Download Eighth Grade Math Book Teacher Edition

Step Up to the TEKS
by GF Educators, Inc.
Eighth Grade
Math Book
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Teacher:
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Teacher Edition
Copyright © 2014
w w w.StepUpT EK S.c om
Step Up to the TEKS
Eighth Grade
Math Book
LE
by GF Educators, Inc.
Table of Contents
Numerical Representations and Relationships
Classifying Numbers (8.2A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Relating Numbers with Number Lines (8.2B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Scientific Notation (8.2C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Comparing and Ordering Numbers (8.2D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Category 1 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Computations and Algebraic Relationships
Developing Foundation of Slope (8.4A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Unit Rate as Slope (8.4B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Determine the Slope and y-intercept (8.4C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Proportional Multiple Representations (8.5A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Non-Proportional Multiple Representations (8.5B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Direct Variation (8.5E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Proportional or Non-Proportional (8.5F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Identifying Functions (8.5G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Applied Proportionality (8.5H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Slope-Intercept Form of Linear Functions (8.5I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Writing Equations and Inequalities (8.8A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Interpreting Equations and Inequalities (8.8B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Solving Equations and Inequalities (8.8C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Identify Solution to Graphed Equations (8.9A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Category 2 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
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Geometry and Measurement
Similar Shapes (8.3A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Dilations (8.3B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Dilations with Unknowns (8.3C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Volume of a Cylinder (8.6A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Models with the Pythagorean Theorem (8.6C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Volume of Cylinders, Cones, Spheres (8.7A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Lateral and Total Surface Area (8.7B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Pythagorean Theorem (8.7C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Distance with the Pythagorean Theorem (8.7D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Angle Postulates (8.8D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Transformations on a Coordinate Plane (8.10A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Properties of Congruence (8.10B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Translations and Reflections (8.10C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Linear and Area Dilations (8.10D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Category 3 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
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Data Analysis and Personal Financial Literacy
Linear Relationships on a Graph (8.4A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Trend Lines (8.5D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Scatterplots (8.11A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Mean Absolute Deviation (8.11B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Cost of Credit (8.12A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Saving Money (8.12C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Simple and Compound Interest (8.12D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
College Education Plans (8.12G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Category 4 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Vocabulary Masters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
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Background
Information
Classifying Numbers
Name:
Numerical Representations and Relationships
Pages 1-4 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to represent and manipulate
numbers and expressions.
1
LE
Student Expectations - Supporting Standard
TEKS
Number and operations. The student applies mathematical process standards to
represent and use real numbers in a variety of forms. The student is expected to extend
8.2A
previous knowledge of sets and subsets using a visual representation to describe
relationships between sets of real numbers.
4th
5th
Student is expected to...
None
None
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Representing and Classifying
Numbers
Vertical Alignment
Grade
6th
6.2A classify whole numbers, integers, and rational numbers
using a visual representation such as a Venn diagram to describe
relationships between sets of numbers
7th
7.2A extend previous knowledge of sets and subsets using a visual
representation to describe relationships between sets of rational
numbers
Algebra
I
None
Vocabulary
A
whole numbers, natural numbers, integers, real numbers, rational numbers, irrational
numbers, square root
Understanding the TEKS
S
This TEKS is looking for describing the relationships between the sets of real numbers.
Students covered rational numbers in 7th grade but have not been introduced to real
numbers. Not only do the students need to create and understand the Venn Diagram, but
students need to be able to describe the different number systems and the relationship
between them.
Essential Question(s)
How are rational and irrational numbers different? Why are integers also rational
numbers? If a number is a whole number, what other categories must it also be
placed in? And why? Are all negative numbers integers? Why or why not?
8th Grade Mathematics
1
© 2014
A
Engaging
Activity
Name:
Classifying Numbers
TEKS 8.2A Supporting
Numerical Representations and Relationships
Have the students complete the visual representation using the words in the word
bank. Then have the students take the number cards and place them on the visual
representation.
1
2
3
9
2
0.3
1
3
5
π
-3
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Word Bank: Real Numbers, Rational Numbers, Irrational Numbers, Integers, Natural
Numbers, Whole Numbers.
Explain the difference between Whole Numbers and Natural Numbers
© 2014
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8th Grade Mathematics
TM
Classifying Numbers
Teaching
Teaching
Model
Model
Numerical Representations and Relationships
I can categorize
...
numbers based on their characteristics
Name:
and understand
the difference between rational and
irrational numbers. TEKS
M
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Place two numbers in each category.
LE
Fill out the Venn diagram below with the following words: Real Numbers, Irrational
Numbers, Rational Numbers, Integers, Whole Numbers
Putting the Pieces Together
A
It is important for students to understand that all the numbers that they “know” are
real numbers. Every number they will use in 8th grade is either a rational or irrational
number. When they are classifying numbers once they have decided that it is a
rational number. If it is an irrational number, it cannot be any other type of number,
except real.
S
Thinking Mathematically
How are rational and irrational numbers different?
Rational numbers are any numbers that can be written as a ratio of two integers. The majority of the
numbers that students will use in 8th grade are rational numbers.
Why are integers also rational numbers?
Integers can be written as a ratio of two integers therefore they are rational numbers, such as, -5 =
-5/1
If a number is a whole number, what other categories must it also be placed in? And why? A
whole number will always be an integer and a rational number.
Are all negative numbers integers? Why or why not? No, only whole negative numbers are
integers.
8th Grade Mathematics
3
© 2014
GP
Classifying Numbers
Guided
Guided
Practice
Practice
Numerical Representations and Relationships
Name:
TEKS 8.2A Supporting
Real Numbers
Rational Numbers
Integers
LE
Whole
Numbers
Irrational
Numbers
1 Where would you place π on the Venn diagram?
Rational numbers
Irrational numbers
Integers
Whole numbers
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A
B
C
D
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
2 Where would you place √ 3 ?
Rational numbers
Irrational numbers
Integers
Whole numbers
Look out for students
confusing negative
numbers as automatically
being integers. Integers
do not include negative
decimals, fractions or
percent.
A
A
B
C
D
7
3 Where would you place 2?
Rational numbers
Irrational numbers
Integers
Whole numbers
Look out for students
placing number as
irrational and in another
set.
S
A
B
C
D
4 Where would you place -2?
A
B
C
D
© 2014
You must show your work.
Walk around and catch,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
Rational numbers only
Irrational numbers only
Integers and rational numbers
Whole numbers, integers, and rational numbers
4
Look out for students
placing number within
real without putting the
number in one of the
sections, either rational or
irrational number.
8th Grade Mathematics
BI
Background
Information
Relating Numbers with Number Lines
Numerical Representations and Relationships
Pages 5-8 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to represent and manipulate
numbers and expressions.
1
LE
Student Expectations - Supporting Standard
TEKS
Number and operations. The student applies mathematical process standards to
represent and use real numbers in a variety of forms. The student is expected to
8.2B
approximate the value of an irrational number, including pi and square roots of numbers less
than 225, and locate that rational number approximation on a number line.
4th
5th
Student is expected to...
4.2G relate decimals to fractions that name tenths and hundredths.
None
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Representing and Relating
Numbers Using Number
Lines
Vertical Alignment
Grade
6th
7th
Algebra
I
6.2C locate, compare, and order integers and rational numbers
using a number line.
None
None
Vocabulary
A
real numbers, rational numbers, irrational numbers, pi, square roots
Understanding the TEKS
S
The TEKS specifies that the numbers students are finding the square roots of numbers less
than 225 and pi. Students need to understand that these are approximations, and can be
represented on a number line. They also need to be able to estimate the value of irrational
numbers as closer to one integer or another.
Essential Question(s)
How can you estimate the value of a square root?
How do I locate approximations on a number line?
How do I use benchmark numbers to compare and order real numbers?
8th Grade Mathematics
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Name:
Relating Numbers with Number Lines
Engaging
Activity
Numerical Representations and Relationships
TEKS 8.2B Supporting
Tape a number line on the floor in the hallway from 0 to 36. Have the students
draw a card and go and stand on where they think the value of the number is on the
number line. You may want to do this activity more than once so that all students can
participate.
√9
√12
√30
√36
√3
√4
√5
√16
√20
√25
π
2
6
4
5
0
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√2
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√1
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1
© 2014
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6
8th Grade Mathematics
I can find the estimated value of an irrational
number and Name:
locate it on a number line.
TM
Relating Numbers with Number Lines
Teaching
Model
Numerical Representations and Relationships
The number line below shows several points plotted on it.
A
B
0.75
D
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0.5
C
Which point on the number line best represents x where
C C
B B
D D
<x<
3
√4
?
