7.NS.A.3: Word Problems with All Four Operations on Rational Numbers
In plain English: 7.NS.A.3 is the Common Core grade 7 math standard that asks students to solve real-world and mathematical problems that add, subtract, multiply and divide rational numbers: positive and negative fractions, decimals and integers. It includes complex fractions, such as a distance of 2 1/2 miles divided by a time of 3/4 hour. Students plan the steps, follow the order of operations and check that each answer makes sense.
Solve real-world and mathematical problems involving the four operations with rational numbers.
Official note: Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
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.3 · Official standard
This lesson brings together all four operations with rational numbers: numbers that can be written as a fraction of two integers (whole numbers and their opposites), such as -7, 2.35, -4/9 and 1 1/2. Students already know the rules for adding, subtracting, multiplying and dividing them (7.NS.A.1 and 7.NS.A.2). Now they use those rules to solve problems about temperatures, depths, money, recipes and speeds, where one problem may need several operations and several number forms.
The standard's official note says that computing with rational numbers extends the rules for fractions to complex fractions. A complex fraction is a fraction with a fraction in its numerator, its denominator or both, such as (3/4)/(1/2). It means the numerator divided by the denominator. Complex fractions come up naturally in rates: 2 1/2 miles in 3/4 hour gives the speed (2 1/2)/(3/4) miles per hour. Throughout the lesson, students follow a four-step plan (understand, plan, solve, check) and estimate before they compute, so they can tell when an answer does not make sense.
Learning Objectives
By the end of this lesson, students will be able to:
Write one expression for a real-world problem that uses several operations with positive and negative rational numbers
Evaluate expressions with fractions, decimals and integers, following the order of operations and the sign rules
Simplify complex fractions, including ones with negative parts, and use them to find rates
Estimate before computing and decide whether an answer makes sense in the story
Prior Knowledge Required
Students should already be comfortable with:
Dividing fractions by fractions 6.NS.A.1
Computing with multi-digit decimals 6.NS.B.3
Using the order of operations to evaluate expressions 6.EE.A.2
Unit rates, such as miles per hour 6.RP.A.2
Adding and subtracting positive and negative numbers 7.NS.A.1
Multiplying and dividing positive and negative numbers 7.NS.A.2
Read the problem aloud. Students work alone for 3 minutes, then compare with a partner.
Warm-Up Prompt
"On a winter morning, the temperature at 6 a.m. was -4.5°C. By noon it had gone up 12°C. By 9 p.m. it had dropped 9.5°C from the noon temperature. What was the temperature at 9 p.m.? Write one expression for the whole story."
Collect expressions on the board, such as -4.5 + 12 - 9.5 and -4.5 + 12 + (-9.5). Both give -2, so it was -2°C at 9 p.m. Ask: "Is it reasonable that the answer is below zero?" Yes: the temperature went up 12 degrees and down 9.5 degrees, a net rise of 2.5 degrees from -4.5. Tell students that today's problems need more than one operation, and some use fractions inside fractions.
Direct Instruction20 minutes
A plan for every problem. Post these four steps and use them for each example.
Understand: what is the question, what are the units, and which amounts are negative (a drop, a debt, a depth)?
Plan: write one expression for the story. Decide whether to work in fractions or decimals.
Solve: follow the order of operations (parentheses first, then multiplication and division from left to right, then addition and subtraction from left to right) and the sign rules.
Check: compare with an estimate (a quick answer with rounded numbers) and ask whether the sign and the size make sense in the story.
Complex fractions. To simplify a complex fraction, divide its numerator by its denominator. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction flipped, so that the two multiply to 1). For example, (3/4)/(1/2) = 3/4 × 2/1 = 3/2. Signs follow the usual rules: one negative part gives a negative value, and two negative parts give a positive value.
Fractions or decimals? Work in decimals when all the numbers are money or measurements with decimals. Work in fractions when a number such as 1/3 or 2/3 would become a repeating decimal. A mixed number such as 2 1/2 is easier to multiply or divide after you rewrite it as a fraction, 5/2.
