I implemented this activity with a class of middle school students who did not have prior experience with mathematical modeling. Before the students started any part of the activity, I went over a list of messages and question that are important at different parts of mathematical modeling. I reminded students of the messages at each stage of the modeling implementation.
I showed students the following video and then they answered three questions.
(1) What clues or evidence can scientists and professional trackers get from footprints?
(2) Can scientific knowledge change over time? Explain
(3) How is mathematics used to give us a better understanding of the natural world?
The problem statement was then gone over with the students.
The Northern Minnesota Bigfoot Society would like your help to make a “HOW TO” TOOLKIT; a step-by-step procedure, they can use to figure out how big people are by looking at their footprints. Your toolkit should work for footprints like the one that is shown on the next page, but it also should work for other footprints.
Materials were made available for the groups to use including rulers, string, scissors, graph paper, graphing calculators, and laptops.
Before beginning the activity, I showed students part of a video on how to make effective group decisions to support students’ social skills when working in a group.
The cooperative learning strategy of numbered heads was also used. In this strategy the teacher lets students know that one member of the group will be randomly picked to present. All students are more likely to be engaged and prepared with this strategy.
I will give a few example group's solution development below. For each group the students in the group are designated as S1- student one, S2-student two, S3-student 3, or S4-student 4.
This group came up with a general notion of how to approach the problem but took awhile to figure out what specific steps to take. They initially had some discussion around looking at the Bigfoot footprints and then decided to measure the length of one of the footprints. They got several different answers for this from 15 inches to 15 and one fourth to 15 and one half. The students tried to use two of the same rulers to help get a more accurate measurement. They also discussed if one of the footprints that they were given was different from the other one, but decided they were equivalent. They ended up using 15 and one-fourth inches as their measurement though the foot length was 15 and a half inches.
During this discussion student one started to put forth an idea of what they could do. “What if we measure our own foot size and then we measure how tall we are and then see if it is the same thing. Then if it is the same thing” (student did not finish the thought). At this point none of the other students responded and continued to discuss measuring the Bigfoot footprints. After a few minutes student two tried to put forth an idea. “What if measure our height and if it is the same amount wouldn’t it be. Like so, say your foot is 10 inches or no.” After this student three went back to the problem statement and read it aloud. “We need to make a step-by-step procedure to figure out how tall people are by looking at their footprints.”
Student one then directed the group on what to do though was met with some resistance on how this would help.
S1: “We should measure our feet and then measure our height.”
Two students then responded:
S4: “How are we supposed to use the information to figure out how tall Bigfoot is?”
S3: “I have no clue.”
Student one went ahead with the measurements, but it was clear not everyone still understood what they would do with these measurements.
S4: “I still don’t get how we are supposed to use feet to measure height.”
S1: “You are basically just going to say what it is based on the footprint.
S3: “He is 15 and ¼ feet?”
Student three thought that the foot length in inches would translate then to the height in feet.
Student one then got the foot length and height of student 4 but had to think for a few minutes on what to do with this information.
S1: “What is the equation again?”
A few minutes passed
S1: “So basically I want to get a fraction. I remember how to get the equation. You guys know how to do this right?” (sets up a proportion)
S4: “So cross multiply”
The final solution that this group described is shown below. When they presented, they had not yet figured out the height of Bigfoot from their proportion but added later that Bigfoot would be around 15 feet tall. Though calculators were available, this group solved the proportion by hand and incorrectly multiplied 59 inches by 15.25 inches to get 137.25. They knew a correct method for solving the proportion, but did the math incorrectly.
Group two went right to the Internet to help them with their solution and were able to quickly identify a method that they were able to check was appropriate. They found the equation x divided by 0.15. Initially, they had measured the Bigfoot footprint at 15 inches but when I asked them to check the measurement again they came up with 15 and a half inches. After the students got the equation, student one questioned, “What if the maximum is the width?” Student two and three quickly responded that the website says to use length and not width.
I questioned the students to see why they would divide by 0.15 in the equation and the group provided a few responses. They tried their own foot lengths in the equation to the see if the height was accurate and also referenced information given to the students before working on the problem that Bigfoot has been estimated to be between 6 and 10 feet.
S2: “Because it says so on this website. And it seemed legit and it actually worked out.
