Activity 1: Think of a Number (GLEs: 5, 8, 9)
Materials List: paper, pencil, calculator (optional)
Number puzzles are an interesting way to review order of operations, properties of a number, and simple algebraic manipulation. Have students answer the following puzzle:
Think of a number. Add 8 to it. Multiply the result by 2. Subtract 6. Divide by 2. Subtract the number you first thought of. Is your answer five?
Create a table with some numbers from the student results like the table below. Ask
students if they know how the puzzle works. Have students visualize the puzzle by using
symbols for the starting number and individual numbers. Then have students use a
variable for the beginning number and write algebraic expressions for each step to
complete the final column of the table.
Starting number

6

13

10

24

x

Add 8

14

21

18

32

x + 8

Multiply by 2

28

42

36

64

2(x + 8)

Subtract 6

22

36

30

58

2(x + 8)  6

Divide by 2

11

18

15

29

[2(x + 8) – 6]

Subtract starting number

5

5

5

5

5

Have the students develop their own puzzles, using spreadsheets if available. Use a math textbook as a reference to provide other opportunities for students to review and practice order of operations and algebraic manipulations. Include expressions with various forms of rational numbers and integer exponents so that students can work to demonstrate computational fluency.
Activity 2: Order of Operations and Solving Equations (GLE 5, 8, 11)
Materials List: Paper, pencil, calculator, Splitpage Notetaking Example BLM
Have students work in groups to review solving onestep and multistep equations. Discuss with students the reason for isolating the variable in an equation and use the comparison of solving an equation to a “balance scale.” Then provide students with examples of equations that require simplification using algebraic manipulations and order of operations before they can be solved. Have students cover up one side of the equation and completely simplify the other, then repeat with the other side of the equation. Use a math textbook as a reference to provide students with other opportunities to practice solving different types of linear equations including literal equations. Include equations with various forms of rational numbers so that students can work to demonstrate computational fluency.
Have students use splitpage notetaking (view literacy strategy descriptions) to show the steps of solving a multistep equation. One of the purposes of splitpage notetaking is to create a record for later recall and application. When students learn to take effective notes, they develop a greater understanding of key concepts and information. Have students show the steps of solving a multistep equation in the left column. In the right column, students should write the operation that was performed and any note that will help them to later solve a similar equation. A good method of demonstrating the use of this strategy is to show the students an example of a poorly organized set of notes and an example of splitpage notetaking. A blackline master of an example of splitpage notetaking is included.
Activity 3: Using a flow chart to solve equations (GLE 5, 8, 11, 34)
Materials List: paper, pencil, Equation Flowchart BLM
Review with students the steps to constructing a flow chart from Unit 1 Activity 2. Have the students construct a flow chart for solving equations in onevariable. Use the Equation Flowchart BLM as a guide. Help students come up with other ways to make decisions about solving equations. For example, some students may choose not to eliminate fractions as the first step in solving equations. After the flow charts have been constructed, have the students use the charts to solve different equations.
Activity 4: Linear relationships – Keeping it “real” (GLEs: 7, 9, 13, 11, 37, 39)
Materials List: paper, pencil, Linear Relationships BLM, calculator
The Linear Relationships BLM provides students with several inputout put data tables that depict direct variation relationships found in realworld applications. For example, the relationship between the number of gallons of gasoline and the total purchase price or the number of minutes on a cell phone and the total monthly bill both depict a linear function. Have students plot the ordered pairs generated by these data tables on a coordinate graph. See that students recognize that the graph is linear. Revisit direct variation from Unit 1 Activity 7, and discuss with the students that linear data through the origin represents a direct variation. Relate the constant of variation to the rate of change (slope) of the line. Have students write the equation to model the situation. Discuss the reallife meaning of the slope and the yintercept for each table of values. (Although students have not been formally introduced to the terminology of slope and yintercept, these examples should provide for a good discussion on the reallife meaning of slope and yintercept). Have students state the rate of change in reallife terms. For example: For every gallon of gasoline purchased, the total cost increases by _____. Give students values that provide opportunities for them to solve the linear equations algebraically. For example, if John wants to spend exactly $20 on gasoline, how many gallons can he purchase?
Activity 5: Direct Variation – Science Connection (GLEs: 7, 9, 37)
Materials List: paper, pencil, math learning log, calculator, Direct VariationUnit Conversion BLM
Using the Direct VariationUnit Conversion BLM, provide students with the following table of data:
Mountain

Height

Location

1

Mount Everest

8,850m

29,035 ft

Nepal

2

Qogir (K2)

