LESSON PLAN (Linda Bolin)
Lesson Title: Squares, Square Roots and Exponential Expressions
Course: Pre-Algebra
Date October Lesson 5
Utah State Core Content and Process Standards:
1.2d, Recognize the inverse operation of computing squares of whole numbers and taking
the square roots of perfect squares.
1.3b Simplify numerical expressions with whole number exponents
Lesson Objective(s): Identify the inverse of a square or square root. Write expressions in
exponential form. Evaluate exponential expressions
Enduring Understanding
Essential Questions:
(Big Ideas):
• How is n² different from 2n, n³ different from 3n?
• How is n² related to √n²
Exponential expressions
• How can I simplify exponential expressions
model real world situations
Skill Focus: writing and
Vocabulary Focus:
evaluating exponential
base, power, exponent, square number, square root, cube,
expressions
exponential form, exponential expression
Materials:
• Color Tiles (25 for each team)
• Foldable Perfect Square and Square Roots (two sided)
• Linker Cubes (100 for each team)
• “Operations and Exponents”, “Exponential Expressions”
• TI-73 graphing calculators
Assessment (Traditional/Authentic): performance task, writing
Ways to Gain/Maintain Attention (Primacy): cooperative group, technology,
manipulatives, writing in journal, game
Written Assignment:
• Building Squares and Square Roots and Foldable for journal
• Operations and Exponents
• Sketches and representations for manipulatives work and record for game
• Building Exponential Expressions
• Exponential Expressions Journal Page
Content Chunks
Starter: Find the answers
Circle the expression(s) or model with the greatest value in each problem below.
1.
2 · 19
19 + 19
19²
2.
7+7
7²
7·7
5·5
5²
3.
4.
5+5
3x3
3+3
Lesson Segment 1: How is n² related to √n²? How is n² different from 2n?
How is n³ different from 3n?
Use “Building A Square Patio” (attached), an investigation with Color Tiles to help
students visualize the inverse relationship between squaring a number and taking the
square root of that perfect square. Student pairs or teams can build each patio using
the Color Tiles. Discuss each step as a class focusing on the relationship between the
side length and the root, between the square and total tiles, and between the root and
the square.
Briefly review with students how to write a base number and an exponent. Students
have used this notation since 5th grade. Show students how to use the 6 or 7 keys
on a Ti-73 to write exponential expressions. Have them use the calculator and the
attached Foldable Perfect Squares and Square Roots to build a table of values. Copy
the foldable to make two sided page that will be folded in thirds on the dotted line.
Have students work with partners to complete the investigation, “Operations and
Exponents”. Discuss possible answers to question # 4 and 5 on the investigation.
Handout the journal page for exponents (attached).
Lesson Segment 2: What are some real-world applications for exponential
notation? How is n² related to √n² ?
Follow the instructions on the attached activity “Building Exponential Expressions With
Color Tiles and Linker Cubes” to help students broaden their understanding and see
real-world application. Complete the Journal Page.
Lesson Segment 3: Practice and application
Journal: Do Mix-Freeze-Pair where students mix around the room until you say freeze.
They find the person closest to them to be their partner. If no partner is immediately
available, they raise their hand high and look for someone else with hand raise high.
During this activity, you will model an example or two for each of the vocabulary words
on the journal page and for items 1-4. Use the graphing calculator to show examples.
Then you will have the students use their TI-73’s to give an example to their partner,
or you will give them an example and they will supply the vocabulary word for it.
Students should mix and find a new partner for each of the words.
The links on the District Math Page have some great examples of where exponents are
used in the real world. Area, Volume, Scientific Notation, Biology, Astronomy,
earthquake (Richtor Scale). You may want to assign students to find a real world
example to bring to class.
Game: Playing With Powers
Two players take turns rolling two dice and deciding which to use as a base and which
to use as an exponent. After five turns, the players find the sum of the five
exponential expressions they created. Player with the greatest sum, wins.
Assign students the attached Exponential Expressions practice attached, or appropriate
text items.
Building Squares and Square Roots
With Color Tiles
Name____________________
Use Color Tiles to build each consecutive square. Write the expression
for each cell in the table. Use the table below to organize your data.
Then, use the patterns to answer the questions.
Sketch
Length
of side
1"
Find total
Squares
Find total using
an exponent
1•1=1
Find length of a side
using a square root
1² = 1
1 = √1
2"
3"
4"
5"
n
1. How many total tiles would there be if the length of the side of the square was 6?
2. What would the length of the square’s side (the square root) have to be if there
were 49 total tiles in the area?
3. If you knew the length of a side was 10, explain how you would find the total
number of squares.
4. Explain how you would find the length of the side of the square patio, if you
knew the total number of tiles was 64.
5. There is an inverse relationship between squaring a number and taking the square
root of that perfect square. Explain what this means.
Find the value for each:
6) 12²
7)
8²
8) √169
9)
√10²
n
n²
√n²
1
1²=1
√1 = 1
2
2² = 4
√4 = 2
3
3² = 9
√9 = 3
To square a number
means to:
To find a square root
of a number means:
Operations and Exponents
Name __________________
1. Circle the two expressions in each of the following sets that have the same
answer?
A.
1 + 1 + 1 + 1=
1•1•1•1=
B.
2+2+2+2+2=
2x5=
2•2•2•2•2=
C.
8•8•8=
8+8+8=
8x3=
D.
1 =
4
1x4=
1•1•1•1=
1x4
3
E.
