what is the rule? how can you represent the...

IM 8 Ch 3.1.3 What Is A Graph & How Is It Useful

CPM Materials modified by Mr. Deyo

What is the rule?

How can you represent the pattern?

Common Core Standard: 8.F.1, 8.F.2, 8.F.3, 8.F.4


By the end of two periods, I differenate between discrete and connuous graphs as well as use a graph to verify a predicon.

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

Learning TargetTitle: IM8 ‐ Ch. 3.1.3 ‐ What Is A Graph & How Is It Useful? Date:


Home Work: Sec. 3.1.3Desc. Date Due

Review & Preview

Day 1: 3 Problems: 3‐23, 3‐24, 3‐27

Day 2: 4 Problems: 3‐28, 3‐29, 3‐31, 3‐32


Vocabulary1) Equation

2) Solution

3) Discrete

4) Continuous


3.1.3 What Is A Graph and How Is It Useful?In the last two lessons, you examined several patterns. You learned how to represent the patterns in a table and with a rule. For the next few days, you will learn a powerful new way to represent a pattern and make predictions.

While working today, focus on these questions:

What is the rule?

How can you represent the pattern?


Draw the 0 figure.

Represent the number of tiles in each figure with:An x → y table.

An algebraic rule: y = ( )x + ( )

Draw the 4th figure.

P 318a&bhttp://www.cpm.org/pdfs/stuRes/CC3/chapter_03/CC3%20Lesson%203.1.3%20RP.pdf

http://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__XbbacRbqbadibqbacRbVbadibVbacRcmbacRcRbadicRbacRdibadidibafadibafacRbafacmbaf6bVbaf6bqbafadNbafaeebaf6eJbagBfabagBeJbafaeJbagBbVbagBbqbafabVbafabqbafFbqbabqhxbabVhxbabVh_bafFbVbaaZhxbaaZh_babqh_baaZg2baaZgBbaaZf6babVfFbabqfFbabqfabaaZfababVfabaaZfFbafFeJbaf6fabafafabafFfaaycJecFigure%201__aya5iFFigure%202__ayfDfYFigure%203__ayaKauCC3%20318__ayaMdtFigure%200__ayiUhBFigure%204__

http://www.cpm.org/pdfs/stuRes/CC3/chapter_03/CC3%20Lesson%203.1.3%20RP.pdfhttp://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__XbbacRbqbadibqbacRbVbadibVbacRcmbacRcRbadicRbacRdibadidibafadibafacRbafacmbaf6bVbaf6bqbafadNbafaeebaf6eJbagBfabagBeJbafaeJbagBbVbagBbqbafabVbafabqbafFbqbabqhxbabVhxbabVh_bafFbVbaaZhxbaaZh_babqh_baaZg2baaZgBbaaZf6babVfFbabqfFbabqfabaaZfababVfabaaZfFbafFeJbaf6fabafafabafFfaaycJecFigure%201__aya5iFFigure%202__ayfDfYFigure%203__ayaKauCC3%203-18__ayaMdtFigure%200__ayiUhBFigure%204__


Make a graph of the data.

P 318b&chttp://www.cpm.org/pdfs/stuRes/CC3/chapter_03/CC3%20Lesson%203.1.3%20RP.pdf


How many tiles will be in Figure 5? Justify your answer in at least two different ways.

1)

2)



What will Figure 100 look like? How many tiles will it have?

How can you be sure?

P 318dhttp://www.cpm.org/pdfs/stuRes/CC3/chapter_03/CC3%20Lesson%203.1.3%20RP.pdf




319

a) Enter the information from your x → y table for problem 3‑18 into your graphing calculator. Then plot the points using a window of your choice. What do you notice?

b) Find another x → y pair that you think belongs in your table. Use your graphing calculator to plot the point. Does it look correct? How can you tell?

c) Imagine that you made up 20 new x → y pairs. Where do you think their points would lie if you added them to the graph?

https://www.desmos.com/calculator/4ac3uzgaxzUse the 319 to 322 Student eTool (Desmos) to analyze the pattern further and make predictions.

https://www.desmos.com/calculator/4ac3uzgaxz


Graph the algebraic equation for the pattern from problem 318.320a,b,c

What do you notice? Why did that happen?

