In addition to solving problems on their own, students of algebra
should analyze the work of others to further their own understanding.
The Principles and Standards for School Mathematics states,
"Students' facility with symbol manipulation can be enhanced if it is
based on extensive experience with quantities in context" (NCTM,
p. 227). The problem in this lesson gives students an opportunity to
consider a proportional relationship and interpret the results of an
Prior to this lesson, prepare a transparency of the Bagel Comparison Chart.
In addition, attempt to understand the mathematics on this chart before
teaching this lesson. You will be better able to facilitate a class
discussion if you have thought about it on your own before discussing
it with students.
At Tysons Bagel Market in Vienna, VA, the owners display a sign
with some information and equations that compare their bagels with
their competitors’ bagels. (To add a little interest, you might want to
tell students that this store really exists, and the sign really did
appear at the Bagel Market in December 2005.) Display the Bagel Comparison Chart
on the overhead projector, and let students think about it for a
minute. This should be a silent time for individual work, as students
try to make sense of the information on their own.
Students will attempt to make sense of the chart during this
lesson, mainly through partner and whole-class discussion. One
discussion that you may want to have right at the start, though,
concerns the first numbers in the table. The weight given for a bagel
is 25‑35 ounces, which is probably an error, since this would mean that
each bagel weighs about two pounds. (A typical bagel weighs less than
five ounces.) You might want to ask students to comment on the
reasonableness of this number. You might also want to ask students what
may have caused this error. [One possibility is missing decimal points;
perhaps the weight ranges are supposed to be 3.3‑3.5 and
2.5‑2.8 ounces. Another possibility is that the units are mislabeled.
Instead of ounces, perhaps the correct units are grams, though this
doesn't seem right, either—33‑35 grams is only a little more than an
ounce, which is too light.] A good discussion can result from students
speculating as to the error that occurred.
After 1‑2 minutes, pair students and allow them to share their
thoughts with a partner about what the information in the table means.
Push students to come to consensus during this conversation. That is,
if the students disagree about what the table is attempting to convey,
they should try to resolve their disagreement with a reasonable
After 2‑3 minutes of partner sharing, bring the whole class
together and allow the entire class to share their thoughts. To help
this discussion, use the following questions:
- What point is Tyson’s Bagel Market (TBM) attempting to make with
this sign? [They are attempting to argue that their bagels are a better
value, even though they may cost more than their competitors.]
- What value does the variable x represent? [The cost of a competitor’s bagel, assuming that price is proportional to weight.]
- What does "their price" refer to? [In the calculation, the last line states that x = 17.25 ÷ 33; converted to a decimal, this gives x = 0.523. Since x represents the cost of a competitor’s bagel, "their price" refers to the price a competitor should charge.]
- What factor is TBM saying contributes to the price of bagels?
Is this the only factor that should be considered when making a
purchase? Explain why or why not. [They are saying that price should be
proportional to weight. However, if weight is the only consideration
when determining the price of food, then retailers would sell rocks or
other heavy objects! Other factors such as taste, nutritional content,
and freshness contribute to the price that consumers are willing to pay
for bagels and other foods.]
- The first step of the calculation uses the ratio 33:25. What
does this ratio represent? What other ratio could be used here? [This
ratio represents the lower end of the range of bagel weights for TBM
and their competitors. Instead, they could have used the ratio 35:28,
which compares the upper ends of each weight range. Or, they could have
used the ratio 34:27.5, which uses the midpoint of each range.]
- National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: NCTM. 2000.
Questions for Students
Refer to the Instructional Plan.
- Did the situation in this lesson result in a high level of student
enthusiasm? If not, what modifications could be made to get students
more excited? Is there a better example than a bagel shop that could be
- How did you challenge those students who quickly understood
what the chart was trying to convey? How did you keep them engaged
throughout the class?
- How did you help those students who were having trouble deciphering the algebra?
- How did the students demonstrate understanding of the
materials presented? How do you know that all students understood the
- Explain a situation that uses proportional reasoning
NCTM Standards and Expectations
- Develop an initial conceptual understanding of different uses of variables.
- Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations.
- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
Common Core State Standards – Mathematics
Grade 6, Ratio & Proportion
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''