7.EE.B.3Common CoreMathExpressions and EquationsGrade 7
7.EE.B.3: Multi-Step Problems with Rational Numbers
In plain English: 7.EE.B.3 is the Common Core grade 7 math standard that asks students to solve multi-step real-life problems with positive and negative whole numbers, fractions and decimals. Students choose a helpful form for each number, use properties of operations to compute, and check that the answer is reasonable with mental math and estimation.
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Also written as 7.EE.3 · Official standard
Students solve real-life problems that take more than one step and use rational numbers: numbers that can be written as a fraction of two integers (whole numbers and their opposites), such as 4, -2.5, 3/8 and -1 1/4. The numbers in one problem often come in different forms, so students learn to convert (rewrite a number in another form, such as 3/4 = 0.75 = 75%) when it makes the work easier. They use properties of operations, the rules that let us reorder, regroup or split numbers without changing the answer, to compute with less effort.
Every problem starts and ends with an estimate, a quick answer found with rounded numbers. Students estimate first, compute exactly, then compare the two to decide whether the exact answer is reasonable (about the right size, with the right sign and units). The lesson works through both official examples: a 10% raise on $25 an hour, and centering a 9 3/4-inch towel bar on a 27 1/2-inch door. Students also choose their tools on purpose: mental math, paper, a ruler, a number line or a calculator.
Learning Objectives
By the end of this lesson, students will be able to:
Solve multi-step real-life problems with positive and negative whole numbers, fractions and decimals
Convert between fractions, decimals and percents, and choose the form that makes a problem easier
Use the commutative, associative and distributive properties to compute mentally
Estimate an answer before computing and use the estimate to judge whether the exact answer is reasonable
Choose a tool on purpose (mental math, paper, a ruler, a number line or a calculator) and explain the choice
Prior Knowledge Required
Students should already be comfortable with:
Adding, subtracting, multiplying and dividing positive and negative rational numbers 7.NS.A.3
Changing a fraction to a decimal by dividing, for example 3/20 = 0.15 7.NS.A.2d
Finding a percent of a quantity, for example 30% of 60 is 18 6.RP.A.3c
Adding, subtracting, multiplying and dividing multi-digit decimals 6.NS.B.3
Show this question and give students one minute with no calculator and no paper:
Warm-Up Prompt
"A pack of juice boxes costs $4.89. Is $20 enough to buy 4 packs? Answer in your head, then explain how you know."
Collect a few explanations. A strong one rounds $4.89 up to $5: 4 × $5 = $20, and each pack costs a little less than $5, so 4 packs cost a little less than $20. Yes, $20 is enough. Then find the exact cost together: 4 × $4.89 = $19.56. Point out that rounding up made the estimate a little too big, and that is why it was a safe check. Tell students this is the habit for the whole lesson: estimate first, then compute, then compare.
Direct Instruction20 minutes
Show the four steps students use on every multi-step problem:
Estimate: round each number to a nearby easy number and find a quick answer in your head.
Choose a form: decide whether fractions, decimals or percents make this problem easier, and convert where it helps.
Compute exactly: work one step at a time, with the right tool (mental math, paper, a ruler or a calculator).
Check: compare with the estimate. Is the answer about the right size, with the right sign and units? Then answer in a sentence.
Three properties of operations make mental math easier. The commutative property says you can add or multiply in any order: 4 × 2.9 × 25 = 4 × 25 × 2.9. The associative property says you can group numbers in any way when you add or multiply: (4 × 25) × 2.9 = 100 × 2.9 = 290. The distributive property says multiplying a sum or difference is the same as multiplying each part: 3 × 19.9 = 3 × (20 - 0.1) = 60 - 0.3 = 59.7. Work through the examples below, and for each one write the estimate next to the exact answer.
Percent raise, the official example
A woman makes $25 an hour and gets a 10% raise. How much more does she make each hour, and what is her new hourly pay?
