I begin the lesson by pairing up students. A quick way of pairing is to go down the roster alphabetically calling students two at a time. Then, I hand each pair a Launch Activity Graphs slip. Students can also be asked to pull out their own graph paper and construct their own T charts for this activity.
I ask students to color squares in with their pen or pencil, adding adjacent squares to form a polygon. The polygons should be regular polygons made up of quadrilaterals. The number of squares filled go into the first column of the table. The perimeter of the polygon should be recorded in the second column. For example, the first line in the table would form the coordinate pair (1, 4). In this video I demonstrate the procedure that the students will follow: Launch Video.
After they complete the table of measurements, I ask students to make a scatter plot of the ordered pairs. I expect students to recognize the pattern as a linear relationship. If the points are not in a line, the student has made a mistake. I stroll around the room identifying and helping students who are having a problem graphing the points.
Once students complete their graphs, I address the class and ask what type of relationship they obtained. I then lead a brief discussion during which I ask learners to analyze their work and explain the reasons why this is a linear function. I am hopeful that after Day 1 of this lesson, the students will see that the y-values (perimeter) increase by a constant value. If not, I carefully lead them towards this conclusion.
When they see the constant increase in y values, I ask: "So, if you see an output column of values with a constant increase, are you then 100% sure it is a linear function?" I let students think a while and avoid calling on the students that pop up immediately with answers, so as to give everyone time to think. Since they are acquainted with slope, I want students to reflect on the fact that the values in the x-column must have a constant increase or decrease as well.
Here is a Scaffold to help students understand this last concept.
As I begin to introduce the students to New Info, I tell the class that the constant value obtained by comparing consecutive values from the output column of a linear relation is called the First Difference. In the first part of this PowerPoint Presentation (Slides 1-7), I demonstrate how mathematicians analyze data tables to find these differences.
My students always like to be called on to read, so I ask a student to read the text on the slides. Slides 4 and 6 present examples for the students to complete. I ask students to solve each before showing the answer on the following slide.
When we get to Slide 8 I introduce Second Differences. I explain that these differences can be used to identifying Quadratic Functions or parabolas. This slide has animations. I click once on Slide 8 to show the constant change in x-values. I then click again to show that the first difference change is not constant and therefore this function is non-linear. Before clicking again I ask students to calculate the difference between "first difference" values. They should see that see that it is always 2. I proceed to show this by clicking the slide a third time. Then, I ask a student to read about finding second differences.
Slides 10 and 12 are guided examples which I go through together with the whole group.
Students continue to work in pairs. I hand each student an Application Linear vs Quad sheet. I tell students that not all points have to show on the given grid. I remind the students that finding the first and second differences confirms the type of function represented in the data table.
Teacher's Note: For this activity learners can simply approximate the plotting of decimal ordered pairs.
To end this lesson, I call on a student to summarize the main points that were covered. Many times, students understand what went on in the lesson, but they cannot always phrase their thinking in clear sentences. I like to insist that they do. I make it a point in class that students verbalize their math ideas and communicate about their work. I may have to call on more than one student to get all the main ideas of the lesson stated.