7th Grade · 7.NS.A.2

Multiplying integers in 7th grade

Multiplying integers follows two sign rules: a positive times a negative is negative, and a negative times a negative is positive. In 7th grade, your child uses these rules for multiplying and dividing integers, fractions and decimals. Your child also turns fractions into decimals with long division.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
7.NS.A.2.a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
7.NS.A.2.b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –( $\frac{p}{q}$ ) = ^(–p)/_q = ^p/_(–q). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.A.2.c Apply properties of operations as strategies to multiply and divide rational numbers.
7.NS.A.2.d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

What it means, in plain words

Integers are the whole numbers and their negatives. Multiplying integers with different signs gives a negative answer: $3 \times (-4) = -12$ , which is three groups of $-4$ . A negative times a negative gives a positive answer. A pattern shows why. $3 \times (-2) = -6$ , $2 \times (-2) = -4$ , $1 \times (-2) = -2$ and $0 \times (-2) = 0$ . Each answer goes up by 2, so the next one, $(-1) \times (-2)$ , is 2. In class, children also see that this is the only rule that keeps the distributive property true. That property says, for example, that $2 \times (3 + 4) = 2 \times 3 + 2 \times 4$ .

Dividing integers uses the same sign rules. Same signs give a positive answer, and different signs give a negative answer: $(-20) \div 4 = -5$ and $(-20) \div (-4) = 5$ . Dividing by zero has no answer. A negative fraction can carry its minus sign in three places, so $-\frac{3}{4}$ , $\frac{-3}{4}$ and $\frac{3}{-4}$ are the same number. With several factors, count the negatives: an even count gives a positive product, and an odd count gives a negative one.

Every fraction can be written as a decimal by dividing the top number by the bottom number. Your child does this with long division and finds two kinds of answer. Some decimals stop, like $\frac{1}{4} = 0.25$ . Others repeat forever, like $\frac{1}{3} = 0.333...$ , which is written $0.\overline{3}$ .

Your child should be able to:

Find the sign of a product or a quotient before working it out
Explain with a pattern why a negative times a negative is positive
Multiply and divide negative fractions and decimals
Turn a fraction into a decimal by long division and tell whether it stops or repeats

Example

What are $(-7) \times 6$ , $(-7) \times (-6)$ and $(-42) \div 6$ ?

$(-7) \times 6$ : the signs are different, so the answer is negative. $7 \times 6 = 42$ , so the product is $-42$ .
$(-7) \times (-6)$ : the signs are the same, so the answer is positive, 42.
$(-42) \div 6$ : the signs are different, so the answer is negative, $-7$ .
The three answers are $-42$ , 42 and $-7$ .

Example

Write $\frac{3}{8}$ and $\frac{5}{11}$ as decimals.

$3 \div 8$ : the long division gives 0.375 and then a remainder of 0, so the decimal stops.
$5 \div 11$ : the remainders go 6, 5, 6, 5 and never reach 0, so the digits 4 and 5 repeat.
So $\frac{3}{8} = 0.375$ , and $\frac{5}{11} = 0.4545...$ , which is written $0.\overline{45}$ .

Where children usually get stuck

Your child writes $(-5) \times (-3) = -15$ .
In adding, two negatives make a bigger negative, so the child carries that rule over to multiplying.
What helps. Before thinking about signs, ask: are we adding or multiplying? When multiplying, same signs give a positive answer.
For $(-2) \times (-3) \times (-1)$ , your child says the answer is 6.
The rule was learned as "negatives make a positive". That works for two negatives. A third negative turns the answer negative again, so the product is $-6$ .
What helps. Ask your child to count the negative signs first. An even count gives a positive answer, and an odd count gives a negative one.
Your child writes $\frac{1}{6}$ as 0.161616..., repeating the wrong digits.
In $\frac{1}{6}$ the repeat starts after the first digit: $1 \div 6 = 0.1666...$ . Guessing the pattern from the first two digits gives the wrong decimal.
What helps. Ask your child to write each remainder beside the long division. When a remainder comes back, the digits from that point repeat.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up

For each product, select whether it is positive or negative. Do not find the actual product.

Product	Sign of the product
$\displaystyle (-3)\cdot 5\cdot 7$
$\displaystyle (-4)\cdot (-6)\cdot 8$

Show the answer and how to solve it

Select Negative for row 1 and Positive for row 2 because the rows have 1 and 2 negative factors respectively.

