Problem Solving Log
Laura Watkins
Reflection Problem #3
This problem was presented to students in a 5th grade classroom at Waller Mill Elementary. These students are placed in the highest achieving math class, because the math classes are split up by achievement level. My cooperating teacher does a lot of problem solving in mathematics with his students, so I found no real issues in presenting this problem to the class. Most students used the heuristic described below in the “possible method” section: they made a picture or a model based on information found in the problem. Of course, the types of pictures that they drew differed, but most had the same general idea. Students figured out right away that this problem dealt with fractions, and my cooperating teacher had already taught them some strategies in problems dealing with fractions. Therefore, this problem did not take them a long time to complete. Most students got the correct answer: that the boys ate the same amount of pizza. In the few instances of wrong answers, it was computational error, not difficulty solving the problem. An extension for this problem could be helpful for future lessons, because it was not too challenging for the students. Perhaps a problem that required more critical thinking would be a good idea as a challenge.
Problem #3
Topic: Computation and estimation: Multistep applications and order of operations – Grade 5 (SOL 5.6)
Expected heuristics: use equations and formulas; use a picture; use logical reasoning; choose the operation; use a model
“Stuffed With Pizza” Problem
Tito and Luis are stuffed with pizza! Tito ate one-fourth of a cheese pizza. Tito ate three-eighths of a pepperoni pizza. Tito ate one-half of a mushroom pizza. Luis ate five-eighths of a cheese pizza. Luis ate the other half of the mushroom pizza. All the pizzas were the same size. Tito says he ate more pizza than Luis because Luis did not eat any pepperoni pizza. Luis says they each ate the same amount of pizza. Who is correct? Show all your mathematical thinking.
Answer: Luis is correct – both boys ate one and one-eighth of a pizza each.
Possible method:
A useful way to solve this problem would be to use a model or a picture. Each pizza would get its own “circle” as a model:
The first pizza is cheese; the second is pepperoni; the third is mushroom. The pizzas would have to be clearly labeled to avoid any confusion. Once the student has drawn these three “pizzas”, he or she can use the information in the problem to gradually figure out how much of a pizza each boy ate. In this example, each boy could be designated with a color to differentiate and easily see how much of the pizza he ate.
Tito ate one-fourth of the cheese pizza, three-eighths of the pepperoni pizza, and one-half of the mushroom pizza. This would look like:
That is nine-eighths total – or changed to a mixed number, one and one-eighth.
What about Luis? He ate five-eights of the cheese pizza, no pepperoni pizza, and one-half of the mushroom pizza. This would look like:
That is nine-eighths total – or changed to a mixed number, one and one-eighth.
Therefore, both boys ate the same amount of pizza!
New York City Department of Education, Grade 5 Math: Stuffed With Pizza. Retrieved from:
Reflection Problem #5
This problem was presented to the same 5th grade classroom as in problem #3. The topic fit in nicely with the current unit of elapsed time that students were learning, but it also combined prior knowledge of fractions. The students found this to be much more difficult than problem #3, and a wider variety of strategies to solve the problem came up. Because this problem was more difficult, I let the students work in pairs to try to come up with a solution. Some used the “possible method” laid out below, some drew pictures to visualize the problem, but the majority used the “guess and check” strategy. I presume that this is because there is no explicitly clear way to solve this problem. For future problem solving activities, I must make my directions clear that students should show their work. In some cases, I could not tell the strategy that students were using to solve the problem. Accuracy in the solution to the problem was not as high as it was for problem #3. In my teaching, I would probably use this problem again, because it really got students to develop their critical thinking skills and work collaboratively to solve problems.
Problem #5
Topic:Patterns, Functions, & Algebra: Use a variable expression (SOL 5.21b)
Expected heuristics: Make an equation; make a table; guess and check; use a formula; draw a picture
“Diminishing Return”
Maxine and Sammie have the same size lawn. Maxine can mow the lawn in 24 minutes, and Sammie can mow the lawn in 36 minutes. At what time will Sammie have twice as much lawn to mow as Maxine? Show your work.
