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2012-13 and 2013-14 Transitional Comprehensive Curriculum

Grade 8Mathematics

Unit 4: Expressions and Equations in Algebra

Time Frame: Approximately five weeks

Unit Description

The unit focus is on determining relationships of patterns. Representations of these relationships are made using tables, graphs and equations. Equation solutions and descriptions of how rates of change in one variable affect the rate of change in the other variable are also explored as graphs are analyzed and slopes are discussed.

Student Understandings

Students show a strong command of working with whole number exponents in evaluating expressions. Students are able to understand the connections between proportional relationships, lines and linear equations. Students are able to analyze and solve linear equations and pairs of simultaneous linear equations. They can discuss rates of change, such as found in the graphs of linear relationships. Students develop an intuitive grasp of slope and will be able to compare and contrast slope in linear settings. They are capable of shifting among representations and discussing the nature of such representations for functions as tables, graphs, equations, and in verbal and written formats.

Guiding Questions

1. Can students apply whole number exponents in evaluating expressions? 2. Can students apply the order of operations in evaluating expressions involving

fractions, decimals, integers, and real numbers along with parentheses and exponents?3. Can students shift among written, verbal, numerical, symbolic, and graphical

representations of functions?4. Can students solve and graph solutions of multi-step linear equations?5. Can students explain and form generalizations about how rates of change work in

linear settings?6. Can students construct a table of values for a given equation and graph it on the

coordinate plane?

Unit 4 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS)

Grade-Level ExpectationsGLE # GLE Text and Benchmarks

7. Use proportional reasoning to model and solve real-life problems (N-8-M)

Grade 8 Mathematics Unit 4 Equations and Expressions in Algebra 4-1


9. Find unit/cost rates and apply them in real-life problems (N-8-M) (N-5-M) (A-5-M)

12. Solve and graph solutions of multi-step linear equations and inequalities (A-2-M)

13. Switch between functions represented as tables, equations, graphs, and verbal representations, with and without technology. (A-3-M) (P-2-M) (A-4-M)

14. Construct a table of x- and y-values satisfying a linear equation and construct a graph of the line on the coordinate plane. (A-3-M) (P-2-M) (A-4-M)

15. Describe and compare situations with constant or varying rates of change. (A-4-M)

CCSS # CCSS Text8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the

graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.Reading Standards for Literacy in Science and Technical Subjects 6-12

3. Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks.

4. Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 6 – 8 texts and topics.

Sample Activities

Activity 1: The Better Buy? (GLEs: 7, 9, 15; CCSS: 8.EE.5)

Materials List: Vocabulary Self-Awareness BLM, The Better Buy BLM, Choose the Better Buy? BLM, pencils, paper, math learning log, grocery ads (optional)

This activity was not changed because it already incorporates the CCSS.

Begin the activity by distributing the Vocabulary Self-Awareness Chart BLM. Students will begin with the vocabulary self-awareness strategy, (view literacy strategy descriptions). The words have been written in the chart. Students should rate their understanding of each of the vocabulary words by placing a plus sign (+) if they are very comfortable with the word, a check mark () if they are uncertain of the exact meaning and a minus sign if the word is completely new (-) to them. Have the students try to write definitions and give examples for the vocabulary. Explain that they may have to make guesses if they have no understanding of the word. Tell the students that each day they should take the chart out and update any definitions that have been developed through the day’s lesson. Through this unit, students should develop an understanding


http://www.louisianaschools.net/lde/uploads/11056.doc


of all of the vocabulary in the chart. The repeated use of the vocabulary self-awareness chart will give the students multiple opportunities to practice and extend their growing understandings of the vocabulary.

Place the transparency of The Better Buy? BLM on the overhead. Cover the bottom portion that gives group directions. Using a modified SQPL, (view literacy strategy descriptions), have students independently write questions about the statement, “One potato chip costs $0.15” that they may need to determine if the statement is true. After about one minute, have the students get into pairs, compare questions and select at least two of their questions to share with the class. Write questions on the board or a sheet of paper and post them.

Discuss proportional relationships by telling the students that an ad in today’s paper showed that a 12-pack of soda was on sale for $3.99. Set up a proportion showing the 12 sodas compared to the 3.99 and ask the students to think about how an equivalent ratio showing the cost of only 1

soda, such as , could be written. Ask students to share methods of finding

the cost of one soda using a proportion. Tell them that finding the cost of one unit is called finding the unit rate. Have students graph the costs of soda and answer various questions such as How much do 6 sodas cost? How many sodas could you buy for $10?

