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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Time Frame: Approximately five weeks



Unit Description

This unit focuses on the various forms for writing the equation of a line (point-slope, slope-intercept, two-point, and standard form) and how to interpret slope in each of these settings, as well as interpreting the y-intercept as the fixed cost, initial value, or sequence starting-point value. The algorithmic methods for finding slope and the equation of a line are emphasized. This leads to a study of linear data analysis. Linear equalities and inequalities are addressed through coordinate geometry. Linear and absolute value inequalities in one-variable are considered and their solutions graphed as intervals (open and closed) on the line. Linear inequalities in two-variables are also introduced.




Student Understandings
Given information, students can write equations for and graph linear relationships. In addition, they can discuss the nature of slope as a rate of change and the y-intercept as a fixed cost, initial value, or beginning point in a sequence of values that differ by the value of the slope. Students learn the basic approaches to writing the equation of a line (two-point, point-slope, slope-intercept, and standard form). They graph linear inequalities in one variable (and) on the number line and two variables on a coordinate system.

Guiding Questions


  1. Can students write the equation of a linear function given appropriate information to determine slope and intercept?

  2. Can students use the basic methods for writing the equation of a line (two-point, slope-intercept, point-slope, and standard form)?

  3. Can students discuss the meanings of slope and intercepts in the context of an application problem?

  4. Can students relate linear inequalities in one variable to real-world settings?

  5. Can students perform the symbolic manipulations needed to solve linear and absolute value inequalities and graph their solutions on the number line and the coordinate system?



Unit 4 Grade-Level Expectations (GLEs)


GLE #

GLE Text and Benchmarks

Number and Number Relations

4.

Distinguish between an exact and an approximate answer, and recognize errors introduced by the use of approximate numbers with technology (N-3-H) (N-4-H) (N-7-H)

5.

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

Algebra

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)

14.

Graph and interpret linear inequalities in one or two variables and systems of linear inequalities (A-2-H) (A-4-H)

15.

Translate among tabular, graphical, and algebraic representations of functions and real-life situations (A-3-H) (P-1-H) (P-2-H)

Measurement

21.

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

Geometry

23.

Use coordinate methods to solve and interpret problems (e.g., slope as rate of change, intercept as initial value, intersection as common solution, midpoint as equidistant) (G-2-H) (G-3-H)

24.

Graph a line when the slope and a point or when two points are known (G-3-H)

25.

Explain slope as a representation of “rate of change” (G-3-H) (A-1-H)

Data Analysis, Probability, and Discrete Math

29.

Create a scatter plot from a set of data and determine if the relationship is linear or nonlinear (D-1-H) (D-6-H) (D-7-H)

34.

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

Patterns, Relations, and Functions

38.

Identify and describe the characteristics of families of linear functions, with and without technology (P-3-H)

39.

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

Sample Activities




Activity 1: Generating Equations (GLEs: 13, 23, 24, 25)
Materials List: paper, pencil, graph paper, geoboard (optional), colored rubber bands
Remind the students that the slope of a line is the ratio of the change in the vertical distance between two points on a line and the change in horizontal distance between the two points. Use a geoboard or graph paper to model the concept. Ask the students to think of the pegs on the geoboard as points in a coordinate plane and explain that the lower left peg represents the point (1,1). Ask the students to locate the pegs representing the pair (1,1) and the pair (3,5) and place a rubber band around the pegs to model the line segment joining (1,1) and (3,5). Ask them to use a different colored rubber band to show the horizontal from x value to x value of the two endpoints and use another colored rubber band to show the distance from y-value to y-value to the endpoints. Ask the students to find the value of the change in y-values (3) and the change in x-values (2) and show that the defined slope ratio is. Ask students to use this procedure to find the slope of the segment from the point (5,2) and (1,4). Lead the students to discover that, because the line moves downward from left to right, the change in y would produce a negative value and the slope ratio is negative. Show the class that if the computations above are generalized, the formula where is not equal to could determine the slope of the line passing through the two points.
When student understanding of slope is evident, ask them to find the slope between a specific point and a general point (x, y). Guide them to the conclusion that this slope would be . Work with the students to algebraically transform this equation into its equivalent form. Explain that this is the point-slope form for the equation of a line and that it may be used to write the equation of a line when a point on the line and the slope of a line are known. Guide the students through the determination of the line with slope 2 and passing through points with coordinates (3, 4).
Have students use split-page notetaking (view literacy strategy descriptions) as they work through the process of finding the equation of the line when given two points on the line. They should perform the calculations on the left side of the page and write a verbal explanation of each step on the right side of the page. An example of what split-page notetaking might look like in this situation is shown below.


