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Students’ Difficulties in Linear Functions. C&I235M, Spring 2004. Part I. Slope. Sample: 252 high school students Response key A, B, C, D : one of which may be correct E: blank or different from A, B, C, D F: do not know. A. 3 (65\%) B. 2 (11\%) C. 3/2 (11\%). D. –2 (3\%) E. (4\%) F. (5\%).

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part i slope
Part I. Slope
  • Sample: 252 high school students
  • Response key
    • A, B, C, D : one of which may be correct
    • E: blank or different from A, B, C, D
    • F: do not know
the slope of the line y 3x 2 is
A. 3 (65%)

B. 2 (11%)

C. 3/2 (11%)

D. –2 (3%)

E. (4%)

F. (5%)

The slope of the line y=3x+2 is
  • 11% said 3/2. This is due to a confusion with the idea that the slope is a ratio. Is it also due to students’ reliance on the slope formula (but plug in the wrong numbers)?
  • 11% said 2. Probably due to a confusion of the meaning of the values of m and b from the general equation y = m x + b.
the slope of the line y 2 3x is
A. 3 (12%)

B. 2 (9%)

C. -3/2 (19%)

D. –3 (50%)

E. (2%)

F. (8%)

The slope of the line y=2-3x is
  • 8% did not even try.
  • 12% said 3. A misuse of the equation or a confusion with negative numbers.
  • 9% said 2. A misuse of the equation.
  • 19% said –3/2. Confusion with the idea that the slope is a ratio.
the slope of the line 2y 6x 5 is
A. 6 (14%)

B. 6/5 (11%)

C. 2 (4%)

D. 3 (52%)

E. (7%)

F. (12%)

The slope of the line 2y=6x+5 is
  • 12% did not even try.
  • 14% said 6. Lack of understanding of slope or a misuse of the equation.
  • 11% said 6/5. Confusion with the idea that the slope is a ratio.
what is the slope of the line ab in the following figure
What is the slope of the line AB in the following figure?
  • A. (x2-x1)/(y2-y1) (36%)
  • B. (x2-y2)/(x1-y1) (18%)
  • C. (x2-y1)/(y2-x1) (3%)
  • D. (y2-y1)/(x2-x1) (33%)
  • E. (1%)
  • F. (9%)
  • 36% said A. Lack of understanding of the definition of slope.
  • 18% said B. Define slope as B/A.
  • Difficulties in math symbols (math language).
slide7
A. 2Δx + 1 (12%)

B. 2(x+Δx) +1 (23%)

C. 2 Δx (20%)

D. Δx (7%)

E. (5%)

F. (33%)

The points P (x, y) and Q(x+Δx, y+Δy) both lie on the line y=2x+1. Which of the following expressions is equal to Δy?
  • 33% did not even try. Only 20% were correct.
  • 12% said A. Just plug in Δx and Δy into the equation.
  • 23% said B. Could have been correct if the question was y+Δy = …
  • Again, difficulties in math symbols.
the points p 2 4 lines on the curve y x 2 q 2 h 4 k also lies on the curve what is the slope of pq
A. k/h (15%)

B. h/k (11%)

C. (4+k) / (2+h) (13%)

D. (2+h) / (4+k) (13%)

E. none of the above (21%)

F. (27%)

The points P (2, 4) lines on the curve y = x^2. Q(2+h, 4+k) also lies on the curve. What is the slope of PQ?
  • Only 15% were correct.
  • 21% said “none of these.” 27% did not even try. Nearly half of the sample did not know how to find a ratio to represent the slope.
  • 13% said C and another 13% said D and ignored point P.
summary of difficulties in slope
Summary of Difficulties in Slope
  • Vague concept of slope. The subtlety of the concept is not realized.
  • Inability to fine the slope of a line when given two points.
  • Confusion with the idea that the slope can be considered as a ratio, but a ratio of what to what?
  • Confusion with the question of “x over y” or “y over x.”
  • Confusion with the m and b given in equation y = m x + b.
part ii intercept
Part II. Intercept
  • Sample: 95 high school students
  • Response key
    • A, B, C, D : one of which may be correct
    • E: blank or different from A, B, C, D
    • F: do not know
the value of the y intercept for the line y 5x 6 is
A. 5 (9%)

B. 6 (81%)

C. 6/5 (0%)

D. 5/6 (2%)

E. (1%)

F. (6%)

The value of the y-intercept for the line y = 5x+6 is
  • Most (81%) were correct.
  • 9% said 5. A confusion with the slope.
  • Only 2% said C or D. It seems that ratios and intercepts are not associated.
the general equation for a straight line is y m x b if the slope of the line is negative b may be
A. positive (8%)