Next, provide students with Choose the Better Buy? BLM. Have students work individually to find the unit rates to determine the better buy in each situation. Students should verify results with a partner. Give opportunities for questions if students have answers on which they do not agree.

Extend the activity by providing grocery ads from different stores which carry the same items to each group of four students. Give students a list of items to purchase and have student groups of four make projections about savings on groceries by shopping at store A versus store B over a year. Have students present their findings to another group or the class.

Distribute a sheet of grid paper and have the students select two items on their ad sheet to create an ‘xy’ table and graph the proportional relationships when buying one, two, and three items of each kind using the unit rate. Students will create a graph showing the relationship illustrated in the table of values. Tell the students to use two different colored pencils to draw the graphs that represent each unit rate. Ask students how the cost changes with each additional item purchased. Have students indicate which of the two items graphed increased cost at a higher rate. Ask students how the graph shows this relationship.




Next, sketch a graph that illustrates two linear equations of cars traveling at different average speed (as in diagram at the right). Ask students if their conclusion about the graphs of unit rate increasing at a higher rate is also true of the distance-time equations shown on this graph. Ask students to predict where a line showing the relationship of y = ½ x would be located on the graph shown at the right. It should be below the lower sloped line which is closer to y = x.

Look back at the initial list of questions or wonderings about one potato chip costs $.15 and ask students if their questions could be answered at this point. Have students answer each of the questions that were posed at the beginning of class.

Challenge the students by having them determine the number of chips in a bag if the bag costs $1.90.

Have the students take out their Vocabulary Self-awareness Chart and update any vocabulary that might have become clearer through the lesson today. Encourage the students to update this chart daily.

Have students record in their math learning log (view literacy strategy descriptions) what they understand about unit prices. Invite random students to share their understandings with the class.

Activity 2: Refreshing Dance (GLE: 7, 9, 15; CCSS: 8.EE.5)

Materials List: Refreshing Dance BLM, pencils, paper


Have students work in groups of four to prepare a cost-per-student estimate for refreshments at an 8th grade party. Distribute Refreshing Dance BLM. Have students complete the chart and determine the total cost of refreshments for each student and the total cost of the dance if they plan for 200 students.

Students will present their proposals and the answers to the questions to the class using a modified professor know-it-all (view literacy strategy descriptions) strategy. Using professor know-it-all, call on groups of students randomly to come to the front of the room and provide “expert” answers to questions from their peers about their proposal. Remind the students to


time (hours)

distance (miles)

Traveling a given distance over time




listen to the questions and to think carefully about the answers received so that they can challenge the “experts” if answers need elaboration and/or amending. Students should be able to justify not only the cost of the refreshments but also the amount that needs to be ordered. This strategy is a good review strategy, and the groups of students develop understanding of the content as they prepare to lead the discussion.

Once the students have completed their professor know-it-all discussion, have them graph the candy bar unit price and the popcorn unit price for at least 3 different values and write a statement as to which of the items shows the greatest rate of change from the graph. Have students turn in these graphs as an exit ticket for today’s class.

Activity 3: My Future Salary (GLE: 15; CCSS: 8.EE.5)

Materials List: grid paper for students, My Future Salary BLM, paper, pencil, Internet access


Introduce SQPL (view literacy strategy descriptions) by posting the statement, “An electrical engineer earns more money in one year than a person making minimum wage earns working for 5 years.” Have students work in pairs to generate questions that they think they need to have answered to be able to determine whether the statement is true or false. Have students share questions with the class and make a class list of questions. Students must make sure that a question relating to a comparison of job salaries is asked. Give students time to research the information needed to answer the question. A site that has recent top salaries can be found at http://www.payscale.com/best-colleges/degrees.asp.

Give students time to research salaries of various jobs. Students should then create proportions to calculate salaries when working a minimum wage job for 40 hours per week. Once the proportional relationship for the 40 hr/wk has been solved, have students work with a partner or group of four to determine method of justifying whether the SQPL statement is “true” or “false.”

Have the student pairs create a graph to illustrate the average salary for two of the careers that they are interested in and make a statement on the graph that provides evidence of their understanding of which of the jobs will help them earn $500,000 in the shortest amount of time if the salary remains constant. Give students time to complete their evidence and then have various groups share their information. Give class members time to ask questions of the students to further the understanding of class and presenters.

