SWBAT derive equations in two variables. SWBAT solve equations for one of the variables. SWBAT graph equations and identify solutions.

How long should an employee's shift be? Students determine possible shift lengths to satisfy an equation.

10 minutes

The task in this lesson is derived from the IMP curriculum. Click here for more information about IMP and to purchase textbooks.

This lesson uses an activity that can be found on page 242 of the IMP Year 1 textbook (2009). The main idea of the task is that two girls and three boys must work shifts that cover ten hours. The activity then asks students a series of questions where they first find ordered pairs that will fit the situation, then solve write an equation for one variable in terms of another. From there, students graph the equivalent equation on a graphing calculator and find more ordered pairs that could represent the length of a shift. I adapted this task to better fit my student population.

I begin the class by introducing the main problem on the Smartboard or whiteboard. I usually write something like:

**A local ice cream stand needs 10 hours of coverage a day during the summer and has 2 girls and 3 boys as employees. Only one employee is needed to work at a time. Girls and boys shifts will be different lengths of time. What are some possibilities for the length of each shift?**

I give students a chance to think about a few options before having them report out. If students are stuck getting started, I might ask, "What if a girl's shift is 1 hour? How long would a boy's shift have to be?" Students will need to work with fractions here, and if we have time you can talk about how fractions of an hour relate to minutes (but I try not to get bogged down here). I elicit a few different options for length of times of the two kinds of shifts. Some students will say each shift should be two hours in length. I let them know that that would work, but remind them that the owner wants boy's shifts and girl's shifts to be different lengths.

30 minutes

Next, I let students get working in small groups or pairs on the activity questions.

If students struggle to write an equation to represent the two kinds of shifts, I ask them to look at the work they did to come up with possible lengths. I might ask them questions like:

- Well when you had the girls work three hours, what did you do to figure out how much time their shifts would take up? Ok, you said you multiplied by 2. Did you multiply by 2 when you had them work 1 hours shifts? Ok, so what I hear you saying is you took the length of the girls shift and multiplied it by 2, the number of girls? Can you do the same for the boys?
- Once you found out how much time the girls' shift used up and then you found out how much time the boys' shift used up, what did you do to see if they covered the 10 hours?

Hopefully, this line of questioning, which draws on **SMP 8: Look for and express regularity in repeated reasoning**, will lead students to see how they can write the equation. Some students will struggle with what B and G represent. Be sure they are clear that B represents the number of boys, NOT the length of a boy's shift. Bringing them back to their own calculations should help them to see this point.

If students struggle with Question 3, I use a similar line of questioning. I might ask,

- So, let's say a girl's shift is 1 hour. If a girl's shift is 1 hour, how would find out what the boy's shift would have to be? What calculations would you have to make?
- Again, this question helps students to "look for and express regularity in repeated reasoning" as they then generalize these computations to write an equation in terms of G.

When students graph the resulting equation on the graphing calculator, I point out that now they have the equation solved for one variable, so it is ready to go into the calculator. If I have used language like "solving for y" I tell them that B would be the y here.

Lastly, when students find other points that would satisfy the shift lengths, they are working on CCSS REI.D.10 "Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane." I want to make sure students realize that all of the ordered pairs on that line are the possible shift lengths.

20 minutes

Depending on how much time I have at the end of class, I might make a big In/Out table on the board and have students write in the ordered pairs they found that represent possible shift lengths. Then I have students make a large graph of all of the points. This will be a nice visual to reference during the discussion. The key points I want to address in the discussion section of this lesson are:

- The repeated calculations students do that can help them generalize to an equation, both in the first question and in the question where they solve for the length of a boy's shift.
- The value of solving for one of the variables. I ask students why they might want to solve for the length of a boy's shift? What does that help them to do in the context of the situation.
- The graph as the representation of the all the possible lengths of boys and girls shift.
- What is happening at the intercepts of the graph. What if the length of a boy's shift was 0 minutes?
- Why we are only looking at the first quadrant, but the the extension of the equation beyond that.
- The relationship between the two variables. I might ask students, what happens to the boy's shift as each girl's shift gets longer?

Closing Advice:

Sometimes I find in a lesson like this one, where students have gone through a series of questions that lead them in a specific direction, students who struggle may be able to get through the "tasks" but won't have a good understanding of the overall theme. I end class today by recreating that arc, so that students can see why they did the different steps of the activity.

I use a graphic organizer like Lesson Review in the Resource section. I project it on the board and have students fill in the sections sequentially. What we want students to see here is that first they derived an equation, then the solved the equation for one of the variables, and then they graphed it to find other solutions to the equation. This is an important sequence that I want to make sure all students understand.

If you are looking for an appropriate homework assignment to give students after this lesson, IMP has a great follow up activity called *Fair Share for Hired Hands* on page 243 of their Year 1 textbook (2009). The only difference that you may want to discuss with students after they do the activity is that their is a restriction on the amount of money experienced and inexperienced workers can be paid. Inexperienced workers cannot make more than experienced workers, so not all points on the line will represent solutions to the problem, though they will satisfy the equation.

In some years I teach this lesson directly from the IMP curriculum or assign work from the curriculum for homework. Here is the citation for the lesson that I use.

IMP. (2008, June 6). *Fair Share on Chores*. Retrieved from the Connexions Web site: http://cnx.org/content/m16404/1.2/