Grade 8
Mathematics
Grade 8
Mathematics
Table of Contents
Unit 1: Real Numbers, Measures, and Models .............................................................. 1-1
Unit 2: Transformations and the Pythagorean Theorem ............................................. 2-1
Unit 3: Transversals, Surface Area and Volume .......................................................... 3-1
Unit 4: Expressions and Equations in Algebra ............................................................. 4-1
Unit 5: Functions, Growth and Patterns, Part 1 ........................................................... 5-1
Unit 6: Functions, Growth and Patterns Part 2 ............................................................ 6-1
Unit 7: Data and Lines of Best Fit .................................................................................. 7-1
Unit 8: Enhancing Understanding and Fluency ............................................................ 8-1
2012 Louisiana Transitional Comprehensive Curriculum
Course Introduction
The Louisiana Department of Education issued the first version of the Comprehensive Curriculum in
2005. The 2012 Louisiana Transitional Comprehensive Curriculum is aligned with Grade-Level
Expectations (GLEs) and Common Core State Standards (CCSS) as outlined in the 2012-13 and 201314 Curriculum and Assessment Summaries posted at http://www.louisianaschools.net/topics/gle.html.
The Louisiana Transitional Comprehensive Curriculum is designed to assist with the transition from
using GLEs to full implementation of the CCSS beginning the school year 2014-15.
Organizational Structure
The curriculum is organized into coherent, time-bound units with sample activities and classroom
assessments to guide teaching and learning. Unless otherwise indicated, activities in the curriculum are
to be taught in 2012-13 and continued through 2013-14. Activities labeled as 2013-14 align with new
CCSS content that are to be implemented in 2013-14 and may be skipped in 2012-13 without
interrupting the flow or sequence of the activities within a unit. New CCSS to be implemented in 201415 are not included in activities in this document.
Implementation of Activities in the Classroom
Incorporation of activities into lesson plans is critical to the successful implementation of the Louisiana
Transitional Comprehensive Curriculum. Lesson plans should be designed to introduce students to one
or more of the activities, to provide background information and follow-up, and to prepare students for
success in mastering the CCSS associated with the activities. Lesson plans should address individual
needs of students and should include processes for re-teaching concepts or skills for students who need
additional instruction. Appropriate accommodations must be made for students with disabilities.
Features
Content Area Literacy Strategies are an integral part of approximately one-third of the activities.
Strategy names are italicized. The link (view literacy strategy descriptions) opens a document
containing detailed descriptions and examples of the literacy strategies. This document can also be
accessed directly at http://www.louisianaschools.net/lde/uploads/11056.doc.
Underlined standard numbers on the title line of an activity indicate that the content of the standards is a
focus in the activity. Other standards listed are included, but not the primary content emphasis.
A Materials List is provided for each activity and Blackline Masters (BLMs) are provided to assist in the
delivery of activities or to assess student learning. A separate Blackline Master document is provided
for the course.
The Access Guide to the Comprehensive Curriculum is an online database of
suggested strategies, accommodations, assistive technology, and assessment
options that may provide greater access to the curriculum activities. This guide is
currently being updated to align with the CCSS. Click on the Access Guide icon
found on the first page of each unit or access the guide directly at
http://sda.doe.louisiana.gov/AccessGuide.
2012-13 and 2013-14 Transitional Comprehensive Curriculum
Grade 8
Mathematics
Unit 1: Real Numbers, Measures, and Models
Time Frame: Approximately four weeks
Unit Description
This unit focuses on number theory and the comparison of rational and irrational numbers.
Comparing the size of these numbers to each other and zero is the focus of the contents of the
unit. Writing very large and very small numbers in scientific notation is also a part of this unit.
Student Understandings
The student will determine the relative size of rational numbers, comparing fractions, integers,
decimals and percents. The student will compare rational and irrational numbers and discuss the
differences. Students will determine which two whole numbers that radicals are located
between.
Guiding Questions
1. Can students compare rational numbers using symbolic notation as well as use
position on a number line?
2. Can students recognize, interpret, and evaluate problem-solving contexts with rational
numbers?
3. Can students determine approximate value of non-square radicals?
4. Can students group numbers into categories of rational and irrational numbers?
5. Can students perform operations with numbers written in scientific notation?
Unit 1 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS)
Grade-Level Expectations
GLE #
GLE Text and Benchmarks
Number and Number Relations
1.
