Benchmark Lesson - Points of Interest
Authors' Name: Emily Nelson
Length of Lesson: 100 minutes
Grade/Topic: 8th Grade Advanced Math
or 9th Grade Algebra I
Source of the Lesson: Florida Math Connects, Teacher Edition. Course 3, Volume 1. Glencoe: McGraw-Hill
Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning by Page D. Keeley and Cheryl Rose Tobey
Day 2 calculator activities adapted from the site below:
http://education.ti.com/xchange/US/Math/AlgebraI/11994/Solving%20a%20Pair%20of%20Linear%20Equations%20by%20Graphing.pdf
Concepts
Linear relationships contain or sometimes overlap with many other concepts such as symbolic relationships, mathematical modeling, and the skill of recognizing patterns and connections. Specifically, recognizing connections such as the relationship of the symbolic representations of algebra to the spatial representations of geometry is an important skill. Science for All Americans suggests three phases of mathematical inquiry, which seem to me especially helpful in understanding the overall intent of the subject throughout a K-12 education: abstraction and symbolic representation, manipulating mathematical statements, and application. These phases are much more complex and meaningful than students sometimes recognize. Solving systems of linear equations really requires algebraic reasoning. Vital to this process is the “crucial understanding that the variables stand for numbers and hence manipulations of them must be governed by the same arithmetic rules that govern numbers” (Beyond Numeracy, pg. 9). Without this foundation, generalizing the basic principles of algebra would be nearly impossible. In the context of this lesson, students will have to solve for a set of variables that define a system of equations (TI-Nspire lesson). Once those foundational pieces are linked to these activities, meaning will attach to this method of graphing linear equations, which is largely visual, and for many this is the cornerstone of understanding what is a solution to a system of linear equations.
Beyond Numeracy
Science for All Americans
(TI-Nspire exploration lesson) http://education.ti.com/xchange/US/Math/AlgebraI/11994/Solving%20a%20Pair%20of%20Linear%20Equations%20by%20Graphing.pdf
Performance Objectives
· Students will be able to…
1. Graph systems of linear equations by hand and find their solution or point of intersection from the graph.
2. Determine if a system of linear equations has only one solution, infinitely many solutions, or no solutions by referencing a graph.3. Demonstrate that parallel lines have no point of intersection using a graph.
4. Graph systems of linear equations using a graphing calculator and approximate points of intersection using the graphing calculator.
5. Verify the solution of a linear system by substitution of the solution into both equations.
Florida State Standards:
MA.912.A.3.13 - Use a graph to approximate the solution of a system of linear equations in two variables with and without technology. Materials List and Student Handouts
Advance Preparations
Safety
There are not many significant concerns. Students should be instructed to handle the graphing calculators carefully. They should not be tossed or thrown – neither the students nor the calculators.
Length of Lesson: 100 minutes
Grade/Topic: 8th Grade Advanced Math
or 9th Grade Algebra I
Source of the Lesson: Florida Math Connects, Teacher Edition. Course 3, Volume 1. Glencoe: McGraw-Hill
Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning by Page D. Keeley and Cheryl Rose Tobey
Day 2 calculator activities adapted from the site below:
http://education.ti.com/xchange/US/Math/AlgebraI/11994/Solving%20a%20Pair%20of%20Linear%20Equations%20by%20Graphing.pdf
Concepts
Linear relationships contain or sometimes overlap with many other concepts such as symbolic relationships, mathematical modeling, and the skill of recognizing patterns and connections. Specifically, recognizing connections such as the relationship of the symbolic representations of algebra to the spatial representations of geometry is an important skill. Science for All Americans suggests three phases of mathematical inquiry, which seem to me especially helpful in understanding the overall intent of the subject throughout a K-12 education: abstraction and symbolic representation, manipulating mathematical statements, and application. These phases are much more complex and meaningful than students sometimes recognize. Solving systems of linear equations really requires algebraic reasoning. Vital to this process is the “crucial understanding that the variables stand for numbers and hence manipulations of them must be governed by the same arithmetic rules that govern numbers” (Beyond Numeracy, pg. 9). Without this foundation, generalizing the basic principles of algebra would be nearly impossible. In the context of this lesson, students will have to solve for a set of variables that define a system of equations (TI-Nspire lesson). Once those foundational pieces are linked to these activities, meaning will attach to this method of graphing linear equations, which is largely visual, and for many this is the cornerstone of understanding what is a solution to a system of linear equations.