Go through each of the questions asked at the beginning of the lesson. Have students give answers that they found or explain discoveries to them. They should either prove or disprove the SQPL statement from the beginning of class. The students can share their information with the class by using professor know-it-all (view literacy strategy descriptions). The student group, randomly selected, will go to the head of the class and report their findings to the class and answer questions from the group about their findings. Give other groups time to share their findings, also.



http://www.payscale.com/best-colleges/degrees.asp



Ask the students why the minimum hourly wage is considered a unit rate (amount of money paid per hour of work). Distribute the My Future Salary BLM and have students make observations about what has happened to the minimum wage in the years since 1960. Lead a discussion with students about how the minimum wage has changed through the years. Have students create a graph of the minimum wage from the information in the chart and predict the minimum wage for the year 2020. Have the students calculate what a person working a minimum wage job working 40 hours per week made in 2011 (make sure the students understand that if the year is not given, the minimum wage is the same as it was in the previous year listed) and what that person would make using their prediction for the year 2020. Discuss how the graph helps with making predictions.

The information on the My Future Salary BLM is also found on the following website: http://www.dol.gov/whd/minwage/chart.htm. The website http://www.salarylist.com/ gives current wages of jobs listed with the labor division of the U. S. government. The Louisiana Board of Regents has an e-portal designed specifically for Louisiana students: https://www.laeportal.com/main.aspx. This portal was designed to be used by eighth grade students as they make a five year academic plan. There is a teacher section which provides links to careers, salaries and other information that would be applicable to this activity.

Activity 4: Proportional Relationships (GLE: 7; CCSS: 8EE.5 )

Materials List: Proportional Relationships BLM, pencils, paper

Begin the lesson by having students analyze the following table of values by predicting what information the table of values could possibly represent. Students should be able to give a reasonable situation that could be represented by the information.

Ask the students how many meters would be traveled in “0” seconds. This ordered pair (0, 0), makes sense because if time does not pass, there is no distance gained. This point is the “origin” and all proportional relationships will pass through the origin. Ask the students to plot the points represented in the table on a coordinate grid, where time is plotted on the horizontal axis as “seconds” and the distance on the vertical axis as “meters.” Make sure that students know that the ordered pairs must be (t, d) to meet this

requirement. The equation represents the

equation for the line, indicating that the y-


t seconds 4 8 12 16 20d meters 5 10 15 20 25

After 1 second, the distance is 5/4 meters

run = 4

rise = 5

run = 8

rise = 10

16

16

14

14

12

1210

10

8

6

4

2

0 8642

Distance(meters)

Time (seconds)

https://www.laeportal.com/main.aspx

http://www.dol.gov/whd/minwage/chart.htm


coordinate can be found by multiplying times the x-coordinate. It is important for the students

to understand that 5 meters are traveled in 4 seconds.

From point (0,0) another point on the line can be found by counting up 5 (rise) and 4 to the right (run) or (4, 5). The rate of change does not vary in the graph at the left and when a constant rate of change is graphed, the graph is linear. A rate of change that is constant is also the slope of the line. All three representations, the table, the graph, and ordered pairs (t, d), show a proportional

relationship. The ordered pairs are also shown as (d is the rise and t the run)

because the line goes through the origin. The proportions represented by the coordinates of

points on the line will be indicating that the data in the table represents a

proportional relationship. Remember that is called the rate of change or constant of

proportionality and the general linear equation y = mx represents a proportional relationship where y represents the rate of change multiplied by the length of time. The rate of change represents the slope of the line.

Put the following chart and graph on the board:

The values along the ‘y’ axis of the graph and have been left off of the graph. This is to help students determine how the rate of change and the slope are related. Have students determine what these numbers should be using the information in the table to ensure the graph represents the data in the table. Ask students how they can determine the “intervals” used along the y axis. The rate of change is 70 miles per hour and after one hour the distance traveled is 70 miles. The graph increases by going up 70 miles each hour. The distance that is traveled is always 70 times the number of hours traveled.


t (hours)

1 2 3 4 5

70t (miles)

70 140 210 280 350

876543210

Distance(miles)

time (hours)


Distribute Proportional Relationship BLM and have students work in small groups to complete the proportional relationships and describe how the slope of the graph of each relationship represents the unit rate. Discuss results with the whole class or have students work together in groups of four and discuss these relationships. Have each group share one point with the class that helped them understand the relationship of the slope of the line shown on the graph and the unit rate.

