1

Algebraic Relationships

2

Geometric and Spatial Relationships

2

Number and Operations

PREVIOUS LEARNING

TARGETED LEARNING

FUTURE LEARNING

•   compare and contrast various forms of representations and patterns

•   represent and describe patterns with tables, graphs, pictures, symbolic rules, or words (A1B6)

•   analyze patterns in various forms of representation

•   represent a mathematical situation as an expression or a number sentence

•   compare various forms of representations to identify a pattern (A1C6)

•   compare and contrast various forms of representations to identify a pattern

•   apply the distributive and associative properties to whole numbers

•   identify functions as linear or nonlinear from a table or graph (A1D6)

•   identify functions as linear and nonlinear from tables, graphs, or equations

•   model problem situations and draw conclusions, using representations such as graphs, tables or a number sentence

•   use variables to represent unknown quantities in expressions (A2A6)

•   generate equivalent forms of simple algebraic equations

•   identify, model, and describe situations with constant or varying rates of change

•   recognize equivalent forms for simple algebraic expressions (associative, distributive properties) (A2B6)

•   model and solve problems using multiple representations such as graphs, tables, expressions, equations or inequalities

•   use coordinate systems to specify location

•   model and solve problems, using multiple representations such as graphs, tables, expressions, and equations (A3A6)

•   compare and contrast various rates of change

•   compare situations with constant or varying rates of change (A4A6)

•   analyze patterns in various forms of representation

a.

b.

1.

2 points

The student correctly identifies all four rates of change: a. constant, b. varying, c. varying, d. constant.

1 point

The student correctly identifies any three rates of change.

0 points

The student provides less than three correct answers or no answers.

2.

2 points

The student identifies a constant rate of change and explains that the rate of change is constant because Chris adds the same amount of money each week to his savings.

1 point

The student identifies a constant rate of change or explains that the rate of change is constant because Chris adds the same amount of money each week to his savings.

0 points

The student provides incorrect or no answers.

3.

2 points

The student identifies a varying rate of change and explains that the rate of change is varying because the number of cards added each time is not the same.

1 point

The student identifies a varying rate of change or explains that the rate of change is varying because the number of cards added each time is not the same.

0 points

The student provides incorrect or no answers.

4.

2 points

The student identifies a varying rate of change and explains that the rate of change is varying because the car did not travel at the same speed, due to the stop.

1 point

The student identifies a varying rate of change or explains that the rate of change is varying because the car did not travel at the same speed, due to the stop

0 points

The student provides incorrect or no answers.

5.

2 points

The student writes the expression 8m – 2 to represent the total cost of tickets for the Smith family and identifies $38 as the total cost of tickets for a family of five.

1 point

The student writes the expression 8m – 2 to represent the total cost of tickets for the Smith family or identifies $38 as the total cost of tickets for a family of five.

0 points

The student provides incorrect or no answers.

10.

2 points

The student identifies x + (3 + 7) as the correct expression and explains that it illustrates the commutative property of addition.

1 point

The student identifies x + (3 + 7) as the correct expression or explains that it illustrates the commutative property of addition.

0 points

The student provides incorrect or no answers.

12.

2 points

The student identifies Austin’s equation as correct and explains that it is correct because the associative property of addition enables him to change the grouping of the numbers that are being added together without changing the answer.

1 point

The student identifies Austin’s equation as correct or explains that it is correct because the associative property of addition enables him to change the grouping of the numbers that are being added together without changing the answer.

0 points

The student provides incorrect or no answers.

13.

2 points

The student identifies Ginny’s equation as incorrect and explains that it is incorrect because Ginny failed to multiply each of the addends by 4. She only multiplied a by 4; she should have also multiplied b by 4.

1 point

The student identifies Ginny’s equation as incorrect or explains that it is incorrect because Ginny failed to multiply each of the addends by 4. She only multiplied a by 4; she should have also multiplied b by 4.

0 points

The student provides incorrect or no answers.

14.

Phase

Number of Ties

1

4

2

7

3

10

4

13

5

16

6

19

2 points

The student provides a correctly completed table and explains that the pattern is a linear function because there is a constant rate of change in both the phases (1 each time) and the number of ties (3 each time) from one term to the next.

