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Mathematics

I.          Title:  The Yo-Yo Problem: Solving Linear Equations

II.         Organization:  Whole Group / Small Group

III.       Objectives:

·        The students will explore linear patterns.

·        The students will write a pattern in symbolic form.

·        The students will solve linear equations using symbolic manipulation, and the graphing calculator.

IV.       Standards Covered:

·        2.1:  Numbers, Number Systems, and Number Relationships

·        2.2:  Computation and Estimation

·        2.4:  Mathematical Reasoning and Connections

·        2.5:  Mathematical Problem Solving and Communication

·        2.6:  Statistics and Data Analysis

·        2.7:  Probability and Predictions

·        2.8:  Algebra and Functions

V.        Materials:

·        Graphing calculator and overhead unit

·        The Yo-Yo Problem Worksheet

·        The Penny Pattern Worksheet

·        Pencil

·        Paper

For each group:

·        31 pennies

·        graphing calculators

VI.       Procedure:

Intro / Motivation:

Introduction of the problem:  Explain the details of The Yo-Yo Problem to the class.

The Yo-Yo Problem

Andy wants to buy a very special yo-yo.  He is hoping to be able to save enough money to buy it in time to take a class in which he will learn how to do many fancy tricks.  The 5-ounce aluminum yo-yo costs \$89.99 plus 6% sales tax.  Andy has already saved \$17.25, and he is earning \$7.20 a week by doing odd jobs and chores.  How many weeks will it take him to save enough money for the yo-yo?

Have the students calculate the total amount of money he will have to save by determining the sales tax and adding that amount to the price of the yo-yo.  Before the students begin to solve the problem, you should review linear patterns and have students practice solving linear equations.

Developmental Activities:  Penny Pattern Exploration

For the second part of the lesson, students create a design in stages.  The first stage is one penny surrounded by six pennies.  For each successive stage, six more pennies are added to the outside of the pattern.  Have students continue to make several more stages of this design with their groups.  They should create a table of values using n for the stage number and p for the number of pennies used.  Finally, have each group determine an algebraic rule representing the relationship between the stage number and the number of pennies used.  Then have each group share its rule with the entire class.

Closure:  Back to the Yo-Yo Problem

Review the basic facts of The Yo-Yo Problem for the students and direct them to work with their groups to solve the problem.  Have students use various methods to determine the solution, including writing a symbolic equation and solving it, using the trial and error method, and using simple arithmetic.  Have students go to the board and present their solutions to the class.

·        You could have the students write problems that are similar to The Yo-Yo Problem.  They could share their problems with their groups or with the class.  You could assign some of these problems as homework.  You could also display the problems on a bulletin board in the classroom.

·        For visual learners, have students bring in examples of linear models.  Ask them to explain why the relationship is a linear model.  You could also have the students bring in examples that are not linear and explain why they are not linear.

·        You could group students who are struggling with the material with students who have a strong understanding.  These stronger students could act as peer tutors for the students who are struggling.

·        For ESL students, you could group them with students who are bilingual or who can help with their language.

VIII.     Evaluation:

Student:

·        This lesson offers many opportunities for ongoing assessment.  As students work in groups and as they make presentations to the class, you can evaluate their mathematical understanding.  This lesson also gives students many opportunities to connect ideas from the various activities and use those ideas as they work to solve The Yo-Yo Problem.

o       Did they see The Yo-Yo problem as another linear pattern?

o       What was the initial value, and what was the rate of change?

o       Could they solve the symbolic representation?

o       Do they have generally good problem solving ideas?

·        The teacher can walk around the room while students work in groups and use what they hear and see as an assessment.

·        The teacher can also use the worksheets as an assessment of how well the students understood the lesson.

Teacher:

·        Was there a missing element to my planning?

·        Did things turn out better or worse than expected?

·        If some students did not seem to work well in the assigned groups, try changing groups for the next cooperative learning assignment.

·        Having the groups work on two activities in one class period seemed to be overwhelming for the students.  Next time assign only one activity per period.

As a follow up activity, ask the students to write an equation for the relationship between the pattern number, n, and the number of pennies required to make the pattern.  Also, have them draw the graph and make a table of values.  Have them do the same thing for the relationship between the number of weeks, w, and the amount of money saved in The Yo-Yo Problem.  Then ask the students to compare the equations, graphs, and tables and describe how they are different and how they are the same.