2013-14Activity 5: Slope and Similar Triangles (CCSS: 8.EE.6)

Materials: Similar Triangles and Slope BLM, pencil, paper

Draw similar triangles on the board similar to the ones at the right.

Review with the students how to determine what information is sufficient to know that triangles are similar. Make sure the students justify similarity using the length of side ratio proportionality and the angle equality as the factors necessary to justify their similarity.

Ask the students to discuss with their shoulder partner the following: whether two triangles will always be similar if two angles of the triangle are congruent. If the proportionality of two triangles is one, and two angles are congruent, are these triangle similar? Yes. What other attribute do these two triangles share? Congruency

Tell them that today’s activity asks that they work as mathematicians to form a conjecture about why the slope (rate of change or constant of proportionality) between any two distinct points along a non-vertical line is equal to the slope between any other two distinct points along the same line. The use of similar triangles will be used to help justify the conjecture.

Distribute the Similar Triangles and Slope BLM. Have pairs of students work through these questions to develop a conjecture. Once the students have had time to complete the BLM, have student pairs get with one other student pair and compare conjectures. Discuss conjectures as a class after students have had time to work in groups of four to compare findings.

Students’ conjectures should be something like this: The slope of a non-vertical line is the same no matter which two points on the line are used to find the slope.

2013-2014Activity 6: Developing the Slope-y-intercept Formula (CCSS: 8.EE.6)

Materials List: Rate of Change Grid 1 & 2 (used with Activity 12), Developing Slope Formula BLM, paper, pencil


15 cm9 cm

3 cm 5 cm 30 30


Instruct students to use the Rate of Change BLMs 1 and 2 from Activity 12 to complete the first part of this lesson. Have students go to their tables of values for their linear equations and compare the changes in x and y. Ask questions that will lead the students to discover the move it takes to get from one point on to the next point. Students might move up or down first, and then right or left (up 2 right 1). Repeat this with .

Distribute Developing Slope Formula BLM and have the students use the points indicated on the graph to determine the rise/run or the rate of change for the line. Have students make a conjecture explaining the proportional relationship between any two sets of points along the line that crosses the origin. Students might set the proportions equal as (y2 – y1)/(x2 – x1) and each will justify proportionality between the ordered pairs. Have students use the ordered pair (0, 0) and any other point along the line and determine the slope of that segment of the line, then have students use two other ordered pairs along the line and determine the slope of this segment of the

equation. Suggest that all students think about the points as for this activity.

Have students write the ordered pair that represents each of these proportions. Remind students that the coordinates of any two points along a linear graph which passes through the origin have a proportional relationship, resulting in a constant rate of change. Set up the following

proportion on the board .

Ask the students to explain how this relates to graph 1, and then have the students solve this

equation for y. Lead students to understand that = m and y = mx are equivalent

equations. = m also indicates that the ratio of the coordinates of any point on a line that passes

through the origin when written as is the slope of that line. Thus, the equation for the line in

graph 1 would be y=2x.

Tell the students that the equation y = mx is the format for linear equations where the line crosses through the origin of the graph. The m represents the slope of the line or the rate of change. Direct the students to look at graph 2 on the Developing Slope Formula BLM. Instruct the students to work through #2 and determine if the conjecture formed for #1 is valid for use with this linear graph, too. After students have time to work through this question, discuss the results. It is important for the students to see that the addition of the y-intercept keeps the points from showing a proportional relationship. The slopes between any two points is the same, but the

ratios of any two points, , are not proportional. It is important to stress that can be used to

determine the slope of a line only when the line passes through the origin.

Have students work with a shoulder partner to discuss this and how the two lines can be parallel when the ordered pairs of the second line are not proportional. For example:



Graph 1: y=2x . ordered pair (1, 2), (2,4), (3, 6) =

Graph 2: y=2x+ 4. ordered pair (1, 6), (2, 8), (3, 10) Calculate the slope using the coordinates

of the y-intercept (0, 4) and other points on the line. or and .

The slope of the second line in Graph 2 is 2. The slope of line in Graph 1 is 2. However, in

Graph 2, = for (1, 6) and = for (2, 8). Because the ratios of the coordinates of points

on the line are not proportional, the line does not pass through the origin. Therefore, the equation y= mx for a proportional relationship is adjusted to become y = mx + b, where b is the y-intercept. The equation y = mx + b will work for any equation, even the equations when the equation passes through the origin, because if b=0, the equation becomes y=mx.

