Mathematics Success – Grade 8
T1
LESSON 7: Cube Roots
[OBJECTIVE]
The student will evaluate perfect cube roots using whole numbers and fractions and
use cube root symbols to represent solutions to equations.
[PREREQUISITE SKILLS]
square roots, volume, prime factorization
[MATERIALS]
Student pages S1−S16
Centimeter cubes (91 per student pair)
Calculator (optional)
[ESSENTIAL QUESTIONS]
1. What is a cube root?
2. How can you determine cube roots of larger numbers?
3. Explain how to solve an equation with a cubed value of x.
[WORDS FOR WORD WALL]
three-­dimensional figures, radical sign, cube root, prime factors, prime factorization, factor tree
[GROUPING]
Cooperative Pairs (CP), Whole Group (WG), Individual (I)
*For Cooperative Pairs (CP) activities, assign the roles of Partner A and Partner B
to students. This allows each student to be responsible for designated tasks within
the lesson.
[LEVELS OF TEACHER SUPPORT]
Modeling (M), Guided Practice (GP), Independent Practice (IP)
[MULTIPLE REPRESENTATIONS]
SOLVE, Algebraic Formula, Verbal Description, Pictorial Representation, Concrete
Representation, Graphic Organizer
[WARM-UP] (IP, I, WG) S1 (Answers on T16.)
• Have students turn to S1 in their books to begin the Warm-Up. Students will
work with area and finding side lengths of figures to review squares and square roots. Monitor students to see if any of them need help during the Warm-Up.
Have students complete the problems and review the answers as a class. {Verbal
Description, Pictorial Representation}
[HOMEWORK]
Take time to go over the homework from the previous night.
[LESSON] [4-5 days (1 day = 80 minutes) - (M, GP, WG, CP, IP)]
T2
Mathematics Success – Grade 8
LESSON 7: Cube Roots
SOLVE Problem
(WG, GP) S2 (Answers on T17.)
Have students turn to S2 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students
that during the lesson they will learn how to find the cube roots of perfect cubes and represent the solution to a cube root equation. They will use this knowledge to
complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description}
Cubes with Whole Numbers Using Volume – Concrete
(M, GP, WG, CP) S2, S3 (Answers on T17 and T18.)
M, WG, GP, CP:
Have students turn to page S2 in their books. Make sure
partners know their designation as Partner A or Partner B.
Pass out centimeter cubes. Calculators are optional.
MODELING
Cubes with Whole Numbers Using Volume - Concrete
Step 1: Have student pairs answer Questions 1 and 2 and be ready to share
answers with the whole group.
• Partner A, how did you find the area of the square in the warm-­up? (Multiply the length times the width.) Record.
• Partner B, how did you write the units when finding the area? (units squared) Record.
• Have students discuss and then share with the group why they used units squared. (You use units squared because area tells how many
squares cover the shape. Also, area is two-dimensional with length
and width.) Record.
Step 2: Have student pairs place 1 centimeter cube in the space below Question 3.
• Have students discuss the attributes of the cube. (bottom or base of the cube is a square, all sides are equal, has six faces, has 3 dimensions,
is a three-­dimensional figure) Record any correct observations as
they record on S2.
• Have student pairs discuss how a cube is different from other rectangular prisms. (A cube is a 3-­dimensional figure with all side lengths congruent.) Record.
• Partner A, what is the third dimension of a cube besides length and width? (height) Record.
• Partner B, what is the length of the cube? (1 unit) Record.
• Partner A, what is the width of the cube? (1 unit) Record.
• Partner B, what is the height of the cube? (1 unit) Record.
Mathematics Success – Grade 8
T3
LESSON 7: Cube Roots
Step 3: Direct students’ attention to the top of S3.
• Have students read the two sentences at the top of S3 and discuss the two missing words. When we work with 2-­dimensional figures, we find the (area) that the shape covers. When we have a 3-­dimensional figure, the space that is occupied by the shape is called the (volume). Record.
