title of lesson points of interest - weebly · 2018-10-04 · florida math connects, teacher...

Title of Lesson: Points of Interest Authors’ Names: Emily Nelson Teaching Date and Time: March 26-27, 2012 at 7:55 Length of Lesson: 100 minutes Grade / Topic: 8th Grade Advanced Math Source of the Lesson: Florida Math Connects, Teacher Edition. Course 3, Volume 1. Glencoe: McGraw-Hill For formative assessment ideas, see the following book: Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning by Page D. Keeley and Cheryl Rose Tobey For Day 2 calculator activities and word problems, see 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

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. o 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.

o 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

Probing/Eliciting Questions Student Responses and

Misconceptions

Begin powerpoint slides. First one will list several equations whose y-intercept is 0. Use to promote discussion about solutions to systems of

What is the solution to this system of linear equations? What do I mean by a “solution”?

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

linear equations. Explain to your students that we will be looking at systems of linear equations and try to solve for them graphically.

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?

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 Assessment Student Outcomes

If 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.

Students should be comfortable and familiar with this idea of solutions. However, I will watch out for furrowed brows and blank stares.

EXPLORATION Time: 18 Minutes

What the Teacher Will Do Probing/Eliciting Questions Student Responses and

Misconceptions

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 and 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 for their partner. Encourage them to move around and talk to one another about their equations and intersection points.

(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

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…

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.

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.


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.

Students should be able to re-write equations in the y = mx +b form, graph them, and determine points of intersection.

EXPLANATION Time: 12 Minutes


Misconceptions

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.

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

[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

If students are still nowhere near finished with their T/F worksheets, let them continue to complete those during the remaining time. If they finish those, begin the following elaboration.

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.

solution exist?

the two lines intersect at every point because they are really the same line]…


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.

Students 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.

ELABORATION Time: 5 Minutes


Misconceptions

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.

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

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

Day 2 Lesson Plan

Depending on evaluations from day one, I will either review those or begin with the following cell phone

problem.

ENGAGEMENT Time: 10 Minutes

What the Teacher Will

Do Probing/Eliciting Questions

Student Responses and

Misconceptions

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.

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

Duh, everyone owns one… Never, of course, I’m even texting as you speak… y = 51 + 28x,[ y = 51 +.28x], I have

.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.

won’t be exactly right because the y-intercept is still different], I’m still not so sure what the question is…


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.

Students should at least see that finding a solution by hand-graphing is really not the best way to approach this problem.

EVALUATION Time: 7 Minutes


Misconceptions

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.

Students will have a sheet of graphing paper should they choose to attempt to hand draw the graph.

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.

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…


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 should be able to write

out each cell phone plans’ rate after

600 texts as a linear equation.

Students should be able to

students and letting students volunteer. 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


Misconceptions

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

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?

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,

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.

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]…


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.

Once most students have

completed the calculator activity

worksheet, we will move into the

explanation.

EXPLANATION Time: 12 Minutes


Misconceptions

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.

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?

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

Move onto shoe problem.

Continue similar questioning until

all of the problems are answered

by the students.

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.

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 x-variable…


Once we have gone through both problems from the calculator

activity and if there is time, we will begin the elaboration

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


Misconceptions

Change cell phone plan rates into

something more easily solved by

elimination or substitution. Begin

discussing other ways of

approaching solutions to linear

equations.

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?

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]….


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.

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


Misconceptions

During the last five minutes of class, pass out evaluation involving graphing linear equations and

answering some quick questions.

title of lesson points of interest - weebly · 2018-10-04 · florida math connects, teacher...

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