Adding and subtracting in context
A scuba diver is at -12.4 m. She rises 5.75 m, then swims down 2.5 m. What is her depth now?
Equation: -12.4 + 5.75 + (-2.5) = -6.65 + (-2.5) = -9.15. She is at -9.15 m, 9.15 m below the surface. Estimate: -12 + 6 - 3 = -9, close to the answer.
All four operations with money
Jaden has $95 in savings. He takes out $12.50 each week for 6 weeks, then adds one third of a $48 birthday gift. How much does he have now?
Equation: 95 + 6(-12.50) + (1/3)(48) = 95 - 75 + 16 = 36. He has $36. Multiply first, then add and subtract from left to right.
Complex fraction, from the official note
Simplify the complex fraction (-3/4)/(5/8).
Equation: (-3/4) ÷ (5/8) = (-3/4)(8/5) = -24/20 = -6/5, or -1 1/5. One part is negative, so the value is negative.
Complex fraction as a rate
Maya walks 2 1/2 miles in 3/4 hour at a steady pace. What is her speed in miles per hour?
Equation: (2 1/2)/(3/4) = (5/2)(4/3) = 10/3 = 3 1/3 miles per hour, a normal walking speed. Diagram 1 shows the same answer on a double number line: two number lines, one above the other, with matching amounts lined up.
Mean of signed numbers
The low temperatures on four days were -3.5°C, 2°C, -6.25°C and -0.25°C. What was the mean low temperature?
Equation: The mean is the sum divided by the number of values: (-3.5 + 2 + (-6.25) + (-0.25)) ÷ 4 = -8 ÷ 4 = -2. The mean low was -2°C.
Diagram 2 shows the diver example on a vertical number line, with depth below the surface as negative numbers. Point out that the order of the moves does not change where she ends up, because addition can be done in any order.
Guided Practice15 minutes
Pairs solve the five problems. Before computing, each pair writes an estimate. After computing, they compare the answer with the estimate and explain any big difference.
Guided practice problems with answers
Problem
Answer
-2/5 + (3/10)(-4)
-8/5 (multiply first: (3/10)(-4) = -6/5, then -2/5 + (-6/5))
(-7.2) ÷ 0.9 - (-3)
-5 (-7.2 ÷ 0.9 = -8, then -8 + 3)
(1 1/4)/(-5/8)
-2 ((5/4)(-8/5) = -40/20)
A 40-gallon tank drains at 2 1/2 gallons per minute. How long does it take to drain 3/4 of the tank?
The temperature fell from 3.5°F to -8.5°F in 4 hours at a steady rate. What was the change per hour?
-3°F per hour ((-8.5 - 3.5) ÷ 4 = -12 ÷ 4)
Watch for students who add before they multiply in the first problem, which gives (-2/5 + 3/10)(-4) = 2/5. In the tank problem, ask what the complex fraction 30/(2 1/2) means: the number of minutes, since gallons divided by gallons per minute gives minutes.
Independent Practice10-15 minutes
Students solve six problems alone, writing one expression per word problem. They circle any answer that does not match their estimate and look for the error.
Independent practice problems with answers
Problem
Answer
(-1/3)(0.6) + 1.5
1.3
-4 1/2 ÷ (3/4) × (-2)
12
(-2/9)/(-4/3)
1/6
A hiker at 1,480 ft goes down 215.5 ft each hour for 3 hours. What is her elevation?
833.5 ft
A recipe uses 3/4 cup of oil for 12 muffins. How much oil is needed for 20 muffins?
1 1/4 cups
A painter covers 5/6 of a wall in 1/3 hour. How many walls does he cover per hour?
2 1/2 walls per hour
Closure5 minutes
Exit ticket: (1) Simplify (3/5)/(-9/10). (Answer: -2/3.) (2) A drone flies at 45 m, goes down 2.5 m per second for 8 seconds, then climbs 6.75 m. How high is it now? (Answer: 45 - 20 + 6.75 = 31.75 m.) (3) In one sentence, say how you check that an answer makes sense.