Ask the students to write the equation for the first graph and beneath it write what is known about the equation for the second graph.Graph 1: y=2xGraph 2: y = 2x+ 4 (Students might see this as y – 4 = 2x; if this happens, have them solve for y by adding 4 to each side of the equation.)

Explain to the students that this form of writing an equation is called the slope-intercept form of an equation. Ask students to explain how the name of the equation describes the line.

Distribute More Exploration with the Slope-intercept Form of an Equation BLM. Have students work through these problems with a partner and discuss as a class their results.

Activity 7: Camping Sounds: (GLE: 12)

Materials List: Grid BLM, Camping Sounds! BLM, Grid for Questions 5 and 6 BLM, pencils, paper

Have the students work in pairs for this activity. Put the table of values on the board or overhead. Have students write in their math learning log (view literacy strategy descriptions) a short paragraph describing a situation that the table represents. Ask students to predict the number of animal sounds they would hear if they camped out ten nights. Have students explain their predictions and encourage them to make some rule for the data in the chart.

Ask the students how the x and y values change. Make sure the students understand that the rate of change is constant because the difference in y values is 3 and the difference in x values is 1; therefore, the rate of change is 3 over 1 or 3 sounds per night. Because the rate of change is 3, y has to equal 3x but 3(1) does not equal 4; therefore, a 1 must be added to the 3x for the equation to be correct. Students should try this with at least 2 of the values in the table to ensure that the y-intercept is correct. The equation will be y =3x + 1. Encourage students to try other values;


Number of nights Number of animal sounds heard

1 42 73 104 13





building 3building 2building 1


discuss the strategy of trying various ordered pairs to determine which of the ordered pairs will fall on the line.

Distribute the Grid BLM and have students plot these four ordered pairs on a coordinate grid and explain the relationship that is shown.

Using your rule, determine the number of animal sounds heard on the twentieth night.

Distribute Camping Sounds! BLM and Grid for Questions 5 & 6 BLM. Give students time to solve each of the problems. Discuss the results.

Activity 8: Beaming Buildings! (GLEs: 12,; CCSS: 8.EE.7)

Materials List: Patterns and Graphing BLM, More Practice with Patterns BLM, Grid BLM, Patterns and Graphing Practice BLM, pencils, paper, toothpicks (15-20 per student pair)

This activity has not changed because it already incorporates this CCSS.

Distribute 15 – 20 toothpicks to each pair of students. Have the students place the toothpicks in the arrangement shown below for buildings one through three.

Have them create a table of values showing the building number represented by the x-value and the number of support beams it takes to build the building as the y-value for this pattern of buildings through building #6.

For example, building one takes 7 beams, building 2 takes 12 beams, etc.

Example: The 3 red beams and 2 blue beams in figure 1 are the beams repeated with each addition.

Encourage students to match the x value as seen in the diagram above (the loops show the five beams added each time a new building is developed). Ask students to use their table values to predict the number of beams it will take for building #10.


x = building # y = # beams1 72 123 174 225 276 327 378 429 4710 52

Building 1 Building 2 Building 3


At this point, the students might notice that the number of beams a) increase by 5 and, b) the ones digit alternates between 7 and 2. How often students have used tables to develop “rules” for patterns may influence what they see in the pattern. Use leading questions if necessary to help them develop these skills. Encourage students to relate the figure number to the input of an input/output table and the total number of beams as the output. Questions such as: If I need to know how many beams will be in the 15th building, how can I determine this? (The input is 15, so encourage the students to determine the function(s) that will give the output if figure 10 has 52 beams and figure 2 has12 beams. What rule can work for both situations?) Once students determine that 5(2) + 2 = 12 and 5(10) + 2 = 52, they should be able to determine that the 15th building will require 5(15) + 2 = 77. Ask if there is a relationship to which buildings will have an odd number of beams and have a “7” in the ones place and those that will require an even number of beams and have a “2” in the ones place? (Odd numbered buildings end in 7, and even numbered buildings end in 2 because 5 is added each time and an odd number (5) added to an even number will always yield an odd number. An odd number added to an odd number will always yield an even number.) Challenge the students by asking such questions as: Which building will require 352 beams? (building number 70)

Engage the class in a discussion about these predictions and what they based their predictions upon. Ask the students to work with a partner and use RAFT writing (view literacy strategy descriptions) to determine which building would take 62 beams and have the students explain their thinking. Have them begin making the connection between the data in the chart to the linear equation. RAFT writing is used once students have gained new content so that they have opportunities to rework, apply and extend their knowledge. The R is used to describe the role of the writer; the A refers to audience or to whom the RAFT is being written; the F is used to give students a form to follow in their writing; and the T is the subject matter or topic of the writing. In today’s assignment,

R = a beamA = to the construction worker ordering the number of beams neededF = paragraphT = how to determine the number of beams needed to build the structure

Pairs of students should share their writing with other pairs of students. Students should listen for accuracy and logic in each others’ RAFTs.