Beyond Numeracy
Science for All Americans
(TI-Nspire exploration lesson) http://education.ti.com/xchange/US/Math/AlgebraI/11994/Solving%20a%20Pair%20of%20Linear%20Equations%20by%20Graphing.pdf
Performance Objectives
· Students will be able to…
1. Graph systems of linear equations by hand and find their solution or point of intersection from the graph.
2. Determine if a system of linear equations has only one solution, infinitely many solutions, or no solutions by referencing a graph.3. Demonstrate that parallel lines have no point of intersection using a graph.
4. Graph systems of linear equations using a graphing calculator and approximate points of intersection using the graphing calculator.
5. Verify the solution of a linear system by substitution of the solution into both equations.
Florida State Standards:
MA.912.A.3.13 - Use a graph to approximate the solution of a system of linear equations in two variables with and without technology. Materials List and Student Handouts
- Set of index cards with linear equations (22)
- Graphing paper (44)
- True/False worksheet (22)
- T/F Evaluation for Day 1 (22)
- TI-Inspire for each student (22)
- TI-Inspire for teacher with a cable to connect to the smartboard. (1)
- Step-by-Step calculator directions for graphing linear equations and questions. (22)
- Evaluation for Day 2. (22)
- PowerPoint Presentation. (1)
Advance Preparations
- PowerPoint will be created and sent to Mentor Teacher a couple days in advance to ensure it loads.
- Class sets of handouts will be copied and organized at least one day prior to teaching.
- Day 1: index cards as well as graphing paper will be on each students’ desktop before they arrive. While they are finding their partners for the exploration activity, I will also pass out the T/F worksheet for their completion.
- Day 2: Each student will have a sheet of graphing paper and a worksheet, turned upside down, at their desk. After engagement, I will pass out graphing calculators with the help of Mr. Hinchman.
- Evaluations will be distributed at five minutes till the end of the period.
- For day 1, students will be working in pairs. To work with their partner, they can move their desks however they choose.
- For day 2, students will mostly be working at their desks individually.
- Make sure that TI-Inspires hook up properly to the smartboard before the day of the lesson.
Safety
There are not many significant concerns. Students should be instructed to handle the graphing calculators carefully. They should not be tossed or thrown – neither the students nor the calculators.
Day 1 Lesson
Engagement Time: 6 Minutes
What the Teacher Will Do
Begin powerpoint slides. First one will list several equations whose y-intercept is 0. Use to promote discussion about solutions to systems of linear equations. Explain to your students that we will be looking at systems of linear equations and try to solve for them graphically. |
Probing/Eliciting Questions
What is the solution to this system of linear equations? What do I mean by a “solution”? Is there even a solution? Is there more than one solution? Would it help if I graphed it? Let’s look at the graph of all of these equations. Can someone tell me what the solution is? How can we double check? Does (0,0) work for every equation? |
Student Responses and Misconceptions
There is no solution because there are too many equations to solve for, there are infinite solutions because they all intersect, [all of them pass through the origin so that must be the solution…] When all of the equations equal the same thing, [the point that all the equations pass through, point of intersection,] the answer to the problem, there is none… [Yes], no, I don’t think so, maybe…because there are only two variables, there is only one solution, [there is not more than one solution because none of the equations are the same line]…. [It is where the intersect, they all pass through the origin], there is no solution…. Yes, no, let’s try it…. |
Evaluation/Decision Point AssessmentIf students quickly assume or know the solution is the origin, it will tell me that they have some basic understanding of solutions to systems of equations being points of intersection. We will move into the card matching activity. If they seem to struggle with this concept, we will brainstorm about the meaning of a solution and I will write cue words on the board for them to refer to throughout the lesson.