• Part A, identify the units we use when determining the volume. (units cubed) Record.
• Partner B, explain why we used units cubed. (Volume tells how many cubes can be used to create the larger cube. The volume has three
dimensions – length, width, and height.) Record.
• Partner A, identify the exponent we use with the units for area. (2)
• Have pairs discuss what exponent they think will be used for volume. (3) What will this represent (the cubed value) Record. Why use a 3?
(because we are multiplying 3 dimensions)
• Discuss Question 4. Step 4: Have student pairs refer back to the information from the cube on S2
and use the plan they just developed to determine the volume of the
cube. (1 unit • 1 unit • 1 unit = 1 unit cubed or 1u3) Record.
Step 5: Have student pairs place 8 centimeter cubes in the space below Question
5 and create a cube using the cubes.
• Partner A, what is the length of the cube? (2 units) Record.
• Partner B, what is the width of the cube? (2 units) Record.
• Partner A, what is the height of the cube? (2 units) Record.
• Partner B, how do you find the volume of a cube? (Multiply the length times the width times the height.) Record.
• Partner A, what is the volume of the cube? (2 units • 2 units • 2 units = 8 cubic units or 8u3) Record.
Cubes with Whole Numbers Using Volume – Concrete to Pictorial
(M, GP, WG, IP, CP) S4 (Answers on T19.)
M, WG, GP, CP:
Have students turn to page S4 in their books. Make sure partners know their designation as Partner A or Partner B.
Students will move from discovery using the centimeter
cubes to a pictorial representation of volume of cubes in
order to determine cube roots. {Concrete Representation,
Pictorial Representation, Algebraic Formula, Verbal Description,
Graphic Organizer}
T4
Mathematics Success – Grade 8
LESSON 7: Cube Roots
MODELING
Cubes with Whole Numbers Using Volume –Concrete to Pictorial
Step 1: Have students use the information at the top of S4 in the graphic organizer to create cubes with side lengths of 3 and then 4. Discuss the number of cubes needed for each and the volume.
• Partner A, how many cubes will be needed to create a cube with a length, width, and height of 3 units. (27) Record.
• Partner B, what is the volume of that cube?
(3 units • 3 units • 3 units = 27u3) Record.
• Partner A, how many cubes will be needed to create a cube with a length, width, and height of 4 units. (64) Record.
• Partner B, what is the volume of that cube?
(4 units • 4 units • 4 units = 64u3) Record.
Step 2: Have students discuss how to use the centimeter cubes to find volume. Will it always be possible to build a 3D figure with the side lengths given? (No.) Record. Have students discuss why that is not possible.
(With larger side lengths, the number of cubes will increase, and there
may not be enough cubes to build the figure.) Record.
Have students read and discuss the next question.
• When we were working with the cubes, what did we do each time in order to find the cubed value of the whole number? (Multiply the value of the whole number three times.) Record.
Step 3: Explain to students that we can also represent a cube with a picture.
Direct students’ attention to the pictorial representation of the cube
and the graphic organizer below the cube.
• Partner A, identify the length, width, and height of the cube. (1 unit each) • Partner B, explain how to find the volume. (Multiply the length times the width times the height.) Record.
• Have student pairs discuss how to determine the cubed value of 1? (Multiply 1 • 1 • 1) What is the cubed value of 1? (1) Record. Why? (Because 13 = 1 • 1 • 1 = 1) Record.
Cubes with Whole Numbers Using Volume – Pictorial to Abstract
(M, GP, WG, IP, CP) S5 (Answers on T20.)
M, WG, GP, CP:
Have students turn to page S5 in their books. Make sure
partners know their designation as Partner A or Partner
B. Students will move from the pictorial representation to
finding the cubed value from information given in a table. {Pictorial Representation, Algebraic Formula, Verbal Description,
Graphic Organizer}
Mathematics Success – Grade 8
T5
LESSON 7: Cube Roots
MODELING
Cubes with Whole Numbers Using Volume – Pictorial to Abstract
Step 1: Direct students’ attention to the cube at the top of S5.
