7.NS.A.2: Multiplying and Dividing Rational Numbers
In plain English: 7.NS.A.2 is the Common Core grade 7 math standard that asks students to multiply and divide positive and negative fractions, decimals and integers. Students see why a negative times a negative is positive, why dividing by zero has no answer, and what products and quotients mean in real situations. They also use long division to show that every fraction becomes a decimal that ends or repeats.
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
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.
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 -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
c.Apply properties of operations as strategies to multiply and divide rational numbers.
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.
Common Core State Standards for Mathematics · Domain: The Number System (NS) · Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Also written as 7.NS.2 · Official standard
Students extend what they know about multiplying and dividing fractions to all rational numbers. A rational number is any number that can be written as a fraction a/b, where a and b are integers (whole numbers and their opposites, such as -4, 0 and 7) and b is not 0. So -3, 0.25, -5/9 and 5 1/2 are all rational numbers. A product is the answer to a multiplication, and a quotient is the answer to a division.
The lesson has four parts, one for each part of the standard. (a) Students see why (-1)(-1) = 1: the distributive property, a(b + c) = ab + ac, must keep working for negative numbers, and that forces the sign rules. (b) Students divide integers, learn why the divisor (the number you divide by) cannot be 0, and see that -(p/q) = (-p)/q = p/(-q). (c) Students use properties of operations to make products easy to compute. (d) Students use long division to turn fractions into decimals and see that the decimal either ends or repeats. Every part includes real situations, such as temperatures, depths and money, so students can say what a product or quotient means. Numbers that are not rational, such as the square root of 2, come in grade 8 (8.NS.A.1).
Learning Objectives
By the end of this lesson, students will be able to:
Explain with the distributive property why (-1)(-1) = 1, and use the sign rules to multiply rational numbers
Divide integers and other rational numbers, explain why the divisor cannot be 0, and write -(p/q) = (-p)/q = p/(-q)
Describe what a product or quotient of rational numbers means in a real situation, with units
Use the commutative (order), associative (grouping) and distributive properties to multiply and divide rational numbers efficiently
Use long division to write a fraction as a decimal and tell whether the decimal terminates (ends) or repeats
Prior Knowledge Required
Students should already be comfortable with:
Seeing a fraction a/b as the division a ÷ b 5.NF.B.3
Dividing fractions by fractions 6.NS.A.1
Multiplying and dividing multi-digit decimals 6.NS.B.3
Opposites, such as -(-3) = 3, on the number line 6.NS.C.6
Adding and subtracting positive and negative numbers 7.NS.A.1
Write this list on the board. Students copy it and fill in the blanks by following the pattern.
Warm-Up Prompt
"3 × (-2) = -6, 2 × (-2) = -4, 1 × (-2) = -2, 0 × (-2) = 0. What should (-1) × (-2), (-2) × (-2) and (-3) × (-2) be? Explain the pattern."
Students know that 3 × (-2) means three groups of -2, so it is -6. Each time the first factor (a number being multiplied) goes down by 1, the product goes up by 2. Continuing the pattern gives (-1) × (-2) = 2, (-2) × (-2) = 4 and (-3) × (-2) = 6. Diagram 1 shows the products on a number line. Tell students: "A pattern is a good hint, but it is not a reason. Today we find the reason."
Direct Instruction20 minutes
Part 1: Why a negative times a negative is positive (standard a). Mathematicians extended multiplication to negative numbers so that the properties of operations still work. The key one is the distributive property. Show this chain on the board, one line at a time:
Start with a fact everyone agrees on: (-1) × 0 = 0.
Write 0 as 1 + (-1), because a number plus its opposite is 0: (-1) × (1 + (-1)) = 0.
Use the distributive property: (-1)(1) + (-1)(-1) = 0.
We know (-1)(1) = -1, so -1 + (-1)(-1) = 0.
The only number that adds to -1 to make 0 is 1. So (-1)(-1) = 1.
