Pathfinder Game Contestants stand in the middle of a five by five grid of numbers and are given the first number in the price of a car, which they stand on. They then have four choices for the 2nd number in the car, three choices for the 3rd number, two or three choices for the 4th number depending on the show, and two choices for the 5th number. Contestants also have the opportunity to win three more guesses by knowing the price of three smaller prizes. Students are asked if they think it is easier to win a car in this game or the temptation game (see the blog post below for the temptation game). If students need more guidance the following questions could be used to help students determine which game is the better chance to win a car.
Punch a Bunch Game In this game, contestants can earn up to four chances to punch on a fiftyhole punchboard arranged in five rows of ten. Contestants earn chances by being shown a price for a prize and correctly stating if the actual price is higher or lower than the price shown. Each hole on the punchboard has a value from $25000 down to $100 and there is a certain amount of each value. After seeing the amount in one punched hole, contestants can keep the money or go on to see what value is in another punched hole. Students answer the following questions on probability and expected value to determine the best strategy for playing this game.
Reference
Stohlmann, M. (2018). The math is right! The Australian Mathematics Teacher, 74(3), 914.
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The Price is Right bidder’s row and games provide many opportunities for mathematical connections. I will describe activities integrated with the mathematics of probability and statistics. Bidder's row First, students are shown an item and guess the price. Like the television show, the goal is to be the closest without going over. Students record their guesses for ten items that are shown to them. The students are then told the actual price and record this. Students use the guessed prices as the independent variable (xaxis) and the actual price as the dependent variable (yaxis) to produce a scatterplot. In doing this activity with middle years students, the following items were used: television, tablets, trampoline, men’s watch, women’s watch, moped, two video game consoles, basketball hoop, and a ping pong table. Any items that could be relevant to a teacher’s student population could be used and prices can be readily found on the internet. Next, students are given a piece of spaghetti and asked to show the line of a person that was a perfect guesser; where every guessed price matched the actual price. Spaghetti is used because it is thin, helps with visualization, and allows students to easily make adjustments to the line of best fit. Students then answer the following questions:
For question 1, the line y = x is the line that students’ spaghetti should be showing. For question 2, it may be counterintuitive to students that points that fall below the spaghetti line are overbids and points above the spaghetti line are underbids. Students can use a few of their points that they know were underbids or overbids to check this as they determine where the points appear. The yintercept for the line of best fit for the students’ data would not have a sensible interpretation in this context as no contestant would bid zero dollars. Temptation Game Reference
Stohlmann, M. (2018). The math is right! The Australian Mathematics Teacher, 74(3), 914. 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. Problem Statement The Northern Minnesota Bigfoot Society would like your help to make a “HOW TO” TOOLKIT; a stepbystep 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, S2student two, S3student 3, or S4student 4. Group 1 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 onefourth 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 stepbystep 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 2 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 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. Group 4 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? S2: nope 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 ModelEliciting Activity. All groups understood the problem context and were able to come up with a method that worked to estimate Bigfoot’s height.
Reference Stohlmann, M. (2017). Middle school students first experience with mathematical modeling. International Journal for Research in Mathematics Education, 7(1), 5671. This task is connected to an attempt to set a world record for the farthest distance traveled rolling across swiss (exercise balls). In order to break the record, Tyler, one of the Dude Perfect members, has to use forward motion only to move his body across exercise balls that are lined up in a row. If he falls off the balls or his hands or feet touch the ground the attempt is not successful. The surfing across the balls is done with Tyler’s body being horizontal lying on the balls so it is similar to lying down on a surfboard. An actual surfboard is not used to move across the balls. My learning goal for this lesson was for students to be able to solve a reallife problem using numerical and algebraic expressions and equations. This activity was used as introduction for this goal. I first had students watch the Youtube video entitled “World Record Exercise Ball Surfing” up until the 1:39 mark of the video (Dude Perfect, 2018). This helps to motivate students in solving the task and also lets them become familiar with the task context. The video is stopped before Tyler’s attempt to break the World Record is shown. If students want more detail on how Tyler will move across the balls a couple seconds of the video starting at the 2 minute and 1 second mark could be shown. I then posed the question for the problem. How many swiss (exercise) balls does Tyler need in order to break the Guinness World Record of 255 feet and 5 inches? Students next provided a number for three prompts. Write down a guess that you think is correct. Write down a number that you know is too low. Write down a number that you know is too high. The guesses ranged from 20 to 128 balls, with most guesses around 50. The majority of students felt that 20 balls or below was too low and that more than 100 balls was too high. Students then wrote down what information they would need to know in order to solve this problem. The most common information that students wanted to know was the distance of the space between the balls and the size or the diameter of the balls. I provided students with the diameter of the balls, 26 inches, and also a picture from the video. The picture shows that Tyler has already started the attempt. The first ball initially started at the strip of black tape. Students worked in groups of three to four. Students individually thought about how they would solve the problem, then discussed in their group a strategy they felt would be best and developed a solution. Before implementing the task with students I thought about possible student strategies as well as questions that I would ask students. For each of the possible student strategies, I thought of assessing and anticipating questions that could be asked for each strategy. Table 1 provides more details on these general types of questions. Assessing questions enable teachers to elicit student thinking and advancing questions position teachers to use student thinking to move students toward the goals of the lesson. Further, the purpose of advancing questions is to support students in moving forward in solving the task beyond their current thinking or to explore underlying mathematics more deeply. There are different solution strategies that I anticipated that students would use. This was possible because the question did not state or imply any solution strategy. In order to answer the question students must justify why their approach makes sense. For each strategy, I have listed assessing and advancing questions that are based on student thinking and how to move this forward. If students just guessed a number of balls the questions focus students on what information then can use to solve the problem mathematically as well as looking at cases with a small number of balls to develop ideas for how to work towards the number of balls needed for 255 feet and 5 inches. In the numerical expression strategy students are not considering that there is one less gap than the number of balls being used. The first advancing questions help students to see this. Students are also asked if they can come up with an equation and what would happen if the World Record distance increased. The questions for the algebraic equation ensure that students know what each part of the equation represents in the real world context as well as considering how different lengths would affect the solution. In the actual video the gaps between the balls are not all exactly the same so the first advancing question has students think about why the actual number of balls might not match the answer. In this strategy students also estimated the length of the gap which could affect the accuracy of the answer. The questions for the proportional reasoning strategy are similar to the algebraic equation questions with some different assessing questions to determine how the length of the gap was calculated.
