Question
Some friends are buying bags of ice for a picnic. The graph shows the cost of the ice, in dollars, in relation to the weight, in pounds. Which statement about a point on the graph is true?
- AThe point (20, 3) means that 3 lbs of ice costs $20.00.
- BThe point (1, 0.15) means that 1 bag of ice costs $0.15.
- CThe point (10, 1.5) means that 10 lbs of ice costs $1.50.
- DThe point (0, 0) means that 0 lbs of ice will be needed if 0 people go to the picnic.
Show the answer
Correct answer: C
The x-axis represents the weight, in pounds (lbs), of ice purchased, and the y-axis represents the cost of the ice, in dollars.
The x-value in the point (10, 1.5) represents 10 lbs of ice, and the y-value represents 1.5 dollars, or $1.50. Therefore, the cost of 10 lbs of ice is $1.50.
Question analysis
Some friends buying ice for a picnic use a graph of the ice's cost in dollars against its weight in pounds, a straight line from the origin. Students pick the statement that correctly explains what a point on the graph means. Each statement names a point that really is on the line, so the units and the order of the coordinates decide.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Explaining what a point on a proportional graph means in its situation is a named grade 7 skill. The graph is clear and starts at the origin, but each statement has to be read closely against the axis labels.
- Estimated difficulty
- Straightforward: two steps or one careful reading.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly understand
- What makes it hard
- Reading a figure, table or graph moderate
- Every choice must be checked minor
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 2, 77 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
Nothing needs computing, but each statement attaches words to two numbers, and three of them attach the wrong words. The point (1, 0.15) is the unit rate students are trained to look for, and the “bags” in the stem’s story make its statement sound right, though the graph measures pounds. Saying each point aloud as so many pounds for so many dollars exposes the swapped coordinates and the extra people as well.
Reading a figure, table or graph moderate Understanding the question
Each statement must be checked against the axis labels: x is the weight of ice in pounds and y the cost in dollars, on a grid ruled every 2 pounds and every $0.30. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Every choice must be checked minor Choosing the answer
All four statements name a point that really lies on the line, (1, 0.15) included, so being on the graph rules nothing out; only the meaning given to each coordinate does. (Embretson & Wetzel, 1987)
Strategy: Read both axis labels and their units first, then for each choice say in words what its x-coordinate and its y-coordinate stand for, and compare that with the statement.
Traps
- Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
- Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
- 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.
The wrong answers, classified
A Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Reads the point (20, 3) backward, taking 20 as the price and 3 as the weight, though x counts pounds of ice and y counts dollars.
B Units or scale (King et al., 2004)
Confirms that (1, 0.15) is on the line, the cost of 1 pound, but accepts “1 bag” because the friends buy bags, though the x-axis measures pounds.
D Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Knows (0, 0) is on every proportional graph and accepts a true-sounding story about people, though the graph never counts people; the point means 0 pounds cost $0.
Worked solution, step by step
Formulas used
- Unit rate
- The cost is in dollars, and the weight is in pounds. The unit rate is the cost of just 1 pound. On a straight line through the origin, it is the same at every point. So you can find it from any point you can read.
Sort out the question
The graph is called ICE COSTS. Read both axes first:
- The x-axis, across the bottom, is the weight of the ice in pounds (lb).
- The y-axis, up the side, is the cost in dollars.
A point is written (x, y), weight first and cost second. So a point means: x pounds of ice cost y dollars. You need the statement that says this correctly.
Find the cost of one pound
The line reaches $3.00 at 20 pounds. Use the unit rate formula:
So 1 pound of ice costs $0.15. This helps you test each point.
Test the point in choice C
Find 10 on the weight axis and go straight up to the line. The line crosses there at 1.5 on the cost axis. The unit rate agrees:
So the point (10, 1.5) means 10 pounds of ice cost $1.50. That is exactly what choice C says.
Check the other statements
Each one names a real point on the line. The problem is the meaning it gives the numbers.
Watch out: choice A reads (20, 3) backward. It means 20 pounds cost $3, not 3 pounds cost $20.
Watch out: choice B is tricky. The point (1, 0.15) is on the line, but the x-axis counts pounds, not bags. So it means 1 pound costs $0.15.
Choice D talks about people, but the graph never counts people. The point (0, 0) means 0 pounds cost $0.
Give the answer
Only choice C gives the right meaning to both numbers.
So the answer is C, 10 lbs of ice costs $1.50.
Check your answer
Use halves. The point (20, 3) says 20 pounds cost $3.00. Half as much ice, 10 pounds, should cost half as much. Half of $3.00 is $1.50. That matches choice C.
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