1 point

The student provides a correctly completed table or explains that the pattern is a linear function because there is a constant rate of change in both the phases (1 each time) and the number of ties (3 each time) from one term to the next.

0 points

The student provides incorrect or no answers.

15.

2 points

The student identifies B as the nonlinear graph and explains that it is nonlinear because the rate of change varies from one term to the next..

1 point

The student identifies B as the nonlinear graph or explains that it is nonlinear because the rate of change varies from one term to the next..

0 points

The student provides incorrect or no answers.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“Today, we’re going to be exploring ways to represent expressions. Please take 30 seconds to think about how you would define ‘expression’ to someone else. At the end of that 30 seconds, I’ll ask you to share your thoughts with a partner.”

Allow time for students to think about the definition and then ask them to share with a partner.

“Now, who can explain to the class what an expression is?” (An expression is a variable or combination of variables, numbers, and symbols that represents a mathematical relationship.)

“One way to represent a mathematical situation is to write an expression. I’m going to write some examples of expressions.”
(Write the following on the overhead)

p + 5

12 – 3

28 ¸ a

6c

6 / 2

x + y

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“How are the expressions alike?” (They all have numbers in them.)

“How are they different?” (Some have operation symbols and letters or numbers; some just have letters and numbers.)

“Bob has 28 extra pieces of candy that he wants to share equally among his friends.

We don’t know the number of friends Bob has. In mathematics, we can use letters to represent unknown quantities in expressions. Since we don’t know the number of friends Bob has, we can use the letter “a” to represent that number. We refer to letters that we use to represent unknown amounts as variables. Variables hold the place of an unknown value, allowing us to represent a mathematical situation without knowing all the numbers.”

“Are there any unknown values in this statement? (number of friends)

“Which expression from the list on the overhead would represent this problem?” (28 ¸ a)

“Do we know the value of letter ‘a’?” (No.)

“Is there more than one value that ‘a’ could represent? (Yes.)

“What numbers could the letter ‘a’ represent? (2, 4, 7, 14)

“We like to write expressions as efficiently as possible. Not only do we use variables in writing mathematical expressions, we also use different representations for multiplication and division. For example, 25 ¸ 5 can also be written as 25 / 5, and
6 ´ c can also be written as 6c.”

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

The teacher should then hold up a brown paper bag with the letter “n” written on the outside and containing some tiles.

“This bag has some tiles in it. I have written the number of tiles on the outside of the bag.”

“Who can tell me how many tiles are in the bag?” (n)

Some students may respond that they don’t know the number of tiles in the bag. Remind them that the label on the bag tells them how many tiles are in the bag.

Have students work with a partner to write in words a situation that uses the number of tiles in the bag.

After students have completed their examples, have them share with the whole class and discuss some of the examples.

There are a variety of examples students could provide, such as

•  the number of tiles in the bag minus 1

•  8 plus the number of tiles in the bag

“Since we know there is “n” number of tiles in the bag, let’s use that information to help us write some expressions to represent the number of tiles in the bag.”

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“Suppose we add 7 tiles to the bag, what expression could we write to represent the number of tiles in the bag?” (n + 7 or 7 + n)

“What expression could be used to represent each situation below? I want everyone to write an answer on their paper.”

(Write the following on the overhead):

•  n decreased by 12

•  the product of n and 15

•  the quotient of n and 9

As students are recording their answers, tour the classroom to check for understanding, and assist students who might be having difficulties. When everyone is done, call upon students with correct answers to respond.

Project the “Paper Bag” transparency, and have students pair up and write an expression to represent each of the four mathematical situations described on the overhead.

Call upon students to report the expressions they

wrote, and discuss whether they are correct or

incorrect.

Answers to “Paper Bag” transparency problems:

•  n – 8

•  n / 4

•  3n

•  12 – n

Distribute “A Bagful of Problems” formative assessment to each student to complete.

Situation:

The product of the number of candies in the bag and 13.

Expression:

Situation:

Sue wants to share her candy equally with some of her friends.

Expression:

Situation:

Expression:

Expression:

13z or 13 ´ z

Expression:

12 ¸ f or 12 / f (any variable can be used for f)

Student answers may vary. Example:

Situation:

John has some candies that he wants to share equally among his five friends. How many candies will each friend get?

