Download 7-9 Problem-Solving Strategy

7-9 Problem-Solving Strategy:
Find a Pattern
Name
Date
Find a pattern to solve.
1. What is the sum of the first 35 odd numbers?
Find a pattern: Sum of first two odd numbers: 1 ⴙ 3 ⴝ 4
Sum of first three odd numbers: 1 ⴙ 3 ⴙ 5 ⴝ 9
Sum of first four odd numbers: 1 ⴙ 3 ⴙ 5 ⴙ 7 ⴝ 16
Pattern shows that each sum is the square of the number of addends.
So the sum of the first 35 odd numbers is 352 or 1225.
2. How many diagonals does a regular 12-gon have?
Find a pattern: Count the number of diagonals for each figure.
0
1
3
2
1
0
2
0
3
Examining the number of diagonals,
the 4-gon has 1 ⴙ 1 ⴝ 2 diagonals,
the
5-gon has 2 ⴙ (2 ⴙ 1) ⴝ 5 diagonals,
0
0
0
1
1
0
the 6-gon has 3 ⴙ (3 ⴙ 2 ⴙ 1) ⴝ
3 sides
4 sides
5 sides
6 sides
9 diagonals.
So an n-gon has this many diagonals: (n ⴚ 3) ⴙ [(n ⴚ 3) ⴙ (n ⴚ 4) ⴙ (n ⴚ 5) ⴙ . . . ⴙ 2 ⴙ 1].
So a 12-gon has 9 ⴙ 9 ⴙ 8 ⴙ 7 ⴙ 6 ⴙ 5 ⴙ 4 ⴙ 3 ⴙ 2 ⴙ 1 ⴝ 54 diagonals
0
0
2
3. If the design seen here (made with toothpicks) is extended to 20 triangles,
how many toothpicks will it require?
10 triangles
Find a pattern:
Copyright © by William H. Sadlier, Inc. All rights reserved.
Number of Triangles
Number of Toothpicks
1
3
2
3ⴙ2ⴝ5
3
3ⴙ2ⴙ2ⴝ7
4
3ⴙ2ⴙ2ⴙ2ⴝ9
n
3 ⴙ 2(n ⴚ 1) ⴝ 2n ⴙ 1
Using the pattern, it can be seen that a figure
with 10 triangles requires 21 toothpicks
(it does), while a figure with 20 triangles will
require 41 toothpicks.
4. This design (made with toothpicks) is 4 layers deep. How many toothpicks
are required to make a similar design that is 12 layers deep?
Find a pattern: 1 layer: 3 toothpicks; 2 layers: 3 ⴙ 6 (9 toothpicks);
3 layers: 3 ⴙ 6 ⴙ 9 (18 toothpicks); 4 layers: 3 ⴙ 6 ⴙ 9 ⴙ 12 (30 toothpicks);
So, 12 layers takes 3 ⴙ 6 ⴙ 9 ⴙ 12 ⴙ . . . ⴙ 36 ⴝ 3(1 ⴙ 2 ⴙ 3 ⴙ . . . ⴙ 12) ⴝ 3(78) ⴝ 234
So 234 toothpicks are required to make a design that is 12 layers deep.
Lesson 7-9, pages 194–195.
Chapter 7
185
For More Practice Go To:
5. Use a calculator. What digit is in the ones place of the standard form of 742?
Find a pattern: 71 ⴝ 7; 72 ⴝ 49; 73 ⴝ 343; 74 ⴝ 2401; 75 ⴝ 16,807; 76 ⴝ 117,649;
77 ⴝ 823,543; 78 ⴝ 5,764,801; 79 ⴝ 40,353,607;
Examining the ones digit shows this repeating pattern: 7, 9, 3, 1. Because it repeats every 4 digits,
it can be used to find the ones digit for any power of 7. Because 40 is a multiple of 4, 740 must have
a ones digit of 1. Likewise, 741 has a ones digit of 7, and 742 has a ones digit of 9.
1
6. The fraction 7 can be represented as a repeating decimal. What is the 87th
1
digit in the decimal expansion of 7 ?
1
ⴝ 1 ⴜ 7 ⴝ 0.142857; So every six digits, the sequence of number repeats.
7
To find which of these digits is the 87th digit, divide: 87 ⴜ 6 ⴝ 14 R 3; The whole number
in the quotient represents the number of times the sequence repeats (14); the remainder 3
represents placement of the 87th digit in the sequence. The third digit is 2, and so likewise,
the 87th digit is 2.
Find a pattern:
7. The patterns of dots seen here are the first 5 terms
of the sequence of successive “hollow triangles
of size n.” How many dots are there in the
hollow triangle of size 30?
Size
size 3
size 4
size 5
Number of dots
1
1
2
3ⴝ2ⴙ1
3
6ⴝ3ⴙ2ⴙ1
4
9ⴝ4ⴙ3ⴙ2
5
12 ⴝ 5 ⴙ 4 ⴙ 3
n
n ⴙ (n ⴚ 1) ⴙ (n ⴚ 2) or 3(n ⴚ 1)
Using this pattern, it can be seen
that a hollow triangle of size 30 will
have 3(30 ⴚ 1) ⴝ 87 dots.
8. Refer to the pattern of dots seen here as a “4-by-3 tree” because it has
4 dots along two lateral sides and 3 dots along its base. How many line
segments can be drawn that connect the dots of a 9-by-8 tree?
(Segments may cross, but may not overlap.)
Find a pattern:
2
3
It can be seen that if n represents the number of dots
along the base, then the top dot gets n segments.
0
0
5
3
Along the right side, each dot gets 2n ⴚ 1 segments,
5
1
1
2
and there are n ⴚ 1 of these. All points on the base
3-by-2
1
3
except the leftmost dot get n segments; there are n ⴚ 1
3
of these. Finally, along the left segment, there are n ⴚ 1
4-by-3
dots that get 1 segment each. There is a last dot that
gets 0 segments. So each of these triangles has this many line segments:
n ⴙ (2n ⴚ 1)(n ⴚ 1) ⴙ n(n ⴚ 1) ⴙ (n ⴚ 1) or n(3n ⴚ 2). So if n ⴝ 8, 8(24 ⴚ 2) ⴝ 176.
So a 9-by-8 triangle has 176 line segments.
186
Chapter 7
4
0
7
1
1
1
7
5-by-4
4
4
7
4
Copyright © by William H. Sadlier, Inc. All rights reserved.
Find a pattern:
size 1 size 2