• Partner A, identify the length, width, and height of the cube. (2 units each). Record.
• Partner B, explain how to find the volume. (Multiply the length times the width times the height.) Record.
• Partner A, if the value for the length, width, and height is 2, what is the cubed value? (8) Record. Why? Because 23 = 2 • 2 • 2 = 8. Record.
Step 2: Have students record the volume and the cubed value in the graphic
organizer for 2 units.
IP, CP, WG:
Have students read the paragraph below the graphic
organizer and then complete the volume and cubed
value of 6 inches, 8 feet and 10 meters. Then come back together as a class and share their results. {Graphic
Organizer, Algebraic Formula, Verbal Description}
Cubes and Cube Roots – Pictorial to Abstract
(M, GP, WG, IP, CP) S5, S6, S7(Answers on T20, T21, and T22.)
M, WG, GP, CP:
Have students continue to work on page S5 in their books.
Make sure partners know their designation as Partner A
or Partner B. Students will work with cube roots of whole
numbers. {Graphic Organizer, Algebraic Formula, Pictorial
Represention, Verbal Description}
MODELING
Cubes and Cube Roots – Pictorial to Abstract
Step 1: Have students look at the section at the bottom of S5 entitled. Finding the
cube root.
Read the three paragraphs with the students. Have the students discuss
how the square root and cube root are alike and different.
• Partner A, what is the opposite of cubing a number? (finding the cube root) Record.
• Partner B, what is the symbol we use to represent a cube root? (a radical symbol with a 3) Record.
*Teacher Note: Remind students that the radical symbol is the same one they
use for square roots, but the number in the radical sign is a 3.
T6
Mathematics Success – Grade 8
LESSON 7: Cube Roots
Step 2: Direct students’ attention to the chart at the top of S6.
• Partner A, what is the value for the first column? (side length of the cube)
• Partner B, explain why only one value is given. (because the length, width, and height are all the same value)
• Partner A, what is the value in the second column? (volume)
• Partner B, explain why we use the exponent of 3 for the volume. (When we are working with a cube to find volume there are 3 dimensions.) Record.
• Partner B, what is the value in the third column? (the value of the number cubed without the units)
• Partner A, what is the value of 1 cubed? (1)
• Partner B, what is the cube root of 1 in the last column? (1) Record.
• Partner A, explain why. (because 1 • 1 • 1 = 1) Record.
• Partner B, what do you notice about the cube root of 1 and the side length of the cube with a volume of 1u3? (They are the same
value.) Record.
Step 3: Have student pairs look at the cube made of 8 centimeter cubes.
Students will use that cube and the graphic organizer to complete
this step.
• Partner A, what is the volume of the cube? (8u3) Record.
• Partner B, explain why. (because 2u • 2u • 2u = 8u3) Record.
• Partner A, look at Column 3 in the graphic organizer. What is the value of the cubed number? (8) Record.
• Partner B, explain why the value of the cubed number is 8. (Because 23 = 2 • 2 • 2 = 8). Record. Partner A, explain how to determine the cube root of 8 if we know the cubed value of 2 is 8.
3
(Find the cube root of 8. √8 = 2) Record. • Partner B, what do you notice about the cube root of 8 and the side length of the cube with a volume of 8u3? (They are both the same
value.) Record.
IP, CP, WG:
Have students work with a partner to complete the graphic
organizer on S6 and on the top of S7 for the values of 6, 8, and 10. Then come back together as a class and share their results. {Graphic Organizer, Algebraic Formula, Verbal Description}
Mathematics Success – Grade 8
T7
LESSON 7: Cube Roots
Cube Root Equations with Whole Numbers
(M, GP, WG, CP, IP)
S7, S8 (Answers on T22 and T23.)
M, WG, GP, CP, IP: Have students turn to page S7 in their books. Make sure
partners know their designation as Partner A or Partner B.