Charles wonders about connecting the points of the “Big C’s” data. When the points are connected with an unbroken line or curve, the graph is called continuous. If the graph of the tile pattern is continuous, what does that suggest about the tile pattern? Explain.

Jessica prefers to keep the graph of the tilepattern data as separate points. This is called a discrete graph. Why might a discrete graph be appropriate for this data?


For the following problems, justify your conclusions with the graph, the rule, and the figure (whenever practical). Your teacher may ask your team to present your solution to one of these problems. Be sure to justify your ideas using all three representations.

Frangelica thinks that Figure 6 in the “Big C’s” pattern has 40 tiles. Decide with your team whether she is correct and justify your answer by using the rule, drawing Figure 6, and adding the point to your graph of the data. Be prepared to show these three different ways to justify your conclusion.

321a If necessary, reenter your data from the “Big C’s” pattern into your graphing calculator. Reenter the rule you found in problem 318 and graph the data and rule in the same window.

Rule: y = ( )x + ( )

Figure 6:

Justification:



Giovanni thinks that the point (16, 99) belongs in the table for the “Big C’s” pattern. Decide with your team whether he is correct, and justify your conclusion by examining the graph and the rule.

321b If necessary, reenter your data from the “Big C’s” pattern into your graphing calculator. Reenter the rule you found in problem 318 and graph the data and rule in the same window.

Rule: y = ( )x + ( )Justification:



Jeremiah is excited because he has found another rule for the “Big Cs” pattern! He thinks that y = x + 8 also works. Prove or disprove Jeremiah’s claim. Be prepared to convince the class that your conclusion is correct.

321c If necessary, reenter your data from the “Big C’s” pattern into your graphing calculator. Reenter the rule you found in problem 318 and graph the data and rule in the same window.




LaTanya has been thinking hard and has found another rule for the same pattern! She is sure that y = 3(2x + 1) is also correct. Prove or disprove LaTanya's position in as many ways as you can.

321d If necessary, reenter your data from the “Big C’s” pattern into your graphing calculator. Reenter the rule you found in problem 318 and graph the data and rule in the same window.



Look back at the prediction you made in problem 318 for Figure 100 in the “Big C’s” pattern. Decide now if your prediction was correct, and be ready to defend your position with all of the math tools you have.

322

Rule: y = ( )x + ( )

Justification:

Our prediction was / was not correct.


323 Find the value of x that makes each equation true.

a) x + 7 = 2

b) 5 = 1 x2

c) 3x = 45

d) 2 = x

f) x2 = 9 (all possible values for x)

( ) e) 5 = x2 ( )

http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/chapter/Ch3/lesson/3.1.3/problem/323

http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch3/lesson/3.1.3/problem/3-23


324a

Explanation

• Combine Like Terms• Add/Subtract ___ to Both Sides• Divide Both Sides by ___• Distributive Property

Draw the Equation Mat for the given equation. Simplify as much as possible and then solve for x. Record your work.


3c − 7 = −c + 1



324b

Explanation




−2 + 3x = 2x + 6 + x



325

Left Expression = Right Expression Explanation


Write the equation represented on the Equation Mat. Simplify as much as possible and then solve for x.

http://www.cpm.org/technology/general/tiles/?tiledata=bcx__boy__NaaabNbSavdGa1axdocyaxa2cDaddVeyavc7eLavcDd0avb6fuavhie8axh4ezavgPbUavhqbaavisbqavg5cGavh3cravi7cnayajaiCC3%20325%20HW%20eTool__


http://www.cpm.org/technology/general/tiles/?tiledata=bcx__boy__NaaabNbSavdGa1axdocyaxa2cDaddVeyavc7eLavcDd0avb6fuavhie8axh4ezavgPbUavhqbaavisbqavg5cGavh3cravi7cnayajaiCC3%203-25%20HW%20eTool__http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch3/lesson/3.1.3/problem/3-25


326Define a variable and then write and solve an equation for the problem below. Remember that you can use the 5D Process (described in the Lesson 1.1.3 Math Notes box) to help you get started. State your solution in a sentence.West High School’s population is 250 students fewer than twice the population of East High School. The two schools have a total of 2858 students. How many students attend East High School?




a)

c)

327 Calculate the following values using the Order of Operations. Show your steps. Verify your answer with your calculator.

