The Math Forum: Problems of the Week
Problem Solving and Communication
Activity Series
Round 18: Change the Representation
All math problems, whether they are word problems, arithmetic problems, equations to solve, etc., come to us in a
particular representation. Word problems are represented in story form, using words and often referencing a
particular context. Arithmetic problems are represented numerically. Equations are represented using mathematical
symbols. Each representation has benefits to the problem solver. For example, word problems allow students to
apply their knowledge of the given context, which can allow them to check that their approaches are reasonable.
Numeric and symbolic representations can make it easy for students to manipulate objects in the problem, and to
quickly see patterns. Visual and physical representations, such as manipulatives, diagrams, and graphs, can often
help students gain new insights into the problem and provide them with additional tools for solving it. Changing the
representation can mean use of a different form of representation (e.g. using a line drawing for a word problem) or it
can mean trying different ways of presenting the information in the same form (e.g. rewriting all of the numbers as
fractions with a numerator of 1). Considering multiple representations and choosing representations that fit the
problem well are important problem-solving skills.
The activities below help students to brainstorm and work with multiple representations, and compare what they
learned about the problem using their different representations.
The activities are written so that you can use them with problems of your choosing. We include a separate section
afterward to show what it might look like when students apply these activities to the current Algebra Problem of the
Week.
Problem-Solving Goals
Changing the representation of a problem can help problem-solvers:
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Strengthen their understanding of the problem.
Gain new insights into the problem or solution.
Provide additional tools for solving the problem.
Find multiple solution paths, leading to deeper mathematical connections.
Communication Goals
Changing the representation of a problem requires that students change how they communicate about the problem
and find different ways to express the same idea or information. They might:
•
•
•
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Paraphrase the problem in terms of a different representation.
Re-tell the story of the problem with a different context.
Organize the numerical and calculation strategies using a table or other organizational method.
Use mathematical symbols to restate the problem succinctly.
Use diagrams to communicate the math in the problem.
Represent the problem graphically.
Activities
I. Brainstorm Representations
Format: students working individually or in pairs, then sharing with groups of 4-6.
There are many ways to represent math problems and mathematical ideas. Math problems are often represented in
words. Math can be represented visually, through graphs, diagrams, and sketches. Tables and expressions can be
used to represent math ideas numerically. Mathematical ideas are often represented symbolically, with operations,
numbers, variables, and functions. Each representation can help you understand and solve the problem in different
ways. Problems represented in words help you to make sense of the problem and use your knowledge of real-world
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AlgPoW: Field Trip
situations. Graphical representations can lead to new insights or problem-solving methods. Numerical
representations can help you find patterns and generate strategies. Symbolic representations represent math ideas
clearly and succinctly, and help you to manipulate the mathematical objects.
Sample Activity
Work individually or in pairs to begin filling in the blanks of the following prompts for just a few minutes. Then
share ideas with the larger group of 4-6 students. The first question asks you to think about the math ideas in the
problem, which might get you thinking about other representations you know. The second asks you to think as
creatively as you can.
1) The main mathematical ideas and relationships in this problem are _________________________________.
2) I could represent this problem by: (drawing a picture, using some blocks, making a graph, writing some
equations, telling a different story, writing the numbers or expressions in a different way).
3) The idea I want to try is ______________________________________.
Share your thoughts with your group. If the ideas you hear spark other ideas, record those too. Try to brainstorm
as many possible representations as you can.
Suggestions
These suggestions can be used as a place to start with a student who is struggling.
• Think of or make up another story or situation that could be used to present a math problem just like this.
• Think of manipulatives like blocks or chips or algeblocks or fraction bars or the number line or a drawing that
you could use to model and explore this problem.
• Think about substitutions you could make for some of the quantities in the problem (rewriting whole numbers as
fractions; express all of the quantities in terms of the smallest item; use a different expression that has the same
value but might make the calculations easier)
Key Outcomes
• Identify key mathematical ideas in the problem that can be represented.
• Think of ways that mathematical ideas are sometimes represented.
• Generate multiple possible representations for the problem to be solved.
