# Algebra I table of Contents Unit 1: Understanding Numeric Values, Variability, and Change 1 Unit 2: Writing and Solving Proportions and Linear Equations 14 Unit 3: Linear Functions and Their Graphs, Rates of Change, and Applications 25 Unit 4: Linear

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## Sample Activities

Activity 1: The Numbers (GLEs: 1, 4, 5)
Materials List: Identifying and Classifying Numbers BLM, paper, pencil, scientific calculator
Use a number line to describe the differences and similarities of whole numbers, integers, rational numbers, irrational numbers, and real numbers. Guide students as they develop the correct definition of each of the types of subsets of the real number system. Have the students identify types of numbers selected by the teacher from the number line. Have the students select examples of numbers from the number line that can be classified as particular types. Example questions could include the following: What kind of number is? What kind of number is 3.6666? Identify a number from the number line that is a rational number.
Discuss the difference between exact and approximate numbers. Have the students use Venn diagrams and tree diagrams to display the relationships among the sets of numbers.
Help students understand how approximate values affect the accuracy of answers by having them experiment with calculations involving different approximations of a number. For example, have the students compute the circumference and area of a circle using various approximations for. Use measurements as examples of approximations and show how the precision of tools and accuracy of measurements affect computations of values such as area and volume. Also, use radical numbers that can be written as approximations such as .
Use the Identifying and Classifying Numbers BLM to allow students extra practice with identifying and classifying numbers.

Activity 2: Using a Flow Chart to classify real numbers (GLEs: 1, 34)
Materials List: Flow Chart BLM, What is a Flow Chart? BLM, DR-TA BLM, Sample Flow Chart BLM, paper, pencil
A flow chart is a pictorial representation showing all the steps of a process. Show the students a transparency of the Flow Chart BLM. Have them list some of the characteristics that they notice about the flow chart or anything that they may already know about flow charts. Record students’ ideas on the board or chart paper.
Use the “What is a flowchart?” BLM as a directed reading-thinking activity (DR-TA) (view literacy strategy descriptions) to have students read and learn about flow charts. DR-TA is an instructional approach that invites students to make predictions and then check their predictions during and after the reading.
Give the students a copy of the What is a flowchart? BLM and the DR-TA BLM. Have students fill in the title of the article. Ask questions that invite students’ predictions. For example a teacher may ask, “What do you expect to learn after reading this article?” or “How do you think flow charts might be used in algebra class?” Have students record the prediction questions on the DR-TA BLM and then answer the questions in the Before Reading box on the BLM.
Have students read the first and second paragraphs of the article, stopping to check and revise their predictions on the BLM. Discuss with students whether or not their predictions have changed and why. Continue with this process stopping two more times during the reading of the article. Once the reading is completed, use student predictions as a discussion tool to promote further student understanding of flow charts.
Emphasize that in most flow charts, questions go in diamonds, processes go in rectangles, and yes or no answers go on the connectors. Guide students to create a flow chart to classify real numbers as rational, irrational, integer, whole and/or natural. Have students come up with the questions that they must ask themselves when they are classifying a real number and what the answers to those questions tell them about the number. A Sample Flow Chart BLM is included for student or teacher use. Many word processing programs have the capability to construct a flow chart. If technology is available, allow students to construct the flow chart using the computer. After the class has constructed the flow chart, give students different real numbers and have the students use the flow chart to classify the numbers. (Flow charts will be revisited in later units to ensure mastery of GLE 34.)

Activity 3: Operations on rational numbers (GLE 5)
Materials List: paper, pencil, scientific calculator
Have students review basic operations (adding, subtracting, multiplying, and dividing) with whole numbers, fractions, decimals, and integers. Include application problems of all types so that students must apply their prior knowledge in order to solve the problems. Discuss with students when it is appropriate to use estimation, mental math, paper and pencil, or technology. Divide students into groups and give examples of problems in which each method is more appropriate; then have students decide which method to use. Have the different groups compare their answers and discuss their choices.
Have students participate in a math story chain (view literacy strategy descriptions) activity to create word problems using basic operations on rational numbers. The process for creating a math story chain involves a small group of students writing a story problem and then solving the problem. Put students in groups of four. The first student initiates the story. The next student adds a second line, and the next student adds a third line. The last student is expected to solve 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. Students can be creative and use information and characters from their everyday interests.
A sample story chain might be:
Student 1:

A scuba diver dives down 150 feet below sea level and a shark swims above the diver at 137 feet below sea level.

