Lesson Plan Guide

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Teacher Name: Jamie Ryan Class and Grade Taught: Algebra II – mostly 11th graders Lesson Date: Tuesday, November 15th, 2011 Unit Topic: Systems of Linear Inequalities Previous Lesson Topic: Solving Linear Systems Algebraically (Review using the substitution and elimination method) Current Lesson Topic: Graphing and Solving Systems of Linear Inequalities Next Day: Test preparation – test covering systems of equations; solving them graphically, and algebraically. Next Unit: Solving Systems of Linear Equations in Three Variables using Matrices. Lesson Objectives: Students will…

o be able to graph a system of two inequalities on a coordinate grid o be able to understand what a system of equations is and understand how it relates to real life situations o be able to decide which area should be shaded/where the solution set lies o be able to pick a point from the solution set and explain what it means within a context and verify it using algebra o understand that a solution to a system of inequalities is located where the shaded regions overlap

How will I know students have met the objectives?

1. Students will be able to successfully graph the inequalities that they create, use the correct line type (dotted or solid), and shade on the appropriate side of the line. The homework will also assess if they can do this or not. The homework is going to be some problems from the book that ask them to solve some systems of equations by graphing.

2. Students will be able to explain the relationship between seeing the transparencies separate and seeing them at the same time. Students will be able to accurately translate the real world problem into an inequality and use it to find the solution set.

3. Students will be able to correctly shade each graph separately and then notice what happens when they are overlapped. This will be addressed when I ask the students for a solution after they have placed the two transparencies together. This will again be assessed during the homework when the students are asked to graph the systems of inequalities.

4. Students will be able to correctly choose a point in the solution set, verify it algebraically and explain what it means in the context of the problem. This will be assessed on the task when I ask “Choose a point that is a possible solution and prove it. Describe what the point represents in the context of the problem.” 5. Students will be able to make the observation that there is an overlapped region when the two transparencies are laid on top of one another. They will also correctly choose a solution point that is located inside the overlapped region. The

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homework will also require them to shade only the region that is a part of the solution, so the students will have to realize that the solution set is in the area that gets overlapped.

Standards Addressed: A-REI.1- Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI.3- Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A-REI.10- Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-REI.12- Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Materials Needed:

o Transparencies (2 per group with grids already on them, scaled and labeled) o Overhead markers (2 different colors per group) o Worksheet/directions o Rulers

Introductory Routines - Students are looking at the premade overhead, listing where their seat is for the day. This is set for their groups for the task. - Bell Work (7 minutes from start till we go over it.) (During this time I will take attendance) - Go over the answers to the homework from the previous day, go over any questions that the students have (No more

than 6 minutes). - Collect the homework

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Lesson Activities (See task on wiki for proper spacing and formatting)

Systems of Inequalities

1.) You are trying to raise money for a school sports team by selling pizza and bread sticks in the commons. You are selling a slice of

pizza for $1.00 and a breadstick for 50¢. In order to break even (make back the money that you spent on the food originally) you need

to make at least $30.00.

a.) Write an inequality that fits the situation.

b.) Using your inequality from above, take one of the transparencies and create a graph to display what is going on.

c.) How much of a profit would you make if you sold 35 slices of pizza and 27 breadsticks? Explain how you know.

2.) Suppose that whoever bought the food only bought 50 items or less (slices of pizza and breadsticks).

a.) Write an inequality that describes this situation.

b.) Using your inequality from number 2, create a graph on the second transparency to display what is going on.

3.) Now suppose both of the above limitations are true; at most 50 items can be bought and you need to make at least 30 bucks, selling

the food at the rate listed in number 1.

a.) What is the system of equations that represents this problem?

b.) What on the graph represents a solution to this system? Explain how you know.

c.) Choose a point that is a possible solution and prove it. Describe what the point represents in the context of the problem.

