G6 m2-d-lesson 19-t

178
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•2Lesson 19
Lesson 19: The EuclideanAlgorithm as an Applicationofthe Long Division
Algorithm
Date: 1/9/15
Lesson 19: The Euclidean Algorithm as an Application of the
Long Division Algorithm
Student Outcomes
 Students explore and discover that Euclid’s Algorithmis a more efficient means to findingthe greatest
common factor of larger numbers and determine that Euclid’s Algorithmis based on long division.
Lesson Notes
Students look for and make use of structure, connecting longdivision to Euclid’s Algorithm.
Students look for and express regularity in repeated calculations leadingto findingthe greatest common factor of a pair
of numbers.
These steps arecontained in the Student Materials and should bereproduced, so they can be displayed throughoutthe
lesson:
Euclid’s Algorithmis used to find the greatest common factor (GCF) of two whole numbers.
1. Dividethe larger of the two numbers by the smaller one.
2. If there is a remainder, divideitinto the divisor.
3. Continue dividingthe lastdivisorby the lastremainder until the remainder is zero.
4. The final divisor istheGCF of the original pair of numbers.
In application,the algorithmcan be used to find the sidelength of the largestsqua rethat can be used to completely fill a
rectangle so that there is no overlap or gaps.
Classwork
Opening(5 minutes)
Lesson 18 Problem Set can be discussed beforegoing on to this lesson.
MP.7
MP.8

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Algorithm
Date: 1/9/15
OpeningExercise (4 minutes)
 There are 3 division warm-ups on your Student Material pagetoday. Please
compute them now. Check your answer to make sure itis reasonable.
Opening Exercise
Euclid’sAlgorithmisused to find the greatest commonfactor (GCF)oftwo whole numbers.
1. Divide the larger ofthetwo numbersby thesmallerone.
2. If there isaremainder, divide itinto the divisor.
3. Continue dividing thelast divisor by thelast remainder until theremainder iszero.
4. The final divisor isthe GCF ofthe originalpair ofnumbers.
𝟑𝟖𝟑÷ 𝟒 = 𝟗𝟓.𝟕𝟓 𝟒𝟑𝟐÷ 𝟏𝟐 = 𝟑𝟔 𝟒𝟎𝟑÷ 𝟏𝟑 = 𝟏𝟏
Discussion(2 minutes)
 Our opening exercisewas meant for you to recall howto determine the quotient of two numbers using long
division. You have practiced the long division algorithmmany times. Today’s lesson is inspired by Euclid,a
Greek mathematician, who lived around 300 BCE. He discovered a way to find the greatest common factor of
two numbers that is based on long division.
Example 1 (10 minutes): Euclid’sAlgorithmConceptualized
 What is the GCF of 60 and 100?
 20
 What is the interpretation of the GCF in terms of area? Let’s take a look.
Project the followingdiagram:
Example 1: Euclid’s Algorithm Conceptualized
Scaffolding:
 Students who arenot
fluent usingthe standard
division algorithmor have
difficulty estimatingcan
use a calculator to test out
multiples of divisors.
100 units
20 units
60 units
20 units

180
Algorithm
Date: 1/9/15
 Notice that we can use the GCF of 20 to create the largestsquaretilethat will cover the rectangle without any
overlap or gaps. We used a 20 × 20 tile.
 But, what if we didn’t know that? How could we start? Let’s guess! What is the biggest squaretile that we
can guess?
 60 × 60
Display thefollowingdiagram:
 It fits,but there are 40 units left over. Do the division problemto prove this.
Teacher note: with each step in this process,pleasewrite out the longdivision algorithmthat
accompanies it.
It is importantfor students to make a record of this as well. The remainder becomes the new
divisor and continues until theremainder is zero.
 What is the leftover area?
 60 × 40
 What is the largestsquaretilethat we can fitin the leftover area?
 40 × 40
60 units
60 units 40 units
100 units
1
60 100
− 60
40

181
Algorithm
Date: 1/9/15
Display thefollowingdiagram:
 What is the leftover area?
 20 × 40
 What is the largestsquaretilethat we can fitin the leftover area?
 20 × 20
 When we divide40 by 20, there is no remainder. We have tiled the whole rectangle!
 If we had started tilingthe whole rectangle with squares,the largestsquarewe could
have used would be 20 by 20.
Take a few minutes to allowdiscussion,questions,and clarification.
Example 2 (5 minutes): Lesson 18 Classwork Revisited
DuringLesson 18, students found the GCF of several pairs of numbers.
 Let’s apply Euclid’s Algorithmto some of the problems from our lastlesson.
Example 2: Lesson 18 Classwork Revisited
a. Let’sapply Euclid’sAlgorithm to someofthe problemsfrom our last lesson.
i. What isthe GCF of 𝟑𝟎and 𝟓𝟎?
𝟏𝟎
ii. Using Euclid’sAlgorithm, we follow thestepsthat are listedin theopening exercise.
When theremainder is zero, thefinal divisor is theGCF.
60 units
60 units 40 units
100 units
2
20 40
− 40
0