Answer:
At 18 minutes, Sammie will have twice as much lawn to mow as Maxine.
Possible method:
You could make a table to solve this problem.
Since Maxine and Sammie have the same size lawn, the only thing that varies is the amount of time taken to mow – and therefore, how much of the lawn is left to mow.
Time / Maxine / Sammie6 minutes / ¾ left to mow / 5/6left to mow
12 minutes / ½ left to mow / 2/3 left to mow
18 minutes / ¼ left to mow / ½ left to mow
24 minutes / DONE / 1/3 left to mow
30 minutes / 1/6 left to mow
36 minutes / DONE
As you can see, since one half is two times the amount of one fourth, Sammie will have twice as much lawn to mow when 18 minutes has passed.
It is also possible to make an equation out of the given information, which is another method.
Inside Mathematics, Problem of the Month: Diminishing Return. Retrieved from:
Reflection Problem #2
Once again, this problem was presented to students in the same high achieving fifth grade class at Waller Mill. I really liked this problem because it involves money and economics, which is valuable to have a skill set in, especially at the fifth grade level and above. It also expands on knowledge of number patterns, and involves comparisons of patterns. I wasn’t sure which kind of strategies students would use at the start of this lesson. Many students chose to make a table, some added numbers together without making a specific table, and some used a calculator. Making an equation seemed out of reach for them, however, as none of them chose to solve the problem this way. This problem took a fair amount of time to complete, but students were more accurate and uniform in their solutions – most of them got the right answer. Reflecting back on my instruction, I wish I had pushed them (after they had solved the problem) to come up with an equation that would describe the relationship between the checking plans at the bank. Making an equation out of a complex problem such as this one is very important in mathematics instruction in every grade level following 5th grade.
Problem #2
Topic: Computation & Estimation: Multistep applications/order of operations – Grade 5 (SOL 5.4)
Expected Heuristics: Make a table or chart; guess and check; look for a pattern; draw a diagram; make an equation
The Bank Problem
A bank that has been charging a monthly service fee of $2 for checking accounts plus 15 cents for every check announces that it will change its monthly fee to $3 and that each check will cost 8 cents. If the customer writes three checks per month, which plan is cheaper?
The bank claims the new plan will save the customer money. How many checks must a customer write per month before the new plan is cheaper than the old plan?
Answer:
First part – the older plan is cheaper (it will cost the customer $2.45, as opposed to $3.24)
Second part - the customer must write at least 15 checks per month in order for the new plan to be cheaper.
Possible method:
A good strategy to solve this problem would be to make a table. By making a table, you can answer both questions, and it is easy to find patterns between the number of checks and the monthly charge.
Number of checks / Cost, old plan ($) / Cost, new plan ($)1 / 2.15 / 3.08
2 / 2.30 / 3.16
3 / 2.45 / 3.24
4 / 2.60 / 3.32
5 / 2.75 / 3.40
6 / 2.90 / 3.48
7 / 3.05 / 3.56
8 / 3.20 / 3.64
9 / 3.35 / 3.72
10 / 3.50 / 3.80
11 / 3.65 / 3.88
12 / 3.80 / 3.96
13 / 3.95 / 4.04
14 / 4.10 / 4.12
15 / 4.25 / 4.20
Now, after making the table, you can scan to find how much the monthly cost would be for each plan, for each amount of checks (starting with 1). The first question to the problem asked us which plan is cheaper if we wrote 3 checks per month. Using our table, we can see that the old plan would cost $2.45 per month, and the new plan would cost $3.24 per month. Therefore, the old plan is cheaper in this scenario!
The second part of the problem is a little trickier. You would need to continue the table until you realize that the cost of the old plan has surpassed the cost of the new plan. Since the cost per check is less for the new plan than the old plan, we can figure that the old plan will eventually surpass the total cost of the new plan. You can see that at check #15, the old plan finally is more expensive than the new plan. Therefore, a customer must write AT LEAST 15 checks for the new plan to be cheaper than the old plan.
Chapter 1: Problem solving. (pp. 13-14). New York, NY: McGraw-Hill. Retrieved from