Problem:

Find the equation of the line that passes through the points (4, 7) and (-2, -11). Write your answer in slope-intercept form.







Find the slope of the line.

Formula:

y – 7 = 3(x – 4)

y – 7 = 3x – 12



+ 7 + 7

y = 3x - 5




Find the equation of the line using the slope and one of the original points.

Point-slope formula:
Slope-intercept form: y = mx + b

Simplify equation to slope-intercept form.



Remind students again about how to use their split-page notes to review by covering content in one column and using the other column to recall the covered information. Students can also use their notes to quiz each other in preparation for tests and other class activities.
Ask the students to use a coordinate grid and graph several non-vertical lines. Guide the students to the discovery that all non-vertical lines will intersect the y-axis at some point and inform them that this point is called the y-intercept. Pick out several points along the y-axis and write their coordinates. Through questioning, allow the students to infer that all points on the y-axis have x-coordinates of 0. Then, establish that a general point of the y-intercept of a line could be expressed as (0, b). Ask the students to write and simplify the equation of the line with slope m and passing through the point (0, b). Using the point-slope form for the equation of a line, , have students insert the point (0, b) and solve for y, producing the slope-intercept form for the equation, y = mx + b. Place the students in small groups and have them work collectively to write equations of lines when given the slope and the y-intercept.
Introduce the standard form of a linear equation, Ax +By = C. Have students practice converting linear equations into point-slope, slope-intercept, and standard form. Use an algebra textbook as a reference to provide students with more practice in finding the equation of a line given a point and the slope and also given two points. Have students write their answer in each of the three forms.

Activity 2: Points, Slopes, and Lines (GLE: 24)
Materials List: paper, pencil, graph paper
Provide students with opportunities to plot graphs using either a known slope and a point or two points. When given a slope and a point, help students start at the given point and use the slope to move to a second point. Have students label the second point. Then have them connect these two points to produce a graph of the line with the given slope which passes through the given point. When given two points, ask students to plot them and then connect them with a line. Next, have students determine the slope of the line by counting vertical and horizontal movement from one of the plotted points to the other plotted point. Repeat this activity with various slopes and points. Then give students an equation in slope-intercept form and provide discussion for graphing a line when the equation is in slope-intercept form. Use an algebra textbook as a reference to provide more opportunities for students to practice graphing linear equations.
Have students complete a RAFT writing (view literacy strategy descriptions) assignment using the following information:
Role – Horizontal line

Audience – Vertical line

Format – letter

Topic – Our looks are similar but our slopes are incredibly different


Have students share their writing with the class, and lead a class discussion on the accuracy of their information. A RAFT writing sample is given in Unit 1 Activity 8.

Activity 3: You Sank my Battleship! (GLE: 23, 24, 29, 38)
Materials List: paper, pencil, Battleship BLM, manila file folder per group
In this activity, students will play a modified version of the game Battleship to practice graphing linear equations. Place students in groups of four and have them form teams of two. Provide each team with the Battleship BLM and a manila file folder to shield the other teams view. Have each team draw four battleships on their Battleship BLM. The four ships should have lengths of 5, 4, 3, and 2 units as indicated at the bottom of the Battleship BLM. Teams should take turns coming up with linear equations that the other team will graph and determine if the line goes through any of the battleships. They should then provide the other team with information as to how many hits were made (i.e. if the line passed through any of the ships) or if the line missed all of the ships. When all of the points on a ship are passed through, the ship sinks. The first team to sink all of the other team’s battleships wins.

