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Unit 2 Linear Equations

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Unit 2 Linear Equations Powered By Docstoc
					Unit 2: Linear Equations and Inequalities                              5 weeks
Unit Overview
Essential Questions:
What is an equation?
What does equality mean?
What is an inequality?
How can we use linear equations and linear inequalities to solve real world problems?
What is a solution set for a linear equation or linear inequality?
How can models and technology aid in the solving of linear equations and linear inequalities?
Enduring Understandings:
To obtain a solution to an equation, no matter how complex, always involves the process of undoing the
operations.
UNIT CONTENTS
Note: The bolded Investigations are model investigations for this unit.
Investigation 1:       Undoing Operations (1 day)
Investigation 2:       Unit Pre-Test and Review (3 days)
Investigation 3:       Two-Step Linear Equations in Context (5 days)
Investigation 4:       Two-Step Linear Inequalities (2 days)
Mid-Unit Test (1 day)
Investigation 5:       Multi-Step Linear Equations and Linear Inequalities (8 days)
Performance Task: iPods (3 days)
End-of- Unit Test (2 days including review)
Appendices:            Investigation 2: Unit Pre-Test, Activity 2.1a Student Worksheet, Activity 2.1b Student
                       Worksheet, Activity 2.1c Student Worksheet Mid-Unit Test, Unit 2 Performance Task,
                       End-of-Unit Test.
                                          Course Level Expectations
                   What students are expected to know and be able to do as a result of the unit
1.2.1 Develop and apply linear equations and inequalities that model real-world situations.
1.3.1 Simplify and solve equations and inequalities.
2.1.1 Compare, locate, label and order integers, rational numbers and real numbers on number lines, scales
        and graphs.
2.2.1 Use algebraic properties, including associative, commutative and distributive, inverse and order of
        operations to simplify computations with real numbers and simplify expressions.
2.2.2 Use technological tools such as spreadsheets, probes, algebra systems and graphing utilities to organize,
        analyze and evaluate large amounts of numerical information.
2.2.3 Choose from among a variety of strategies to estimate and find values of formulas, functions and roots.
2.2.4 Judge the reasonableness of estimations, computations, and predictions.
3.3.1 Select and use appropriate units, scales, degree of precision to measure length, angle, area, and volume
        of geometric models.
                                                  Vocabulary
algebraic expression                   distributive property                literal equations
associative property                   evaluate                             order of operations
coefficient                            integers                             properties of equality
constant                               inverse operations                   real numbers
cummutative property                   linear inequalities                  simplify
                                                                            variable
                                             Assessment Strategies
                Performance Task(s)                                         Other Evidence
         Authentic application in new context                     Formative and Summative assessments
CT Algebra One for All                                                         Page 1 of 23
Unit Plan 2, 8 13 09
iPODS                                                      Warm-ups, class activities, exit slips, and homework
Students will work on a two-day task that has them         have been incorporated throughout the investigations.
investigating file storage size and cost for various       In addition, the students will take a Unit Pre-Test,
models of iPods™. Students will share their findings       Mid-Unit Test, and Formal End-of-Unit Test.
with the class.
                               INVESTIGATION 1 – Undoing Operations (1 day)
 Students will discover the underlying algebra and see why number puzzles always work. In the process, they
 will represent real world situations with algebraic expressions, evaluate an algebraic expression for a given
 value of a variable, apply real number properties to simplify algebraic expressions, and describe reasonable
 values that the variable and/or expression may represent.

Suggested Activities
1.1 You may use a number puzzle such as the one below to launch the lesson. Have the students perform the
    following steps on a calculator or by hand:
    Step 1: Enter the number of the month in which you were born
    Step 2: Multiply by 5.
    Step 3: Add 6
    Step 4: Multiply the sum by 4.
    Step 5: Add 9
    Step 6: Multiply the sum by 5.
    Step 7: Add the number of the day you were born.
    Step 8: Subtract 165.

    Ask the students, “What do you notice? Is your observation true for everyone in class?” The result should
    be the month and day of their birthday. Ask the students, “Why did this happen? Can you prove that the
    month and day of your birthday will always be the result when performing these steps? How?” Consider
    having the students work in pairs or small groups to try to figure out how to justify this result. Have the
    students share their work with the class. Emphasize the idea of “undoing” the steps. This, along with the
    concept of a variable, will be an extremely vital concept throughout this unit.

