Find the Change

Task

1. The table below shows two coordinate pairs $(x, y)$ that satisfy the equation $y=mx+b$ for some numbers $m$ and $b$.

   $x$	$y$
   2	$y_1$
   5	$y_2$

   1. If $m = 7$, determine possible values for $y_1$ and $y_2$. Explain your choices.

   2. Find another pair of $y$-values that could work for $m = 7$. Explain why they would work. How do these $y$-values compare to the first pair you found for $m = 7$?

   3. Use the same $x$-values in the table and find possible values for $y_1$ and $y_2$ if $m=3$. Explain your choices.

   4. Find another pair of $y$-values that could work for $m = 3$. Explain why they would work. How do these $y$-values compare to the first pair you found for $m = 3$?

2. Each of the three tables below shows two coordinate pairs $(x, y)$ that satisfy the equation $y=mx+b$ for some numbers $m$ and $b$. If $m = 3$ in each case, find possible values for $y_1$ and $y_2$ for each pair of $x$-values given.

   1. $x$	$y$
      4	$y_1$
      9	$y_2$

   2. $x$	$y$
      2	$y_1$
      13	$y_2$

   3. $x$	$y$
      -1	$y_1$
      14	$y_2$

   4. Suppose we take all six $x$-values from the three tables above. Can you find six corresponding $y$-values so that all the coordinate pairs satisfy the same equation if $m=3$? Fill out the table below and explain how you know they will all work with the same equation.

      $x$	$y$
      4
      9
      2
      13
      -1
      14

IM Commentary

This task is adapted from one I have used with my eighth grade students to help them solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and $y$-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

With some instructional effort, this task can produce some nice classroom discussion. If adapted slightly (see comments in the solution), the task could be used as an assessment if placed properly in the curriculum. This task can also be extended so that the unknown values are $x$-values rather then $y$-values. There is also opportunity to discuss vertical and horizontal shifts if desired since all values that satisfy the tables for a given slope will be on lines that are parallel to one another.

Although this task provides information about linear relationships represented by tables, a strong connection can be made to both the equations and the graphs. By pressing students to create multiple representations and then focusing on their graphs, similar triangles can be used to verify the slopes and the equations. Students become comfortable with tables and representing the differences along the sides of the tables; maintaining the connection to what those differences mean graphically can assist them in building their understanding of slope, lines, and linear equations.

This task was submitted by Travis Lemon for the first IMP task writing contest 2011/12/12-2011/12/18.

Solution

1. This task allows for multiple approaches and solutions:

   • Students often will choose 14 and 35 when $m=7$ and 6 and 15 when $m=3$. This comes from a direct relationship with an initial value ($y$-intercept) of zero.

   • Others chose $y_1$ to be zero in both tables and then determine the second $y$-value by multiplying the slope with the difference in the $x$-values. This becomes their second $y$-value.

   • Yet more fluent students realize that the difference in $x$-values multiplied by the slope would equal the difference in the $y$-values. Therefore, the $y$-values on the first table could be any values that have a difference of 21 ($y_2 > y_1$), while the $y$-values used to satisfy a slope of 3 could be any values with a difference of 9 ($y_2 > y_1$).

    

   Part (ii) asks students to compare the pairs of $y$-values they found in parts (i) and (ii); this question is open-ended enough that it doesn't make a good assessment question, but it does make a good opener for discussion. While students might come up with a number of different ways of comparing them, the goal is to highlight that regardless of the values they found, the difference will always be the same. If no students make this observation by the time they get to part (iv), the teacher can point it out. Ideally, the discussion of the different approaches can lead to a solidification of the following relationship:

   $$(\text{change in x–values})m = (\text{change in y-values})$$ and then a more formal $$(x_2 - x_1)m=(y_2 - y_1)$$

2. There are many different solutions for parts (b)(i)–(iii); in all cases the difference between the $y$-values is 3 times the difference in the $x$-values. The final question (part iv) presses students once again to further solidify their understanding of slope by asking them to pick all six $y$-values so that all the pairs will satisfy the same equation.

   Instructional opportunities arise as students justify their choices. Helping students focus on the ratio of the change in $y$ ("rise") to the change in $x$ ("run") and encouraging them to draw triangles and explain why they are similar can facilitate students' understandings. Teachers should capitalize on any opportunity to use multiple representations; strategically posed questions allow for students to make more connections. For example,

   • "Can you simply take the values used in the first three parts and use them to fill in the table in the last part? If so, why? If not, why not?"

   • "What would change if we used a negative slope?"

   • "If someone decided to adjust one of the $y$-values they chose by 5, would any of the other values need to change? Why or why not? How would they need to change?"