Students will work with cube roots in equations. {Graphic
Organizer, Algebraic Formula, Verbal Description}
MODELING
Cube Roots Equations with Whole Numbers
Step 1: • Partner A, explain what we did each time for the values to find the cube root of the whole number cubed.
(Found the cube root of the whole number by determining what number
multiplied by itself 3 times would be the volume of the cube.) Record.
• Partner B, identify the information given in the first column of the last row. (a) Record.
• Partner A, explain the meaning of the variable. (A variable can represent any number.)
• Partner B, explain why we can use a variable for this example. (The same process can be used to determine the cube and cube root of any value
even if we do not know the side measure of the cube.) Record.
Step 2: Have student pairs discuss how they could use the variable (a) to find the volume of cube with a side length of (a). (Multiply length times width
times height: a units • a units • a units = a3 units3) Record in the chart
in the volume column.
• Partner A, explain how the value of the cubed number can be written using the variable, a. (a • a • a = a3)
• Partner B, describe how we can find the value of the cube root of a in
the last column. (Find the cube root of a by determining what variable
multiplied by itself 3 times would be the volume of the cube.)
• Partner A, what is the value of the cube root of a3? Explain how you
3
determined that. (a • a • a = a3 so √a3 = a) Record the cube root in
the graphic organizer.
Now that we have explored cubing and cube roots with a variable, let’s look at
how we can use that information to solve equations with cube roots.
Step 3: Have student pairs discuss the two basic concepts they need to remember
when solving any equation.
• Partner A, what is the first goal when solving equations. (Isolate the variable.) Record.
• Partner B, what is the second goal when solving equations. (Balance the equation.) Record.
T8
Mathematics Success – Grade 8
LESSON 7: Cube Roots
Step 4: Direct students’ attention to the sample cube root equation in the graphic
organizer at the bottom of S7.
• Partner A, identify the equation. • Partner B, how can we isolate the variable? (by finding the cube root of x3.) Record.
• Partner A, what is the cube root of x3? (x). Explain how you determined
this. (because x • x • x = x3 )
• Partner B, how can we balance the equation? (by finding the cube root of 8)
• Partner A, what is the cube root of 8? (2) Explain how you determined this. (because 2 • 2 • 2 = 8)
• Partner B, what is the value of x? (2) Record.
• Model how to check the equation in the last column by substituting in the value of 2 for the x.
Step 5: Have student pairs work together to find the solution for the cube root equations on the top of S8. Remind students to also check their equation
by substituting back in the value for x. Once student pairs have completed
the 2 equations, have them share and defend their responses.
Step 6: Direct students’ attention to the graphic organizer at the bottom of the
page with the equation.
• Partner A, identify the equation. • Partner B, how can we isolate the variable? (by finding the cube root of x3)
• Partner A, how can we balance the equation? (by finding the cube root of 729)
*Teacher Note: At this point in time you may need to review how to find the prime factorization of a number.
Step 7: Have students discuss what may be challenging about this equation.
(Possible responses may include identifying 729 as a large number to
find the cube root.)
• Have student pairs brainstorm possible strategies to break down 729 or methods they can think of to find the cube root.
• Have students share their ideas and then focus in on prime factorization.
Step 8: Direct students’ attention to the last graphic organizer on S8. Model and
explain the prime factorization of 729. Students may suggest using the
factor tree or the ladder method.
• Partner B, What are you looking for when you find the prime factorization of a number to find the cube root? (three numbers that are the same so you can find the cube root)
• Partner A, what are the prime factors of 729? (3 • 3 • 3 • 3 • 3 • 3)
• Partner B, how many groups of factors do we want for the cube root? (3)
• Partner A, how can we write the prime factors as three values?
(32 •32 •32) or (9 • 9 • 9)
• Partner B, what is the value of x?
(9) Record.
Have student pairs complete the list of steps next to the factor tree and
then share answers as a whole group.
