Question
Mr. Chan’s lawn grows inches every 2 weeks. He mows his lawn every 2 weeks and cuts off the top inches of lawn. If Mr. Chan’s lawn was 4 inches tall at the beginning of the season, how many inches tall, in decimal form, is Mr. Chan’s lawn after 8 weeks?
Grid-in question: work out the answer and enter it as a number.
Show the answer
Correct answer: 5.5
The lawn grows inches every 2 weeks. Find the total number of inches it grows in 8 weeks.
The lawn was 4 inches tall at the beginning of the season.
Mr. Chan mows his lawn every 2 weeks and cuts off the top inches. Find the total number of inches that are cut off in 8 weeks.
The height after 8 weeks is the total number of inches grown minus the total number of inches cut off.
Question analysis
This multistep word problem follows Mr. Chan's lawn over 8 weeks: it grows a mixed number of inches every 2 weeks, and he mows a mixed number of inches off on the same schedule. Students combine the growth, the cuts and the starting height and give the final height as a decimal. Leaving out one of those three pieces is the usual error.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Grade 7 students solve multi-step problems with mixed numbers and convert between fractions and decimals. This one strings four such steps together, including counting the 2-week periods, which is typical grade 7 work done carefully.
- Estimated difficulty
- Moderate: several steps, or a close reading with one strong distractor.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Many steps moderate
- Error-prone arithmetic moderate
- Turning words into math minor
- An answer in an unexpected form minor
- Solution
- 4 steps
- Reading level
- Question and choices: about grade 3, 49 words.
- Corrected from the DOE handbook
- A misspelled name in the explanation was corrected.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
No single step is hard, but three parts must all reach the final height: the starting 4 inches, four rounds of growth and four rounds of mowing. Leaving out any one of them gives a clean decimal that can be gridded without a second thought. Working one 2-week period at a time, in which the lawn gains 3/8 inch net, keeps the bookkeeping short.
Many steps moderate Working it out
Four pieces have to be combined: count the four 2-week periods in 8 weeks, total the growth, total the mowing, and start from the lawn's first 4 inches. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)
Error-prone arithmetic moderate Working it out
The amounts are mixed numbers with unlike denominators, 2 1/8 and 1 3/4, which must be multiplied by 4 or subtracted and then gridded as a decimal. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
Turning words into math minor Understanding the question
The phrase "every 2 weeks" governs both the growing and the mowing, so 8 weeks means four rounds of each, not eight. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
An answer in an unexpected form minor Choosing the answer
As a grid-in it gives no choices, so a height that leaves out the starting 4 inches, or leaves out the mowing, looks as finished as the right one. (Enright, Morley & Sheehan, 2002)
Strategy: Find the net change for one 2-week period by subtracting the cut from the growth, multiply by the number of such periods in 8 weeks, and add the starting height.
Traps
- Stops one step early: A choice is a number from partway through the solution.
- Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
The wrong answers, classified
1.5 Stops one step early (King et al., 2004; Gierl et al., 2017)
Finds the net gain of 3/8 inch per period and multiplies by the 4 periods, 1.5 inches, but stops before adding the lawn's starting height of 4 inches.
12.5 Stops one step early (King et al., 2004; Gierl et al., 2017)
Adds four periods of growth, 8.5 inches, to the starting 4 inches and stops there, never subtracting the 1 3/4 inches cut at each of the four mowings.
7 Misreads the question (Hegarty, Mayer & Monk, 1995)
Applies the net gain of 3/8 inch to each of the 8 weeks instead of each 2-week period, 8 × 3/8 = 3, then adds the starting 4.
What was corrected from the DOE handbook, and why
- Explanation: The handbook’s explanation says “Mr. Chen mows his lawn every 2 weeks.” The question calls him Mr. Chan throughout, so the site now reads “Mr. Chan.”
Worked solution, step by step
Formulas used
- Mixed number to decimal
- A mixed number has a whole part, a, and a fraction part, b over c. To write it as a decimal, divide b by c to get the fraction part as a decimal. Then add the whole part. For example, one and a half is 1 + 0.5 = 1.5.
- Repeated amount
- n is how many times something happens. a is the amount each time. Adding the same amount n times is the same as multiplying it by n.
Sort out the question
The lawn starts at 4 inches tall. Every 2 weeks it grows inches. Every 2 weeks Mr. Chan also cuts off inches. You need the height after 8 weeks, as a decimal. Three things matter: the start, the growing and the cutting.
Count the 2-week periods
Both the growing and the cutting happen every 2 weeks. So count how many 2-week periods fit in 8 weeks.
There are 4 periods. The lawn grows 4 times and is cut 4 times. Watch out: do not count 8 rounds, one for each week. That leads to the wrong answer 7.
Change the mixed numbers to decimals
The answer must be a decimal, so change both amounts now.
Each period, the lawn grows 2.125 inches and loses 1.75 inches.
Add up the growing
The lawn grows 2.125 inches, 4 times.
It grows 8.5 inches in all. Add the starting height of 4 inches.
Without mowing, the lawn would be 12.5 inches tall. Watch out: do not forget the starting 4 inches. Leaving them out leads to 1.5.
Add up the cutting
Mr. Chan cuts off 1.75 inches, 4 times.
He cuts off 7 inches in all.
Subtract the cutting
The lawn is 5.5 inches tall after 8 weeks. Watch out: 12.5 is not the answer. It leaves out the mowing. So you would grid 5.5.
Check your answer
Work one period at a time. Each period the lawn gains inch. In 4 periods that is inches. Add the starting 4 inches: . Same answer.
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The original DOE question