II. Representing
Format: Individually and then in pairs or teams.
The focus in this activity is on using writing to organize the problem solving activity, to notice patterns or ideas that
make the solution possible, and to ask specific questions that need to be answered in order to make progress,
Sample Activity
Step 1: Pick the representation that seems most useful or stimulating and play it out in detail. Use your new
representation to explore the problem and solve it if you can.
Step 2: Explain to your partner how your new representation works:
o What did you notice about the problem when you changed the representation? What new information,
relationships, patterns, and approaches occurred to you?
o What have you been able to figure out for a solution so far?
o If you are stuck, which representation shows best where and why you are stuck? Ask your partner(s) if they
have a way of representing the problem that helps you get unstuck.
Step 3: Check your solution with the original problem to make sure it fits with all of the information and
constraints of that situation.
Key Outcomes
• Play out a particular representation as fully as possible.
• Consider multiple representations when you get stuck.
• Work with others to see multiple perspectives and fresh ideas.
III. Comparing
Format: Students working individually or pairs and then sharing with the whole group.
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AlgPoW: Field Trip
Sample Activity:
Step 1: Within your group of 4-6, compare the different solution paths you generated. Did different
representations lead to different insights? In what ways are the different solutions similar?
Step 2: Select the representations that provided the most insight into the problem or key steps in the solution.
Prepare a presentation for the class on how each representation helped you get to a key insight into the
problem.
Step 3 (optional): Submit your write-up to the PoW online.
Step 4 (optional): Use a jigsaw or gallery walk format to share your explanation with classmates and to
appreciate their insights.
Key Outcomes:
• Compare insights generated by multiple representations. Identify the similarities and the differences in the
contributions from each representation.
• Evaluate how well different representations fit a given problem. Figure out how to recognize when a particular
way of changing the presentation of the problem will be useful.
Examples: Field Trip (AlgPoW)
The goal of these lessons is for the students to reflect on their own process in developing simpler versions of a
problem. While it’s tempting to steer them towards certain key ideas, we want students to experience the gain in
confidence that comes from being able to rely on their own resources in order to get going. As a result, we tend to
hold back on suggestions and focus on supporting the student’s own thinking. If students are stuck, or we feel their
ideas need further probing and clarifying, we might help with facilitating questions that reinforce the problem-solving
strategies. Check out the “algpow-teachers” discussion group (http://mathforum.org/kb/forum.jspa?forumID=528)
for conversations about this problem in which teachers can share questions, student solutions, and implementation
ideas.
If we do facilitate by asking some strategy questions, then at the end of the session we often ask students to notice
the questions and suggestions we asked so that they can begin to do that for themselves: Which were helpful?
Could you see how you could use these with other problems? Which questions would you put on a class list of
“Ways to get Unstuck in Changing the Representation”?
I. Brainstorm Representations
• The mathematical ideas in this problem are:
o Total cost: $189
o Total Cost=# of items * cost for an item
o Constant total cost: $189
o Changing number of items (students):
o Original #: n
o New #: n-12
o As # of items is reduced, cost per item increases to keep total cost constant:
o Original cost: c
o New cost: c+1
• I could represent this problem by:
o Drawing a picture.
o Acting the problem out.
o Making a graph.
o Writing equations using variables.
• My representation might look like:
o An area diagram with the dimensions equal to the # of students and cost per student.
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AlgPoW: Field Trip
o Taking 189 chips and put them into equal-sized piles.
o Graphing the two total cost relationships.
o A system of two equations with two variables.
II. Representing
Student 1: In this problem you have two numbers multiplied together (# students * cost per student) and this
product is always 189. That made me think of two rectangles that have different dimensions but the same area.
I made an area model for the multiplication relationships in this problem:
I started with a rectangle that is n by c. When the number of students drops by 12 and the cost goes up by $1
per student, a new rectangle is formed that is (n-12) by (c+1). The red 1 by (n-12) piece is added and the blue
12 by c piece is subtracted. If the area (total cost) is the same ($189), then the areas of those two pieces have
to be the same: 1(n-12) = 12c.