Differentiation Strategies
For Struggling Students
Give a planning frame with four boxes: question and units, expression, work, estimate check
Start each complex fraction by rewriting it as a division sentence, such as 3/4 ÷ 1/2, before multiplying by the reciprocal
Let students solve a word problem first with whole numbers, then with the real fractions and decimals
For Advanced Students
Ask students to write a word problem whose expression uses all four operations and has a negative answer that makes sense
Give complex fractions with sums in both parts, such as (1/2 + 1/3)/(1/4 - 1/6), and ask for the simplest strategy
Ask students to explain when a decimal answer should be rounded in a story and when it should not
Assessment Guidance
What to Look For
Look for one clear expression per word problem, with negative numbers for drops, debts and depths. Check the order of operations, especially multiplication before addition, and the sign of every product and quotient. For complex fractions, students should rewrite the fraction as a division and multiply by the reciprocal of the denominator. Every answer should end with a sentence and units, and students should compare it with an estimate.
02
Classroom Activities
3 Activities
1
Problem-Solving Stations
20 minGroups of 3-4
Set up 4 stations, each with one problem card. Groups spend 5 minutes at each station: they write an estimate, one expression and the answer on the station's chart paper, then rotate. The data on the cards are invented.
Station Cards
Station 1 (weather): In a northern town, the temperature was -15.5°F at 7 a.m. It rose 2.25°F each hour until 3 p.m. What was the temperature at 3 p.m.?
Station 2 (money): The school store starts the week with $120 in its cash box. It sells 36 pens at $1.25 each and pays $62.40 for new supplies. Then it splits the money in the box equally among 3 clubs. How much does each club get?
Station 3 (cooking): A recipe for 8 servings uses 2 2/3 cups of flour. How much flour is needed for 6 servings?
Station 4 (elevation): A hiker starts at -175 ft, below sea level, and climbs to a lookout at 1,325 ft in 2 1/2 hours. What was her average climb per hour?
Station 4: (1,325 - (-175)) ÷ 2 1/2 = 1,500 ÷ 2 1/2 = 600 ft per hour
Discussion Questions
At Station 1, the temperature starts below 0°F and ends above it. Between which two whole hours did it pass 0°F?
At Station 4, why do we subtract -175 instead of adding 175? Do both give the same answer?
Which station's expression is a complex fraction when you write it as one fraction?
Modification for Distance Learning
Put each station on its own slide. Groups work in breakout rooms, type their estimate, expression and answer under the problem, and move to the next slide every 5 minutes.
2
Complex Fraction Match
15 minPairs
Pairs get 16 cards: 8 complex fraction cards and 8 value cards. They simplify each complex fraction on scrap paper and match it to its value card.
Complex Fraction Cards
M1: (1/2)/(1/4)
M2: (-3/8)/(3/4)
M3: (2/5)/(-4)
M4: (-6)/(2/3)
M5: (1 1/3)/(2/9)
M6: (-5/12)/(-5/6)
M7: 0.6/(-3/10)
M8: (-2 1/2)/(5/6)
Value Cards (answer key)
M1 = 2, M2 = -1/2, M3 = -1/10, M4 = -9
M5 = 6, M6 = 1/2, M7 = -2, M8 = -3
Discussion Questions
M2 and M6 have values that are opposites. What is different about their signs?
M3 and M4 each have a whole number as one part. How did you write the whole number as a fraction?
M7 mixes a decimal and a fraction. Which form did you change, and why?
Challenge Variation
Each pair picks one card and writes a word problem whose answer is that complex fraction, for example a rate such as miles per hour or cups per batch.
3
Find the Mistake
15 minPairs
Pairs read five student solutions. Each one has exactly one mistake. Pairs find the mistake, explain it in a sentence and write the correct answer.