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A A
6
7
Putting the Pieces Together
A
Given √ 37 , student can use the table to identify that this answer must be between
6 and 7, being closer to 6 because 37 is one more number than 36. From this
information students should be able to place √ 37 on a number line.
The other irrational number that students need to be able to identify is π; with an
approximate value of 3.14.
Students need to understand that number lines do not have to have 0 in the middle
because the arrows mean that the numbers go one forever in both directions.
S
Thinking Mathematically
How can you estimate the value of a square root?
By determining the two number the square root is between, students can begin the
estimation of the square root. Once the two numbers are determined, then you can
determine if the square root is closer to one of the numbers.
How do I locate approximations on a number line?
√ 37 is between 6 and 7. Since 37 is closer to 36 than to 49, the square root of 37
would be closer 6 than to 7.
How do I use benchmark numbers to compare and order real numbers?
Benchmark numbers allow for us to approximate quickly where to place the number.
8th Grade Mathematics
7
© 2014
GP
Relating Numbers with Number Lines
Guided
Practice
Numerical Representations and Relationships
Name:
TEKS 8.2B Supporting
1 Which point on the number line best represents √ 5 ?
K
1
L M
2
3
4
A J
C L
B K
D M
5
6
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
Look out for students
always placing the square
root numbers in the middle
between two numbers.
M
P
2 Tammy has four cards with irrational numbers on
them. She is placing them on a number line. She is
determining which card is closest to 7. Which card
should she choose?
0
Walk around and catch,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
LE
J
You must show your work.
1
2
3
4
5
6
7
8
9
A √ 23
Look out for students creating
number lines that always
include zero and encourage
them to make number lines
that reflect the constraints of
the problem.
B √ 38
A
C √ 50
D √ 67
S
3 Thomas wants to put √ 9.6 on the number line below.
Which pair of consecutive integers should Thomas
put √ 9.6 between?
0
1
2
3
4
5
6
A 2 and 3 C 8 and 9
B 3 and 4 D 9 and 10
© 2014
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8th Grade Mathematics
BI
Background
Information
Scientific Notation
Numerical Representations and Relationships
Pages 9-12 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to represent and manipulate
numbers and expressions.
1
LE
Student Expectations - Supporting Standard
TEKS
Number and operations. The student applies mathematical process standards to
represent and use real numbers in a variety of forms. The student is expected to convert
8.2C
between standard decimal notation and scientific notation.
4th
Student is expected to...
4.2A know the value of each place-value position as 10 times the
position to the right and as one-tenth of the value of the place to its
left
4.2B represent the value of the digit in whole numbers through
1,000,000,000 and decimals to the hundredths using expanded
notation and numerals
4.2E represent decimals, including tenths and hundredths, using
concrete and visual models and money
M
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Representing and Classifying
Numbers
Vertical Alignment
Grade
5th
5.2A represent the value of the digit in decimals through the
thousandths using expanded notation and numerals
6th
None
7th
None
Algebra None
I
Vocabulary
A
standard decimal notation, scientific notation, power
Understanding the TEKS
S
Scientific notation is used to express very large numbers and very small numbers. Scientific
notation is not used to express negative numbers.
When converting to scientific notation it is important for students to understand that the
value of the number in front of the decimal point must be greater than or equal to 1 and
less than 10. In addition to converting between the two number notations, students should
be asked to compare within the scientific notation and make simple calculations.
Essential Question(s)
Why is scientific notation used? What types of fields of study use scientific notation
the most? What values does the constant have to be between in order to be in
scientific notation? What does a negative in the exponent mean?
8th Grade Mathematics
9
© 2014
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Name:
Scientific Notation
Engaging
Activity
TEKS 8.2C Supporting
Numerical Representations and Relationships
Word Notation
1 Astronomical Unit
149 million, 598
thousand km
Integer
Speed of Light
3.0 x 108 m/sec
Nanosecond
0.000000001 sec
Cells in the Human
Body
M
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1.0 x 1014
Distance from Earth
to Sun
Diameter of a grain
of sand
Scientific Notation
LE
Real-Life Examples
93,000,000
24 ten thousandths
5.88 x 1012
Density of oxygen
1.332 x 10-3
S
A
Miles in a light-year
© 2014
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8th Grade Mathematics
I can convert between standard and scientific
notation.
TM
Scientific Notation
Teaching
Model
Numerical Representations and Relationships
The diameter of a human hair is about 0.00067 inches in length. How is this length
expressed in scientific notation?
A 6.7 × 104 in.
C6.7 × 10–4 in.
M
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D67 × 10–4 in.
LE
B 67 × 104 in.
Putting the Pieces Together
A
Scientific notation is understanding the rules and counting. One digit is allowed
before the decimal point, so the start is to place the number with that format. Next,
count how far the decimal needs to be moved, this is the exponent for the problem.
If the number is a large number the exponent will be positive and if the number is
small the exponent will be negative.
Thinking Mathematically
S
Thinking Mathematically
Why is scientific notation used? Scientific notation is used when number are
extremes, really large or really small so that the numbers can be represented more
efficiently.
What types of fields of study use scientific notation the most? The sciences
such as biology and astronomy use scientific notation the most because they work
with really small numbers and really large numbers.
What values does the constant have to be between in order to be in scientific
notation? The constant needs to be greater than or equal to 1 and less that 10.
What does a negative in the exponent mean? When there is a negative in
the exponent, it means the number is very small, it does not mean the number is
negative.
8th Grade Mathematics
11
© 2014
GP
Scientific Notation
Guided
Practice
Numerical Representations and Relationships
Name:
1 What is the correct way to express 0.00000499 in
scientific notation?
TEKS 8.2C Supporting
You must show your work.
A 35,000,000,000 mi
When converting between
standard form (rational
numbers) and scientific
notation, students need to
understand magnitude of
multiplying or dividing by a
power of 10.
A scientific number is always
written as a number between
1 and 10 times a power of
10.
M
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B 3,500,000,000 mi
LE
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
______________________
reinforce errors.
If all of the students are
having the same difficulty,
2 If the distance from the sun to Pluto is approximately then it is time to go back into
class instruction.
3.5 × 109 miles, what is this distance expressed in
standard notation?
C 0.0000000035 mi
D 0.00000000035 mi
3 Which is another way to express 2.31 × 105 miles?
A
HINT: Graphing calculators
will convert numbers between
standard and scientific
notation.
______________________
S
4 A computer can perform 9 × 109 arithmetic
operations per second. How many operations could
be performed in one minute?
A 54 × 1011
B 5.4 × 1011
C 9 × 108
D 9 × 1010
© 2014
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8th Grade Mathematics
BI
Background
Information
Comparing and Ordering Numbers
Numerical Representations and Relationships
Pages 13-18 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to represent and manipulate
numbers and expressions.
1
LE
Student Expectations - Readiness Standard
TEKS
Number and operations. The student applies mathematical process standards to
represent and use real numbers in a variety of forms. The student is expected to order a set
8.2D
of real numbers arising from mathematical and real-world contexts.
4th
Student is expected to...
4.2B represent the value of the digit in whole numbers through
1,000,000,000 and decimals to the hundredths using expanded
notation and numerals.
4.2F compare and order decimals using concrete and visual models
to the hundredths.
5th
5.2B compare and order two decimals to thousandths and represent
comparisons using the symbols >, <, or =.
6th
6.2D order a set of rational numbers arising from mathematical and
real-world contexts.
M
P
Comparing and Ordering
Numbers
Vertical Alignment
Grade
7th
None
Algebra None
I
Vocabulary
A
order, place value
Understanding the TEKS
S
It is important to understand that this TEKS is about ordering real number which includes
rational and irrational numbers. The second part of this TEKS is that both real-world
problems and numbers are needed to be used for the students to order. With students
using scientific calculators in 8th grade, expect some of the numbers used to be very
difficult. Practice in the classroom entering square roots of various forms in the calculator is
a must.
Essential Question(s)
Why does order matter?
Why is it helpful to write numbers in different ways?
How do you compare and order real numbers?
8th Grade Mathematics
13
© 2014
A
Engaging
Activity
Name:
Comparing and Ordering Numbers
Numerical Representations and Relationships
TEKS 8.2D Readiness
Maria would like to paint a design on her wall. However, she doesn’t have a lot of
paint. She can paint her design in all squares that have an area of √ 12 ft2 or all in
circles that have an area of 1.2π ft2, or all in triangles that have an area of 3.4 ft2.