Expression:

c ¸ 5 or c / 5

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Divide the class into small groups (Orange and Green), and distribute one blue, one red, and one yellow cup to each group, along with 100 counters.

The colors of the cups do not affect the activity. Any color may be used.

Each group needs to place the blue cup on the left, red cup in the middle, and yellow cup on the right.

Students should sit facing the overhead so they can see the responses to record in their group journal.

“I would like each group to put 15 counters in the blue cup, 8 counters in the red cup, and 9 counters in the yellow cup.”

Remind students that they should track the steps they are following by drawing pictures in their journal as the teacher records on the overhead.

The teacher keeps track of student responses on the overhead while students keep track of the responses in their journal.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“How can we find out the total number of counters in all three cups?” (Pour them out and count them or add the three numbers in each cup.)

“Now, I’d like the Orange groups to pour their counters from the red cup into the blue cup and the Green groups to pour their counters from the red cup into the yellow cup.”

“How can we find the total number of counters in the cups that have been combined?” (Pour them out and count them or add the numbers from the two cups together.)

“Suppose we didn’t have cups to pour our counters into. How could we show mathematically that we want to add the counters in those two cups first?” (We could use parentheses to indicate that we want to add those two cups together.)

“How would you write that as an algebraic expression?” (15 + 8) + 9 and 15 + (8 + 9)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“Yes, when you write the expressions, use parentheses to indicate what you want to combine together. So let’s put parentheses in our problems and see what we get.”

Orange group should add:

(15 + 8)   +   9

        23   +   9

Green group should add:

15   +  (8 + 9)

15   +  17

“Do you now have the same values to add?” (No. Orange has 23 + 9, and Green has 15 + 17.)

“What do you get as the total of all three cups?” (32)

“So would everyone agree that

(15 + 8) + 9 = 15 + (8 + 9)?” (Yes, because both algebraic expressions are equal to 32.)

Be sure to insert the equal sign between the two expressions on the overhead.

“This is an example of the associative property of addition.”

“What does the word ‘associate’ mean?” (to hang out with someone, to do things with someone, etc.)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“The associative property of addition states that the sum stays the same when the grouping of the addends (numbers being added) is changed.”

“Why is our problem an example of the associative property of addition?” (because we grouped together different numbers to add first, but it did not change our sum)

“Do you think this would work for decimals?” (Yes.)

“Can you give me an example?” (A common example would be a money problem.)

“Can you give me an example with a variable?” [Example: 5 + (x + 2) =
(5 + x) + 2]

“Do you think the associative property would work for multiplication?” (Yes.)

“Work with your group to write two mathematical expressions that would illustrate the associative property of multiplication.”

Allow groups time to write their

expressions, then as a whole class check

the examples.

“The associative property of multiplication states that the product stays the same when the grouping of the factors is changed.”

Remind students that only the grouping of the numbers can be changed, not the order.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“We are going to try one more problem with your cups. You will need only your blue and red cups.”

Have each group count 25 counters in the red cup and 43 counters in the blue cup.

“Suppose we want to double all the counters but don’t want to count them all out. Let’s look at how we can do that.”

“Orange groups, I want you to add the counters in your cups then double the total.

“Green groups, I want you to double the counters in the red cup, double the counters in the blue cup, then add the two totals together.”

Give students time to accomplish their task.

“Orange Groups, what was your answer?” (136 counters.)

“Green Groups, what was your answer?” (136 counters.)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“Both groups got the same answer. However, they used different steps, so let’s think about how we could write each group’s steps mathematically. Remember, you can use parentheses to indicate what you want to do first in a problem.”

“What algebraic expression would you use to write the mathematical steps you took?”

Orange group 2(25 + 43);

Green group 2(25) + 2(43)

Allow time for each group to work on writing its steps mathematically, then record the steps on the overhead.

So the steps would look like this:

2(25 + 43)    =    2(25) + 2(43)

2(68)            =    50 + 86

136              =    136

“Since both groups got the same answer, we can write 2(25 + 43) = 2(25) + 2(43)?” (Yes.)

“Who can describe the correct calculations for this equation? (On the left side, you would add 25 + 43 before multiplying the total by 2, and on the right side, you would multiply 2 times 25 and 2 times 43 then add those two values together.)