When I implemented this activity, there were a few errors that students made that I had not planned questions for that could be added to Table 2. Some students used a numerical expression strategy but were working in different units, feet for the World Record distance and inches for the gap and diameter of the ball. The groups that did this were able to identify their error when finding the number of balls they calculated was not reasonable. If a group did not notice this they could be asked if working in different units affects their answer. One group decided to work in feet, and wrote 255.5 feet as the current World Record distance. I asked this group what half a foot was in inches and they were able to notice the error. Most of the groups used a numerical expression strategy and tended to use an estimation of 36 inches for the length of the gap between the balls. These groups felt that the gap looked like the size of a yard stick. For the groups that used a numerical expression strategy, I used my advancing questions so they would consider that the number of gaps should be one less than the number of balls. Students were able to understand this part, but had difficulty in translating this to how they would adjust their numerical expression. The groups were unsure if they should add or subtract the length of the gap to the current World Record distance. The rationale to subtract the length was due to the fact that there was an extra gap that needed to be removed from consideration. To do it correctly, the extra gap gets added because the last ball is at the World Record length and the last gap would then go 36 inches beyond this length. In order to look at this more I decided to build on the numerical expression strategy used by many groups to discuss how this strategy could be transitioned to be shown with an algebraic equation. There were not any groups that developed an algebraic equation. I used 36 inches for the gap since many groups had done this. Students agreed that we were trying to find the number of balls so we defined x as the number of balls. One group’s strategy that I had not anticipated was to take the World Record distance and divide this by 26 inches to find the number of balls without any gaps so I started with this equation, 26x = 3065. I then asked how we could incorporate the estimated 36 inch gap knowing that there is one less gap than the number of balls. I followed up with asking if there are 50 balls, how many gaps would there be? If there were 42 balls how many gaps? If there were 40 balls, how many gaps would there be? If there were x balls how many gaps? Students were able to generalize that there would be x1 gaps. I then connected that students had added the diameter of the ball to the gap estimate in their strategy so the equation would be 26x + 36 (x 1) = 3065. In solving the equation, it was seen that the length of the gap is added to the current World Record length. Next, we watched the rest of the video to see that Tyler ended up breaking the World Record. To tie the World Record there were 41 balls used so 42 balls broke the record. In total Tyler used 47 balls as he went 290 feet. Since no groups had used a proportional reasoning strategy and this was not directly connected to my learning goal, this strategy was not discussed. If students do not get the correct number of balls, it can be discussed what could affect this including that the gaps were not set up with the same length or that a better estimate of the gaps could be used. References Bill, Victoria, and Margaret S. Smith. “Characteristics of Assessing and Advancing Questions.” University of Pittsburgh: Institute for Learning, 2008. Dude Perfect. 2018. “World Record Exercise Ball Surfing.” https://www.youtube.com/watch?v=i25i6vyjIpg&t=146s “I love these! I want to do this!” “You love watching us struggle!” “Let’s make sure we help each other.” The following student quotes are music to a teacher’s ears and occurred during the implementation of the mathematical escape room that I will describe. Incorporating escape rooms is one way to engage students, encourage productive struggle and teamwork. An escape room is a game in which teams solve multiple puzzles using clues, hints, and strategy in order to figure out how to escape from a locked room. Students must work together in their teams in order to solve challenges. Setting up a mathematical escape room can be a great way for students to apply and practice mathematics they have learned. I will highlight important principles and ideas for teachers who want to develop their own escape room. These include a unifying theme and a brief backstory, challenges that involve students using and connecting mathematical representations (National Council of Teachers of Mathematics, 2014), the inclusion of hints if needed, and a compelling twist or twists. I will also describe insights from implementing the escape room with a classroom of students. Unifying Theme and a Brief Backstory The theme that I selected for my escape room was lines. All of the challenges involved lines or line segments in some way with the mathematics mostly focused on linear equations. The backstory that I created involved a student, Luna, who knew everyone in the room. To introduce the escape room I projected the backstory for students to read (See "Math Activities" tab for all escape room handouts) I went over a few things with students before starting the escape room. I emphasized to students the importance of teamwork and ensuring everyone was being utilized in a team. I tried to put students in teams with other students who they did not know as well. This can be to done to develop classroom community and let students know the importance of helping each other. In actual escape rooms, a team of players are locked in a room and given 60 minutes to try to solve the puzzles in order to determine how to leave the room. I did not lock the door but instead placed the seven locks on tables in a part of the room where students could try combinations to open the locks. It is