Sign rules for multiplying and dividing
Signs of the two numbers
Sign of the product or quotient
Example
positive and positive
positive
(2.5)(4) = 10
positive and negative
negative
(3)(-1/2) = -3/2
negative and negative
positive
(-0.5)(-8) = 4
Count the negative factors: an even number of negative factors gives a positive product, and an odd number gives a negative product.
Part 2: Dividing integers (standard b). Division undoes multiplication: -12 ÷ 3 = -4 because 3 × (-4) = -12. So division follows the same sign rules. Ask: "What is 6 ÷ 0?" We would need a number that gives 6 when multiplied by 0, and there is none. So division by 0 is undefined (it has no answer). But 0 ÷ 6 = 0, because 6 × 0 = 0. Every quotient of integers with a non-zero divisor, such as -7 ÷ 2 = -7/2, is a rational number, because it is a fraction of integers. The negative sign can go in three places: -(3/4) = (-3)/4 = 3/(-4). All three equal -0.75, because each has exactly one negative sign.
Part 3: Properties as strategies (standard c). The commutative property lets you change the order of factors (ab = ba). The associative property lets you change the grouping ((ab)c = a(bc)); it works for addition too. The identity property says a number times 1 stays the same. Use them to pick easy pairs first, or to split a mixed number with the distributive property.
Part 4: Fractions to decimals (standard d). A fraction a/b means a ÷ b, so long division turns it into a decimal. If a remainder (the amount left over at a step) of 0 appears, the decimal terminates: it ends, and only 0s follow. If not, a remainder must come back, because each remainder is smaller than the divisor. Then the digits repeat. We write a repeating decimal with a bar over the repeating digits, as in Diagram 2.
Interpreting a product (standard a)
On a desert evening, the temperature changes by -3°F each hour from 6 p.m. to midnight. Compared with 9 p.m., what is the temperature at midnight, 3 hours later? At 7 p.m., 2 hours earlier?
Equation: (-3)(3) = -9: at midnight it is 9°F cooler than at 9 p.m. (-3)(-2) = 6: going back 2 hours undoes two drops, so at 7 p.m. it was 6°F warmer than at 9 p.m.
Interpreting a quotient (standard b)
A research submarine goes from the surface down to -180 m in 6 minutes at a steady speed. What is its change in depth per minute?
Equation: -180 ÷ 6 = -30, and -(180/6) = (-180)/6 = 180/(-6) = -30. The submarine's depth changes by -30 m each minute: it goes 30 m deeper every minute.
Properties as strategies (standard c)
Compute (-2.5)(-7)(4) and -8 × 4 1/4 without a calculator.
Equation: 70 ÷ 8 = 8 remainder 6, 60 ÷ 8 = 7 remainder 4, 40 ÷ 8 = 5 remainder 0. The remainder is 0, so 7/8 = 0.875, and only 0s follow (0.875000...).
Repeating decimal (standard d)
Use long division to write 5/11 as a decimal.
Equation: 50 ÷ 11 = 4 remainder 6, 60 ÷ 11 = 5 remainder 5, then remainder 6 comes back. So 5/11 = 0.4545..., written 0.45 with a bar over 45.
Guided Practice15 minutes
Pairs work each problem. Partner A computes and Partner B says the sign rule or property that was used, then they switch. Stop after problems 2 and 4 to compare answers as a class. Before problem 4, recall the reciprocal: the number you multiply by to get 1. The reciprocal of 4/15 is 15/4, and the reciprocal of -2/3 is -3/2. Dividing by a fraction is the same as multiplying by its reciprocal, as in grade 6.
Guided practice problems with answers
Problem
Answer
(-7)(4.5)
-31.5 (one negative factor: negative)
(-3/4)(8/9)
-2/3 (multiply across: -24/36, then simplify)
-56 ÷ (-8)
7 (two negatives: positive)
(-2/5) ÷ (4/15)
-3/2 (multiply by the reciprocal: (-2/5)(15/4) = -30/20)
Write 3/16 as a decimal
0.1875 (remainder 0 after four steps: terminates)
Write 2/9 as a decimal
0.222... (remainder 2 every step: repeats)
Listen for students who flip the wrong fraction, or who lose the negative sign when they simplify.
