Students will get back into their groups from the previous lesson and revisit the tower task. My directions for students are to: (1) Discuss the recursive formula that you determined for the tower task and why it works. (2) Discuss the explicit formula that you determined and why it works. (3) Name one thing that you saw on another groups chart that stuck with you. Opening in this way serves to reframe the conversation and get all members of the class back up to speed with where they had left off the day before. I go around to each group and have them share out one of the three items (I choose which item will be shared in order to hold all groups accountable and to ensure that each topic gets proper treatment).
Students work in their groups to extend and organize their thinking from the tower task. In completing this work the groups are learning how to use multiple representations to display the same idea (algebraically, graphically, numerically in a table).
What to watch out for:
-Watch to see that each group is evaluating the equation B = 5H – 4 at each value of H. The step is crucial in understanding how to evaluate functions later on in the lesson. If groups are merely coping both sides of the table and leaving the center column blank, encourage them to validate their answers by evaluating the equation at each point.
-The “think” question that goes along with the graph was included very intentionally here. While students have not really been exposed to discrete data in middle school it is important that they start to think about the domain and range of the function in terms of the context it describes. The decision of whether or not to include the words “domain” and “range” in this lesson can be made based on student dialog around this particular question.
-Students may need some guidance on the term “rate of change.” While this term is used very heavily in the eighth grade standards, you may need to revisit this concept either with individual students or, if necessary, with the whole class.
Once each group has a chance to work through all of the prompts on the investigation, I bring the whole class together to discuss the “think” question and the “rate of change” question. When I structure a whole class discussion like this, I like to create an environment where after one person shares, each subsequent person should try to build off of and add to the initial idea. This forces the students to listen to each other and not just wait for their turn to share their own idea. It is only when we have exhausted an idea that we move on and start another thread with a new idea.
We have now been using the word function for several days. The next portion of the lesson is going to tie a few concepts together and uses the tower task as the common thread. I show the following video and while students are watching I stop the video at certain times so that students can do a think-pair-share with one other member of their group. Stop times and good questions to ask are shown below:
0:54-In the tower task, what is the input and what is the output?
2:57-Let students know that f and g are not the only names for functions. Any letter can stand for a function.
3:20-Pause video here and ask students to explain to their partner where the 26 and the -9 came from to ensure they are understanding the evaluation step.
-How could we use function notation to write the following: How many blocks does it take to build a tower that is 5 blocks tall? (This may need to be shared out whole class so that students can see that it can be written as either B(5) = 5(5) – 4 or f(5) = 5(5) – 4.
6:05-Ask students to look back at the table that they filled in for the tower task and their answer to “is this relation a function?” Have them discuss whether or not their answer is correct based on this definition or if they need to change their answer. Give an opportunity for groups to share out as needed so that all students are convinced that the tower task is a function.
8:17-Ask students to look at the graph that they created. Does this graph pass the vertical line test?