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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Real problems rarely arrive in one tidy number form. A training plan might list one run as 2 3/4 miles and another as 3.2 miles, a raise is given as a percent, and a recipe mixes cups and fractions of cups. Here students solve multistep problems that combine positive and negative numbers written as whole numbers, fractions and decimals, deciding along the way which form makes each step easiest.
Two habits sit at the heart of the work. The first is converting strategically: 1/4 is easier to use as 0.25 when money is involved, while 1/3 is better left as a fraction because its decimal repeats. The second is checking reasonableness with mental math and estimation before trusting an answer. If an estimate says a total should be about 7.5 miles and the calculation gives 73.25, something went wrong. Students are also expected to use tools strategically, knowing when a calculator helps and when mental computation is faster.
Adding 2 3/4 and 3.2 as '5 3.2' or similar shows students working with two forms at once. Choose one form first, for example 2.75 + 3.2.
Students who calculate straight away often accept answers that are off by a factor of ten. Requiring an estimate line before the exact work catches misplaced decimal points.
Rounding 1/3 to 0.3 in the middle of a problem builds up error. Keep exact fractions until the final step, then round the answer if the context calls for it.
Ana runs 2 3/4 miles on Monday and 3.2 miles on Tuesday. On Wednesday she runs half of Monday's distance. How far does she run in total? Estimate first.
Answer: Ana runs 7.325 miles in total, a little over 7 1/4 miles.
Model a think-aloud in which you deliberately pick the form for each step and say why. Then give problems where the same numbers appear in different forms in different places, so students practice the choosing, not only the computing.
A quick routine is 'estimate, solve, compare': students write an estimate, solve exactly and then write one sentence comparing the two. This matches what test items increasingly ask, which is to identify a reasonable answer or explain an error.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 18.90 (also accepted: 18.9, $18.90)
5% of $18 is 0.05 × 18 = $0.90. The new wage is 18 + 0.90 = $18.90.
Answer: B) About 50
4 7/8 is close to 5 and 9.9 is close to 10, so the product is about 5 × 10 = 50. The exact answer is about 48.3.
Answer: 2.75 (also accepted: 2 3/4, 11/4)
8 1/4 = 8.25, and 8.25 ÷ 3 = 2.75. Each piece is 2 3/4 feet long.
Answer: D) $11.75
The books cost 3 × 12.75 = $38.25. Then 50 - 38.25 = $11.75 is left.
Answer: -3.5 (also accepted: -3 1/2, -7/2)
Multiply first: 0.4 × (-15) = -6. Then -6 + 2.5 = -3.5.
Answer: B) $18
31% is close to 30% and $59.80 is close to $60. 30% of $60 is $18. The exact value is about $18.54.
A full lesson with slides, activities and an exit ticket on multistep problems with numbers in any form, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.EE.B.3 with an answer key, ready in about a minute.
Make a worksheet →Turn multistep problems with numbers in any form into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Problems can include whole numbers, fractions, mixed numbers, decimals and percents, positive or negative, often within the same problem.
Estimating gives a target to compare against, so students can catch slips such as a misplaced decimal point and judge whether their exact answer makes sense.