Independent Practice10-15 minutes
Students work alone. For each product or quotient, they first predict the sign, then compute. For each fraction, they show the long division and circle the remainders.
Independent practice problems with answers
Problem
Answer
(-1)(-1)(-1)(-1)(-1)(-1)
1 (six negative factors, an even number)
(-0.4)(-0.25)
0.1
7 ÷ (-1/3)
-21
Write -(11/5) in two other ways, then as a decimal
(-11)/5 and 11/(-5); -2.2
Which has a value: (-15)/0 or 0/(-15)?
0/(-15) = 0; (-15)/0 is undefined
Write 5/12 as a decimal
0.41666..., with a bar over the 6 (remainder 8 repeats)
Closure5 minutes
Exit ticket: (1) Compute (-6)(-2.5). (Answer: 15.) (2) Compute -24 ÷ (-1.5). (Answer: 16.) (3) Use long division to write 2/11 as a decimal. (Answer: 0.1818..., with a bar over 18.) (4) In one sentence, explain why 9 ÷ 0 has no answer. (Answer: no number times 0 gives 9.)
Differentiation Strategies
For Struggling Students
Give a sign-rule card: count the negative factors; even means positive, odd means negative
Compute the answer without signs first, then decide the sign as a separate step
Use a place-value chart for long division so each remainder and brought-down 0 has its own column
For Advanced Students
Ask students to prove (-2)(-3) = 6 with the distributive property, starting from (-2)(3 + (-3)) = 0
Ask which unit fractions (numerator 1) from 1/2 to 1/20 terminate, and to describe what their denominators have in common (a challenge beyond this standard)
Ask students to write a story in which a negative number divided by a negative number makes sense
Assessment Guidance
What to Look For
Check that students decide the sign by counting negative factors, and that they explain (-1)(-1) = 1 with the distributive property, not just "two negatives make a positive". In division, look for the rule that the divisor cannot be 0, while 0 divided by a non-zero number is 0. In context problems, students should say what the answer means with units, for example "30 m deeper each minute". In long division, students should circle the remainders and stop when one repeats or reaches 0.
02
Classroom Activities
3 Activities
1
Sign Sort
15 minPairs
Pairs get 12 cards. They predict the sign of each answer, sort the cards into four groups (positive, negative, zero, undefined), and then compute each card to check.
C9 has three negative factors and C11 has four. How does the count decide the sign?
C7 and C8 use the same two numbers. Why does one have a value and the other not?
Only one card is undefined. Could a multiplication card ever be undefined?
Modification for Distance Learning
Put the 12 cards on a shared slide with four labeled boxes. Pairs drag each card into a box, then type the value next to it.
2
Story Match
15 minPairs
Pairs match 6 story cards to 6 expression cards, compute each one, and write a sentence that says what the answer means, with units.
Story Cards
S1: A diver has been swimming down for 10 minutes, and her depth changes by -2 m each minute. Where will she be 5 minutes from now, compared with now?
S2: The same diver: where was she 3 minutes ago, compared with now?
S3: Sam's lunch card balance changes by -$4.50 each school day. How much does it change in 5 days?
S4: A store's cash box went down by $84 over 7 days, the same amount each day. What was the change per day?
S5: The front edge of a glacier moved back 36 m in 12 years at a steady rate. What was the change in its position per year?
S6: A football team lost 18 yards in 3 plays, the same amount on each play. What was the change per play?
