Estimating Roots – Grade Nine
Ohio Standards
Connection:
Number, Number Sense,
and Operations
Benchmark G
Estimate, compute and
solve problems involving
real numbers, including
ratio, proportion and
percent, and explain
solutions.
Indicator 4
Demonstrate fluency in
computations using real
numbers.
Benchmark I
Estimate, compute and
solve problems involving
scientific notation, square
roots and numbers with
integer exponents.
Indicator 5
Estimate the solutions for
problem situations
involving square and cube
roots.
Mathematical Processes
Benchmarks
A. Formulate a problem
or mathematical model
in response to a
specific need or
situation, determine
information required
to solve the problem,
choose method for
obtaining this
information, and set
limits for acceptable
solution.
Lesson Summary:
In this lesson, students develop strategies to solve problems
involving the estimation of square and cube roots. Students
use lists of square and cube numbers, the Pythagorean
Theorem and other mathematical tools to solve problem
situations. Working with partners provides students an
opportunity to share learning responsibility, use
mathematical terms and solve problems in a variety of
methods. Pre- and post-assessments support instruction by
demonstrating evidence of prior knowledge and student
learning.
Estimated Duration: Two to three hours
Commentary:
Students use the Pythagorean Theorem, introduced in grade
seven in the Academic Content Standards, to solve problems
involving right triangles. In this lesson, students estimate
(without calculators) the square root of numbers to solve
problem situations. Students should already have experience
with perfect squares and square roots. It is important for
students to be able to estimate the whole number nearest to a
given square root.
Mathematics involves more than computing with numbers. It
includes having a sense of what numbers and answers mean,
the relationship of the answer to the context of the problem
situation. Students must sense the reasonableness of the
solution to the problem situation.
Pre-Assessment:
 Pose the following problem to students:
A new swimming pool has been proposed for the community
park. The pool is square in shape and will cover an area of
400 square meters. What are the measurements of the sides
of the proposed pool?
 Students discuss the problem with partners and determine the
measurements. Provide grid paper and ask questions to
determine which students need assistance. Observe students’
strategies and determine which students knew that 400 was a
perfect square and used this to solve the problem. Do not use
calculators to find the lengths for the sides of the pool.
 Discuss the problem and students’ strategies. After discussing
the idea of perfect squares, direct students to write the squares
for the numbers one to 20. After students complete the task,
have them compare answers with partners or use a calculator.
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Estimating Roots – Grade Nine
Ohio Standards
Connection:
C.
G
Recognize and use
connections between
equivalent
representations and
related mathematical
concept; e.g., zero of a
function and xintercept of the graph
of the function, apply
proportional thinking
when measuring,
describing functions,
and comparing
probabilities.
Write clearly and
coherently about
mathematical thinking
and ideas.
Scoring Guidelines:
Observe partner and small group discussions. Check strategies
students use to find the lengths for the sides of the pool. Use
the rubric below to assess student prior knowledge.
Two Points
 Identifies lengths of side of pool using perfect square.
 Writes square numbers through 20 without calculator.
One Point
 Identifies length of the pool, using calculator or with
assistance.
 Writes square numbers through 12 without calculator.
Zero Points
 Does not identify length of the pool with assistance.
 Writes zero to 10 square numbers.
Post-Assessment:
Part One assesses student ability to estimate square roots; Part
Two assesses student ability to estimate cube roots.
Part One
 Distribute Estimating Roots Post-Assessment, Attachment
A. See answers on Attachment B.
Scoring Guidelines:
Three Points
 Locates Wes’s house, the fairgrounds and Jackson Road,
and determines the distances both ways (10 miles and
about seven miles).
 Uses mathematical or everyday language and/or logical
algebraic steps to display the correct method for solving
the problem.
Two Points
 Displays a logical method for solving the problem, but
does not arrive at the correct solution due to errors in
locating Wes’s house, the fairgrounds or Jackson Road.
 Minor calculation errors.
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Estimating Roots – Grade Nine
One Point
 Displays an incorrect or incomplete method for solving the problem arriving at an incorrect
solution.
OR
 Arrives at the correct solution while displaying only a partial method or no method for
solving the problem.
Zero Points
 Displays no understanding of how to approach solving the problem.