8,611m

28,250 ft

Pakistan

3

Kangchenjunga

8,586m

28,169 ft

Nepal

4

Lhotse

8,501m

27,920 ft

Nepal

5

Makalu I

8,462m

27,765 ft

Nepal

6

Cho Oyu

8,201m

26,906 ft

Nepal

7

Dhaulagiri

8,167m

26,794 ft

Nepal

8

Manaslu I

8,156m

26,758 ft

Nepal

9

Nanga Parbat

8,125m

26,658 ft

Pakistan

10

Annapurna I

8,091m

26,545 ft

Nepal

Have the students graph the heights of the mountains using meters as the independent variable and feet as the dependent variable. Have the students use the graph to determine the rate of change of the line formed by the points. Lead them to discover that the rate of change is the conversion factor for the two units of measure. Have students write the equation of the line. Have students determine if the equation represents a direct variation. Discuss with students that the rate of change is also the constant of variation. Remind students that direct variation relationships will always go through the origin.
In their math learning logs (view literacy strategy descriptions) have students reflect on the following statement:
All unit conversions are direct variation relationships.
Have students write a paragraph explaining why they agree/disagree with the statement, and include examples to justify their position.
Activity 6: Lines and Direct Proportions (GLEs: 9, 11, 37, 39)
Materials List: paper, pencil, Direct Proportion Situations BLM, calculator
Have students identify some relationships that are direct proportions. For example, they could state that distance traveled is directly proportional to the rate of travel, or the cost of movie tickets is directly proportional to the number purchased, or their total earnings are directly proportional to the hours they work.
After some discussion and sharing, divide students into groups and distribute the Direct Proportion Situations BLM. Assign each group of students one of the direct proportion situations. Have students create an inputoutput table, plot the ordered pairs, and draw the line connecting the ordered pairs. Have students write equations to model each direct proportion. Have students determine the constant of proportionality of each relationship, and have each group present their graphs to the entire class. Discuss with the students that the constant of proportionality is the slope (rate of change) for each of the proportions graphed. Have the students state the rate of change in reallife terms. Discuss with students the idea that direct variation and direct proportion are both linear relations passing through the origin. (Other proportional data sets that could be used: The total cost for a bunch of grapes is directly proportional to the number of pounds purchased, the number of miles traveled is directly proportional to the number of kilometers traveled, or if the width of a rectangle is kept constant, then the area of the rectangle is directly proportional to the height.)
Have students participate in a math story chain (view literacy strategy descriptions) in their groups to create a problem for each of the direct proportion situations discussed in class and included on the Direct Proportion Situations BLM. The first student initiates the story and passes the paper to the next student who adds a second line. The next student adds a third line, until the last student solves the problem. All group members should be prepared to revise the story based on the last student’s input as to whether it was clear or not.
Example:
1^{st} student writes: Katherine got paid on Friday from her job at Cheesy Joe’s Pizza.
2^{nd} student writes: Her paycheck was $35 and she wants to use half of it to bring her friends to see Spiderman XIV.
3^{rd} student writes: If movie tickets are $6.50, how many friends can she bring to a movie?
4^{th} student solves the problem. (Since Katherine has to pay for herself, she can only bring one friend with her.)
Activity 7: Solving Proportions (GLEs: 7, 8, 9, 22)
Materials List: paper, pencil, calculator
Students were exposed to proportional reasoning and solving proportions in 7^{th} and 8^{th} grade. In 8^{th} grade, students used proportions to find the missing sides of similar triangles.
Review with students the concept of solving proportions. Have students set up and solve proportions that deal with reallife scenarios. For example, many outboard motors require a 50:1 mixture of gasoline and oil to run properly. Have students set up proportions to find the amount of oil to put into various amounts of gasoline. Recipes also provide examples for the application of proportional reasoning. Finding the missing side lengths of similar figures can allow students to set up a proportion as well as find measures by indirect measurement. For example, students can set up and solve a proportion that finds the height of an object by using similar triangles. Use a math textbook as a reference to provide students with more opportunities to practice solving application problems using proportional reasoning.
Activity 8: Using proportions and direct variation (GLEs: 7, 8)
Materials List: paper, pencil, calculator
Review with students the idea that direct variation and direct proportion are both linear relations passing through the origin and that the constant of variation is also called the constant of proportionality. Present students with the following direct variation problem that can be solved using a proportion: The cost of a soft drink varies directly with the number of ounces bought. It cost 75 cents to buy a 12 oz. bottle. How much does it cost to buy a 16 oz. bottle? Have students set up a proportion to solve the problem (). Provide students with other direct variation problems that can be solved using a proportion.
Activity 9: How tall is the flagpole? (GLEs: 21, 22)
Materials List: meter sticks or tape measures, Stadiascope Template BLM, How Tall is the Flagpole? BLM, paper, pencil, calculator, 8 ½ by 11 card stock, 4 inch squares of clear acetate ( ex: overhead transparency)
In Grades 7 and 8, students studied similar triangles and found the parts of missing triangles using proportions. In Grade 8, students found the height of a structure using similar triangles and shadow lengths. In this activity, students will use a more complex form of proportional reasoning and indirect measurement to find the height of a flagpole, light pole, or any other structure.
Have students build a stadiascope and use it to find the height of the flagpole. A stadiascope is a tool that was used by the ancient Romans to measure the height of very tall objects. Have students work in groups of 3 or 4. Students will need an” x 11” sheet of card stock and a 4inch square of clear acetate, such as an overhead transparency. (A round potato chip can could also be used instead of the card stock.) Have students draw equally spaced parallel lines on the acetate about onehalf centimeter apart or make copies of the Stadiascope Template BLM on transparencies and distribute to students. Roll the card stock sideways (not lengthwise) to make a viewing tube (See diagram below). Then tape the acetate to one side of the tube, being careful that the bottom parallel line is just at the bottom of the tube.
Have students use the What is the Flagpole? BLM to complete this activity. Have students measure the distance they are standing from the flagpole and view the entire flagpole through the stadiascope, carefully lining up the bottom of the flagpole with the bottom of the tube. Have students decide the appropriate units to use when measuring the stadiascope and the distance to the flagpole. They will then use similar triangles and proportions to find the height of the flagpole. (Similar triangles are formed with the length of the bottom of the stadiascope corresponding with the distance the student is from the flagpole and the height of the top of the sighting of the flagpole in the stadiascope corresponding with the height of the flagpole)
Activity 10: Using inequalities to problem solve (GLE: 11)
Materials List: paper, pencil
In 7^{th} and 8^{th} grade students learned to solve inequalities. Review the basics of solving onestep and multistep inequalities. Present students with the following problem for class discussion:
Trashawn wants to order some DVDs from Yomovies.com. DVDs cost $17 per DVD plus $5.50 for shipping and handling. If Trashawn wants to spend at most $75, how many DVDs can he buy? How much money will he have left over? Have students give more examples of vocabulary that may be used in solving inequalities, such as at least, not more than, or not to exceed. Use a math textbook as a reference to provide students with more opportunities to solve application problems using inequalities.
Sample Assessments
General Assessments Performance and other types of assessments can be used to ascertain student achievement. Here are some examples.