    An alternative puzzle to use is a simpler version for students who would benefit from starting with a less
    complex problem. The solution to this puzzle will always be 5.
    Step 1: Pick any number
    Step 2: Take the number you picked and double it
    Step 3: Add 10
    Step 4: Divide the new number in half
    Step 5: Subtract your original number

1.2 You may choose to have the students create their own number puzzles and try them out on each other. Or,
    you may choose to use the website http://thinkofanumber.net/ which has a nine-step number puzzle which,
    no matter what number you start with, will always result in the number 350. This number is the “red line”
    for humans, the carbon footprint that proponents think is necessary to sustain the earth. A video about
    carbon footprints can be seen at: http://www.350.org/ .

Assessment
By the end of this investigation students should be able to:
 represent real world situations with algebraic expressions;
 evaluate an algebraic expression for a given value of a variable;
 apply real number properties to simplify algebraic expressions; and
 describe reasonable values that the variable and/or expression may represent.
CT Algebra One for All                                                          Page 2 of 23
Unit Plan 2, 8 13 09
                           INVESTIGATION 2 - Unit Pre-Test and Review (3 days)
Students will take a pre-test on the prerequisite skills required for success in this unit. You may use the results
to plan any needed review or adjust the amount of instructional time needed as students work on the activities in
this unit.

Suggested Activities
2.1 Administer the Unit Pre-Test, particularly if you have noticed that some students had difficulty writing the
    expressions that described the explicit rules in Unit 1.

You may find from the results of this diagnostic assessment that some students may require more review of
prerequisite skills and concepts than other students. These skills include work with integers, order of operations,
evaluating expressions, simplifying expressions, the commutative, associative and distributive properties,
solving one-step equations, using properties of equality, solving simple literal equations and solving word
problems that involve one-step equations.

You may use some of the following ideas to keep those students who do not require an extensive review
challenged and engaged in relevant mathematics while you plan and implement a review of the concepts
identified as weaknesses with the remainder of the class. Note: If you recognize a weakness in solving two-step
linear equations, then do not review this topic here. The next investigation in this unit focuses on solving two-
step linear equations.

You may choose to have students who are ready for enrichment work individually, in pairs, or in small groups
to complete one or more activities. One option is Activity 2.1a Student Handout - New Cell Phone Plan.
After students complete the worksheet they may research (either via the web or the newspaper) various cell
phone plans. They might then use the data to compare plans and select the best buy.

Another option is to have students work individually, in pairs, or in small groups and complete either the
standard Activity 2.1b Student Handout - Recycling Activity or the differentiated Activity 2.1c Student
Handout - Recycling Activity. The differentiated version includes some more challenging questions.

Assessment
By the end of this investigation students should be able to:
    perform integer operations;
    combine like terms;
    evaluate expressions;
    use the distributive property;
    solve one-step linear equations;
    solve one-step linear inequalities; and
    solve one-step linear equations and inequalities in context.
                   INVESTIGATION 3 – Two-Step Linear Equations in Context (5 days)
Students will be able to write a linear equation that models a real world scenario, solve two-step linear
equations, and justify their steps.
See Model Investigation 3.
                         INVESTIGATION 4 – Two-Step Linear Inequalities (2 days)
Students will be able to write a linear inequality that models a real world scenario, solve two-step linear
inequalities and justify their steps.

Suggested Activities
4.1 Students may have worked with inequalities and solved linear inequalities in one step. However, when
CT Algebra One for All                                                            Page 3 of 23
Unit Plan 2, 8 13 09
    dividing or multiplying both sides of an inequality by a negative, some students do not understand why you
    have to change the direction of the inequality. If students have not developed a understanding of this prior
    to this course, then the teacher may choose how to teach this concept. One suggestion is to use the TI-84
    program LINEQUA (This program can be found at
    http://education.ti.com/educationportal/sites/US/homePage/index.html and typing 8773 into the search box.
    Or, you can download ideas about how to use this program in your classroom.). Another suggestion is to use
    the more traditional approach of testing values for the variable in the original inequality and the simplified
    inequality (if they are equivalent, then the solution set should be the same).

