TEACHER GUIDE TO CLARIFICATION
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.3 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.
Solve Multi-Step Real Life Problems with Rational Numbers
To assist students’ assessment of the reasonableness of answers, especially problem situations involving fractional or decimal numbers, use whole-number approximations for the computation and then compare to the actual computation
Connections between performing the inverse operation and undoing the operations are appropriate here. It is appropriate to expect students to show the steps in their work. Students should be able to explain their thinking using the correct terminology for the properties and operations.
Use whole number approximations for the computation and then compare to the actual computation.
Connections between performing the inverse operation and undoing the operations are appropriate here.
Continue to build on students’ understanding and application of writing and solving one-step equations from a problem situation to multi-step problem situations. This is also the context for students to practice using rational numbers including: integers, and positive and negative fractions and decimals.
Students are expected to practice using rational numbers.
As students analyze a situation, they need to identify what operation should be completed first, then the values for that computation. Each set of the needed operation and values is determined in order. Finally an equation matching the order of operations is written. For example, Bonnie goes out to eat and buys a meal that costs $12.50 that includes a tax of $.75. She only wants to leave a tip based on the cost of the food. In this situation, students need to realize that the tax must be subtracted from the total cost before being multiplied by the percent of tip and then added back to obtain the final cost. C = (12.50 - .75)(1 + T) + .75 = 11.75(1 +T) + .75 where C = cost and T = tip.
Provide multiple opportunities for students to work with multi-step problem situations that have multiple solutions and therefore can be represented by an inequality. Students need to be aware that values can satisfy an inequality but not be appropriate for the situation, therefore limiting the solutions for that particular problem.
Explanations and Examples
7.EE.3 Students solve contextual problems using rational numbers. Students convert between fractions, decimals, and percents as needed to solve the problem. Students use estimation to justify the reasonableness of answers.
Estimation strategies for calculations with fractions and decimals extend from students’ work with whole number operations. Estimation strategies include, but are not limited to:
front-end estimation with adjusting (using the highest place value and estimating from the front end making adjustments to the estimate by taking into account the remaining amounts),
clustering around an average (when the values are close together an average value is selected and multiplied by the number of values to determine an estimate),
rounding and adjusting (students round down or round up and then adjust their estimate depending on how much the rounding affected the original values),
using friendly or compatible numbers such as factors (students seek to fit numbers together - i.e., rounding to factors and grouping numbers together that have round sums like 100 or 1000), and
using benchmark numbers that are easy to compute (student’s select close whole numbers for fractions or decimals to determine an estimate).
Estimation Strategies include:
Using friendly or compatible numbers
Using benchmark numbers
Expose students to a variety of strategies.
When assessing formally and informally, allow students to choose their strategy.
* The youth group is going on a trip to the state fair. The trip costs $52. Included in that price is $11 for a concert
ticket and the cost of 2 passes, one for the rides and one for the game booths. Each of the passes cost the same
price. Write an equation representing the cost of the trip and determine the price of one pass.
Coherence and Connections: Need to Know
By Grade 7 students start to see whole numbers, integers, and positive and negative fractions as belonging to a single system of rational numbers, and they solve multi-step problems involving rational numbers presented in various forms.
Students use mental computation and estimation to assess the reasonableness of their solutions. For example, the following statement appeared in an article about the annual migration of the Bartailed Godwit from Alaska to New Zealand: She had flown for eight days—nonstop—covering approximately 7,250 miles at an average speed of nearly 35 miles per hour. Students can make the rough mental estimate:
8 X 24 X 35 = 8 X 12 X 70 < 100 X 70 = 7000
to recognize that although this astonishing statement is in the right ballpark, the average speed is in fact greater than 35 miles per hour, suggesting that one of the numbers in the article must be wrong.
Common Core Standards Writing Team. (2011, April 22).
Progressions for the CommonCore State Standards in Mathematics(draft). 6-8, Expressions and Equations. Tucson, AZ: Institute for Mathematics and Educations, University of Arizona.
Illinois Assessment of Readiness Mathematics Evidence Tables. Retrieved from:
Evidence Statement Text
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.
Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1 + 4 = 5 + 7 = 12, even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions. Content Scope: Knowledge and skills articulated in 7.EE.3
Also check out Student Achievement Partners Coherence Map.
Number talk on What’s the Savings?
Fences Task This is a multi-step problem with possible extensions. It allows students to engage in a real life problem of planning for a fence.
Yogurt Packaging This is a multi-step problem with possible extensions. Students work with rational numbers to determine percentages, fat content and container sizes of yogurt.
You are looking to tile your basement floor. Your basement is 15 feet by 30 feet. Your options are below.
12 inch by 12 inch tiles
9 inch by 9 inch
6 inch by 6 inch
Part A: Order the options by cost.
Option 2 is cheapest at approximately $0.01/square inch, then option 1 at approximately $0.02/square inch, then option 3 at approximately $0.03/square inch.
Estimating, students should understand that the area of the option 3 tile is ¼ the area of option one tile, yet the price is not equivalent to ¼ the price.
Part B: What is the total cost to tile the floor for the most expensive option?
15 feet = 180 inches, 30 feet = 360 inches. The area of the basement floor in inches is 64,800 square inches. Each tile in option 3 is 36 square inches. 64,800÷36 = 1800. You will need 1800 tiles. 1800 • $0.99 =$1782 for a total cost.
Estimating the total cost could be 1800 tiles • $1/tile = $1800
Maribel makes $9.75 per hour at her job. Her rent for a month is $700. What is the minimum number of hours Maribel must work each week, just to pay for her rent?
$700 ÷ 9.75 = 71.79… hours. Maribel needs to work at least 72 hours each month to pay the rent. This means she needs to work 72 ÷ 4 = 18, 18 hours a week.
To estimate, students can round her pay to $10. 700 ÷ 10 = 70. Maribel needs to work about 70 hours each month. Then rounding 70 up to 72 as 72 is the next number divisible by 4 weeks give approximately 18 hours each week.
The Fouts twins have $400.00 to spend on school clothes. They have a coupon for 20% off their order of $460.00. They buy two packs of socks, 4 shirts and 4 pants. The shipping is 15% of their checkout price. Will they have enough money to purchase all the items? If not, how much more money will they need?
$460 x 0.80 = $368
$368 x 1.15 = $423.20
Inside Mathematics - Problem of the Month
Level B https://www.insidemathematics.org/sites/default/files/materials/diminishing%20return_49.pdf
Inside Mathematics - Performance Task
Math Assessment Resource