SHSAT Sunset

A graph shows the proportional relationship

2024 Handbook

2024 Test A Math · Question 46 of 57

Question

A graph shows the proportional relationship between the number of test questions a student gets correct, x, and the student’s test score, y. The ordered pair (1,54)\left(1, \frac{5}{4}\right) is on the graph. What does the y-coordinate of the ordered pair represent in this relationship?

  • AThe test will last 1141\frac{1}{4} hours.
  • BEach test question is worth 1141\frac{1}{4} points.
  • CAn average student can answer 5 questions in 4 minutes.
  • DA student who answers 5 questions correctly will earn 4 points.
Show the answer

Correct answer: B

The graph represents a proportional relationship, which means the y-intercept of the graph is (0, 0).

This means that the point (1,54)\left(1, \frac{5}{4}\right) represents the unit rate for this situation.

In this situation, x represents the number of questions answered correctly, and y represents the number of points received.

Therefore, for each question answered correctly, the student receives 54\frac{5}{4}, or 1141\frac{1}{4}, points.

Question analysis

A proportional relationship links the number of test questions a student gets right, x, to the test score, y, and the point (1, 5/4) lies on its graph. Students decide what the y-coordinate of that point means in the situation. The choices reuse the numbers 5, 4 and 1 1/4 in different ways, so the meaning of each coordinate must be kept straight.

NYS standards
  • 7.RP.2d points on proportional graphs
  • 7.RP.2b constant of proportionality
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Explaining what a point on the graph of a proportional relationship means is named in the grade 7 standards. This item is a direct case, with the graph described in words rather than drawn and only one point to interpret.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
2 steps
Reading level
Question and choices: about grade 7, 73 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 question checks one idea: when x is 1, the y-coordinate is the score for one correct answer, the unit rate. The pitfall is reading 5/4 as two separate numbers, 5 and 4, and attaching them to questions and points the wrong way round. Ruling out the choices about hours and minutes is quick, since the relationship never mentions time.

  1. Variables instead of numbers minor Understanding the question

    No graph is drawn: the relationship and the point (1, 5/4) exist only in words, so students must say for themselves that x counts correct answers and y counts points. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)

  2. An answer in an unexpected form minor Choosing the answer

    The point gives 5/4, but two choices write it as 1 1/4 and two split it into 5 and 4, so the right reading must be recognized in another form. (Enright, Morley & Sheehan, 2002)

Strategy: Say in words what x and y stand for, then read the ordered pair as a sentence: when x has the first value, y has the second.

Traps

  • Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
  • Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Proportional relationship
y=k×xy = k \times x
x and y are two amounts that change together, here right answers and the test score. k is the constant of proportionality: how much y goes with 1 of x. In a proportional relationship, y is always the same number times x. So when x is 1, y equals k.

  1. Sort out the question

    x is the number of questions a student gets right. y is the student’s test score, in points. An ordered pair such as (1,54)\left(1, \frac{5}{4}\right) lists x first and y second. The y-coordinate is the second number, 54\frac{5}{4}. You need what that number means.

  2. Read the point as a sentence

    In the point, x is 1 and y is 54\frac{5}{4}. Put those into words: a student who gets 1 question right scores 54\frac{5}{4} points.

  3. Use the proportional rule

    The relationship is proportional. So the score is always the same number times the questions right.

    y=k×xy = k \times x

    54=k×1\frac{5}{4} = k \times 1

    So k is 54\frac{5}{4}. Every right answer adds the same 54\frac{5}{4} points. That makes 54\frac{5}{4} the points for each question.

  4. Write it as a mixed number

    A mixed number is a whole number plus a fraction. Four quarters make 1 whole, so five quarters are 1 whole and 1 quarter.

    54=114\frac{5}{4} = 1\frac{1}{4}

    Each question right is worth 1141\frac{1}{4} points.

  5. Match the choices

    Choice B says each question is worth 1141\frac{1}{4} points. That matches. Watch out: choice D splits 54\frac{5}{4} into 5 and 4. But 5 right answers earn 5×54=6145 \times \frac{5}{4} = 6\frac{1}{4} points, not 4. Choices A and C talk about time, which this relationship does not measure.

    So the answer is B, each test question is worth 1141\frac{1}{4} points.

Check your answer

Try another point. A student with 4 right answers scores 4×54=54 \times \frac{5}{4} = 5 points. That fits B, since four questions at 1141\frac{1}{4} points each make 5 points. It does not fit D, which says 5 questions earn 4 points.

More questions like this

The original DOE question

Question 103 as printed on page 88 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 88, question 103. Open page 88 of the handbook (PDF)