1Pair the negative factors. Each pair has a positive product, and positive factors keep it positive. An even count uses all the negatives in pairs, so the product is positive. An odd count leaves one negative factor, so the product is negative.
2Row 1 has 1 negative factor: −3. Since 1 is odd, the product is negative, so select Negative.
3Row 2 has 2 negative factors: −4 and −6. Since 2 is even, the product is positive, so select Positive.

Common mistake. A common mistake is selecting Negative for row 2 simply because it contains negative factors. Row 2 has 2 negative factors, and an even count of negatives makes the product positive.

The idea underneath. For a product with many factors, the sign depends only on whether the number of negative factors is odd or even, not on the sizes of the numbers.

2On level
A judge needs to show the opposite of 3 gains of 5 points each. Drag the tiles to build the product in this order: opposite of the number of gains × points in each gain.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
×
(-5)
4
3
5
(-3)
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
The answer is (-3) × 5 because the situation is the opposite of 3 gains, and each gain is +5 points.
1. 1The phrase 'opposite of 3 gains' makes the count negative, so the first factor is (-3).
2. 2Each gain is 5 points, so the value of each gain is positive 5. That makes the second factor 5.
3. 3Write the factors in the required order: opposite of the number of gains × points in each gain.
Common mistake. A common mistake is choosing $3 \times (-5)$ . That shows 3 negative gains, but the situation says the opposite of 3 positive gains, so the negative sign belongs on the count, not on the point value.
The idea underneath. The sign on the first factor can come from 'opposite of,' while the sign on the second factor comes from whether each repeated change is a gain or a loss.
3On level
Drag < or > into the blank to make the comparison true.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
$\displaystyle \left(-\frac{1}{2}\right)\left(3\right)$
0
<
>
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
The answer is $<$ because $-\frac{1}{2}$ and $3$ have opposite signs, so their product is negative and therefore less than $0$ .
1. 1The factors are $-\frac{1}{2}$ and $3$ . One factor is negative and one factor is positive.
2. 2When two factors have different signs, the product is negative.
3. 3A negative number is less than 0, so the correct symbol is $<$ .
Common mistake. A common mistake is noticing the fraction and thinking the product must be small but positive. The sign of $-\frac{1}{2}$ is negative, and since the two factors have different signs, the product is negative.
The idea underneath. You do not need the product's actual value to compare it with 0; the signs of the factors tell you the sign of the product.
4Stretch
Use the tiles to build the distributed form. Keep the terms in the same left-to-right order, and do not simplify.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
$\displaystyle (-\frac{1}{2})(8 + 4) =$
$\displaystyle (-\frac{1}{2})(12)$
$\displaystyle (-\frac{1}{2})(8)$
+
$\displaystyle (-\frac{1}{2})(4)$
-
$\displaystyle (\frac{1}{2})(8)$
$\displaystyle (-\frac{1}{2})(3)$
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
The answer is $(-\frac{1}{2})(8) + (-\frac{1}{2})(4)$ because the factor $-\frac{1}{2}$ is distributed to both 8 and 4, and the addition sign stays in the middle.
1. 1Identify the factor outside the parentheses. The factor $-\frac{1}{2}$ must be attached to both numbers inside, which are 8 and 4.
2. 2The first inside number is 8, so the first distributed term is $(-\frac{1}{2})(8)$ . The factor stays $-\frac{1}{2}$ , not $\frac{1}{2}$ .
3. 3The inside operator is +, so the middle operator stays +. The second distributed term is $(-\frac{1}{2})(4)$ , again keeping the factor as $-\frac{1}{2}$ .
4. 4Check both forms: $(-\frac{1}{2})(8+4)=-6$ . The distributed form gives $-4+(-2)=-6$ . Both forms have the same value.
Common mistake. A common mistake is choosing $(\frac{1}{2})(8)$ as the first term by dropping the negative sign from the factor. The factor is $-\frac{1}{2}$ for both distributed terms, and the middle sign comes from the original + inside.
The idea underneath. The distributive property says that multiplying a sum or difference gives the same value as multiplying each number and then adding or subtracting the products. The outside factor, including its negative sign, multiplies both numbers.