S3: “The paper says it is between 6 and 10 feet.”
Group 2’s final solution is shown below.
Group 3 used the Internet to assist in their solution development and determined two possible heights for Bigfoot that both made use of ratios. The group started by having a discussion of their own heights and doing general Internet searches on Bigfoot. They then had a discussion on the length of the Bigfoot footprint that was provided to them because they got different measurement lengths. They ended up deciding on 15.5 inches.
On a website they found an average height to foot ratio and student one explained how they could use this. If the height to foot ratio is 6.6 to 1 then the foot size would be 1 and the height would be 6.6 so write 6.6 to 1 ratio. So then you would multiply 6.6 times 15.5…So that is 102.3 and then this is in inches. You divide this by 12. You get 8.525. Bigfoot is 8 feet tall.”
Student two then questioned, “How tall is the tallest man in the world?” This led student three and four to investigate this. They were able to determine that the tallest man ever was Robert Pershing Wadlow at 8 feet and 11 inches. The group decided to use the height to foot ratio for this person to help determine the height of Bigfoot as well.
S1: 8 times 12 is 96 and 96 plus 11 is 107. So it would be 107 to 17.5. The tallest man, wait nevermind let me simplify this…So it is about 6.1, so the actual. Listen to me. We are going to erase.
S3: You have to keep all your evidence.
Student three convinced the group members to keep all of their work and they calculated what the height of Bigfoot would be based on the height to foot ratio of Robert Wadlow. The Figure has this group’s final solution. After all the groups had presented, this group added in the table that had their own heights and foot lengths as well.
This group struggled for awhile to figure out what to do but were able to us the Internet to find an idea. Initially, this group measured the length of the Bigfoot footprint and got 15.5 inches. Student three then thought about using an additive method, “15.5 plus 50 that is about 66. That is not true.” There was no explanation for selecting 50 to add and the answer did not seem to fit for the actual height of Bigfoot. The group then measured the width of the foot, but was unsure what to do.
As the group tried to think of ideas, student one kept pushing for an idea but had trouble knowing exactly how to explain it. “We could think about our foot size and our height. I have tiny feet so I am a tiny person.” Later she stated, “How about we measure our height and then measure our foot length and then if we add this foot length and multiply our foot length. Student three responded that this would not work because “A lot of people could be his height and have big shoes and a lot of people could be his height and have small shoes.” Student one responded that it does work “because usually tall people have big feet. I am guessing we can figure it out by multiplying.”
The group then proceeded to look on the Internet for several minutes to see what they could find but were unsure what to do until student 4 was able to find information on a proportion for foot length and height. “Divide the length of each person’s height by their foot so we are going to set up a proportion.”
Student four then proceeded to explain what she came up with to the other three group members.
S4: A proportion is a over w equals p over 100 and since it is 15% of his foot size, what would a and w be though? I don’t know his height, wait no just kidding.
S1: Why would you put 15 and not 15.5?
S4: Because it is 15%
S1: Okay I get it.
S4: So he is 8 foot 6. Do you understand how I got it? You know what proportions are right?
S4: So proportions are basically ways to find percentages. A is something. Basically it is a over w equals p over 100. The p equals percent. These two are just different numbers that you can put down.
S1: So it doesn’t matter what numbers?
S4: It does. If you give them the numbers it does.
S1: Is it height, width?
S4: No. The percentage of your
S1: foot to your entire body
S4: It is 15%. 15 over 100 right? That equals
S3: 0.15 right?
S4: It equals that. With this one, if you don’t have two numbers and you have one, put that number on a and leave w. So you would multiply that it would be 1,550. Then you divide by 15.
While student four could have given a better explanation on how to decide which number is a or w, this group was able to use the proportion correctly and come up with an estimate for Bigfoot’s height. The figure shows this group’s final solution.
In total, students were able to use their other group members and the Internet to develop solutions to the Bigfoot Model-Eliciting Activity. All groups understood the problem context and were able to come up with a method that worked to estimate Bigfoot’s height.
Stohlmann, M. (2017). Middle school students first experience with mathematical modeling. International Journal for Research in Mathematics Education, 7(1), 56-71.
Christian, author, and professor of mathematics education.