Teaching Unit

Drew Cronic and Chris Hood

Inequalities and the Number Line

Ordered Pairs

The Coordinate System

Equations with Two Variables

Graphing Equations

Slope

Intercepts

Assumed Knowledge: The students should already know how to solve equations. They should also be familiar with <,>,>=,<= signs for inequalities and know how to solve inequalities.

Homework Rubric: Homework will be graded for effort and counted for 25% of final grade.

Completion = 100

Complete 3/4 = 75

Complete half or less = 0

Goal: The students should be able to graph solutions of equations on
a number line.

Objectives:

1. The students should be able to solve one variable equations and graph
the solution.

2. The students should understand the possibilities for representing data
on a number line.

Lecture: I would begin the lesson having a number line created in G.S.P..
I will remind the students that this demonstration is a number line.

What do the arrows mean ?

I will tell the students that the numbers represent a coordinate of a point
and its direction from zero. The students have learned that when they solved
equations they found a solution, so now I will teach them to graph the coordinates
on a number line.

Examples:

1. y+4=5 check: y+4=5 2. 9x+2=5 check: 9x+2+5

-4 -4 1+4=5 -2 -2 9(1/3)+2=5

y=1 5=5 9x=3 3+2=5

9 9 5=5

x=1/3

A good idea would be to have the students come up and point with the mouse
where the points should go.

1)

2)

Example:

3. 2a+4a-1=19 check: 2a+4a-1=-19

6a-1=-19 2(-3)+4(-3)-1=-19

+1 +1 -6-12-1=-19

6a=-18 -19=-19

6 6

a= -3

3)

Now, I will ask the students to write an equation for the following graph
and explain why you know your equation is right.

4 Use a number line to express the low temperature last night was 30 degrees.
Tell the students that for this problem we are expressing data on a number
line.

Ask the students does it matter how we mark our number line. Tell them that
the increments of our number line must be equal and that the numbers should
increase from left to right.

A possible misconception is the graphing of decimals and fractions.

Homework will be given as classwork. Problems will include real world problems
and the same problems used above in the examples.

First, I will ask some of the students to put particular problems on
the board from the homework and while they are doing this I will go around
and check the homework.

Goal: The students should be able to graph the solutions of inequalities
on a number line.

Objectives:

1. The students should recognize the difference in the open and closed point
or another alternative is the use of parentheses as a way of expressing
the open dot.

2. The students should recognize where to bold the line in the appropriate
direction.

3. The students should appreciate the possibilities of using a graph to
represent data.

First, I will begin by reviewing the different symbols used for inequalities.
I will start the class with an example representing the speed limit on interstate
85 in Georgia. I will prompt the students to produce an inequality that
represents the speed limit on this interstate. I will let s stand for speed.

So s<=70.

( Because G.S.P does not allow closed dots I would represent inclusive with
black point and noninclusive with red point)

The graph:

Then we would do an inequality that represents all illegal speeds. s>70

Notice that 70 is not included in the solution.

Do some more examples of graphing inequalities with open and closed dots.

Another example: Graph x>=0 or x<2

Notice that 0 is included and 2 is not included.

Possible Misconceptions:

1. Forgetting to change the direction of the inequality symbol when dividing
and multiplying by a negative number.

2. Understanding of the open and closed dots..

Remaining classwork and homework: I will choose problems from the book for
them to do. I will make sure that some challenging problems are given.

Same routine for homework is done.

Goal: The students should be able to graph points in the Cartesian Plane.

Objectives:

The students should understand:

1. how to recognize an ordered pair (x,y) with being the horizontal shift
and y being the vertical shift.

2. to always move horizontally first then vertically.

3. to label the axes and origin.

4. the location of all four quadrants and their appropriate signs.

First, I will introduce the students to the Cartesian Plane and explain
who it was named after ( Rene' Descartes ). I will also explain that the
Cartesian Plane is just one horizontal number line and one vertical number
line that intersect at 0. I will explain that this point is called the origin
and the horizontal number line is the x-axis and the vertical the y-axis.

I will now begin to graph points in the Cartesian Plane to explain how to
graph ordered pairs (x,y).

I would also mark some points on the Cartesian Plane and ask the students
to give me the ordered pair. A worthwhile task that could be implemented
for graphing ordered pairs is the game of battleship. This would be enjoyable
for the students and they would be getting familiar with graphing ordered
pairs in the Cartesian Plane. No homework, but a quiz will be given on the
next day.