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Algorithm
Date: 1/9/15
b. What isthe GCF of 𝟑𝟎and 𝟒𝟓?
𝟏𝟓
Example 3 (5 minutes): Larger Numbers
Example 3: Larger Numbers
GCF (𝟗𝟔,𝟏𝟒𝟒 GCF (𝟔𝟔𝟎,𝟖𝟒𝟎
Example 4 (5 minutes): Area Problems
Example 4: AreaProblems
The greatest common factor hasmany uses. Among them,the GCF letsusfind out the maximum size ofsquaresthat will
cover arectangle. Whenever wesolve problemslikethis, we cannot haveany gapsor any overlapping squares. Of
course, the maximumsize squareswillbe theminimum number ofsquaresneeded.
A rectangular computer table measures 𝟑𝟎inchesby 𝟓𝟎inches. We need to cover itwith squaretiles. What isthe side
length ofthe largest square tile wecan use to completely cover the table, so that thereisno overlap or gaps?
50 50
30 30

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Algorithm
Date: 1/9/15
Direct students to consider the GCF (30,50 , which they already calculated. It should
become clear thatsquares of 10 inches by 10 inches will tilethe area.
c. If we use squaresthat are 𝟏𝟎by 𝟏𝟎, how many will we need?
𝟑 · 𝟓, or 𝟏𝟓squares
d. If thiswere agiant chunk ofcheese in afactory, would it changethethinking or the
calculationswe just did?
No.
e. How many 𝟏𝟎inch × 𝟏𝟎inch squaresofcheese could be cut from thegiant 𝟑𝟎inch × 𝟓𝟎
inch slab?
𝟏𝟓
Closing(3 minutes)
 Euclid’s Algorithmis used to find the greatest common factor (GCF) of two
whole numbers. The steps arelisted on your Student Page and need to be
followed enough times to get a zero remainder. With practice,this can be a
quick method for finding the GCF.
 Let’s use the steps to solvethis problem: Kiana is creatinga quiltmadeof
squarepatches. The quiltis 48 inches in length and 36 inches in width. Whatis
the largestsizeof squarelength of each patch?
 Dividethe larger of the two numbers by the smaller one. What is the quotient?
 1
 If there is a remainder, divideitinto the divisor. Whatis theremainder?
 12
 What is the divisor?
 36
 Divide36 by 12. What is the quotient?
 3
 Continue dividingthe lastdivisorby the last remainder until the remainder is
zero. Is the remainder zero?
 Yes
 The final divisor istheGDF of the original pair of numbers. Whatis the final
divisor?
 12
Exit Ticket (5 minutes)
Scaffolding:
 Ask students to compare
the process of Euclid’s
Algorithm with a
subtraction-only method
of findingthe GCF:
1. Listthe two numbers.
2. Find their difference.
3. Keep the smaller of the
two numbers; discard the
larger.
4. Use the smaller of the
two numbers and the
difference to form a new
pair.
5. Repeat until the two
numbers are the same.
This is the GCF.
Teacher resource:
http://www.youtube.com/watc
h?v=2HsIpFAXvKk
Scaffolding:
 To find the GCF of two
numbers, give students
rectangular pieces of
paper usingthe two
numbers as length and
width (e.g., 6 by 16 cm.)
 Challengestudents to
mark the rectangles into
the largestsquares
possible,cutthem up, and
stack them to assuretheir
squareness.
 GCF(6,16)=2

184
Algorithm
Date: 1/9/15
Name ___________________________________________________ Date____________________
Lesson 19: The Euclidean Algorithm as an Application of the Long
Division Algorithm
Exit Ticket
Use Euclid’s Algorithmto find the Greatest Common Factor of 45 and 75.

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Algorithm
Date: 1/9/15
Exit Ticket Sample Solutions
Use Euclid’sAlgorithmto find thegreatest common factor of 𝟒𝟓and 𝟕𝟓.
The GCF of 𝟒𝟓and 𝟕𝟓is 𝟏𝟓.
Problem Set Sample Solutions
1. Use Euclid’sAlgorithmto find the greatest common factor ofthe following pairsofnumbers:
a. GCF of 𝟏𝟐 and 𝟕𝟖.
GCF of 𝟏𝟐and 𝟕𝟖 = 𝟔
b. GCF of 𝟏𝟖 and 𝟏𝟕𝟔
GCF of 𝟏𝟖and 𝟏𝟕𝟔 = 𝟐
2. Juanitaand Samuel are planning apizzaparty. They order a rectangular sheet pizzawhich measures 𝟐𝟏inchesby
𝟑𝟔inches. They tell the pizzamaker not to cut itbecause they want to cutit themselves.
a. All piecesofpizzamust be square withnoneleft over. What isthe length ofthe sideofthe largest square
pieces into which Juanitaand Samuelcan cut the pizza?
LCM of 𝟐𝟏and 𝟑𝟔 = 𝟑. They can cut thepizza into 𝟑by 𝟑inchsquares.
b. How many piecesofthissize will there be?
𝟕 · 𝟏𝟐 = 𝟖𝟒. Therewill be 𝟖𝟒pieces.
3. Shelly and Mickelle aremaking aquilt. They have apieceoffabric that measures 𝟒𝟖inchesby 𝟏𝟔𝟖inches.
a. All piecesoffabric must besquare withnoneleft over. What isthe lengthofthe side ofthelargest square
pieces into which Shelly and Mickelle can cut thefabric?
GCF of 𝟒𝟖and 𝟏𝟔𝟖= 𝟐𝟒.
b. How many piecesofthis size will there be?
𝟐 · 𝟕 = 𝟏𝟒. Therewill be 𝟏𝟒pieces.

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