Dakota has four yardsticks

2022 Handbook

2022 Test A Math · Question 29 of 57

Question

Dakota has four yardsticks and two 12-inch rulers. If she lays all four of the yardsticks and both rulers end-to-end, what is the maximum number of feet she can measure accurately at once?

A12 ft
B14 ft
C16 ft
D18 ft

Show the answer

Correct answer: B

A yardstick is 1 yard in length.

$1\text{ yd} = 3\text{ ft}$

$4\text{ yd} = 12\text{ ft}$

A 12-inch ruler is 1 foot in length.

Two 12-inch rulers are 2 feet in length.

$12\text{ ft} + 2\text{ ft} = 14\text{ ft}$

Question analysis

Dakota lays four yardsticks and two 12-inch rulers end to end and wants to know how many feet she can measure at once. Students convert yards and inches to feet and add the lengths. Mixing up the units, or leaving out one kind of measuring stick, is the main risk in this unit conversion question.

NYS standards: 6.RP.3d converting measurement units
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: Below grade 7 Converting yards and inches to feet is grade 6 measurement work. With friendly numbers and only two conversions to add, this is easier than typical grade 7 tasks, though the wording about measuring at once may slow some readers.
Estimated difficulty: Routine: one familiar step, little to misread.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: Something you must already know minor
Solution: 2 steps
Reading level: Question and choices: about grade 3, 37 words.
Corrected from the DOE handbook: A grammatical slip 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

This is a unit conversion dressed as a measuring story: four yardsticks and two rulers are laid end to end, so the total is simply a sum of lengths. Points are lost by forgetting that the rulers count too, or by giving the rulers the yardsticks' length. Writing every piece in feet before adding, 3 + 3 + 3 + 3 + 1 + 1, shows the total at a glance.

Something you must already know minor Working it out
Neither conversion is given: a yardstick is 1 yard, or 3 feet, and each 12-inch ruler is 1 foot, so 4 yards and 24 inches become 12 + 2 feet. (Anderson & Krathwohl, 2001)

Strategy: Change every length to feet before adding: a yard is 3 feet and 12 inches is 1 foot. Then total the six pieces laid end to end.

Traps

Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

A Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: strong
Converts the four yardsticks to 12 feet and stops, forgetting to add the 2 feet of the two 12-inch rulers laid at the end.
C Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
Reveals a surface strategy Expected pull: weak
No common working gives 16 feet; it would need each 12-inch ruler to be 2 feet long, so it mostly draws guesses between nearby totals.
D Units or scale (King et al., 2004)
Reveals a slip Expected pull: weak
Converts the rulers with the yard factor too, counting all six sticks as 3 feet long, 6 × 3 = 18, though a 12-inch ruler is 1 foot.

What was corrected from the DOE handbook, and why

Explanation: The handbook says “Two 12-inch rulers is 2 feet in length.” (p. 142, question 86), repeating the verb of the line before it, “A 12-inch ruler is 1 foot in length.” The site now reads “Two 12-inch rulers are 2 feet in length.” The working and the answer, 14 ft, are unchanged.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Yards to feet: $f = 3 \times y$; y is a length in yards. f is the same length in feet. One yard is 3 feet, so each yard adds 3 feet. A yardstick is exactly 1 yard long.
Inches to feet: $f = \frac{i}{12}$; i is a length in inches. f is the same length in feet. One foot is 12 inches, so divide the inches by 12. A 12-inch ruler is 1 foot long.

Sort out the question
Dakota lays 4 yardsticks and 2 rulers end to end, in one long line. End to end means each piece starts where the last one stops. So the total length is the sum of all six pieces. You need it in feet. A yardstick is 1 yard long. Each ruler is 12 inches long.
Change the yardsticks to feet
The four yardsticks make 4 yards.
$f = 3 \times y$
$f = 3 \times 4 = 12$
The yardsticks make 12 feet. Watch out: don’t stop here. 12 feet is choice A, but the rulers count too.
Change the rulers to feet
Two rulers of 12 inches make 24 inches.
$f = \frac{i}{12} = \frac{24}{12} = 2$
The rulers make 2 feet. Watch out: a ruler is not as long as a yardstick. Counting all six pieces as 3 feet gives $6 \times 3 = 18$ feet, choice D.
Add the lengths
$12 + 2 = 14$
Dakota can measure 14 feet at once. So the answer is B, 14 ft.

Check your answer

List each piece in feet: $3 + 3 + 3 + 3 + 1 + 1 = 14$ . That matches. Or use inches. A yard is 36 inches, so 4 yards is $4 \times 36 = 144$ inches. Add the rulers: $144 + 24 = 168$ inches. Then $168 \div 12 = 14$ feet.

The original DOE question

Question 86 as printed on page 86 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2023 Admissions, page 86, question 86. Open page 86 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D). This is the DOE’s version, before the corrections listed in the analysis.