Mathematics Success – Grade 8
T9
LESSON 7: Cube Roots
Exploring Cube Roots with Fractions
M, WG, GP, CP:
(M, GP, WG, CP)
S9 (Answers on T24.)
Have students turn to page S9 in their books. Pass out 8
centimeter cubes to each student pair. Make sure partners
know their designation as Partner A or Partner B. Calculators
are optional. {Graphic Organizer, Algebraic Formula, Pictorial
Representation, Verbal Description}
MODELING
Exploring Cube Roots with Fractions
*Teacher Note: Have students build a cube with 8 centimeter cubes. Explain that
we will represent the side measures as one unit.
Step 1: Direct students’ attention to the model of the cube on S9.
• Partner A, identify the width of the cube. (1 unit) Record.
• Partner B, identify the length of the cube. (1 unit) Record.
• Partner A, identify the height of the cube. (1 unit) Record. • Partner B, explain how we can determine the volume of the cube.
(1 unit • 1 unit • 1 unit = 1 u3) Record.
Step 2: Direct students’ attention to Row 2 of the graphic organizer.
• Partner A, examine the one layer of the cube and describe to your partner what we now have. (Answers may vary: two rectangular
prisms, two congruent halves)
• Partner B, explain to your partner what happens to the measurements of the length, width, and height when we divide the cube in half. (The
width, and length are still 1 unit, but the height is 1 unit.) Record in
2
Row 3.
• Partner A, explain how we can determine the volume of 1 of the
2
original cube.(1 unit • 1 unit • 1 unit = 1 u3) Record.
2
2
Step 3: Direct students’ attention to Row 3 of the graphic organizer.
• Partner B, explain to your partner what happens to the measurements of the length, width, and height when we divide the one-half of the
cube in half. (The length is still 1 unit, but the height and width are
1
unit.) Record in Row 3.
2
• Partner A, explain how we can determine the volume of 1 of 1 of the
2
2
original cube. (1 unit • 1 unit • 1 unit = 1 u3) Record.
2
2
4
Step 4: Direct students’ attention to Row 4 of the graphic organizer.
• Partner B, explain to your partner what happens to the measurements of the length, width, and height when we divided the one half of half the cube
in half. (The length, height, and width are all 1 unit.) Record in Row 7.
2
• Partner A, explain how we can determine the volume of 1 of 1 of 1
2
2
2
of the original cube.( 1 unit• 1 unit • 1 unit = 1 u3) Record.
2
2
2
8
T10
Mathematics Success – Grade 8
LESSON 7: Cube Roots
Cubes with Fractions Using Volume – Pictorialto Abstract
(M, GP, WG, IP, CP) S10 (Answers on T25.)
M, WG, GP, CP:
Have students turn to page S10 in their books. Make sure partners know their designation as Partner A or Partner
B. Calculators are optional. {Graphic Organizer, Algebraic
Formula, Pictorial Representation, Verbal Description}
MODELING
Cubes with Fractions Using Volume – Pictorial to Abstract
Step 1: Direct students’ attention to the model of the cube on S10.
• Partner A, what do you notice about the model? (One of the pieces or smaller cubes is separated.)
• Partner B, what fraction can we use to represent the width of the smaller cube in relationship to the larger cube? (The width of the smaller cube
is 1 the width of the larger cube.)
2
• Partner A, what fraction can we use to represent the length of the smaller cube in relationship to the large cube? (The length of the
smaller cube is 1 the length of the larger cube.)
2
• Partner B, what fraction can we use to represent the height of the smaller cube in relationship to the larger cube? (The height of the
smaller cube is 1 the height of the larger cube.) Record the answer in
2
Question 1.
Step 2: Have students use the fractional information from the smaller cube to find the volume of the smaller cube in relationship to the larger cube.
• Partner A, how do we find the volume of a cube? (Multiply the length times the width times the height.) Record the answer in Question 2.
*Teacher Note: It may be beneficial here to share a “think aloud” for the process of multiplication of fractions as a review for students.