important to let students know to leave the locks at the table and for only one person in each group to check combinations. This ensures that all groups will have access to the locks because students will not be taking locks back to their group. I designed the escape room so it would be challenging but so that students could finish in under 60 minutes. In actual escape rooms a record of the fastest time to escape the room is kept so a timer could be set when students begin. Use and Connect Mathematical Representations It is important for students to have the opportunity to make connections between different representations. The Lesh Translation Model emphasizes five categories of representations (a) Representation through realistic, realworld, or experienced contexts, (b) Symbolic representation, (c) Language representation, (d) Pictorial representation, and (e) Representation with manipulatives (concrete, handson models). The understanding of concepts depends on the learner’s ability to represent concepts and situations through the five modes of representation, and the ability to translate between and within representations (Lesh and Doerr, 2003). I choose the escape room challenges with these ideas in mind. To begin the escape room, students are given a folder with the information that they will need. The folder contains the following pages: (a) message from Luna, (b) table to record the combinations for the seven locks, (c) the seven challenges labeled with the corresponding lock number, and (d) six pages that contain 96 possible answers, 96 lock combinations, and 96 sets of three characters. Groups need to figure out that when they solve a challenge, they need to refer to the possible answers to then identify the correct combination. I included 96 possible answers so that students would not try to just guess the correct combination. I also included answers choices based on common mistakes that students could make. For example, in determining the slope of x = 1/2 y + 10 students might think that the slope is ½, but should recognize the equation is not in slope intercept form. I included the following answer choice for the challenge to calculate 8 slopes represented in different ways, {6, 3, 2, ½, 1, 3, 4, 5}, which is correct except for the slope of ½. Escape rooms often have a sequential or asynchronous design. My escape room uses both designs. The seven challenges do not have to be solved in any set order, but all must be solved correctly to move on to the next part of the escape room. The seven challenges all involve lines in some fashion. Inclusion of Hints if Needed The escape room is set up for students to be able to check if they correctly solved the challenges and hints can be provided as well if groups are struggling. Students can check if they correctly solve the challenges by trying the corresponding combinations to open the locks. If groups are struggling on a challenge or falling behind other groups, below are some possible hints to provide to groups. Compelling Twist Real life escape rooms often have hidden objects, keys that need to be found, and/or objects placed in the room that end up being integral to escape the room. Once a group has correctly identified all seven lock combinations to open all of the locks, they have to figure out what to do next. In addition to the correct seven lock combinations groups must find out what everyone has in common. Once the groups got to this point they were unsure of what to do next. I reminded them of the question they needed to answer and if they had used all of the information they were given. This led groups to look back over the handouts they were given. In the six pages of answers and combinations there were three characters under each combination that gave the next clue. The three characters placed in order for the correct combinations spelled out the following: L o o k u n d e r t h e k e y t a b l e ! Before students arrived in the classroom, I had taped a key along with a note under the table where the locks were placed. In the clue “key” meant important and also that there was a key under the table to be found. Along with the key a small scroll of paper was found with the following clue: What is the meaning of “life”? The group started to search the room. Before class started I had placed a stack of books in a different part of the room from the locks but in plain sight. One of the books appeared to be a dictionary, but when opened had a locked compartment. This item can be purchased for about ten to fifteen dollars through Amazon. I reminded the group of the phrase on the scroll and after several minutes of searching I asked where do you find the meaning of words? This lead one group to be able to find the dictionary which the key opened. The locked compartment of the dictionary provided the last challenge. On the inside of the dictionary I had placed a note for each group to take one folded up piece of paper. When the piece of paper was unfolded, there were two questions. The first question was what do all of the following numbers have in common? 11, 37, 2, 101, 71, 73, 41, 29 Students had to notice that all of the numbers were prime numbers. Underneath this was the question to finish the escape room. What do we all have in common? Below this question was the series of characters and numbers in the figure below. In order to answer the last question students had to circle all of the prime numbers and use the associated letters above the numbers to spell out the answer. WE ALL LOVE MATH! In summary, the following are important elements in designing a mathematical escape room: a unifying theme and a brief backstory, challenges that involve students using and connecting mathematical representations, the inclusion of hints if needed, and a compelling twist. An escape room is an excellent way to increase engagement, teamwork, and provide students with a memorable experience!

Micah StohlmannChristian, author, and professor of mathematics education. Archives
May 2020
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