Find the Change

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 $y_2$ 
If $m = 7$, determine possible values for $y_1$ and $y_2$. Explain your choices.
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 $m = 7$?
Use the same $x$values in the table and find possible values for $y_1$ and $y_2$ if $m=3$. Explain your choices.
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 $m = 3$?


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.

$x$ $y$ 4 $y_1$ 9 $y_2$ 
$x$ $y$ 2 $y_1$ 13 $y_2$ 
$x$ $y$ 1 $y_1$ 14 $y_2$ 
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

Comments
Log in to commentHaley says:
over 3 yearsIn part b of the solution should it be "the difference between the yvalues is 3 times the difference in the xvalues"?
Cam says:
over 3 yearsIndeed, thanks!
Dana says:
over 5 yearsIsn't pointslope form in the HS standards? Although the activity is good, the commentary and solution are not aligned with the 8th grade standards.
Kristin says:
over 5 yearsThis task is definitely about "understanding the connections between proportional relationships, lines, and linear equations" (8.EE.B). It is true that in this task there is an equation that is sometimes called the "point slope formula," but I don't believe this language is specified anywhere in the standards either in middle or high school. The Algebra Progressions document talks about students being able to move between different forms of a linear equation rather than memorizing them all as separate things, and I think we might say something analogous about the formulas: students should understand where they come from (and that they all come from the relationship between proportional relationships, lines, and linear equations) before they are expected to memorize and call upon them in different problem situations.
In Common Core, most of the hard work that goes into developing the ideas around linear equations happens in 8th grade, and this task suggests an approach to helping students dig deeply into those ideas. In high school, students are expected to have most of those ideas down fairly solidly and to use that knowledge in the context of more sophisticated mathematical and realworld problems.