That’s interesting. It makes sense, but I wouldn’t have seen it that way without this representation. You charge
everyone who is still going $1 more: $1(n-12). That has to make up what the 12 players would have paid: 12c.
Now, how do I use that to find the number of students and how much they paid? I haven’t used the $189 yet. I
know that the original cost per student is $189/n. I can use that in my equation:
n-12=12c.
I’ll just substitute 189/n for c and solve for n.
 189 
n − 12 = 12

 n 
Hmmm, that’s kind of scary.
€
Student 2: I divided 189 counters into 21 piles with 9 counters in each pile.
This represents 21 students paying 9 dollars each. Then I took away 12 piles, which is 108 counters. If I took
those extra counters and gave 1 to each of the 9 piles remaining, I would still have a lot of extra counters.
I need a lot more piles if the additional $1 is going to make up the difference.
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AlgPoW: Field Trip
What is the next whole number of students I should try? 189 can be factored as:
189 * 1
63 * 3
27 * 7
21 * 9
27 or 63 are the other whole number it makes sense to try, but I know I need a lot more piles, so 67 is the next
number it makes sense to try. I imagined 63 piles. Each had 3 counters in it. I took away 12 piles, which is 36
counters. I have 51 piles left. Now I don’t have enough extra counters to give 1 to each remaining pile.
Let me see if I can turn this into an equation, since working with whole numbers doesn’t seem to be effective:
First I make n piles with c counters in them: n*c = 189
Then I take away 12 piles and this is my extra counters: 12*c= e
Then I see if the number of extra counters is just enough to give 1 to each of the n-12 remaining piles:
Does e/(n-12) = 1?
From the last equation I see that e has to equal n-12 for this to work. I can substitute n-12 for e in the
equation above:
12c=n-12
From the first equation I know n=189/c
12c = 189/c – 12
I can make that a more normal-looking equation by multiplying both sides by c. I also can take a factor of 3
out of each term, which simplifies the numbers. I get an equation I can factor:
2
4c + 4 c − 63 = 0
(2c + 9)(2c − 7) = 0
€
Setting each factor to 0 and solving it, I can see that the cost per student originally was $3.50. The number
of students was 54. The new cost for the 42 students who actually go is $4.50.
In terms of my representation, I could make 54 piles with 3.5 counters in each pile. Then I could take away
€ 12 piles, with a total of 42 counters in them. I could give each of the 42 counters to one of the remaining 42
groups.
Student 3: I figured I could make graphs of the total cost for both situations and see where the graphs cross,
because they are supposed to equal $189. I let x represent the number of students originally going on the trip
and y be the original cost per student.
It looks as though the number of students should be 54 and the cost per student should be $3.50.
Check:
54*3.5 = 189
(54-12)(3.5+1) is 42*4.5 which is $189.
That works! So 54 – 12 = 42 students actually got to go, and they paid 3.50 + 1 = $4.50 each.
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AlgPoW: Field Trip
Student 4:
n = number of students originally
c = cost per student originally
$189 is the total cost.
189/n = c. I can use this to substitute for the cost.
When the 12 students go to baseball instead, 189/(n-12) is the new cost per student
The cost goes up a $1 per student from the original price.
189/(n-12) = 189/n + 1
Simplifying:
189(n-12) + n(n-12) = 189n (multiply both sides by n and n-12 to clear the denominators)
2
189n − 189 * 12 + n − 12n = 189n
2
n − 12n − 189 * 12 = 0
I wonder if I can find factors of 12*189 that differ by 12. Let me break it down to more factors.
12*(9*21)
€
(3*2*2)*(3*3)*(7*3)
I need to make two factors that differ by 12. So I can try making factors that both have 6 in them or that
both have 3 in them. I’ll try 6.
The factors could by 6*7 and 6*9. 189 = 42*54. 54 is 12 more than 42 because 6*9 has two more sixes
than 6*7.
(n + 42)(n − 54) = 0
So, the number of students originally was 54 (can’t be -42). Which means the cost per student was
$3.50. The 42 students went on the field trip ended up paying $4.50 each.