Compare rational numbers using symbols (i.e., <, <, =, >, >) and
position on a number line (N-1-M) (N-2-M)
2.
Use whole number exponents (0-3) in problem-solving contexts (N1-M) (N-6-M)
CCSS#
CCSS Text
8.NS.1
Know that numbers that are not rational are called irrational, and
approximate them by rational numbers. Understand informally that
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8.NS.2
8.EE.1
8.EE.2
8.EE.3
8.EE.4
every number has a decimal expansion; for rational numbers show
that the decimal expansion repeats eventually, and convert a decimal
expansion which repeats eventually into a rational number.
Use rational approximations of irrational numbers to compare the
size of irrational numbers, locate them approximately on a number
line diagram, and estimate the value of expressions (e.g., 2). For
example, by truncating the decimal expansion of 2 , show that
2 is between 1.4 and 1.5, and explain how to continue to get better
approximations.
Know and apply the properties of integer exponents to generate
1
1
equivalent numerical expressions. For example, 32 x 3-5 = 3 
3
27
Use square root and cube root symbols to represent solutions to
equations of the form x2 = p, where p is a positive rational number.
Evaluate square roots of small perfect squares and cube roots of
small perfect cubes. Know that 2 is irrational.
Use numbers expressed in the form of a single digit times an integer
power of 10 to estimate very large or very small quantities, and to
express how many times as much one is than the other.
Perform operations with numbers expressed in scientific notation,
including problems where both decimal and scientific notation are
used. Use scientific notation and choose units of appropriate size for
measurements of very large or very small quantities (e.g., use
millimeters per year for seafloor spreading). Interpret scientific
notation that has been generated by technology.
Sample Activities
Activity 1: Compare and Order (GLE: 1 , 2)
Materials List: Rational Number Line Cards - student 1 BLM, Rational Number Line Cards student 2 BLM, Rational Number BLM, Compare and Order Word Grid BLM, calculators,
paper, pencil
Have students work in pairs. Provide a number line showing only the integers –1, 0, and 1. Give
each student a deck of cards containing rational numbers including some negative rational
numbers. Use the Rational Number Line Cards - Student 1 BLM and the Rational Number Line
Cards – Student 2 BLM to make both sets of cards for each pair of students. Student 1 should
get a deck of rational numbers in fraction form, made by using Rational Number Line Cards Student 1 BLM, and Student 2 should get a deck of rational numbers in decimal form, made
using Rational Number Line Cards - Student 2 BLM. Have each student select a card from
his/her deck and compare the cards. The comparison can be done using a calculator, mental
math, or paper/pencil. Ask students to correctly place both rational numbers on the number line
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and then write a correct comparison statement using symbols. For example, if the two rational
numbers were 12 and 0.05 , they would place a mark at the 12 point and the .05 point on their
number line; then they would write a correct statement like “0.05< 12 ” or “0.05≤ 12 .” Continue the
activity having students place these on the number line. Distribute the Rational Number Line
BLM to students for additional practice with comparing and ordering rational numbers.
Once the students have placed the numbers correctly on the number line and discussions ensure
that students understand the size of the various numbers. Put 40 on the board and ask the students
to discuss with their partner where this might be located along the number line.
44
256
This concept can be taught by making a table of values for 4 as shown in the
3
4
64
table at the right. Ask the students to determine a method of determining the
42
16
value of 40. If students need a hint to get started, ask them to think of how each
1
4
4
of the values changes as the exponents decrease. Students will determine that to
40
1
get from one to the next, the number is divided by four. Thus, 4 divided by 4 is
1. Ask the students to try another base and raise the power to four and decrease
the exponent by one each time until the base is raised to the zero power to determine if the
method works every time. A second table has been drawn at the right to
(-3)4
81
illustrate negative bases raised to different powers. This also demonstrates that
3
-27
dividing by the base gives the next value or if going up, multiplying by the base; (-3)2
(-3)
9
these are done going in a downward direction to illustrate that the Power of zero
1
(-3)
-3
yields 1. Once this has been established, explain to the students that any
0
(-3)
1
number raised to the zero power results in 1. The division by the base rule
discussed above will work for integers except “0” because any number divided by 0 is undefined.
