# The Craft of Word Problems

## Word Problems Dealing with Ratio and Proportion

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## Procedure

Students should be heterogeneously grouped in clusters of 4 to 5 students. Desks should be set up in such a way as to afford every student an unimpeded view of the board or overhead. I prefer the desks to be set up where the students all face each other and need to turn their heads to see the board, because it is far more important for me to witness
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their
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interactions, than for them to watch
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me
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every second of the math period. Because I like for the students to work collectively on math concepts, there are often heated discussions and disagreements on which method is best employed in specific situations, it is helpful for the students to know that there are rules and procedures that must be followed even in those times of discord.

Math classes in my school run about 42 minutes and it is important that we rigidly adhere to the schedule in order for the time to be used as productively as possible. The 42 minutes are broken down in the following manner. (Teachers should feel free to alter the schedule according to their own objectives and their students’ abilities.)

Tentative Schedule:

- 2 minutes - students gather and prepare themselves
- 5 minutes - students copy and discuss what they feel they are being asked to accomplish, strategies that they will need to utilize to fulfill their responsibilities
- 5 minutes - students are given an opportunity to write their findings, in their notebooks and answers
- 10 minutes - students share their responses and strategies with the teacher and the class at large, with the teacher acting as facilitator and clarifier of information
- 10 minutes - an alternative problem is posed, copied, and discussed, building on the information given by the class
- 5 minutes - the homework assignment is given and explained
- 5 minutes - wrap-up and last minute questions

When the students walk into the class there is always a Problem of the Day on the board. I have a timer that I use, and when it goes off the students realize that it is time to copy the problem into their notebooks and to begin the process of solving the problem. The students are given 2 minutes when they walk in to get themselves together, to finish the conversations they were having on the way to class and to sit themselves in their seats. Once they have seated themselves, and the timer goes off, the students finish their conversations and begin copying the Problem of the Day.

The timer is reset for 5 minutes and soon I begin to hear conversations as to what strategies will need to be employed in the process of solving the Word Problem. When the timer goes off again, I give my students a verbal cue to let them know that their group sharing should conclude and that they should now decide on the methodology that they will utilize to determine the answer. They then are required to take 5 minutes to write the method chosen and the result acquired. 5 minutes later, the timer goes off again, and we now begin the process of sharing the methods utilized and the answers obtained with the class at large.

For 10 minutes, the students are encouraged to participate. Because I try to make the answers open-ended with a variety of distinct correct responses, disagreement is encouraged, and consistently validated. I want the students to try to identify the word clues in the word problem and to verbalize the methods they used. I try to be as encouraging as possible, while at the same time I try to hear from as many different students as possible in order to get a wider view of who understood the problem and how did they go about attempting to find a resolution. I try to get at least one member of each group to say something that will contribute to the lesson, hoping to authenticate, endorse, and validate the work of each group. At the end of the ten minutes, a further problem is posed. This assists me in understanding how many of my students not only understood the last problem, but can repeat the process. For me, it is far more important that the student can duplicate the process of the first problem, than for the student to give me a correct response. This final problem is generally designed to reveal any glitches in the thinking of the students. If I find that many of the students did not quite comprehend, then I utilize this information. Because it indicates that the majority of the students did not understand, then I realize that I must reteach the lesson.

After further discussion, I give a homework assignment, and explain my expectations. Sometimes clarifications still need to be made. During the final five minutes of the class we do some wrap-up exercises, pose some final questions, and look forward to the following day’s lesson.

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Sample Lesson - Problem Solving Vocabulary
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Problem of the Day: “Juanita is your best friend. You have known her for years. Like all good friends you sometimes have fights. However, school has just begun, and you have made a new friend named Luz. Luz tells you that Juanita is talking about you behind your back. What do you do?”

On the first day that we begin math, it is important that we continue the procedures that had been outlined in the previous few days of the year. The Problem of the Day (or the POD as it will be referred to from here on in) is up on the board to reinforce the procedures. Verbal cues (as well as manual cues like me setting the timer) will be used during the first few lessons to continuously give the lesson an ebb and flow that will hopefully allay boredom, promote excitement, and above all stimulate thinking.

Further, the POD gets the students working and prepared for math. It is especially important that on the very first few days that I let the students recognize that we will be working on vocabulary as well as procedures so that I reinforce the idea that every time they use math terms they will make me happy.

For this particular lesson, there are many graphic organizers that can be used to help the student understand that problem solving is a process that people do instinctively. However, what needs to be made explicit is that even though problem solving is something that all people do instinctively, it is a process that requires some thought. We do not just simply go and solve a problem without pursuing several avenues of inquiry. There is a progression that must be followed. If the sequence is followed adequately and reasonably, then problem solving becomes easier because thought was involved, alternatives were pondered, and the best response was chosen.

The graphic organizer that I generally use contains the four basic shapes: square, circle, triangle, and rectangle. Each shape has an arrow leading from one shape to the next to show that one step progresses to another. Each shape also has a word that communicates the process of problem solving: explore, plan, resolve, examine. The children intuitively realize which space requires what response, but there are subtle variations that need to be explained. A significant note to remember is that despite teacher initiated practice, the lesson is really student directed. I consciously have to continuously remind myself that I want
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them
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to figure out the correct vocabulary to use, I want
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them
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to provide the effort, and ultimately
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they
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should find the result. Although I always try to allow my students the opportunity to consider and achieve for themselves the results, I always have a list of some important ideas that I want my students to ponder, as a consequence I sometimes ask some leading questions in order to guide the students to a specific frame of reference. (Understand that if the students arrive at a specific important concept, these leading questions do not have to be asked).

Some important questions that I might ask about problem solving include:

1. Explore - What is it that I am looking for? What information do I have? What information do I need? What possible strategies might I employ to arrive at a solution??

2. Plan - Should I estimate a solution? Which one strategy would help me to find a response? Are there any alternatives that might be easier or better? Are there any difficulties that I might anticipate?

3. Resolve - Which is the best way to find a solution? How do I do it?

4. Examine - Can I check my answer? How do I check my answer? Why is it important to check my work?

After my students have taken their first ten minutes to discuss what it is that is expected of them, they decide, and begin to fill-in the graphic organizer that I have provided for them. Many of the students try to skip the initial process of discussion, because they have the misconception that schoolwork is all about writing the responses to problems as opposed to the actual working through of the problem. In order to make sure that they do not write before they discuss, I do not hand out the graphic organizer until the second 5 minutes is about to begin.

In this exercise, as in most of the word problems we do, there is no single answer but a number of possible solutions. By giving this word problem as the initial exercise, the students can see math in a situation that most of them have had some experience with. In fact, one of the questions that will invariably arise (generally at the end of the lesson) is, “What does this problem have to do with math?” That question is the opening that I look for to express the fact that math is not so much about solution as it is about process.