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The Yo – Yo Problem  Worksheet 1

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Andy wants to buy a very special yo-yo.  He is hoping to be able to save enough money to buy it in time to take a class in which he would learn how to do many fancy tricks.  The 5-ounce aluminum yo-yo costs \$89.99 plus 6% sales tax.  Andy has already saved \$17.25, and he is earning \$7.20 a week by doing odd jobs and chores.  How many weeks will it take him to save enough money for the yo-yo?

PART ONE:

1. How much sales tax will Andy have to pay?

1. What will be the total cost of the yo-yo, including tax?

PART TWO:

1. Let w be the number of weeks that it will take Andy to save enough money to buy the yo-yo.  Write an algebraic equation that will help solve the problem.

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The Penny Pattern: Exploring Linear Models Worksheet 2

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1.   Create a pattern using pennies.  Stage one of the pattern is shown next to the title above – one penny surrounded by six additional pennies.  To create each additional stage of the design, place more pennies extending out from the six that surround the center penny.  Continue making this design until you have used up all of your pennies.  On the back of this sheet, sketch the first four stages of the pattern.

1. Using your penny pattern or the sketches of your penny pattern, create a table of values.

Stage Number, n                             1            2                3              4                5

Number of Pennies Required, p

1. How many pennies are needed to make stage 6, stage 7, and stage 8 of the penny pattern?  How did you determine your answer?

1. Write an algebraic model that expresses the relationship between the stage number n, and the number of pennies required to make that design, p.

1. Use your model to determine how many pennies are needed to make stage 80, stage 95, and stage 100 of the penny pattern.

1. Using your graphing calculator, make a scatter plot of the table of values from problem 2.  Graph your model from problem 4 to determine if it is correct, and then use the graphing calculator to create a table of values to check your answers to problems 3 and 5.

1. If you use 127 pennies to make the penny pattern, how many pennies will be in each spoke coming out from the center penny?  Can you find this answer three different ways?

Daily Lesson Plan 2

Mathematics

I.      Title:  Absolute-Value Equations and Inequalities

II.      Organization:  Whole group / small groups

III.      Objectives:

·        The students will be able to solve absolute-value equations.

·        The students will be able to solve absolute-value inequalities.

·        The students will be able to express solutions as a range of values on a number line.

IV.      Standards Covered:

·        2.1: Numbers and Operations

·        2.2: Computation and Estimation

·        2.3: Measurement and Estimation

·        2.4: Reasoning and Connections

·        2.5: Problem Solving and Communication

·        2.8: Patterns, Functions, and Algebra

V.      Materials:

·        Pencil

·        Paper

·        Textbook: Holt Algebra I

·        Calculator

·        2 Worksheets: Absolute Value Equations and Absolute Value Ineq.

VI.      Procedure:

Intro / Motivation:

·        Assign Pre-class/Warm-up, walk around room, take roll, and monitor students.

·        Review questions from homework for the previous night.

·        Give 3 or 4 absolute value practice questions for review.

o       | 13 – 24 | = 11

o       | 1 – 27 | = 26

o       | 5 – 10 | = 5

o       | 11 – 3 | = 8

Developmental Activities:  The problems that we just solved involve using absolute values.  Today we will be looking at absolute-value equations and absolute-value inequalities.  Before we begin, let’s recall the definition of the absolute value of a number.  In section 6.4 we learned that:

For a ³ 0, | x | = a is equivalent to x = a or x = -a.

When solving absolute-value equations, you must consider two cases.  Let’s look at an example.

| 3x – 2 | = 10

Case 1:  | 3x – 2 | = 10                                     Case 2:  | 3x – 2 | = -10

3x – 2 = 10                                                      3x – 2 = -10

3x = 12                                                             3x = -8

x = 4                                                              x = -8/3

Let’s check our work to see if this is correct!

Case 1:  | 3(4) – 2 | = 10                                  Case 2:  | 3(-8/3) – 2 | = 10

| 12 – 2 | = 10                                                  |-8 – 2 | = 10

| 10 | = 10                                                    | -10 | = 10

TRUE!                                                          TRUE!

