Section 3.8 Reasoning Strategies
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The document discusses a reasoning strategy for simplifying problems by breaking them down into smaller sections. It provides examples, such as a snail escaping from a well, determining how many digits to write for page numbering, calculating matches in a tennis tournament, and counting triangles in a figure. Each example illustrates the process of simplifying complex problems to reach a solution more effectively.
Section 3.8 Reasoning Strategies
- 1. Section 3.8: Reasoning Strategy: Simplifying a Problem By: Ms. Dewey-Hoffman
- 2. Breaking Problems Down to Simplify Them Sometimes it helps to break down problems into small sections, and solve each section, one at a time. Sometimes when you solve a problem, it helps to solve other problems that have similar conditions. Breaking down problems is a great way to solve WORD PROBLEMS.
- 3. Try This: Word Problems A snail is trying to escape from a well 10feet deep. The snail can climb 2feet each day, but each night, it slides back 1foot. How many days will the snail take to climb out of the well?
- 4. The Snails Slow Journey…Read What do we know about the snail? How far the snail will the snail travel in one day? 2 feet. How far will the snail slide back at night? 1 foot. What is the total distance the snail will move after one day and one night? 1 foot.
- 5. The Snails Slow Journey…PLAN What is the total distance the snail will move after one day and one night? 1 foot. With this information, we could guess that it would take 10 days to climb the well. 10 feet in 10 days…Right? WRONG!
- 6. The Snails Slow Journey…PLAN 10 feet in 10 days…Right? Think about a simpler 3 foot well. After Day 1: 2 feet. After Night 1: 1 foot. 1 foot total. After Day 2: 2 feet. Moved a total 3 feet. And out of the well.
- 7. The Snails Slow Journey…PLAN 10 feet in 10 days…Right? Think about a simpler 4 foot well. After Day 1: 2 feet. After Night 1: 1 foot. After Day 2: 3 feet. After Night 2: 2 feet. After Day 3: 4 feet. Now the snail is out!
- 8. The Snails Slow Journey…Observations From what we’ve observed… It takes one day less then we thought for the snail to get out of the well.
- 9. The Snails Slow Journey…Solve So, for every foot of the well, it takes one less day for the snail to get out of it. So…10 feet? It’ll take the snail 9 days to climb out of the well.
- 10. Lets Try Another: Pages Of A Book You decide to number the 58 pages in your journal from 1 to 58. How many digits do you have to write? 1) Read…What is a digit? 2) Plan…What is a simpler version of this this problem? 3) Solve…for the more complicated problem.
- 11. Pages Of A Book: Plan You decide to number the 58 pages in your journal from 1 to 58. How many digits do you have to write? Lets simplify the problem. How many digits will I write for a 9 page book? 1, 2, 3, 4, 5, 6, 7, 8, 9 = 9 digits. 9 digits for the first 9 pages.
- 12. Pages Of A Book: Plan You decide to number the 58 pages in your journal from 1 to 58. How many digits do you have to write? 1, 2, 3, 4, 5, 6, 7, 8, 9 = 9 digits. 9 digits for the first 9 pages. But what about the remaining pages?
- 13. Pages Of A Book: Solve 9 digits for the first 9 pages. But what about the remaining pages? There are 2 digits per page for the remaining 58 – 9 pages. Or 49 pages. So 49(2) + 9 = the total number of digits. 107 total digits you have to write out by hand.
- 14. This One: Tennis Tournament. In a tennis tournament, each athlete plays one match against each of the other athletes. There are 12 athletes scheduled to play in the tournament. How many matches will be played? Read : So we have to get 12 people to play each other once.
- 15. Tennis Tournament. Plan : Lets make it simpler! With only 4 players. Player A: A-B, A-C, A-D. Player B: B-C, B-D. Player C: C-D. Player D: has already played everyone. So, for each pattern, player One starts will 11 matches. Player Two starts with 10 matches. Player Three starts with 9 matches. Etcetera.
- 16. Tennis Tournament. Solve : With 12 Players. So, for each pattern, player One starts will 11 matches. Player Two starts with 10 matches. Player Three starts with 9 matches. Etcetera. So, 11 matches + 10 matches + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 66 matches.
- 17. Last One: Triangle What is the total number of triangles in the figure? Read: Figure out how many triangles there are. Plan: Count the triangles. There are 9 small triangles. There is 1 big triangle.
- 18. Triangle: Plan and Solve. There are 9 small triangles. There is 1 big triangle. There are 3 medium triangles made up of 4 small triangles. So: 9 + 1 + 3 = 13 total triangles.
- 19. Assignment #19 Page 161-162: 5-11 odd and 13-20 all.