4.2 To begin work with inequalities you might start with a problem of interest and practical use to students. For
    example, students may go to a specific website such as, www.prepsportswear.com and look up the cost of a
    popular item that might be a good item for a fundraiser or sports booster sale. They might pick out the shirt
    they think everyone should wear at the annual pep rally. Have students find the cost of one of the shirts.
    Suppose the student council has set aside $6,000 to purchase the shirts. (They plan to sell them later at
    double the price.) Then ask them to determine how many shirts they can buy at the price they found online
    if the shipping costs are $14. At this point, you might want to remind students that schools are tax exempt.
    Students might want to set up an equation at this point. You can have a discussion with the students about
    how you don’t have to spend all $6,000 dollars but you definitely can’t spend more than that. If the T-shirts
    are $21.96 each, then the inequality would be $21.96x + 14.00  $6000, where x is the number of T-shirts.
    Be sure that students define the variable. You can then have students explain how they would solve for x
    and have them show their steps and check their answer. In this case, x is about 272.6. Facilitate a discussion
    with the class regarding this answer and why the solution shows that they can buy up to 272 T-shirts. Have
    them discuss why rounding down in this case is necessary versus rounding to the closer value of 273 shirts.

    For differentiation, you can solve the equation $21.96x +14.00 = $6,000 and then try to solve the inequality
    again using the equation as a model. Students should also use a number line to get a visual of the solution
    set.

    For more practice, solve similar problems after finding the costs for uniform shirts for the football team,
    baseball hats, or something else of interest to students.

    An alternative problem would be to have students visit a cellular phone company website and find a plan
    that they think fits the needs of their family based on the number of minutes they use a cell phone per
    month. Then have them pick out the texting option that they think would fit the number of texts they send
    per month. Lastly, determine if they need to add Internet as an option also. With this data, you can have
    students set up an inequality that will help them to determine the number of months they can afford the cell
    phone. For example, a cell phone family plan has 1,400 minutes for $89.99/month for the first two lines and
    every line after that is $9.99, unlimited texting messages for $30.00/month, and Internet for $10.00/month
    per line. Suppose the family has allotted $200.00 per month for cell phone lines for the family. Have
    students write and solve an inequality to determine the number of cell phone lines that the family can have
    with a $200.00 allotment. Remind them that they must define the variable. Possible solution: 200  89.99
    + 9.99x + 30 +10(x + 2), where x is the number of additional lines to the plan. In this case, 3 additional
    lines, for a total of 5 lines, can be on the plan.

    As an extension, students can also go back and calculate how much it would cost each individual member of
    the family to have an equivalent individual cell phone plan. Students can then determine how much a
    family would possibly save by getting a family plan versus an individual plan for each family member.

    For differentiation, you can have more advanced students also write and solve the inequality to include tax.
    For students who are struggling with this concept, you can remove the fact that the $89.99 covers the first
CT Algebra One for All                                                           Page 4 of 23
Unit Plan 2, 8 13 09
    two people. Instead make it so that $90.00 is the base price and every member costs $10.00/month. For
    students who are still struggling, it may be helpful to use play money to act out the algebraic steps.

    Also, for students who are really having a difficult time, you may want to find a couple of cellular plans
    ahead of time and ask them simpler questions about the plans. This way, they will be dealing with the same
    context, but won’t have to contend with some of the trickier constraints.

Assessment
By the end of this investigation students should be able to:
    Write and solve two-step linear inequalities in context;
    justify why you flip the inequality sign when multiplying or dividing by a negative number; and
    justify the steps in solving linear inequalities.

                                              Mid-Unit Test (2 days)
           INVESTIGATION 5 – Multi-Step Linear Equations and Linear Inequalities (8 days)
Students will be able to write a linear equation that models a real world scenario, solve multi-step linear
equations and linear inequalities, and justify their steps.
See Model Investigation 5.
                                     Unit 2 PERFORMANCE TASK – iPods
iPods provides students an opportunity to apply what they have learned in Unit 2. They will investigate how
linear equations and inequalities can be used to help them make decisions. (See Unit 2 Performance Task-
Sample Student Handout.) Students may work in pairs or small groups during this performance task. Each
group will share their findings and recommendations with the class.

Suggested Activities
In a whole class discussion, let students know that they will assume the role of a writer for a local newspaper
who is investigating information about iPods. The reporter wants to determine whether or not the data is
accurate. His goal is to make recommendations to the marketers and consumers of this technology. Students
may work in pairs or small groups over the next two class periods to complete a variety of contextual problems
and share findings with the class. You may use a third day if students need more time to discuss ideas and write
their group’s responses. After all groups have reported, you may have students assess the response of their
group and suggest ways that they may improve their responses.