LE
1 How do you determine if the value of √ 12 ft2 is greater than or less than 3.4 ft2?
M
P
2 How do you determine if the value of 1.2π ft2 is greater than or less than 3.4 ft2?
3 Since Maria doesn’t have a lot of paint which shape should she choose?
A
4 What shape would be Maria’s last choice to paint on her wall?
S
5 How much paint does Maria need if she wants to paint 5 triangles on her wall?
© 2014
14
8th Grade Mathematics
I can put real numbers in order.
TM
Comparing and Ordering Numbers
Teaching
Model
Numerical Representations and Relationships
Name:
The points P, Q, R, S can be arranged in order from greatest to least. The values for
the points are given below.
P = 72%
Q = √ .875
5
R = -7
S=
8
9
A R, P, S, Q
B R, S, Q, P
C P, Q, S, R
M
P
D Q, S, P, R
LE
Which answer choice has these points in correct order?
Putting the Pieces Together
A
When ordering numbers it is important to convert the numbers into the same form.
Frequently converting to decimals is the easiest method to order numbers. When we
are doing square roots the number used is an approximation. Students should be
allowed to use calculators to help with the conversion.
P = 72% = .72
Q = √ .875 = .935
R = -5/7 = -.71
S = 8/9 = .888
So the smallest is R, then P, then S and lastly Q.
S
Thinking Mathematically
Why does order matter? In mathematics and real world situations, putting things in
order creates a logical situation to analyze.
Why is it helpful to write numbers in different ways? Many times different
forms of numbers make the problem easier to do. Not all fractions convert easily to
decimals, so it is important to work with all types of numbers.
How do you compare and order real numbers? Converting number to the same
format allows for number to be ordered easily. When comparing numbers visual
representations, such as a number line, can be helpful.
8th Grade Mathematics
15
© 2014
GP
Comparing and Ordering Numbers
Guided
Practice
Numerical Representations and Relationships
Name:
1 Which rational number is between √ 10 and √ 22 ?
A10.3
B π
D 5.01
2 Mrs. Munoz kept track of the portion of problems
missed by 5 students in her classroom.
Portion
Missed
Tommy
22%
John
2
7
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
Hint: Use a calculator to
approximate the square root
of the numbers and place
them on a number line.
M
P
Student
You must show your work.
LE
C3.3
TEKS 8.2D Readiness
Doug
35%
Staci
1
4
Kim
1
4
List the students in order from highest grade to
lowest grade.
S
A
3 De’andre, Will, Brice, and Derion scored an average
70 points a game during last season. Derion scored
2
5 of those points, Brice scored 10% of the points and
De’andre averaged 4 fewer points than three times
the points Brice averaged. Which list shows the
athletes in order from greatest points per game to
fewest points per game?
Hint: convert each of the
numbers to decimals and
then rank the numbers from
the largest to smallest. Use
a calculator in converting the
factions and square roots.
You want to calculate to
decimals because percent are
easy to convert to decimals
and square roots are always
approximations.
Hint: Make a table of the
four player and convert their
scores to decimals.
A Derion, Will, De’andre, Brice
B Will, Brice, De’andre, Derion
C De’andre, Derion, Will, Brice
D Brice, Will, Derion, De’andre
© 2014
16
8th Grade Mathematics
AK
Answer Key
Category 1 Answer Key
Numerical Representations and Relationships
TEKS 8.2C
Guided Practice pg 10
1 4.99×10-6
2 B
3231,000
4 B
Independent Practice
pg 3-4
1 B
2D
3C
4 A
5 A
Independent Practice
pg 11-12
1C
2 7.03×108
3C
4 B
5 1.2×108
6 A
7 9.3×109
8 7.8×10-4
9C
10 B
Independent Practice
pg 15-18
1C
π
2 ⅕ , 22%, 0.25, 8, √ 35
3 B
4D
5D
6 Raiders, Mavericks,
Warriors, Tigers
7 B
8D
9 12.97, 13.08, 13.09,
13.22, 13.37
10 B
11C
12 B
13D
M
P
TEKS 8.2B
Guided Practice pg 6
1D
2C
3 B
TEKS 8.2D
Guided Practice pg 14
1C
2 Tommy, Staci, Doug,
Kim, John
3 A
LE
TEKS 8.2A
Guided Practice pg 2
1 B
2 B
3 A
4C
S
A
Independent Practice
pg 7-8
1 B
2D
3C
4 A
5 A
6D
8th Grade Mathematics
17
© 2014
BI
Background
Information
Developing Foundation of Slope
Computations and Algebraic Relationships
Pages 19-22 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to perform operations and represent
algebraic relationships.
2
LE
Student Expectations - Supporting Standard
TEKS
Proportionality. The student applies mathematical process standards to explain
proportional and non-proportional relationships involving slope. The student is expected
8.4A
to use similar right triangles to develop an understanding that slope, m, given as the rate
comparing the change in y-values to the change in x-values, (y2 – y1)/(x2 – x1), is the same
for any two points (x1, y1) and (x2, y2) on the same line.
4th
5th
Student is expected to...
None
None
M
P
Developing Foundations of
Slope
Vertical Alignment
Grade
6th
7th
None
None
A.3A determine the slope of a line given a table of values, a graph,
Algebra two points on the line, and an equation written in various forms,
I
including y = mx + b, Ax + By = C, and y – y1 = m(x – x1).
Vocabulary
A
slope, rate of change, similar right triangles
Understanding the TEKS
S
This TEKS is very explicit in how the concept of slope is to be addressed. Using similar right triangles,
the formula is developed. The formula should not be introduced prior the student graphing similar
triangles on a graph. It is not uncommon to graph two points on a coordinate plane and count for
the rise and the run between the points. This TEKS requires that the triangle is drawn to connect
the points. As the concept of rise/run is discussed, this should also be explained as the change in
y compared to the change in x. Finally the formula can be an extended to the formula. It is not
uncommon for students to confuse the formula, so it important they have plenty of time to practice
counting on a graph prior to the formula is introduced. Students need to be able to find y1, y2, x1, and
x2 on the graph and understand that the difference in the y-values divided by the difference in the
x-values is constant for any two points on the line.
Essential Question(s)
How do the lengths of the two legs of a triangle relate to the slope of a line?
How do similar triangles relate to the slope of a line?
How can you graph proportional relationships on a coordinate plane?
© 2014
18
8th Grade Mathematics
A
Engaging
Activity
Name:
Developing Foundation of Slope
Computations and Algebraic Relationships
TEKS 8.4A Supporting
Directions: Match the correct graph with the correct proportion that determines the
similarity of the right triangles. Use the slope of a line formula to determine that the
change in the y values to the change in the x values, are the same for any two points on
the same line. Then find the slope of each line.
y
A
9
8
4
3
2
1
1 2 3 4 5 6 7 8 9
x
-2
-3
-4
-5
-6
=
-3 - -5
6 - 8
LE
6
5
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-3 - 9
6 - -6
7
Slope = ______________
-7
-8
Matches Letter: ____
-9
y
B
M
P
9
8
0 - 3
=
3 - 7
7
6
5
0 - -6
3 - -5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
1 2 3 4 5 6 7 8 9
x
-3
-4
-5
-6
-7
-8
-9
Matches Letter: ____
y
9
8
Slope = ______________
C
5 - 1
3 - -9
7
6
5
=
5 - 7
3 - 9
A
4
3
2
1
1 2 3 4 5 6 7 8 9
x
Slope = ______________
Matches Letter: ____
S
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
y
D
9
8
7
6
5
1 - 7
4 - -5
=
1 - -1
4 - 7
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
1 2 3 4 5 6 7 8 9
8th Grade Mathematics
x
Slope = ______________
Matches Letter: ____
19
© 2014
I can find the slope of a line by using similar
Name:
right triangles.
TM
Developing Foundation of Slope
Teaching
Model
Computations and Algebraic Relationships
In the figure shown, ΔABC and ΔBDE are similar.
y
9
8
7
6
5
LE
4
3
2
A
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
D
B
C
1 2 3 4 5 6 7 8 9
x
E
M
P
You can use the properties of similar triangles to show the ratios of the change in
vertical length to the change in horizontal length are equal. Write a proportion
comparing the vertical change to the horizontal change for each of the similar
triangles shown above. Then determine the numeric value.
Putting the Pieces Together
A
You can use the properties of similar triangles to show the ratios of the change in
vertical length to the change in horizontal length are equal.
Write a proportion comparing the vertical change to the horizontal change for each of
the similar triangles shown above. Then determine the numeric value.
S
Thinking Mathematically
How do the lengths of the two legs of a triangle relate to the slope of a line? The ratio
of the lengths of the two legs is proportional to the slope the hypotenuse of a right
triangle.