OR
 Does not attempt to solve the problem.
Part Two
Distribute Estimating Cube Roots Post-Assessment, Attachment C. See answers on Attachment
D.
Scoring Guidelines:
Three Points
 Solves the problem correctly.
 Uses mathematical or everyday language and/or logical algebraic steps to display the correct
method for solving the problem.
Two Points
 Displays a logical method for solving the problem, but does not arrive at the correct solution
due to calculation errors.
One Point
 Displays an incorrect or incomplete method for solving the problem arriving at an incorrect
solution.
OR
 Arrives at the correct solution while displaying only a partial method or no method of solving
the problem.
Zero Points
 Displays no understanding of how to approach solving the problem.
OR
 Does not attempt to solve the problem.
Instructional Procedures:
1. Complete the pre-assessment activity.
2. Pose the following problem to students:
The proposed swimming pool was too large for the space set aside in the park. A new
swimming pool is being proposed. The pool will remain a square, but
will only cover an area of 350 square meters.
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Estimating Roots – Grade Nine
The answer is about 18.7 meters.
3. Students discuss the problem and determine the approximate measurements for the sides of
the pool. Students may refer to their list of perfect squares. Observe students as they estimate
the length and record strategies or reasoning students used.
4. Discuss the problem with students. Ask several students to explain the strategies used for
estimating the measurements of the sides for the pool. Students should notice that 350 is
between 324 or 18 squared, and 361 or 19 squared. Students verify their answers using a
calculator. Some students may state that the answer is closer to 19 because 350 is closer to
361. This may be the case for this situation, but it is not always true.
5. Provide exercises for student to practice estimating the square roots of numbers. Students
supply the two whole numbers that the estimate falls between and reasoning for where to
place the root between the two whole numbers (is it close to one of the whole numbers, halfway between them, etc.). Only students needing assistance should use calculators. Students
show their reasoning in written or oral explanations. Facilitate a class discussion or assemble
small groups to check estimations and strategies.
115
250
31
750
475
6. Present students the following scenario:
Karma said that her grandmother’s house was 12 miles from her home. She explained she
drives five miles east on Route 224 and then turns north and drives seven miles on County
Road 30. Do you agree with Karma that her grandmother’s house is 12 miles from hers?
 Divide students into pairs and distribute a sheet of grid paper to each pair. Direct the students
to answer the question for the scenario. Observe strategies students use. Strategies may
include using the Pythagorean Theorem, counting squares or other invented non-routine
methods. Make note of student methods. Students should estimate between eight and nine
miles, with a more accurate estimation of 8.5 to 8.7 miles.
8. As a class, discuss the methods used to solve the problem. Ask the class which methods may
give the best estimate. Review the Pythagorean Theorem. Ask questions which clarify when
an estimate is sufficient, such as:
 Is it important to know the exact distance or is an estimate sufficient for this problem?
(An estimate is sufficient because the actual answer is not relatively close to the
estimate.)
 When using the Pythagorean Theorem, do we get an estimate or an exact answer?
(Exact answer)
9. Divide students into groups of four and number the groups. Distribute a piece of butcher
paper, chart paper or bulletin board paper to each group. Have groups draw four triangles
accurately on the papers, labeling the lengths of the sides (but not the hypotenuse) of each
different triangle. Groups should assign letters A through D to the problems.
10. At various locations around the room, groups tape their papers to the wall. As groups rotate
around the room, they estimate the lengths of the hypotenuses. Students record their
estimated measurements in journals or notebooks.
11. After completing the problems, each group presents the estimates of the hypotenuse for their
triangles. The class evaluates the reasonableness of the estimations. Develop consensus by
asking questions to encourage students to use the Pythagorean Theorem and the squares of
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Estimating Roots – Grade Nine
numbers.
12. Closure: Pose the following problem to students:
Suppose a fire truck and rescue helicopter left a fire station at the same time and traveling at
the same speed. They headed to a house located three blocks south and five blocks east of the
station. Which one got there first? How do you know?
Tell students to solve the problem and show their work. Allow time for students to share
answers.
13. Assign Attachment E, Estimating Square Roots, for homework. Use the answer key on
Attachment F, Estimating Square Roots Answer Key. Collect the paper and use it to
informally assess student progress.