Student Solutions
E1: -3/4 + 1/2 ÷ 2 = -1/4 ÷ 2 = -1/8
E2: (2/3)/(4/9) = 2/3 × 4/9 = 8/27
E3: A kite is 12.5 m high. It drops 4.75 m and then rises 2.2 m. 12.5 - 4.75 - 2.2 = 5.55 m
E4: -18 ÷ (-1 1/2) = -18 ÷ (-3/2) = -12
E5: The mean of -5, 3 and -7 is 9 ÷ 3 = 3
Answer Key
E1: divide before adding: 1/2 ÷ 2 = 1/4, so -3/4 + 1/4 = -1/2
E2: multiply by the reciprocal of 4/9: 2/3 × 9/4 = 3/2
E3: a rise is added: 12.5 - 4.75 + 2.2 = 9.95 m
E4: two negatives give a positive quotient: 12
E5: the sum is -9, not 9, so the mean is -3
Discussion Questions
Which mistakes would an estimate have caught? Try E3 and E5.
Which mistakes are about signs, and which are about the order of operations?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Complex Fraction as a Unit Rate
A double number line for Maya's walk. The solid part shows what we know: 2 1/2 miles in 3/4 hour. Splitting 3/4 hour into three quarter hours splits the distance into three equal parts of 5/6 mile. One more quarter hour reaches 1 hour and 3 1/3 miles. That is the unit rate (the distance for 1 hour), and it is the same value as the complex fraction (2 1/2)/(3/4). Drawn to scale.
Diagram 2: The Diver on a Vertical Number Line
Depths below the surface are negative. The diver starts at -12.4 m, rises 5.75 m to -6.65 m, then swims down 2.5 m to -9.15 m. The estimate -12 + 6 - 3 = -9 is close to the exact answer. Drawn to scale, 22 pixels per meter.
04
Homework Assignment
~30 min
7.NS.A.3 Homework: Solving Problems with Rational Numbers
Directions: Show all work. For each word problem, write an estimate first, then one expression, then the answer in a sentence with units. Simplify fractions fully.
Part 1: Computing with Rational Numbers (Problems 1-2)
The temperature at midnight was 4.5°F. It fell 1.75°F each hour for 6 hours, then rose 3 1/4°F by 9 a.m. Write one expression and find the temperature at 9 a.m.
Leo jogs 3/4 mile in 1/8 hour. Write a complex fraction for his speed and simplify it. At that speed, how many minutes does it take him to jog 2 1/4 miles?
Part 3: Multi-Step Problems (Problems 5-6)
A class account has $250. The class buys 8 pizzas at $18.75 each, then earns $6 for each of 26 cars at a car wash. Then it sets aside one fourth of the money in the account for a field trip. How much is set aside?
Over 6 summer days, with light rain on two of them, a pond's water level changed by these amounts (inches): -0.4, 0.25, -1.1, -0.75, 0.6, -0.4. Find the mean daily change. If the level kept changing at that mean rate, what would the total change be over 30 days? (The data are invented.)
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Expression
One correct expression per problem, with negatives for drops and debts
Expression mostly correct, one sign or step missing
No expression, or it does not match the story
Computation
Order of operations and sign rules correct; fractions simplified
One computation error
Several errors
Complex Fractions
Rewritten as division and simplified correctly
Method correct, arithmetic error
Multiplied instead of divided, or missing
Estimate and Meaning
Estimate shown; answer in a sentence with units that makes sense
Answer has units but no estimate
No sentence or units
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
Evaluate -2.5 + 4(-0.75).
Answer: C
Multiply first: 4(-0.75) = -3. Then -2.5 + (-3) = -5.5. Choice A adds first and then multiplies: (-2.5 + 4)(-0.75) = -1.125. Choice B makes the product positive: -2.5 + 3 = 0.5. Choice D drops the negative sign of -2.5: 2.5 + 3 = 5.5.
Question 2 of 20 · Multiple Choice
Simplify the complex fraction (-4/5)/(2/15).