How do similar triangles relate to the slope of a line? The ratio of the vertical change
and the horizontal change of the hypotenuse is proportional to the slope.
How can you graph proportional relationships on a coordinate plane? The graph of a
proportional relationship will always pass through the origin.
© 2014
20
8th Grade Mathematics
GP
Developing Foundation of Slope
Guided
Practice
Computations and Algebraic Relationships
Name:
You must show your work.
y
9
8
7
6
5
1
1 2 3 4 5 6 7 8 9
-2
-3
-4
-5
-6
-7
-8
-9
A
B
2
3,
x
Find the points of the triangle
that intersect the line
M
P
1 What is the slope of the line?
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
LE
4
3
2
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
TEKS 8.4A Supporting
because the leg of one triangle is 2 and the leg of
the other triangle is 3.
5-2
5-2
=
2-0
2-0
=
3
3
=
2
2
=1
Calculate proportional
because the leg of one triangle is 3 and the leg of relationships
the other triangle is 2.
3
2,
C 1, because
2
2
=
3
3
=1
Be sure the students
understand the ratio of
vertical change to the
horizontal change. Then
compare the two ratios.
D 0, because 2 – 2 = 0 and 3 – 3 = 0
y
A
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1 2 3 4 5 6 7 8 9
x
S
-2
-3
-4
-5
-6
-7
-8
-9
2 What is the slope of the line shown by the similar
triangles in the graph above?
A
1
2,
B 2,
C
D
3
2,
2
3,
2
3
1
4 = 6 = 2
4
6
because 2 = 3 = 2
because
because 6 – 3 = 3 and 4 – 2 = 2.
because 4 – 2 = 2 and 6 – 3 = 3.
8th Grade Mathematics
21
© 2014
BI
Background
Information
Unit Rate as Slope
Computations and Algebraic Relationships
Pages 23-28 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to perform operations and represent
algebraic relationships.
2
LE
Student Expectations - Readiness Standard
TEKS
Proportionality. The student applies mathematical process standards to explain
proportional and non-proportional relationships involving slope. The student is expected
8.4B
to graph proportional relationships, interpreting the unit rate as the slope of the line that
models the relationship.
4th
5th
Student is expected to...
None
None
M
P
Developing Foundations of
Slope
Vertical Alignment
Grade
6th
7th
None
None
A.3B calculate the rate of change of a linear function represented
Algebra tabularly, graphically, or algebraically in context of mathematical
I
and real-world problems.
Vocabulary
A
proportional, non-proportional, unit rate
Understanding the TEKS
S
When working with proportional relationships, students need to understand that origin is
always included in the relationship. The slope of the line is related to the unit rate, but may
not be the unit rate because the slope can be fractional and the unit rate is for one whole
unit.
It is important for students to understand that the line is a model of the data in a problem
situation.
It may be beneficial for students to graph from a table of data.
Essential Question(s)
How do you find slope or the rate of change from a graph?
How do you know data is proportional on a coordinate plane?
© 2014
22
8th Grade Mathematics
A
Name:
Unit Rate as Slope
Engaging
Activity
TEKS 8.4B Readiness
Computations and Algebraic Relationships
Read the situation below and determine the graph that has the same unit rate. In
order to find the unit rate, k, you will need to ________________ the dependent
variable, ____, by the independent variable, ____.
LE
1 Rebecca can read her library book at a rate of 20 pages in 15 minutes. Determine
which graph would display this information with the same unit rate.
Y
Y
10
10
9
9
8
8
7
A
7
6
B
5
4
6
5
4
3
2
M
P
3
2
1
1
1
2
3
4 5
6 7
8 9 10
X
1
2
3
4 5
6 7
8 9 10
X
2 Look at the graph below and determine which situation below matches the unit rate
given in the situation.
A At the movies popcorn cost $8 a bucket.
Y
A
B A bike is traveling 4 miles every 2 hours.
50
45
40
C Aiden receives $20 for 10 chores.
35
S
D Kristen spent $40 on 10 pizzas at the grocery
store.
30
25
20
15
10
5
0
8th Grade Mathematics
23
1
2
3
4 5
6 7
8 9 10
© 2014
X
A
Engaging
Activity
Name:
Unit Rate as Slope
TEKS 8.4B Readiness
Computations and Algebraic Relationships
Every month you owe $145 for your phone service that offers 4 gigs of data. Fill in
the graph below to display the amount of money paid for your phone overtime.
Title of graph
LE
Y
9
8
7
6
5
4
M
P
Dependent Variable
10
3
2
1
0
1
2
3
4 5
6 7
8 9 10
X
Independent Variable
A
What is the slope,k, of the line? ___________________________________
Write an equation in the form of y = kx to represent this situation.
S
___________________________
© 2014
24
8th Grade Mathematics
I can graph a proportional relationship.
TM
Unit Rate as Slope
Teaching
Model
Computations and Algebraic Relationships
Name:
Travis Middle School is having a dance on Friday night. The tickets costs $6.
The equation y = 6x can be used to find the total cost y for any number of tickets x.
Find the rate of change.
y
What do you know?
42
Total Cost ($)
LE
36
What do you need to find?
30
24
18
12
6
0
2
4
6
8
10
12
14
x
M
P
Number of Dance Tickets
Putting the Pieces Together
Use your calculator to help.
Enter the equation. Press Y = 6 X, T, 0, n .
Graph the equation. Be sure you have a standard window.
Press 2nd TABLE
Choose any two points from the table
change in total cost
A
change in number of tickets
=
$
-
tickets
-
=
tickets
So, the rate of change or unit rate is
S
Thinking Mathematically
How do you find slope or the rate of change from a graph? You need to find two points
on a line and calculate the vertical change divided by the horizontal change
How do you know data is proportional on a coordinate plane? When any linear
proportional data is graphed, it will go through the origin.
8th Grade Mathematics
25
© 2014
GP
Unit Rate as Slope
Guided
Practice
Computations and Algebraic Relationships
Name:
You must show your work.
m
100
90
80
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
70
60
50
40
30
LE
Distance (miles)
TEKS 8.4B Readiness
20
10
0
t
Time (hours)
1 Which of the following best describes the meaning of
the slope of the line representing this situation?
B
C
D
M
P
A
When interpreting data from
a graph the students must
The cruise ship travels at a speed of about 3 miles
first check the data at 1 on
per hour.
the x-axis to determine the y
The cruise ship travels at a speed of about 12 miles
point. This will give you the
per hour.
unit rate or slope of the line.
1
The cruise ship travels at a speed of about 1 2
miles per hour.
The cruise ship travels at a speed of about 15 miles
per hour.
2 Look at the graph below.
y
14
A
12
10
8
6
4
S
2
0
10
20
30
40
50
60
70
x
Which is the best interpretation of the graph?
A Danny drinks 10 glasses of water every day.
B Sammy uses 4 packages of nails for every 20 feet
of boards he puts down.
C Tommy gets 2 hits for every 5 at-bats during the
baseball season.
D Bobby makes 8 free throws for every 20 attempts
in basketball.
© 2014
26
8th Grade Mathematics
BI
Background
Information
Determine the Slope and y-intercept
Computations and Algebraic Relationships
Pages 29-34 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to perform operations and represent
algebraic relationships.
2
LE
Student Expectations - Readiness Standard
TEKS
Proportionality. The student applies mathematical process standards to explain
proportional and non-proportional relationships involving slope. The student is expected to
8.4C
use data from a table or graph to determine the rate of change or slope and y-intercept in
mathematical and real-world problems.
4th
5th
6th
Student is expected to...
None
None
None
M
P
Developing Foundations of
Slope
Vertical Alignment
Grade
7th
None
A.3B calculate the rate of change of a linear function represented
tabularly, graphically, or algebraically in context of mathematical
and real-world problems.
Algebra
I
A.3C graph linear functions on the coordinate plane and identify
key features, including x-intercept, y-intercept, zeros, and slope, in
mathematical and real-world problems.
Vocabulary
A
domain, range, rate of change, slope, y-intercept
Understanding the TEKS
S
Students often believe that if there is a constant rate of change, then the relationship between
the data point must be proportional. The difference between proportional and non-proportional
relationships is not the constant rate of change, but the proportional relationship must go through the
point (0, 0) and have a constant rate of change.
Students will need to determine proportional and non-proportional from a table and a graph. When
looking at a graph, students should identify the y-intercept (the value of y when x=0) to determine
proportional or non-proportion first, then look for the constant rate of change.
When working from a table, students can graph the points, or they can look for the common
differences to determine the slope. If the table does not include the x-intercept, students need to
determine the y-intercept.
