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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 grade 6-8 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.
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.
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.
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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:
a. | Answers will vary. Most people draw a diagram showing the number of trips to get 1 or 2 adults across the river and then generalize to 8 adults. |
b. | Answers will vary. The problem requires students to find a way to represent the problem, look for a pattern, and generalize. The problem can be thought of as a recursive pattern. |
c. | Answers will vary. Most students draw a diagram with arrows showing people crossing the river in each direction. This arrow representation allows students to “see” the number of trips required for each adult. |
d. | Answers will vary. For example: What sequence of events must happen to get 1 adult across the river? Where are the 2 children at the end of this sequence? How many trips does the sequence require? What must happen so that 1 adult and 2 children are across the river? How does this change if there are 2 adults? Think of “undoing” your sequence to find out the number of adults with 2 children that required 13 trips to cross the river. |
Problem C2
The answer is 25 trips: 4 trips to bring over each adult, plus 1 more trip for the 2 children.
a. | Answers will vary. Most people draw a diagram showing the number of trips to get 1 or 2 adults across the river and then generalize to 8 adults. |
b. | Answers will vary. The problem requires students to find a way to represent the problem, look for a pattern, and generalize. The problem can be thought of as a recursive pattern. |
c. | Answers will vary. Most students draw a diagram with arrows showing people crossing the river in each direction. This arrow representation allows students to “see” the number of trips required for each adult. |
d. | Answers will vary. For example: What sequence of events must happen to get 1 adult across the river? Where are the 2 children at the end of this sequence? How many trips does the sequence require? What must happen so that 1 adult and 2 children are across the river? How does this change if there are 2 adults? Think of “undoing” your sequence to find out the number of adults with 2 children that required 13 trips to cross the river. |
Problem C3
The answer is 61 trips: 4 trips to bring over each adult, plus 1 more trip for the 2 children.
a. | Answers will vary. Most people draw a diagram showing the number of trips to get 1 or 2 adults across the river and then generalize to 8 adults. |
b. | Answers will vary. The problem requires students to find a way to represent the problem, look for a pattern, and generalize. The problem can be thought of as a recursive pattern. |
c. | Answers will vary. Most students draw a diagram with arrows showing people crossing the river in each direction. This arrow representation allows students to “see” the number of trips required for each adult. |
d. | Answers will vary. For example: What sequence of events must happen to get 1 adult across the river? Where are the 2 children at the end of this sequence? How many trips does the sequence require? What must happen so that 1 adult and 2 children are across the river? How does this change if there are 2 adults? Think of “undoing” your sequence to find out the number of adults with 2 children that required 13 trips to cross the river. |
Problem C4
The answer is 93 trips: 4 trips to bring over each adult, plus 1 more trip for the 2 children.
a. | Answers will vary. Most people draw a diagram showing the number of trips to get 1 or 2 adults across the river and then generalize to 8 adults. |
b. | Answers will vary. The problem requires students to find a way to represent the problem, look for a pattern, and generalize. The problem can be thought of as a recursive pattern. |
c. | Answers will vary. Most students draw a diagram with arrows showing people crossing the river in each direction. This arrow representation allows students to “see” the number of trips required for each adult. |
d. | Answers will vary. For example: What sequence of events must happen to get 1 adult across the river? Where are the 2 children at the end of this sequence? How many trips does the sequence require? What must happen so that 1 adult and 2 children are across the river? How does this change if there are 2 adults? Think of “undoing” your sequence to find out the number of adults with 2 children that required 13 trips to cross the river. |
Problem C5
The answer is 401 trips: 4 trips to bring over each adult, plus 1 more trip for the 2 children.
a. | Answers will vary. Most people draw a diagram showing the number of trips to get 1 or 2 adults across the river and then generalize to 8 adults. |
b. | Answers will vary. The problem requires students to find a way to represent the problem, look for a pattern, and generalize. The problem can be thought of as a recursive pattern. |
c. | Answers will vary. Most students draw a diagram with arrows showing people crossing the river in each direction. This arrow representation allows students to “see” the number of trips required for each adult. |
d. | Answers will vary. For example: What sequence of events must happen to get 1 adult across the river? Where are the 2 children at the end of this sequence? How many trips does the sequence require? What must happen so that 1 adult and 2 children are across the river? How does this change if there are 2 adults? Think of “undoing” your sequence to find out the number of adults with 2 children that required 13 trips to cross the river. |
Problem C6
The answer is (4N + 1) trips: 4 trips to bring over each adult, plus 1 more trip for the 2 children.
a. | Answers will vary. Most people draw a diagram showing the number of trips to get 1 or 2 adults across the river and then generalize to 8 adults. |
b. | Answers will vary. The problem requires students to find a way to represent the problem, look for a pattern, and generalize. The problem can be thought of as a recursive pattern. |
c. | Answers will vary. Most students draw a diagram with arrows showing people crossing the river in each direction. This arrow representation allows students to “see” the number of trips required for each adult. |
d. | Answers will vary. For example: What sequence of events must happen to get 1 adult across the river? Where are the 2 children at the end of this sequence? How many trips does the sequence require? What must happen so that 1 adult and 2 children are across the river? How does this change if there are 2 adults? Think of “undoing” your sequence to find out the number of adults with 2 children that required 13 trips to cross the river. |