Students should start out working individually on tables_linear_functions_warmup and then work with a partner after about 3-5 minutes. The warmup should follow the Think-Pair-Share model. While students work make note of how students attempt to find the 10th term in the sequence. Some students will use additive reasoning and others will use multiplicative reasoning. The formula for finding the nth term of an arithmetic sequence can be given to students. However, they are much more apt to know how to use the formula if they can derive it themselves. When students share out their answers to the whole class, try to start with those that used additive reasoning and then have students who used multiplicative reasoning show their thought process (MP2). Both of these techniques will get you to the correct answer. One is just more efficient than the other (think about finding the 100th term).
The students who used the multiplicative process may notice that there is one less common difference being added than there are terms (simple example: 1, 3, 5, 7, n. If we wanted to know the 5th term we know we are adding four 2's to the first term. So 1 + 8 = 9) In our example 2, 5, 8, 11,...n we will be adding nine 3's to the first term to come up with the 10th term, so 27+2 = 29. Make sure to bring this observation to the attention of the whole class.
See tables_linear_functions_launch Slide 3
In this slide students are going to list their inputs and outputs in a table format to organize their work. Students should work with their partner to determine the following:
1) What are the first 8 terms of the sequence?
2) Put the first 8 terms into a table of values.
3) Find both a recursive and explicit rule for this function.
4) Graph the coordinates in the coordinate plane and draw a line to model this sequence using the function.
When students work on this launch/investigation they will be continuing to increase their understanding of the rate of change between values of x and f(x). For the first two examples, they will also be building their recursive and explicit formulas from a concrete model that is easier to understand.
The investigation will begin by having students find the same information for Slide 4. This is also a concrete model. Students can use manipulatives to help them understand the pattern and rate of change (MP6).
In this slide, the concept becomes more abstract as students are not given a picture to help them understand the pattern. Students will need to use the given table to help them determine the rate of change.
Teaching point: Some students may have difficulty seeing that the rate of change is constant. They will see that the changes in f(x) vary. They may not realize how this is connected to the value of x changing at a proportional rate. When working with pairs of students, guide them to fill in the missing values so that they can see that the rate of change is, in fact, a constant.
When discussing these three examples with the class, draw the three tables on the board. Have students fill in their answers for the input and output values. Add a third column to each table to show the common difference for each jump from one value to the next. Next, have students write down their explicit formulas for each of the three examples. Then have them do a think-pair-share around the following question: "What do you notice about the explicit formula and the table of values that you made?" Give students a couple of minutes to write something down and then share their ideas with their partner (MP3). When students share out, guide their thinking towards the understanding that the common difference is the number that preceeds the variable in their explicit formula (the slope of the line). For example, in the first tile question the explicit formula is f(x) = 2x - 1. The common difference between each of the subsequent terms is 2. This idea will continue to develop in future lessons as we investigate the idea of slope of a line.
Students who are more concrete thinkers can also connect to this by looking at how the common difference is presented in the diagrams in the first two examples. They can see how many tiles or discs are being added each time.
Being able to see the same concept from multiple perspectives is key in mathematics. Tables_linear_functions_close is designed to help students make a connection between two variable equations and functions. Have students make a table of simple x values* (imputs) and find the corresponding y-values (outputs). They can then graph their solutions on a piece of graph paper.
You can also encourage students to be strategic in the x-values that they choose so as to have y-values that are simpler to graph. For example, question 1 has x + 2y = 4. An x-value of 2 would give a y-value of 1. In contrast, and x-value of 1 would give a y-value of 1.5. Obviously, these are all values that students can handle but integer values are easier to graph than rational values.
*Simple x-values simply refers to integer values. Student can certainly choose rational values if they would like but it may slow down their computation.