Answer: B
Divide by multiplying by the reciprocal: (-4/5)(15/2) = -60/10 = -6. Choice A multiplies the two fractions instead of dividing: (-4/5)(2/15) = -8/75. Choice C drops the negative sign. Choice D flips the first fraction instead of the second: (-5/4)(2/15) = -1/6.
Question 3 of 20 · Multiple Choice
A cyclist rides 4 1/2 miles in 1/3 hour at a steady speed. What is her speed?
Answer: C
Speed is distance divided by time: (4 1/2)/(1/3) = (9/2)(3) = 27/2 = 13 1/2 miles per hour, a normal bike speed. Choice A multiplies the distance by 1/3. Choice B treats 1/3 hour as a full hour. Choice D divides the time by the distance: (1/3)/(9/2) = 2/27.
Question 4 of 20 · Multiple Choice
At 5 a.m. the temperature was -7.5°F. By 1 p.m. it had risen 13.25°F, and by 8 p.m. it had fallen 9°F from there. What was the temperature at 8 p.m.?
Answer: A
-7.5 + 13.25 - 9 = 5.75 - 9 = -3.25°F. Choice B adds the 9°F drop: -7.5 + 13.25 + 9 = 14.75. Choice C starts at 7.5 instead of -7.5: 7.5 + 13.25 - 9 = 11.75. Choice D treats the rise as a drop: -7.5 - 13.25 - 9 = -29.75.
Question 5 of 20 · Multiple Choice
Ava has $40. She buys 3 notebooks at $2.90 each, then splits the rest of her money equally between 2 savings jars. How much goes in each jar?
Answer: D
(40 - 3(2.90)) ÷ 2 = (40 - 8.70) ÷ 2 = 31.30 ÷ 2 = $15.65. Choice A subtracts only one notebook: (40 - 2.90) ÷ 2 = 18.55. Choice B divides only the cost of the notebooks: 40 - 8.70 ÷ 2 = 35.65. Choice C adds the cost instead of subtracting it: (40 + 8.70) ÷ 2 = 24.35.
Question 6 of 20 · Multiple Choice
Which expression gives the mean of -3.2, 1.8 and -4.6?
Answer: B
The mean is the sum divided by 3, so the sum needs parentheses: (-3.2 + 1.8 - 4.6) ÷ 3 = -6 ÷ 3 = -2. Choice A divides only -4.6 by 3, because division comes before addition. Choice C drops the negative signs. Choice D multiplies the values instead of adding them.
Question 7 of 20 · Multiple Choice
Simplify the complex fraction (-3 1/3)/(-5/6).
Answer: D
Rewrite the mixed number: -3 1/3 = -10/3. Then (-10/3)(-6/5) = 60/15 = 4. Two negative parts give a positive value. Choice A gives the value a negative sign. Choice B multiplies the parts instead of dividing, and drops the signs: (10/3)(5/6) = 25/9. Choice C divides the denominator by the numerator: (-5/6)/(-10/3) = 1/4.
Question 8 of 20 · Multiple Choice
Simplify (5/6 - 1 1/3)/(-3/4).
Answer: A
Numerator first: 5/6 - 4/3 = 5/6 - 8/6 = -1/2. Then (-1/2)/(-3/4) = (-1/2)(-4/3) = 4/6 = 2/3. Choice B makes the numerator +1/2 by subtracting in the wrong order, so the value becomes negative. Choice C multiplies instead of dividing: (-1/2)(-3/4) = 3/8. Choice D adds in the numerator: (5/6 + 4/3)/(-3/4) = (13/6)(-4/3) = -26/9.
Question 9 of 20 · Multiple Choice
A diver goes from -3 m to -21 m in 7.5 minutes at a steady rate. What is her change in depth per minute?
Answer: B
The change is -21 - (-3) = -18 m, and -18 ÷ 7.5 = -2.4 m per minute: she goes 2.4 m deeper each minute. Choice A loses the negative sign, which would mean she is rising. Choice C subtracts 3 instead of adding it: (-21 - 3) ÷ 7.5 = -3.2. Choice D multiplies instead of dividing: (-18)(7.5) = -135.
