I use this Warm Up to introduce Quadratic Functions, by making comparisons to Linear and Exponential Functions that students have studied in previous lessons. I intend for this Warm Up to take about 10 minutes. My goal is for students to recognize the pattern of a quadratic function in a table, and to be able to graph the coordinates. As a class, we will discuss the second common difference between output values when reviewing the warm up.
Another purpose of this warm up is to remind students that functions found in the real world may have a restricted domain or range. For example, in the area of a square problem, the domain is all positive numbers (x>0).
I review the Warm Up in the video below:
After the Warm Up, I plan to use a collaborative paper folding activity to enable my students to further explore linear, exponential, and quadratic functions.
I give each student an 8.5 X 11 inch piece of paper to begin the paper folding activity. I label the independent variable x as the number of folds, and the dependent variable y as the number of rectangles formed. I ask the students, "What is the y intercept in this problem?" At zero folds, there is one rectangle formed. At one fold, there are two rectangles. At two folds, there are four rectangles. At three folds, there are eight rectangles, and so on..... After five folds, and 32 rectangles, it becomes difficult, and eventually impossible to fold.
The paper folding activity is one of the functions given in the matching cards resources in the lesson, Functions and Everyday Situations, from the Mathematics Assessment Project. For my lesson, I changed the output variable of the paper folding activity to be the number of rectangles (instead of the thickness of the paper) because it is easier to visualize.
I post the instructions from the card activity in this Screen Shot . I use the Feedback Questions to discuss the paper folding function as a class, as I randomly call on students. As a class, we create the t-table of values, plot the graph, and write the equation for the function. I ask the students if they recognize the function from a previous problem that we have done in class. It is the same function as the penny problem introduced in lesson one of this unit. The paper folding function's domain is restricted to whole numbers, from 0 to 8 folds. The penny problem's domain is the set of all positive integers greater than or equal to one. The graph shows a continuous curve for this function, and would be more recognizable if it was showing a discrete graph with distinct points at (0,1),(1,2),(2,4),(3,8).......(8,64).
I have provided a complete copy of this lesson from the Mathematics Assessment Project in the resource section. However, in this lesson, I only use the cards for the collaborative activity from pages S-3 through S-8 for this lesson. There are 11 functions excluding the paper folding situation. Each function is represented with a situation in words, a graph, and an equation. I assign three students to each group, I select homogeneous groups based off of the previous grades in this unit. The answers for the collaborative activity are posted on pages T-10 through T-12 of the lesson. This collaborative activity is meant for students to study as many of the situations as possible and use the key characteristics of each of the functions in depth to match the cards. All of the cards do not have to be completed. If students have difficulty with any of the situations, it will be clarified for them in the class presentation and discussion after the collaborative activity.
Finally, I hand the students the other 11 situations, graphs, and equations for each group to match. I set the timer for 20 minutes for the students to match the cards, tape to the poster paper, and write reasons for the matches on the poster. I emphasize for students to focus on the relationship between the independent and the dependent variable. Again, the goal is for each group to discuss the functions in depth, even if all of the functions are not complete. While groups are working, I monitor, provide questioning, and select 3 groups to present one of each of the functions. I select one linear, one exponential, and one quadratic for the groups to present in depth. The class is expected to listen to the presentation, ask questions, provide extra input, and show respect. I allow students to talk in between presentations as other groups set up, but my students are expected to be quiet during the presentation. I gain their attention when groups are ready to present by saying "RESPECT".
To complete the activity, I have students individually select a function that was not presented and write a paragraph in detail about that function. I also have them evaluate the function for the specific question on the situation card. I grade this writing assignment as a formative assessment to check for student understanding on the key characteristics of the function that they selected. Here is a student example of the cooling kettle situation. The student explained the starting point and it as an exponential function, but did not explain that the 20 in the equation represented the room temperature. Some students made the mistake of identifying this problem as a decreasing linear function.
Source URL for the Functions and Everyday Situations activity:
http://map.mathshell.org/materials/download.php?fileid=1259 (accessed April 11 2014)
I usually collect exit slips before the class ends if time permits. Due to the time that this lesson took, I assign today's Exit slip for homework.
I selected this problem on exponential decay because I wanted my students working more problems involving percents. On a recent assessment, several of my students struggled with percents and fractions. I also thought that the problem would attract the interest of those students interested in the medical field.
I demonstrate the Exit Slip task in the video below.