Student 2:

The diver dives down 125 more feet.

Student 3:

How far apart are the shark and the diver?

Student 4:

138 feet
Have the groups share their story problems with the rest of the class, and have the class solve the problems.

Activity 4: Comparing Radicals (GLE 6)
Materials List: Investigating Radicals BLM, paper, pencil
This activity is a discovery activity that students will use to observe the relationship between a non-simplified and simplified radical. Have students work with a partner for this activity using the Investigating Radicals BLM. Have them draw a right triangle with legs 1 unit long and use the Pythagorean theorem to show that the hypotenuse is units long. Then have them repeat with a triangle that has legs that are 2 units long, so they can see that the hypotenuse is or units long. Have them continue with triangles that have legs of 3 and 4 units long. For each hypotenuse, have them write the length two different ways and notice any patterns that they see. This activity leads to a discussion of simplifying radicals. Give students examples of other equivalent radicals, some that are simplified and some that are not simplified. Guide students to discover the relationship between the equivalent radicals and the process for simplifying a radical. After students have observed the modeling of simplifying additional radicals, provide them with an opportunity for more practice..

Activity 5: Basic Operations on Radicals (GLEs: 6, 8)
Materials List: paper, pencil
Review the distributive property with students and its relationship to combining like terms. (i.e.) Provide students with variable expressions to simplify. Give the following radical expression to students: . Guide students to the conclusion that the distributive property can also be used on radical expressions, thus . Provide radical expressions for students to simplify. (Note: Basic operations on radicals in Algebra I are limited to simplifying, adding, subtracting and multiplying.)

Activity 6: Scientific Notation (GLEs: 2, 3)
Materials List: Scientific Notation BLM, paper, pencil, calculator
Have students use a calculator and the Scientific Notation BLM to make a chart with powers of 10 from –5 to 5. Discuss the patterns that are observed and the significance of negative exponents. Provide students with real-life situations for which scientific notation may be necessary, such as the distance from the planets to the sun or the mass of a carbon atom. Have students investigate scientific notation using a calculator. Allow students to convert numbers from scientific notation to standard notation and vice versa. Relate the importance of scientific notation in the areas of physical science and chemistry.

Activity 7: Independent vs. Dependent Variable (GLE: 10)
Materials List: paper, pencil, Independent and Dependent Variables BLM
Discuss the concept of independent and dependent variables in reference to real-world examples. For example:

• The area of a square depends upon its side length

• The distance a person travels in a car depends upon the car’s speed and the length of time it travels

• The cost of renting a canoe at a rental shop depends on the number of hours it is rented

• The number of degrees in a polygon depends on the number of sides the polygon has

• The circumference of a circle depends upon the length of its diameter

• The price of oil depends upon supply and demand

• The fuel efficiency of a car depends upon the speed traveled

• The temperature of a particular planet depends on its distance away from the sun

Present students with ten different pairs of variables used in real-world contexts and have the students work in groups to determine which of the variables is the dependent variable and which is the independent variable. Discuss each situation as a class.

Explain that a two-dimensional graph results from the plotting of one variable against another. For instance, a researcher might plot the concentration in a person’s bloodstream of a particular drug in comparison with the time the drug has been in the body. One of these variables is the dependent and the other the independent variable. The independent variable in this instance is the time after the drug is taken, while the dependent variable is the thing that is measured in the experiment—the drug concentration. Explain to students that conventionally the independent variable is plotted on the horizontal axis (also known as the abscissa or x-axis) and the dependent variable on the vertical axis (the ordinate or y-axis). Relate this all pictorially with graphs.
The Independent and Dependent Variables BLM is provided for student practice of identifying independent and dependent variables.

Activity 8: Variation (GLEs: 7, 9, 10, 15, 28, 29)
Materials List: paper, pencil, meter sticks, algebra tiles, Foot Length and Shoe Size BLM, Dimension of a Rectangle BLM, calculator
Part 1: Direct variation

Have the students collect from classmates real data that might represent a relationship between two measures (i.e., foot length in centimeters and shoe size for boys and girls) and make charts for boys and girls separately. Discuss independent and dependent variables and have students decide which is the independent and which is the dependent variable in the activity. Instruct the students to write ordered pairs, graph them, and look for relationships from the graphed data. Is there a pattern in the data? (Yes, as the foot length increases, so does the shoe size. Does the data appear to be linear? Data should appear to be linear.) Help students notice the positive correlation between foot length and shoe size. Have students find the average ratio of foot length to shoe size. This is the constant of variation. Have students write an equation that models the situation (shoe size = ratio x foot length). Following the experiment, discuss direct variation and have the students come up with other examples of direct variation in real life.