Equation: Convert: 10% = 10/100 = 1/10. One tenth of $25 is 25 ÷ 10 = $2.50 more each hour. New pay: $25 + $2.50 = $27.50 an hour. Another way: the new pay is 110% of the old pay, and 1.10 × 25 = 27.50. Estimate check: 10% of $25 is a little less than $3, and $27.50 is a little less than $28.
Fractions and an estimate check, the official example
You want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide. How far from each edge of the door does the bar go?
Equation: Estimate: 27 1/2 is about 28 and 9 3/4 is about 10, so (28 - 10) ÷ 2 = 9 inches. Exact: 27 1/2 - 9 3/4 = 27 2/4 - 9 3/4 = 17 3/4 inches of door is not covered. Split it in half: 17 3/4 ÷ 2 = (71/4) × (1/2) = 71/8 = 8 7/8 inches from each edge. The estimate of about 9 inches is close to 8 7/8, so the exact answer is reasonable.
Negative numbers and a mixed number (a whole number and a fraction together)
At 6 a.m. the temperature was -6.5°F. It rose 3 1/4°F every hour for 4 hours. What was the temperature at 10 a.m.?
Equation: Convert: 3 1/4 = 3.25, so every number is a decimal. Total rise: 4 × 3.25 = 13°F. New temperature: -6.5 + 13 = 6.5°F. Estimate: -7 + 4 × 3 = 5°F, close to 6.5°F, and both are positive, so the answer is reasonable (see Diagram 2).
Distributive property and sales tax
Omar buys 4 T-shirts at $12.95 each. The sales tax is 8%. He has $55. Is that enough?
Equation: Shirts: 4 × 12.95 = 4 × (13 - 0.05) = 52 - 0.20 = $51.80. Tax: 0.08 × 51.80 = 4.144, about $4.14. Total: 51.80 + 4.14 = $55.94. No, $55 is not enough; he is 94 cents short. Estimate: $52 plus 8% of about $50 ($4) is about $56, which also says no.
Fractions and decimals in one problem
A trail is 5.6 miles long. Mia hikes 2/5 of the trail before lunch and 1 3/4 miles after lunch. How much of the trail is left?
Equation: Convert: 2/5 = 0.4 and 1 3/4 = 1.75. Before lunch: 0.4 × 5.6 = 2.24 miles. Hiked in all: 2.24 + 1.75 = 3.99 miles. Left: 5.6 - 3.99 = 1.61 miles. Estimate: 2/5 of 5.6 is a little more than 2, so she hiked about 2.2 + 1.8 = 4 miles, and about 5.6 - 4 = 1.6 miles are left.
After the towel bar example, show Diagram 1 and ask: "Why is the estimate a little more than the exact answer?" (Rounding the door up adds 1/2 inch, and rounding the bar up takes 1/4 inch away from the uncovered part, so the uncovered part grows by 1/4 inch and each side by 1/8 inch.) After the temperature example, show Diagram 2: each jump is +3 1/4, and the fourth jump lands at 6.5.
Discuss when to convert. Fractions like 1/10, 1/4 and 1/2 are easy to use in mental math, while money and measurements with a decimal point are easier as decimals. A mixed number such as 9 3/4 can stay a mixed number when you subtract, or become 9.75 when you use a calculator. Remind students of a quick test for any answer: a problem about something that got colder should not end with a warmer temperature, and a sale price should be less than the original price.
Guided Practice15 minutes
Pairs solve four problems. For each one, Partner A writes an estimate before Partner B computes. Then they swap roles. Circulate and ask each pair: "Which form did you use, and why?"
Guided practice problems with answers
Problem
Answer
A phone plan costs $35 a month. The price goes up 20%. What is the new monthly price?
20% = 1/5, and 35 ÷ 5 = 7, so the new price is $35 + $7 = $42
A curtain rod 32 3/4 inches long is centered over a window 45 1/2 inches wide. How far does the window extend past each end of the rod?
