Common Core Math 8
Unit 1, Days 5-6
Topic: Estimating Square Roots
Common Core State Standard(s):
Instructional Resources:
8.NS.2 Use rational approximations of irrational
 Estimating Square Roots
numbers to compare the size of irrational numbers,
 Approximate and Check
locate them approximately on a number line diagram,
 Holt 3-9 Resources
and estimate the value of expressions (e.g., π2). For
 Simplifying Square Roots Supplement
example, by truncating the decimal expansion of √2,
 Estimating Square Roots without a
show that √2 is between 1 and 2, then between 1.4 and
Calculator
1.5, and explain how to continue on to get better
 Comparing Square Roots
approximations.
 Evidence of Student Learning Exit Ticket
Essential Vocabulary:
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Mathematical Practice Standard(s):
MP.3 Construct viable arguments and critique the
reasoning of others.
MP.5 Use appropriate tools strategically
Essential Understandings:
Students will understand that…
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Some numbers have integer square roots and
cube roots
They can use perfect squares and perfect cubes
to evaluate small perfect square roots and
small perfect cube roots
The square roots of non-perfect square
numbers fall between the square roots of
perfect squares on the number line
March 2015
Integer
Irrational Numbers
Perfect Cube
Perfect Square
Radical
Radicand
Rational Numbers
Root
Square Root
Whole Number
Warm-up:
 Day 5: Display the Day 5 warm up questions
on the overhead or under the document
camera. Allow students time to respond to the
questions then discuss.
 Day 6: WITHOUT a calculator…Estimate
√39 to the nearest tenth. Compare your
answer with your partner’s and discuss why
you chose the estimate you did. As a class,
discuss partner estimates. Call on students to
explain their reasoning. Use this as a
springboard for today’s discussion about
estimating square roots without a calculator.
Common Core Math 8
Unit 1, Days 5-6
Topic: Estimating Square Roots
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They can use the number line to order and
estimate irrational numbers
Essential Questions:
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How can you use inverse operations to find a
perfect square root or perfect cube root?
How can you approximate the square root of a
non-perfect square?
Instruction:
Day 5:
 Write √15 on the board and ask for a volunteer to tell you what it equals. Help students understand that
√15 will not be a whole number because 15 is not a perfect square. Ask students what number is a
perfect square that is close to 15. Show how students can use their knowledge of perfect squares to
determine which two integers √15 falls between. Then have students enter √15 into their calculator.
You may have to teach them how to do this if they have never done it before. Discuss how √15 is a
non-terminating, non-repeating decimal. (At this point, students do not need to be able to classify √15
as an irrational number. They will do this in Unit 2. Just make sure they understand that all of the
square roots of non-perfect squares will be non-terminating, non-repeating decimals.)
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Facilitate students’ completion of Estimating Square Roots.
Facilitate partner Approximate and Check activity.
Guided Practice Strategies or Activities:
 Holt Reteach 3-9
 Simplifying Square Roots Supplement
 Holt Problem Solving 3-9
Independent Practice:
 Holt Exploration 3-9
Day 6:
 Discuss at least one method for finding an approximation of a square root without the use of a
calculator. One method straight from the DPI unpacked standards is the following:
Find an approximation for √28.
 Determine the perfect squares √28 is between, which would be 25 and 36.
 The square roots of 25 and 36 are 5 and 6 respectively, so we know that √28 is
between 5 and 6.
 Since 28 is closer to 25, an estimate of the square root would be closer to 5. One
method to get an estimate is to divide 3 (the distance between 25 and 28) by 11 (the
March 2015
Common Core Math 8
Unit 1, Days 5-6
Topic: Estimating Square Roots
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distance between the perfect squares of 25 and 36) to get 0.27.
 The estimate of √28 would be 5.27 (the actual approximation to two decimals is
5.29).
Have students use this method to determine √39 then compare their answer to their estimate at the
beginning of class. Using this method, students should get 6.23. (the actual approximation to two
decimals is 6.24)
Guided Practice Strategies or Activities:
 Facilitate Estimating Square Roots without a Calculator
 Partner – Comparing Square Roots activity
Independent Practice:
 Holt Practice 3-9 B – Alter the instructions to have the students do the middle section without a
calculator utilizing the method for estimating the square roots discussed in class
Evidence of Student Learning:
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Have students complete the Evidence of Student Learning Exit Ticket to see if they are able to graph
the square roots of perfect and non-perfect squares on the number line.
Comments / Notes:
 Differentiate with grouping strategies for the partner activities. Activities could be modified so that
Partner B’s problems were all significantly easier that Partner A’s.
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Optional calculator activity
(http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=11730&addactivity
=Y )
Other methods for calculating square roots without a calculator:
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
March 2015