“I know that when I multiply fractions, I multiply the numerators and then the denominators.”
• Partner B, what is the volume of the smaller cube in relationship to the 3
larger cube? ( 1 units • 1 units • 1 units = 1 units cubed or = 1 u )
2
2
2
8
8
Record the answer in Question 3.
*Teacher Note: Remind students that we are finding the volume of the smaller cube in relationship to the larger cube.
• Partner A, explain the process we used with the whole numbers to find the cubed value. (Multiply the value three times.)
Mathematics Success – Grade 8
T11
LESSON 7: Cube Roots
Step 3: Direct students’ attention Question 4.
• Partner
A, explain the number equation we use to determine the cubed
1
1
1
1
1
value of 2 . ( 2 • 2 • 2 = 8 ).
• Have student pairs discuss Question 4.
• Partner B, if we compare the volume of the single cube in relationship
1
to the whole cube the volume of the single cube is ( 8 ) of the volume
of the whole cube.
• Partner
A, what does this mean? (The volume of the individual cube
3
is 1u and the volume of the larger cube is 8u3. When we compare
1
them, 1 out of the 8 cubes is 8 .) Record.
IP, CP, WG:
Have students work with a partner to complete the graphic
1 1
organizer on the bottom of S10 for the values of 6 , 8 , and
1
. Then come back together as a class and share
10
their results. {Graphic Organizer, Algebraic Formula,
Verbal Description}
Cube Roots with Fractions
M, WG, GP, CP:
(M, GP, WG, IP, CP)
S11, S12 (Answers on T26 and T27.)
Have students turn to page S11 in their books. Make sure
partners know their designation as Partner A or Partner B.
Students will work with cube roots of fractions. {Graphic
Organizer, Algebraic Formula, Verbal Description}
MODELING
Cube Roots with Fractions
Step 1: Have students read the two paragraphs, filling in the blanks, on finding the cube root of a fraction and then discuss answers as student pairs.
• Partner A, what is the opposite of cubing a number? (finding the cube root) Record.
• Partner B, what is the symbol we use to represent a cube root? (a 3
radical symbol with √ ) Record.
T12
Mathematics Success – Grade 8
LESSON 7: Cube Roots
Step 2: Direct students’ attention to the graphic organizer at the bottom of
the page.
• Partner A, what is the value for the first column? (side length of the cube)
• Partner
B, explain why only one value is given. (because the length,
width, and height are all the same value)
• Partner A, what is the value in the second column? (volume)
• Partner B, explain why we use the exponent of 3 for the volume. (When we are working with a cube to find volume, there are 3 dimensions.) Record.
• Partner B, what is the value in the third column? (the value of the number cubed without the units)
1
1
• Partner A, what is the value of cubed? ( 8 ) Record.
2
1
1
• Partner B, what is the cube root of in the last column? ( 2 )
8
1
1
1
1
• Partner A, explain why. (because • 2 • 2 = 8 ) Record.
2
*Teacher Note: Review the process of multiplication of fractions as needed.
1
• Partner B, what do you notice about the cube root of 8 and the side length
1 3
of the cube with a volume of 8 u ? (They are the same value.) Record.
IP, CP, WG:
Have students work with a partner to complete the graphic
1 1
1
organizer on the top of S12 for the values of 6 , 8 , and 10 .
Then come back together as a class and share their results.
{Graphic Organizer, Algebraic Formula, Verbal Description}
Cube Root Equations with Fractions
M, WG, GP, CP:
(M, GP, WG, CP)
S12, S13 (Answers on T27 and T28.)
Have students continue to work on S12 in their books.
Make sure partners know their designation as Partner A or
Partner B. Students will work with cube roots in equations.
{Graphic Organizer, Algebraic Formula, Verbal Description}
Mathematics Success – Grade 8
T13
LESSON 7: Cube Roots
MODELING
Cube Roots Equations with Fractions
Step 1: Direct students’ attention to the last row of the graphic organizer.