Performance Task: The student will find something that can be paid for in two different ways, such as admission to an amusement park or museum (Some museums will charge for each admission or sell a yearround pass, or an amusement park will sell a payoneprice ticket or a perride ticket) and compare the costs. The student will explain the circumstances under which each option is better and justify the answers with a table, graph, and an equation, using inequalities to express their findings.

The student will find the mistake in the solution of the following equation, explain the mistake, and solve the equation correctly:

The student will solve constructed response items such as this:
The amount of blood in a person’s body varies directly with body weight. Someone weighing 160 lbs. has about 5 qts. of blood.
a. Find the constant of variation and write an equation relating quarts of blood
to weight. (, )
b. Graph your equation.
c. Estimate the number of quarts of blood in your body.

The student will use proportions to solve the missing parts of similar figures.

The student will determine if the following situations represent direct variation and explain why or why not:

The amount of a gas in a tank in liters and the amount in gallons (yes)

The temperature in Fahrenheit degrees and in Celsius degrees (no. Although this relationship is linear, the line does not go through the origin.)

The price per pound of carrots and the number of pounds (no)

The total price of tomatoes and the number of pounds (yes)

The student will submit a portfolio containing artifacts such as these:

daily student journals

teacher observation checklists or notes

examples of student products

scored tests and quizzes

student work (inclass or homework)

The student will respond to the following prompts in their math learning logs:

Write a letter to a friend explaining order of operations.

Explain how solving an inequality is similar to solving an equation? In what ways is it different?

Describe a situation from your experience in which one variable is:

Explain why the graph of a direct variation always goes through the origin. Give an example of a graph that shows direct variation and one that does not show direct variation.
ActivitySpecific Assessments

Activity 4: The student will solve constructed response items such as this:
The drama club is selling tickets to their production of Grease for $4 each.

Make a table and a graph showing the amount of money they will make if 0, 5, 10, …, 100 tickets are sold.

Identify the variables and write an equation for the total amount the club will make for each ticket sold. ()

Use your equation to show how much money the club will make if 250 people attend their production. ($1000)

The club spent $500 on their production. How many tickets must they sell to begin to make a profit? Justify your answer. (125 tickets)

Activity 6: The student will choose one of the direct proportion situations and write at least two application problems that can be solved using a linear equation. The student will then write the equation for each application problem and solve it algebraically.

The student will determine the constant of proportionality for a direct proportion by relating it to the slope of the line they obtain from inputoutput data.

Activity 9: The students will write a lab report describing the procedure for finding the height of the flagpole. The student will include diagrams and detailed work for justifying the solution as well as the conclusions in the report.

Activity 10: Given an inequality such as , the student will write an application problem for the inequality.