We accept that 00 will also have a value of 1.
Have a student make a rational number card with a zero exponent and place the card on the
number line. Students will place this on their individual number line.
A modified word grid (view literacy strategy descriptions) will
be used to encourage higher order thinking through comparing
>3
<1
=2
and contrasting mathematical characteristics of numbers. The
purpose of the Compare and Order Word Grid BLM is to
(0.5)(-11)
develop an understanding of the relative size of a number when
using the four operations as they make comparisons of the
0.5/0.25
numbers. The Compare and Order Word Grid BLM can be
given as a homework assignment and a modified word grid
(view literacy strategy descriptions) will be used to encourage higher order thinking through
comparing and contrasting mathematical characteristics of numbers. This grid is modified to
help the students apply their understanding of number relationships and how the function
performed affects the outcome, whereas a word grid is used to help students relate terms and
concepts. Use a simple modified word grid like the one shown and model how it is constructed
and what they will need to do to complete each cell. The example has the student apply what
he/she knows about multiplying and dividing decimals. Students should check each cell and
determine whether the given operation in the vertical column will produce an outcome given in
the horizontal row. If it is determined that this will be true, the student will check the
corresponding cell, if not, leave the cell blank. Have the student justify the choice by showing
Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models
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proof. The purpose of the Compare and Order Word Grid BLM is to develop an understanding of
the relative size of a number when using the four operations as students make comparisons of the
numbers using inequality symbols. The Compare and Order Word Grid BLM can be given as a
homework assignment and returned the following day for discussion.
A question such as the followings can lead to rich classroom discourse and should be responded
to in their math learning logs (view literacy strategy descriptions): Is the rule you discovered the
same for any two numbers? Why or why not? (Encourage students to think of cases in which
they can challenge the answer). A learning log is a notebook or some other tool used to record
reflections or understandings that have been experienced with mathematics. Explain to the
students that their math log will be used all year to record new learning, and they should write
questions that they want to understand through math class. This math learning log should be kept
either in a separate notebook or a section in the binder used for reflection of mathematical
throughout the year.
Activity 2: Fraction or Not? (CCSS: 8.NS.1; 8.NS.2; 8.EE.2)
Materials List: Real Number BLM cards for each pair of students, Squares and Square Root
BLM, grid paper, unlined paper, pencil, calculator
Teacher information: This is the first introduction to irrational numbers for the students.
Explore the meaning of square root and discover that there are some radical numbers that do
not have an exact square root. The activity is an exploration activity and the formal definitions
will be discussed in the next activity. Rational numbers are all of the numbers that can be
written as fractions and do not have a denominator of zero. Rational numbers include natural
numbers, whole numbers, integers, fractions and decimals that repeat or terminate. Students will
find that some decimals do not repeat and, therefore, they do not end – these are irrational
22
numbers. Pi is an irrational number and the approximation of
represents the approximate
7
value of pi. It is critical to stress that whatever is used for pi (22/7, 3.14) is all approximation
only; they are not, in fact, pi.
Begin the class by using SQPL (view literacy strategy descriptions). This strategy, Student
Questions for Purposeful Learning, begins with a statement or question that pertains to the
content. Write the statement on the board or read it to the students, and then pair them up.
Begin with the statement, “Every number can be written as a fraction” (students have explored
repeating and terminating decimals in earlier grades and have written whole numbers as fractions
in early grades). Have the students pair up and generate 2 – 3 questions that they would like to
have answered about the statement. Give the groups 2 – 3 minutes to generate their questions.
Students will then share their questions with the whole class, and the teacher will record the
questions as the students share. After all questions have been shared, look over the list and add
any questions that need to be answered during the lesson. Tell the students that another class had
these questions.
Possible questions:
What can you say about repeating decimals?
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Can the denominator be a zero?
Distribute grid paper to each student. Instruct students to sketch a square
with an area of 9 square units. Discuss the concept that each side of the
square has a length of 9 units or 3 units in length. Have them draw
squares with areas of 1, 4, 16, 25, 36 and 49 square units. Discuss the
length of the sides in square root and whole numbers. Have the students
take out their grid paper once more and make a sketch of a square with an
area ½ square unit. Give students time to find the length of the side. On
1
grid paper this will be easily seen. Explain to the students that
can also
4
1/4 square unit
1/2 unit
1/16 square unit
1/4 unit
1
and students can find the square root of both 1 and 4 to get ½. Discuss the idea
4
of the relationship of ¼ of a square unit and that the length of the side of the square is ½ unit.