As extensions to the activity sheet, students may act like reporters and keep a list of additional questions they
have as they do their “research” on iPods. Then, challenge them to find the solutions.

Students may write a newspaper column highlighting their findings. This would appeal to students who enjoy
creative writing. It might serve as a way to get them excited about mathematics as a vehicle for narrative or
persuasive writing.

Yet another extension is to have the students create a newspaper advertisement for the 16GB iPod Nano or the
1TB iPod to accompany the newspaper column. Students should include the relevant solutions they found as
they create the advertisement. Going even further, the advertisement shared with the class as a PowerPoint
presentation or an audio or video commercial. The presentations should include references, sources of their
data, and important data and calculations.

Students may notice that the cost of iPods is going down and that storage size and performance are increasing.
They may wish to do an internet search of recent models and costs. Their search might start with Moore’s Law.

                                            End-of-Unit Test (1 day)
CT Algebra One for All                                                            Page 5 of 23
Unit Plan 2, 8 13 09
                             Technology/Materials/Resources/Bibliography
Technology:
    Classroom set of graphing calculators
    Graphing software
    Whole-class display for the graphing calculator
    Computer
    Overhead projector with view screen or computer emulator software that can be projected to whole
       class, and interactive whiteboard
On-line Resources:
    NUMB3RS “Burn Rate” Activity:
       http://education.ti.com/educationportal/activityexchange/Activity.do?aId=7998&cid=US
    350 Number puzzle: http://thinkofanumber.net/
    350 Video: http://www.350.org/
    Information on Electoral College: http://www.270towin.com/
    Information on Presidential Pets: http://www.presidentialpetmuseum.com/whitehousepets-1.htm
    Mathematics and the Police: http://mathcentral.uregina.ca/beyond/articles/RCMP/traffic.html
    IRS: www.irs.gov
    Create a class blog page at: https://www.blogger.com/start
    LINEQUA: http://education.ti.com/educationportal/sites/US/homePage/index.html (Search 8773)
    Look-up your school colors here: www.prepsportswear.com
    Body Mass Index (BMI) info: http://www.cdc.gov/healthyweight/assessing/bmi/index.html
    Video from NUMB3RS TV show:
       http://vids.myspace.com/index.cfm?fuseaction=vids.individual&videoid=9926688
    NUMB3RS “Burn Rate” activity:
       http://education.ti.com/educationportal/activityexchange/Activity.do?aId=7998&cid=US
    Poll Everywhere site: www.polleverywhere.com
    Build a backyard regulation-size volleyball court:
       http://www.popularmechanics.com/home_journal/how_to/4218238.html?page=2
    Cool theaters: http://www.essential-architecture.com/TYPE/TYPE-10.htm
Materials:
    Algebra Tiles
    Coins or counters
Bibliography: (Helpful print resources)
Horak, V.M. (2005). Biology as a Source for Algebra Equations: The Heart. Mathematics Teacher: 99(4).
Horak, V.M. (2005). Biology as a Source for Algebra Equations: Insects. Mathematics Teacher: 99(1).
Kunkel, P., Chanan, S., & Steketee, S. (2006). Exploring Algebra 1 with The Geometer’s Sketchpad. CA: Key
Curriculum Press.




CT Algebra One for All                                                     Page 6 of 23
Unit Plan 2, 8 13 09
                                           UNIT 2 PRE-TEST
NAME ________________________________________________ DATE ____________________

Directions: This pre-test is designed to find out how much you know about several math topics that will be part
of the algebra course. Try and answer every problem on this pre-test and show the work you did to complete the
problem.

1. Calculate the value of each math expression below:

    a. 3 + 2 • 4 ____________               b. 5 – 2(9 – 5) _________


    c. 4 • 9 – 12 ÷ 6 __________            d. 5 •32 + 3 • 3 +10 __________


2. Find the answer for each of the problems below:

    a. 8 + (- 5) = _______                  b. (- 3) – (- 5) = ______

    c. 14 ÷ (- 2) = _______                 d. (- 6) • (- 7) = _______

3. If water is pouring into a tank at the rate of 15 gallons every 4 seconds, how long will
   it take to completely fill a tank that holds 900 gallons? ______________ Show your work.