Part 2: Inverse variation

Have students work with a partner. Provide each pair with 36 algebra unit tiles. Have students arrange the tiles in a rectangle and record the height and width. Discuss independent and dependent variables. Does it matter in this situation which variable is independent and dependent? (No, but the class should probably decide together which to use.) Have students form as many different sized rectangles as possible and record the dimensions. Instruct the students to write ordered pairs, graph them, and look for relationships in the graphed data. Help students understand that the constant of variation in this experiment is a constant product. Have them write an equation to model the situation (height (or dependent) = 36/width (or independent))

Provide students with other data sets that will give them examples of direct variation, inverse variation, and constant of variation. Ask students to write equations that can be used to find one variable in a relationship when given a second variable from the relationship.
Have the students complete a RAFT (view literacy strategy descriptions) writing assignment . This form of writing gives students the freedom to project themselves into unique roles and look at content from unique perspectives.

RAFT is an acronym that stands for Role, Audience, Format, Topic:.
To connect with this acitivity the parts are defined as

Role – Direct variation

Audience – Inverse Variation

Format – letter or song

Topic – Why I am linear and you are not.
Help students to understand that they are going to take the Role of a direct variation and write to (speak to) an Audience that is an inverse variation. The Format of the writing may be either a letter or a song with the Topic entitled, “Why I am linear and you are not!” Once RAFT writing is completed, have students share with a partner, in small groups, or with the whole class. Students should listen for accurate information and sound logic in the RAFTs.
A sample RAFT might look like this:
Dear Izzy the inverse variation,

I understand that there may be some confusion about my linear characteristics that seem to be annoying you. “What makes me linear,” you ask? Well, I will tell you.

In my relationships, as one value increases, the other will increase also at a constant rate. For example, if you buy one candy bar at the store, you will pay 75 cents. If you buy two candy bars, you will pay \$1.50. The amount that you pay increases at a constant rate.

In your relationships, my friend, the two values will have a constant product. So as one value increases, the other will decrease, but not at a constant rate. For example, suppose I am driving to New Orleans which is 55 miles away. If I drive 55 miles per hour, I will arrive in New Orleans in one hour. But if I drive 65 miles per hour, I will arrive in approximately .846 hours or 51 minutes. The distance stays constant, but the relationship between the speed and the time is an inverse variation.

I hope this clears things up for you.

Dennis the direct variation

Activity 9: Exponential Growth (GLEs: 2, 9, 10, 15, 29)
Materials List: paper, pencil, 1 sheet of computer or copy paper, Exponential Growth and Decay BLM
Give each student a sheet of ” by 11” paper. Have the students complete the Exponential Growth and Decay BLM similar to the one shown below as they work through this activity. Instruct students to fold the paper in half several times, but after each fold, they should stop and fill in a row of the table.

 Number of Folds Number of Regions Area of Smallest Region 0 1 1 1 2 or 2 4 or 3 8 or . . . . . . . . . N or

Have the students complete a graph of the number of folds and the number of regions. Have them identify the independent and dependent variables. Is the graph linear? This is called an exponential growth pattern. Have the students also graph the number of folds and the area of the smallest region. This is called an exponential decay pattern. Include the significance of integer exponents as exponential decay is discussed.

Activity 10: Pay Day! (GLEs: 9, 10, 15, 29)
Materials List: math learning log, Pay Day! BLM, paper, pencil
Have students use the Pay Day! BLM to complete this activity.
A math learning log (view literacy strategy descriptions) is a form of learning log. This is a notebook that students keep in math classrooms in order to record ideas, questions, reactions, and new understandings. This process offers a reflection of understanding that can lead to further study and alternative learning paths.
In their math learning logs have students respond to the following prompt:
Which of the following jobs would you choose?

• Job A: Salary of \$1 for the first year, \$2 for the second year, \$4 for the third year, continuing for 25 years

• Job B: Salary of \$1 million a year for 25 years

Have the students compare the two options and give reasons for their answer.