Expression Cards (answer key)
S1: (-2)(5) = -10: she will be 10 m deeper
S2: (-2)(-3) = 6: she was 6 m higher, closer to the surface
S3: (-4.50)(5) = -22.50: the balance goes down $22.50
S4: -84 ÷ 7 = -12: the cash box went down $12 each day
S5: -36 ÷ 12 = -3: the edge moved back 3 m each year
S6: -18 ÷ 3 = -6: the team lost 6 yards on each play
Discussion Questions
Only S2 has two negative factors. What does the negative time, -3, mean in the story?
In S4, S5 and S6, the quotient is negative. What does the negative sign tell you each time?
Write your own story for (-5)(-4) and share it with another pair.
3
Long Division Detectives
20 minGroups of 3
Each group uses long division to write the unit fractions 1/2 through 1/12 as decimals. A unit fraction has 1 as its numerator. Groups record every remainder, then sort the 11 decimals into "terminates" and "repeats".
Procedure
Split the 11 fractions among the three group members (4, 4 and 3 fractions)
For each division, write every remainder in a list and stop when a remainder is 0 or comes back
Write the decimal, with a bar over the repeating digits when it repeats
Check each decimal with a calculator, then sort all 11 on a two-column poster
For 1/7, the remainders are 3, 2, 6, 4, 5, 1, and then 3 again: all six possible remainders appear before one comes back
Discussion Questions
Why can the division for 1/7 have at most six different remainders before one repeats?
1/6 and 1/12 start with digits that do not repeat. Do they still count as repeating decimals?
Is 0.25 the same as 0.25000...? What does "terminates in 0s" mean?
Challenge Variation
Look at the denominators of the terminating fractions: 2, 4, 5, 8 and 10. Their only prime factors (the prime numbers that multiply to give the number) are 2 and 5. Predict whether 1/16, 1/15 and 1/20 terminate, then check with long division. (This pattern goes beyond this standard.)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Pattern for Multiplying by -2
The products 3 × (-2), 2 × (-2), 1 × (-2) and 0 × (-2) are known facts: -6, -4, -2 and 0. Each time the first factor goes down by 1, the product moves 2 to the right. Continuing the pattern puts (-1) × (-2), (-2) × (-2) and (-3) × (-2) at 2, 4 and 6. The number line is drawn to scale.
Diagram 2: Long Division Shows Why 5/11 Repeats
Long division of 5 by 11. The circled numbers are the remainders: 6, 5, 6, 5. When remainder 6 comes back, the division starts over, so the quotient digits 4 and 5 repeat forever: 5/11 = 0.4545...
04
Homework Assignment
~30 min
7.NS.A.2 Homework: Multiplying and Dividing Rational Numbers
Directions: Show all work. Decide the sign of each answer before you compute. For word problems, write a sentence that says what your answer means, with units. For long division, list every remainder.
Part 1: Sign Rules and Why They Work (Problems 1-2)
Use the distributive property to show that (-3)(-4) = 12. Start with (-3)(4 + (-4)) and explain every step.
Part 2: What Products and Quotients Mean (Problems 3-4)
A candle has been burning for 5 hours, and its height changes by -0.8 cm each hour. Compute (-0.8)(3) and (-0.8)(-1.5), and explain what each answer means for the candle.
A class fundraiser account changed by -$126 over 9 weeks, the same amount each week. Find the change per week. Then write the quotient in three ways, using -(p/q) = (-p)/q = p/(-q), and explain why all three are equal.
Part 3: Properties and Decimals (Problems 5-6)
Compute without a calculator, and name the property you used: (a) (-25)(-3.7)(4) (b) -12 × 3 1/4
Use long division to write each number as a decimal: (a) 3/40 (b) 7/12 (c) -8/11. Say which ones terminate and which repeat, and name the repeating digits.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Signs and Computation
All signs and values correct
One sign or arithmetic error
Several errors
Reasoning
Distributive property proof complete, every step explained
Steps shown but one step not explained
Only states the rule
Meaning in Context
Each answer explained in a sentence with units and direction
Answers correct but meaning unclear or units missing
No interpretation
Long Division
Remainders listed; terminating and repeating decimals named correctly
Decimals correct but remainders not shown
Division incorrect or missing
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Compute (-6)(-9).
