Illustrative Mathematics Modeling
Friends Around a Table
Composed by Don, Maile, and Nia
CCSS addressed:
7.SP.C.8.a: Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound events occurs.
7.SP.C.8.b: Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams For an event described in everyday language, identify the outcomes in the sample space which compose the event.
Materials Needed:
• 4 miniature dolls per “table”
• 1 “table” (this can be a piece of cardboard, a book, or another square object)
• Paper and pencils
Lesson:
1. Hook:
Penguin, Tiger, Bear, and Frog are seated at random around a square table, one person to each side. What is the theoretical probability that Penguin and Tiger are seated opposite from each other?
2. To do:
Students need to draw pictures or use manipulatives of the problem. They should list all the possibilities and record that number as the total. Then they should list all the outcomes where Angie and Carlos are seated opposite each other. They should list all the combinations were this happens and record that number as the desired outcomes. This fraction is the theoretical probability. Students should make a chart of their data. Students can work in small groups to solve this problem.
3. Discussion Questions:
• What are some ways we can figure out how to answer this question?
• How can we set up what we know in a way to help us solve this?
• What information do we know? What do we need to know?
• How many ways can these 4 friends sit around the table (with no duplicates of course)?
• What if Penguin decided that he did not want to give up his seat, how would this change the amount of possible sitting arrangements?
4. Wrapping up:
Students will discuss and share their totals (how many total sitting arrangements there are). Seeing if there are any “outliers” or numbers that just don’t seem to fit with the rest of the classes data (1 number just does fit with the rest). If there is an outlier then that number is thrown out of the “group” (that data is not used in the resulting discussion). Next, ask the students how many times they found that Tiger and Penguin sat across from each other. Students should look to their lists and come up with a single digit answer. Ask the students to use this number and create a fraction that incorporates their total number of sitting arrangements as a whole. Finally, ask the students, “what do you notice about the fraction you created and why do you think this fraction makes sense?” Students should have a variety of answers and it should lead to a valuable discussion about probability, and combinations.
Possible problems:
• Students may not list all possible combinations of seating arrangements.
• Students may accidentally list a sitting combination multiple times (having a double) and thus their resulting answer will be inaccurate.
• Students may not be responsible with the dolls, and thus they may not have the opportunity to work with them (these dolls are manipulatives and thus should be tried like manipulatives, with a purpose, and not as a toy).
Extensions:
• More “students” or dolls can be added to increase the difficult, “now there are 6 friends wanting to sit at the table, what are the new possible sitting arrangements?”
• Adding other conditions such as Frog needs to be seated to Tiger’s right.
• Giving Penguin and Bear fixed seats will decrease the amount of sitting arrangements, while still illustrating the same concepts/learning targets.
Concepts:
This activity is designed to show how modeling and theoretical probability can be used to solve compound event problems. Students can generate a list of all desired outcomes and divide that by the total possible outcomes. This lesson can also show how permutations can be used to find all possible outcomes.
Students will need to be able list and understand sample space, desired outcomes, and fractions.
Aspects of mathematical modeling:
1. Making assumptions and approximations to simplify a complicated situation.
3. Identify important quantities and organize their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas.
4. Analyze those relationships mathematically to draw conclusions.
5. Interpret their mathematical results in the context of the situation and reflect on whether the results make sense.
Math Practices used:
1. Make sense of problems and persevere in solving them.
2. Model with mathematics.
3. Use appropriate tools strategically.
4.Reason abstractly and quantitatively.
Resources:
• https://www.illustrativemathematics.org/illustrations/885