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Student OutcomesStudents should be comfortable and familiar with this idea of solutions. However, I will watch out for furrowed brows and blank stares.
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Exploration Time: 18 minutes
What the Teacher Will Do
Each student will be given an index card with two equations written on it. Their task is to rewrite their linear equations in slope-intercept form and the graph their equations on graphing paper provided to find the point of intersection. Once they have the intersection or “solution,” they must then find another person in the class who has the same equations on their note card and compare your solutions to each other. Once you agree on a solution then split up and find someone who has a different set of equations but has the same solution. Exemplify the process if they do not follow what is happening. As students begin to work on their graphs, circulate the room. Give them a minute or two to consider the equations and try graphing them. Continue to ask questions about the process until they walk you through re-writing the equation in the y=mx + b form. Let them finish graphing on their own and finding the points of intersection. I will help any students that are still struggle to re-write the equations. Once there, they can start looking for another person with the same original equations on their note card. Encourage them to move around and talk to one another about their equations and intersection points. After comparing solutions, split up again and find one other person who has a different set of equations but the same solution. Once their partner has been found, the students will work in pairs to complete the T/F worksheet. |
Probing/Eliciting Questions
(Teacher will give explicit directions before beginning. Before letting students begin, ask students what they might need to do in order to graph these equations.) How might re-writing these equations in slope-intercept form help us graph them? Think back to how we graphed yesterday. What made it easy? What would be my first step if I want to get it into y = mx + b form? Once you have found a solution to your equations, find one other person who has the same original equations on their card and compare your answers. Once you have found your partner who has the same point of intersection but different original equations, double check with me or Mr. Hinchman, and we will then hand out the next step. Begin working on the T/F worksheet with your partner. |
Student Responses and Misconceptions
Just try plugging in numbers until you have something that works and then graph those points, [isolate or solve for y and then graph it like that regular y = mx +b form,] find out what the slope is and plug in numbers till you find the y-intercept, it isn’t graph-able, I have no idea… [Add the 3 to both sides, divide everything by 6], get rid of 4x… |
Evaluation/Decision Point If students get this relatively quickly, we will have move on into the explanation. If students still are demonstrating difficulty re-writing the equations, I will go through a few more examples of isolating y.
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Student OutcomesStudents should be able to re-write equations in the y = mx +b form, graph them, and determine points of intersection.
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Explanation Time: 12 minutes
What the Teacher Will Do
Once students have found their partners, get the class focused at the front again, and ask them some questions to review some formal vocabulary, and then go over their findings. Have a slide with these formal terms to remind them as we go along. Begin referring to solutions as ordered pairs more often. Have students help you fill in a graphic organizer on the Smartboard that generalizes how many, if any solutions there are to a particular system of linear equations. At this point, each student should be able to graph two or more linear equations, and find the solution to that system of linear equations or explain why there is no solution or infinitely many solutions. |
Probing/Eliciting Questions
It seems like everyone has found their partner now, and has had some time to work on their T/F worksheet. We were solving for a particular point. What is another term we use to describe a point on the graph? Why do we call them systems of equations? What do we sometimes call x and y in an equation? Why are they called variables? Will a solution to a system of linear equations always involve a specific value for x and y, or an ordered pair? Did everyone find a point of intersection for their system of equations? No? [Names of students], what did your graphs look like? What were your equations? Without even graphing, how might we know that they are not going to have a point of intersection? What does it mean for two lines to be parallel? Did anyone else have trouble finding a solution? [Student name], what problems did you run into? How can you find a solution to two equations that are the same thing? Does a solution exist? |
Student Responses and Misconceptions
[An x and y, the independent and dependent variable], I have no idea, [an ordered pair]… Because they both are linear, because they always work together and depend on each other, because they are the same thing, but written differently, I have no idea, to confuse us, [because they both have a similar set of variables, x and y represent numbers, x and y are numbers that can be changed – they can “vary”]… Maybe, [not always], sometimes, I’m not sure, no because a solution is just one number, [no because they could be parallel]… Yes, [no], I think so, we found an infinite number of points…both of us had graphs that were just parallel lines so they never intersected… [y = (1/3)x + 2, y = (1/3)x + 1, 3y – 9 = x, 3y-x = 3, the last two written in y = mx + b form: y = (1/3)x + 3 and y = (1/3)x -1, we cannot know unless we graph it, they all have the same slope so they will all be parallel,] [it means they never intersect, it means they are always the same space apart]… Yes, my equations were not different. It was the same line written in two different ways, there is no solution because the equation is the same thing, [there are infinitely many solutions because the two lines intersect at every point because they are really the same line]… |
Evaluation/Decision Point As a quick check, on a few PowerPoint slides I will individually list simple systems of linear equations either by graphs or simple equations. Students will give a fist for no solutions, one pointer finger for one solution, or jazz hands for infinite solutions. We will move onto the elaboration or continue to review depending on their answers.