Next, instruct students to plot the ordered pairs of building numbers and number of beams on the Grid BLM to determine if the relationship is linear. Challenge pairs of students to create a “what-if” question for another pair of students and be able to justify their answers. Give students time to share questions with other pairs of students.

Distribute Patterns and Graphing BLM and have students work independently to complete the questions about the patterns on this activity sheet. Once students have had time to complete the activity sheet, have them work with a partner to discuss their answers. Provide time for students to ask questions of other students if needed. If students still need more practice, distribute Patterns and Graphing Practice BLM. This BLM might also be used for homework practice.

Activity 9: From Table to Graph to Conjecture (GLE:14, 15)





Materials List: Circles and Patterns BLM, Grid BLM, pencils, paper

Have students create a table of values for the area of a circle using Circles and Patterns BLM. Have students complete the table for circles with radii of 1 through 5 units. Have students plot the ordered pairs on the Grid BLM and compare the formula for finding the area of a circle with the graph of the points from the table of points. Have students make observations about the shape of the graph (linear or non linear). Lead students to make a conjecture about the effects of doubling or tripling the radius on the area of the circle by examining the table of values and the graph. Have students share their conjectures and justify their reasoning as to why they think their conjectures are true using the professor know-it-all strategy (view literacy strategy descriptions). Select students at random to share their thinking and justify their reasoning as to why their conjectures are true. Discuss the shape of the graph and whether the relationship is linear.

Activity 10: Speed, Time and Distance (GLE: 13, 14; CCSS: 8.EE.5)

Materials List: one stop watch per group, tape measures or meter sticks, paper, pencils, Grid BLM, colored pencils


Instruct students to work in groups of four. One student in each group should have a stop watch or second hand on a watch to be used as a timer. If possible, borrow stop watches from the science or P. E. department. Have students mark off a distance of 10 meters and take turns walking the distance and gathering data about the time it takes each student in the group to walk the distance.

Students should then each take their time from the 10 meter walk and work independently to create a table of values representing the time it takes them to walk distances of 15, 20, 25, and 30 meters if the rate stays the same as 10 meters.

Have students determine the equation that represents their speed (unit rate). Next, have students plot the coordinates from their tables on a coordinate grid with one other group member so that the values for 10, 15, 20, 25 and 30 meters are used for each of the two students. Each student’s graph should be done with a different color pencil for comparison. Challenge groups of students to develop a conjecture as to the relationship of time and distance shown on the graph. This is a good time to have the students think about the independent and dependent variables and where these are placed on the graph. Discuss graphs from the different students’ data, and discuss whether the graphs are linear and why they are or are not. Students should calculate their speed and relate this to the rate of change on the graph. Lead a discussion to help the students begin to see the connection between their speed, the coefficient of x and the slope of the line. Have students describe in their math learning logs (view literacy strategy descriptions)






which of the two people represented by the graphs walked faster and how the graph shows this relationship.

Give students an opportunity to share their understanding with another student who is not part of their original group. Use a timer. One student will be the speaker and one the listener. The listener cannot speak at this time. The first speaker gets 1 minute to explain his/her graph to his/her listener. The listener will then become the speaker. The new speaker will be given 30 seconds to restate what his/her partner said about the graph. Then the speaker will be given 1 minute to explain to the new listener the relationship of speeds shown on his/her graph. Lastly, the first speaker should be given 30 seconds to repeat what his partner shared.

Activity 11: Linear Equations—Fuel Consumption (GLE: 11, 14, CCSS: 8.EE.5)

Materials List: paper, pencil, newsprint, markers, Internet, spreadsheet, graphing calculator (optional)


Have students research the fuel (natural gas, electricity, gasoline) consumption for various types of furnaces, refrigerators, water heaters, and automobiles. Using the fuel consumption data for automobiles, have students use a spreadsheet to construct a table of x- and y-values where x represents the gallons of gasoline and y represents miles driven. This website gives many models of cars and their fuel consumption ratings: http:// www.fueleconomy.gov/ . Make sure to have some fuel consumption ratings to use in class if the computer lab is not available for research. Once the table of values is complete, have students plot the (x, y) coordinates from the table onto the coordinate plane. If graphing calculators are available, have the students set up a table of values and plot the points to determine if the points are linear. If not, have pairs of students create the graph on paper. Have students prove that the rate of change between any two points on the line are equal, therefore, showing a constant rate of change, making it linear.

