product
(−3/4)(−2/5)
Simplify your answer.
Signed rate
A stock price drops $1.45 each hour for 5 hours.
What rational number represents the total change?
Practice only what matters
Error analysis: explain what Ming did wrong.
#20 is select all that apply. More than one choice may be correct.