Part Two
14. Present the following scenario to students:
Devon was given the job of building shelves in the company’s warehouse. The shelves are to
hold boxes which are cubes that have a volume of 27 cubic feet. What is the minimum width
he can make the shelf so the box fits completely on the shelf?
15. Have students discuss the problem with partners and determine the approximate
measurements. Provide unit cubes and clarifying questions for students who need assistance.
Observe students strategies and determine which students knew that 27 was a perfect cube
and used this to solve the problem. The shelf needs to be at least three feet or 36 inches wide
(accept one yard).
16. Discuss the problem and the strategies students used. After the idea of perfect cubes has been
shared, direct students to write the cubes for the numbers one to 10. After students complete
the task, they compare answers with partners or use calculators.
17. Pose the following situation:
The Choco-Sweet Company used cubed-shaped containers to package their specialty treats.
The original container was 125 cubic inches. In order to raise the company’s profit, a new
cubic container is being developed to hold 100 cubic inches of the specialty treats. What will
the approximate dimensions of the new container be?
18. Have students discuss the problem and determine the approximate measurements for the
sides of the new container. Students may refer to their list of perfect cubes. Observe students
as they estimate the length and record strategies or reasoning students used.
19. Discuss the problem with students. Ask several students to explain their strategies for
estimating the measurements of the sides for the new containers. Students should notice that
100 is in between the 64 (43) and 125 (53) but closer to 125. Estimates should be around 4.6
to 4.7. Students can verify the answer using a calculator.
20. Provide exercises to students to practice estimating the cube roots of numbers. They should
give the two whole numbers that the estimate falls between and reasoning for where to place
the root between the two whole numbers. Use calculators only for students needing
assistance or to verify answers. Students also show their reasoning in written or oral
explanations.
3
3
3
3
160
850
230
41
21. Distribute Attachment G, Estimating Cube Roots. See answers on Attachment H, Estimating
Cube Roots Answer Key. Students perform a paired reading. While reading, they ask each
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Estimating Roots – Grade Nine
other questions to clarify the problem, determine appropriate strategies to estimate and
identify the whole numbers the estimate falls between.
22. Observe students as they solve the problems and record the strategies students use to estimate
square and cube roots. Assist students who demonstrate little understanding of the task and
concept.
23. Ask several pairs to share their answers and lead the class to a consensus about the correct
solutions and the method(s) used to find them. Questions for discussion include:
 Between what two whole numbers is the estimate? What is your support?
 What determined the placement in relationship between the two whole numbers?
Differentiated Instructional Support:
Instruction is differentiated according to learner needs, to help all learners either meet the intent
of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified
indicator(s).
 Provide students who need assistance or do not recall square or cube numbers with a list of
the squares of natural numbers through 20 and the cubes of natural numbers through 10.
Provide calculators to verify their estimations. Additional materials such as grid paper and
unit cubes will help develop understanding of perfect squares and perfect cubes. Students can
draw or build models of numbers which make the perfect squares and cubes.
 Direct students who demonstrate understanding of the concept to create number lines
containing perfect squares, perfect cubes and the square and cubed roots found in this lesson.
Give exercises that go beyond 20-squared and 10-cubed. Challenge these students to develop
methods for estimating to the nearest tenth or hundredth.
 Students’ paired readings appeal to the verbal/linguistic learner. Reading graphs involving
directions satisfy the visual/spatial learner. Student work in pairs and small groups requires
interpersonal skills. Working through the problems uses the logical/mathematical
intelligence.
Extensions:
 This lesson is a great pre-assessment for teaching the distance formula.
 Ask students how they would estimate roots of larger numbers to lead them into simplifying
radicals.
Home Connections:
Students could look at maps at home to estimate distances when going on family outings. Using
the Pythagorean Theorem, students can determine the actual mileage driven versus the actual
distance from point A to point B.
Materials and Resources:
The inclusion of a specific resource in any lesson formulated by the Ohio Department of
Education should not be interpreted as an endorsement of that particular resource, or any of its
contents, by the Ohio Department of Education. The Ohio Department of Education does not
endorse any particular resource. The Web addresses listed are for a given site’s main page,
therefore, it may be necessary to search within that site to find the specific information required
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Estimating Roots – Grade Nine
for a given lesson. Please note that information published on the Internet changes over time,
therefore the links provided may no longer contain the specific information related to a given
lesson. Teachers are advised to preview all sites before using them with students.
