Download Using Variables to Describe Number Patterns

Using Variables to
Describe Number Patterns
Objectives To describe general number patterns in words; and
to
t write special cases for general number patterns.
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Ongoing Learning & Practice
Key Concepts and Skills
Math Boxes 3 1
• Apply a general pattern to find 10%
of a number. Math Journal 1, p. 85
Students practice and maintain skills
through Math Box problems.
[Number and Numeration Goal 2]
• Extend numeric patterns. [Patterns, Functions, and Algebra Goal 1]
• Write a number sentence containing a
variable to describe a general pattern. [Patterns, Functions, and Algebra Goal 1]
• Apply general patterns to explore
multiplicative and additive inverses. [Patterns, Functions, and Algebra Goal 4]
Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problem 2. [Patterns, Functions, and Algebra Goal 1]
READINESS
Connecting General Number Patterns
and Special Cases
Math Masters, p. 71
Students use the “What’s My Rule?” routine
to relate general number patterns and
special cases.
ENRICHMENT
Math Masters, p. 74
Students practice and maintain skills
through Study Link activities.
Math Masters, pp. 72 and 73
Students describe patterns and relationships
among triangular numbers, square numbers,
and rectangular numbers.
Ongoing Assessment:
Informing Instruction See page 182.
Key Vocabulary
general pattern variable special case
Materials
Math Journal 1, pp. 82–84
Student Reference Book, p. 105
calculator
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 278–289
Variables, Formulas, and Graphs
Differentiation Options
Exploring Number Patterns
Students describe a general number pattern
in words and write examples or special
cases of it. They are given special cases for
a general pattern and describe it with a
number sentence having one variable.
Unit 3
Interactive
Teacher’s
Lesson Guide
Study Link 3 1
Key Activities
180
Curriculum
Focal Points
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, pp. 157, 240, and 242
Students write rules to describe
numeric patterns.
Mathematical Practices
SMP2, SMP3, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
6.NS.6a, 6.NS.6c, 6.EE.2, 6.EE.2a, 6.EE.2b, 6.EE.2c, 6.EE.6
Mental Math and Reflexes
Math Message
Students rename mixed numbers and whole numbers
as fractions. Suggestions:
5
1 _
33
1 _
1 _
22
1_
8_
3_
1. Write the number that is the opposite of
a. 15 -15 b. -8 8 c. π -π d. 5.25 -5.25
2. Add.
a. 5 + -5 0
b. -2.6 + 2.6 0
c. -(-11) + (-11) 0
d. y + (-y) 0
4 4
8
2 _
2_
3 3
4 4
5
5_
1
7 7
64
64 _
1
1 Teaching the Lesson
▶ Math Message Follow-Up
Interactive whiteboard-ready
ePresentations are available at
www.everydaymathonline.com to
help you teach the lesson.
WHOLE-CLASS
DISCUSSION
Draw a number line on the board. As students share their
answers, plot the numbers and their opposites on the number line.
Ask: What do you notice about the positions of a number and its
opposite in relationship to 0? They are on opposite sides of 0 on the
number line and are each the same distance from 0 on the number
line. What does this tell you about the opposite of 0? 0 is its own
opposite. Is any other number its own opposite? no What is the sum
of any number and its opposite? 0
▶ Describing General Number
7 7
e - _8 _8
PROBLEM
BL
BLE
LEM
LE
LEM
SOLVING
LVIN
V NG
Patterns with Variables
Adjusting the Activity
To review the concept of opposites,
use the number line to model the three
properties of opposites at the top of Student
Reference Book, page 105.
AUDITORY
KINESTHETIC
TACTILE
VISUAL
PARTNER
ACTIVITY
ELL
(Math Journal 1, pp. 82 and 83)
Algebraic Thinking Read and discuss the text at the top of
journal page 82. Have students complete Problem 1 and discuss
their solutions. Be sure to cover the following points:
Rules that describe patterns are sometimes called general
patterns.
A general pattern may be described in words.
Student Page
Date
Time
LESSON
3 1
Patterns and Variables
103
Study the number sentences at the right. All three sentences
show the same general pattern.
10
10% of 50 100 50
10
10% of 200 100 200
This general pattern may be described in words: To find 10%
10
1
of a number, multiply the number by 100 (or 0.10, or 10 ).
10
10% of 8 100 8
The pattern may also be described by a number sentence
0
that contains a variable: 10% of n 1
n.