4. Solve each of the following equations for x. Show your work.

    a. x + 12 = 23 _____                    b. x – 6 = 34 ________



                                                  x
    c. 6x = 38 _______                      d.      7
                                                  4




5. Solve each of the following equations. Show your work.

                                                 x
    a. 3x + 5 = 20       _________          b.      7  2 __________
                                                 3




CT Algebra One for All                                                        Page 7 of 23
Unit Plan 2, 8 13 09
6. Your high school had a tag sale to raise funds for a local charity. It cost five dollars to get in and three
   dollars for every item you purchased. If you spent twenty-three dollars, how many items did you buy?
   Write an equation that models the problem, and then solve it. Show your work.




7. Use the formula M = 5T + 3 to fill in the values of M in the chart, given the
   values of T.

                                                  T           M
                                                  1
                                                  2
                                                  3
                                                  4
                                                  5

8. a. Give an example of the Commutative Property of Addition:


    b. Give an example of the Associative Property of Multiplication:         __________


9. Use the Distributive Property to rewrite the following expression without parentheses:

        3(4x – 8) = ____________


10. Simplify: 4x + 7 + 3x – 10 + x


11. Solve each equation for the designated variable:

        a. d = rt                              b. P = a + b + c

        Solve for t. ____________              Solve for b. ____________




CT Algebra One for All                                                             Page 8 of 23
Unit Plan 2, 8 13 09
Unit 2, Investigation 2
Activity 2.1a, p. 1 of 2
                                             New Cell Phone Plan

Name: ___________________________________________ Date: ______________________

Suppose you are shopping for a new cell phone plan. The table represents the various plans that you can
purchase from a local communications store.

                                           x = # of minutes     Cost of plan
                                  Plan
                                           used per month       per month
                                    A     x< 450                  $39.99
                                    B     450  x < 900           $59.99
                                    C     x  1350                $79.99
                                    D     Unlimited               $99.99

    a. Describe two different months of possible minutes that someone could use in Plan A and not be charged
       overage fees.


    b. Would all numbers less than 450 be possible under plan A?


    c. Could a customer use exactly 450 minutes in plan A and pay $39.99?


    d. If we were to graph all of the possible values that would work under plan A, how many points would
       you have to plot?


    e. On the number line, shade all of the possible minutes that could be used in plan A without being charged
       overage fees. Be sure to label points of reference on your number line.

                                           Plan A Possible Minutes



    f. On the number line below, graph all of the possible minutes for plan B.

                                           Plan B Possible Minutes




CT Algebra One for All                                                           Page 9 of 23
Unit Plan 2, 8 13 09
Unit 2, Investigation 2
Activity 2.1a, p. 2 of 2


    g. On the number line below, graph all of the possible minutes for plan C.

                                           Plan C Possible Minutes



    h. On the number line below, graph all of the possible minutes for plan D.

                                           Plan D Possible Minutes




    i. You need to choose a plan. During the school year you use your cell phone less frequently than in the
       summer. You estimate that, on average, you use about 800 minutes per month. But in June, July and
       August you use about 1200 minutes per month.
       For any of these plans, you have to pay $60.00 extra for any month in which you exceed the plan’s limit.
       Which plan should you choose if you have to sign up for a one-year contract?
       Show your work.




CT Algebra One for All                                                       Page 10 of 23
Unit Plan 2, 8 13 09
Unit 2, Investigation 2
Activity 2.1b, p. 1 of 1
                                                    Recycling

Name: ___________________________________________ Date: ________________________

You are collecting aluminum cans to raise funds for a local dog shelter. Should you bring the cans to the
supermarket or the recycling center?

If you bring the cans to the supermarket, you receive 5 cents per can as a return on the deposit.

If you bring the cans to the recycling center, you receive 6 cents per can but also must pay a flat $15 recycling
fee.

Suppose you collect 5000 cans.

    a. How much would you get if you took the cans to the supermarket?


    b. How much would you get if you took the cans to the recycling center?


    c. If you took some cans to the supermarket and got $6.50, how many cans did you take?


    d. Write an equation to solve c.

    e. What does the variable you used represent?



    f. Solve the equation.




    g. Write an equation to find the number of cans you brought to the recycling center if you got $18.60?
       Solve the equation.




    h. How many cans would you have to collect in order to receive the same amount at the recycling center or
       the supermarket?

    i. Describe how you found your answer.




