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 algebraic solution.
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.
Bagel Comparison Chart
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 conversation.
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, there are questions located in the Questions tab.
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: NCTM. 2000.
Questions for Students
1. 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.]
2. What value does the variable x represent?
[The cost of a competitor’s bagel, assuming that price is proportional to weight.]
3. 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.]
4. 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.]
5. 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.]
- 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 used?
- 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 end‑of‑class discussion?
- 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.''
Grade 7, Ratio & Proportion
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.