• Partner A, explain what we did each time for the other values to find the cube root.
(Find the cube root of the whole number by determining what number
multiplied by itself 3 times would be equal to the value of the numerators
and then by determining what number multiplied by itself 3 times
would equal to the value of the denominator.) Record.
• Partner B, identify the information given in the first column of the last 1
row. ( a ) Record.
• Partner A, explain the meaning of the variable. (A variable can represent any number.)
• Partner B, explain why we can use a variable for this example. (Because, the same process can be used to determine the cube and cube root of
any value.) Record.
1
Step 2: Have student pairs discuss how they could use the variable a to find 1
the volume of cube with a side length of a . (Multiply length times width
1
1
1
13
times height: a units • a units • a units = a3 units3) Record in the
chart in the volume column.
*Teacher Note: Remind students that when we are using fractions we are
comparing the smaller cube to the larger cube. Also, because the numerators
we are working with are 1, the cube or the cube root of 1 is always 1.
• Partner A, explain how the value of the cubed number can be written 1
1
1
1
13
using the variable, a . ( a • a • a = a3 ) Record.
• Partner B, describe how we can find the value of the cube root of a
in
13
the last column. (Find the cube root of a3 by determining what variable
multiplied by itself 3 times would be the volume of the cube.)
3
• Partner A, what is the value of the cube root of a13 ? Explain
3
3
how you determined your answer. ( 1
• 1
• 1
= a13 so a13 = 1 ) Record
a
a
a
a
the cube root in the graphic organizer.
Now that we have explored cubing and cube roots with a variable,
let’s look at how we can use that information to solve equations with
cube roots.
Step 3: Have student pairs discuss the two basic concepts they need to remember
when solving any equation. Have students share answers.
• Partner A, what is the first goal when solving equations. (Isolate the variable.) Record.
• Partner B, what is the second goal when solving equations. (Balance the equation.) Record.
T14
Mathematics Success – Grade 8
LESSON 7: Cube Roots
Step 4: Direct students’ attention to the sample cube root equation in the graphic
organizer at the bottom of S12.
1
3
• Partner A, identify the equation (x
= 8 ).
• Partner B, how can we isolate the variable? (by finding the cube root of x3)
3
• Partner A, what is the cube root of x
? (x) Explain how you determined
3
this. (Because x • x • x = x )
• Partner A, how can we balance the equation? (by finding the cube root 1
of 8 )
1
1
• Partner A, what is the cube root of 8 ? ( 2 ) Explain how you determined
1
1
1
1
this. (Because 2 • 2 • 2 = 8 )
1
• Partner B, what is the value of x?
( 2 ) Record.
Step 5: Have student pairs work together to find the solution for the cube root equations on the top of S13. Remind students to also check their equation
by substituting back in the value for x.
Once student pairs have completed the 2 equations, have them share and
defend their responses by solving and showing the check of the equation.
Step 6: Direct students’ attention to the graphic organizer with the equation
1
x3 = 729 .
1
3
• Partner A, identify the equation x
= 729 .
• Partner B, how can we isolate the variable? (by finding the cube root of x3)
• Partner A, how can we balance the equation? (by finding the cube root 1
of 729 )
Step 7: Remind students that when they are finding the cube root of a fraction they must find the cube root of the numerator and the denominator.
• Have student pairs brainstorm possible strategies to break down 729 or methods they can think of to find the cube root.
• Have students share their ideas and then focus in on prime factorization.
Step 8: Model and explain the prime factorization of 729. Students may suggest
using the factor tree or the ladder method.
• Partner A, what are the prime factors of 729 (3 • 3 • 3 • 3 • 3 • 3 )
• Partner B, how many groups of factors do we want for the cube root? (3)
• Partner A, how can we write the prime factors as three values.
(32 • 32 • 32) or (9 • 9 • 9)
1
• Partner B, what is the value of x? ( 9 ) Record.