Question 10 of 20 · Multiple Choice
Which is the best estimate of (-19.8)(0.51)?
Answer: C
Round: -19.8 is about -20 and 0.51 is about 1/2, so the product is about -20 × 1/2 = -10. The exact value is -10.098. Choice A doubles -20 instead of halving it. Choice B has the wrong sign. Choice D is off by a factor of 10, as if 0.51 were 0.051.
Question 11 of 20 · Multiple Choice
Evaluate -1/2 ÷ 1/4 + 3.
Answer: D
Divide first: -1/2 ÷ 1/4 = -1/2 × 4 = -2. Then -2 + 3 = 1. Choice A adds first: -1/2 ÷ (1/4 + 3) = -1/2 ÷ 13/4 = -2/13. Choice B makes the quotient positive: 2 + 3 = 5. Choice C multiplies instead of dividing: -1/8 + 3 = 2 7/8.
Question 12 of 20 · Multiple Choice
A recipe uses 2/3 cup of sugar for 16 cookies. How much sugar is needed for 24 cookies?
Answer: B
Sugar per cookie: (2/3) ÷ 16 = 1/24 cup. For 24 cookies: 24 × 1/24 = 1 cup. Check: 24 cookies is 1 1/2 times 16, and 1 1/2 × 2/3 = 1. Choice A uses the ratio upside down: (16/24)(2/3) = 4/9. Choice C multiplies 2/3 by 24 and forgets to divide by 16. Choice D adds the 8 extra cookies to the 2/3 cup.
Question 13 of 20 · Multiple Choice
Which story matches the expression 80 + 4(-15)?
Answer: C
Spending $15 is -15, and doing it four times is 4(-15) = -60. Starting from $80 leaves 80 - 60 = $20. Choice A adds $15, which would be 4(15). Choice B switches the starting amount and the amount spent. Choice D is 80 - 15 - 4 = 61, not 80 + 4(-15).
Question 14 of 20 · Multiple Choice
Ella uses 3/10 gallon of paint to cover 3/4 of a fence. How much paint does the whole fence need?
Answer: A
Paint for the whole fence is (3/10)/(3/4) = (3/10)(4/3) = 12/30 = 2/5 gallon. It makes sense: the whole fence needs a little more than 3/10 gallon. Choice B multiplies instead of dividing: (3/10)(3/4) = 9/40. Choice C divides the fence by the paint: (3/4)/(3/10) = 5/2 = 2 1/2. Choice D adds the two numbers: 3/10 + 3/4 = 1 1/20.
Question 15 of 20 · Short Answer
Evaluate (-0.8)(2 1/2) - 6 ÷ (-1.2). Show the order of your steps.
Multiply and divide first: (-0.8)(2.5) = -2 and 6 ÷ (-1.2) = -5. Then subtract: -2 - (-5) = -2 + 5 = 3.
Question 16 of 20 · Short Answer
Simplify the complex fraction (-7/12)/(-7/4).
(-7/12) ÷ (-7/4) = (-7/12)(-4/7) = 28/84 = 1/3. Two negative parts give a positive value.
Question 17 of 20 · Short Answer
A freezer is switched on when it is at room temperature, 21°C. Its temperature drops at a steady rate and reaches -18°C after 6 1/2 hours. Find the change in temperature per hour.
The total change is -18 - 21 = -39°C. The change per hour is -39 ÷ 6 1/2 = -39 ÷ 13/2 = -39 × 2/13 = -6°C per hour. The temperature drops 6°C each hour.
Question 18 of 20 · Short Answer
Mia owes her brother $15. She pays back 2/5 of what she owes, then borrows $4.50 more. Write one expression using -15 for the debt, and find what she owes now.
-15 + (2/5)(15) + (-4.50) = -15 + 6 - 4.50 = -13.50. Mia now owes her brother $13.50.
Question 19 of 20 · Short Answer
Sam reads 12 1/2 pages in 1/4 hour. Write his reading rate as a complex fraction and simplify it. At that rate, how many hours does he need for 175 pages?
