For the Entry Ticket: Finding the Rate of Change over a Specified Interval, students work on calculating and estimating the slope for different linear functions represented as graphs. The intent of this entry ticket is to provide students with additional practice on skills taught in a previous lesson (see Tablet Wars lesson for ways to engage students in calculating and interpreting the slope of linear functions in different forms).
I like to use Google or Fooplot as technology resources to graph linear functions. One advantage of using Google (just type the equation in, y = 2x+3 for example) and the graph will be one of the first results) is the website gives the user the value of different points along a line as you scroll across it. I would have students determine the slope for different graphical representations of lines - positive slope, negative slope, a slope that is not an integer, a line with no slope and a line with a slope of 0 - some examples include:
1. positive slope: f(x) = 3x + 6
2. Non-integer slope: f(x) = 0.75X
3. No Slope: f(x) = 3/0(x) + 7
4. Negative Slope: f(x) = -4X + 11
In terms of implementation, the teacher can write the functions to graph on the whiteboard and students can use appropriate technology (Google, Graphing calculator or fooplot, as three examples) to calculate the slope. As an alternative, the teacher can project the graphs of each function, one at a time, on a Smart Board or projector.
In addition, I want students to be comfortable calculating the slope of a linear function because it is a prerequisite skill they will need to become more automatic with for this lesson on calculating the rate of change over different intervals for functions that are do not perfectly fit a linear function.
For the last 5 minutes of this section I have students complete a Turn and Talk to compare and contrast their answers.
I then go through a brief Powerpoint Presentation that provides an overview of calculating the rate of change for specific intervals for two examples: a graph of maximum heart rate compared to age and Apple stock prices over the past 5 years. The lecture also covers how to interpret the domain of a function to the context describes by the problem.
After completing the Entry Ticket, students are asked to take out their notebooks and we review the main concepts for the day's lesson as well as review some example problems together. During this time students are taking active notes, which means they are listening, speaking, writing and reading during the time - I like to use Two Column Note format as it lends itself nicely to taking visually appealing notes in mathematics. The end game for note-taking is to teach students strategies that they can individualize and use as tools so that they can learn how to be better learners over time.
The heart rate example provides students with a chance to work on interpreting the rate of change and domain of a linear function (the equation used is Y = 220 - X, with Y = max. heart rate and X = age in years). A great extension to this example is to have students derive the function from the graph.
The Apple stock example pushes students' thinking because the data is not strictly linear. Ask students, "What is different about this graph? How can we use what we know about slope to estimate the rate of change over different time periods?"
The class engages in the Jigsaw Activity for Rate of Change and Domain for the next two sections of class. Students begin the work in “expert” groups diving deep into either the set of problems about 1. estimating the rate of change or 2. relating the domain of a function to its graph and relationship.
In this segment, students are all assigned to an Expert Group. One way to arrange students is by their current understanding of the material.
The beauty of the Jigsaw Activity is students will all have a chance to teach each other. All students have access to the different tasks of this assignment, and get to hear multiple perspectives on strategies to solve the different types of problems. I also thoroughly enjoy running this activity. My role as a teacher shifts to that of a facilitator as the students take on the role of teaching each other.
In this section, there is a transition where students go from their expert groups to mixed groups for the Jigsaw Activity for Rate of Change and Domain.
I would plan for this transition to take a few minutes, especially if students are not used to the jigsaw format. One way to keep the learning going is to have pre-assigned group names that students can connect to (local sports teams, community hangouts, etc.). When students are in expert groups give them a number so there will be a number 1, 2, 3, …. For each expert group. That way all of the number 1’s can be a mixed group and you can assure each mixed group will have at least one expert from each area. In the mixed groups students are asked to go throught the worksheet as a group, taking the time for peers to ask and answer questions of each other. Note, students are still working on the Jigsaw Activity from the previous section.
During this time, the teacher is making rounds checking in and providing cues and tips to keep each group on track. Once groups are finished I take a few minutes to have the class reflect and engage in some in-process metacognitive skills. This can be as easy as asking students what the big picture task is and where we are in the process. This is also a great opportunity to ask students to reflect on what they have done well and what they need to work on to be successful for the remainder of the class.
When groups are finished I ask them to consider the two reflection questions at the end of the activity:
-Are any of the functions modeled well by a linear function? Explain your reasoning.
-How could we assign values for the domain of the different functions? Using this assignment, what values would not make sense for the domain given the scenario being modeled in these graphs?
Students then work on the Idea Organizer: Interpreting Rate of Change and Domain. The intent of this assignment is to help students consolidate and summarize their thinking. in addition, I want students to critique an argument, in line with Math Practice MP.3, a skill that many of the students I work with have relative difficulties with.
The benefit to this Idea Organizer lies in its flexibility. It allows students at various levels of mathematical understanding to access and critique the argument. The assignment is also flexible in how it can be used in the classroom. For example, a teacher might assign the Idea Organizer as an exit ticket to wrap up this lesson or students ca come into class the next day, with the Idea Organizer completed for homework. They can then work on writing up their response in a well-polished multi-paragraph response.
Some students are reluctant to write in the math classroom, even saying they do not think they need to be able to write about math - my response is writing is a skill that might be the best mortar or glue around - it gives each of us the power to connect and communicate to people with various backgrounds and skill sets. Writing also can help students better understand the content because the process requires students to translate their ideas and understanding into another form.
Once students are finished with the Idea Organizer, I have them share their critiques with a partner and conclude with a whole class share out of ideas using the following reflection prompts:
Critique the following argument: The scenarios in this assignment are best modeled by linear functions. Provide evidence to back up your reasoning.
Pair - 4 minutes
Share - 4 minutes
For the remaining 5 minutes or so of class I hand out the Homework: Rate of Change and Domain for Heart Rate and VO2 and give students a chance to begin the assignment - I find tat giving a few minutes at the end of class to begin the homework provides a nice quiet transition to the end of class and also gives students a chance to ask me critical questions about the assignment, that could be major barriers to completing the assignment if they waited to look at the assignment until they got home.