Answer: B
Two negative factors give a positive product: (-6)(-9) = 54. Choice A uses the wrong sign rule. Choice C adds the numbers instead of multiplying: -6 + (-9) = -15. Choice D adds the numbers and drops both signs: 6 + 9 = 15.
Question 2 of 20 · Multiple Choice
Compute (-3/5)(10/21).
Answer: C
Multiply across: (-3 × 10)/(5 × 21) = -30/105 = -2/7. One factor is negative, so the product is negative. Choice A drops the negative sign. Choice B gives the reciprocal of the right answer. Choice D divides instead of multiplying: (-3/5)(21/10) = -63/50.
Question 3 of 20 · Multiple Choice
Compute 42 ÷ (-7).
Answer: A
42 ÷ 7 = 6, and a positive divided by a negative is negative: -6. Check: (-7)(-6) = 42. Choice B uses the wrong sign. Choice C adds instead of dividing: 42 + (-7) = 35. Choice D multiplies: 42 × (-7) = -294.
Question 4 of 20 · Multiple Choice
Which expression is NOT equal to -(8/3)?
Answer: D
(-8)/(-3) has two negatives, so it equals positive 8/3. Choices A, B and C each have exactly one negative sign, so each equals -(8/3), about -2.67. A student who picks A or B may think the place of the negative sign changes the value, but -(p/q) = (-p)/q = p/(-q).
Question 5 of 20 · Multiple Choice
Which statement is true?
Answer: B
0 ÷ (-9) = 0, because (-9) × 0 = 0. Choices A and C divide by 0, which is undefined: no number times 0 gives -9. Choice D fails the multiplication check: it would need (-9)(-9) = 0, but (-9)(-9) = 81.
Question 6 of 20 · Multiple Choice
During a two-week dry spell, the water level of a pond changes by -0.75 inches each day. What does (-0.75)(-4) = 3 mean?
Answer: C
The -4 means 4 days in the past. Going back undoes 4 drops of 0.75 inches, so 4 days ago the level was 3 inches higher than now. Choice A describes (-0.75)(4) = -3, 4 days in the future. Choice B gets the time right but reads the positive 3 as lower. Choice D reads the -4 as the future and keeps the positive 3.
Question 7 of 20 · Multiple Choice
A phone's battery level changed by -18% over 4 hours of video, the same amount each hour. What does -18 ÷ 4 = -4.5 mean?
Answer: A
The quotient is the change per hour, and the negative sign means the level went down: 4.5% of a full charge each hour. Choice B ignores the negative sign. Choice C reads the total change as the change per hour. Choice D multiplies instead of dividing: 18 × 4 = 72.
Question 8 of 20 · Multiple Choice
Nina computes (-4)(17)(25) as (-4)(25)(17) = (-100)(17) = -1,700. Which property lets her change the order of the factors?
Answer: D
Changing the order of factors, 17 × 25 to 25 × 17, is the commutative property of multiplication. Choice A splits a sum, as in a(b + c) = ab + ac, which Nina did not do. Choice B is about grouping a sum, but Nina multiplied. Choice C is about multiplying by 1.
Question 9 of 20 · Multiple Choice
Use the distributive property to compute -6 × 3 1/2.
Answer: A
-6 × (3 + 1/2) = (-6)(3) + (-6)(1/2) = -18 + (-3) = -21. Choice B makes (-6)(1/2) positive: -18 + 3 = -15. Choice C multiplies only the 3 by -6 and keeps the 1/2: -18 - 1/2. Choice D treats a negative times a positive as positive: 18 + 3 = 21.
Question 10 of 20 · Multiple Choice
Use long division to write 5/8 as a decimal.
Answer: D
50 ÷ 8 = 6 remainder 2, 20 ÷ 8 = 2 remainder 4, 40 ÷ 8 = 5 remainder 0. The remainder 0 means the decimal terminates: 0.625. Choice A divides 8 by 5. Choice B writes the numerator and denominator side by side. Choice C stops after two steps, while the remainder is still 4.