A pond is 2.4 feet below its normal level (-2.4 ft). It rises 3/4 ft in each of 3 storms, then drops 0.6 ft in a dry week. Where is the water level now?
-2.4 + 3(0.75) - 0.6 = -0.75, so the pond is 3/4 ft below normal
Listen for these errors: forgetting to split the leftover space in half on the curtain problem, adding the storms without the negative start, and writing the 20% increase as the new price ($7) instead of adding it to $35. Ask: "Does -0.75 make sense? The pond rose 2.25 ft but started 2.4 ft low, so it should still be a little below normal."
Independent Practice15 minutes
Students solve six problems on their own. For each, they write an estimate first, then the exact answer, then one word: "close" or "far". If the answer is far from the estimate, they look for the mistake.
Independent practice problems with answers
Problem
Answer
A $64 video game has 7.5% sales tax. What is the total cost?
0.075 × 64 = 4.80, so $68.80
(-3/4) × 12.8 + 5
-9.6 + 5 = -4.6
A recipe uses 2 1/4 cups of flour per batch. How many full batches can you make with 10 cups, and how much flour is left?
4 batches use 9 cups, so 4 full batches and 1 cup left
-18.6 + 7.25 - (-4 1/2)
-18.6 + 7.25 + 4.5 = -6.85
Write 7/8 as a decimal and as a percent
0.875 and 87.5%
Estimate, then compute: 19.8 × 5.1
Estimate 20 × 5 = 100; exact 100.98
Closure5 minutes
Exit ticket: (1) A $40 sweater is 15% off. Estimate the sale price, then find it exactly. (Answer: 10% of $40 is $4 and 5% is $2, so the discount is $6 and the sale price is $34.) (2) Is -2 3/4 + 1.5 positive or negative? Decide before you compute, then compute. (Answer: negative, because the negative number is farther from 0; -2.75 + 1.5 = -1.25.) (3) In one sentence: how does an estimate help you check an exact answer?
Differentiation Strategies
For Struggling Students
Give a conversion strip that shows common forms side by side, for example 1/4 = 0.25 = 25% and 1/10 = 0.1 = 10%
Start each problem with whole numbers (such as a $20 price and a 10% raise), then change one number to a fraction or decimal once the steps feel easy
Use a number line, like Diagram 2, for every problem with negative numbers, and draw one jump per step
For Advanced Students
Ask students to write a multi-step problem where the estimate and the exact answer lead to different decisions, for example a budget that the estimate says is enough but the exact total is not
Compare two raises: 10% now, or 5% now and 5% more in six months. Which gives the higher hourly pay at the end of the year, and which pays more money over the whole year?
Ask students to find a subtraction where rounding both numbers up gives an estimate that is too small, and to explain why
Assessment Guidance
What to Look For
Check that every answer has an estimate written next to it, and that students can say why the two are close or far apart. Look for sensible conversions (1/10 as 0.1 for money, 3/4 kept as a fraction when halving an inch measurement), correct signs in problems with negative numbers, and a final sentence with units. When students use a property of operations for mental math, ask them to name it.
02
Classroom Activities
3 Activities
1
Estimate First Relay
15 minPairs
Each pair gets 6 problem cards face down. Partner A turns over a card and has 20 seconds to say an estimate out loud. Partner B writes it down, then both solve the card exactly on paper. They mark the card "close" if the exact answer is within about 10% of the estimate, and swap roles for the next card.
