The purpose of today's word problem is to get students engaged in problem solving, to show there are multiple ways to solve the same problem, and to get students thinking about how they can express their mathematical thinking in writing.
I begin today's class by handing out cafeteria actions and reactions. I read the problem aloud with my students. There is a lot of written information in this problem, so I make sure to go through it slowly. I encourage students to highlight or underline important information. NOTE: In today's class we only use the first problem about the trays. We work on the second problem about the number of students at the front table in the next lesson.
When we finish reading the problem, I ask students what the problem is asking, "What are they looking to find?"
Now, I let students get working on the cafeteria actions and reactions problem. I think it's best for them to work individually, but I might have have students work in pairs. Many students will struggle with not knowing where to start. I will have manipulatives available so students can "act out" or model the story.
If they are stuck, I ask students to pick a random number of trays in one carton to start with. Then, I have them go through the various events in the problem. For example, a student might chose to start with 20 trays to a carton. If so, they will have to go through the different parts of the problem, multiplying 20 by 4 to get the total number of trays, then subtracting 24, etc. If they started with 20, students will realize that they have too many trays left at the end, and therefore should adjust their starting point accordingly.
I make sure to look out for students who work the problem in reverse order. That is, they may think about those twelve trays in the three lines and go backwards through the story. If they work this way, they will have to multiply by three first and then add. This is an interesting reversal that I will highlight in the discussion if it occurs during the lesson.
DIFFERENTIATION: If some students solves the problem easily or early and are ready for an extension activity, I will ask them to try to substitute a variable for the unknown in the problem. For example, Elvira wants to know how many trays are in a carton (and therefore, how many trays she started with); we can use x to express this number. I challenge students to come up with an equation that would represent this problem situation. This equation involves complex fractions and my students may need a review on how to work with them.
This task also has a second part that students can move on to if they quickly solve the first task. The second task asks students a similar problem, and additionally focuses on whether the order of the steps in the problem matter.
It is likely students in my class will solve the problem in different ways. So, I want to give students the opportunity to highlight their thinking and the diversity of strategies. I will also try to make connections to the different ways of solving the task as explicit as possible. For example, a student who works backwards through this problem will do the steps in the opposite order of the student who uses a guess-and-check strategy.
If a majority of students used guess-and-check to solve the problem, I will show them a strategy for turning their work into an algebraic expression. Guess-check-generalize is a method that helps students to see their repeated calculations and translate that repetition into algebra. For example, each time a student uses a new guess to try in the problem, s/he multiplies that number by 4, subtracts 24 and then divides by 3 to see if the answer is 12. I typpically show students that they can replace their guess with a variable and write an equation to represent the problem.
Before concluding, I emphasize to students that ALL ways of solving are equally valid. I make sure to give praise to students who took non traditional approaches. And, I always ask students HOW they know their answer is correct. This is an important part of MP3: Construct viable arguments and critique the reasoning of others.
To continue to make progress around writing about math, I provide a sample student Write Up for this problem. I hand out copies of the writeup to students. I ask the students to read through the sections either aloud or on their own. As they read, I ask them to highlight or underline any good mathematical thinking that they see in writing. When the class has finished reading the write-up, I ask students to share out what they underlined/highlighted and WHY. I highlight the following points:
I ask students to write up the Process section of their own experience with Cafeteria Actions and Reactions. I explain to students that the Process section should include how they got started on the problem, where they got stuck, what they tried, how they moved forward, and if anyone helped them.
I remind students that they started this kind of work on the Broken Eggs problem. I may assign the completion of the write-up for the Broken Eggs problem at this time. No further time in class will be spent on that problem.
Cafeteria Actions and Reactions is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.