Question
This graph shows y, the weight of a pole of length x.
What is represented by the point with coordinates (1, 4)?
- AThe unit rate is 4 ounces per foot.
- BThe y-intercept is 4.
- CA pole 4 feet long weighs 1 ounce.
- DThe length increases 4 feet for each 1 ounce of weight.
Show the answer
Correct answer: A
The x-axis represents the length of the pole, in feet, and the y-axis represents the weight of the pole, in ounces.
The point (1, 4) means that a 1-foot pole would weigh 4 ounces.
Since the graph is linear, the pole weighs 4 ounces for every additional 1 foot of length.
So the unit rate is 4 ounces per foot.
Question analysis
A graph of a pole's weight in ounces against its length in feet is a straight line through the origin, with the point (1, 4) marked. Students decide what that point says about poles, choosing among statements about rates, intercepts and particular poles, so the meaning of each coordinate has to be pinned down.
- 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 the situation is a named grade 7 skill. This is a direct, one-step example with a clear graph, typical of grade 7 classwork.
- 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
- Solution
- 1 step
- Reading level
- Question and choices: about grade 2, 45 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; the question tests whether a point on a graph can be put into words. Each wrong statement misuses the same two numbers, as an intercept, with the coordinates swapped, or as a rate turned upside down. Saying the point aloud with its units, a 1-foot pole weighs 4 ounces, and comparing that sentence with each statement settles it quickly.
Reading a figure, table or graph moderate Understanding the question
The axes carry the meaning: length in feet runs along x and weight in ounces up y, so (1, 4) describes a 1-foot pole, and the line through O makes that a rate. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
A very close wrong answer minor Choosing the answer
Both rate statements pair 4 with 1, one as ounces per foot and the other as feet per ounce; only the axis labels say which way the rate runs. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Something you must already know minor Working it out
The choices use the terms unit rate and y-intercept, which the question does not define, and two statements hang on knowing what each one means. (Anderson & Krathwohl, 2001)
Strategy: Note which quantity each axis measures, say in words what the x-coordinate and the y-coordinate of the marked point stand for, then compare that sentence with each choice.
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).
- Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
- 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
B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Takes the 4 in the point's label for the y-intercept, though the line starts at the origin, where a pole of no length weighs 0 ounces.
C Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Reads the coordinates the wrong way round, taking 1 as the weight in ounces and 4 as the length in feet.
D Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)
Turns the rate upside down, describing feet of length per ounce of weight instead of ounces of weight per foot of length.
Worked solution, step by step
Formulas used
- Unit rate
- For a point on the line, y is the weight in ounces and x is the length in feet. The unit rate is the weight of 1 foot of pole. This works for a straight line that starts at the origin, the point (0, 0).
Read the axes
The graph has two axes (number lines). The x-axis, across the bottom, shows length in feet. The y-axis, going up, shows weight in ounces.
A point is written (x, y). The across number comes first, then the up number. So in (1, 4), x is 1 and y is 4.
Say what the point means
x = 1 means a length of 1 foot. y = 4 means a weight of 4 ounces. So a pole 1 foot long weighs 4 ounces.
Watch out: do not swap the numbers. “A pole 4 feet long weighs 1 ounce” reads them the wrong way round. That is choice C.See that this is a unit rate
A unit rate is an amount for 1 unit, here the ounces in 1 foot. The line starts at O, the origin (0, 0), and it is straight. So each extra foot adds the same weight.
The unit rate is 4 ounces per foot. Watch out: choice D turns this upside down. It gives feet for each ounce, not ounces for each foot.
Rule out the y-intercept
The y-intercept is where the line crosses the y-axis. This line crosses it at O, where y is 0. So the y-intercept is 0, not 4, and choice B is wrong.
So the answer is A, the unit rate is 4 ounces per foot.
Check your answer
Try another point on the line, such as (3, 12). A pole 3 feet long weighs 12 ounces. , so it is still 4 ounces per foot. The rate stays the same all along the line.
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