Problem Cards (answer key)
E1: A school buys 28 chairs at $19.75 each. What is the total? (Estimate 28 × 20 = $560; exact $553)
E2: A hot-air balloon is 1,240 ft above the ground and descends 85.5 ft each minute for 3 minutes. How high is it now? (Estimate 1,240 - 270 = 970 ft; exact 983.5 ft)
E3: At 7 a.m. the air in a mountain town is -17.5°C. It warms 2 1/2°C per hour for 5 hours. What is the temperature at noon? (Estimate -18 + 15 = -3°C; exact -5°C)
E4: A $56 jacket is 30% off. What is the sale price? (Estimate 30% of 60 = 18, so about $38; exact $39.20)
E5: A ribbon 6 1/4 yards long is cut into 5 equal pieces. How long is each piece? (Estimate 6 ÷ 5, a little more than 1 yard; exact 1 1/4 yards)
E6: Jin's savings account has $86.40. He withdraws $12.60 each week for 4 weeks. How much is left? (Estimate 86 - 4 × 13 = $34; exact $36)
Discussion Questions
On which cards was your estimate a little too high, and on which was it too low? What did your rounding do?
Card E3 has a negative answer and card E2 has a positive one. How did your estimate tell you the sign before you computed?
Which card was the easiest to do exactly in your head, and which property or conversion did you use?
Modification for Distance Learning
Put the 6 cards on a shared slide with a timer. Each student types an estimate in the chat before anyone computes, then the class compares estimates with the exact answers.
2
Pick Your Tool
15 minGroups of 3
Each group sorts 6 problems into three piles by the best tool: mental math, paper and pencil, or a calculator. Then each student solves two problems with the chosen tool, and the group checks every answer with an estimate. The goal is to practice "using tools strategically": choosing a tool on purpose, not always the calculator.
T4: 6.5% sales tax on a $38.40 order (calculator: 0.065 × 38.40 = 2.496, about $2.50)
T5: 14.7 × 3.18 (calculator: 46.746)
T6: -48 ÷ (-1.5) (paper or mental math: 480 ÷ 15 = 32, and the quotient, the answer to a division, is positive)
Procedure
Sort the 6 cards into three piles and write one reason for each choice
Each student solves two cards with the tool the group chose
Pass your answers to the right; the next student checks each one with an estimate
Discuss any card where group members disagree about the tool
Discussion Questions
Cards T1 and T2 are in the mental math pile. What makes them easy?
T4 comes out to $2.496. How should you round it for money, and why?
Is a calculator answer always right? How did the estimate protect you?
Challenge Variation
Groups write one new card for each pile and trade with another group, who must agree or disagree with the pile.
3
Center It
20 minPairs
This activity follows the official towel bar example with real measuring. Pairs choose two objects in the classroom, such as a poster to center on a bulletin board or a strip of tape to center on a desk. They measure to the nearest 1/8 inch, estimate the distance from each edge, compute it exactly, then place the object and check with the ruler.
Procedure
Measure the wide object (the board or desk) and the object you will center on it, to the nearest 1/8 inch
Estimate the distance from each edge with rounded numbers, and write it down
Compute exactly: subtract the two widths, then divide by 2
Mark the distance with painter's tape, place the object, and measure both sides to check that they match
A strip of tape 10 3/8 inches long on a desk 23 1/4 inches wide: estimate (23 - 10) ÷ 2 = 6 1/2 inches; exact 12 7/8 ÷ 2 = 6 7/16 inches
Discussion Questions
In the poster sample, the estimate is 1/4 inch more than the exact answer. Which rounding made it bigger?
Would you rather work in fractions or decimals when your ruler is marked in eighths? Why?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Centering the Towel Bar
The official towel bar example drawn to scale across the door (20 pixels per inch): a 9 3/4-inch bar centered on a 27 1/2-inch door leaves 8 7/8 inches on each side. The estimate, about 9 inches, is close to the exact answer, so the exact answer is reasonable.
Diagram 2: Temperature Jumps on a Number Line
Worked example 3 on a number line drawn to scale: the temperature starts at -6.5°F and rises 3 1/4°F each hour. Four jumps pass through -3.25 and 0 and land at 6.5°F at 10 a.m. The estimate, 5°F, is close and has the same sign.
