How Many Chip Bags Will There Be?

 
Frito-Lay produces packages of chips that contain a variety of flavors like the package below.  The circles on the bags’ lower right corner normally displays the quantity of each type of chips.  With 20 bags in the package and four flavors, it would be natural to assume that there would be 5 bags of each flavor.

 

However, that would be wrong.  This is the actual quantity of each type of chips.

 

• How many of each bag of chips will there be?

 

These questions may be useful in helping students down the problem solving path:

• How many of each bag of chips would you choose if the chips were all for you?
• How would the amount of each bag of chips change if you had to take your classmates’ preferences into consideration?
• What assumptions are you or Frito-Lay making about chip eaters’ preferences?

 

This lesson gives students an opportunity to think about how a company like Frito-Lay determines the quantity of chips they will include in each package and compare that to their own class’ or school’s preferences.  The idea would be to present students with one of the chip packages (with numbers blanked out) and find a strategy for determining how many of each type of chip they would need.  Strategies may include:

• Divide the total number of bags by the number of flavors in the package.  This may lead to a discussion as to how to handle a remainder (i.e. which flavor gets an extra bag or can you split the remainder across the flavors).
• Let people choose their own combination of flavors each time.  This may seem like a good idea in theory but is not practical for a company that mass produces their products.
• Survey their classmates and use the results to distribute the bags according to their classmates’ preferences.

 

My hope is that after students have a conversation about each strategies’ pros and cons, they will conclude that surveying their classmates will enable the fairest results (and conveniently incorporates CCSS 7.SP.1 and CCSS 7.SP.2).  Here is an example of how this might go in class once they know they need to take a survey.  Start by showing them a picture of the chip package with the quantities blanked out:

 

Next, have students vote for their favorite type of chips and collect the results.  Let’s say that in a class of 36 students you get:

• 12 votes for Cheetos
• 6 votes for Fritos
• 7 votes for Doritos (Cool Ranch)
• 3 votes for Lay’s
• 8 votes for Doritos (Nacho Cheese)

 

Accordingly, the favorites breakdown as:

• 33.3% (12/36) for Cheetos
• 16.7% (6/36) for Fritos
• 19.4% (7/36) for Doritos (Cool Ranch)
• 8.3% (3/36) for Lay’s
• 22.2% (8/36) for Doritos (Nacho Cheese)

 

With 54 bags in a package, a distribution of chips that matches the class’ preferences (based on the voting percentages) would be:

• 18 bags of Cheetos
• 9 bags of Fritos
• 10.5 bags of Doritos (Cool Ranch)
• 4.5 bags of Lay’s
• 12 bags of Doritos (Nacho Cheese)

 

It is important to note that the breakdown above includes a half bag of Doritos (Cool Ranch) and a half bag of Lay’s.  It is worth having a conversation about there being an extra bag and how to decide what happens to it.  The reality is that there would not likely be half bags and a decision would have to be made about which flavor would get an extra bag.

Once students have reached their conclusion for how many of each type of bag there should be, show them the actual number of bags in the package:

 

In this example, all of the flavors had a different quantity of bags.  It is important that kids have time to reflect on why the answers are different.  Some may think that they have miscalculated or had a mathematical error.  Make sure to facilitate the conversation towards realizing that their classroom may not be a representative sample of the preferences of people buying the chips.

 

Here are two extensions to consider:

• As it is most likely the case that a class’ preference will not completely match up with the package’s distribution, consider comparing each class’ results against the combined results of all classes and/or the whole school to determine if their “sample is representative of that population” as stated in CCSS 7.SP.1.
• Students might feel that voting only for their favorite flavor does not provide a complete picture.  For example, a flavor may be the favorite of few but the second choice of many and should therefore have a larger share of the bags.

 

• Classic Mix – 20 bags

 

• Flavor Mix – 20 bags

 

• Variety Mix – 30 bags

 

• Flavor Mix – 32 bags

 

• Classic Mix – 32 bags

 

• Classic Mix – 54 bags

• CCSS 6.RP.3 – Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
• CCSS 6.RP.3c – Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
• CCSS 7.RP.2 – Recognize and represent proportional relationships between quantities.
• CCSS 7.RP.3 – 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.
• CCSS 7.SP.1 – Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
• CCSS 7.SP.2 – Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

 

• Frito-Lay