Learning Cycle Lesson

Lesson : Systems of Equations

Grade: 9th (Algebra 1)

Concept: Be able to solve systems of equations by graphing

Expected Prior Knowledge: Be able to solve and graph linear equations. Be able to write linear equations to represent word problems.

Alabama Course of Study :#8 Solve systems of linear equations and inequalities in two variables graphically or algebraically

Exploration: I will give put the students in a group of two and they will do the activity on the next slide. They will be given a copy of the next 2 slides on a word document. I will also give them a ruler when they graph.(30-40 minutes)

Concept Application: We will go over the activity, then I will show them a power point presentation. (25 minutes)

Extension: I will give them two real-life word problems to solve using systems of equations. We will do one together, then they will do one on their own. (45-55 minutes)

Airplanes Activity

You are an air traffic controller. You have to make sure that no airplanes crash into each other. To prevent this, you have to know where the potential danger spots (where they could collide) are. Each plane has its own route given by an equation. The planes only fly in straight paths.

Airplane A has an air path of Airplane B has an air path of y = x -1

y = x + 432

1 2 3 4 5 6 7 8 9 10 11 12–1–2–3–4–5–6–7–8–9–10–11–12 x

1

2

3

4

5

6

7

8

9

10

11

12

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

–12

–13

y

AB

1.) How many potential danger spots are there for Airplane A and B?

2.) How do you know for sure that there are no more danger spots?

3.) Find the exact location of each danger spot?

4.) What happens when you plug your danger spot(s) into each equation?

5.) Is there ever a time that 2 airplanes will not have any danger spots? If so, when?

Teacher Notes 1) There will be one danger zone. 2) There is only one spot that the two planes

could cross when you draw their paths because 2 different lines can only intersect in one place.

3) (3,2) is where their paths intersect 4) It makes both sides of both equations

equal. 5) When one is flying directly over or under

each other.

1 2 3 4 5 6 7 8 9 10 11 12–1–2–3–4–5–6–7–8–9–10–11–12 x

1

2

3

4

5

6

7

8

9

10

11

12

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

–12

–13

y

AB

(3,2)

y = x - 1y = x + 4

32

Teacher Questions to Students Why did the planes only have one danger zone? (Looking for them to

see that 2 different lines can only intersect at one point.) You got an ordered pair as your location for the danger spot. That

ordered pair is on both lines so it is a solution when you plug it in to both equations. Is there any other ordered pair that you could plug into both equations and it be a solution? (You want them to see since there is only one point that they intersect, that is the only ordered pair that is a solution to both equations.)

Was there ever a time that you found where two planes would not crash? (Hopefully a group figured out if one plane was flying directly above or below the other and flying horizontally, they would not crash.)

If we think about these 2 planes and lines, what type of lines would they represent if they do not have a danger zone? (Wanting them to see parallel lines will never intersect.)

Is there ever a situation where 2 planes would have more than one danger zone? ( You want them to think if they are flying the same route, that they could crash in a lot of spots. You them lead them to understand that if you have 2 of the same line, there is infinitely many points that they would intersect.)

Solving Systems by Graphing

Objective

The student will be able to:

solve systems of equations by graphing.

What is a system of equations?

A system of equations is when you have two or more equations using the same variables.

The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair.

When graphing, you will encounter three possibilities.

Intersecting Lines

The point where the lines intersect is your solution.

The solution of this graph is (1, 2)

(1,2)

Parallel Lines

These lines never intersect!

Since the lines never cross, there is NO SOLUTION!

Parallel lines have the same slope with different y-intercepts.

2Slope = = 21

y-intercept = 2y-intercept = -1

Coinciding Lines

These lines are the same!

Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS!

Coinciding lines have the same slope and y-intercepts.

2Slope = = 21

y-intercept = -1

What is the solution of the system graphed below?

1. (2, -2)2. (-2, 2)3. No solution4. Infinitely many

solutions

1) Find the solution to the following system:

y = -2x + 4y = x - 2

Graph both equations.

Graph the equations.

y = -2x+4

m=-2 and b=4

y = x - 2

m=1 and b=-2Where do the lines intersect?

(2, 0)

y = -2x+4

y = x-2

Check your answer!

To check your answer, plug the point back into both equations.

y = -2x + 4 (0) = -2(2) + 4

y = x - 2(0) = (2) – 2 Nice job…let’s try another!

2) Find the solution to the following system:

y = 2x – 3-2x + y = 1

Graph both equations. Put both equations in slope-intercept form.

y = 2x – 3y = 2x + 1

Graph using slope and y-intercept

Graph the equations.y = 2x – 3m = 2 and b = -3y = 2x + 1m = 2 and b = 1Where do the lines intersect?

No solution!

Notice that the slopes are the same with different y-intercepts. If you recognize this early, you don’t

have to graph them!

Check your answer!

Not a lot to check…Just make sure you set up your equations correctly.

I double-checked it and I did it right…

What is the solution of this system?

1. (3, 1)2. (4, 4)3. No solution4. Infinitely many

solutions

3x – y = 82y = 6x -16

Hint: Solve both for y.

Solving a system of equations by graphing.

Let's summarize! There are 3 steps to solving a system using a graph.

Step 1: Graph both equations.

Step 2: Do the graphs intersect?

Step 3: Check your solution.

Graph using slope and y – intercept. Be sure to use a ruler and graph paper!

This is the solution! LABEL the solution!

Substitute the x and y values into both equations to verify the point is a solution to both equations.

Extension Read the following 2 word problems. Write a system of

equations (2 equations) to represent the word problem, then solve them by graphing using the graphing calculator.

1.) Corey scored 41 points without shooting any free throws. He made 16 baskets. The school record for made 3-point baskets in one game is 7. Did Corey break the school record?

What do they want to know? How do we find out? What are our unknowns? Let x and y represent our unknowns.

x = # of 2 pointers made y = # of 3 pointers madeWrite an equation for the number of shots scored.x + y = 16Write an equation for points scored.2x + 3y = 41

Our system of equations is:x + y = 162x + 3y = 41

We have to solve for y, then plug into our y= on our calculator.

y = -x + 16y = (-2/3)x + (41/3)

We need to adjust our window. Do we need to look at any quandrant but 1? Can # of shots made be negative? What are our y-intercepts?

If will adjust each axis starting at 0 going to 25, should we see everything we need to?

Now press graph

Now what will our solution be? Now press 2nd Trace, and tell me which

one will we choose?

Press 5

Press Enter

Press Enter

Press Enter

What does this mean?What did x and y represent?So Corey made 7 two pointers and 9 three

pointers.Is that the answer?The question was did he break the school

record?Your final answer is: Yes, because he

made 9 three pointers to break the previous record of 7.

Now work with your partner and go through the same process to answer the following problem.

2.) Brian has $3.70 in quarters in dimes. If he has 22 coins, how many dimes does he have?

By allowing them to do the airplanes activity, it promotes the purpose of schools because it forces the students to think.

It reflects the nature of science, because it was an act of doing.

It aligns with how students learn. It gives an opportunity for exploration and disequilibrium. Then the term induction provides accomodation, and the extension provides organization.

Reflections:This lesson would usually be taught by the teacher standing in the

front and telling them the definition of systems of equations. They would then tell them you can graph to find an answer. Then they would work through examples of I do, we do, and you do. This would not have engaged the students nor given the students an experience and chance for exploration. They would not have had to critically think at all. It helps the students to gain conceptual understanding and not just procedural understanding of systems of equations.