CT Algebra One for All                                                          Page 11 of 23
Unit Plan 2, 8 13 09
Unit 2, Investigation 2
Activity 2.1c, p. 1 of 1
                                                    Recycling

Name: ____________________________________________ Date: ___________________________

You are collecting aluminum cans for a fund-raising drive. Should you bring the cans to the supermarket or the
recycling center?

If you bring the cans to the supermarket, you receive 5 cents (.05) per can as a return on the deposit.

If you bring the cans to the recycling center, how much will you receive?

       They pay a base amount of 4 (.04) cents per can.
       Plus an additional 0.3 (.003) cents per can to 1.2 (.012) cent per can depending on the current value of
        scrap metal.
       Plus a bonus of 0.35 (.0035) cents per can for every can over 1000 cans.
       They charge a flat fee of $7.50 for each lot of cans brought in (because small lots require nearly as much
        handling as large lots).

Suppose you collect 10,000 cans.

    a. How much money would you need to receive from the recycler in order to make the trip equal to what
       you could get at the local supermarket? Explain your thinking.



    b. Assign a variable to the current scrap metal value per can. _________________________

    c. Write an expression containing the variable which tells the money received from the recycler for 10,000
       cans.



    d. How much would you get if you took the cans to the supermarket? __________________


    e. Use your answers to problems c. and d. to write an equation. Solve the equation for the variable.




CT Algebra One for All                                                          Page 12 of 23
Unit Plan 2, 8 13 09
Unit 2 Performance Task
Sample Student Handout, p. 1 of 3

                                                      iPods!

Name: ________________________________________________ Date: ______________________

As a part-time job, you write the Consumer Watch column for your local newspaper. This week’s column is
going to be on the 16 GB iPod nano.

Someone wrote in to your column and attached a printout from a website that claimed that the 16 GB iPod nano
has “16 GB capacity for 4,000 songs or 16 hours of video.” This seemed interesting to you so you decided to
investigate.

You found this information:

                Downloading               Downloading              iPod nano          iPod touch
              songs on iTunes:          movies on iTunes:          8 GB - $149        8 GB - $229
               Each song costs          Each movie costs           16 GB - $199       16 GB - $299
                   $0.99                     $14.99                                   32 GB - $399
                and takes up              and takes up
            approximately 5 MB        approximately 1.5 GB

You also did some research and found that 1,024 MB = 1 GB.

For each of the following questions, be sure to define the variables you used in each question and show your
work.

1. If the website’s claim for the songs is accurate, how large (in MB) is the average song? How does this
   compare to the data you found?



2. Check information from the site: http://forums.macrumors.com/archive/index.php/t-192709.html (or any
   other site you find) to see if the claim in the ad seems reasonable. Another site you may want to investigate
   is: http://support.apple.com/kb/HT1906?viewlocale=en_US#faq5


3. It seems reasonable that the length (L) of a song is related to its size in MB (S). That is, the longer a song,
   the larger its size in MB should be. Use the website information from question #2, or another site you find,
   to develop a formula that relates L and S.



4. Assuming you can fit 4,000 songs on the 16 GB iPod nano, how much would it cost to buy it and fill it with
   songs? Justify your answer.




CT Algebra One for All                                                          Page 13 of 23
Unit Plan 2, 8 13 09
Unit 2 Performance Task
Sample Student Handout p. 2 of 3

5. How long would it take you in days and hours to listen to all 4,000 songs? Round to the nearest hour if
   appropriate. Also, be sure to carefully explain your reasoning.




6. One of the perks of the job is that you often get to test the products you are investigating. Your editor has
   given you an iPod and $250 to purchase songs and videos to test it out. You want to give your readers some
   idea of their downloading options. You assume that most people have, on average, 45 times more songs than
   they do videos. Complete the following chart:

                     # of Videos       # of Songs          Total Cost        Total Size of Files
                          1
                          2
                          3
                          4
                          5


7. Let us assume that you bought a 16 GB iPod nano and downloaded two movies for it. Construct an
   inequality to determine the number of songs that will fit on it. As always, be sure to define your variables
   and solve the inequality.




8. You hear that Apple is planning to introduce a 24 GB iPod nano this summer. What price do you estimate it
   will sell for? Be sure to explain your reasoning.




9. The ad that you found also mentioned that a fully charged 16 GB nano can play “Up to 24 hours of music
   when fully charged.” About how many songs can you play before it runs out of power?