Repeat this part of the activity by having students make a sketch of a square with the area of
1
square unit. A sketch can be made on a grid as shown in the diagram.
16
be written as
Distribute unlined paper and ask the students if it is possible to draw a square with the area of 2
square units. Have students draw a square with an area of 2 square units. Ask students what
they know for sure about the length of the side of the square with an area of 2 square units. The
students can relate that the 1  1 and 4  2 , justifying that the 2 is between 1 and 2. Have
them write the length of the side of the square as 2 . Lead a discussion about the square root of
two and have students use their calculators to find the square root of two. Encourage a
discussion about how they know that the square root of 2 is closer to 1 than to 2 on the number
line. Students can refer to the original comparison between 1  1 and 4  2 and explain that
2 is closer to the square root of 1, so it will be less than half-way between 1 and 2 on the
number line. Students will see that it is a decimal that does not terminate or repeat. Explain that
the side length is the square root of 2 ( 2 ).
Make a sketch of a number line on the board with only “0” placed and have students determine
the position of the 2 on the number line. Ask students if this is the only place that the 2 can
be correct. Discuss the position and if discussion does not elicit the idea that the unit has not
been established, explain that the position of the 2 could be any position along the line. Place
a “1” on the number line and have the students determine the position of 2 and 2 . Have
1
students place the
on the number line and discuss the placement.
4
Distribute the Exploring Squares and Square Root BLM. Give the students about 3 minutes to
explore the questions in pairs. Discuss findings by having the groups share with the class.
Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models
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Give each student pair a set of cards with Real Numbers, BLM. Some cards will have a rational
number and others irrational. Give the students time to determine which of their numbers can be
written as a fraction and which numbers do not repeat or terminate. Give the students time to
explore. Some frustration is okay because they have to learn how to persevere in their
explorations. Monitor groups, however, so that questions can be asked as needed.
Once the students have completed their explorations, have them group the cards into those that
can be written as fractions and those that cannot be written as fractions Have groups share their
findings with the class. Have students place cards along a number line on the board. Students
should justify the placement of each of their numbers. To ensure whole class participation, have
students draw a number line on paper and place the points justified at the board on their paper.
Refer to the questions that were generated at the beginning of the class and have the students
determine which of these can now be answered.
Have students respond in their learning logs (view literacy strategy descriptions) as they reflect
on what they learned today about rational and irrational numbers.
Activity 3: Real Numbers (CCSS: 8.NS.1)
Materials List: Venn Diagram BLM, paper, pencil
Discuss the different groupings of real numbers by placing a list of the following numbers on the
board: -1, 0, ½, 0.122344456666377777. . . , 0.04, and -4.
a
Tell the students that “A rational number is a number that can be represented by , where a and
b
b are integers and b ≠ 0. Rational numbers are sometimes referred to as rationals, which does not
mean the same as when a person is referred to as being a “rational person”. It means that the
5
numbers represent a ratio. Some examples of rational numbers are , 1.3 , 7.5, -5, and 9 .” Ask
8
the students which of the numbers listed on the board will fit into this category (all but the
0.122344456666377777… will fit into the rational number category).
Next, ask the students why 0.122344456666377777…will not fit this definition of a rational
number (it cannot be written as a fraction).
Put the following definitions on the board and ask the students to determine which of the rational
numbers listed fit into each of these categories.
Natural or counting numbers are the set of numbers used to count objects. 1, 2, 3, 4, 5, . .
Whole numbers are natural numbers including zero. 0, 1, 2, 3, 4, 5. . .
Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models
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Integers are whole numbers and their opposites. -2, -1, 0, 1, 2, . . 
All rational and irrational numbers form the set of real numbers.
Have the students organize the information on the BLM into a graphic organizer (view literacy
strategy descriptions). Graphic organizers are used to assist students as they organize
information to make it easier to understand. Since this is the first graphic organizer of the year,
it might be best to suggest a type of organizer to use with the information they just read. A flow
chart might be an example of a graphic organizer to use for this information.
Any graphic organizer can be used that will help
the student make sense of the information.
