As students enter, I hand each a copy of ENTRANCE SLIP. I ask my students to complete this slip independently. If anyone has difficulty, however, I encourage discussion among elbow partners. After Part 1 of the lesson yesterday, I expect that my students will handle the questions pretty smoothly.
After giving students a reasonable amount of time to complete the worksheet, I will call on volunteers to share their answers. Some possible explanations are:
1. Because there is one cost (c) for a given number of muffins, the "cost is a function of the number of muffins bought".
2. m is the independent variable and c is the dependent variable because c depends on m.
3. The domain is the set of integers greater of equal to 0. The range is any positive even integer
Students may say the domain is the set of real numbers. If this occurs, I ask if it is possible to buy a negative amount of muffins, or part of a muffin, to which they will likely answer "no". Questioning like this often helps students to better understand why the domain is an important concept.
For part 2 of this lesson there is no New Info section. We have defined a function in two ways. A relation involving two sets, input and output, where each component of the input determines exactly one output component. We also stressed the correspondence between independent and dependent variables in which each member or element of the independent variable corresponds to one element of the dependent variable set.
Before beginning this section, I ask students to get up and walk around to find one or two students whose birthday is in the same month as theirs. I limit the size of the groups to three students. It’s always a good idea to have students get up and move around, especially after a direct instruction section where everyone sat for a long while.
Then, I make sure each group receives an Application Functions Rule worksheet. Although many times I give students their own worksheet, today I only give one sheet per group to encourage discussion and peer assistance.
Questions 1 to 3 ask students to create their own real world situation using the given image. They write their situation in the first line and proceed to describe what the independent and dependent variables are in the context they provide. Again, they are asked to write a function statement. An example would be, "the distance traveled is a function of the time spent biking."
I expect that describing the domain and range will probably be the most challenging part of the work. Students may have the idea that the domain and range are the set of all real numbers. But, in many real world contexts, the domain is restricted. For example, negative numbers may not be valid input values. So, I ask that to test various possible values from the domain and make sure that they can explain why the values are in the domain. I want them to focus on the idea that the numbers in the domain are "feasible" or "not feasible."
For Questions 4 and 5, student responses will vary. Expected responsed are;
#4 Time is input and temperature is output (temperature depends on time)
#5 Time is input and height is output (height depends on time)
For Questions 6 through 9, I motivate students to discuss each situation carefully among the members of their group. In math textbooks, one of the first things that students learn about functions is that there are sets of ordered pairs that are, and are not functions. This is fine, but the nature of using real world circumstances necessarily opens discussions and thinking, about restrictions to the domain and range, whether there really is only one output for each input value, or whether one set (dependent variable) depends on the other set (independent variable). Students should realize that questions 7 and 8 are not functions. Number 7 because two middle school students can have the same final math grade. Number 8 because the number of marine soldiers does not depend on the number of soldiers in the navy.
One of the informal ways I like to close is by asking how students feel about their mastery of concepts. When I do this I ask my students to draw a happy, straight, or sad face representing how much they understood certain questions or concepts.
I ask each group to draw the faces right next to the questions in their application worksheets.
Happy face = "I really understood this idea"
Straight face ï»¿= I understand but still have questions about this..
Sad face = I really don't understand this idea or how to answer these questions.
This usually gives me an idea of which ideas I should address again, or if there are simply a few students that I need work with more, maybe after class and so forth.
For homework this evening I plan to ask my students to complete the Homework Functions Rule 2 handout.