(12 1/2)/(1/4) = (25/2)(4) = 50 pages per hour. Then 175 ÷ 50 = 3 1/2 hours.
Question 20 of 20 · Short Answer
A hot-air balloon is at 1,200 ft. It goes down 150 ft per minute for 4.5 minutes, then rises 1/3 of the distance it went down. What is its height now?
Distance down: 150 × 4.5 = 675 ft. Rise: (1/3)(675) = 225 ft. Height: 1,200 + (-675) + 225 = 750 ft.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.NS.A.3 mean?
7.NS.A.3 means students can solve word problems and number problems that use addition, subtraction, multiplication and division with rational numbers. The numbers can be positive or negative, and they can be integers, fractions, mixed numbers or decimals. A single problem often needs several operations, such as a temperature that rises for some hours and then falls.
Are complex fractions part of 7.NS.A.3?
Yes, complex fractions are part of 7.NS.A.3: the standard's official note says computing with rational numbers extends the rules for fractions to complex fractions. A complex fraction has a fraction in its numerator, its denominator or both, such as (2/3)/(5/9). Students simplify it by dividing the numerator by the denominator.
How do you simplify a complex fraction?
Rewrite it as a division and multiply by the reciprocal of the denominator. For (2/3)/(5/9), write 2/3 ÷ 5/9 = 2/3 × 9/5 = 18/15 = 6/5. If the numerator or denominator is a sum or difference, such as 1/2 + 1/3, work it out first. Use the sign rules for division at the end.
What grade is 7.NS.A.3, and what comes next?
7.NS.A.3 is a grade 7 standard in The Number System domain, and it closes the cluster on operations with rational numbers. In the same grade, students use these skills to solve multi-step problems and equations (7.EE.B.3 and 7.EE.B.4). In grade 8, they solve linear equations with rational coefficients (8.EE.C.7).
Should students use fractions or decimals?
Either form is correct, and students should choose the one that keeps the work simple. Money and measured lengths are usually easier in decimals. Thirds, sixths and other fractions that become repeating decimals are easier to keep as fractions. Converting mid-problem is fine, as long as students do not round too early.
How can students check that an answer makes sense?
Students should estimate before they compute and compare. Round each number to a friendly value, work the problem quickly and see if the exact answer is close. Then ask whether the sign and the size fit the story: a depth below the surface should be negative, and a person walking should not move 40 miles per hour.
What is the difference between 7.NS.A.2 and 7.NS.A.3?
7.NS.A.2 is about why the rules for multiplying and dividing signed numbers work, and 7.NS.A.3 is about using all four operations to solve problems. In 7.NS.A.2, students explain, for example, why a negative times a negative is positive. In 7.NS.A.3, they use that rule, along with the addition and subtraction rules, inside real-world problems.
How can parents help with 7.NS.A.3 at home?
Parents can ask everyday questions that use negative numbers and fractions. For example: "The bank balance was $20, we spent $32.50, then got $15 back. What is it now?" or "This recipe serves 6, but we need 4 servings. How much of each ingredient?" Ask your child to estimate first and to explain the sign of the answer.
Why does the order of operations matter with negative numbers?
The order of operations decides which numbers are multiplied, and that changes both the size and the sign of the answer. For example, -6 + 2 × (-5) is -6 + (-10) = -16, but working left to right gives (-4)(-5) = 20. Students should multiply and divide before they add and subtract, unless parentheses say otherwise.
What are common mistakes with rational number word problems?
Common mistakes are writing a drop or a debt as a positive number, adding before multiplying, and multiplying the parts of a complex fraction instead of dividing them. Another is forgetting that subtracting a negative number is the same as adding its opposite, as in 1,000 - (-50) = 1,050. An estimate catches many of these errors.
07
Related Standards
6 standards
These standards connect to 7.NS.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.NS.A.1Prerequisite
Interpret and compute quotients of fractions and solve word problems with them