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Lesson

Students are working … (details about how students are configured, what work they are doing and how they are recording their work)

Anticipated Student Thinking/Questions

Teacher Moves

Launch Students are sitting in rows (not too different from the arrangement that we have in TE 802, just a smaller room). Students are turning in their homework from the previous night

Walk down the aisle and collect the homework. I will set up a desk with transparencies, worksheets, rulers, and markers. “Alright so today we are moving on to solving systems of inequalities. This is the last section that we are going to go over before the test. Tomorrow we will wrap up this section, go over the test topics, and I’ll hand out the review. So today we are going to be working on another task, this is not your assignment, this actually needs to be done at the end of the hour. Today we are going to be working on a problem dealing with a system of inequalities. (I have preassigned some groups,

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Note: For third hour, there will need to be two groups of four so that we can fit at all of the tables. This is the only hour with that problem.

Students are probably looking around at each other, trying to figure out who is going to be in charge of what. One student from each group is getting up and grabbing the materials. Then slowly making their way back to their seat.

the students should already be sitting in their assigned groups). “I have assigned you different seats today because this is the group that I would like you to work with today. So the people in your row are going to be working with you today. You are going to need to assign someone in your group to be the recorder on the worksheet, someone to present the work if called on, and someone to be in charge of graphing. Everyone needs to participate and help in solving the task, you just need to all have different roles for your group. This worksheet is due at the end of the hour. Right now I would like you to send one representative from each group to come up and collect one worksheet, two transparencies, two markers that are different colors, and a ruler. Then I’ll give you some more directions. Make it snappy, you have 30 seconds.“

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Students begin chatting about who is going to be in charge of what and begin to look at the worksheet.

“Okay, so we are looking at a fundraiser in the commons. Some of the sports teams sell pizza to earn a profit. We are going to be looking at a couple of equations that describe the situation and then we will look at them together as a system of equations. Again, this is due at the end of the hour. You actually only have until ____ (8:42- 1st hour, 10:58- 3rd hour, 2:45- 7th hour) before we talk about this as a class and some groups present what they did.” “First you are going to need to translate to make an equation, then graph it, and make some observations. If the question asks you to explain, please take some time to explain how you got your answer.”

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Explore – Description of Task(s)

1.) You are trying to raise

money for a school sports

team by selling pizza and

bread sticks in the commons.

You are selling a slice of

pizza for $1.00 and a

breadstick for 50¢. In order

to break even (make back the

money that you spent on the

food originally) you need to

make at least $30.00.

a.) Write an inequality that

fits the situation.

“I don’t get it, what are we supposed to do?” “The price? Like of the pizza and the breadsticks…” “Like… _______?” Common Misconceptions:

1p+50b=30

1p+50b<=30

1p+50b>=30

1p+.5b=30

1p+.5b<=30 “Dollars.” “Oh yeah! I got you Ms. Ryan, it’s cents.”

Exact question that you will pose to students to begin the exploration. “Write a inequality to describe what is going on.” “What do you know?” “Yeah okay, so how can we write that into an equation?” For those that forget the breadstick is $0.50, not $50… “What are the units on this again? What are the units on the pizza?” “And the breadsticks?” For the incorrect inequality sign… “So what does your equation mean in the context of the problem?”

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“…It means that the number of breadsticks times one dollar plus the number of breadsticks times 50 cents is equal to 30.” “That’s what we need to make a profit.” “More… oh. Yeah, so it would be 1p+.5b>=30.” “Yeah, because we want to make a profit.” “We break even.” “We need at least… $30.50. Do I need to change my equation?” “Make it strictly greater than.”

“Where does the 30 come from?” “So what does that mean? Do we need to make more or less money to make a profit?” “Does that make sense?” “So in your equation what happens if we make exactly 30 dollars?” “How much do we need to make to get a profit?” “No, as long as you know what it means. If you were looking for the solutions that allowed you to make a profit, how would you change it?” “Good.”

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Different Correct Answers: 1p+0.5b>30 1p+0.5b>=30 (Depends on the way that they define/think about the problem) “Yeah. It costs $1 for pizza, 50 cents for a breadstick, and we need to make more than 30 bucks to get a profit, so I set it up as 1 times the number of slices of pizza, plus 50 cents times the number of breadsticks.” “yeah, if you make 30 bucks you break even, so you make $0.” “It would be greater or equal to.”