1 00
A variable is a symbol, such as n, x, A, or
. A variable can stand for any one of
many possible numeric values in a number sentence.
10
10
Number sentences like 10% of 50 100 50 and 10% of 200 100 200 are
0
examples, or special cases, for the general pattern described by 10% of n 1
n.
100
A general numeric pattern may be described with symbols, at
least one of which represents a number. Symbols that represent
numbers are called variables.
To write a special case for a general pattern, replace the variable with a number.
Example:
General pattern
Special case
1.
A variable can have any one of many possible numeric values.
A common misunderstanding of variables is that a variable
always stands for one particular number.
10
10% of n n
1 00
10
10% of 35 35
1 00
Here are 3 special cases
for a general pattern.
10
10
1
725
725
1
2
1
2
1
1
Sample answers:
a. Describe the pattern in words.
Any number divided by itself equals 1.
b.
2.
Give 2 other special cases for the pattern.
9
9
1
Here are 3 special cases
for another general pattern.
15 (15) 0
50
0
500
3 (3) 0
1
4
1
1
(4) 0
Sample answers:
a. Describe the pattern in words.
Any number added to its opposite equals zero.
b.
Give 2 other special cases for the pattern.
0.75 (0.75) 0
100 (100) 0
Math Journal 1, p. 82
Lesson 3 1
181
Student Page
Date
Time
LESSON
Patterns and Variables
3 1
3.
103
A spider has 8 legs. The general pattern is: s spiders have s 8 legs.
Write 2 special cases for the general pattern.
10 8 80 legs
a.
4.
continued
Sample answers:
22 8 176 legs
b.
There are many ways to describe the same pattern using
n = 1, _
b = 1, and
= 1 all describe
variables. For example, _
n
b
the pattern in Problem 1 on journal page 82.
Study the following special cases for a general pattern.
Sample answers:
6
The value of 6 quarters is 4 dollars.
10
The value of 10 quarters is 4 dollars.
33
The value of 33 quarters is 4 dollars.
Describe the general pattern in words.
a.
The value of n quarters is n4 dollars.
b.
Give 2 other special cases for the pattern.
The value of
The value of
To support English language learners, discuss the everyday
meaning of variable and of special case, as well as their
meanings in this context.
15 quarters is 145 dollars.
0
dollars.
100 quarters is 10
4
Have partners complete journal pages 82 and 83. They may
use calculators.
Sample answers:
Write 3 special cases for each general pattern.
5.
pp2p
6.
c 1c 1
7.
p p (3 p) 5 p
8.
s 2 s (s 1) s
4 4 2 4
1.8 1.8 2 1.8
20 20 2 20
2 2 (3 2) 5 2
3.8 3.8 (3 3.8) 5 3.8
16 16 (3 16) 5 16
When a particular number is substituted for the variable in a
general pattern, the result is called an example, or a special
case, of the general pattern.
1
2
1
33
1
6.4
2 1
33 1
6.4 1
NOTE There are no actual calculations required on these pages, but some
students may want to verify that a particular number sentence is true. This may
) .
involve the use of parentheses keys (
52 5 (5 1) 5
102 10 (10 1) 10
92 9 (9 1) 9
When most students have completed the journal pages, ask
volunteers to share their solutions.
Have students describe each general pattern in their own words
before giving special cases for the pattern. There is often more
than one way to describe a pattern in words. Two ways of
n =
describing the general pattern _
1 follow.
n
Math Journal 1, p. 83
1. If the numerator and denominator of a fraction are the same
number (except 0), the fraction is equivalent to 1.
NOTE General patterns may be described
in words or by an open number sentence.
Whenever Everyday Mathematics asks
students to write a general pattern, they
should write a number sentence that describes
the pattern, unless the directions specifically
state that they are to describe it in words.
Time
LESSON
3 1
Writing General Patterns
103
Following is a method for finding the general pattern for a group
of special cases.
8 /1 8
0.3 / 1 0.3
Solution Strategy
Step 1 Write everything that is the same for all of the special cases.
Use blanks for the parts that change.
/ 1
Each special case has division by 1 and an equal sign.
Step 2 Fill in the blanks. Each special case has a different number, but the number is
the same for both blanks, so use the same variable in both blanks.
N
/1
N
, or
x
Write a general pattern for each group of 3 special cases.
1.
