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# Section 3.8 Reasoning Strategies

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Section 3.8: Reasoning Strategies. Breaking Word Problems Down into Smaller Problems. October 6 Lecture Notes.

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### Section 3.8 Reasoning Strategies

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