Next, have students write an equation for the fuel consumption. For example, if a refrigerator uses 180 kwh per month, then an equation that depicts this situation is where x is the number of months and y is the total kilowatts used. Lead a discussion about the equations developed for the data so that students understand the applicability of their use of algebra. Have students brainstorm other situations with constant or varying rates of change (i.e., altitude and barometric pressure, number of rotations made with the pencil sharpener and the length of the pencil). After generating a list of these situations, have the students use a graphic organizer (view literacy strategy descriptions) to organize their thinking. A Venn diagram would be a method to use for classifying these situations representing constant or varying rates of change. To use the Venn diagram, have students sketch two large overlapping circles on newsprint or other large sheet of paper. Have students write above one of the circles constant rates of change, and above the other varying rates of change. Give students time to classify these situations. Some of their situations might be classified into both categories, such as driving or riding in a car (in town it would probably be varying and on the highway it might be constant). Give time for groups to plan their presentations to justify their diagrams using professor know-it-all (view




http://www.autosite.com/content/research/index.cfm?Action=search&id=22041%3BASITE&Search=fuel+economy&Go=Go


literacy strategy descriptions). Using this strategy, do not take volunteer groups but randomly select groups of students to justify their graphic organizer classification. The groups will explain how their Venn diagram represents the relationships described in the situations. The group presenting information will answer questions from the class, clarifying any questions that might arise.

Activity 12: Rate of Change (GLE: 14, 15; CCSS: 8.EE.5)

Materials List: paper, pencil, Rate of Change Grid 1 and 2 BLM (see Activity 6), colored pencils, graphing calculators (optional)

Provide students with Rate of Change Grid 1 or graphing calculators. Have them create a table of at least five values, including and then two opposite x values, and plot coordinates for on a coordinate grid for the three equations at the right.

Have them plot and connect the points using different colors for the lines on their graphs. Pair students and give them time to create conjectures about the relationships of these equations and share conjectures with the class.

Distribute Rate of Change Grid 2 BLM, and have students complete the tables and graphs of these equations as they did with grid 1.

Have students discuss the observations that can be made by comparing the equations and the graphs. Be sure that students’ observations include the following:

Differences in the rates of change for y = ½ x, y =2x and y =3x The graph for y = x2 is not linear. The graphs in grid 2 do not cross at the origin; they all have a y-intercept. There is one negative rate of change. Be sure the students understand that y = (-x) +2

shows a negative rate of change. Negative slopes can be identified in the equations by the negative rate of change

(coefficient of the x value). These are the equations with a y-intercept other than “0.”

Make sure the students begin to understand that as the absolute value of the rate of change (the coefficient of x) increases, the steepness of the line increases. The students also need to see that the y-intercept can also be seen in the second set of equations, and this places the intersection of the graph and the y-axis at different points on the y-axis. The first equations are arranged so that as the students graph them in order, the steepness of the graph increases. The second set looks at the points of intersection on the y-axis and the negative slope.

Use the professor know- it- all strategy (view literacy strategy descriptions) as students are required to describe their conjectures to the class. Pairs of students should develop situations that represent at least two of the equations graphed. Have students discuss the rate of change and whether this rate of change is constant. During the professor know-it-all discussion, ask the


y = ½ x y = 2x y =3x y= x2




students questions such as the following: Which of the equations appear to have a linear relationship? How can you tell? Does one of the linear relationships look as if it changes at a faster rate than another? Discussion should evolve to the slope or slant of is steeper than

. Compare the graphs of the equations y = -x + 2 and y = x - 2.After the BLM is completed, challenge the students by asking them to decide whether an equation with the variable squared would ever be a linear equation? Students will provide different examples to justify that a squared variable does not increase at the same rate, thus the rate of change is not constant. Students need the graphs from these activities for Activity 13.