Solve and graph:

1.2y+4=16

2. Write an equation that would provide the following solution:

Graph:

3. y>3

4. x<=-5

5. Write an inequality that describes:

From the following set of axes, write the ordered pair that corresponds
to each point and tell what quadrant it is in.

Graph the following points ( remember to label your axes and points ).

A( -4,3) B(2,-1) C(0,0)D(-6,-2)

After quiz is finished a review will be given.

Goal: Our goal is for the students to know how to solve equations with
two variables, and how to graph these equations using coordinates.

Objectives for 8.5:

1) We want our students to recognize that putting values in for x is the
easiest way to determine y-values.

2) We want our students to understand that at least two points must be determined
before a line can be graphed.

3) We want our students to understand that the final solution to an equation
is not one number but is expressed as an ordered pair.

4) We want them to understand that there are infinitely many solutions to
an equation with two variables.

5) We want to introduce them to the technique of using tables to organize
their thinking and solutions.

Objectives for 8.6:

1) We want to review graphing ordered pairs.

2) We want our students to understand that graphing is a step by step process.

3) We want the students to recognize other solutions to the equations from
the graphs.

4) We want to introduce the term " linear equation", and we want
them to be able to recognize what a linear equation looks like.

Lesson Plans:

**Day 5**

- We will begin discussing solutions to equations.

- Introduce tables.

- We will let the students create a table from an example equation. (An
interesting and real world example that will call for them to create their
own equation). We will provide coaching during this time. In addition, we
may let some students write their tables on the board.

Real World Example:

Forests serve as the only source of timber. Sometimes the production of
timber conflicts with the need to conserve the environment and wildlife.
In an effort to conserve both the forests and the wildlife, foresters replace
the tress that have been cut down. About half of the seedlings planted survive
until they are full grown.

- Plot points on a graph. Connect points. Point out that it forms a line.

- Conclude: So class we can see that there is not just one solution to an
equation. There may be infinitely many. In fact, every point that is on
the line we just graphed is a solution.

y=(1/2)x

x 1/2x y

-5 -2.5 -2.5

-3 -1.5 -1.5

-1 -0.5 -0.5

1 0.5 0.5

3 1.5 1.5

5 2.5 2.5

7 3.5 3.5

9 4.5 4.5

11 5.5 5.5

13 6.5 6.5

15 7.5 7.5

17 8.5 8.5

19 9.5 9.5

21 10.5 10.5

23 11.5 11.5

25 12.5 12.5

We would have the students to create tables either on Microsoft Excel or
TI-82 graphing calculator.

Possible Questions to ask on Day 1

- Why is (2, -5) a solution to the equation y= 3 - 4x ?

- If x=4, what is the value of y in the equation y=x/4 + 1 ?

- Why does an equation have an infinite number of solutions ?

Possible Misconceptions on Day 1

- substitution for y instead of x.

- the difference between a solution for the whole equation being a point and a solution for just x.

Possible use of Technology on Day 1

- no need for technology

Assignments

- questions 1-4, 6, 8-12,,24,30 pp. 297-298

- worksheet attached (maybe not an assignment)

- More practice with tables and plotting points. Practice worksheet included.

- Clarify that the graphs that have been made are lines and that every solution to the equation is on that line.

- Introduce the term " linear equation ".

- Review by doing an example that calls for the students to recall what they have learned from the past two days.

Graph the equation Y=x/2-5.

Possible Questions to ask on Day 2

- Is a certain point on the line ? If it is not on the line, is it a solution to the equation ?

- What do we call an equation that looks like y = mx + b ?

Possible Misconceptions on Day 2

- the fact that all possible solutions are on the line

Possible Use of Technology

- let the students use tables on the TI-82.

Assignments

- pp 300-301 1-3, 4-20 even.

- Write in your own words how to get a solution or solutions to an equation with two variables.

- practice worksheet on graphing linear equations and linear inequalities.

( Slope will be planned for a two day period )

Goal: The students should understand the concept of slope.

Objectives:

1) Review the graphing of linear equations.

2) The students should understand how to find the slope of a line.

3) The students should be able to find the slope of a line given two points on a line or choosing any two points on a line.

4) The students should be able to graph a line using slope and a point.

Lesson Plans:

- Review: We will get the students to graph the equation y = 3x + 1 through tables. Does everyone's graph look like this ?

- Introduce the term " slope " . A good example is found in the book with the idea of a roller coaster.

- Give another example using the idea of a handicapped ramp.

