I start today's lesson by projecting Bridge Weight and my asking students to determine an equation or rule that represents the relation described. Each bridge is really 1, 2, 3, or 4 layers of wood which support different weights (measured in pounds). The weight exhibited is the maximum weight that the corresponding number of layers can hold before failing or breaking.
I ask that each student copy and complete the table at their desks and study the relation. As usual, I motivate discussion among classmates sitting beside each other. Some students may begin to give up after a couple of failed attempts at finding the correct equations. I look out for these students and motivate them to try using other values in their equations, or asking that they use more than one operation in their equation if they are not doing so already. My experience is that if allowed sufficient time, many of my students will figure out a rule and write an equation that models the relation accurately.
Before beginning this section, I draw the table of values for the Bridge weights on the whiteboard for all to see. I like to ask students to think of the different ways in which they can name both columns. Some possible answers are “input/ output”, “x/y”, “domain/range”.
I indicate that they will learn one more way of naming these columns. I say, "Comparing data in tables is really important so there are a lot of different ways of talking about the tables." Then, I ask:
Which value is dependent on the other? Does the number of wooden boards depend on the maximum weight they can hold, or does the bridge maximum weight depend on the number of wooden boards?
Most of my students will likely say the latter. Then I write on the board:
The maximum weight depends on the number of boards.
I tell the class that another way of naming the columns is using the terms independent and dependent variable and proceed to ask which is which in our bridge case. If students seem confused I will ask, "Which part of the bridge task can you manipulate or change?"
Most of the time, my students will find it logical that these measurements (number of boards, maximum weight) represent the independent variable and that the weight the bridge can hold is the dependent variable.
I annotate my notes on the board by writing the Function Statement and stating the dependent variable and the independent variable.
To practice determining the independent and dependent variables in a given situation, I pair up my students randomly. Then, I provide each pair with a copy of the PPt Independent Dependent Variables.pptx. This resource contains seven slides, each presenting a real world circumstance where students must identify the independent and dependent variables. Then, they should work together to write a function statement for each situation. I ask that students write their answers on paper, or use the D and I table.
Once students are done, I will display my answers using D and I table Answers. I plan to give students time to discuss any wrong answers. My students usually have questions about the cases. For this activity I want to let classmates answer, before any intervention on my part so our review will take the form of a class discussion.
I've found that it is easier for my students to correctly identify the independent and dependent variables than to write a correct function rule. As we discuss function statements, I refer back to the New Info section of the lesson. I encourage students to interpret the phrase "is a function of" as "depends on". So, if the maximum weight depends on the number of boards used, the function statement would be "The maximum weight is a function of the number of boards used."
This CLOSING EXIT SLIP is a good way to end and assess the first part of this lesson. Students must complete the two column table using the vocabulary they know, and they should come up with their own real-world function relation, identify the dependent and independent variables and write a “function sentence” for their relation.