Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Why should (-1)(-1) equal 1? Seventh graders answer this not by memorizing a slogan but by insisting that the familiar properties of multiplication keep working for negative numbers. Since -1 × (1 + (-1)) must be -1 × 0 = 0, the distributive property forces (-1)(1) + (-1)(-1) = 0, and the only value that makes this true is (-1)(-1) = 1.
From there come the sign rules: a product or quotient of two numbers with the same sign is positive, and with different signs it is negative. Division works as long as the divisor is not zero, and every quotient of integers is a rational number, so -(3/5), (-3)/5 and 3/(-5) all name the same value. Students also convert fractions to decimals with long division and discover that the decimal either stops (3/8 = 0.375) or falls into a repeating pattern (2/9 = 0.222...). Contexts help: losing $4 a day for 3 days is 3 × (-4) = -12 dollars.
Some students think -3 × -4 is -12 because 'both are negative'. Relating products to repeated losses removed (taking away 3 debts of $4) shows why the answer is +12.
Writing -(3/5) as (-3)/(-5) gives a positive value. Only one negative sign belongs in the fraction, in front, in the numerator or in the denominator.
Students who stop long division after two places write 2/9 as 0.22. Continuing the division shows the remainder repeats, so the decimal never ends.
The even-or-odd count of negative signs decides the sign of a product, not a sum. -2 + -3 + -4 is negative even though there are three negatives.
Find (-2/3) × (9/4) and write the answer as a fraction and as a decimal.
Answer: (-2/3) × (9/4) = -3/2 = -1.5.
Patterns are a gentle entry: list 3 × -2, 2 × -2, 1 × -2, 0 × -2 and ask what comes next as the first factor keeps dropping. Students see the products climbing by 2 and predict -1 × -2 = 2 before you prove it with the distributive property.
For decimals, let calculators show 1/7 to many places, then do the long division by hand so students see exactly when a remainder repeats. Assessment items commonly test sign rules inside word problems and ask which fractions have repeating decimals.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 42
Two negative factors give a positive product: 6 × 7 = 42, so (-6) × (-7) = 42.
Answer: -8
The signs are different, so the quotient is negative: 48 ÷ 6 = 8, so the answer is -8.
Answer: D) (-3)/5
One negative sign in the fraction keeps it negative, so (-3)/5 = -(3/5). (-3)/(-5) and -(-3/5) are both positive 3/5.
Answer: A) 0.222... (the 2 repeats forever)
Long division of 2 by 9 gives a remainder of 2 every step, so the digit 2 repeats forever: 0.222...
Answer: -6
1.5 × 4 = 6, and the signs are different, so the product is -6.
Answer: C) Always negative
Negative factors pair up to make positives. Five negatives make two pairs plus one left over, so the product is negative.
Answer: -3/2 (also accepted: -1.5, -1 1/2)
Dividing by 1/2 is multiplying by 2: (-3/4) × 2 = -6/4 = -3/2, which is -1.5.
A full lesson with slides, activities and an exit ticket on multiplying and dividing rational numbers, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.NS.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn multiplying and dividing rational numbers into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Because the distributive property must keep working. Since -1 × (1 + -1) = 0, the products -1 and (-1)(-1) must add to 0, so (-1)(-1) has to be 1.
In simplest form, a fraction terminates only if its denominator has no prime factors other than 2 and 5. Any other prime factor, such as 3 or 7, makes the decimal repeat.