8+8+8=
8•8•8=
8
F.
2x5=
2+2+2+2+2=
2
3
2. Which is equal to 6 :
4
3. Which is equal to 5 :
6+6+6
5x4
or
6•6•6?
or
5•5•5•5?
5
4. What is the difference between the meaning of n² and the meaning 2n?
5. What is the difference between the meaning of n³ and the meaning of 3n?
Exponential Expressions Journal Page
Name _________________
1. Some real world situations that are represented using exponents might be:
2. The exponent in an expression tells how many times the base will be used
as a factor. Write an expression that shows any base used as a factor 2 times.
Write another that shows a base being used as a factor 3 times. Then write a
third expression showing a bas being used as a factor 4 times.
3. Any number to the zero power is _____.
Ex. 6^0 = _______
4. If the exponent of a number is a negative integer, this tells us to write the
___________________ of the product.
Word
Base
Exponent
Power
Exponential Form
Square number
Square root
Exponential expression
Example
Sketch the meaning. OR
explain the meaning.
Building Exponential Expressions with Color Tiles and Linker Cubes
Objective: Students will use build models for exponential expressions, will write
correct mathematical notation and connect that notation to the models.
Materials: Color Tiles for building a square models, Linker Cubes for building cubed
and other exponential models.
In small groups have student take turns building models so that one person is
the builder, one person is the coach, one person is the checker, and the other is the
encourager. All group members will sketch the diagrams and label the dimensions.
With each successive model, students rotate the roles.
1a. First, ask students to build the smallest possible square using Color Tiles. Q.
“What do we know about the relationship between length of the sides of any square?
(They must be congruent.) Have them sketch, label and write symbolically the
measure of the area. 1” x 1” = 1² = 1 in. ²
Explain that 1 is the length of a side
and that the exponent, 2, might represent the fact that this model is a two dimensional
figure. Have them write π1 = 1 asking, “What would the length of a side, or the
square’s root be if there was only one square in the model?” TELL them a square root
is considered to be the inverse of the square.
1b. Next, have students use the Linking Cubes to build the smallest possible cube so
that the length, width and height are congruent. Have them sketch, label and write
symbolically the measure of the volume.
1” x 1” x 1” = 1³ = 1 in. ³. Q. “What do we know about the relationship between the
length, width, and height in a cube?” Explain that 1 is the length, the width and the
height, and that the exponent, 3 might represent the fact that this model is a three
dimensional figure. Extension connection: Have them write ³ π1 = 1 asking, “What
would the length of a side, or the square’s root be if there was only one square in the
model?”
2a. Ask students to build the next smallest possible square (2 x 2). Have them
sketch, label
follow the same procedure as with 1² making the connection between squaring, square
root and side length.
2b. Then, ask them to build a cube with length of side being 2 and follow the same
procedure as with 1³ making the connection between the symbolic representation and
the congruent length, width and height. Next have them follow the same procedure
building a square with length of sides equal to 3 and then a cube with length of sides
equal to three.
3a. Ask students to build the next smallest possible square (3 x 3). Have them follow
the procedure for squares.
3b. Then, ask them to build a cube with length of side being 3 and follow the
procedure for cubes.
4a. Ask students to build the next smallest possible square (4 x 4). Have them follow
the procedure for squares
4b. Then, ask them to build a cube with length of side being 4 and follow the
procedure for cubes.
5. Ask them to discuss with group members what 2^4 power would look like. They will
probably struggle with an idea. One way to create a physical model for this is two
cubes containing 2^3 linking cubes:
(2 x 2 x 2)2. Then, what would 2^5 power look like (twice as many as 2^4), and 2^6
would have twice as many linkers as 2^5, doubling every time. Have students write
descriptions of what they think these would look like.
6. Have students suggest what 3^4, 3^5, 3^6 would look like.
7. Finally, help them write a variable expression by ask them to think of the same
pattern with any length of side or “s” length. Get them to write expressions for s²,
s³, s^4, and s^5. As they write the expression, have them also write the factors out
for each expression.
Exponential Expressions
Name____________________
1. How many square units would be needed to build a square with 8-inch sides?
With 10-inch sides?
2. What would the length of the square’s side (the square root) have to be if there
were 49 square units in the area?
3.
Explain how a square root is related to the square of that root.
4.
How many cubes would be needed to build a cube with a height of 3 units?
5.
What would the height of a cube be if there were 125 cubic units in the cube?
Name the base and the exponent in each of the following exponential
expressions:
6.
4²; base
8.
6 ; base
0
exponent
7.
exponent
9.
5³; base
exponent ____
2¹; base
exponent
Rewrite each of the following expressions as a product of factors. The first
one has been done for you.
10.
4³ = 4 x 4 x 4
11.
7² =
12.
0
6
=
13.
8¹ =
Evaluate each of the following expressions
4³ =
14.
15.
7² =
0
16.
6
=
17.
8¹ =
Rewrite each of the following expressions using a base and an exponent, then
find the value. Ex: 6 x 6 x 6 = 6³ = 216
18.
3•3•3•3
19.
5x5x5
20.
(9)(9)
21.
1•1•1•1•1
Rewrite each expression using exponential form
22. a•a•a
23. mmmmm
24. 2•2•2•2(kkk)
25.
1
Use the calculator to find the value for each
26. 5
-2
27. 3
-3
28.
10
-4
29.
100
-1
28. Describe how your thinking about exponential expressions has changed during this
lesson. Include what you understand better or what you now know that you didn’t
know before.