Question 11 of 20 · Multiple Choice
Use long division to write 5/6 as a decimal.
Answer: B
50 ÷ 6 = 8 remainder 2, then 20 ÷ 6 = 3 remainder 2 again and again. So only the 3 repeats: 0.8333..., with a bar over the 3. Choice A writes the numerator and denominator side by side. Choice C repeats the 8 as well, but the 8 appears only once. Choice D divides 6 by 5.
Question 12 of 20 · Multiple Choice
Why must the long division for 4/7 repeat?
Answer: A
Each remainder is less than 7. None is 0 here, so only 1, 2, 3, 4, 5 and 6 are possible, and within seven steps one must come back. Then the digits repeat: 4/7 = 0.571428... Choice B is false: 5 is odd, and 1/5 = 0.2 terminates. Choice C is false: 3 is not a multiple of 8, but 3/8 = 0.375 terminates. Choice D contradicts the long division, where the six digits 571428 repeat.
Question 13 of 20 · Multiple Choice
Which of these has no value?
Answer: C
(-12)/0 divides by 0, which is undefined: no number times 0 gives -12. Choice A is 3. Choice B is 0, because 0 divided by a non-zero number is 0. Choice D is -1. Each of them is a quotient of integers with a non-zero divisor, so each is a rational number.
Question 14 of 20 · Multiple Choice
Compute (-2)(-3)(-1)(4).
Answer: B
There are three negative factors, an odd number, so the product is negative: 2 × 3 × 1 × 4 = 24, and the answer is -24. Choice A treats the product as positive. Choice C adds the numbers: -2 + (-3) + (-1) + 4 = -2. Choice D adds the numbers without signs: 2 + 3 + 1 + 4 = 10.
Question 15 of 20 · Short Answer
Compute (-2 1/4) ÷ (3/8). Show your steps.
Write the mixed number as a fraction: -2 1/4 = -9/4. Multiply by the reciprocal of 3/8: (-9/4)(8/3) = -72/12 = -6. Check: (-6)(3/8) = -18/8 = -2 1/4.
Question 16 of 20 · Short Answer
Use the distributive property to explain why (-2)(-5) = 10.
Start with (-2)(5 + (-5)) = (-2)(0) = 0. Distribute: (-2)(5) + (-2)(-5) = 0, so -10 + (-2)(-5) = 0. The only number that adds to -10 to make 0 is 10, so (-2)(-5) = 10.
Question 17 of 20 · Short Answer
Use long division to write -17/20 as a decimal. Does it terminate or repeat?
Divide 17 by 20: 170 ÷ 20 = 8 remainder 10, then 100 ÷ 20 = 5 remainder 0. So 17/20 = 0.85 and -17/20 = -0.85. The remainder reached 0, so the decimal terminates.
Question 18 of 20 · Short Answer
Use long division to write 4/15 as a decimal. Name the repeating digit.
40 ÷ 15 = 2 remainder 10, 100 ÷ 15 = 6 remainder 10, and remainder 10 keeps coming back. So 4/15 = 0.2666..., and the repeating digit is 6 (a bar over the 6). The 2 appears only once.
Question 19 of 20 · Short Answer
A hiker walks down a mountain trail, and her elevation changes by -600 feet each hour. Write and evaluate a product for the change in elevation after 2.5 hours, and say what the sign means.
(-600)(2.5) = -1,500. After 2.5 hours, her elevation has changed by -1,500 feet: the negative sign means she is 1,500 feet lower than when she started.
Question 20 of 20 · Short Answer
A water tank lost 45 liters in 6 minutes at a steady rate. Write the change per minute as a quotient in three ways, find its value, and say what it means.
