Thought for the day: Patterned lists of equations or expressions can help students shift their focus from procedures and answers to relationships and meaning.
Concepts: Solutions of linear equations (possibly graphs of linear equations, x-intercepts, translations of graphs, solutions of non-linear equations)
Examples of noticing and wondering
I notice that all three equations look the same except for the number subtracted from x.
I notice that the solutions will not be whole numbers.
I notice that all of the equations are linear.
I notice that each solution increases by 1 as you go down the list.
I notice that the graph of each left side moves one unit to the right as you go down the list.
I wonder why the solutions increase by 1 each time.
I wonder If the solutions would still increase by 1 if you changed the 5 (or the 12) to some other number (but still kept it the same in every equation).
I wonder if I could create a list of linear equations whose solutions increase by some other number each time.
I wonder how the patterns I have seen would change (or stay the same) if (1) the parentheses were removed, or (2) the subtraction were replaced by addition, or (3) the x were replaced by x squared.
Creating something new
Students may create and explore their own patterned lists of equations. Many of their creations may flow out of things that they have wondered about. Examples:
Create patterned lists of equations like the ones in this prompt but without the parentheses.
Create patterned lists of equations like the ones in this prompt but using addition instead of subtraction.
Create patterned lists of equations using x squared instead of x.
Create patterned lists of equations whose solutions follow other predetermined patterns.
Create stories or word problems to fit the equations in this prompt or lists of equations of your own.
The solutions to the equations are 5.4, 6.4, and 7.4, respectively, from top to bottom. Students' first inclination is often to solve each one separately from scratch, rather than to use the relationships between the equations to help them predict a solution or find a more efficient approach.
The pattern in the solutions mirrors the pattern in the numbers that are subtracted from x. Thinking about why this happens can (at least eventually) shift students' focus from the procedure to the meaning of the word "solution." Since, in each case, 5 times some number (which happens to be 2.4) must equal 12. When the amount subtracted from x increases by 1, the value of x must also increase by 1 in order to compensate and leave the value within the parentheses at 2.4.
Students who draw graphs to represent the equations may recognize that n in the expression 5(x – n) is always equal to the x-intercept of the graph of the left side and that this intercept must therefore increase by 1 when n increases by 1. This rightward shift in the graph also shifts the point of intersection of the graph with the horizontal line y = 12 one unit to the right, which offers another way to understand (and visualize) the increase in the solutions.
You can easily adapt this prompt to different needs by adjusting the complexity of the equations. For example, the equations will have simpler solutions if you replace 5 by 6 in each one.