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Student OutcomesStudents should be able to explain their and their partner’s solution set. They should also be able to determine how many solutions a system of equations has based on their slope.
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Elaboration Time: 5 minutes
What the Teacher Will Do
Explain text messaging plan and have them consider the following systems of equations: y = .28x + 52 and y = .21x + 61. Clearly, graphing these by hand is not very simple. However, I will push students to approximate the slopes and first consider the graphs that way. Consider first when x = 0, that is, I did not go over 600 text messages. Have students graph y = .2x and y = .3x, and then discuss if that will indicate whether it is a better plan or not. |
Probing/Eliciting Questions
How many of you have a cell phone? Do you ever text message? I counted the number of times I texted anyone just yesterday, and it was 28! I’ve been considering switching to a new cell phone plane that has the following rates. For Plan A, the base cost is $51 per/month. However, it only includes 600 text messages for the month. After 600 text messages, each text costs 28 cents. After 600 texts, how might we write the cost of Plane A as an equation? Why? What if Plan B’s cost after 600 text messages is represented by the equation: y = .21x + 61? What does that mean? Which is the better plan to go with? Go ahead and graph it by hand, and then tell me if it is more clear, which is cheaper. Why is that difficult? Let’s look at the slope of each by itself first. What if we rounded .28 to .3 and then .21 to .2? Would that be easier to graph? Will that give us the answer to our question? Is there another way to approach this kind of problem? Think about it, and we’ll begin class with this problem next time. |
Student Responses and Misconceptions
Duh, everyone owns one… Never, of course, I’m even texting as you speak… y = 51 + 28x,[ y = 51 +.28x], I have no idea, [we have that the plan starts at $51 but then you each additional text over 600 is another 28 cents, so that is 51 + .28(multiplied by the number of texts over 600 = x), which is the same as writing y = 51 +.28x]…. That means that the base cost is 21 dollars but then you have to add 61 cents per text message after, [that means that the base cost is 61 dollars per month, but then you have to add 21 cents for each additional text over 600], I have no idea, [it is not clear right away because they are not whole numbers], the first plan because it costs less initially and I never text anyway… Because we’re not working with whole numbers or pretty fractions, I’m not sure how to graph the slope of .28 and .21… [We might be able to graph it, but it won’t be exactly right because the y-intercept is still different], I’m still not so sure what the question is… |
Evaluation/Decision Point Considering we may or may not get to this problem, the decision point will depend mostly on the time. Regardless, at least five minutes prior to the end of class, we will begin the evaluation.
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Student Outcomes Students should at least see that finding a solution by hand-graphing is really not the best way to approach this problem.
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Evaluation
Students will do another short T/F table that requires graphing two linear equations and then referring to the graph to determine which statements are true and which are false.
