Also in 2025 Practice Test A Math, question 42.
Question
When a student buys 4 notebooks, the total cost is $12. Each notebook costs the same amount. Select each point on the graph that best represents y, the total cost of x notebooks.
Select all the correct answers.
On the test you click the points in the figure; here, choose them below. The DOE has not published how answers with several parts are scored; this site counts one right only when every part is right.
- A(1, 1)
- B(1, 3)
- C(2, 5)
- D(3, 7)
- E(3, 9)
- F(4, 12)
- G(5, 14)
Show the answer
Correct answer: B, E, and F
Solution:
There are three correct points: (1, 3), (3, 9), and (4, 12).
Find the cost per notebook: .
The equation of the line is , where y is the total cost of x notebooks.
Check each point by plugging the x-value into the equation . If the y-value from the equation matches the y-value of the point, then the point is on the line .
Check all seven points:
(1, 1) y-values do not match; incorrect point
(1, 3) y-values match; CORRECT point
(2, 5) y-values do not match; incorrect point
(3, 7) y-values do not match; incorrect point
(3, 9) y-values match; CORRECT point
(4, 12) y-values match; CORRECT point
(5, 14) y-values do not match; incorrect point
Question analysis
A graph titled Buying Notebooks shows seven points, and students select every point that fits the total cost of buying notebooks, given the cost of a few notebooks and that each costs the same. It tests finding a unit rate and recognizing points of a proportional relationship. Some points line up in tempting patterns, and the item does not say how many to select.
- NYS standards
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Finding a unit rate and deciding which points fit a proportional relationship are grade 7 skills, and the arithmetic here is easy. What adds demand is testing all seven points with no count given, since one miss or extra pick loses the item.
- Estimated difficulty
- Straightforward: two steps or one careful reading.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 5, 36 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
Finding the price is one division, 12 ÷ 4 = 3, so the question is really about checking: each chosen dot must fit y = 3x exactly, and the item does not say how many do. The graph punishes judging by eye, since a straight row of dots climbs from (1, 3) at $2 a notebook and two dots sit a single gridline off the true line. Multiplying each x by 3 and reading each y from the axis, not from the pattern, settles all seven.
Every choice must be checked moderate Choosing the answer
Seven dots must each be tested against $3 a notebook, and the item never says how many fit; one missed point or one extra loses the whole item, so no dot can be skipped. (Embretson & Wetzel, 1987)
Reading a figure, table or graph minor Understanding the question
The graph stacks two dots above 1 notebook and two above 3, and each total cost has to be read off a grid ruled at every dollar up to 15. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
A very close wrong answer minor Choosing the answer
(5, 14) sits one gridline below (5, 15), where 5 notebooks at $3 would be, and (2, 5) sits one below (2, 6), so judging by the line's direction cannot rule either out. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Strategy: Work out the cost of one notebook from the total given in the question, then multiply each point's x-coordinate by that price and keep only the points whose y-coordinate matches exactly.
Traps
- Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
- Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
The wrong answers, classified
A Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
No working from 4 notebooks for $12 makes one notebook cost $1, so this lowest dot mostly catches guesses, such as taking the dot nearest the origin as a start.
C Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Takes the straight row (1, 3), (2, 5), (3, 7) as the graph because its dots line up, though it climbs $2 a notebook and would meet the y-axis at $1, not the origin.
D Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Prefers the lower of the two dots above 3 notebooks because it continues the evenly spaced row from (1, 3) and (2, 5), instead of checking that 3 notebooks cost $9.
G Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Finds that 5 notebooks cost 5 × $3 = $15, then reads the dot above 5 as sitting on the top gridline, 15, when it is one line lower, at 14.
Worked solution, step by step
Formulas used
- Unit price
- The unit price is what one item costs. Divide the total cost by the number of items bought. This works because every notebook costs the same, so the total splits into equal parts.
- A proportional relationship
- y is the total cost in dollars, and x is the number of notebooks. k is the price of one notebook, the number you always multiply by. A point fits only if its y is exactly k times its x. The graph is a straight line through (0, 0).
Sort out the question
Four notebooks cost $12, and each one costs the same. On the graph, x is the number of notebooks, and y is the total cost in dollars. The graph shows seven points. You must select every point that fits. The question does not say how many fit, so test all seven.
Find the price of one notebook
Split the $12 equally among the 4 notebooks:
Each notebook costs $3. So the total cost is 3 times the number of notebooks:
Test each point
For each point, multiply its x by 3. Keep the point only if the result equals its y.
Choice Point 3 × x Fits? A (1, 1) 3 No B (1, 3) 3 Yes C (2, 5) 6 No D (3, 7) 9 No E (3, 9) 9 Yes F (4, 12) 12 Yes G (5, 14) 15 No Watch for look-alike points
Watch out: the dots (1, 3), (2, 5) and (3, 7) sit in a neat straight row. But that row climbs $2 per notebook, not $3. Lining up is not enough: (2, 5) and (3, 7), choices C and D, do not fit.
Watch out: 5 notebooks cost dollars. The dot above 5 sits one gridline lower, at 14. Read the y-axis carefully, and you will skip choice G.
Name the points
Three points fit: (1, 3), (3, 9) and (4, 12). On the test, you click each of these dots on the graph. On this site, you check their boxes. So the answers are B, E and F.
Check your answer
All three points fit : , and . They also line up with (0, 0), since 0 notebooks cost $0. And (4, 12) is the fact the question gave you.
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The original DOE question