> “Can you explain to me how you got your answer?” “Okay, so let me get this straight, you chose greater than because you need to make more than 30 dollars to get a profit?” “Okay cool. Your equation is great. Just wondering… what would the equation look like if we included solutions that allowed us to break even?” “Cool, nice thinking.”

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“Yeah. It costs $1 for pizza, 50 cents for a breadstick, and we need to make 30 bucks to break even, so I set it up as 1 times the number of slices of pizza, plus 50 cents times the number of breadsticks.” “You need to make at least 30 bucks… otherwise you are losing money.” “…It would be just >. Do I need to change my equation?”

>= “Can you explain to me how you got your answer?” “Okay, so you chose >= because you need to what?” “Okay yeah, that makes sense. What would the equation look like if we were only including answers where we actually made money?” “Nope! Yours is great as long as you say what it represents, I was just wondering. I’ve seen the equation written both ways while walking around today.”

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b.) Using your inequality

from above, take one of the

transparencies and create a

graph to display what is

going on.

“How do we graph this?” “I know we use slope intercept but what is x and what is y?” “What do you mean?” “It was time.” “Okay…” “Is the pizza independent?” “Because… most people are going to get pizza…”

“How do we normally graph lines?” “Good question. Do you think one is independent and one is dependent as a variable?” “Like when we plot things against time, like the zombie activity that we did where we plotted time vs. distance, which variable was x?” “Yeah, because time is going to move on no matter what, the distance that we were from the starting point changed depending on how much time had passed.” “So again, do you think that one variable is dependent while the other is independent?” “Explain why you think that.” “Do you think that the number of breadsticks that a

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“I don’t know… yes… because they aren’t going to get a lot of pizza and a lot of breadsticks… unless they are really fat.” “Yeah, I guess.” “No.” “No.” “No… not really.”

person buys is dependent on how much pizza they get?” “Haha true, but they could be buying for multiple people. What if I only want a breadstick, am I allowed to buy just a breadstick?” “So is the number of breadsticks I get dependent on the number of slices of pizza I get?” “Is the number of slices of pizza that I get dependent on the number of breadsticks I get?” “Okay, so do we have a independent/ dependent variable?” “So do you think it matters which variable we choose to be x and which one we choose to be y?”

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“well… yeah…” “Well… our graph is going to look different depending on which variable we use to represent x and y.” “I don’t know.” “Okay, so I can just choose?”

“How come?” “That’s true. But do you think your solution set is going to change?” “As long as you label your variables it shouldn’t make a difference. Selling n slices of pizza and m breadsticks is the same as selling n slices of pizza and m breadsticks. No matter what axis we put it on.” “Yep. Some other groups might do it differently than you but that is okay, as long as your group defines your variables and explains what is going on, it’ll work out.”

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“Ms. Ryan, what are we supposed to be doing with this problem?” “Yeah we got ____” “Yeah but… we’re stuck.” “The y-intercept and the slope…” “find the x and the y intercept.”

NOTE: Per suggestion of Ms. Kruger, I have provided the students with graph paper where the scale is pre-labeled, so we should have no misconceptions about having two different scales on the two different transparencies. “Well, did you already make your equation?” (If incorrect return back to previous questioning, if correct continue with…) “Okay, so how do we graph that? Remember what we did a few weeks ago to graph inequalities?” “How do we graph lines? What do we need to know?” “that would work, what else could we do?” “Okay, so do one of those!”

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“Ms. Ryan! That’s why we called you over here, we don’t know how!” “Should we put it into the slope intercept form?” “How do we do that?” “y=mx+b” “Solve for y.” “Does this look right?” “It’s (depends on which variable they make y and which one they make x…if pizza is y then the y-intercept should be 30 and if breadsticks is y then 60 is)”

“Sure you do, you just told me. How can we do those things? Find the intercepts or find the slope?” “That would work right! Then you would have the slope and the y-intercept. Try it!” “What does the slope intercept form look like?” “Okay, so what do we have to do to this equation to get it to look like that?” (Walk away) “Let’s see… what should your y-intercept be?”

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c.) How much of a profit

would you make if you sold

35 slices of pizza and 27

breadsticks? Explain how

you know.