Activity 13: From Equation to Situation and Situation to Equation: (CCSS: 8.EE.5)

Materials: Rate of Change Grid 2, pencil, paper

Instruct students to work with a shoulder partner and take their Rate of Change Grid 2. Ask the students to work with the shoulder partner and write at least three observations about the relationship of the equations and the graphs that were completed in Activity 12. It is important that the students compare the graphs y = x + 2 and y = x – 2 to derive the idea that rates of change are the same and the lines are parallel; the only difference is the y-intercept. The same conclusion can be drawn from the equations y = -x +2 and y = -x - 2. Ask questions of the students to ensure the understanding of the y-intercept and same slope is recognized from the equations.

If no students bring up the observation that y = x – 2 and y = -x – 2, and also, y = x + 2 and y = -x + 2 look to be perpendicular, ask the students to use their graphs to form a conjecture that explains that the intersection of all of the graphs in the center of the page form a square. The students can relate the distance from the origin to the vertices of the shape as being equal. This information can lead to the deduction that the shape is square. The idea that the angles are 90 might be derived by using the triangle formed with vertices at (-2, 0) (0, 2) and (0, 0). Observe side lengths to derive that the triangle is a right triangle.

Have the students work with their shoulder partner to test their conjecture with other equations.

Put the equation y = 2x + 5 on the board. As a class, develop a word problem that would represent the equation. Start the story with a possible statement that Joe wants to buy a new MP3 player. Ask different students to add one sentence at a time to develop a word problem to represent the situation. A possible situation might be, “Joe has saved $5 and each week he saves $2 so that he can buy a new MP3 player. How much money will Joe have saved after 10 weeks?”

Use the text chains (view literacy strategy descriptions) strategy to make connections of equations to real-life situations. Write the equation y = 4x + 3 on the board. Explain to the students that to create a text chain, they will work in groups of 4. Each person in the group will begin a text chain by writing one sentence of a situation that can be modeled with this equation. Tell them that you are going to model the writing of a text chain from the equation, then write on the board the example:




1) Jerry is older than his sister. The equation shows the relationship between their current ages in years. (Ask the students to speculate as to which part of the equation represents Jerry’s age and which represents his sister’s age).

Each person in the group writes the first sentence of a situation to represent the equation. Each group member should begin his/her own situation. Then each paper is passed to the person on the right, and the students think of a second sentence to add to the situation started by the 1st person – perhaps for the example starter,

2) “Jerry is not 4 times as old as his sister.”

Discuss how this relates to the equation – if he were 4 times as old as his sister, the equation would be y = 4x. Again, pass the paper to the right and everyone writes a third sentence that relates to the situation for the equation:

3) Jerry is currently 3 years more than 4 times as old as his sister. Again, to the right, and everyone writes a sentence 4) Jerry is 11 if his sister is 2; Jerry is 15 if his sister is 3.

The paper goes back to the original author, and the author reads the situation making sure that the situation contributed to by each of the group members is consistent and accurate. Assign groups of four students to one of the four equations to describe using a real-life situation that could be modeled with the equation using the text chain. Allow students from each group to share their situations with another group or the class, depending upon time.

Sample Assessments

Performance assessments can be used to ascertain student achievement. For example:

General Assessments The student will prepare a brochure comparing mileage of different cars. The student

will include graphs of at least three cars and their mileage and explain the relationship of the mileage and the slope of the line. A website that the students can use to find different mileage comparisons is http://www.fueleconomy.gov/feg/FEG2004_GasolineVehicles.pdf

The student will prepare a presentation using number sequences or pattern sequences and describe when the sequence results in a linear relationship and how they determine this.

The teacher will provide students with a list of situations that can be represented with an algebraic expression. The student will write the expression that represents the situation.

The teacher will provide the student with a list of expressions involving variables with whole number exponents up to three. The student will evaluate the expressions using a given set of values for the variables.


http://www.fueleconomy.gov/feg/FEG2004_GasolineVehicles.pdf


The teacher will provide the student with a table of values that describe a linear situation (i.e., a constant rate of change) and the student will determine the rate of change.

Whenever possible, the teacher will create extensions to an activity by increasing the difficulty or by asking “what if” questions.

The student will create portfolios containing samples of experiments and activities.

Activity-Specific Assessments

Activity 4 : Give students a list of rates, have the students graph the proportional relationship and describe the unit rate based on the graph.

Activity 7 : Give the students an equation such as y = 2x + 3 and have them draw a tile pattern to represent the equation, graph the equation and find the slope.

Activity 12 : Give students some situations and have them determine if the situations represent a constant or varying rate of change.

Activity 13: Have students write a description for the hexagon pattern and compute the perimeter if each side of the hexagon measures 3 units.


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