Transcript BB Chapter 1

Thinking Critically
1.1 An Introduction to Problem Solving
1.3 More Problem-Solving Strategies
1.4 Algebra as a Problem-Solving Strategy
1.5 Additional Problem-Solving Strategies
1.6 Reasoning Mathematically
1.1
An Introduction to Problem
Solving
Slide 1-2
Example 1.1
Toni is thinking of a number. If you double the
number and add 11, the result is 39. What number
is Toni thinking of?
Solution 1: Guessing and checking
Guess 10: 2(10) + 11 = 20 + 11 = 31 too small
Guess 20: 2(20) + 11 = 40 + 11 = 51 too large
Guess 15: 2(15) + 11 = 30 + 11 = 41 a bit large
Guess 14: 2(14) + 11 = 28 + 11 = 39 This checks!
Toni’s number must be 14.
Slide 1-3
Example 1.1-continued
Toni is thinking of a number. If you double the
number and add 11, the result is 39. What number
is Toni thinking of?
Solution 2: Make a table and looking for a pattern
We need to get to 39, jump
by 2 each time.
39  27 12

 6 more steps
2
2
Guess 8 + 6 = 14.
Slide 1-4
Problem-Solving Strategy 1
Guess and Check
Make a guess and check to see if it
satisfies the demands of the problem. If
it doesn’t, alter the guess appropriately
and check again.
When the guess finally checks, a
solution has been found.
Slide 1-5
Example 1.3 Using Guess and Check
In the first diagram the numbers
in the big circles are found by
adding the numbers in the two
smaller adjacent circles.
Complete the second diagram
so that the same pattern holds.
Slide 1-6
Problem-Solving Strategy 2
Make an Orderly List
For problems that require
consideration of many possibilities,
make an orderly list or a table to
make sure that no possibilities are
missed.
Slide 1-7
Example 1.4 Make an Orderly List
How many different
total scores could
you make if you hit
the dartboard shown
with three darts?
Slide 1-8
Problem-Solving Strategy 3
Draw a Diagram
Draw a diagram or picture that
represents the data of the
problem as accurately as
possible.
Slide 1-9
Example 1.5 Draw a Diagram
In a stock car race the first five
finishers in some order were a
Ford, a Pontiac, a Chevrolet, a
Buick, and a Dodge.
a)
The Ford finished 7
seconds before the
Chevrolet.
b)
The Pontiac finished 6
seconds after the Buick.
c)
The Dodge finished 8
seconds after the Buick.
d)
The Chevrolet finished 2
seconds before the
Pontiac.
Slide 1-10
1.3
More Problem-Solving
Strategies
Slide 1-11
Problem-Solving Strategy 4
Look for a Pattern
Consider an ordered sequence of
particular examples of the general situation
described in the problem.
Then carefully scrutinize these results,
looking for a pattern that may be the key to
the problem.
Slide 1-12
Example 1.6 Look for a Pattern
For the following numerical sequence,
fill in the blanks.
1, 4, 7, 10, 13, ___,
16 ___
19
+3
+3
+3
+3
+3
+3
Slide 1-13
Example 1.6 continued
For the following numerical sequence,
fill in the blanks.
19, 20, 22, 25, 29, ___,
34 ___
40
+1
+2
+3
+4
+5
+6
Slide 1-14
Example 1.6 continued
For the following numerical
sequence, fill in the blanks.
36 ___
49
1, 4, 9, 16, 25, ___,
+3
+5
+7
+9
+11
+13
Slide 1-15
Example 1.6 continued
OR
notice the pattern of perfect squares:
1, 4, 9, 16, 25,
12, 22, 32, 42, 52,
36 ___
49
___,
62,
72
Slide 1-16
Problem-Solving Strategy 5
Make a Table
Make a table reflecting the data in the
problem. If done in an orderly way, such a
table will often reveal patterns and
relationships that suggest how the problem
can be solved.
Slide 1-17
Example 1.7 Applying Make a Table
a. Draw the next two diagrams to
continue this sequence of dots:
b. How many dots are in each figure?
c. How many dots would be in the onehundredth figure?
Slide 1-18
Example 1.7 continued
a. Draw the next two diagrams to
continue this sequence of dots:
The arrays of dots are similar, each
array has one more two-dot column
than its predecessor.
Slide 1-19
Example 1.7 Applying Make a Table
b. How many dots are in each figure?
Count the dots in each array.
1, 3, 5, 7, 9, 11, …
c. How many dots would be in the onehundredth figure?
Slide 1-20
Example 1.7 Applying Make a Table
c. How many dots would be in the one-hundredth
figure?
Create a table.
The number of 2s added is one less than the
number of the term. The one-hundredth term
is 1 + 99 × 2 = 199.
Slide 1-21
Problem-Solving Strategy 6
Consider Special Cases
In trying to solve a complex problem,
consider a sequence of special cases.
This will often show how to proceed naturally
from case to case until one arrives at the
case in question.
Alternatively, the special cases may reveal a
pattern that makes it possible to solve the
problem.
Slide 1-22
1.4
Algebra as a Problem-Solving
Strategy
Slide 1-23
Problem-Solving Strategy 7
Use a Variable
A variable is a symbol (usually a letter) that
can represent any of the numbers in some
set of numbers.
Often a problem requires that a number be
determined. Represent the number by a
variable, and use the conditions of the
problem to set up an equation that can be
solved to ascertain the desired number.
Slide 1-24
Example 1.9 Gauss’s Insight

Find the sum of the whole numbers from 1 to 100.