6
10
6
1 1
0
1
2
General pattern
x 1x
General pattern
T 00
General pattern
c cats have c 4 legs
00
5 cats have 5 4 legs.
6 6 62
1
2
2
1
1
2 (2)
0.7 0.7 (0.7)2
General pattern
2
Math Journal 1, p. 84
182
Unit 3
with Number Sentences
(Math Journal 1, p. 84)
Algebraic Thinking As a class, read and discuss the problem and
solution strategy at the top of journal page 84. Ask students to
decide which parts stay the same in all of the special cases and
which parts change from case to case.
Circulate and assist as needed. When most students have
completed the page, ask volunteers to share their solutions.
Ongoing Assessment: Informing Instruction
1 cat has 1 4 legs.
2 cats have 2 4 legs.
4.
/1
, or
Sample answers:
600
78.7 0 0
3.
x
18 1 18
2.75 1 2.75
2.
/1
PARTNER
ACTIVITY
12.5 / 1 12.5
Example: Write the general pattern for the special cases at the right.
Possible solutions:
A general pattern that is written with a variable looks the same as
any of the special cases for that pattern. The only difference is
that the variable has been replaced by a specific value.
(See margin.)
▶ Describing General Patterns
Student Page
Date
2. If a nonzero number is divided by itself, the result is equal to 1.
Variables, Formulas, and Graphs
Watch for students who do not recognize what the special cases have in
common. Some students may benefit from circling all the numbers and symbols
that stay the same from one special case to the next.
Student Page
Date
2 Ongoing Learning & Practice
Time
LESSON
Math Boxes
3 1
1.
Complete the “What’s My Rule?” table.
2.
Rule: Subtract 1.32
▶ Math Boxes 3 1
INDEPENDENT
ACTIVITY
in
out
8
6.68
0.83
0.48
2.15
1.8
(Math Journal 1, p. 85)
4.89
7.33
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 3-3. The skills in Problems 5 and 6
preview Unit 4 content.
Writing/Reasoning Have students write a response to the following:
Explain how you know where to place the decimal point in the
quotient of Problem 4. Sample answer: I estimated. 90 divided by
6 is 15, so 93.6 divided by 6 is about 15.
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 2
3.
3.57
6.01
32 33
253
103
4.
Divide.
6冄9
9
苶3
苶.6
苶
200,000
200 thousand, or 0.2 million
b.
16,900,000,000
c.
58,400,000,000,000
16.9 billion
58.4 trillion
5.
Use Math Boxes, Problem 2 to assess students’ ability to write special cases
for a general pattern. Students are making adequate progress if they are able to
write three special cases. Some students may recognize the general pattern as
illustrative of the Distributive Property of Multiplication over Addition.
a (b + c) (a b) + (a c)
Sample answers:
1 (2 3)(1 2) (1 3)
5 (10 5) (5 10) (5 5)
7 (20 9)(7 20) (7 9)
Write each number in number-and-word
notation.
a.
Write 3 special cases for the
general pattern.
93.6 6 Add or subtract.
5
1
4
7
a. 7
7
5
7
2
9
b. 9
9
10
3
7
c. 11
11
11
1
5
6
d. 1 6.
3
a. 10
83
b. 100
4
c. 15
19
d. 20
6
15.6
42–45
Compare each pair of fractions.
Write < or >.
5
10
81
100
5
15
20
20
83
75
Math Journal 1, p. 85
[Patterns, Functions, and Algebra Goal 1]
▶ Study Link 3 1
INDEPENDENT
ACTIVITY
(Math Masters, p. 74)
Home Connection Students practice describing general
patterns with words and number sentences having one
variable. They write special cases for general patterns.
Teaching Master
Study Link Master
Name
Date
STUDY LINK
7
8
43 0 43
36.09 0 36.09
(2 24) 24 3 24
0.5 0.25 0.25 0.5
For each set of special cases, write a general pattern.
5.
Sample answers:
7
7 0.1 6.
10
52 53 55
3
3 0.1 132 133 135
4 0.1 4
10
in
13
12
11
16
21
7
20
25
60
20
105
110
300
100
2.
out
4
Rule: Divide the in number by 3.
You are writing special cases for a general number pattern
when you complete a “What’s My Rule?” table.
s 0.1 Rule: Add the opposite of the number.
(x x 0)
out
in
out
0
8
1
25
0
0
0
9
1
1
1
53
m0 1
Rule: Divide by the number.