CT Algebra One for All                                                         Page 14 of 23
Unit Plan 2, 8 13 09
Unit 2 Performance Task
Sample Student Handout p. 3 of 3

10. Since computers are continually getting smaller and smaller and can hold more and more information, there
    is a possibility that one day there will be a one terabyte (TB) iPod. Find out how large a terabyte is and use
    that information to estimate what it would cost to load a one-TB iPod with movies.




11. An important aspect of iPods is their ability to play both audio and video podcasts. For example, check out
    the podcast on Muslim Contributions to Mathematics: http://www.youtube.com/watch?v=goI-GnPj3DQ.
    In it, there is mention of al-Khawarizmi, often considered the Father of Algebra. He named the study aljabr,
    which obviously sounds very similar to the word algebra.

    Because of all the numbers and variables involved in exploring the iPod, it would be very helpful for your
    readers if they had somewhere to go to “brush up” on the algebra needed to understand your column. Find a
    podcast that helps people understand variables or equations. List the website or the source of the podcast
    and write a brief description of the podcast.




Note: A website that is helpful for product reviews is www.consumerreports.org




CT Algebra One for All                                                          Page 15 of 23
Unit Plan 2, 8 13 09
                                         UNIT 2 MID-TEST
                                            After Investigation 3
                                             EQUATIONS

NAME ________________________________ DATE ________

1. Simplify the following expressions by combining like terms. Show all work.

    a. (3x – 7) + (x + 9)                   b. 12x – (4x – 3)


    c. 3(2x + 5) + 4(x – 8)                 d. 2x – 5(2x + 1)




2. Evaluate each expression if x = 5. Show all work.

                                                                     1
    a. 14 – 4x ÷ 10                         b. –(6 – 12) ÷ 3 + 10(     )
                                                                     x



3. Solve the following equations. Show your work and state which Property of Equality you used.
                                                                   x
   a. x + 3 = -2                    b. 5x = 32                  c.   6
                                                                   7



4. Solve the following inequalities. Show work and show solutions on the given number lines.
    a. x – 2 > 5                                  b. 5x  15




5. Pedro and Janelle are given the task by their families of getting the rubbish ready for pickup each week
   during the summer vacation. Pedro is paid $4 per week while Janelle is paid $15 at the beginning of the
   summer and $2 each week. Choose variables and explain what they represent.




CT Algebra One for All                                                       Page 16 of 23
Unit Plan 2, 8 13 09
    Write an equation for each situation.

    Pedro: ________________________          Janelle: ___________________________

    Which method of payment would you prefer? Explain your choice.



    6. A store could use the equation C = 7.50 + 1.75w to calculate the price C it charges to ship
       merchandise that weighs w pounds.
       a. Find the price of mailing a 5-lb package. ______________

        b. What is the real world meaning of 7.50?

        c. What is the real world meaning of 1.75?

7. Solve the equation, 45 = 4c – 11, for c. Explain what Property of Equality was used in each step in your
   solution.
       45 = 4c - 11



8. Oscar and his band want to record and sell CDs. There will be a set-up fee of $400, and each CD will cost
   $3.75 to burn. The recording studio requires bands to make a minimum purchase of $1000.

        a. Write an equation that relating the total cost to the number of CDs burned.

                ______________________________

        b. Write and solve an inequality to determine the minimum number of CD’s the band can burn to meet
           the minimum purchase of $1000.




9. Solve the following equations. Show your work.
                                                                 x
    a. – 6 – 9x = 3           b. –3x + 10 = 10              c.      13  15
                                                                 6




CT Algebra One for All                                                         Page 17 of 23
Unit Plan 2, 8 13 09
10. Solve the following inequalities. Show your work.
    a. –2x – 10 < -8                       b. 14 10x-26




11. At the Berkshire Balloon Festival, a hot air balloon is sighted at an altitude of 400 feet and appears to be
     descending at a rate of 25 feet per minute.
    a. Write an equation that relates the height of the balloon to the rate the balloon is descending per minute.



    b. How high is the balloon 10 minutes after it is spotted at 400 feet? Show your work.



    c. How high is the balloon 5 minutes before it is spotted at 400 feet? Show your work.



    d. How long will it take the balloon to land after it is spotted at 400 feet? Show you work.