After the students are done, have a discussion about their responses.
At the end of 25 years, which job would produce the largest amount in total salary?
Have the students use the chart on the BLM to explore the answer. They should organize their thinking using tables and graphs. Have the students represent the yearly salary for both job options using algebraic expressions. Have them predict when the salaries would be equal. Return to this problem later in the year and have the students use technology to answer that question. Discuss whether the salaries represent linear or exponential growth.

#### Activity 11: Linear or Non-linear? (GLEs: 10, 15, 29)

Materials List: paper, pencil, poster board or chart paper, markers, Linear or Non-linear BLM, Sample Data BLM, Rubric BLM

Divide students into groups. Give each group a different set of the sample data from the Sample Data BLM. Have each group identify the independent and dependent variables of the data and graph on a poster board. Let each group investigate its data and decide if it is linear or non-linear and present its findings to the class, displaying each poster in the front of the class. After all posters are displayed, conduct a whole-class discussion on the findings. As an extension, regression equations of the data could be put on cards, and the class could try to match the data to the equation. The Linear or Non-Linear BLM has a sample list of directions. The Linear or Non-Linear Rubric BLM can be used with this activity. The data sets on TVs, Old Faithful, Whales, and Physical Fitness are linear relationships.

Activity 12: Using Technology (GLEs: 10, 15, 29)
Materials List: paper, pencil, graphing calculator, Calculator Directions BLM
Have students enter data sets used in Activity 11 into lists in a graphing calculator and generate the scatter plots using the calculator. The Calculator Directions BLM has the directions for entering data into the graphing calculator.

Activity 13: Understanding Data (GLEs: 5, 10, 28, 29)
Materials List: paper, pencil, Understanding Data BLM

Have students complete the Understanding Data BLM with a partner.

After students have completed the activity, lead a class discussion to ensure student understanding of GLE 28. Students should be able to identify trends in data and support conclusions by using distribution characteristics such as patterns, clusters, and outliers.

Sample Assessments

General Assessments

• The students will explore patterns in the perimeters and areas of figures such as the “trains” described below.

Train 1
Train number 1 2 3 4 5 … n 1 2 3 4 5

Area 1 4 9 16 25

Perimeter 4 8 12 16 20

Describe the shape of each train. (square)

What is the length of a side of each square? (n)

Compare the lengths of the trains with their areas and perimeters. (length-n, area-, perimeter-4n)

Train 2
Train Number 1 2 3 4 5 … n 1 2 3 4 5

Area 1 3 6 10 15

Perimeter 4 8 12 16 20

Formulas: area - , perimeter – 4n

• The students will solve constructed response items, such as these:

1. Cary’s Candy Store sells giant lollipops for \$1.00 each. This price is no longer high enough to create a profit, so Cary decides to raise the price. He doesn’t want to shock his customers by raising the price too suddenly or too dramatically. So, he considers these three plans,

• Plan 1: Raise the price by \$0.05 each week until the price reaches \$1.80

• Plan 2: Raise the price by 5% each week until the price reaches \$1.80

• Plan 3: Raise the price by the same amount each week for 8 weeks, so that in the eighth week the price reaches \$1.80.

1. Make a table for each plan. How many weeks will it take the price to reach \$1.80 under each plan? (Plan 1 – 16 weeks, Plan 2 – 12 weeks, Plan 3 – 8 weeks)

2. On the same set of axes, graph the data for each plan.

3. Are any of the graphs linear? Explain.

4. Which plan do you think Cary should implement? Give reasons for your choice. (Answers will vary.)

2. The table below gives the price that A Plus Car Rentals charges to rent a car including an extra charge for each mile that is driven.

Car Rental prices
 Miles Price 0 \$35 1 \$35.10 2 \$35.20 3 \$35.30 4 \$35.40 5 \$35.50

1. Identify the independent and dependent variables. Explain your choice.

2. Graph the data

3. Write an equation that models the price of the rental car. ()

4. How much would it cost to drive the car 60 miles? Justify your answer. (\$41)

5. If a person only has \$40 to spend, how far can he/she drive the car? Justify your answer. (50 miles)

• The students will complete writings in their math logs using such topics as these:

• Describe the steps used in writing .000062 in scientific notation

• How can you tell if two sets of data vary directly?