Once the students have completed a graphic
organizer, have them take the number cards from
Activity 2. Tell the pairs of students to group their
cards as rational numbers or irrational numbers.
This should take only about 2 -3 minutes. Have
students share their groupings and justify why the
number is either rational or irrational. Distribute
the Venn Diagram BLM and using their graphic organizer, have the students take their rational
numbers and place the numbers in the correct circle. Use the Venn Diagram as a formative
assessment to monitor the pairs of students and observe their level of understanding.
Activity 4: Yes or No (CCSS: 8.NS.1)
Materials: Word Grid BLM, Paper, Pencil
Have a volunteer share his or her method of organizing real numbers into a graphic organizer.
Lead a discussion with students about how the numbers are related. The word grid (view literacy
strategy descriptions) used in this activity provides students with an organized framework for
comparing real numbers. Before beginning the activity, put a simple word grid on the board and
discuss a concept of classifying real numbers. For students to
understand the use of a word grid, students should complete the
2
5
simple word grid. With the example, students should justify
why 2 is considered rational, an integer, a whole number and a
rational
natural number while the square root of 5 is simply an irrational
number. Good vocabulary review with this example.
irrational
Distribute the word grid BLM with the unit vocabulary
comparing real numbers. Have students work independently to
complete the word grid. This can be used as a formative
assessment of student understanding of classifying rational and
irrational numbers.
integer
whole
number
natural
number
Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models
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2012-13 and 2013-14 Transitional Comprehensive Curriculum
Activity 5: Let’s do Radicals! (CCSS: 8.NS.2)
Materials: Real Number Cards BLM, Number Line BLM, paper, pencils, calculators
Begin the lesson by asking students to make a list of the first ten perfect squares or square
numbers. Review square roots by having the students write the square roots of the square
numbers. Ask students to discuss with a partner how to estimate the square root of a number that
is not a square number. Have the students share their ideas with another pair of students. Ask for
volunteers to explain their ideas to the class. Ask the students to determine which two whole
numbers it will fall between and to which of the two numbers it is closer. Have the class
determine the position of 5 on a number line. Have students write the two square numbers
closest to 5 and instruct students to find the square roots of these two numbers. ( 4 = 2 9 =3).
Using this information, have students justify whether the 5 will be > 2.5 or < 2.5(it is less than
2.5 because 5 is closer to 4 than it is to 9).
Distribute cards with real numbers, Real Number Cards BLM, to groups of four students.
Explain to the students that they will work to put the numbers in order from least to greatest. If a
group of students is “stuck,” guide them to look at their square root list that was prepared at the
beginning of the lesson.
After the students have had time to put the numbers in order, distribute number line BLM and
have students place the numbers from the cards in correct position along the number line. Have
students indicate which numbers on the number line are irrational numbers and have someone
explain again the definition of an irrational number. Discuss results as a class.
Challenge the students by asking the question: What do you think 3 27 symbolizes? This may be
the first time the students have seen the cube root symbol. Have the students use a “factor tree”
to find the prime factorization of 27 so that they will get the exponential representation of 33.
This gives a hint without telling the students that it represents something with a cube. At this
time, tell the students that when a number is a perfect cube, the result of finding the cube root
will be a whole number, as was done with the square roots. Have the students make a list of the
first five perfect cube numbers. (1, 8, 27, 64, 125) Tell the students to write equations to show
the cube root of each of these perfect cubes. ( 3 1  1; 3 8  2; 3 27  3; 3 64  4; 3 125  5 )
Activity 6: Radically! More or Less (CCSS: 8.NS.2)
Materials: Paper, pencils
Students should write the following radical numbers on their paper:
5, 12, 111, 67 , 55, 99 , 43, 67 . Have the students work in groups of four to
determine the two consecutive whole numbers that are on either side of the numbers and which
of these whole numbers is closest to the radical number. Students should be able to justify their
answers. Give the class time to complete the assignment.
Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models
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Explain to the class that professor know-it-all (view literacy strategy descriptions) is a method of
reviewing content. One group of four students will be in the front of the room and answer
questions about determining the approximate value of radical numbers that are not square
numbers. The group at the front will be the “experts,” and the class members can ask questions
they have about approximating the value of radical numbers. The group at the front will
“huddle” after a question is asked so that they can agree upon an answer. Since this is the first
time they have used this strategy this year, it might help to have a list of questions prepared and
distributed so that the students have an idea as to what type of questions will be beneficial for
review. Questions such as 1) How do you determine where to start? 2) How do you decide if the
radical is closer to one whole number or the other? 3) Is it possible to have a radical number that
is halfway between two whole numbers?
Activity 7: Computing Using Scientific Notation (CCSS:8.EE.3)
Materials List: paper, pencil, Scientific Notation BLM, calculators
Have the students write one hundred twenty-three billion (123,000,000,000). Lead a discussion
about how numbers this large become cumbersome and not easy to record accurately. This is
why scientific notation is used. This same number can be written as:
1.23 x 1011
The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than
10.
The second number is called the base. It must always be 10 in scientific notation. The base
number 10 is always written in exponent form. In the number 1.23 x 1011, the number 11 is
referred to as the exponent or power of ten.
To write a number in scientific notation, place the decimal point in the original number so that a
number >1 and <10 is created. Moving the decimal to the left is the same as dividing by a power
of 10. To determine what power of 10 was used in the division, count the number of decimal
places that the decimal would have to be moved to get back to the original number. Use this
number of decimal places as the power of 10. The zeros to the right of 3 are no longer needed as
they would be eliminated when the division was made.
Example: 1.23000000000 x 1011 becomes 1.23 x 1011 after the zeros are dropped.
There are also numbers that are extremely small. Suppose you were to measure the length of an
ant. The ant is 0.0625 inch long. The decimal will be moved to the right so that a number >1
and <10 is created.
Teacher Note: The negative exponents might be a new idea for the students; be sure the
discussion clarifies any confusion. Point out that to multiply by products of 10, move the decimal
to the right and the inverse would be to divide by products of 10 so move the decimal to the left,
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and likewise, the exponent will be negative to indicate that the number in scientific notation is
smaller than 1 but greater than 0.
Example: 0.0625 would become 6.25 x 10-2, with the negative exponent indicating the
original number was less than 1.
Activity 8: Trip to Mars! (CCSS: 8.EE.4)
Materials List: Trip to Mars BLM, paper, pencil, calculator
Begin the class by using SQPL (view literacy strategy descriptions).
The distance from Earth to Mars changes every minute with the difference between the closest
and farthest distance being more than 300,000,000 kilometers.
Distribute Scientific Notation BLM and give students time to complete the situations. The
information to answer questions generated by the SQPL statement should be answered using the
information on the BLM. After students have completed the BLM, refer to the SQPL questions
at the beginning of the lesson and have students determine if the questions have been answered.
Questions may arise that are more science related; challenge the students to research these and
report their findings tomorrow and share with the class.
2013-2014
Activity 9: Powerful numbers (CCSS: 8.EE.1)
Materials List: Powerful Numbers BLM, paper, pencil, calculator
Have students use a calculator and the Powerful Numbers BLM to complete the chart with
powers of 10 from –4 to 4. Discuss the patterns that are observed and the significance of negative
exponents. Discuss the idea developed earlier when looking at the Power of zero, that as the
exponent increases or decreases they are multiplying by the base or dividing by the base.
Challenge the students by asking them to complete an exponent chart using the powers of 2 from
-4 to 4. Ask the students to work with a partner and develop a conjecture describing the effect of
the negative exponent on the value of the number.
Ask the students what a number like 4.5 x 10-3 would be before having them complete the
Powerful Numbers BLM. After the students have completed the scientific notation portion of
the BLM, discuss any problems the students may have encountered.
Provide students with real-life situations for which scientific notation may be necessary, such as
the distance from the planets to the sun or the mass of a carbon atom. Have students investigate
scientific notation using a calculator. Allow students to convert numbers from scientific notation
to standard notation and vice versa. Relate the importance of scientific notation in the areas of
physical science and chemistry.
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Activity 10: Exponential Growth (CCSS: 8.EE.1)
Materials List: paper, pencil, 1 sheet of computer or copy paper, Exponential Growth and Decay
BLM
Explain to the students that will look at exponential decay. Give each student a sheet of 8 12 ” by
11” paper. Have students look at the number of regions on the unfolded paper. There is one
region (8 ½” x 11”). Have them make one fold in the paper so that the two areas or regions are
equivalent. Ask students to determine the number of regions and the area of the smallest region
using the sheet of paper as the unit.