Activity 4: Applications (GLEs: 4, 5, 11, 13, 21, 23, 24, 25, 38, 39)
Materials List: paper, pencil, tape measures, graph paper, a piece of uncooked spaghetti, Applications BLM, Transparency Graphs BLM, graphing calculator (optional), Data Collection BLM from Activity 5
This activity includes an investigation that will involve applying the concepts learned in Activities 1 and 2. Students will investigate the linear relationship between a person’s foot length and length of the arm from the elbow to fingertip. They will also collect and organize data, determine line of best fit, investigate slope and y-intercept, and use an equation to make predictions.
Initially this is done as an in-class activity. Place students in groups of four and have them use the Applications BLM to record their data collection. Have the students measure their foot length and arm length to the nearest millimeter (a class discussion of measurement techniques and of rounding measurements is appropriate). The foot length should be measured from the heel to the end of the big toe. The arm length should be from the elbow to the tip of the index finger. Have the students agree on a measuring technique so that all measures are somewhat standardized. Have students take measurements and compile their data into the tables where foot length is the independent variable and arm length is the dependent variable. Use the Transparency Graph BLM and have each student graph his/her personal data on the overhead coordinate system. After all points are plotted, discuss what occurs. Ask questions like, “Looking at the graph, do you see any interesting characteristics? Does there appear to be a relationship? What happens to the y-values as the x-values increase?”
Talk about the line of best fit. The piece of spaghetti will be used as a tool to estimate the line of best fit. Allow the students to make suggestions as to where it will be placed on the graph. Once the line is placed, review the ideas of slope of a line, y-intercept, point-slope form of a line, dependent and independent variables, etc. Determine two points that are contained in the line of best fit, find the slope of the line, and use the point-slope formula to write the equation. Have students state the real-life meaning of the slope of the line. Explain that this equation could be used as a means of estimating the length of a person’s arm when the length of his or her foot is known. Have the students take foot and arm measures of an individual not yet measured (often the teacher is a good candidate for these measures). Place the newly found foot length into the equation to estimate foot length and to compare the actual value with the measured value.
Conduct another linear experiment such as timing students in the class as they do the wave where the number of students would be the independent variable and time in seconds would be the dependent variable. Assign a student to be the timer. Have 5 students do the wave and have the student time them. Continue to increase the number of students doing the wave by five until the entire class has participated. Students may use the Data Collection BLM from Activity 5. After the entire class has conducted the experiment and collected the data, put students in small groups and have each group create the scatter plot, derive the linear equation for the data, state the real-life meaning of the slope, and calculate how long it would take 100 students to do the wave. Compare each group’s lines of best fit. Have students identify the characteristics of the different lines that are the same or different. Also have them compare and contrast the linear functions they obtained algebraically in terms of their rates of change and y-intercepts. Many graphing calculators are programmed to use statistical processes to calculate lines of best fit. Students might find it interesting to input class data into the calculator and compare the calculator’s estimate with theirs.
In their math learning logs (view literacy strategy descriptions), have students respond to the following prompt:

Describe some other examples that could be modeled with a scatter plot and a line of best fit. Give reasons for your choice and explain why you believe they could be linear models.

After students have completed their entries, have them share their explanations with the class. Guide a class discussion of each entry and have the class decide if the examples are truly indicative of linear examples.

Activity 5: Linear Experiments (GLEs: 13, 15, 23, 25, 39)

Materials List: paper, pencil, Experiment Descriptions BLM, Data Collection BLM, rubber ball, measuring tape or meter stick, spring, paper cups, pipe cleaner, peppermints, birthday candle, jar lid, matches, rulers, stopwatch, marbles, glass with water, uncooked spaghetti, paper clips


Place students in groups and have them complete a variety of experiments. Copy the Experiment Description BLM, cut the descriptions so they are on separate strips of paper, and give each group a different linear experiment. Provide each student with a copy of the Data Collection BLM. For each experiment, have the groups collect, record, and graph the data using the Data Collection BLM. Have the group discuss the meaning of the y-intercept and slope, identify independent and dependent variables, explain why the relationship is linear, write the equation, and extrapolate values. The sample experiments listed on the BLM include:
Bouncing Ball

Goal: to determine how the height of a ball’s bounce is related to the height from which it is dropped

Materials: rubber ball, measuring tape

Procedure: Drop a ball and measure the height of the first bounce. To minimize experimental error, drop from the same height 3 times, and use the average bounce height as the data value. Repeat using different heights.