• Explain the error in the following work:

• Explain how one might use a flow chart to help with a process.

• Is it true that a person can do many calculations faster using mental math than using a calculator? Give reasons to support your answer.

• The student will complete assessment items that require reflection, writing and explaining why.

• The student will create a portfolio containing samples of their activities.

Activity-Specific Assessments

• Activity 1: Given a set of numbers, A, (similar to the set on problem 15 of the Identifying and Classifying Numbers BLM) the student will list the subsets of A containing all elements of A that are also elements of the following sets:

• natural numbers

• whole numbers

• integers

• rational numbers

• irrational numbers

• real numbers

• Activity 2: The students will use the Internet to find other examples of flow charts. The student will print a flow chart and write a paragraph explaining what process the flow chart is showing and how the different boxes indicate the steps of the process. If Internet access is not available to students, the teacher will provide the student with different examples of flow charts to choose from and write about.

• Activity 7: The students will complete a writing assignment explaining how to tell if an equation is that of an inverse variation or that of a direct variation.

• Activities 8 and 9: The student will graph the following sets of data and write a report comparing the two, including in the report an analysis of the type of data (linear or non-linear).

 Males in the U.S. Year Annual wages 1970 9521 1973 12088 1976 14732 1979 18711 1985 26365 1987 28313
 Average income

 Professional baseball players Year Annual wages 1970 12000 1973 15000 1976 19000 1979 21000 1985 60000 1991 100000

(Linear) (Non- Linear)

• Activity 12: Provide the student with (or assign the student to find) similar statistics from the school basketball team, a favorite college team, or another professional basketball team. The student will study the data and develop questions that could be answered using the data. The student will submit the data set, questions, and graphs that must be used to complete the assignment.

Algebra I

Unit 2: Writing and Solving Proportions and Linear Equations

#### Time Frame: Approximately three weeks

Unit Description
This unit includes an introduction to linear equations and inequalities and the symbolic transformation rules that lead to their solutions. Topics such as rate of change related to linear data patterns, writing expressions for such patterns, forming equations, and solving them are also included. The relationship between direct variation, direct proportions and linear equations is studied as well as the graphs and equations related to proportional growth patterns.

Student Understandings
Students recognize linear growth patterns and write the related linear expressions and equations for specific contexts. They see that linear relationships have graphs that are lines on the coordinate plane when graphed. They also link the relationships in linear equations to direct proportions and their constant differences numerically, graphically, and symbolically. Students can solve and justify the solution graphically and symbolically for single- and multi-step linear equations.

Guiding Questions

1. Can students graph data from input-output tables on a coordinate graph?

2. Can students recognize linear relationships in graphs of input-output relationships?

3. Can students graph the points related to a direct proportion relationship on a coordinate graph?

4. Can students relate the constant of proportionality to the growth rate of the points on its graph?

5. Can students perform simple algebraic manipulations of collecting like terms and simplifying expressions?

6. Can students perform the algebraic manipulations on the symbols involved in a linear equation or inequality to find its solution and relate its meaning graphically?

GLE #

GLE Text and Benchmarks

#### Number and Number Relations

5.

Demonstrate computational fluency with all rational numbers (e.g., estimation, mental math, technology, paper/pencil) (N-5-H)

Algebra

7.

Use proportional reasoning to model and solve real-life problems involving direct and inverse variation (N-6-H)

8.

Use order of operations to simplify or rewrite variable expressions (A-1-H) (A-2-H)

9.

Model real-life situations using linear expressions, equations, and inequalities (A-1-H) (D-2-H) (P-5-H)

11.

Use equivalent forms of equations and inequalities to solve real-life problems (A-1-H)

13.

Translate between the characteristics defining a line (i.e., slope, intercepts, points) and both its equation and graph (A-2-H) (G-3-H)

Measurement

21.

Determine appropriate units and scales to use when solving measurement problems (M-2-H) (M-3-H) (M-1-H)

22.

Solve problems using indirect measurement (M-4-H)

Data Analysis, Probability, and Discrete Math

34

Follow and interpret processes expressed in flow charts (D-8-H)

Patterns, Relations, and Functions

37.

Analyze real-life relationships that can be modeled by linear functions (P-1-H) (P-5-H)

39.

Compare and contrast linear functions algebraically in terms of their rates of change and intercepts (P-4-H)