Essential Question(s)
How do you find slope or the rate of change from a graph or table?
How is slope interpreted in the real world?
How is the y-intercept represented in the real world?
8th Grade Mathematics
27
© 2014
A
Engaging
Activity
Name:
Determine the Slope and y-intercept
Computations and Algebraic Relationships
TEKS 8.4C Readiness
f(x)
4
7
10
13
16
x
1
3
5
7
f(x)
11
10
9
8
S
A
M
P
x
-2
-1
0
1
2
LE
Use the information provided to determine the slope of each linear
function.
Given
Slope
© 2014
28
8th Grade Mathematics
A
Name:
Determine the Slope and y-intercept
Engaging
Activity
Computations and Algebraic Relationships
TEKS 8.4C Readiness
Cut out the cards below and match the equation, graph and verbal description.
Graph
Equation
Verbal Description
120
100
80
LE
140
y = 10x + 20
60
40
20
0
2
3
4
5
6
7
M
P
1
Lyn ordered several reams
of paper. Each ream costs
$10 with a shipping fee of
$20.
140
120
100
80
60
40
y = 5x + 10
Lisa is buying school
supplies. She buys several
binders at $5 each with a
backpack that costs $10.
A
20
y = 20x + 10
Darrell is planning a party
and has calculated that it
would cost $20 per person
for food with a $10 cost for
the drinks.
0
1
2
3
4
5
6
7
70
S
60
50
40
30
20
10
0
1
2
3
4
8th Grade Mathematics
5
6
7
29
© 2014
A
Name:
Determine the Slope and y-intercept
Engaging
Activity
Computations and Algebraic Relationships
Graph
TEKS 8.4C Readiness
Equation
Verbal Description
70
60
40
y = 10x + 5
30
20
10
1
7
6
5
4
3
2
1
1
3
4
5
6
7
2
3
4
5
6
y = 0.5x + 1.5
A Taxi cab ride in New York
City cost $0.50 a mile with
an initial fee of $1.50.
y = 1.5x + 0.5
A taxi cab ride in London
England cost $0.50 as an
initial fee and $1.50 a mile.
7
A
0
2
M
P
0
Wendy is buying gifts for
her friends. She buys
scarves at $10 each for all
her friends, but one. Her
gift for her last friend cost
$5.
LE
50
7
6
S
5
4
3
2
1
0
© 2014
1
2
3
4
5
6
7
30
8th Grade Mathematics
I can find the slope and y-intercept from a table or
graph
TM
Determine the Slope and y-intercept
Teaching
Model
Computations and Algebraic Relationships
x
y
-2
-8
-1
-4
2
8
4
16
5
20
6
24
B
x
y
-8
-2
-4
-1
8
2
16
4
20
5
24
6
C
x
y
-2
-6
-1
-5
2
6
4
8
5
9
6
10
D
x
y
-6
-2
-5
-1
6
2
8
4
9
5
10
6
M
P
A
LE
Each table below lists ordered pairs of numbers. Which table identifies points
contained on a line with a slope of 4?
Putting the Pieces Together
When determining the slope from a table, you must find the ratio of the differences in
your y data and the differences in your x data. This ratio must be constant for all
data.
A
When finding the y-intercept, you will find the y value when the x value is equal to 0.
S
Thinking Mathematically
How do you find slope or the rate of change from a graph or table? You need to find a
constant ratio of the difference in y values divided by the difference in x values.
How is slope interpreted in the real world? Slope in the rate of change.
How is the y-intercept represented in the real world? The y-intercept is you starting
point.
8th Grade Mathematics
31
© 2014
GP
Determine the Slope and y-intercept
Guided
Practice
Computations and Algebraic Relationships
Name:
1 What is the slope of the linear function shown in the
graph?
Y
10
9
8
7
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 X
5
A –4 5
4
C
4
D
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
4
5
When determining slope from
a graph, you find two integer
ordered pairs to calculate the
vertical change divided by the
horizontal change.
To determine the y-intercept
on a graph, find the
interception point of on the
y-axis.
M
P
B –5 You must show your work.
LE
6
TEKS 8.4C Readiness
2 What is the y–intercept of the line graphed below?
y
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
1 2 3 4 5 6 7 8 9
x
-3
-4
-5
-6
-7
A
-8
-9
A (–7, –7)C (–7, 0)
B (0, 0)D (0, –7)
S
3 What is the y-intercept of the line containing the
points shown below?
© 2014
x
y
-6
-8
-3
-7
0
-6
3
-5
32
8th Grade Mathematics
BI
Background
Information
Proportional Multiple Representations
Computations and Algebraic Relationships
Pages 35-38 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to perform operations and represent
algebraic relationships.
2
LE
Student Expectations - Supporting Standard
TEKS
Proportionality. The student applies mathematical process standards to use proportional
and non-proportional relationships to develop foundational concepts of functions. The
8.5A
student is expected to represent linear proportional situations with tables, graphs, and
equations in the form of y = kx.
4th
5th
Student is expected to...
4.5B represent problems using an input-output table and numerical
expressions to generate a number pattern that follows a given rule
representing the relationship of the values in the resulting sequence
and their position in the sequence.
5.4C generate a numerical pattern when given a rule in the form y
= ax or y = x + a and graph.
M
P
Applying Multiple
Representations for
Foundations of Functions
Vertical Alignment
Grade
6th
6.4A compare two rules verbally, numerically, graphically, and
symbolically in the form of y = ax or y = x + a in order to
differentiate between additive and multiplicative relationships.
7th
7.4A represent constant rates of change in mathematical and realworld problems given pictorial, tabular, verbal, numeric, graphical,
and algebraic representations, including d = rt.
Algebra A.2D write and solve equations involving direct variation.
I
Vocabulary
A
linear, proportional, y = kx
Understanding the TEKS
S
Given a problem situation, student should be able to determine the constant rate of change,
which can be translated to the slope of the equation. Students should be able to determine
this rate from the table and the graph. When looking at linear proportional situations, it is
often a good idea to have the students create a four corners graphic organizer.
Essential Question(s)
How can you represent linear relationships in various forms?
How can you model proportional relationships between quantities?
How are table, graphs, and equations related?
8th Grade Mathematics
33
© 2014
A
Proportional Multiple Representations
Engaging
Activity
Computations and Algebraic Relationships
Name:
TEKS 8.5A Supporting
Notes:
Linear proportional situations are wrote in the form of y = kx.
.
LE
y
To determine k, the constant rate, you need to divide x
Directions: Determine k, the constant rate, for each graph and table. Then write the
equation in the form of y = kx.
Amir bought 12 light bulbs and paid $15 for ten light bulbs. Fill in the table below to
that match this situation and write an equation in the form of y = kx.
Light
Money
bulbs (x) Spent (y)
M
P
Equation:
4
5
11
17
Jacob printed pictures and paid $56 for four 8 inch by 10 inch photos. Fill in the table
for this situation and graph the points in the table. Then write the equation for this
information in the form of y = kx. Make sure to label the graph.
A
Y
(x)
(y)
50
45
40
Equation:
1
35
30
4
S
25
20
15
7
10
5
12
X
© 2014
34
8th Grade Mathematics
TM
Proportional Multiple Representations
Teaching
Model
Computations and Algebraic Relationships
I can use tables, graphs and equations to represent
Name:
linear proportional
relationships.
Jane has a job babysitting the girl next door. She makes $6.50 per hour.
The equation t = 6.50h can be used to find the total amount t she makes for each hour
h that she works. Represent this as a table and a graph.
t
h
M
P
h
LE
t
Putting the Pieces Together
A
When representing a proportional relationship, you will want to choose values for x
and y to create a table of ordered pairs that satisfy the equation. You can graph the
ordered pairs to create a line. You will then be able to convert between the table,
graph or equation depending on what information is given. If given a graph first,
create the table of ordered pairs from the graph. You can find the slope from the
graph to write the equation.
S
Thinking Mathematically
How can you represent linear relationships in various forms? A linear relationship can
be represented in an equation, table, graph or verbal form.
How can you model proportional relationships between quantities? Models of
proportional relationships can be done in tables and graphs.
How are tables, graphs, and equations related? All of these forms can represent the
same set of data.
8th Grade Mathematics
35
© 2014
GP
Proportional Multiple Representations
Guided
Practice
Computations and Algebraic Relationships
TEKS 8.5A Supporting
You must show your work.
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
LE
1 An equation can be used to find the total distance
driven by Todd during his vacation. Using the table
below, find the equation that best represents y, the
total miles driven, as a function of x, the amount of
time spent driving.