Stretched Spring

Goal: to determine the relationship between the distance a spring is stretched and the number of weights used to stretch it

Materials: spring, paper cup, pipe cleaner, weights, measuring tape

Procedure: Suspend a number of weights on a spring and measure the length of the stretch of the spring. A slinky (cut in half) makes a good spring; one end can be stabilized by suspending the spring on a yard stick held between two chair backs. A small paper cup (with a wire or pipe cleaner handle) containing weights, such as peppermints, can be attached to the spring.


Burning Candle

Goal: to determine the relationship between the time a candle burns and the height of the candle.

Materials: birthday candle (secured to a jar lid), matches, ruler, stopwatch

Procedure: Measure the candle; mark the candle in 10 cm or 1/2 in. units. Light the candle while starting the stopwatch. Record time burned and height of candle.


Marbles in Water

Goal: to determine the relationship between the number of marbles in a glass of water and the height of the water.

Materials: glass with water, marbles, ruler or measuring tape

Procedure: Measure the height of water in a glass. Drop one marble at a time into the glass of water, measuring the height of the water after each marble is added.


Marbles and uncooked spaghetti

Goal: to see how many pieces of spaghetti it takes to support a cup of marbles

Materials: paper cup with a hook (paper clip) attached, spaghetti, marbles

Procedure: place the hook on a piece of uncooked spaghetti supported between two chairs, drop in one marble at a time until the spaghetti breaks, repeat with two pieces of spaghetti, and so on. (number of pieces of spaghetti is ind. and number of marbles is dep.)



Activity 6: Processes (GLE: 34)
Materials List: paper, pencil, Processes BLM
Have students follow the steps in a flow chart for putting a linear equation expressed in standard form into slope-intercept form. A sample flow chart that could be used is included as the Processes BLM. Next, have students work in pairs to create a flow chart of steps an “absent classmate” could use to convert a linear equation written in slope-intercept form to standard form. Review the following procedures: questions go in the diamonds; processes go in the rectangles; yes or no answers go on the connectors. Have a class discussion of the finished flow charts, and then have students construct another flow chart individually to convert a linear equation from point-slope form to standard form. Have them exchange charts with another student and follow them to perform the conversion.

Activity 7: Inequalities (GLEs: 11, 14)
Materials List: paper, pencil
Provide students with real-life scenarios that can be described by an inequality in one variable. Have students graph the inequality and interpret the solution set. Make sure students are given inequalities to interpret that include both weak inequalities (i.e., < or >) and strict inequalities (i.e., < or >), as well as absolute value inequalities. An example follows:

When Latoya measured Rory’s height, she got 172 cm but may have made an error of as much as 1 cm. Letting x represent Rory’s actual height in cm, write an inequality indicating the numbers that x lies between. Write the equivalent inequality using absolute value. )
Activity 8: Is it Within the Area? Interpreting Absolute Value Inequalities in One Variable (GLEs: 5, 14)
Materials List: paper, pencil
Review with students the idea of being within a certain distance of a location. For example, ask what it means to be within 25 miles of their home. Have students graph simple absolute value inequalities in one variable on the number line. (Example: )The location point would always be the number that makes the expression inside the absolute value bars zero. For example, if is given, then the “location” is 3 because is zero at . The “area” the inequality encompasses is from –2 to 8. This “area” is found simply by moving 5 units away from the “location” in both directions. Repeat this activity several times. Extend this idea to solving absolute value inequalities like .

Activity 9: Graphing Inequalities in Two Variables (GLE: 14)
Materials List: paper, pencil, chart paper, colored pencils

Introduce the activity by asking students if (5, 3) and (3, 1) are solutions to the inequality . Ask how many other points are solutions? Have students work with a partner and make a large coordinate grid on chart paper. Both axes should extend from –4 to 4. Have students write the value of on each coordinate point (i.e., on the point (3, 2) the student would write (3 – 2) or 1). Have students circle with a colored pencil several values that satisfy the inequality . Question students about points that lie between the points (ex. 2.5, 4.5). Have students shade all the solutions to the inequality. Use the students’ conclusions about this inequality to guide a discussion on graphing all inequalities in two variables.


Sample Assessments



General Assessments

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