Solve an Easier Related Problem

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Solve an Easier Related Problem

• Problem Solving Strategy

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Solving an easier version of the same problem
will sometimes reveal a method of solving a
problem.

• For instance, find the sum of the numbers 1 to
  100 might seem like a long problemnot
  necessarily long, but a bit time consuming. When
  Carl Friedrich Gauss, a famous mathematician, was
  given this problem in fourth grade when his
  teacher wanted to keep the students busy for 30
  minutes, he was able to solve it in record time.
  
• 1 2 3 3 3 6 6 4 10 10 5
  15 Gauss decided this was going to take too
  long, so he started looking at the problem
  differently. How can you find the sum easier?
• Try this and see if you see a pattern
• 1 2 3 4 5 6 7 8 9 10 ______
• How is the previous sum similar to
• 11 12 13 14 15 16 17 18 19 20
  ______
• Use what youve seen to predict the sum of the
  next 10 numbers and then the sum of all of the
  numbers from 1 to 100.
• 1 2 3 4 5 6 7 8 98 99 100
  ______

55
155
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Solve an Easier Problem

• Or look at it this way
• 1 100 ______
• 2 99 ______
• 3 98 ______
• Use what youve seen to predict the sum of all of
  the numbers from 1 to 100.
• 1 2 3 4 5 6 7 8 98 99 100
  ?
• After trying several different strategies, the
  second way is the way Gauss solved the problem
  for his 4th grade teacher. He was pretty smart,
  wasnt he!

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Solve an Easier Related Problem

• Here are some things to try to find an easier
  related problem
• Use a number instead of a variable
• Use a smaller or easier number in order to
  develop a process for solving the problem.
• Do a set of easier examples (for the above
  example, find the sum of the first 10 or 20
  numbers) and look for a pattern.
• Do an easier example and figure out how to use
  the same process to solve the harder problem.
• Change, fix, or get rid of some conditions.
• Eliminate unnecessary information.

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Quiz Scores Averages
The average of a group of quiz scores is 31.8.
There are k quiz scores in the group. The average
of 10 of these quiz scores is 24.3. Find the
average of the remaining quiz scores in terms of
k.
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Use a smaller or easier number.

• The average of a group of quiz scores is 31.8
  . There are k quiz scores in the group. The
  average of 10 of these quiz scores is 24.3
  . Find the average of the remaining quiz scores
  in terms of k.

30
25
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Use a number instead of a variable.

• The average of a group of quiz scores is 30.
  There are k quiz scores in the group. The
  average of 10 of these quiz scores is 25. Find
  the average of the remaining quiz scores in terms
  of k .

50
50
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Solve the simpler problem.

• Sum of all scores is 30 x 50 1500.
• Sum of 10 scores is 25 x 10 250.
• Sum of other scores is 1500 250 1250.
• Average of those 40 scores is

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Apply the process to the original problem.
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Lets try another problem.The Election

• In this election, there are 29 issues and
  candidates. In the last election, there were
  28,311 voters, representing 18,954 households,
  and they voted at 14 polling places. This time
  there will be 34,892 voters. How many polling
  places will be needed?

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Use smaller or easier numbers.
PROBLEM In this election, there are 29 issues
and candidates. In the last election, there were
28,311 voters, representing 18,954 households,
and they voted at 14 polling places. This time
there will be 34,892 voters. How many polling
places will be needed?

• Polling places (last election) 15
• Voters (last election) 30,000
• Households 20,000
• Issues 30
• Voters (this election) 35,000
• Polling places (this election) ?

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Eliminate unnecessary information.
PROBLEM In this election, there are 29 issues
and candidates. In the last election, there were
28,311 voters, representing 18,954 households,
and they voted at 14 polling places. This time
there will be 34,892 voters. How many polling
places will be needed?

• Polling places (last election) 15
• Voters (last election) 30,000
• Households 20,000
• Issues 30
• Voters (this election) 35,000
• Polling places (this election) ?

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Solve the simpler problem.
PROBLEM In this election, there are 29 issues
and candidates. In the last election, there were
28,311 voters, representing 18,954 households,
and they voted at 14 polling places. This time
there will be 34,892 voters. How many polling
places will be needed?
voters/polling place
polling places (this election)
14
Apply the process to the original problem.
PROBLEM In this election, there are 29 issues
and candidates. In the last election, there were
28,311 voters, representing 18,954 households,
and they voted at 14 polling places. This time
there will be 34,892 voters. How many polling
places will be needed?
voters/polling place
polling places (this election)
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Lets investigate divisors
The divisors of 360 add up to 1170. What is the
sum of the reciprocals of the divisors of 360?
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Do an easier problem to develop a process.
Problem The divisors of 360 add up to 1170.
What is the sum of the reciprocals of the
divisors of 360?

• Use divisors of 24, which are 1, 2, 3, 4, 6, 8,
  12, and 24. Their sum is 60.
• The sum of reciprocals is

17
Try another easier problem.

• The divisors of 10 are 1, 2, 5, and 10. Their sum
  is 18.
• The sum of reciprocals is

18
Apply the process to the original problem.
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How many squares?

• How many squares are there on a checkerboard?

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Do a set of easier examples and find a pattern.
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Generalize the pattern to solve the original
problem.

• 1 8x8 square
• 4 7x7 squares
• 9 6x6 squares
• 16 5x5 squares
• 25 4x4 squares
• 36 3x3 squares
• 49 2x2 squares
• 64 1x1 squares

total 204 squares
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Next Train East

• A train leaves Roseville heading east at 600
  a.m. at 40 miles per hour. Another eastbound
  train leaves on a parallel track at 700 a.m. at
  50 miles per hour. What time will it be when the
  two trains are the same distance away from
  Roseville?

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Change, fix, or get rid of some conditions.
Problem A train leaves Roseville heading east
at 600 a.m. at 40 miles per hour. Another
eastbound train leaves on a parallel track at
700 a.m. at 50 miles per hour. What time will it
be when the two trains are the same distance away
from Roseville?

• One possibility Change the condition so that the
  first train travels for an hour and then stops,
  and the second travels at 10 miles per hour.

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Solving the simpler problem solves the original
problem.
Problem A train leaves Roseville heading east
at 600 a.m. at 40 miles per hour. Another
eastbound train leaves on a parallel track at
700 a.m. at 50 miles per hour. What time will it
be when the two trains are the same distance away
from Roseville?
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Strategy Solve an Easier Related Problem

• Eliminate unnecessary information.
• Use a number instead of a variable.
• Try smaller or easier numbers.
• Change, fix, or get rid of some conditions.
• Look for a pattern in easier examples.
• Find a process in an easier example.

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Solve an Easier Related Problem

• Open your problem solving book and go to the
  section on Solve an Easier Related Problem (pages
  23-24).
• Print off a problem solving sheet and do a
  complete solution to a problem from this
  sectiondo one and submit it or do more for
  extra fun. (Twenty-five Man Roster, TV Truck,
  and/or Good Luck Goats)
• Email your complete solution, with your steps
  clearly shown to NCAMath_at_district87.org or put
  them in Nancy Powells mail box!
• Have fun and Problem Solve!