Example: A ramp installed to give handicapped people access to a certain building has a base of 36 meters long and an elevation of 3 meters. What is the slope of the ramp?

horizontal change

slope = change in vertical units/change in horizontal units. Explain.

slope = 3/36 = rise/run

- Questions to ask at this point are:

1) What other objects have slope ? ( hills, roads, ladders )

2) Can we measure these slopes ?

3) How would you measure these slopes ?

- At this point we will refer back to the graph of y = 3x+1.

Does this line have a slope ?

- Tell the students that slope is constant for linear equations.

(Demonstration from G.S.P )

- Then draw triangles representing rise and run:

- Reinforce this by looking at table and unit changes of x and y (always
stays constant).

- Show the example y= 3x/2 + 2.

- Show that slope is the difference in y-coordinates over the difference
of corresponding x coordinates. Make sure that the students know the formula
will not work if the variables are changed around.

- Introduce lines with slope zero and no slope.

Misconceptions for slope

- only two points on a line determine slope

Possible uses of technology

- use the TI-82 to show graphs

Assignments

- pp 306-307 1-3, 7-17 odd. 22, 32

- possible practice worksheet

Worksheet on Slope ( to begin the period )

We will divide the students into groups of two or three.

Graph a line with a slope of 2 that contains the point (2,4).

Using your Ti-81 graphing calculators or Algebra Expressor, graph any line of the form y=mx+b.

What happens when you change the m coefficient for the general equation y=mx+b?

What happens to the graph when m is made greater?

What happens to the graph when m is made smaller?

What happens when m is negative?

What happens when m is zero?

Next, we will review the worksheet.

1)

This is the graph of y=2x. It has a slope of two and passes through the
point (2,4).

2)

This is the graph of y=4x. We can see that the graph is increasing at
a greater rate.

3)

This is the graph of y=(1/2)x. We can see that the graph is increasing
at a lesser rate.

4)

This is the graph of y=-5x. We can see the graph is now decreasing.

5)

This is the graph of y=0x+3. We can see that a graph with a slope of
0 is horizontal.

(one to two day period)

Goal: The students should be able to graph a linear equation by using the x and y intercept.

Objectives:

1) Review the concept of slope.

2) For the students to become familiar with the terms x and y intercept.

3) For the students to learn and apply the steps for finding the x and y intercepts. Find the value of x when y=0. Find the value of y when x=0.

Lesson Plans

- Give the daily quiz.

- Review the formula for slope.

- Give general equation of the line y=mx+b.

- Introduce x and y intercept

- show the steps for finding the x and y intercepts and how to graph a line with only the intercepts ( two points determine a line).

- show how to graph linear equations with the use of slope and the y intercept

Possible Questions to ask

- How many points determine a line ?

- What is the y-value when the x intercept is _ ?

- What is the x-value when the y-intercept is _ ?

Possible Misconceptions

- where to substitute 0 for finding the x or y intercept

- why does the m variable stand for slope and the b stand for the y-intercept

Possible Group Activity

- have the students graph some equations using slopes and intercepts and ask them some conceptual short answer questions. Maybe even use the computer lab.

Possible Uses of technology

- definitely TI-82.

Assignments

-pp 308-309 1-4, 5,8,11, 14-22 even, 27,28

- worksheet on graphing lines using slope and the y-intercept

- quiz 2 tomorrow

Day 10

Using your TI-81 calculators or Algebra Expressor, complete the following.

1) Graph the equation y=2x+6. What is the y-intercept.

2) Graph a line with a slope of 3 that passes through the point (0,-4).

What is the y-intercept?

3) Explore what happens when b (in y=mx+b) is made greater.

4) Explore what happens when b (in y=mx+b) is made lesser.

5) Explore what happens when b (in y=mx+b) is negative.

6) Explore what happens when b (in y=mx+b) is zero.

Next we will go over the worksheet.

1)

We can see that the y-intercept is 6.

2)

This is the graph of y=3x-4. We can see that the y-intercept is -4.

3)

This is the graph of y=3x+1. We can see that the graph shifted up.

4)

This is the graph for y=3x-7. We can see that the graph shifted down.

5)

This is the graph for y=3x-3. We can see from this and the other graphs
that when b is negative, the y-intercept is negative.

6)

This is the graph for y=3x. We can see that it passes through the origin.

1) Graph the equation y=2x-3 using a table with four solutions. SHOW ALL WORK!!!!! (5 points)

2) Find the slope of the line that contains the pair of points A(5,3), B(-1,1).

( 3 points )

3) Explain in your own words what you think slope is. (10 points)

4) Graph the equation y=2x+3 by finding the x and y intercepts.(7 points )

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