Name:
Amount of
Time (x)
Number of
Miles (y)
2.5
150
4
240
5.1
306
In order to write the
equation, you must
first calculate the slope or
rate of change of the data.
You will find the ratio of the
difference in the y values
divided by the difference in
the x values.
A y = 0.125x
M
P
B x = 60y
C x = 0.125y
D y = 60x
CAUTION: Make sure the
students are using the y
values as the numerator.
A
2 Ginger works at the local putt-putt golf course. Her
weekly wages, y, are $7.25 per hour, x, she works.
Which equation best represents this relationship?
A yx = 7.25
C x = 7.25y
B y = 7.25x
D
CAUTION: For a proportional
relationship, you must always
use the form y = kx.
S
x
= 7.25
y
© 2014
36
8th Grade Mathematics
BI
Background
Information
Non-Proportional Multiple Representations
Name:
Computations and Algebraic Relationships
Pages 39-42 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to perform operations and represent
algebraic relationships.
2
LE
Student Expectations - Supporting Standard
TEKS
Proportionality. The student applies mathematical process standards to use proportional
and non-proportional relationships to develop foundational concepts of functions. The
8.5B
student is expected to represent linear non-proportional situations with tables, graphs, and
equations in the form of y = mx + b, where b ≠ 0.
4th
5th
Student is expected to...
None
None
M
P
Applying Multiple
Representations for
Foundations of Functions
Vertical Alignment
Grade
6th
6.6A identify independent and dependent quantities from tables and
graphs.
7th
7.7A represent linear relationships using verbal descriptions, tables,
graphs, and equations that simplify to the form y = mx + b.
A.2B write linear equations in two variables in various forms,
Algebra including y = mx + b, Ax + By = C, and y – y1 = m(x – x1), given
I
one point and the slope and given two points.
Vocabulary
A
non-proportional, y = mx + b
Understanding the TEKS
S
When graphing a non-proportional linear equation, it is important to identify the y-intercept
to have a starting place to graph the line. Using the same techniques as working with
proportional relationships should be used when working with non-proportional linear
situation once the y-intercept is identified.
Essential Question(s)
How can you represent linear relationships in various forms?
How can you model non-proportional relationships between quantities?
What makes a function non-proportional?
8th Grade Mathematics
37
© 2014
A
Engaging
Activity
Non-Proportional Multiple Representations
Name:
TEKS 8.5B Supporting
Computations and Algebraic Relationships
Activity for Multiple Representations
Complete the table.
LE
Use toothpicks to continue the pattern represented below:
y (number of
toothpicks)
3
5
M
P
x (number of
triangles)
1
2
3
4
5
x
S
A
Graph the ordered pairs.
Write an equation that represents the data.
____________________
© 2014
38
8th Grade Mathematics
TM
Non-Proportional Multiple Representations
Teaching
Model
Computations and Algebraic Relationships
I can use tables, graphs and equations to represent
linear non-proportional relationships.
Paul cuts the yard. He earns $3.50 per hour plus an extra $10 to trim the hedges.
Write an equation to determine t, the total earnings, Paul will earn if he works for h,
hours.
Make a table to determine his total earnings after working 4, 5, 6 and 7 hours.
$40
Y
$3.50
5
$3.50
6
$3.50
7
$3.50
$36
$34
$32
$30
$28
$26
$24
$22
$20
M
P
4
Total Earnings (t)
Total Hours Worked
Hours Earnings
Total
(h)
per hour Earnings (t)
LE
$38
Putting the Pieces Together
Hours(h)
X
A
When representing a non-proportional relationship, you will want to choose values for
x and y to create a table of ordered pairs that satisfy the equation. You can graph
the ordered pairs to create a line. You will then be able to convert between the table,
graph or equation depending on what information is given. If given a graph first,
create the table of ordered pairs from the graph. You can find the slope and the
y-intercept from the graph to write the equation.
S
Thinking Mathematically
How can you represent linear relationships in various forms? A linear relationship can
be represented in an equation, table, graph or verbal form.
How can you model non-proportional relationships between quantities? You can
model a non-proportional relationship using tables or graphs.
What makes a function non-proportional? The ratio of the difference in y values
divided by the difference in x values will not be constant.
The graph will not intersect the origin.
8th Grade Mathematics
39
© 2014
GP
Non-Proportional Multiple Representations
Guided
Practice
Computations and Algebraic Relationships
1 Which graph best represents the
function y = –0.75x + 3?
y
5
4
3
2
4
3
2
1
-5 -4 -3 -2 -1
-1
1 2 3 4 5
-2
-3
-4
-5
C
x
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
y
y
5
5
4
3
2
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1 2 3 4 5
D
x
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
x
3 Principal Theisen wanted to personally
meet with every senior student prior
to graduation. The table shows the
remaining number of students, S, Mrs.
Theisen still needs to meet after each
week.
1 2 3 4 5
x
Number of
weeks, w
Students
remaining, S
1
385
2
378
3
371
4
364
5
357
Which function can be used to describe
this relationship?
M
P
B
1 2 3 4 5
TEKS 8.5B Supporting
LE
A
y
5
Name:
A S = 392 – 7w
B S = 385 – 7w
2 Which table identifies points on the line
produced by the equation y = 5x + 7?
x
y
-4 -13
-1
2
1
12
6
32
9
47
x
y
C
y
1
12
5
32
6
37
8
47
9
52
x
y
A
A
x
-5 -18
-5 -18
-4 -13
-3 -8
-1
2
1
13
2
22
S
B
© 2014
D
-1
2
2
12
4
27
C S = 385 + 7w
D S = 378 + 7w
You must show your work.
Walk around and catch mistakes, then
correct errors as they happen. Do not give
the student the opportunity to reinforce
errors. If all of the students are having the
same difficulty, then it is time to go back
into class instruction.
Hint: Use the graphing calculator to graph
the equation of the line. Toggle between
the graph and the table to find the match.
40
8th Grade Mathematics
BI
Background
Information
Direct Variation
Computations and Algebraic Relationships
Pages 43-46 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to perform operations and represent
algebraic relationships.
2
LE
Student Expectations - Supporting Standard
TEKS
Proportionality. The student applies mathematical process standards to use proportional
and non-proportional relationships to develop foundational concepts of functions. The
8.5E
student is expected to solve problems involving direct variation.
Student is expected to...
4th
4.5B represent problems using an input-output table and numerical
expressions to generate a number pattern that follows a given rule
representing the relationship of the values in the resulting sequence
and their position in the sequence.
5th
5.4D recognize the difference between additive and multiplicative
numerical patterns given in a table or graph
6th
6.4A compare two rules verbally, numerically, graphically, and
symbolically in the form of y = ax or y = x + a in order to
differentiate between additive and multiplicative relationships.
M
P
Applying Multiple
Representations for
Foundations of Functions
Vertical Alignment
Grade
7th
7.4C determine the constant of proportionality (k = y/x) within
mathematical and real-world problems.
Algebra A.2D write and solve equations involving direct variation.
I
Vocabulary
A
direct variation
Understanding the TEKS
S
When working with direct variation, students should understand it is a proportional
relationship with a constant rate of change. The phrase “when y varies directly with x” will
be used to distinguish the problem as direct variation. The TEKS does not specifically say
within context, but students should be able to solve problems in both mathematical and real
life problems. Direct variation includes prediction and comparison problem situations.
Essential Question(s)
What determines direct variation? Why does finding the rate of change tell you whether
the relationship is a direct variation? How do you determine the constant rate in a direct
variation problem? How is the constant rate of variation related to slope?
8th Grade Mathematics
41
© 2014
A
Name:
Direct Variation
Engaging
Activity
TEKS 8.5E Supporting
Computations and Algebraic Relationships
Jada babysat for 7 hours and earned a total of $$87.50. The amount she makes varies
directly with the hours that she babysat. How much would Jada make if she only
worked for 3 hours?
LE
The phrase “varies directly” means there is a __________________ rate at
which Jada is getting paid. There is a direct relationship between the amount of
____________ she earns and the __________ she works.
To find this rate, set up a proportion. Place labels on your proportion to ensure the
correct numbers get placed in the correct spot.
What you
know
=
What you
need to find
M
P
Labels
=
Find x, the amount Jada makes for 3 hours of babysitting.
S
A
To write an equation for this direct variation you will need to find how much Jada
makes each hour. Divide to find the amount she makes per hour. Let y = the amount
of money she earns and x = hours she babysits.
© 2014
Ratio of
what
you know
Constant
k
=
y=
x
42
8th Grade Mathematics
I can solve problems using a constant rate of change.
