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# Classroom Case Studies, Grades 3-5 Part C: Patterns That Illustrate Algebraic Thinking (25 minutes)

## Session 10: 3-5, Part C

Many students find that a context or situation helps them think about algebraic ideas. In the next problem, we’ll look at different representations of a situation that lead to a solution. Note 6

In this section we’ll analyze problems at the 3-5 grade level for their algebraic content. For each problem, find a mathematical solution, then answer questions (a)-(d) listed below. If it is helpful, print this page and refer to the questions as you work on each problem.

 a. How would you solve the problem? b. What is the algebraic content in the problem? c. How do you think your students might solve the problem? What different representations might they use? d. What question or questions might you ask to get at “doing and undoing”?

Here’s the problem:

A group of 8 adults and 2 children needs to cross a river. They have a small boat that can hold either 1 adult, 1 child, or 2 children.

### Problems

Problem C1

How many one-way trips does it take for the entire group of 8 adults and 2 children to cross the river? Tell how you found your answer.

Problem C2

How many trips in all for 6 adults and 2 children?

Problem C3

How many trips for 15 adults and 2 children?

Problem C4

How many trips for 23 adults and 2 children?

Problem C5

How many trips for 100 adults and 2 children?

Problem C6

Tell how you would find the number of one-way trips needed for any number of adults and 2 children to cross the river.

### Notes

Note 6

Groups: Work on the crossing-the-river problem in small groups. Use concrete representations of the passengers, since “performing” the trips gives some insight into the solution of the problem.

Read “Patterns As Tools for Algebraic Reasoning,” by Kristen Herbert and Rebecca Brown, in Teaching Children Mathematics (February 1997), focusing on different student approaches to solving the problem.

### Solutions

Problem C1

The answer is 33 trips: 4 trips to bring over each adult plus 1 more trip for the 2 children.

Possible responses for questions (a)-(d), which apply to all the problems in part C, are as follows:

Problem C2

The answer is 25 trips: 4 trips to bring over each adult plus 1 more trip for the 2 children.

Problem C3

The answer is 61 trips: 4 trips to bring over each adult plus 1 more trip for the 2 children.

Problem C4

The answer is 93 trips: 4 trips to bring over each adult plus 1 more trip for the 2 children.

Problem C5

The answer is 401 trips: 4 trips to bring over each adult plus 1 more trip for the 2 children.