Download Making Sense of Irrational Roots Lesson Plan

5E Lesson Plan
Teachers: Uhling, Escobar, Scherer, Moore
Date: 9/11/2014
Subject / grade level: 8th Grade Math
Materials:
1. Double Sided counters
2. Grid paper
3. Square Roots Go Rational Student Worksheet
4. Calculators
5. Clothesline
6. Clothes pins
TEKS:
1. 8.2.B approximate the value of an irrational number, including π and square roots of
numbers less than 225, and locate that rational number approximation on a number
line;
KEY UNDERSTANDINGS:
1) That squaring a number and finding the square roots are inverse or opposite operations.
2) That irrational square roots can only be approximated.
3) We can start with figuring out which two perfect squares that they are between.
ENGAGEMENT
1. Teacher will put up the multiplication chart with the diagonal highlighted.
2. Ask students:
 What numbers are highlighted?
 What do you notice about the highlighted numbers?
 Do you see any patterns in these numbers?
EXPLORATION
1. Teacher tells students that today we will be exploring making perfect and not so perfect squares.
2. Ask students:
 What is a perfect square?
 What is a square root?
 What is a natural number? What is a rational number? An irrational number? (want to get
across that irrational numbers can only be approximated, they are non-repeating decimals that
CAN’T be expressed as a fraction.
3. Tell students that we are going to make models of perfect and not so perfect squares to help us understand this
concept.
4. Pass out double sided counters (one bag per student).
5. Have students create a square using 4 counters, 9 counters, 16 counters.
6. Next challenge students to create a square using 7 counters: They can’t do this. Model for them the process
described below of estimating. Instead of using x’s and o’s, use yellow for the 7 and then red for the two that
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5E Lesson Plan
they would need to fill in to get to the perfect square of 9:
Main Activity
The key to the rational roots students will be exploring during the lesson lies in the outer column and row of each perfect
square. Ask students to predict the square root of 7.
7 X's cannot fill a square, so 7 is not a perfect square. However, the square root of 7 can be estimated. The square
root must be between 2 and 3 because 7 lies between the perfect squares 4 and 9.
From this model, the largest perfect square is a square with side lengths of 2 with three remaining units in the outer column
and row. 5 X's are needed in the outer row and column to make the next larger perfect square. The missing units from the
outer column and row are filled in with O's to reinforce the counting of these elements. Therefore, an approximation to the
square root of 7 is very close to 2 3/5. The 3 represents the extra number of units (X's) in the outer row and column; and the
5 represents the total number of units (X's and O's) in the outer row and column. The meaning of a fraction is enforced with
this model. The fractional part of the square root is the part of units we have divided by the whole number of units needed in
the outside column and row.
7.
Challenge students to approximate the 11 using the same process. Give them a few minutes to work on their
own and then check in with them on their progress.
8. Allow the students to work on the first column, ‘Rational Root Expressed as a Fraction,’ independently. Circulate
around the room assisting individual students as needed. If students want to consult with a partner, they may
but each student is expected to complete their own worksheet.
EXPLANATION
1. After the majority of the class has completed the first column, invite individuals to share out their results.
2. Ask students to comment on any patterns they noticed.
ELABORATION
1. Tell the students that the next step will be to convert their fractional approximations to decimals. Tell them that
you are going to let them use the calculator to do this step BUT they need to know how to convert fractions to
decimals. Call on a student to explain how to do this.
2. Pass out calculators. Complete the remaining columns, ‘Rational Roots Expressed as a Decimal,” and
‘Calculator Square Root Expressed as a Decimal,’ for the numbers 2 and 3 together.
3. Allow students to complete the rest of the sheet independently using calculators.
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4. If time allows, you can ask students this challenge question: What natural number would have 8
as its
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rational root?
EVALUATION
1. Tell students that their ‘ticket out the door’ is going to be to arrange a variety of real numbers (which includes
both rational and irrational numbers’ in order on the clothesline. No one leaves until the numbers are arranged
properly. Students may help each other out, but they are to explain to their classmates their reasoning as they
do it (not just snatch the number out of their hands).
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5E Lesson Plan
2. Pass out numbers and allow students to work on ordering the numbers. Provide coaching/questioning as
needed.
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