TM
Direct Variation
Teaching
Model
Computations and Algebraic Relationships
Name:
While on her way to school, Susan noticed that a gallon of gasoline was $3.50 per
gallon. The total cost of a tank of gasoline varies directly with the price per gallon.
LE
Write the direct variation equation.
M
P
What is the total cost for a tank of gasoline if Susan purchased 18 gallons?
Putting the Pieces Together
A
These problems will involve a constant ratio. You will need to be able to calculate that
rate of change and apply it to a real world or mathematical situation. Prediction of
future values and comparison of values will be asked.
S
Thinking Mathematically
What determines direct variation? Direction variable has a constant rate of change.
Why does finding the rate of change tell you whether the relationship is a direct
variation? All proportional relationships have a constant rate of change.
How do you determine the constant rate in a direct variation problem? The constant
rate of change is calculated by y/x.
How is the constant rate of change related to slope? The constant rate of change is
the slope of a proportional relationship.
8th Grade Mathematics
43
© 2014
GP
Direct Variation
Guided
Practice
Computations and Algebraic Relationships
Name:
A–2
C 11.5
B2
D 92
2 If x and y vary directly, and x = 2 when y = 24,
which of the following represents this situation?
A xy = 24
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
Hint: Use the form
y = kx. To find k, divide y/x.
Substitute the values back
into y = kx to solve for x.
M
P
B y = 48x
You must show your work.
LE
1 If y is directly proportional to x, and
y = 46 when x = 4, what is the value of x when
y = 23?
TEKS 8.5E Supporting
C y = 12x
Hint: Divide y/x to calculate
k. Use the form y = kx.
D xy = 48
A
3 Emilio drives a truck. He puts 15,000 miles on his
truck in a two-month period. If Emilio purchases a
brand new truck, how long will it take Emilio to put
300,000 miles on the truck?
C 40 months
B 20 months
D 200 months
S
A 2 months
4 At the frozen treats company they can produce
12,000 popsicles each 4-hour shift. If they have
three 4-hour shifts each day, how many popsicles will
they produce in a 6-day work week?
A288,000
C 72,000
B216,000
D 36,000
© 2014
44
8th Grade Mathematics
BI
Background
Information
Name:
Proportional or Non-Proportional
Computations and Algebraic Relationships
Pages 47-50 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to perform operations and represent
algebraic relationships.
2
LE
Student Expectations - Supporting Standard
TEKS
Proportionality. The student applies mathematical process standards to use proportional
and non-proportional relationships to develop foundational concepts of functions. The
8.5F
student is expected to distinguish between proportional and non-proportional situations
using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0.
4th
5th
Student is expected to...
None
None
M
P
Applying Multiple
Representations for
Foundations of Functions
Vertical Alignment
Grade
6th
6.6C represent a given situation using verbal descriptions, tables,
graphs, and equations in the form y = kx or y = x + b.
7th
7.7A represent linear relationships using verbal descriptions, tables,
graphs, and equations that simplify to the form y = mx + b.
A.2C write linear equations in two variables given a table of values,
Algebra a graph, and a verbal description.
I
Vocabulary
A
proportional, non-proportional
Understanding the TEKS
S
When students are determining if situations are proportion or non-proportional, there are three situations they
need to be able to use: tables, graphs and equations.
Tables: The students should look at the table and determine the y value when x=0. If the x value is not on the
table, the student should determine the constant rate of change and find the x value. If when x=0, then y=0, is a
proportional relationship, if when x=0, y ≠ 0, then the equation is non-proportional.
Graphs: The student should look at the graph and determine the y-intercept. If the y-intercept is (0, 0) and there
is a constant rate of change (the graph is a line), then the graph represents proportional, if the y-intercept ≠ (0,
0), then the graph is non-proportional.
Equations: The student should be able to identify two parts to an equation, one is the slope and the second is the
y-intercept. Students should be able to recognize that y=mk is proportional and y=mk+b, when b≠ 0 is nonproportional.
Essential Question(s)
How do you determine if a function is proportional or not? Why are multiple
representations important? How does understanding proportionality help me to
interpret relationships that exist in the real-world?
8th Grade Mathematics
45
© 2014
A
Engaging
Activity
Name:
Proportional or Non-Proportional
Computations and Algebraic Relationships
TEKS 8.5F Supporting
Proportional verses Nonproportional Data
There are several different ways to look at data; verbally, tables, graphs and
equations. When looking at data, you will need to be able to determine if it is
proportional or non proportional.
Verbal situations: Proportional
situations cannot have any
additional factors in the
situation. The variables need to
display a direct variation.
Tables: The values in the table
will have a constant rate of
change that can be determine by
using the slope of a line formula
y2 - y1. Then determine if the
x2 - x1
y-intercept is at (0, 0).
Another option is to enter the list
of values in your calculator by
pressing the LIST button. Once
the list is entered then press the
STAT button and go over to CALC
and select the option LinReg.
This will give you the slope and
the y - intercept.
M
P
For Example:
LE
Here are some things to consider so that you are able to determine if
something is proportional or nonproportional.
Proportional situation: The cost
of garbage pickup is $45 each
month.
Nonproportional Situation: The
cost of garbage pickup is $45
each month after a $120 deposit
is paid.
A
If the y - intercept is 0 and the
slope is a constant rate of
change, then the table is a set of
proportional data.
Graphs: Look for two things on
graphs when determining if the
data displayed in the graph is
proportional or nonproportional:
1. The line must be straight
(showing a constant rate of
change).
2. The line must contain the
point (0, 0), which means
the line must go through
the origin.
S
Equations: Since we know that
the line has to be straight and go
through the origin proportional
equations will be written in the
format y = mx = b where b = 0.
Fore example:
Proportional: y = 3x or y =
Nonproportional: y = 3x + 2 or
y=
© 2014
5
x
7
46
3
x-5
2
8th Grade Mathematics
A
Name:
Proportional or Non-Proportional
Engaging
Activity
TEKS 8.5F Supporting
Computations and Algebraic Relationships
Sort the verbal descriptions, tables, graphs, and equations into the correct category.
Put all letters that match the proportional data in the table under proportional and the
letters that match non-proportional data under non-proportional.
Non-proportional
A Lucas bought tickets to a college
basketball game and spent $28 for
each ticket.
B y =
3
8
x
LE
Proportional
Y
90
80
70
E
60
50
40
30
C Dawson pays $50.30 each month for
his cell phone service and paid $349
for the cell phone.
M
P
20
10
Y
90
80
70
60
D
50
40
30
A
20
10
F
0
10 20 30 40 50 60 70 80 90
S
8th Grade Mathematics
0
10 20 30 40 50 60 70 80 90
x
y
1
1
7
49
11
121
X
7
G y = -8 x + 4
X
H
I
47
x
y
2
51
5
127.5
13
331.5
Jessica pays for 10 one dollar items
and 7% sales tax on each dollar she
spends.
© 2014
I can distinguish between proportional and nonproportional Name:
situations.
TM
Proportional or Non-Proportional
Teaching
Model
Computations and Algebraic Relationships
1 The cost of purchasing a fountain drink
at a restaurant is $3 for the first glass
and then 50 cents for each refill after
the first drink.
100
90
80
70
60
50
LE
Proportional; y = kx
Non-proportional; y = mx + b
Equation: _____________
Y
40
30
2
20
10
y
-3
-2
1
6
3
10
6
16
3
10 20 30 40 50 60 70 80 90 100 X
Proportional; y = kx
Non-proportional; y = mx + b
Equation: _____________
M
P
x
4 The price for a piece of furniture is x
dollars. A 8.25% tax is charged on the
price of the furniture. How much is the
total cost?
Proportional; y = kx
Non-proportional; y = mx + b
Equation: _____________
Proportional; y = kx
Non-proportional; y = mx + b
Equation: _____________
A
Putting the Pieces Together
S
When given a table, is the ratio of the differences of y/x constant or not constant?
When given a graph, does the line go through the origin?
When given an equation, what form is the equation y = kx or y = mx + b?
Thinking Mathematically
How do you determine if a function is proportional or not? Is there a constant ratio or
not?
Why are multiple representations important? Data comes in all forms.
How does understanding proportionality help me to interpret relationships that exist in
the real-world? Proportionality gives constant rates to real-world situations.
© 2014
48
8th Grade Mathematics
GP
Proportional or Non-Proportional
Guided
Practice
Computations and Algebraic Relationships
Name:
x
y
1
-1
3
1
5
3
You must show your work.
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
LE
1 The table below shows a relationship between x and
y. Which equation best represents this relationship?