Understand the problem.
 Find 1 + 2 + 3 + … + 100.

Devise a plan.
 Let S = 1 + 2 + 3 + … + 100. Note that we
could also write S = 100 + 99 + 98 + … + 1.
Add these together.
Slide 1-25
Carry out the plan.
S = 1 + 2 + 3 + … + 100
S = 100 + 99 + 98 + … + 1
2S = 101 + 101 + 101 + … + 101
2S =
100 × 101
100  101
S
2
S  50  101
S  5050
Slide 1-26
The Steps in Algebraic Reasoning
Slide 1-27
Example 1.12 Setting Up and Solving
an Equation: Can I Get a C?
Larry has exam scores of 59, 77, 48,
and 67. What score does he need on
the next exam to bring his average for
all five exams to 70?
Let s denote Larry’s minimum score
needed on the fifth exam.
59  77  48  67  s
5
Slide 1-28
Example 1.12 Setting Up and Solving
an Equation: Can I Get a C?-continued
Larry has exam scores of 59, 77, 48, and 67. What score
does he need on the next exam to bring his average for all
five exams to 70?
59  77  48  67  s
5
59  77  48  67  s
 70
5
251  s  350
s  350  251  99
Multiply each side by 5 and add the sum
of the first four test scores.
Subtract 251 from both sides to solve for s.
Larry must hope for a 99 or 100 on the last test.
Slide 1-29
Example 1.13 Solving a Rate Problem
Two years ago, it took Tom 8 hours to whitewash a
fence. Last year, Huck took just 6 hours to
whitewash the fence. This year, Tom and Huck
have decided to work together so that they’ll have
time left in the afternoon to angle for catfish. How
long will the job take the two boys?
Let T denote the time, in hours, that Tom and Huck
need together to whitewash the fence.
Tom can whitewash the fence in 8 hours, he can
whitewash 1/8 of the fence per hour.
Slide 1-30
Example 1.13 Solving a Rate Problem
continued
In T hours, Tom will have whitewashed T/8 of the
fence.
Huck can whitewash the fence in 8 hours, he can
whitewash 1/6 of the fence per hour. Therefore he
can whitewash T/6 of the fence in T hours.
Working together they boys will whitewash the
entire fence when
T T
 1
8 6
Slide 1-31
Example 1.13 Solving a Rate Problem
continued
Multiply both sides by 48 gives the equivalent
equation.
6T  8T  48
14T  48
48
T
14
3
T  3 hours
7
Tom and Huck can whitewash the fence in just
less than three and a half hours.
Slide 1-32
1.5
Additional Problem-Solving
Strategies
Slide 1-33
Problem-Solving Strategy 8
Work Backward
Start from the desired result and
work backward step-by-step until
the initial conditions of the
problem are achieved.
Slide 1-34
Problem-Solving Strategy 9
Eliminate Possibilities
Suppose you are guaranteed that a
problem has a solution. Use the data
of the problem to decide which
outcomes are impossible. Then at
least one of the possibilities not ruled
out must prevail.
If all but one possibility can be ruled
out, then it must prevail.
Slide 1-35
Example 1.16 Eliminating Possibilities
Beth, Jane, and Mitzi play on the basketball team.
Their positions are forward, center, and guard.
Given the following information, determine who
plays each position.
a. Beth and the guard bought a milk shake for
Mitzi.
b. Beth is not a forward.
Slide 1-36
Example 1.16 Eliminating Possibilities
a. Beth and the guard bought a milk shake for Mitzi.
b. Beth is not a forward.
Beth
Jane
forward
Use a table.
center
guard
X
0
X
Mitzi
X
X
0
0
X
X
We conclude that Mitzi plays forward, Beth plays center
and Jane plays guard.
Slide 1-37
Problem-Solving Strategy 10
The Pigeonhole Principle
If m pigeons are placed into n
pigeonholes and m > n, then
there must be at least two
pigeons in one pigeonhole.
Slide 1-38
Example 1.17 Using the Pigeonhole
Principle
A student working in a tight space can barely
reach a box containing 12 rock CDs and 12
classical CDs. Her position is such that she cannot
see into the box. How many CDs must she select
to be sure that she has at least 2 of the same type
of CDs?
Make an orderly list:
Two rock CDs and zero classical CDs
One rock CD and one classical CD
Zero rock CDs and two classical CDs
Slide 1-39
Example 1.17 Using the Pigeonhole
Principle
Make an orderly list:
Two rock CDs and zero classical CDs
One rock CD and one classical CD
Zero rock CDs and two classical CDs
Two CDs are not enough; she might get one of
each kind. But if she selects a third CD, she will
end up with a third rock CD, a second rock CD, a
second classical CD, or a third classical CD. In any
case, she will have two CDs of the same type and
the condition of the problem will be satisfied.
Slide 1-40
1.6
Reasoning Mathematically
Slide 1-41
Inductive Reasoning
Inductive reasoning is drawing a
conclusion based on evidence obtained
from specific examples.
The conclusion drawn is called a
generalization.
An example this disproves a statement is
called a counterexample.
Slide 1-42
Problem-Solving Strategy 11
Use Inductive Reasoning
• Observe a property that holds in several
examples.
• Check that the property holds in other
examples. In particular, attempt to find an
example where the property does not hold.
• If the property holds in every example,
state a generalization that the property is
probably true in general.
Slide 1-43
A generalization that seems to be
true, but has yet to be proved, is
called a conjecture. Once a
conjecture is given a proof, it is
called a theorem.
Slide 1-44