Day 2 Lesson
Engagement Time: 5 minutes
What the Teacher Will Do
Explain text messaging plan and have them consider the following systems of equations that define the cost of two different cell phone plans: y = .28x + 52 and y = .21x + 61. Students will have a sheet of graphing paper should they choose to attempt to hand draw the graph. |
Probing/Eliciting Questions
How many of you have a cell phone? Do you ever text message? I counted the number of times I texted anyone just yesterday, and it was 28! That means I send about 200 text messages every week. That’s about 800 text messages per month. I’ve been considering switching to a new cell phone plane that has the following rate. For Plan A, the base cost is $51 per/month. However, it only includes 600 text messages for the month. After 600 text messages, each text costs 28 cents. After 600 texts, how might we write the cost of Plane A as an equation? Why? What if Plan B’s cost after 600 text messages is represented by the equation: y = .21x + 61? What does that mean? Which is the better plan to go with? At what point would the plans cost the same? How could we find out at which point the plans cost the same? Will the plans ever cost the same thing? Without graphing either, how can we tell whether or not they will intersect? Go ahead and graph it by hand, and then tell me if it is more clear, which is cheaper. Why is that difficult? Today we are going to learn a faster way to graph equations that are not easy to graph by hand. |
Student Responses and Misconceptions
Duh, everyone owns one… Never, of course, I’m even texting as you speak… y = 51 + 28x,[ y = 51 +.28x], I have no idea, [we have that the plan starts at $51 but then you each additional text over 600 is another 28 cents, so that is 51 + .28(multiplied by the number of texts over 600 = x), which is the same as writing y = 51 +.28x]…. That means that the base cost is 21 dollars but then you have to add 61 cents per text message after, [that means that the base cost is 61 dollars per month, but then you have to add 21 cents for each additional text over 600], I have no idea, [it is not clear right away because they are not whole numbers], the first plan because it costs less initially and I never text anyway,[we need to find where they intersect like we did the other day, we look at the graph and find the point, we need a solution that’s the same for both of them]… No because they both start with .2 for the slope, I’m not sure, [yes because the slope for each is not the same]… It’s not hard to graph - you just have to go up 28 units and then to the right 100 units, because we’re not working with whole numbers or pretty fractions, I’m not sure how to graph the slope of .28 and .21… |
Evaluation/Decision Point
Since this engagement spills into the exploration activity, formative assessment will just be in the questions I ask the students. I will mix up questioning by randomly calling on students and letting students volunteer. |
Student Outcomes
Students should be able to write out each cell phone plans’ rate after 600 texts as a linear equation. Students should be able to determine if the equations will intersect at some point or not just from looking at the equations. If not, I will try to get other student’s to explain their understanding, or approach the explanation in a different way. |
Exploration Time: 18 minutes
What the Teacher Will Do
Students will already have the calculator activity printed out on their desks. We will walk through the steps together, noting any technical difficulties with the calculator. However, we will not go through the worksheet questions. Students will explore the activity in that sense. Students will work through the worksheet individually. If they seem to be having difficulty working through it, I will allow them to discuss some of the problems with their nearby classmates. Begin shoe problem. Again, after a little discussion have students first attempt to answer the questions, and then return to them during the explanation portion of the lesson. Let students complete the first activity, and then go over their answers as a class. Move onto the next calculator worksheet problem. Make sure students understand the task by asking them the probing questions listed. Give them the rest of the exploration portion to answer the remaining questions. |
Probing/Eliciting Questions
Does everyone have their calculator on? Okay, now begin the first steps on the screen. What are our two linear equations again? Work through the steps to use the calculator to find the point of intersection. What does that point of intersection mean? What does the x-variable represent? What does the y-variable represent? Will someone please read the following word problem? What do x and y represent in this equation? Why would we write it this way? What are we solving for? What other piece of information do we need? What are we missing? |
Student Responses and Misconceptions
Yes, no, mine has no batteries, we have y = .28x + 52 for Plan A and y = .21x +61 for Plan B. It’s the point where Plan A becomes more expensive than Plan B, [it means they actually cost exactly the same at that point, it is the solution],[ the x- variable represents the number of text messages after 600], the x-variable represents the total cost of the plan, the x-variable represents the total number of text messages, [the y-variable represents the total cost of the plan after x- many text messages]… [The number of each type of shoe ordered], they do not mean anything, since we know the price of each one, [we can let x and y represent the number of each ordered multiplied by their price, this will tell us how many of each, the second equation tells us the total number of shoes ordered, together, we can solve for an (x,y) value that satisfies both equations and it will tell us how many of each shoe the Basketball team ordered]… |