“Because we wrote it in y=mx+b form and b is equal to __” “…from m…” “Yeah.” “Is this right?” Misconception: 48.50 – total amount of money brought in, forgetting about how much we spent on the food in the first place “I just plugged it into the equation.” “How much money we made.” “Oh yeah… no! We need to subtract 30, right?”

“How did you know that?” “Alright cool, how about the slope, how did you figure that out?” “Haha, alright, so you went up ___ and over ___?” “Looks good to me.” “Can you explain to me how you got your answer?” “Okay, so what does that number represent?” “Okay, is that all profit?”

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“We got… 18.50” other misconceptions dealing with an incorrect equation misconception with the students forgetting which variable they assigned as y and which one they made x. “It’s the number of slices of pizza.” “Yea…. Oh. Whoops!”

“yep! So what do you get when you do that?” “Yeah, sounds good to me.” Go back to the questioning for misconceptions on the equations. “So what did you define x to be again?” “Oh okay, so you have 50 cents for every slice of pizza?”

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2.) Suppose that whoever

bought the food only bought

50 items or less (slices of

pizza and breadsticks).

a.) Write an inequality that

describes this situation.

“Ms. Ryan, we don’t get it.” “Why did they only get 50 items? Does it matter what the 50 are?” “But that’s dumb because pizza costs a different amount than breadsticks. Why would they differentiate?” “So what do we do?” “We have… x slices of pizza plus y breadsticks and it has to be… less than or equal to 50?”

“Well, what is going on?” “Nope, not in this problem.” “not sure, maybe they just can’t carry more than 50 breadsticks and 50 slices of pizza… maybe it’s like the bread that you get from hungary howies, where the breadsticks come in a box like the pizza does, so it takes up the same amount of space.” “Well, how do we translate that into an equation?” “Yeah, that sounds good. Be confident in your answers!”

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Misconceptions -Students use incorrect inequality notation -Students multiply x and y together. “It is the number of slices times the number of breadsticks, and it has to be less than or equal to 50, because that’s the number of items we are allowed.” “I don’t know.. because we are looking at the total number of items…” “40..” “50-10…Okay. So is it x+y?”

“Can you explain to me what your equation means?” “Can you explain to me what your equation means?” “Okay, so what told you to multiply x and y together?” “What if I only got 10 breadsticks, how many slices of pizza would I be able to get?” “how did you get that?” “Yeah, because we are looking at the number of breadsticks plus the number of slices of pizza. That will be our total number of items.”

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b.) Using your inequality

from number 2, create a

graph on the second

transparency to display what

is going on.

“Ms. Ryan, how do we do this??” Extra Misconception -Students somehow switch the variables and plot pizza as x on the first graph and as y on the second graph. “breadsticks” “…breadsticks, oh wait… it was pizza.” “yeah, probably. So… our equation would then be… (resolves for the other variable, gets the same equation) the same thing?”

return to the questioning for number 1 graphing. (Doesn’t really matter since it’ll be the same line.. but..) “What did you define as x again?” “Okay, what was x on the last graph?” “do you think it would be useful to keep our variables consistent throughout this task?” “yep. ”

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3.) Now suppose both of the

above limitations are true; at

most 50 items can be bought

and you need to make at least

30 bucks, selling the food at

the rate listed in number 1.

a.) What is the system of

equations that represents this

problem?

“how do we figure this out?” “two…” “No, because now we can only have 50 items.” “Do we just have to lay these on top of each other? Is that all you want us to do?” “because we have to look at both equations. So we need to still make more than 30 bucks but we also need to buy less than 50 items, so it has to be between the two lines.”

“Think about it… how many constraints do we have on our solutions now?” “So can we still include the solution that we could buy 345 breadsticks and 23442 slices of pizza?” “So how can we combine the two solution sets to get the new one?” “Well, please explain to me why you think that you would do that?” “That sounds pretty reasonable to me. Why don’t you try it and let me know what you end up getting.”

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b.) What on the graph

represents a solution to this

system? Explain how you

know.