TEKS 8.5F Supporting
A y = 2x
B y = x – 2
C y = x + 2
D y = -2x
Hint: Determine the rate of
change for the data. Is it
constant or not?
M
P
2 Sadie saves $4.50 every week. Her parents started
her saving account with $150.00. Write an equation
that represents the balance in Sadie’s savings
account if there were not withdrawals.
3 Does Sadie’s equation represent a proportional
relationship? Why or why not?
Hint: Determine which form
of the equation to use: y= kx
or y = mx + b?
A
4 The table shows the amount of money Coach
Hagemann spends on different quantities of practice
jerseys.
Cost, C
10
$60
25
$142.50
40
$225
50
$280
S
Number of
Jerseys, j
Which equation best represents the relationship
between C, the cost of j, jerseys?
AC = 6j
BC = 5.5j
CC = 5.5j + 5
DC = 0.18j + 5
8th Grade Mathematics
49
© 2014
BI
Background
Information
Identifying Functions
Computations and Algebraic Relationships
Pages 51-57 in SE
Texas Essential Knowledge & Skills
Category The student will demonstrate an understanding of how to perform operations and represent
algebraic relationships.
2
LE
Student Expectations - Readiness Standard
TEKS
Proportionality. The student applies mathematical process standards to use proportional
and non-proportional relationships to develop foundational concepts of functions. The
8.5G
student is expected to identify functions using sets of ordered pairs, tables, mappings, and
graphs.
4th
5th
Student is expected to...
None
None
M
P
Applying Multiple
Representations for
Foundations of Functions
Vertical Alignment
Grade
6th
7th
Algebra
I
None
None
None
Vocabulary
A
relation, function, mappings, domain, range
Understanding the TEKS
S
A function is a relation in which each member of the domain is paired with
exactly one member of the range. We call this a 1:1 function. By using
multiple representations, this relationship can be demonstrated. This
SE, however, is still focused on proportional and non-proportional linear
relationships. Students should use their previous knowledge of input and
output tables to focus their knowledge. Students may also be asked to know
the difference between a relation and a function.
Essential Question(s)
How do you identify a function? How do you determine if a mapping is a function?
How do you determine if a graph is a function? How do you determine if a table is a
function?
© 2014
50
8th Grade Mathematics
A
Engaging
Activity
Name:
Identifying Functions
TEKS 8.5G Readiness
Computations and Algebraic Relationships
Identify which of the following are functions by writing function or non-function next to
each example.
1 {(-2, 3), (4, 5), 5, -10), (9, -3), (9, 3)}
LE
2 {(-2, 3), (4, 5), 5, -10), (9, -3), (9, 3)}
3 {(-2, 3), (4, 5), 5, -10), (9, -3), (9, 3)}
4 {(-2, 3), (4, 5), 5, -10), (9, -3), (9, 3)}
y
0
0
2
3
4
x
0
6
-2
3
4
4
6
7
8
y
0
4
-6
10
-8
y
2
0
2
4
3
6
4
6
S
9
8
A
x
x
y
-2
0
M
P
5
x
-1
2
0
5
5
7
x
y
0
0
2
2
5
5
7
7
x
y
0
0
2
11
5
5
7
x
y
x
y
-2
-2
0
0
2
2
3
3
4
4
8th Grade Mathematics
12
51
2
5
5
7
© 2014
A
Name:
Identifying Functions
Engaging
Activity
TEKS 8.5G Readiness
Computations and Algebraic Relationships
y
y
9
8
7
6
5
9
8
7
6
5
4
3
2
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
1 2 3 4 5 6 7 8 9
x
4
3
2
LE
13
1
1
16
1 2 3 4 5 6 7 8 9
x
M
P
y
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
9
8
7
6
5
4
3
2
14
1 2 3 4 5 6 7 8 9
x
A
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
S
y
15
© 2014
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
1 2 3 4 5 6 7 8 9
x
52
8th Grade Mathematics
I can represent a function in multiple ways.
TM
Identifying Functions
Teaching
Model
Computations and Algebraic Relationships
Name:
The scoring system at the district track meet states the a runner receives 10 points
for first place, 8 points for second place, 6 points for third place and 4 points for
fourth place. This scoring system is a relation that can be shown with ordered pairs
(1, 10), (2, 8), (3, 6) and (4, 4).
Express this relation as a table, as a graph and as a mapping diagram.
10
9
8
7
x
6
x
y
LE
Y
5
4
3
2
1
2
3
4 5
6 7
8 9 10
X
M
P
1
y
Putting the Pieces Together
A
Functions can be represented in various forms. Given the function in one form, students should be able
to represent the function in the other forms. Whatever the form, the representation matches the x and
y values.
S
Thinking Mathematically
How do you identify a function? Determine if the x value is paired with exactly one y
value.
How do you determine if a mapping is a function? Follow the arrows of the mapping
to determine if each domain value is paired with only one range value.
How do you determine if a graph is a function? Draw a vertical line to see if the line
intersects only one time.
How do you determine if a table is a function? Determine if the x value is paired with
only one y value.
8th Grade Mathematics
53
© 2014
GP
Identifying Functions
Guided
Practice
Computations and Algebraic Relationships
Name:
TEKS 8.5G Readiness
1 Which set of coordinates does not describe a y as a
function of x?
A{(–2, –2), (0, 0), (–2, 0), (0, –2)}
B{(–2, 1), (0, 2), (–3, 3), (1, 4)}
D{(–2, 0), (0, –1), (–5, –2), (3, –3)}
2 In which table is y as a function of x?
y
1
-1
3
1
5
x
1
B
3
3
x
1
C
Hint: For all representations
of a function, determine if
there is only one y value for
every x value. (The x values
cannot repeat)
y
-1
1
1
M
P
A
x
Walk around and catch
mistakes,
then correct errors as they
happen. Do not give the
student the opportunity to
reinforce errors.
If all of the students are
having the same difficulty,
then it is time to go back into
class instruction.
LE
C{(–2, –2), (0, 0), (–4, –4), (2, 2)}
You must show your work.
3
y
-1
D
1
3
5
3
x
y
3
-1
3
1
5
3
3 Which relations show y as a function of x?
y
A
x
-1
I
0
III
0
1
S
x
II
0
IV
0
4
y
-1
2
0
0
1
4
x
y
-1
2
0
0
1
4
y
2
x
A I and III
B II and IV
C I, II and IV
D All of the mappings are functions
© 2014
54
8th Grade Mathematics
GP
Identifying Functions
Guided
Practice
Computations and Algebraic Relationships
Name:
TEKS 8.5G Readiness
4 Which is NOT a function?
y
y
9
8
7
6
5
4
3
2
A
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4 5 6 7 8 9
x
-5
-6
-7
-8
-9
C
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
y
9
8
7
6
5
4
3
2
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
1 2 3 4 5 6 7 8 9
D
x
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
1 2 3 4 5 6 7 8 9
You must show your work.
S
Hint: When determining a function from a graph, use the vertical line test.
8th Grade Mathematics
x
y
9
8
7
6
5
A
B
1 2 3 4 5 6 7 8 9
-5
-6
-7
-8
-9
M
P
LE
9
8
7
6
5
55
© 2014
x
Word
Description
Set
V
Illustration
Word
Example
LE
Vocabulary
Description
M
P
Subset
V
Vocabulary
A
Illustration
Word
Example
Description
S
Counting Numbers
Illustration
© 2014
V
Vocabulary
192
Example
8th Grade Mathematics
Word
Description
Whole Numbers
V
Vocabulary
Example
LE
Illustration
Word
Description
M
P
Rational Numbers
V
Vocabulary
A
Illustration
Word
Example
Description
S
Irrational Numbers
Illustration
8th Grade Mathematics
V
Vocabulary
193
Example
© 2014
Word
Description
Real Numbers
V
Illustration
Word
Example
LE
Vocabulary
Description
M
P
Amortization
V
Illustration
A
Vocabulary
Word
Example
Description
S
Credit
Illustration
© 2014
V
Vocabulary
194
Example
8th Grade Mathematics
Word
Description
Annual Percentage
Rate (APR)
V
Vocabulary
Example
LE
Illustration
Description
M
P
Word
Principal
V
Vocabulary
A
Illustration
Word
Example
Description
S
Collateral
Illustration
8th Grade Mathematics
V
Vocabulary
195
Example
© 2014
Word
Description
Compound
Interest
V
Illustration
Example
LE
Vocabulary
Word
Description
M
P
Compound
Interest Formula
V
Illustration
A
Vocabulary
Word
Example
Description
S
401(k)
Illustration
© 2014
V
Vocabulary
196
Example
8th Grade Mathematics