SHSAT Sunset

The outdoor temperature was 4.2° F

2023 Handbook

2023 Test B Math · Question 48 of 57

Question

The outdoor temperature was 4.2° F at 10 p.m. After 3123\frac{1}{2} hours, the outdoor temperature was −0.7° F. Which value best represents the average change in temperature per hour?

  • A−1.4° F
  • B−1.0° F
  • C1.0° F
  • D1.4° F
Show the answer

Correct answer: A

First, recognize that the temperature decreased. This means that the order of subtraction matters and that the result is a negative number:

−0.7−4.2=−4.9-0.7-4.2=-4.9

The temperature decreased 4.9 degrees. Divide by 3.5 to get the average change per hour:

−4.9÷3.5=−1.4-4.9\div 3.5=-1.4

Question analysis

The temperature was 4.2° F at 10 p.m. and −0.7° F three and a half hours later. Students find the average change in temperature per hour. It tests subtracting and dividing with negative decimals, and the choices pair two sizes of change with both signs, so the sign has to be right as well as the size.

NYS standards
  • 7.NS.3 four operations with rational numbers
  • 7.NS.1c rational number subtraction
  • 7.NS.2b rational number division
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Signed-number problems about temperature are standard grade 7 work. This one asks for an average rate, so a change that crosses zero must be divided by a mixed-number time, with the sign kept through both steps.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
2 steps
Reading level
Question and choices: about grade 5, 31 words.
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

The arithmetic is short, but signs decide every choice: two choices come from treating −0.7 as 0.7, and one comes from getting the size right and the direction wrong. Students who think of the drop in two parts, 4.2 degrees down to zero and 0.7 more, find the 4.9-degree fall; dividing by 3.5 hours and remembering the temperature fell finishes it.

  1. Error-prone arithmetic moderate Working it out

    The change crosses zero, −0.7 − 4.2 = −4.9, and then a negative decimal is divided by 3.5, from the 3 1/2 hours, with the sign kept through both steps. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Turning words into math minor Understanding the question

    The phrase “average change in temperature per hour” means the change from the 4.2° start to the −0.7° end divided by the time, final minus initial, so a drop is negative. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  3. A very close wrong answer minor Choosing the answer

    The answer, −1.4° F, and the choice 1.4° F share every digit and differ only in sign, so a correct division followed by a careless sign cannot be caught by estimating. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Subtract the starting temperature from the ending one, keeping track of the sign, then divide that change by the 3.5 hours.

Traps

  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Average change per hour
average change per hour=end−startnumber of hours\text{average change per hour} = \frac{\text{end} - \text{start}}{\text{number of hours}}
Start is the first temperature, and end is the last one. End minus start is the total change. It is negative when the temperature drops. Dividing by the number of hours shares that change equally over each hour.

  1. Sort out the question

    At 10 p.m. it was 4.2° F. After 3123\frac{1}{2} hours, it was −0.7° F. That is below zero. The temperature went down, so the change is negative. You need the average change for each hour.

  2. Find the total change

    Subtract the start from the end.

    −0.7−4.2=−4.9-0.7 - 4.2 = -4.9

    Picture a thermometer. Going from 4.2 down to 0 is a drop of 4.2 degrees. Going on to −0.7 is 0.7 more. In all, it fell 4.9 degrees.

    Watch out: 4.2 − 0.7 = 3.5 treats −0.7 as if it were above zero. That leads to choice B or C.

  3. Divide by the hours

    Write 3123\frac{1}{2} hours as 3.5 hours. Use the formula.

    average change per hour=−0.7−4.23.5=−4.93.5=−1.4\text{average change per hour} = \frac{-0.7 - 4.2}{3.5} = \frac{-4.9}{3.5} = -1.4

    To divide, think 49 ÷ 35 = 1.4, since 35 × 1.4 = 49. The change is negative, so it is −1.4 degrees per hour.

  4. Keep the sign

    The temperature fell, so the answer must be negative.

    Watch out: 1.4 has the right size, but it says the temperature went up. That is choice D.

    So the answer is A, −1.4° F.

Check your answer

Work forward. Dropping 1.4 degrees each hour for 3.5 hours is 1.4×3.5=4.91.4 \times 3.5 = 4.9 degrees. Then 4.2−4.9=−0.74.2 - 4.9 = -0.7. That is the temperature in the problem.

More questions like this

The original DOE question

Question 105 as printed on page 205 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 205, question 105. Open page 205 of the handbook (PDF)