-(45/6) = (-45)/6 = 45/(-6) = -7.5. Each form has one negative sign, so all three are equal. The amount of water in the tank changes by -7.5 liters per minute: it loses 7.5 liters each minute.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.NS.A.2 mean?
7.NS.A.2 means students can multiply and divide positive and negative rational numbers and explain why the rules work. Rational numbers include integers, fractions and decimals, such as -4, 3/5 and -0.8. The standard has four parts: (a) the sign rules and what products mean, (b) division of integers and what quotients mean, (c) using properties of operations, and (d) writing fractions as decimals with long division.
Why is a negative times a negative positive?
A negative times a negative is positive because that is the only rule that keeps the distributive property true. Start with (-1)(1 + (-1)) = (-1)(0) = 0. Distributing gives -1 + (-1)(-1) = 0, so (-1)(-1) must be 1. A real-world picture helps too: if a tank loses water every hour, then some hours ago it had more water, not less.
What grade is 7.NS.A.2, and what comes after it?
7.NS.A.2 is a grade 7 standard in The Number System domain. In the same grade, 7.NS.A.3 asks students to solve word problems with all four operations on rational numbers. In grade 8, students learn about numbers that are not rational (8.NS.A.1), and in high school they explain why sums and products of rational numbers are rational (HSN.RN.B.3).
Why can't you divide by zero?
You can't divide by zero because no number works as the answer. Division undoes multiplication: 6 ÷ 2 = 3 because 2 × 3 = 6. For 6 ÷ 0, we would need a number that gives 6 when multiplied by 0, and every number times 0 is 0. So division by 0 is undefined. Zero divided by a non-zero number is fine: 0 ÷ 5 = 0.
What is the difference between a terminating and a repeating decimal?
A terminating decimal ends, and a repeating decimal has a group of digits that repeats forever. 3/4 = 0.75 terminates: its long division reaches a remainder of 0. 1/3 = 0.333... repeats: the remainder 1 keeps coming back. The standard asks students to know that every fraction of integers gives one of these two kinds, which it calls terminating "in 0s" (0.75 = 0.75000...) or "eventually" repeating.
Why is -(p/q) the same as (-p)/q and p/(-q)?
They are equal because each one has exactly one negative sign, and that always gives a negative quotient of the same size. For example, -(10/4), (-10)/4 and 10/(-4) all equal -2.5. Students can check by multiplying: (-2.5)(4) = -10 and (-2.5)(-4) = 10.
How can I help my child remember the sign rules?
Have your child count the negative factors: an even count gives a positive answer and an odd count gives a negative one. Ask them to find the answer without signs first, then decide the sign. Everyday examples help too, such as money spent each day (a negative rate) over past days (negative time).
Does every fraction turn into a decimal that ends or repeats?
Yes, every fraction of integers with a non-zero denominator gives a decimal that terminates or repeats. In long division, each remainder is smaller than the divisor, so either a remainder of 0 appears and the decimal ends, or a remainder comes back and the digits repeat. Decimals that never end and never repeat are not rational numbers; students meet them in grade 8.
How is 7.NS.A.2 tested?
Tests on 7.NS.A.2 usually ask students to compute products and quotients of signed fractions and decimals, to pick the meaning of a product or quotient in a story, and to convert a fraction to a decimal. Some items ask students to explain a rule, such as why (-1)(-1) = 1 or why division by 0 is undefined, so practice explaining, not only computing.
What are common mistakes with multiplying and dividing negative numbers?
Common mistakes are using the addition rules for multiplication and losing a negative sign while simplifying. For example, some students write (-4)(-5) = -20 because they think "negative and negative stay negative", which is true for adding, not multiplying. Others flip the wrong fraction when dividing, or stop long division before the remainder is 0 or repeats.
07
Related Standards
6 standards
These standards connect to 7.NS.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.NF.B.3Prerequisite
Interpret a fraction as division of the numerator by the denominator
Lesson coming soon
6.NS.A.1Prerequisite
Interpret and compute quotients of fractions and solve word problems with them