04
Homework Assignment
~30 min
7.EE.B.3 Homework: Multi-Step Problems with Rational Numbers
Directions: Show all work. Write an estimate before every exact answer, and say whether the exact answer is reasonable. Answer word problems in a full sentence with units.
Part 1: Computing with Numbers in Any Form (Problems 1-2)
Compute each one, and name the property or conversion you used: (a) -2.4 + 5 3/5 (b) (1/4) × (-36) × 0.5 (c) 6 × 4.98, using the distributive property (d) Write 7/20 as a decimal and as a percent.
Estimate first, then compute exactly: (a) 3 7/8 × 4.2 (b) -12.15 ÷ 2.7 (c) 49.6 - 18 3/4. For each one, say whether the exact answer is close to your estimate.
Part 2: Real-Life Problems (Problems 3-4)
A bakery pays its workers $16.40 an hour and gives everyone a 5% raise. What is the new hourly pay? How much does a worker earn for a 6 1/2-hour shift at the new pay?
A bulletin board is 72 1/4 inches wide. You want to center a banner that is 40 5/8 inches long. Estimate, then find exactly, how far each end of the banner is from the edge of the board.
Part 3: Negative Numbers and Reasonable Answers (Problems 5-6)
At 5 a.m. in Fairbanks, Alaska, the temperature was -14.5°F. It rose 2 3/4°F each hour for 6 hours, then fell 1.5°F each hour for 2 hours. Estimate, then find the temperature at the end of the 8 hours.
Sam says a $48 pair of shoes at 25% off, plus 6% sales tax, costs $50.88. Without an exact computation, explain why his answer cannot be right. Then find the correct cost and describe the step Sam left out.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Estimation
Estimate written first for every problem and compared with the exact answer
Estimates written but not compared
No estimates
Accuracy
All computations correct, with the right signs
One computation or sign error
Several errors
Forms and Properties
Conversions and properties used and named correctly
Correct work but property or conversion not named
Forms mixed up or conversions wrong
Answer in Context
Full sentence with units that answers the question
Correct number without units or sentence
Answer missing or does not fit the problem
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Estimate before you choose an answer. 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
Which fraction is equal to 37.5%?
Answer: B
37.5% = 37.5/100 = 0.375, and 0.375 = 3/8 (check: 3 ÷ 8 = 0.375). Choice A moves the decimal point only one place (3.75 = 15/4). Choice C moves it three places (0.0375 = 3/80). Choice D drops the .5 and uses 37%.
Question 2 of 20 · Multiple Choice
A worker earns $18 an hour and gets a 15% raise. What is the new hourly pay?
Answer: C
15% of $18 is 0.15 × 18 = $2.70, so the new pay is $18 + $2.70 = $20.70. Choice A is only the raise, not the new pay. Choice B uses 1.5% (0.015 × 18 = 0.27) instead of 15%. Choice D adds $15 instead of 15%.
Question 3 of 20 · Multiple Choice
Which is the best estimate of 7.9 × 3 1/8?
Answer: D
7.9 is close to 8, and 3 1/8 is close to 3, so 8 × 3 = 24 (the exact answer is 24.6875). Choice A rounds 7.9 down to 7, which is farther away than 8. Choice B rounds 3 1/8 up to 4. Choice C puts the decimal point in the wrong place.
Question 4 of 20 · Multiple Choice
Compute -3.6 + 1 1/2 × 4.
Answer: B
Multiply first: 1 1/2 × 4 = 1.5 × 4 = 6. Then add: -3.6 + 6 = 2.4. Choice A adds first and then multiplies: (-3.6 + 1.5) × 4 = -8.4. Choice C subtracts 6 instead of adding it. Choice D drops the negative sign of -3.6.
Question 5 of 20 · Multiple Choice
Which expression is equal to 4 × 7.3 × 25 and easy to compute mentally?
