Present orally to students The other day I was standing in line at the grocery store when I noticed a display of little cardboard school busses filled with miniature M&M's. I bought them thinking I could create a fun math problem with them and sure enough, I did! You will notice that these mini-buses have 10 boxes/seats. My challenge to you is to mathematically determine how many of each color M&M® is riding on that mini-bus without counting all of the M&M's®. You may count the contents of up to 5 of the boxes/seats. At least 5 of the boxes/seats must remain closed until the end of the activity. At the end of the activity you may open all of the boxes/seats and evaluate the accuracy of your determination.
This task was presented to a class of low-performing fourth grade students after a unit on estimation. Students had experiences with sampling, graphing and different estimation strategies prior to being presented with this activity. Students had also been reading The Math Curse by Jon Scieszka and Lane Smith which is a terrific book to read to students to get them thinking about how everything in life can be thought of as a math problem. This task allowed the classroom teacher to see which concepts of estimation the students have internalized and can apply to an appropriate mathematical situation. The mastery of concepts such as determining a sample size and applying a ratio were necessary in successfully solving this task. The student products allowed us to determine the degree to which students have mastered these skills. We also required that students determine the reasonableness of their results (a goal stated in the NCTM Standards). The standards also encourage that students identify a range for "good estimates," and that students should always check their solutions against their original estimates so that students can use the feedback to refine their estimating skills. Some students emptied out five boxes in a pile, sorted by color, counted them and then doubled these numbers to make their determination. Other students opened one or two boxes, sorted them by color, counted them and then multiplied by five or ten. Neither group tended to discuss sample size in making their decision to open the number of boxes that they did. This was disappointing. We had a follow-up discussion about this with students after the activity and realized that although they had the concept of "the larger the sample, the more accurate your results are likely to be," students had difficulty explaining this idea. Students who have had a unit on probability before undertaking this activity may take a different approach by determining the typical number of each color candy per box and then using those results to predict the number of each color in ten boxes. This was how I approached this activity when I solved it myself and my results tended to be more accurate. Three 45 - 60 minute periods This includes time for introducing the task and discussing expectations, as well as allowing for time for students to assess their own performance. Estimation is a mathematical topic that cuts across many fields of knowledge. Park Rangers often estimate the numbers of animals in their forests based on a sample. Samples of blood are drawn from patients to make a determination of an entire patient's blood. It would be a terrific idea to keep a running list in your classroom of real-world applications of estimation and sampling in the real world. Although this task was presented orally to students, supplemental print information was provided as well. On charts in the classroom were listed the following: Task: Mathematically determine the number of each color of M&M's® there are in the bus without counting all of the M&M's®. Guidelines:
- You may count the contents of up to 5 boxes.
- At least 5 of the boxes must remain closed until the end of the activity.
- When you have gotten the "OK", open all the boxes and evaluate your determination.
Must Haves:
- Explain/show what you did and why;
- Mathematical representation (chart, table, graph, etc.);
- Good use of mathematical language;
- A mathematical conclusion; and
- An "I noticed..." statement.
- Evaluate the reasonableness of your determination before looking at all 10 open boxes.
- Compare your determination to the actual number of M&M's®. How close were you? Why do you think so?
Students also brainstormed a list of mathematical terminology they might use when communicating their solution. This was kept up so students could easily refer to it when writing up their solutions. Students also brainstormed ideas for "I noticed..." statements such as finding the range of each color of M&M's®, the mode, the median, etc. In order to make sure students did not eat their data before the task was completed and to avoid other potential disruptions, we had students sign the following contract before they were given their M&M's®: Candy Eating Contract: I understand that I may not eat any M&M's® until I have been given permission to do so. I will not ask about when I can eat my candy or I am breaking this contract and will need to write a new contract with the teacher. If I follow the rules above, I will get 5 mini-boxes of M&M's® to eat. This contract worked very well, with only one student needing to make a new contract. Mini M&M® Buses or another candy/object that has a sorting attribute and comes as a whole broken into smaller parts.** **Halloween candy bags of M&M's® could work, books of Life Savers®, etc. Our students worked in partners, eliminating the need to buy a bus of candy for every student. Solutions will vary, but the reasoning behind the solution should be valued more than an actual numerical answer. When I did this activity, these were the results I obtained: | Color | Box 1 | Box 2 | Box 3 | Box 4 | Box5 | | Green | 6 | 5 | 5 | 5 | 3 | | Brown | 9 | 7 | 11 | 5 | 7 | | Blue | 10 | 11 | 9 | 13 | 11 | | Red/Pink | 15 | 14 | 13 | 13 | 17 | | Orange | 6 | 8 | 5 | 10 | 9 | | Yellow | 6 | 7 | 8 | 6 | 6 |
I then took the average of each color in each box and multiplied by 10 to make my estimate: | Average in each box | Candy Color Times 10 | Equals Prediction | Actual Number
in 10 Boxes | Difference between
estimate and actual | | 4.8 | Green x 10 | 48 | 47 | 1 | | 7 | Brown x 10 | 70 | 66 | 4 | | 12 | Blue x 10 | 120 | 115 | 5 | | 14.4 | Red x 10 | 144 | 154 | -10 | | 7.6 | Orange x 10 | 76 | 77 | -1 | | 6.6 | Yellow x 10 | 66 | 62 | 4 |
A Novice will tend to have little or no reasoning behind his/her approach. The Novice will make a guess with no mathematical logic or consistency. The Novice will use little or no mathematical language and will not clearly document his/her work.
An Apprentice will have some mathematical reasoning behind his/her approach, but the reader will question whether or not the student's final conclusion has a mathematical basis. The Apprentice will confuse the "answer" with being the total number of candies and not the number of each color candy. The Apprentice may use some limited mathematical language and representation to communicate his/her solution.
A Practitioner will obtain a solution that has a mathematical basis. The Practitioner will document his/her work so that one can clearly tell what was done to solve the task and why. A Practitioner will use a variety of mathematical language and representation and will attempt to make some mathematically relevant observation, although it will be rudimentary in nature.
An Expert will obtain a solution that is supported with a mathematical argument. Procedures and reasoning will be explained, all work documented and the student will make a mathematically relevant observation. The Expert will use a variety of mathematical language and representation to clearly communicate his/her solution.
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