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The Cat Food Problem

-by Val Barnes (Grade 5 Math Teacher) and Kevin Sonico (Math Learning Coach)

The Grade 5 students were posed with the task of determining which store, Petland or PetSmart, gave a better deal on the same brand and same size of cat food (Figure 1). Before proceeding with their solutions, students must hypothesize through estimation and rounding which of the two would be the better deal. Students then worked in groups to solve this problem. Each group was also assigned a Grade 8 student who facilitated the discussion. The facilitators’ responsibilities did not include providing hints or strategies, rather to ask questions such as “How do you know this is true?” or “How do we prove this?” and make statements such as “Let’s show proof of this.”

 Figure 1 - The problem

Along with the task of solving the question, students were also given the task of verifying their answers. Instead of solving the question using one methodology, students were encouraged to find alternate strategies or ways to solve the problem. Most students were able to show a visual representation of their process, where they solved the unit value or price, i.e. price per can. Some groups solved it by “doling out” the largest amount they could divide the amounts into the groups, that is by $1, then looked at the remaining amount and students divided them in smaller “chunks” afterwards. Their ability to use this method tells us that the students are aware of place value.

 Figure 2 - Students doling out the dollar-value by the dollar, the time and then the penny

 Figure 3 - Values chunked into dollar, dime and nickel

 Figure 4 - Another strategy that students employed to determine the unit value (price per can) was the use of the long division alorithm

Figure 5 - Long division algorithm with the diagram as strategies to determine

An interesting strategy arose from one of the groups. Instead of determining the unit price from the two stores, they determined the price of unit price for one and then multiplied that number by the quantity for the other store, very reminiscent of common denominators and equivalent fractions, see Figure 6.
Figure 6 - Making both quantities equal to 20 cans

As their alternate strategies, most students employed the working backwards method. They did this by multiplying the unit price by the respective quantities – 12 for PetSmart and 20 for Petland. I think this process does not concretely provide another way of tackling the problem, rather it is a method to verify whether or not the division strategy they employed was accurate. For instance: if a/b = c; then they checked it by b*c=a (See Figures 7 and 8)

Figure  7 - Multiplication algorithm to verify their division strategy 

 Figure 8 - Another multiplication strategy to check the quotient from the original problem.

Interestingly one of the groups used repeated addition (or “skip counting”) as an alternate strategy. However, they did not use it necessarily as a different method. Rather than look at the prices from each store for the same quantity, they used this strategy again as a way to verify that the quotient, i.e. unit price, was indeed the correct amount. (See Figure 9).

 Figure 9 - Skip Counting, but could have been used to compare the prices at 5 cans, for instance

When students presented their solutions along with their strategies, we observed one common mistake. Many were mis-reading the long division algorithm. Instead of reading the dividend divided by the divisor, many were reading it from left to right, i.e. divisor divided by the dividend – a problem with the algorithm’s set up. Upon closer examination of their work, however, we know that it was not a conceptual understanding problem but a “reading” problem.

Although students were able to visually represent their process (Figure 2), manipulatives could have been provided to help with discussions and strategies. In all, the experience has been a positive one for the Grade 5 students and the 5 Grade 8 facilitators. The former enjoying the guidance of the latter, while the latter enjoying leadership responsibilities. In addition the facilitators also expressed being challenged by this experience as they had to rephrase questions in a language that the Grade 5 students could understand without so much coaching them to the solution. In all, both groups indicated enjoying the collaboration. My experience and conversations with Mrs. Barnes during this lesson study have also been positive. We were able to discuss themes such as common threads in strategies, common misconceptions and errors in student work, etc.

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