2012-13 and 2013-14 Transitional Comprehensive Curriculum
Grade 8
Mathematics
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)
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 Text
8.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 (ü) 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 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
This activity was not changed because it already incorporates the CCSS.
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 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
This activity was not changed because it already incorporates the CCSS.
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.
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.