B. negative (21%)

C. positive or negative (20%)

D. positive, negative, or zero (45%)

E. (1%)

F. (4%)

The general equation for a straight line is y=mx + b. If the slope of the line is negative, b may be:
  • 8% said A, and 21% said B. Confusion with the relationship between m and b.
  • 20% said C. Seemed to neglect the possibility that the line might pass the origin.
  • Some students are not sure about zero.
y intercept vs x intercept
Y-intercept vs. X-intercept
  • Some students describe lines as moving to the right or to the left as a result of changing the b in a linear equation.
discussion
Discussion
  • 59% were correct.
  • 26% chose A, suggesting that the slope is negative.
  • 8% chose B. Probably had correct idea about the slope, but the wrong idea about the intercept.
  • Do students consider a graphical representation of an equation when they are given an equation or do students just see an equation and nothing else?
question plot the points 2 5 3 7 5 11 these points lie on a straight line draw the line
Question: Plot the points (2,5), (3,7), (5,11). These points lie on a straight line. Draw the line.
  • A. Find some other points on the line and write them down.
  • B. The point (4.6, 10.2) also lies on the line, mark its position approximately.
  • C. Plot the point (1 ½,4).
  • D. How many points do you think lie on the line altogether?
  • E. Are there any points on the line between the points (2,5) and (3,7)? If so, how many?
are there any points on the line between the points 2 5 and 3 7 if so how many
Are there any points on the line between the points (2,5) and (3,7)? If so, how many?

Percentage Summary

some responses
Some Responses
  • Student 1(age 14)
    • D: 6 points altogether
    • E: No, there aren’t any.
  • Student 2 (age 14)
    • D: 8
    • E: Yes, 1 comes between, at (2 1/2,6)
  • Student 3 (age 14)
    • D: I think that 6 points lie on the line.
    • E: 4
some responses con t
Some Responses (Con’t)
  • Student 4 (age 14)
    • D: 15 ½ points lie on the line
    • E: Yes, 1.
  • Student 5 (age 14)
    • D: I think there can be as many as possible.
    • E: There is one main one.
  • Student 6 (age 15)
    • D: They are many.
    • E: 10.
some responses con t22
Some Responses (Con’t)
  • Student 7 (age 14)
    • D: There are too many to count.
    • E: Yes, 10.
  • Student 8 (age 14)
    • D: There are too many to count.
    • E: There is one main one.
  • Student 9 (age 12)
    • D: 18 if you just take the whole numbers, but more if you can go into fractions and extend the graph.
    • E: Too many to count if you count fractions.
what is the equation of the following line
What is the equation of the following line?

Percentage of correct responses:

what is the equation of the following line24
What is the equation of the following line?

Percentage of correct responses:

what is the equation of the following line25
What is the equation of the following line?

Percentage of correct responses:

what is the equation of the following line26
What is the equation of the following line?

Percentage of correct responses:

questions
Questions
  • Find an x and y that make y = x – 1 true.
  • Find how many pairs of values of x and y are there that make y = x – 1 true?
  • Find an x and y that make x + y = 8 true.
  • Find an x and y that make both y = x – 1 and x + y = 8 true.
  • How many pairs of values of x and y are there that make both y = x – 1 and x + y = 8 true at the same time?
percentage of correct responses
Percentage of correct responses

* Number of students:459 13-year-olds755 14 -year-olds584 15 -year-olds

some observations
Some Observations
  • Generally, students have difficulties understanding that an equation and its graph are two sides of the same coin.
    • Most secondary students do not understand a solution for an equation represents a point on its graph, and vice versa.
    • The concept of continuity: Majority HS students still do not have the idea that there are infinite points on a line or line segment.
    • Another study asserted that only 25% of the students with one year of algebra were able to produce a correct graph corresponding to a linear equation. With 2 years of algebra, still less than 50% could do it.
    • Given a graph of a straight line with indicated intercepts (-3,0) and (0,5), only 20% of the students with two years of algebra could write the equation.
some observations continued
Some Observations (continued)
  • The equations of lines parallel to the axes are difficult to obtain for many students.
  • A picture is supposed to be worth a thousand words, but this does not prove to be quite true for secondary students looking at the graph.
  • Fractions, decimals, negative numbers, and zeros make it even more difficult for students’ learning of these representations.
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