Answer: A
The commutative and associative properties let you multiply 4 × 25 first: 100 × 7.3 = 730. Choice B splits 7.3 into 7 + 0.3 but multiplies only the 7 by 4 and 25, which gives 700.3. Choice C adds 7.3 instead of multiplying by it (107.3). Choice D changes a product into a sum, which no property allows (129.2).
Question 6 of 20 · Multiple Choice
Which expression equals 5 × 3.98 and is easy to compute mentally?
Answer: D
3.98 = 4 - 0.02, and the distributive property multiplies both parts by 5: 5 × 4 - 5 × 0.02 = 20 - 0.10 = 19.90. Choice A multiplies only the 4 by 5. Choice B multiplies only the 3 by 5. Choice C adds 0.02 when 3.98 is less than 4.
Question 7 of 20 · Multiple Choice
A picture frame 14 1/4 inches wide is centered on a wall panel 36 1/2 inches wide. How far is each side of the frame from the edge of the panel?
Answer: C
The uncovered width is 36 1/2 - 14 1/4 = 22 1/4 inches, and half of it is on each side: 22 1/4 ÷ 2 = 11 1/8 inches. Estimate: (36 - 14) ÷ 2 = 11. Choice A forgets to split the space in half. Choice B halves the panel and ignores the frame. Choice D adds the widths instead of subtracting: (36 1/2 + 14 1/4) ÷ 2 = 25 3/8.
Question 8 of 20 · Multiple Choice
At 7 p.m. the temperature was 4.5°C. It dropped 1 1/4°C each hour for 6 hours. What was the temperature at 1 a.m.?
Answer: A
The total drop is 6 × 1.25 = 7.5°C, so the temperature is 4.5 - 7.5 = -3°C. Estimate: each drop is a little more than 1°C, so the total is a little more than 6°C, and 4.5 - 6 = -1.5, so the answer is a little below -1.5°C. Choice B adds the drop instead of subtracting. Choice C is the total drop, not the temperature. Choice D subtracts only one hour's drop.
Question 9 of 20 · Multiple Choice
Ana computes 0.48 × 52 and gets 249.6. Which estimate shows that her answer is not reasonable?
Answer: B
0.48 is close to 1/2, so the product should be close to half of 52, which is 26. Her answer is about 10 times too big (the exact answer is 24.96). Choice A rounds 0.48 to 5, which makes the same decimal-point mistake. Choice C adds instead of multiplying. Choice D rounds 0.48 to 0, which is too far off to check anything.
Question 10 of 20 · Multiple Choice
A water tank is 5/8 full. Then water filling 20% of the tank is added. How full is the tank now?
Answer: D
Convert: 5/8 = 5 ÷ 8 = 0.625 = 62.5%. Then 62.5% + 20% = 82.5%. Choice A rounds 5/8 to 0.63 (63%) before adding. Choice B adds the numerator 5 to 20. Choice C multiplies the two amounts: 0.625 × 0.20 = 0.125, or 12.5%.
Question 11 of 20 · Multiple Choice
An 11-foot board is cut into pieces that are each 1 3/4 feet long. Ignore the width of the saw cuts. How many full pieces can be cut, and how much board is left?
Answer: A
11 ÷ 1.75 is about 6.29, so only 6 full pieces fit. They use 6 × 1 3/4 = 10 1/2 feet, leaving 11 - 10 1/2 = 1/2 foot. Choice B rounds up, but 7 pieces would need 12 1/4 feet. Choice C treats the 0.29 of a piece as 0.29 foot. Choice D multiplies 11 × 1.75 = 19.25 instead of dividing and reads the result as 19 pieces and 1/4 foot.
Question 12 of 20 · Multiple Choice
Maya's lunch account was at -$14.25 because she owed the cafeteria. Her parents made three deposits of $22.50 each, and then she bought $31.80 of lunches. What is her balance now?