“It is the shaded regions.” “it’s all of this shaded stuff (talking about the stuff that is shaded only once also, for each separate equation).” “oh no.. so it is the overlapping region. That’s what I meant.” “Because it was a solution for both of the equations. We can’t include stuff that wasn’t shaded and we can’t keep the solutions that satisfy only one equation.” “(__)” “Because it is in the overlapped region…” “Plug the point into the equations…”

“Can you say more about that?” “Including this stuff way up here? We can buy 803 breadsticks and still satisfy both equations?” “Why is it the overlapped region?” “Sounds good to me. So give me an example of a point that is in the solution region (part c).” “Okay, and how do you know?” “How else could we check?”

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c.) Choose a point that is a

possible solution and prove

it. Describe what the point

represents in the context of

the problem.

“Does this point work?” (Points at an area that is in a shaded region but is not a part of the system) “It means that we sold 40 slices of pizza and 15 breadsticks.” “Yeah, we made 40 dollars off the pizza and another 7.50 on the breadsticks, so 47.50 total, we only needed 30 dollars to break even.” “…No.” “We have too many items.” “….No?”

“Well let’s take a look at the graph… where does it lie?” “So what does that point mean in the context of the problem?” “Okay, so does that mean we made the profit that we were looking for, using the first equation?” “Nice, okay, so how about the other equation. Does this solution work for that one?” “Why not?” “So is it a solution to the whole system?” “What do we need in order

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“It needs to be true for both.” “No.” “In the double shaded region.” “Ms. Ryan, is this right?” (Correct solution) “It is in the double shaded region.” “It’s a solution to both of the equations, it works in both.” “…No. Do we really have to?”

for it to be a solution to the system?” “So is it a solution to the system?” “So where else do you think we might find a solution that will work for both?” “Yep, so why don’t you pick something from in there and check it algebraically to make sure that it works.” (Walk away) “Well, how did you decide upon this point?” “Good thinking, and what does that mean, if it is in the double shaded region?” “yeah, very good. Have you tried to prove your point algebraically?” “Yeah, that would be a great way to check it and prove that it works!”

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Describing what the point

means in the context of the

problem

“it means that we bought 50 slices of pizza and no breadsticks… so we make 50 bucks and had 50 items.” “Um… 20 bucks… since it cost 30 to buy the food.” “How do we explain the point in the context of the problem? Do we really have to?”

“Okay, so what does that point mean in the context of the problem?” “cool, so how much profit did you make?” “Nice work. How about you write that down, I won’t be able to remember all of the groups that I talked to and I’d like to have a record of how much you all got done today and see how you approached the problem.” “Yes! I’d like to know what the point means! It does the team no good to just have a solution point, they want to know how much money they made!”

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Summarize/Share and Discuss

Presentation mode – Sharing solutions, teacher-led discussion, student led discussion, etc. Group puts their two transparencies on the overhead

“Okay, so we got the two equations, the first one was 1p+0.5b>=30. This is because it costs one dollar per slice of pizza, which is p, and 50 cents per breadstick which is b. We said that it was greater than or equal to because we knew that when we make 30 dollars we break even. So we are looking at solutions that

What will you say or do to set up the discussion of the big math ideas? “Alright, let’s bring it together and go over this!” (Pause while students stop talking and look up). “So I saw a lot of good work going on. Now I would like to call up some groups to share their answers. Could group 3 please come up (a group that had the first equation written as >=) “Could you please show your graphs and talk specifically about how you got your equations?”

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Group two is now showing their transparency

break even or make us a profit.” “…” “Yeah, but all we did is say it was greater than 30, because we were looking at the solutions when we actually make a profit, not break even.” “The >= has a solid line and the > has a dotted line.” “Yeah.” “because it is still greater than… only if it was less than would we shade differently.”

“did anyone do this differently?” “How about your, group 2, wasn’t your equation slightly different?” “Okay, nice. So can someone remind me again how we graph those inequalities? What makes them different?” “Yeah, so we use a different kind of line to represent the equation. Do we shade on the same side for both equations?” “How come?” “Okay good. So does everyone remember these properties that we use when we graph inequalities?”

