Problem Solving Log
Grade: 1
Topic: Computation (Powers of 10)
Expected Heuristics:
 Act it out/Use manipulatives
 Make a table
 Make a list
 Guess and check
Problem:
There are 38 children going on a field trip. A small bus can carry 10 children. How many small
buses will be needed?
Solution:
This is a helpful problem for primary-grade students to practice selecting and using appropriate
problem-solving strategies. They can reach a solution using any of the above heuristics. For
example, they can make a list or table to help visualize and organize the data. They should find
that 3 buses can hold 30 students. However, there are still 8 children left over, so the class will
need one more bus to carry the remaining children. In all, the class will need 4 buses.
Teaching Log:
I taught this problem to a student, Brice, in a one-on-one session. Without prompting, Brice
immediately used a guess and check strategy to solve the problem. His initial guess of 3 buses
was an impressive start, showing that he had some prior knowledge of powers of 10. He then
chose to draw a picture to check his estimation. While this was a heuristic I had not expected him
to use, it was appropriate and it proved effective. He started by drawing one bus and then filling
it with 10 dots to represent the 10 students it could carry. He then drew a second bus, adding an
additional 10 students. At this point, he paused to count the total number of students drawn. This
interruption suggests that while Brice is able to skip count by 10s, he does not consistently
recognize when this skill can and should be applied. After counting 20 students, Brice moved on
to draw a third bus. At this point, he was able to check to see if his initial estimation of 3 buses
was correct. As I expected, he quickly found that his estimation was off and that a fourth bus
would have to be drawn to accommodate the remaining 8 students. Using a combination of these
two heuristics, Brice was able to solve this problem without my assistance.
To extend the problem further, I then asked Brice a series of similar, yet challenging
scenarios to assess his problem solving strategies. For example, once he had found the correct
response, I asked him to solve the same problem, but with 28 students this time. With only a
quick glance at his paper, he was able to remember that the number 28 is exactly 10 less than 38,
the initial problem, and so the answer would be 3 buses (1 less than the previous answer, 4). This
type of practice helped reinforce his computation skills and flex his higher-level reasoning.
Overall, I feel that this problem was well-selected for this student’s needs, and I expect
that it would likewise go over well if taught to the rest of the class. In the future, I would build in
additional scenarios (as I had improvised with Brice) to accommodate a range of performance
levels and learning styles.
Grade: 1
Topic: Addition
Expected Heuristics:
 Act it out
 Use manipulatives
 Write an equation
 Draw a picture
 Guess and check
Problem:
Thomas needs 30 apples to make a pie. Which two boxes of apples does he buy?
A
9 apples
B
12 apples
C
18 apples
D
22 apples
Solution:
This is a helpful problem for primary-grade students to practice selecting and using appropriate
problem-solving strategies. They can reach a solution using any of the above heuristics. In order
to help students begin to think about the possible approaches, ask them: “What do you know?
What information is given? How can we eliminate options?” Using one of the above heuristics,
they should find that boxes B and C equal 30, so Thomas will buy these two boxes. All other
combinations are incorrect because they result in either more or less than 30 apples.
Teaching Log:
I taught this problem to a student, MacKenzie, in a one-on-one session. Rather than use the
picture above, I printed and laminated clip-art pictures of baskets with numbers #1-20 overlaid
on them to signify the number of apples in each basket. This modification allowed me to quickly
create numerous scenarios for the student to solve based on her demonstrated needs.
First, I began by presenting the problem using the numbers 5, 8, 7, and 4, for which I
asked MacKenzie to find the two numbers that combine to equal 12. MacKenzie approached the
problem by using guess and check. She pulled aside two numbers, 7 and 5, as her initial guess.
She then checked her answer by using unifix cubes. The cubes were particularly helpful for this
student because she has not yet internalized her #1-10 addition facts. They allowed her to check
her estimation through a concrete representation of the problem. In this way, the problem seemed
to help her computation skills, as she was encouraged to make more careful and precise
estimations.