Answer: C
Deposits: 3 × 22.50 = $67.50. Balance: -14.25 + 67.50 - 31.80 = $21.45. Estimate: -14 + 68 - 32 = 22. Choice A treats the starting balance as +$14.25. Choice B counts only one deposit. Choice D adds the $31.80 of lunches instead of subtracting it.
Question 13 of 20 · Multiple Choice
Which problem is best to solve with mental math instead of a calculator?
Answer: A
25% is 1/4, and 1/4 of $80 is $20, which you can find in your head. Choices B, C and D have numbers that do not round to easy values without losing accuracy, so paper or a calculator (with an estimate as a check) is the better tool. For example, choice B is about 17% of $86, roughly $15.
Question 14 of 20 · Multiple Choice
A diver goes from the surface (0 m) down to -22.8 m in 6 equal steps. What is the change in depth for each step?
Answer: D
Divide the total change by the number of steps: -22.8 ÷ 6 = -3.8 m, so each step goes 3.8 m down. Estimate: -24 ÷ 6 = -4. Choice A drops the negative sign, which would mean going up. Choice B multiplies by 6 instead of dividing. Choice C adds 6 instead of dividing by 6 (-22.8 + 6 = -16.8).
Question 15 of 20 · Short Answer
A $36 video game is on sale for 1/3 off. Then 8% sales tax is added to the sale price. Estimate the total, then find it exactly.
Estimate: 1/3 off $36 leaves $24, and 8% of $24 is about $2, so about $26. Exact: 36 - (1/3)(36) = 36 - 12 = $24, and 1.08 × 24 = 25.92. The total is $25.92, close to the estimate.
Question 16 of 20 · Short Answer
Compute -5 1/4 + 3.9 - (-2.35). Show how you converted the numbers.
Convert -5 1/4 to -5.25, and change subtracting -2.35 to adding 2.35: -5.25 + 3.9 + 2.35. Add the positives first (commutative property): 3.9 + 2.35 = 6.25. Then -5.25 + 6.25 = 1.
Question 17 of 20 · Short Answer
Use the distributive property to compute 7 × 9.97 in your head. Show the steps.
9.97 = 10 - 0.03, so 7 × 9.97 = 7 × 10 - 7 × 0.03 = 70 - 0.21 = 69.79. Check: 7 × 10 = 70, and the answer should be a little less than 70.
Question 18 of 20 · Short Answer
A shelf is 32 3/4 inches wide. A clock 11 1/2 inches wide is placed in the center. Estimate, then find exactly, how far each side of the clock is from the end of the shelf.
Estimate: (33 - 12) ÷ 2 = 10 1/2 inches. Exact: 32 3/4 - 11 1/2 = 21 1/4, and 21 1/4 ÷ 2 = 85/8 = 10 5/8 inches. The exact answer is close to the estimate, so it is reasonable.
Question 19 of 20 · Short Answer
A worker who earns $22.40 an hour gets a raise equal to 1/8 of her hourly pay. How much is the raise, and what is her new hourly pay? Show one way to check it.
The raise is 22.40 ÷ 8 = $2.80, and the new pay is 22.40 + 2.80 = $25.20 an hour. Check with a decimal: 1/8 = 0.125, and 0.125 × 22.40 = 2.80. Or estimate: 1/8 of about $24 is $3, so a raise a little less than $3 is reasonable.
Question 20 of 20 · Short Answer
Leah needs 6 pieces of ribbon, each 4 3/8 feet long. She estimates 6 × 4 = 24 and buys 24 feet. Is that enough? Find the exact length and explain what her estimate missed.
Exact: 6 × 4 3/8 = 6 × 35/8 = 210/8 = 26 1/4 feet, so 24 feet is not enough; she is 2 1/4 feet short. She rounded each piece down, so her estimate was too small. When you buy supplies, round up (6 × 4 1/2 = 27 feet) so the estimate is safe.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.EE.B.3 mean?