Point out to students that 2(25) and 2(43) is another way to show multiplication without using the “´” (times) symbol.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Here we have used the distributive property which states that when one of the factors of a product is written as a sum, multiplying each addend before adding does not change the product.”

“Who can give me an example with a variable?” [Example: 2(x + 5) = 2x + 2(5)]

“Would this work if we wanted to triple the number of counters?” (Yes.)

“How would we write that?”

3(25 + 43)   =    3(25) + 3(43)

3(68)          =    75 + 129

204            =    204

“Would this work for decimals?” (Yes.)

“Work in your groups to write a decimal example that that illustrates the distributive property.”

Allow groups time to work on their examples, and then check the examples as a whole class.

Distribute the Properties-Check Exit Assessment to students to complete before leaving class.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“Today we will be making human graphs. We will be going outside for the first part of the lesson. Before we go out, let’s do some reviewing.”

Prior to the lesson, the teacher should have constructed a 25 ´ 25 coordinate grid on a shower curtain.

Distribute the blue, green, and white index cards (one card with one coordinate to one student) and one Human Graph and Table Recording Sheet to each student. (Each student who receives an index card will have a separate value of x.)

Remind students to take something hard as a writing surface. Assign one student to complete the Human Table transparency and one student to complete the Human Graph transparency. (They take turns entering their data on the transparency.)

Also distribute three pencils (in three different colors) to each student.

“Who can tell me what a coordinate grid is?” (It is a 2-dimensional system in which a location is described by its distance from two perpendicular lines called axes.)

“How do we write the locations for points on a coordinate grid?” (We write them as ordered pairs. The first number x, tells the location on the x-axis and the second number y, tells the location on the y-axis.)

“How do we write the ordered pairs?”

(x, y)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“Some of you have a blue card with an ordered pair written on it that looks like this:

“What is the x value?” (0)

“How would you find the y value?” (Take 0 and add 1 to it for a value of 1.)

“So what would the ordered pair be?”
(0, 1)

Ask nine students with the blue cards to stand on the shower curtain on the x-axis value of their card. Other students should be watching where the students stand.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Allow time for the nine students to determine their y value.

“Now when I say go, walk to the value of your
y-axis.”

“How do you determine the y value?” (Add 1 to x.)

“How should you walk in relation to the
y-axis?” (You must walk parallel to the
y-axis.)

Human Table 1 Coordinates

(as per blue index cards)

Students will line up on the x-axis value on their card. The teacher should be monitoring to see who has difficulty with the x value.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

After all nine students have walked to their y value, have them explain how they know they are in the correct position on the coordinate grid. Other students should record in Human Table 1 the x and y values for each person standing on the shower-curtain grid and plot the points in blue pencil on the Human Graph in the Human Graph and Table Recording Sheet.

“To move from the (0, 1) to (1, 2) how many units up on the grid must you move?” (1)

“How many units over do you have to move?” (1)

“To move from (1, 2) to (2, 3), how many units up must you move?” (1)

“How many units over do you have to move?” (1)

“To move from (3, 4) to (4, 5), how many units up on the grid must you move?” (1)

“How many units over do you move?” (1)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“What pattern do you recognize in your movements?” (You always move one unit up and one unit over to go from one point that we have plotted on the grid to the next.)

Since the moves up and over on the grid to get from one point to the next are always the same, we refer to this as a constant rate of change.

“What do you notice about where you are standing on the grid in relationship to the others?” (We are standing in a line.)

“The ordered pairs you’ve graphed form a straight line. We refer to your graph as a linear function because there is a constant rate of change from each x to the next x and a proportional constant change in the value from one y to the next.”

“Can anyone describe a constant rate of change for x and y?” (As x increases by 1, y increases by 1.)

Make sure that all the students who were observing have the x and x + 1 values recorded in their tables before the next card group lines up on the x value.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Have students move off the grid. Now, ask those that have the green cards to position themselves on their x value.

Human Table 2 Coordinates

(as per green index cards)

“How do you determine the y value?” (Multiply x times 2 or double x.)

Allow time for students to determine their y value.

“Now when I say go, walk to the value of your y-axis.”

