Section 3.8 Reasoning Strategies

Section 3.8: Reasoning Strategy: Simplifying a Problem By: Ms. Dewey-Hoffman

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.

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?

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.

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!

Think about a simpler 3 foot well.
After Day 1:  2 feet.
After Night 1:  1 foot. 1 foot  total.
Moved a total  3 feet.
And out of the well.

Think about a simpler 4 foot well.
After Night 1:  1 foot.
After Night 2:  2 feet.
Now the snail is out!

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.

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.

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.

Pages Of A Book: Plan
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.

Pages Of A Book: Plan
1, 2, 3, 4, 5, 6, 7, 8, 9 = 9 digits.
9 digits for the first 9 pages. But what about the remaining pages?

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.

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.

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.

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.

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.

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.

Assignment #19
Page 161-162: 5-11 odd and 13-20 all.

Section 3.8 Reasoning Strategies

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