7.EE.B.3 means students can solve real-life problems that take several steps and use positive and negative whole numbers, fractions and decimals. They convert numbers to a helpful form, use properties of operations to compute, and check each answer with an estimate. For example, if the temperature is -6.5°F at 6 a.m. and rises 3 1/4°F every hour, students convert 3 1/4 to 3.25 and find 6.5°F at 10 a.m., then check with the estimate -7 + 4 × 3 = 5.
What grade is 7.EE.B.3, and what comes after it?
7.EE.B.3 is a grade 7 standard in the Expressions and Equations domain. It uses the operations on rational numbers from 7.NS.A.3 in longer problems. Next, students write and solve equations for word problems (7.EE.B.4), then solve equations with rational coefficients in grade 8 (8.EE.C.7). In high school, choosing a sensible level of accuracy for measurements continues in HSN.Q.A.3.
How do you solve the 7.EE.B.3 towel bar example?
Subtract the bar from the door width, then split the leftover space in half. The door is 27 1/2 inches and the bar is 9 3/4 inches, so 17 3/4 inches are uncovered, and 17 3/4 ÷ 2 = 8 7/8 inches go on each side. The standard's estimate, about 9 inches, comes from (28 - 10) ÷ 2 and shows that 8 7/8 is reasonable.
What does "using tools strategically" mean in this standard?
It means choosing the tool that fits the problem, not always the same one. Mental math is best for easy numbers such as 25% of $240. Paper works well for fractions, a ruler or tape measure for real lengths, a number line for negative numbers, and a calculator for messy decimals such as 14.7 × 3.18. Students should still estimate when they use a calculator, because a typing mistake gives a wrong answer that looks exact.
When should students convert fractions to decimals or percents?
Convert when a different form makes the next step easier. Money and metric measurements usually work best as decimals. Fractions such as 1/4, 1/3 and 1/10 are often easier for mental math: 1/3 off $45 is $15, which is harder to see as 33.3%. Inch measurements marked in eighths often stay as fractions. Students should be able to explain why they chose a form.
Which properties of operations help most with mental math?
The commutative, associative and distributive properties do most of the work. The first two let you reorder and regroup a product, as in 5 × 1.7 × 20 = (5 × 20) × 1.7 = 170. The distributive property lets you split a number near a whole number: 3 × 19.9 = 3 × 20 - 3 × 0.1 = 59.7. Adding opposites first also helps: -3.5 + 8.2 + 3.5 = 8.2.
How can students tell if an answer is reasonable?
They compare it with an estimate and ask three questions: Is it about the right size? Does it have the right sign? Does it make sense in the story? A sale price must be less than the original price, a temperature after warming must be higher than before, and a length cannot be negative. If the exact answer is far from the estimate, students look for a slipped decimal point or a missing step.
What mistakes should teachers watch for in 7.EE.B.3?
A common mistake is using a percent as the final answer, such as giving the $2.50 raise as the new pay. Other frequent errors are forgetting to split leftover space in half when centering, dropping a negative sign, moving the decimal point the wrong number of places when converting a percent, and rounding down when buying supplies, which gives an estimate that is too small.
Does 7.EE.B.3 include solving equations?
Not mainly. 7.EE.B.3 is about working through multi-step problems with numbers in any form, and many students solve them with arithmetic alone. Writing and solving equations such as px + q = r for word problems is the next standard, 7.EE.B.4. Students may still work backward from an answer, which prepares them for equations.
How can parents help with 7.EE.B.3 at home?
Ask for an estimate before anyone reaches for a calculator. At the store, ask "About how much will these 3 items cost with tax?" and then compare with the receipt. When hanging a picture, ask your child to measure and find how far it must be from each end of the wall to be centered. Cooking with half or double a recipe is good practice with fractions.
07
Related Standards
6 standards
These standards connect to 7.EE.B.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.NS.A.3Prerequisite
Solve real-world problems with the four operations on rational numbers