Number of Folds
0
1
Number of Regions
1
2
2
4
3
...
N
8
...
2n
Area of Smallest Region
1 sheet of paper
1
1
sheet of paper
2 or 2
1
4
or 2 2 sheet of paper
or 2 3 sheet of paper
...
n
1
or 2 sheet of paper
2n
1
8
Distribute the Exponential Growth and Decay BLM and have students complete the table folding
the paper until they can confidently complete the table information. Instruct students to fold the
paper in half several times, but after each fold, they should stop and fill in a row of the table.
After students have completed the BLM, have them discuss how they were able to identify the
independent and dependent variables. The independent variable is the number of folds; the
dependent variable is the number of regions. Ask, will the graph be graph linear? This is called
an exponential growth pattern.
The third column gives the area of the smallest region. When comparing the number of folds to
the area of the smallest region, the pattern becomes one of exponential decay. Include the
significance of integer exponents as exponential decay is discussed.
2013-2014
Activity 11: Properties of exponents (CCSS: 8.EE.1)
Materials: Vocabulary Awareness BLM, Exponent BLM, pencil, paper, calculator
Begin this activity by explaining that today’s activity involves the literacy strategy called
vocabulary self-awareness (view literacy strategy descriptions). Distribute the Vocabulary
Awareness BLM. This chart has the vocabulary necessary already inserted. It can also be given
to the students without vocabulary so that they can write the vocabulary needed themselves.
Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models
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2012-13 and 2013-14 Transitional Comprehensive Curriculum
Have the students rate each vocabulary word (in this case the properties of exponents) according
to their level of familiarity and understanding. A (+) sign indicates a high degree of comfort and
knowledge, a check () indicates uncertainty, and a minus sign (-) indicates the word is brand
new to them. Students should also provide a definition and example for each word; some of
these will be guesses. Over the course of the activity, students will develop an understanding of
these properties.
After students have had time to complete the Exponent BLM, have them take out their
Vocabulary Awareness BLM and make additions or corrections to those vocabulary terms that
were unfamiliar or were not completely understood. Have students get with a shoulder partner
and discuss each of the properties. Ask students to share these properties with the class to further
develop a complete understanding of the properties of exponents.
Sample Assessments
Performance assessments can be used to ascertain student achievement.
General Assessments
 Give the student a list of about fifteen rational numbers including fractions, decimals and
percents, making sure that some of the values are equivalent (i.e. 41 and 25%). The
student will make a number line and place all fifteen rational numbers along the number
line in the correct position. To complete the assessment, the student will write at least 10
inequality statements using the symbols <, >, , and .
 Give the student a list of real numbers and have him/her place the numbers in order from
least to greatest.
 Challenge the student to write one scientific notation real-life problem by researching
scientific facts that result in very large or very small numbers.
 Whenever possible, create extensions to an activity by increasing the difficulty or by
asking “what if” questions.
 Have the student create a portfolio containing samples of the experiments and activities.
Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models
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2012-13 and 2013-14 Transitional Comprehensive Curriculum
Activity-Specific Assessments
 Activity 3: The students will be given a list of real numbers, and they willclassify them
using a graphic organizer.
 Activity 4: Provide students with a situation such as, Tell whether the following can be
found between the numbers 4 and 5. If it is possible, give an example. Provide
explanations for each answer.
o
o
o
o
A real number
A rational number
An irrational number
An integer
 Activity 6: Provide students with a list of radicals similar to the ones used in this activity.
Have students give the whole number values on either side of the radical and state
whether it is closer to one of them or the other.
Example: 6 is between whole number square roots of 4 and 9 , so the square root
is between 2 and 3. 6 – 4 is 2 and 9 – 6 is 3, so the square root of 6 is closer to the
square root of 4 than the square root of 9.

Activity 7: Provide students with sets of real numbers and have them put them in order
from least to greatest.
 Activity 8: Have students determine the maximum number of people that could travel on
the trip to Mars if the food for one person will occupy 0.00098 cubic feet of the available
cargo space and they can use no more than 0.05 cubic feet of the available cargo space
for food.
0.05/0.00098 = 51.02 so 51 people could store enough food
Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models
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