I also taught this same problem to a higher performing student, Jacob. The student was
quickly able to solve the same problems administered to MacKenzie, without the help of
manipulatives, and so I made many modifications to accommodate his needs. I began by
increasing the number of distracters in the set of numbers given. I also threw in problems that
contained more than one number combination. For example, he was asked to find a sum of 14,
where the options included 6, 9, 5, 4, and 10. In this case, Jacob found both solutions (10+4 and
9+5). While he did not use manipulatives, he occasionally referred to a number line to check
those answers than exceeded 10. Once it was clear that Jacob had mastered many of the addition
facts tested by this problem, I then allowed him to take charge and use the cards to create a
difficult problem for me to solve! This proved to be a great learning experience, and it was
evident that he was excited for the opportunity to try to “trick” the teacher. In addition, it also
tested Jacob’s critical thinking skills. He had to decide on an appropriate problem, identify
several possible solutions, and create clever distracters to challenge me. Turning the problem
around also allowed me to model appropriate problem solving tactics, such as guess and check
and acting it out. For each problem I was given, I walked Jacob through my thought process
aloud until I reached a solution. I then asked Jacob to confirm my answer by using his own
heuristic (in this case, he chose to use manipulatives).
I was pleased with this problem. It was easy to manipulate to meet the needs of a range of
students and it provided an engaging and worthwhile learning experience. Both students truly
seemed to enjoy and benefit from the activity, and so I would definitely use it again with future
students.
Grade: 1
Topic: Skip counting (by 2’s)
Expected Heuristics:
 Act it out
 Use manipulatives
 Draw a picture
 Create a graph
Problem:
Mario gets 2 new dinosaur models each week. How many dinosaur models does he have after 6
weeks?
Solution:
This is a helpful problem for primary-grade students to practice selecting and using appropriate
problem-solving strategies. They can reach a solution using any of the above heuristics. They can
create a graph (like the one below) write the number 2 in each column to represent the number of
dinosaur models collected each week. When the graph is completed, the students can then
compute the total number of dinosaurs by skip counting by 2’s. For those students who are not
yet ready for skip or rote counting, they may alternatively choose to draw a figure of each
dinosaur model in the columns and simply count the total number of pictures to reach a solution.
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Using any one of the above heuristics, the students should find that Mario will have 12 dinosaur
models at the end of the 6 weeks.
Teaching Log:
I taught this problem to a high-performing student, Shaia. The problem was appropriately
selected, as the students were currently learning about graphs. While many of the students had
not yet learned how to create a graph on their own using the information given in a problem, I
was confident that Shaia would be able to tackle the problem with some guidance. I began by
reading aloud the problem. Together, we isolated the important information. I then asked Shaia
to explain how she would approach this problem. She suggested using manipulatives, which I
agreed was a helpful strategy. Using unifix cubes, she quickly found the solution to the problem.
I then asked her to consider other methods for solving the same problem that may be more
efficient. I pointed out our current lesson on graphs, reminding Shaia that we can use graphs to
simplify data into an organized, easy-to-interpret manner. I then posed the question: “How can
we create a graph to solve this problem?” With some guidance, Shaia set about creating a graph
with six columns to represent the 6 weeks in the problem. Within each column, Shaia began
organizing manipulatives to represent the dinosaur models. After completing two columns in this
fashion, she realized this was just as tedious as her first approach and so she abandoned the
manipulatives in favor of simply writing the number 2 in each column. Once Shaia finished, I
then asked her to calculate the total number of dinosaurs collected over the 6 weeks. Anticipating
that she may turn to rote counting, I asked her to consider: “which method of calculating might
be easier than counting by 1’s, such as ‘1… 2…3…4…’ and so on?” This question prompted
Shaia to recognize that skip-counting by 2’s was more effective. By modeling these alternate
problem-solving strategies, Shaia again reached the correct solution.
For this student at this point in the year, this problem required considerable instruction
that I had not initially anticipated. I feel that other students in the class would be overwhelmed
by this type of problem, as many of them are not even completely comfortable with basic
addition facts. For many students, the abstract representation of data in a graph is too difficult to
understand. I would then encourage them to use manipulatives, as this type of extended practice
can help them begin to bridge the connection between concrete and abstract representations.
Despite these complications, I feel that teaching this problem was a worthwhile experience. It
helped me assess Shaia’s needs in a one-on-one setting and guide her to an appropriate solution
through modeling and open-ended questioning strategies.