Evaluation/Decision Point
Many students may not get everything on the worksheet. I will circulate to figure out when most students have finished or are close to finishing. It is okay if not everyone is finished because at this point, I just want them to have tried some of these problems out on their own. |
Student Outcomes
Once most students have completed the calculator activity worksheet, we will move into the explanation. |
Explanation Time: 12 min
What the Teacher Will Do
Have students compare their answers with those of their classmates near them if they haven’t done that already. Get everyone focused at the front again and go over the worksheet. Ask students first to give any explanations before chiming in with any explanation. (This may work better if we go back and forth between each problem and the explanation). Students will be able to approximate solutions to systems of linear equations using a graphing calculator. Move onto shoe problem. Continue similar questioning until all of the problems are answered by the students. |
Probing/Eliciting Questions
Starting with the cell phone plan question, what did your graphs look like? Can I have a volunteer come and put the graphs into this calculator (hooked up to smartboard). Does anyone have questions on how to do that? I need someone to tell me the first step for finding the point of intersection… (Continue to go through steps until you find the point of intersection.) After how many text messages after 600 does Plan A become more expensive than Plan B? Does that number make sense? Why or why not? How could we check to make sure? So, if I send about 200 text messages per week, which is about 800 text messages per month, which plan is better for me? Why? For the second problem, what were the equations we were dealing with? What did you do next? Did you just plug those equations into your graphing calculator? No? Why not? How did you re-write them? Continue similar questioning until all of the problems have been answered by the student. |
Student Responses and Misconceptions
At 128.5719 text messages, at 88.0001 text messages, it’s really more like 128 text messages because you can’t have half a text message, it doesn’t really matter as long as you know not to go over 128,[ you could approximate and see if when x = 129 Plan A or Plan B is more expensive], you cannot check because the numbers are too obscure and you will continue to get estimates… Neither are good plans, you’re paying too much, [since you send over 128 extra text messages per month, Plan B will be a little cheaper], it is hard to tell because you may go under 600 in which case, Plan A would be better because of the base price, Plan A is cheaper for you… [x + y = 10, 89.95x + 123.99y = $998.47], we couldn’t find the equations… [No, because you need to re-write them in terms of y or else you’ll be working with equations that cannot be graphed by the calculator so clearly that won’t help you], it works - just replace the y by whatever is in front of it and then you can put it into your calculator, all you need is the |
Evaluation/Decision Point
Once we have gone through both problems from the calculator activity and if there is time, we will begin the elaboration |
Student Outcomes
Students should be able to use the graphing calculator to graph functions and find their points of intersections. If not, we will continue practicing during whatever time remaining. |
Elaboration Time: 5 minutes
What the Teacher Will Do
Change cell phone plan rates into something more easily solved by elimination or substitution. Begin discussing other ways of approaching solutions to linear equations. |
Probing/Eliciting Questions
What if Plan A and Plan B’s text messaging rates changed a little bit? Consider the following equations. Is there an easier way to solve for these other than graphing? |
Student Responses and Misconceptions
Graphing is the easiest because you just have to find a point, well, [if the both equal y, we can set them equal to each other and solve for x. The slope of the second equation is half that of the first, maybe we can subtract them from each other and solve for x]…. |
Evaluation/Decision Point
Considering we may or may not get to this problem, the decision point will depend mostly on the time. Regardless, at least five minutes prior to the end of class, we will begin the evaluation. |
Student outcomes
Students should at least see that finding a solution by hand-graphing or using the calculator may not always be the easiest. |
Evaluation Time: 5 minutes
During the last five minutes of class, pass out evaluation involving graphing linear equations and answering some quick questions.
For accompanying worksheets and pdf versions of this lesson see the following links:
points_of_interest_lesson.pdf | |
File Size: | 680 kb |
File Type: |
pre-postassessmentsworsheets.pdf | |
File Size: | 516 kb |
File Type: |
benchmarklessonfinalpwpt.pptx | |
File Size: | 309 kb |
File Type: | pptx |