For the teacher: Overhead transparency of each handout including grid paper, overhead
marker, supply of grid paper
For the student:
Grid paper, straight edge, tape, markers, calculators
Vocabulary:
 cube root
 hypotenuse
 perfect cube
 perfect square
 Pythagorean Theorem
 square root
Technology Connections:
 Use geometric software to draw triangles.
 Use word processing software to keep journal entries.
Research Connections:
"BSCS Science: An Inquiry Approach." BSCS Biological Sciences Curriculum Study. 23 Dec.
2003 http://63.225.114.218/bscsdotorg/curriculum/InquiryFAQs.htm
Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works:
Research-Based Strategies for Increasing Student Achievement, Alexandria, VA: Association for
Supervision and Curriculum Development, 2001.
National Council of Teachers of Mathematics (2000), Principles and Standards for School
Mathematics, Reston, Va.
Sousa, David A. How the Brain Learns: A Classroom Teacher’s Guide. Reston, Va: NASSP,
1995.
Attachments:
Attachment A, Estimating Square Roots Post-Assessment
Attachment B, Estimating Square Roots Post-Assessment Answer Key
Attachment C, Estimate the Length of the Side of a Cube Post-Assessment
Attachment D, Estimate the Length of the Side of a Cube Post-Assessment Answer Key
Attachment E, Estimating Square Roots
Attachment F, Estimating Square Roots Answer Key
Attachment G, Estimate the Length of the Side of a Cube
Attachment H, Estimate the Length of the Side of a Cube Answer Key
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Estimating Roots – Grade Nine
Attachment A
Estimating Roots Post-Assessment
Name:_____________________________
Date:_________________________
Directions: Read the scenarios and solve.
The carnival is in town at the county fairgrounds, and Wes is going on Saturday after his baseball
game. He’s not sure how far the fairgrounds are from the baseball field, but he knows it’s a
straight path southwest down Jackson Road. When he goes home to take a shower, he has to
drive five blocks west to his house, then another five blocks south to the fairgrounds.
1. Use the grid below to plot and label accurate locations for Wes’s house and the fairgrounds.
Draw and label Jackson Road.
When he arrived at the carnival, Wes realized he had left his wallet at the baseball field.
Being a wise young man, he couldn’t “knowingly” drive without his license, so he decided to
go back and get it by walking up Jackson Road.
2. About how many fewer miles did Wes travel when he went back to the baseball field than
when he went to the carnival the first time?
3. Justify your answer for Question Two. Show your method.
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Estimating Roots – Grade Nine
Attachment B
Attachment A Post-Assessment Answer Key
The carnival is in town at the county fairgrounds, and Wes is going on Saturday after his baseball
game. He’s not sure how far the fairgrounds are from the baseball field, but he knows it’s a
straight path southwest down Jackson Road. When he goes home to take a shower, he has to
drive five blocks west to his house, then another five blocks south to the fairgrounds.
1.
Use the grid below to plot and label accurate locations for Wes’s house and the
fairgrounds. Draw and label Jackson Road.
When he arrived at the carnival, Wes realized he had left his wallet at the baseball field. Being a
wise young man, he couldn’t “knowingly” drive without his license, so he decided to go back
and get it by walking up Jackson Road.
2. About how many fewer miles did Wes travel when he went back to the baseball field than
when he went to the carnival the first time? Three miles
3. Justify your answer for question number 2. Show your method. Wes walked 10 miles the
first time. He walked about seven miles from the fairgrounds to the baseball field.
52+ 52 = c2
25 + 25 =
50
50 is about 7, because 72 =
49
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Estimating Roots – Grade Nine
Attachment C
Estimating the Length of a Side of a Cube Post-Assessment
Name:_______________________________
Date:_____________________
Directions: Read the situation with a partner and answer the questions.
When Wes got back to the carnival, he and a friend went to the dunking booth. It gave Wes an
idea to put a dunking booth at the baseball field. That way, pitchers could work on their accuracy
while cooling off the position players (everyone else).