(−4)(−2) − 6(2 − 5) 23 − (17 − 3 · 4)2 + 6


b)

d)14(2 + 3 − 2 · 2) ÷ (42 − 32) 12.7 − 18.5 + 15 + 6.3 − 1 + 28.5



Use your pattern skills to complete the table.328

a) Explain in words what is done to the input value, x, to produce the output value, y.

b) Write the process you described in part (a) in algebraic symbols.

https://www.desmos.com/calculator/8xs7tjssubhttp://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/chapter/Ch3/lesson/3.1.3/problem/328

https://www.desmos.com/calculator/8xs7tjssubhttp://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch3/lesson/3.1.3/problem/3-28


329a

Explanation



−3x + 7 = −x − 1


http://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__aayamakCC3%20329a%20HW%20eTool__aybaaSDirections%3A%20Use%20the%20tiles%20on%20the%20left%20to%20place%20on%20the%20mat%20and%20use%20%22legal%22%20moves%20to%20simplify%20and%20solve__aybsbmfor%20the%20variable.%20%20Equation%20A%3A%203x%20%2B%207%20%3D%20x%20%201__

http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch3/lesson/3.1.3/problem/3-29http://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__-aayamakCC3%203-29a%20HW%20eTool__aybaaSDirections%3A%20Use%20the%20tiles%20on%20the%20left%20to%20place%20on%20the%20mat%20and%20use%20%22legal%22%20moves%20to%20simplify%20and%20solve__aybsbmfor%20the%20variable.%20%20Equation%20A%3A%20-3x%20%2B%207%20%3D%20-x%20-%201__


329b

Explanation



1 − 2p + 5 = 4p + 6


http://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__aayamanCC3%20329b%20HW%20eTool__aya9aUDirections%3A%20Use%20the%20tiles%20on%20the%20left%20to%20place%20on%20the%20mat%20and%20use%20%22legal%22%20moves%20to%20simplify%20and%20solve__aybrbnfor%20the%20variable.%20%20Equation%20B%3A%201%20%202p%20%2B%205%20%3D%204p%20%2B%206__

http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch3/lesson/3.1.3/problem/3-29http://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__-aayamanCC3%203-29b%20HW%20eTool__aya9aUDirections%3A%20Use%20the%20tiles%20on%20the%20left%20to%20place%20on%20the%20mat%20and%20use%20%22legal%22%20moves%20to%20simplify%20and%20solve__aybrbnfor%20the%20variable.%20%20Equation%20B%3A%201%20-%202p%20%2B%205%20%3D%204p%20%2B%206__


330

Left Expression = Right Expression Explanation


Write the equation represented on the Equation Mat. Simplify as much as possible and then solve for x.

http://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__aayamanCC3%20330%20HW%20eTool__abeobIaabScNavcMbzavhjb4avi2cbaxdJfOaxd9gOacb0hDacgWfBachQgv


Evaluate both the left side and the right side using your solution from part (a). Remember that if your solution is correct, both sides should have the same value.

http://www.cpm.org/technology/general/tiles/?tiledata=a2x__boy__-aayamanCC3%203-30%20HW%20eTool__abeobIaabScNavcMbzavhjb4avi2cbaxdJfOaxd9gOacb0hDacgWfBachQgvhttp://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch3/lesson/3.1.3/problem/3-30


331 Tickets to 50 home baseball games would cost $1137.50. How much would it cost to get tickets for all 81 home games?

How many games could you go to for $728?

What is the cost of one game?




332 Combine like terms in each part below.

Liha has three x2 tiles, two xtiles, and eight unit tiles, while Makulata has five x2tiles and two unit tiles. At the end of class, they put their pieces together to give to Ms. Singh. Write an algebraic expression for each student's tiles and find the sum of their pieces.

Simplify the expression 4x + 6x2 − 11x + 2 + x2 − 19.


Write the length of the line below as a sum. Then combine like terms.


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