Equal Differences over Equal Intervals 1

Task

1. Complete the table. In the third column, show your work as demonstrated. What do you notice about the 3rd column?
    

   $x$	$y=2x+5$	$\Delta y$
   1	7	----
   2	9	9 - 7 = 2
   3
   4
   5

2. Complete the table, showing your work as above. What do you notice about the 3rd column? What is the graphical interpretation of this?

   $x$	$y=ax+b$	$\Delta y$
   1	$a\cdot 1 +b$	---
   2	$a \cdot 2 +b$	$a\cdot 2+b - (a\cdot 1 + b) = a$
   3
   4
   5

3. Let $y=ax+b$. Let $x_0$ be any particular $x$-value. Show that if $x_0$ is increased by 1, the corresponding $\Delta y$ is a constant that does not depend on $x_0$. What is this constant?
4. Does (a) serve as an example of the result in (c) ? Explain.

IM Commentary

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope. In F.LE Equal Differences over Equal Intervals 2, students prove the property in general (for equal intervals of any length).

An alternative approach to this task would be to restructure it as a guided discovery lesson. Instead of asking students to examine first differences (as modeled by the three-column table provided) for one linear function and then jumping right to the general form, students could first examine such differences for a variety of linear functions with varying slopes (including negative slopes) and $y$-intercepts. From this they will be able to ascertain the relative effects of both the slope and the $y$-intercept and hopefully deduce a viable conjecture. At this point, challenge individuals or small cooperative groups of two or three to construct an algebraic argument in support of their conjecture. It would also be instructive to ask students how their conclusions might change with increases in $x$ values greater than or less than one, say increases of 2, 3, one-half, or $k$, segueing nicely into the follow-up problem mentioned above.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This task is linked to Standard for Mathematical Practice #8, “Look for and express regularity in repeated reasoning.”  This task serves as a guide for students to examine the same repeated process and make observations from that process that can be generalized.  It also serves as an example for teachers about the types of connections that can be made by generating a repeated process in a table.  This task format, with a specific table, a more general table, and a couple of questions connecting the two processes could be used with other repeated processes to uncover regularities.  This discovery of regularity from repeating the same steps, is the skill that Standard for Mathematical Practice #8 attempts to foster in student’s thinking.

Solution

1. The table of values of $y = 2x+5$ and the corresponding $\Delta y$ values are given in table below:
   
   

   $x$	$y=2x+5$	$\Delta y$
   1	7	---
   2	9	$9-7=2$
   3	11	$11-9=2$
   4	13	$13-11=2$
   5	15	$15-13=2$

   The numbers in the third column of the table are all twos. In this case, $y = 2x + 5$ is the equation of a line with slope two; therefore each increase of one unit in the $x$-value produces an increase of two units in the $y$-value.

2. The table of values for $y = ax + b$ is given below:
   
   

   $x$	$y=ax+b$	$\Delta y$
   1	a+b	---
   2	2a+b	(2a+b) - (a+b) = a
   3	3a+b	(3a+b) - (2a+b) =a
   4	4a+b	(4a+b) - (3a+b)=a
   5	5a+b	(5a+b) - (4a+b)=a

   The numbers in the third column of the table are all $a$'s. In this case, $y = ax + b$ is the equation of a line with slope $a$; therefore each increase of one unit in the $x$-value produces an increase $a$ in the $y$-value.
3. When $x = x_0+1$, $y= a(x_0 + 1) +b$.
   
   When $x = x_0$, $y = ax_0 +b$.
   
   The difference between these $y$-values is: $$[a(x_0+1) + b] - [ax_0 + b] = (ax_0 +a + b) - (ax_0+b) = a. $$ So the change in the value of $y$ when $x_0$ is increased by one does not depend on the value of $x_0$: it is always $a$, the slope of the line described by the equation $y=ax+b$.
4. In a), as the $x$-values each increased by 1, the $y$-values increased by the 2, which was the coefficient of the $x$ in $y=2x+5$. So a) is an example of the result in c), treating the cases where $x_0 = 1,2,3,4$.