III.€ Comparing
Writing up the Combined Solution
One thing we noticed in all four solutions is that we ended up writing equations in order to solve the problem.
The solution involved non-whole numbers, so it was harder to use guess and check or acting the problem out to
come to a solution. Guess and check and acting the problem out did help us to set up equations, understand the
problem, and check our solutions, though.
The simplest representation to write up is using equations to model the situation, so we chose that. We included
a graphical and a symbolic representation of our equations (because once you have an equation it’s easy to
graph it and helps you see more relationships).
Student 1 asked if we could use her equation, since she wasn’t sure how to solve it, so we did.
Setting up the equation
We based our equations on Student 1’s noticing that the amount of money that the 12 baseball players would
have paid was equal to the additional amount of money that the remaining students contributed when they each
put in another dollar.
We let n represent the total number of students in the class, and c represent the original cost of the trip.
12c (amount baseball players would have paid) = (n – 12)*1 (amount remaining students contributed).
Since nc = 189 (number of students * cost per student = total price), we can replace c with
189
.
n
To think about the equations graphically, we need to get them in “y =” form. We decided to isolate c, the total
cost.
€
189
n − 12
Equation 1: c =
Equation 2: c =
n
12
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€
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AlgPoW: Field Trip
So one way to find the solution is graphically. The original number of students was 54, and the original cost was
$3.50
To solve the equation algebraically, we set the two equations equal to each other.
Equation:
189 (n − 12)
=
n
12
To make the equation less “scary” we decided to get rid of the fraction by multiplying each side by 12n.
12
€(189) = n(n − 12)
12(189) = n 2 − 12n
€
This looks like a quadratic equation, so we put it in standard form:
€
0 = n 2 − 12n − 2268
We thought about using the quadratic formula to solve it, but 2268 is a really big number to work with. Then we
thought that the equation should be factorable, since we know the solution doesn’t involve irrational numbers.
€
We need to find factors on 2268 that differ by 12. We looked back at Student 4’s work and realized that he had
found those factors by thinking systematically about factors of 12 and 189.
So we factored n 2 − 12n − 2268 into (n − 54)(n + 42) .
0 = (n − 54)(n + 42) .
€ 0 or n + 42 = 0, so€n = 54 or -42.
n – 54 =
€
Clearly, n has to be 54, since it represents the original number of students.
54 students were in the class, so each would have paid $189/54 = $3.50.
Instead, 42 students went on the trip (54 – 12 = 42) and paid $4.50 each. Not a bad deal, but it’s too bad the
whole class couldn’t go.
What we Learned from each Representation
Representation 1: Area Model
This gave us a new insight into what we could compare in the problem. It also helped us see the constant in the
problem and how we could express the constant two different ways, once for before the baseball players found
out they couldn’t go on the trip and once for after.
Representation 2: Acting out the Problem
Acting out the problem gave us a slightly different focus, which actually resulted in an equation that was easier to
factor!
Acting out the problem focused Student 2 on the money that the baseball players would have contributed. She
thought about how the money needed to be divided in order for the number of students left to equal the number
of dollars the ball players should have contributed. This is similar to what Student 1 did, but focusing on cost
rather than number of students happened to result in an equation that factored more easily.
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AlgPoW: Field Trip
Representation 3: Graphical
We could represent this situation graphically because we could show two different ways that cost per student
and number of students were related. The graphs show the only ways that both relationships can be satisfied at
the same time (and that there’s only one way that makes sense for this problem with positive numbers of
students and positive costs).
It took a little work to set up the two different relationships, but then the solution was easy because we could use
technology to find the intersection point.
Representation 4: Equations
This problem would have been hard to solve without equations. Student 4 was able to translate directly from the
problems into equations, but ultimately each representation ended up either helping us to do that translation or
required that we do that translation.
If the dollar amount had been a whole number, then Student 2 might have been able to solve the equation using
her acting out method. Student 1 might have been able to use guess and check and logical reasoning to work
out the solution using her visual. But since the solution wasn’t in integers, guess and check would have been
rough, and so equations, however we came to them, were very helpful.
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