Back to Apply and extend previous understandings of multiplication and division to multiply and divide rational numbers

Multiplying and Dividing Rational Numbers

Grade 7·Common Core Math - Grade 7·standard·7-ns-a-2

Work through problems with immediate feedback

Fluency Practice

$A 2-by-2 sign-rules grid showing that same-sign products are positive and different-sign products are negative, with the negative-times-negative cell highlighted.$

Compute $(-6)(-7)$ .

Compute $(-48) \div 6$ .

Compute $\left(-\frac{3}{4}\right) \times \frac{8}{9}$ . Express your answer as a fraction in simplest form.

Compute $\left(-\frac{5}{6}\right) \div \left(-\frac{1}{3}\right)$ . Express your answer as a fraction in simplest form.

Which expression is NOT equivalent to $-\frac{7}{4}$ ?

$\frac{-7}{4}$

$\frac{7}{-4}$

$\frac{-7}{-4}$

$-\left(\frac{7}{4}\right)$

Varied Practice

What is the value of $(-2)(-3)(-4)$ ?

$-24$

$24$

$6$

$-9$

Use long division to convert $\frac{5}{11}$ to a decimal. The digits that repeat form a block; write that repeating block: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . This decimal ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ (write 'terminates' or 'repeats').

repeating block:

terminates or repeats:

The temperature drops 4.5 degrees per hour. After 3 hours, what is the total temperature change, and what does the sign of the result mean?

$-13.5$ degrees; the temperature decreased by 13.5 degrees

$13.5$ degrees; the temperature increased by 13.5 degrees

$-7.5$ degrees; the temperature decreased by 7.5 degrees

$-13.5$ degrees; the temperature is currently at $-13.5$ degrees

Compute $\left(-\frac{5}{6}\right) \times 18$ . Look for a way to simplify before multiplying.

Word Problems

A stock loses $6 in value each day during a 5-day trading slump.

What is the total change in the stock's value over the 5 days? Express your answer as an integer.

A submarine descends at a steady rate of $\frac{3}{4}$ meter per second.

What is the submarine's position, in meters, relative to its starting point after 8 seconds?

During a cold snap, the temperature drops 2.5 degrees Celsius per hour.

What is the total temperature change after 4 hours? Express as a decimal.

A total temperature drop of 10 degrees Celsius occurs at a slower rate of $-1.25$ degrees per hour. How many hours does this take? Express as a decimal.

Zara measures a soil temperature change of $-\frac{7}{8}$ degree Celsius during an experiment.

Convert this measurement to a decimal. Express your answer to the nearest thousandth.

Error Analysis

$Two-column error contrast card. Left (yellow): student incorrectly writes (−5)(−3) = −15, saying any negative gives a negative result, with a red X. Right (teal): correct method showing (−5)(−3) = +15 because neg × neg = positive, with a checkmark.$

Alex computed $(-5)(-3)$ :

Step 1: $(-5)(-3)$
Step 2: $= -15$

Alex said: "Any multiplication involving a negative sign gives a negative result."

What error did Alex make?

Alex applied the wrong sign rule — two negative factors produce a positive product, not negative

Alex's sign is correct; the error is arithmetic — $5 \times 3 \neq 15$

Alex should have used the distributive property before multiplying

Alex should have added the numbers instead of multiplying them

$Two-column error contrast card. Left (yellow): student incorrectly multiplies by 2/5 (no flip), getting −3/10, with a red X. Right (teal): correct method multiplies by 5/2 (reciprocal of divisor), getting −15/8, with a checkmark.$

Bella computed $\left(-\frac{3}{4}\right) \div \frac{2}{5}$ :

Step 1: Change ÷ to ×
Step 2: $\left(-\frac{3}{4}\right) \times \frac{2}{5}$
Step 3: $= -\frac{6}{20} = -\frac{3}{10}$

What error did Bella make?

Bella did not flip the divisor — the correct second fraction should be $\frac{5}{2}$ , not $\frac{2}{5}$

Bella applied the wrong sign rule — the result should be positive

Bella should have found a common denominator before dividing

Bella made an arithmetic error: $(-3) \times 2 \neq -6$

Challenge

Compute $\left(-\frac{2}{3}\right)\left(-\frac{9}{4}\right)(-2)$ . Express your answer as a fraction or integer.

Use the distributive property to explain why $(-1)(-1) = 1$ . Start with the fact that $(-1) \times (1 + (-1)) = 0$ . Show each step of your reasoning and explain what you conclude at each step.