While he was standing in line thinking, he called his dad from his cell phone to ask if he could
get about 1000 gallons of water donated from his dad’s pool supply company. The tanks come in
whole-foot sizes only. Due to cost, Wes needed to come up with the dimensions of the smallest
cube-sized tank that would hold that amount of water.
1. Using the fact that one gallon equals about .1336 cubic feet of water, determine about how
many cubic feet of space he needs. Show your work.
2. What are the dimensions of the smallest cube-shaped tank he could get?
3. Describe how you arrived at your solution for Question Two.
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Estimating Roots – Grade Nine
Attachment D
Estimating Cube Roots Post-Assessment Answer Key
1. Using the fact that one gallon equals about .1336 cubic feet of water, determine about how
many cubic feet of space he needs. Show your work.
1000 * 0.1336 = 133.6 cu.ft.
2. What are the dimensions of the smallest cube-shaped tank he could get?
6 x 6 x 6 = 216 cu. ft., because 5 x 5 x 5 =125 would be too small.
3. Describe how you arrived at your solution for Question Two.
Accept reasonable answers and solutions.
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Estimating Roots – Grade Nine
Attachment E
Estimating Square Roots
Name _______________________________ Date
____________________
Carolyn heard there is a free contest going on at Mike’s Dollar Store and she wants to go there as
fast as possible right after school. Her school is at the corner of Middle Road and Town
Boulevard. The store is several blocks away at the corner of Timothy Drive and Ethan Avenue.
Please use this information and the grid below to complete the following exercises.
1. On the grid, locate and label Carolyn’s school and Mike’s Dollar Store.
Carolyn wants to beat classmates to the store, so she takes a shortcut and rides her bike through
the field, directly to the store. The others use the roads. If she rides at the same rate as the others,
prove she will arrive at the store before they will.
2.
How far will Carolyn’s classmates travel?
3.
Estimate, using the Pythagorean Theorem, how far Carolyn will travel (in “city blocks”).
Show work here.
4. What are some ways to know if your estimate is reasonable for this situation?
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Estimating Roots – Grade Nine
Attachment F
Estimating Square Roots Answer Key
1. On the grid locate and label Carolyn’s school and Mike’s Dollar Store.
2. How far will Carolyn’s classmates travel?
3.
8 city blocks
Estimate, using the Pythagorean Theorem, how far Carolyn will travel (in “city blocks”).
about 6 city blocks
Show work here.
4. What are some ways to know if your estimate is reasonable for this situation?
Answers vary.
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Estimating Roots – Grade Nine
Attachment G
Estimating the Length of a Side of a Cube
Name:_____________________________ Date:_____________________________
Directions: Read the situation with a partner and answer the questions.
Carolyn arrives at Mike’s Dollar Store ahead of her friends. She enters the store and sees this
sign on the counter:
HELP WANTED!
Jelly Bean Packaging
Job Description: Design cube-shaped containers
Solve this problem and you get the job!!!
Jelly beans are ordered in bulk by the case. I want to make cubeshaped containers filled with 500 jelly beans to sell for $1.
Containers come in whole inch sizes. You need to create the
smallest possible container that will hold 500 jelly beans.
Carolyn knows she has only a few minutes head start, so she gets right to work. Next to the sign
is a sample of six jelly beans. She observes that the paper cup they are in is about 1inch by 1inch
by 1inch. Carolyn is confident she can solve the problem before the others even get started - and
without a calculator.
Follow Carolyn’s thought process by answering the following questions:
1. How many jelly beans are in one cubic inch?
2. What is the approximate volume of 500 jelly beans?
3. Estimate the size of the smallest cube-shaped jar that would hold 500 jelly beans.
4. Explain why this is a reasonable estimate.
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Estimating Roots – Grade Nine
Attachment H
Estimating the Length of a Side of a Cube Answer Key
Follow Carolyn’s thought process by answering the following questions:
1. How many jelly beans are in one cubic inch?
about 6
2. What is the approximate volume of 500 jelly beans?
500
~83cu.in.
6
3. Estimate the size of the smallest cube-shaped jar that would hold 500 jelly beans.
5 x 5 x 5 = 125 cu. in.
4. Explain why this is a reasonable estimate.
3
4 = 64, but that is not big enough to hold the jelly beans. You have to go to the next
larger size to hold them all.
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