(y y 1)
3
7
1
(2)0 1
s
10
1
4
100
Use the values from the table above to write special cases for the following
general number patterns:
x x 0.
y y 1.
Special cases
Special cases
Practice
Sample answers:
Complete.
3
10. 4
out
8
in
20 1
1460 1
10
x2 x3 x5
103, 253
Complete.
100 0.25 0.25 100
1
7. 10
in
Rule: Add 5 to the in number.
(2 10) 10 3 10
s 0.25 0.25 s
32 33 35
General Patterns and Special Cases
You are describing a general number pattern for a special case
when you write a rule for a “What’s My Rule?” table.
Write a rule for each table shown below.
Sample answers:
(2 m) m 3 m
Time
8
52 0 52
For each general pattern, give 2 special cases.
4.
103
7
0 Give 2 other special cases for the pattern.
b.
3.
Describe the general pattern in words. Sample answers:
The sum of any number and 0 is equal to the original
number.
a.
2.
3 1
1.
Following are 3 special cases representing a general pattern.
17 0 17
Date
LESSON
Variables in Number Patterns
31
1.
Name
Time
10
100
75
100
0.10
0.75
1
8. 4
4
11. 5
25
—
— 0.
—
25
1
9. 5
80
7
12. 10
100
80
100
Math Masters, p. 74
0.
20
100
70
100
0.20
0.
70
Example: 3 3 0
Example: 8 8 1
25 25 0
7 7 0
53 53 0
9 9 1
1
1
1
4
4
100 100 1
Math Masters, p. 71
Lesson 3 1
183
Teaching Master
Name
Date
LESSON
Time
3 Differentiation Options
Number Patterns
31
Triangular, square, and rectangular numbers are examples of number patterns
that can be shown by geometric arrangements of dots. Study the number
patterns shown below.
Triangular Numbers
Square Numbers
READINESS
1st
2nd
3rd
4th
1st
2nd
3rd
4th
Rectangular Numbers
▶ Connecting General Number
INDEPENDENT
ACTIVITY
5–15 Min
Patterns and Special Cases
1st
1.
2nd
3rd
(Math Masters, p. 71)
4th
Use the number patterns to complete the table.
Number of Dots in Arrangement
2.
1st
2nd
3rd
4th
Triangular
Number
1
3
6
10
Square
Number
1
4
9
16
25 36 49 64 81 100
Rectangular
Number
2
6
12
20
30 42 56 72 90 110
What is the 11th triangular number?
5th
6th
7th
8th
9th
10th
15 21 28 36 45 55
To provide experience with algebraic notation, have
students use the “What’s My Rule?” routine. By
completing “What’s My Rule?” tables, some students may
more easily make the connection between general number
patterns and special cases in Part 1 of this lesson.
66
How does the 11th triangular number compare to the 10th triangular number?
It is 11 more than the 10th triangular number.
ENRICHMENT
▶ Exploring Number Patterns
Math Masters, p. 72
PARTNER
ACTIVITY
5–15 Min
(Math Masters, pp. 72 and 73)
To apply students’ understanding of general patterns, have
them work with a partner to discover some relationships among
figurate numbers—special numbers associated with geometric
figures. For example, the sum of two successive triangular
numbers is a square number; and the sum of a rectangular
number and its corresponding square number is
a triangular number.
EXTRA PRACTICE
▶ 5-Minute Math
Teaching Master
Name
Date
LESSON
31
3.
Number Patterns
continued
Describe what you notice about the sum of 2 triangular numbers that
are next to each other in the table.
Add the second square number and the second rectangular number; the
third square number and the third rectangular number. What do you notice
about the sum of a square number and its corresponding rectangular number?
The sum is a triangular number.
5.
Describe any other patterns you notice.
Sample answer: Rectangular numbers are twice
the corresponding triangular numbers.
6.
You can write triangular numbers as the sum of 4 triangular numbers when repetitions
are allowed. For example: 6 1 1 1 3
Find 3 other triangular numbers that can be written as sums of exactly
4 triangular numbers.
Sample answers:
10 1 3 3 3
15 3
21 3
3
6
3
6
6
6
Math Masters, p. 73
184
Unit 3
5–15 Min
Time
Sample answer: The sum is a square number.
4.
SMALL-GROUP
ACTIVITY
Variables, Formulas, and Graphs
To offer more practice extending and describing numeric patterns,
see 5-Minute Math, pages 157, 240, and 242.