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“yes!” “When it is < or >.” “If it is > we shade above, if it is < we shade below.” “The one about making a profit… >!” “Because we are doing a fundraiser, so we are looking at when we make money!” “Yeah, what if the fundraiser doesn’t go well. Then we are going to want to know if we broke even or lost money… so we want to include when we make 30 bucks.” “Yeah but we can just look at that line separately can’t we? When we check it…”

“How do we know when to use a dotted line?” “Very good, and how do we know what side to shade?” “Cool. So what does everyone think about these two different equations. Which one do you think is more accurate?” “What makes you think that?” “Okay, so that is an interesting thought. Does anyone want to argue for the other side?” “Don’t look at me.”

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Group 5 comes up and shows their transparencies

“We could do that for both then. If we wanted to know when we broke even we would just be looking at that line.” “it’s when it’s equal to 30.” “x+.5y=30” “No. it depends on what we are looking for.” “if we are looking for a profit then we are going to be looking at when it is greater than 30… if we are just looking to break even then we are going to look at when it is 30 or greater.” “Sure…”

“What is that line?” “I got that, but what does it look like as an equation?” “Nice. So do you think we should actually consider these things differently?” “Can you say more about that?” “Okay, thank you group two.” “Let’s move on to talk about number three. Group 5, could you please come up and share your answers?” “Let’s start with part a. What was the system of equation that you came up with?”

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“Well, we said that it was the two equations. So we used the one where it was greater than 30, and then the second equation.. y=50-x.” “Well, we know that we can have at most 50 items, so we had x+y=50. Then to graph it we had to solve for y, so we subtracted x from both sides. Then we just left it like that.” “Ours looks different… flipped kind of.” “Did we mess up?” “The number of slices of pizza is y and the number of breadsticks was x.” “Oh! We used the number of breadsticks as y. Does it matter?” “I don’t know.”

“Can you say something about that second equation, since we didn’t go over it. How did you get it?” “Does anyone have any questions for how they derived that second equation?” “Why do you think that is?” “Not necessarily. What did you define as y?” “Do you think it should?”

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“No.. Because 5 breadsticks is 5 breadsticks… no matter how you look at it.” “the areas that are colored in twice.” “because the clear areas aren’t solutions for either equation…and the areas that are shaded just once are points that work for just one equation…” “Because it’s not consistent. To be a solution point it needs to hold true for both equations.”

Do you think that there are points in your solution set (group a) that don’t exist in your solution set (group b)?” “Good thinking. So your graphs look different just because you flip flopped x and y.” “What on these graphs represents a solution point?” “Why isn’t it the clear areas? Or what about the areas that are shaded in once?” “Why isn’t that good enough?” “Okay, lastly, how can we show that a point is a solution to the system?”

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“Look at the graph…” “Check it algebraically! Plug it into both of the equations to see if it works.” “(50 pizza slices, 0 breadsticks)” “It is 1(50)+.5(0)>30 50 > 30 which is true, so it works for the first equation. Then 50+0 = 50, which is the most items we are allowed, so it is just barely a solution for the second equation.” “20 bucks.”

“How else?” “Good thinking. Remember what we talked about last week when we were solving systems and we proved or checked our points by plugging them back into the original equation. Would someone like to share a point that worked?” “Alright, so how do we plug that in?” “Very good, so how much of a profit did we make?”

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Summary Statement: (May change based on what actually happens in class) “Okay,so today we looked at a system of inequalities. We decided that the solution set was the area that was shaded twice due to the overlap in solution sets. On your homework you are going to be asked to graph these systems of inequalities. Some books show the solution set by only shading in the region that is shaded twice and erasing the parts that were only shaded once. Make sure that when you do your homework you make sure that I know what your solution set is. We will finish up talking about this tomorrow. Now get out your assignment sheets and I’ll give you the homework.” (Pending – still deciding on bookwork or a worksheet)

Homework: Pending. Either way it will be straight equations that ask for a graph, probably no more than 5 systems (all systems only have two

equations, I might throw in a problem with three equations, just to see what they can do before we get to that section (next)).

Graphs:

X=breadsticks Y=pizza X=pizza Y=breadsticks

Note: Graphs will look different depending on the line used (dotted or solid) and the scale on the transparencies will be evenly spaced

on both the x and y axis, in increments of 5.