“Since you do not want to change the value of your x-axis, how should you walk in relationship to the y-axis?” (You will walk parallel to the y-axis.)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Ask the students who are observing to record in Human Table 2 the x and y values of each person standing on the grid and to plot the points with a green pencil on the Human Graph in the Human Graph and Table Recording Sheet.

“How many units must you move up and over on the grid to get from the person standing at (0, 0) to the person standing at (1, 2)?” (Move up two units and over 1.)

“If you begin at (2, 4) and move two units up and one over on the grid, where will you be standing?” (3, 6)

“Is there a constant rate of change from one point on the grid to the next in terms of moves up and over?” (Yes.)

“What is the constant rate of change?” (Move up two units and over one.)

“What do you notice about where you are standing on the grid in relationship to the others?” (We are standing in a line.)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“Why is this another example of a linear function?” (because the points graphed form a line, and there is a constant change from each x to the next x and a constant rate of change from each y to the next y)

“How would you describe the constant rate of change for x and the constant rate of change for y?” (As x increases by 1, y increases by 2.)

Have the group with the white index cards stand on the grid on their x value.

Make sure that all the students who were observing have the x and 2x values recorded in their tables before the next group lines up on their x value.

Human Table 3 Coordinates

(as per white index cards)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“How do you determine the y value?” (Multiply x times itself or square x.)

Allow time for students to determine their y value.

“Now when I say go, walk to the value of your y-axis. Since you do not want to change the value of your x-axis, you must walk parallel to the y-axis.”

Ask the students who are observing to record in Human Table 2 the x and y values of each person standing on the grid and to plot the points with a black pencil on the Human Graph in the Human Graph and Table Recording Sheet.

“To move from (0, 0) to (1,1) on the grid, how many units up and over must you move?” (one unit up and one unit over)

“To move from (1, 1) to (2, 4) on the grid, how many units up and over must you move?” (three units up and one unit over)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“To move from (2, 4) to (3, 9) on the grid, how many units up and over must you move?” (five units up and one unit over)

“Do you always move the same number of units up and over to move from one point on the grid to the next?” (No.)

“We refer to this as a varying rate of change.”

“What do you notice about where you are standing on the grid in relationship to the others?” (We are not in a line.)

“If we call a graph that forms a line linear, what name do you think we would call this graph?” (nonlinear)

“Nonlinear graphs represent varying rather than constant rates of change. And since the change in this graph varies from one set of coordinates to the next, and the points do not make a line, it is a nonlinear graph.”

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Review the graphs (for the blue, green, and white index cards) on the Recording Sheets. Take some time to discuss the constant changes in the x and y values in the three tables. Also discuss that while there is a constant change in the y value on the nonlinear graph, the change in the x value is not constant and that to be a linear function, there must be a constant change in both.

Distribute the Exit Assessment to students to complete before leaving class.

Human Table 1

Human Table 2

Human Table 3

      (x

,

          y   )

      (x

,

    y )

      (x

,

    y )

      (x

,

  x + 1)

      (x

,

   2x )

      (x

,

   x² )

Human Table 1

Human Table 2

Human Table 3

      (x

,

          y   )

      (x

,

    y )

      (x

,

    y )

      (x

,

  x + 1)

      (x

,

   2x )

      (x

,

   x² )

0

1

0

0

0

0

1

2

1

2

1

1

2

3

2

4

2

4

3

4

3

6

3

9

4

5

4

8

4

16

5

6

5

10

5

25

6

7

6

12

6

36

7

8

7

14

7

49

8

9

8

16

8

64

Human Table 1

Human Table 2

Human Table 3

      (x

,

          y   )

      (x

,

    y )

      (x

,

    y )

      (x

,

  x + 1)

      (x

,

   2x )

      (x

,

   x² )

Human Table 1

Human Table 2

Human Table 3

      (x

,

          y   )

      (x

,

    y )

      (x

,

    y )

      (x

,

  x + 1)

      (x

,

   2x )

      (x

,

   x² )

0

1

0

0

0

0

1

2

1

2

1

1

2

3

2

4

2

4

3

4

3

6

3

9

4

5

4

8

4

16

5

6

5

10

5

25

6

7

6

12

6

36

7

8

7

14

7

49

8

9

8

16

8

64

This is an example of a linear expression because there is a constant change in the
x value from one point to the next and a constant change in the y value from one point to the next.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Distribute one carnival ticket to each student as he or she enters the room. On an overhead transparency, display the following advertisement:

Carnival music and/or posters could be used to enhance the theme as students enter the room.