CT Algebra One for All                                                          Page 18 of 23
Unit Plan 2, 8 13 09
                                              UNIT 2 TEST
                                              EQUATIONS

NAME: ________________________________ DATE: ____________
1. The University of Connecticut (UCONN) defeated Louisville University in the finals of the NCAA
   Women’s Basketball Tournament in 2009 by a score of 76 to 54.

    a. There are three ways to score points in college basketball:
                     score a basket within 20 feet (actually 20 feet 9 inches) for two points
                     score a basket farther than 20 feet for three points,
                     or score a foul shot for one point

        Assign a variable for each scoring method.




    b. Write an equation for determining the final score, F, for a team if you know the number of two-point
       baskets, three-point baskets and foul shots.




    c. Suppose UCONN scored 25 two-point baskets and 14 foul shots. Write an equation which will allow
       you to find the number of three-point baskets the team scored.

       Solve the equation.




    d. In their last game, Louisville scored a total of 54 points, of which 42 points were foul shots or 2-point
       baskets. In the equation, 54 = 3g + 42, g represents the number of three-point baskets Louisville scored.

        Solve the equation for g.




CT Algebra One for All                                                         Page 19 of 23
Unit Plan 2, 8 13 09
2. The Bridgeport Bluefish management discovered that the profit they make on their concession stands may
   be found by using the formula: P = 4V – 300, where P represents the profit and V represents the attendance
   at the game.

    a. Find the profit if the attendance is 1,200 fans. Show your work.




    b. Find the profit if the attendance is 3,500 fans. Show your work.



    c. What does the coefficient 4 represent in the equation?



    d. What is the real world meaning of the 300 in the equation?




3. Two different formulas used to find the area of a trapezoid are below.
   Show that the expressions in the formulas are equivalent.



                         1                1     1
                A=         h(a+b)    A=     ah + bh
                         2                2     2




CT Algebra One for All                                                       Page 20 of 23
Unit Plan 2, 8 13 09
4. In Vermont the speed limit on some major highways is 75 miles per hour. To find the
   fine a speeder has to pay when he travels over the speed limit, the State of Vermont uses the following
   procedure:

    • Take the speed of the car and subtract 75
    • Multiply the difference by $45

    a. Select a variable to represent the fine __________________ and a variable to represent the miles per
       hour the speeder is traveling ________________________.

    b. Write an equation that shows the relationship between the fine and the miles per hour the speeder is
       traveling.




    c. Explain what values the variable may have for the miles per hour




    d. Find the fine if the rate of the speeder is 95 miles per hour.




    e. Find how many miles per hour the speeder was traveling if the fine is $360.




CT Algebra One for All                                                        Page 21 of 23
Unit Plan 2, 8 13 09
5. Latisha is traveling to Australia to study aquatic life along the Great Barrier Reef. She is planning her trip
   and needs to know how much money to bring with her. This table shows recent Australian-dollar
   equivalents of various U.S.-dollar amounts.
                                              Dollar Conversion Table
                                      U.S. Dollars       Australian Dollars
                                           15                  18.9982
                                           25                  31.6514
                                           35                  44.3120
                                           50                  63.2999

    a. Use the data in the table show how you could use the data in the table to find the number of Australian
       dollars you would get for one U.S. dollar.



    b. Write an equation for finding the number of Australian dollars, y, equivalent to x U.S.
       dollars.____________________________________

    c. Use your equation to find how many Australian dollars Latisha would receive for $1,400 U.S. dollars.



    d. At the end of her stay, Latisha has $180 Australian dollars. Use your equation to find how many U.S.
       dollars she will receive in exchange.



6. Solve the following equations and inequalities. Show all your work. For the two inequalities, graph the
   solutions on the given number line.

                a. 3(2x – 7) + 2x = 51                        b. 3x – 2(x – 5) = 10- x



                c. 6x + 3 = 2x + 15                           d. 2(x + 3) + 4 = 30


                                                                   2        1
                e. 2x + 3  3x – 5                            f.     (2x-3)+ (x  4)  11
                                                                   3        2




CT Algebra One for All                                                           Page 22 of 23
Unit Plan 2, 8 13 09
7. This expression describes a number trick:

        (6(N  3)  12)
                        N
              6



    a. Test the number trick with two different starting numbers. Show your work.




    b. Did you get the same result for both numbers?



    c. Tell which operations undo each other in the number trick.




CT Algebra One for All                                                      Page 23 of 23
Unit Plan 2, 8 13 09

				
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