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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%)

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.

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.

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?

- 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).

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.

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

- 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

- 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

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.

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

- 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

- 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.

- 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?

How many points do you think lie on the line altogether?

Percentage Summary

Are there any points on the line between the points (2,5) and (3,7)? If so, how many?

Percentage Summary

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)

- 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’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?

Percentage of correct responses:

What is the equation of the following line?

Percentage of correct responses:

What is the equation of the following line?

Percentage of correct responses:

What is the equation of the following line?

Percentage of correct responses:

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

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

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)

- 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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