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IM 8 Ch 5.2.3 When are they the same (Part 3) CPM Materials modified by Mr. Deyo How can you find a rule? How can you compare two rules? How can you use what you know about solving? Common Core Standard: 8.EE.8b, 8.EE.8c

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Page 1: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

CPM Materials modified by Mr. Deyo

How can you find a rule?

How can you compare two rules?

How can you use what you know about solving?

Common Core Standard:   8.EE.8b, 8.EE.8c

Page 2: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

By the end of the period, I will solve systems of equations algebraically when both equations are 

in  y = mx + b form.

I will demonstrate this by completing Four‑Square notes and by solving problems in a pair/group activity.

Learning TargetTitle:  IM8 ‑ Ch. 5.2.3 When are they the same   (Part 3)? Date:

Page 3: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

Home Work: Sec. 5.2.3Desc.         Date Due

Review & Preview 

3 Problems:     5‑47, 5‑48, 5‑50

Page 4: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

Page 5: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

Vocabulary1)  System of Equations

2)  Point of Intersection

3)  Equal Values Method

4)  Slope­Intercept Form

Page 6: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

Page 7: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5.2.3 When Are They The Same (Part 3)?So far in Section 5.2, you have solved systems of equations by graphing two lines and finding where they intersect.  However, it is not always convenient (nor accurate) to solve by graphing.

Today you will explore a new way to approach solving a system of equations.  Questions to ask your teammates today include:

How can you find a rule?

How can you compare two rules?

How can you use what you know about solving?

Page 8: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5­41. CHUBBY BUNNY Use tables and a graph to find and check the solution for the problem below.

Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year.  Her cat weighs 19 pounds and gains 1 pound per year.  When will the bunny and the cat weigh the same amount? 

   What is the rule?       y = (    )x + (    )

   What is the rule?       y = (    )x + (    )

Bunny Cat

The bunny and the cat weigh the same amount _____________________

_____________________________________________________________. 

Page 9: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5­42. SOLVING SYSTEMS OF EQUATIONS ALGEBRAICALLY

c)  How much do the cat and bunny weigh at this time?

In problem 5­41, you could write rules like those shown here to represent the weights of Barbara’s cat and bunny.  For these rules,  x  represents the number of years and  y  represents the weight of the animal.

http://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__LaaabtcPaaaXb3avg1bxavg1b_avg1ctavhwctavhwbxavhwb_avh­ctavhwcYavh­b_avh­cYavg1cYavh­bxavisbxavisb_avisctaviscYaviXb_aviXctaviXcYabj_b.avdXcUavc.clabdYa6avbFbnavcDaTavc4bkayalai5­42__ayb­gDWeight%20of%20the%20Bunny__ayhigEWeight%20of%20the%20Cat__

a)  Problem 5­41 asked you to determine when the weight of the cat and the bunny are the same.  Therefore, you want to determine when the expressions on the left (for the bunny) and the right (for the cat) are equal.  Write an equation that represents this balance.

b)  Solve your equation for  x, which represents years.  According to your solution, how many years will it take for the bunny and the cat to weigh the same number of pounds?  Does this answer match your answer from the graph of problem 5­41?

Page 10: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5­43 CHANGING POPULATIONSPost Falls High School in Idaho has 1160 students and is growing by 22 students per year.  Richmond High School in Indiana has 1900 students and is shrinking by 15 students per year.

a)  Without graphing, write a rule that represents the population at Richmond High School and another rule that represents the population at Post Falls High School.  Let  x  represent years and  y  represent population.

Post Falls:  y = (    )x+(       )b)  Graphing the rules for part (a) is challenging because of the large numbers involved.  Using a table could take a long time.  Therefore, this problem is a good one to solve algebraically, the way you solved problem 5­42. Use the rules together to write an equation that represents when these high schools will have the same population.  Then solve your equation to find out when the schools’ populations will be the same.

Richmond:  y = (    )x+(       )

c)  What will the population be at that time?

Page 11: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5­44. PUTTING IT ALL TOGETHERFind the solution to the problem below by graphing and also by solving an equation.  The solutions using both methods should match, so be sure to review your work carefully if the results disagree.Imagine that your school planted two trees when it was first opened.  One tree, a ficus, was 6 feet tall when it was planted and has grown 1.5 feet per year.  The other tree, an oak, was grown from an acorn on the ground and has grown 2 feet per year.  When will the trees be the same height?  How tall will the trees be when they are the same height? 

Oak:    y = (    )x+(       )Ficus:  y = (    )x+(       )

Page 12: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

In your Learning Log, describe how to find the intersection point or solution of two equations when they are written in the form y = . . .  Create an example and show how to solve it. 

5­46. LEARNING LOG ­ “Equal Values Method”    Date ________

5­45. Ms. Harlow calls the method you have been using today to solve equations the Equal Values Method.  Explain why this name makes sense. 

Page 13: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5­47   Ariyonne claims that (3, 6) is the point of intersection of the lines y = 4x − 2 and y = 0.5x + 5.  Is she correct?  Yes / No  How do you know?

y = 4x ­ 2

http://homework.cpm.org/cpm­homework/homework/category/CC/textbook/CC3/chapter/Ch5/lesson/5.2.3/problem/5­47

y = 0.5x + 5

Page 14: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5­48 Graph the lines y = 2x − 3  and  y = −x + 3.a)  Where do they intersect?  Label the point on the graph. (    ,    )b)  Find the point of intersection using the Equal Values Method.  That is, start by combining both equations into one equation that you can solve for  x.

http://homework.cpm.org/cpm­homework/homework/category/CC/textbook/CC3/chapter/Ch5/lesson/5.2.3/problem/5­48

2x ­ 3 = ­x + 3

c)  Which method is easier,    graphing    or     using algebra    to solve?

Page 15: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

a)  Solve for y  

5­49    Solve each of the following equations for the indicated variable.  Show all of your steps.  Then check each solution.

b)  Solve for a

7y 8

3y 5­ = a+4

  3a7­ =

http://homework.cpm.org/cpm­homework/homework/category/CC/textbook/CC3/chapter/Ch5/lesson/5.2.3/problem/5­49

a+7  5

11 2

Page 16: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5­50 Graph the lines y = 2x − 3  and  y = 2x + 1.a)  Where do they intersect?  

b)  Solve this system using the Equal Values Method.

2x ­ 3 = 2x + 1

c)  Explain how your graph and algebraic solution relate to each other. 

https://www.desmos.com/calculator/sizmtknysv http://homework.cpm.org/cpm­homework/homework/category/CC/textbook/CC3/chapter/Ch5/lesson/5.2.3/problem/5­51

Page 17: How can you find a rule? rules? know about solving?helpmeteach.weebly.com/uploads/1/7/1/0/17102056/... · IM 8 Ch 5.2.3 When are they the same (Part 3) In your Learning Log, describe

IM 8  Ch 5.2.3 When are they the same (Part 3)

5­51    Valeria came to bat 464 times in 131 games.  At this rate, how many times should she expect to have at bat in a full season of 162 games? (Answer in a complete sentence.)

http://homework.cpm.org/cpm­homework/homework/category/CC/textbook/CC3/chapter/Ch5/lesson/5.2.3/problem/5­51