Students will complete this warm up individually in the first few minutes of class. The task is straight forward and will allow you to make judgments about which students are still having difficulty evaluating functions. The skill of evaluating a function will continue to be important in the functions unit and in the remainder of the course.
Before asking students to perform any mathematical work on the launch of this lesson, just ask them to think about the vocabulary words recursive and explicit. Have students do a Think-Pair-Share with their partners to try to define these words and the types of formulas they represent. When students are sharing out, take one term at a time and have as many students as necessary add to the meaning of that term before moving on to the second term.
Next, ask students to determine both the recursive and explicit formulas for the table given. Continue to emphasize the fact that while the recursive formula is additive (adding two each time) the explicit formula will be multiplicative (repeated addition = multiplication). This will require students to examine the structure of the table (MP7) before determining their two formulas.
As a follow up, ask students if they wanted to determine f(102), which formula would they use and why? Have students do a turn and talk around this question with their partners. Call on one or two students to share out their answers with the class.
The mirror task can either be projected or students can be given individual copies of the task. Ask students to read the problem to themselves at least twice and then individually try to pick up on the pattern for each of the mirrors that is given. This will attempt to slow down student thinking and allow them to take the necessary time to understand the problem (MP1).
Scaffold: For more kinestetic learners or those students that need a concrete structure to work off of, you can give them blocks or some other manipulative (to represent the tiles) that will allow them to actually build at least the first three frames for this question. Ask students to pay attention to how many tiles they are adding each time as this will help them come up with their recursive formula.
As students are working, listen in for those pairs that are able to reason abstractly (MP2) to notice how the figure is changing each time. Many students will pick up on the fact that the four corners are always there and that the remaining tiles are based on the length of the side of the square mirror.
The purpose of this lesson is to allow students to see that different ways of writing a function can all be equivalent. In my experience, these different ways of writing the function come out naturally, however you can engineer a conversation if all of the students in your class happen to go about finding the explicit formula in the same way. Some examples of differences in thinking can be seen in this video.
Allow students an opportunity to share their way of thinking about the problem and more importantly, allow others in the class to discuss this way of thinking so that they can make sense of it as well. It is good for students to have time to see how others approach a problem in order to improve their own thinking and sense making. This process will be a good way to attend to mathematical practice 3.
After working through the majority of the functions unit, students are getting very good at spotting a linear pattern and naming it explicitly. You can see, however, from the student work that they are doing so with the use of a table. This is, of course, a completely viable method but I also wanted the students to see some structure in the situation itself. I really forced my students to show some perseverance and stamina by asking them to find 4x+4 in another way in the picture. I have noticed for years that students are not happy about finding "another way" to solve a problem. Isn't one right answer good enough? But I keep pushing them to find different ways as a way to increase their understanding.
Most groups of students (after some prodding) were able to see that the four corners of the mirror were constant and that the length and width of the mirror were the same length as the square (x). Because of this they could see the 4x+4 reflected in the picture but most groups had trouble being able to put this observation down in words. Some were still getting caught up with what the variable, x, represented in the picture. Many groups wanted to say at first that the mirror sizes were 1, 4, 9, etc. rather than 1, 2, and 3, etc. This misconception made it hard from them to generalize by looking at the picture.
This closure requires a lot of thinking so if time runs short allow students to think as much as they can an it may be something you want to revisit in a future lesson. When determining the domain and range of this function, the first thing students will need to notice is that the data is discrete not continuous. Also, students will need to notice the smallest possible output value is 8 and the remaining output values are all multiples of 4: 8, 12, 16, 20, etc.
The closure activity could also be extended if needed by asking students to graph the corresponding input/output values for this function.