“Suppose we want to figure out the cost of attending the carnival and going on different numbers of rides.”

“What information from the advertisement on the transparency would we need to know and use?”

(admission cost and cost per ride)

“What kind of graphic organizer could we use to show the cost of attending the carnival and going on different numbers of rides?” (chart, graph, table)

“T-charts are often used to display two pieces of information.”

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“Could we use a t-chart to show the information that we want to find out?” (Yes.)

Draw a t-chart on a transparency.

“How would we label this t-chart?” (number of rides and cost)

Label the t-chart, and ask students to draw one
t-chart each on a sheet of paper.

“How much will it cost just to attend the carnival?” ($5)

“How would we record that information on our t-chart?” (Number of rides would be 0, and cost would be $5.)

“How much would it cost to attend the carnival and go on one ride?” ($7)

“How would we record that information on our t-chart?” (1, $7)

“Now, with a partner, extend the chart to show the total cost for attending the carnival and going on two, three, four, five, and six rides.”

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Allow time for partners to complete their work, then have students share their results. On the t-chart, record the results that students share.

“What happens to the cost each time one more ride is added?” (It goes up by $2 or costs $2 more.)

“How would you describe in words how to find the cost for attending the carnival and going on rides?” (the number of rides times $2 plus $5)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“How would you represent with symbols or as a symbolic expression how to find the cost for attending the carnival and going on r number of rides?” ($2r + $5)

It may be necessary to guide students through questioning to arrive at the correct symbolic representation. If they provide an incorrect symbolic representation, use questions to make them look carefully at the t-chart and determine if the representation works for all the rides and costs recorded on the t-chart.

“Can we use this expression to find the cost of attending the carnival and going on three rides?” (Yes.) “Six rides?” (Yes.)

Use the expression to demonstrate to students that it does work for finding the cost of both three and six rides.

“Does it work to find the cost for attending the carnival and going on any number of rides on the t-chart?” (Yes.)

If some students are still not convinced that it works, demonstrate with other examples (of numbers of rides) from the
t-chart.

“Now, work with a partner and use the expression to find the cost of 10 rides, 15 rides, and 25 rides.”

Allow time for students to figure out the costs.

10 rides   =  $25

15 rides   =  $35

25 rides   =  $55

“We’ve represented information about the carnival on a t-chart and as expressions. Now, let’s look at representing the information on a coordinate plane.

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

In lesson 3, we discussed how we can use a coordinate grid to graph values for x and y. Let’s draw a new t-chart and relabel number of rides as x and cost as y.”

Draw the following t-chart on the transparency.

“Looking at the relabeled t-chart, what would be our first ordered pair?” (0, 5)

“How many ordered pairs would you have from our t-chart?” (seven pairs—one pair each for rides 0 through 6)

(0, 5) (1, 7) (2, 9) (3, 11) (4, 13) (5, 15) (6, 17)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

Distribute the Quadrant 1 Grid to students.

“Graph the ordered pairs on your grid paper.”

“What title could we give this graph?” (Carnival Costs or Cost of Attending the Carnival, etc.)

“How should we label the y-axis?” (Cost of Admission)

“What numbers should we use as the scale on our y-axis?” (1 through 18 so that we have one value higher than our highest y value of 17)

“How should we label the x-axis?” (Number of Rides)

“What numbers should we use as the scale on our x-axis?” (1 through 7 so that we have one value higher than our highest x value of 6)

Allow time for students to graph the ordered pairs on their grid paper.

“What pattern is represented by the change in the x-value from one point to the next?” (The x-value goes up by 1.)

LEARNING ACTIVITIES

QUESTIONS FOR STUDENTS

TEACHER SUPPORT

“What pattern is represented by the change in the y value each time?” (The
y value goes up by 2.)

“Do the ordered pairs you’ve graphed represent a linear or nonlinear function?”

(linear because there is a constant rate of change in the x and y values.)

Distribute Triangle Trains Assessment for students to complete by the next class period.