Let’s look at another example:

| 2x – 4 | = 8

Case 1:  | 2x – 4 | = 8                                       Case 2:  | 2x – 4 | = -8

2x – 4 = 8                                                        2x – 4 =  -8

2x = 12                                                             2x = -4

x = 6                                                                x = -2

Let’s check our work to see if this is correct!

Case 1:  | 2(6) – 4 | = 8                                    Case 2:  | 2(-2) – 4 | = 8

| 12 – 4 | = 8                                                                 | -4 – 4 | = 8

| 8 | =  8                                                         | -8 | = 8

TRUE!                                                          TRUE!

Next we will take a look at some absolute-value inequality problems.  Solving absolute-value inequalities is very similar to solving absolute-value equations.  When working with absolute-value inequalities we must consider two cases once again.  Let’s look at an example:

| x – (-5) | £ 2

Solve –2 £ x – (-5) £ 2 by solving each part of the inequality separately.

Case 1:  -2 £ x – (-5)                                       Case 2:  x – (-5) £ 2

-2 £ x + 5                                                        x + 5 £ 2

-7 £  x                                                           x £ -3

Then combine the solutions in the form of a conjunction:

-7 £ x £ -3

The graph of this solution will look like:

Let’s look at another example:

| x - 6 | > 2

Case 1: The quantity x - 6 is positive

| x – 6 | > 2

x – 6 > 2

x > 8

Case 2: The quantity x – 6 is negative

| x – 6 | > 2

-  (x – 6) > 2

-         x + 6 > 2

-         x > -4

x < 4

We must change the direction of the sign because we divided by a negative sign.

The graph for this solution looks like:

After going over these example problems, the students will be given two worksheets to work on in small groups.  One worksheet contains problems on absolute-value equations and the other worksheet contains problems on absolute-value inequalities.  The students will be given the remainder of the period to work in small groups on these worksheets.

Closure:  Students who have not completed the assigned worksheets in class will be expected to finish them for homework.

·        For ESL students, allow them to work on problems with a buddy who can help with the language.

·        For special needs students, repeat key concepts.

VIII.      Evaluation:

Student:  The students will be evaluated using the worksheets given in class.

Teacher:

·        Was the class enthusiastic about the lesson?  Was there a lot of participation?

·        Did the students understand the difference between absolute-value equations and absolute-value inequalities?

·        Did the students understand how to graph absolute-value inequalities?

·        Have students complete the worksheets given in class for homework.  Review these worksheets tomorrow at the beginning of class.

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ABSOLUTE VALUE EQUATIONS

Worksheet 1

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Quick Review:

The expression within the absolute value symbols can be negative or positive.

Solve the equations below.

1.    | x | = 7          x = ______________

2.    | 2x | = 6       x = ______________

3.    | x – 3 | = 1    x = _____________

4.    | x + 1 | = 3    x = _____________

5.    | x + 2 | = 3    x = _____________

6.    | x – 2 | = 1    x = _____________

7.    | x + 2 | = 2    x = _____________

8.    | x – 3 | + 1 = 5    x = ____________

9.    | 4x – 2 | = 10    x = ____________

10.  | 3x + 6 | = 12    x = _____________

11.  | 2x – 3 | = 4       x = ____________

12.  | x – ½ | = 9/2     x = ____________

13.  | 3x – 1 | = 8       x = ____________

14.  | 6x + 2 | - 3 = 5    x = ___________

15.  | 5x – 7 | = 3     x = _____________

16.  | 2x + 1 | = 3    x = ______________

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ABSOLUTE VALUE INEQUALITIES

Worksheet 2

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Quick Review:

1. The expression within the absolute value symbols can be negative or positive.
2. Reverse the direction of the inequality for the negative case.

Solve each inequality.  Sketch the graph next to each problem.

1.  | x + 1 | > 1       x <  _______ or x >   ________

2.  | x + 1 | < 1         ________   < x <   __________

3.  | x – 1 | ≥ 2         x ≤ ________ or x ≥  ________

4.  | x + 1 | ≤ 2         ________ ≤  x    ___________

5.  | x – 2 | ≥ 2         x ≤ ________ or x ≥  ________

6.  | x + 2 | ≤ 3         ________ ≤  x    ___________

7.  | x – 4 | < 1        __________  < x <   __________

8.  | 2x – 1 | ≥ 3       x ≤ ________ or x ≥  __________

9.  | 3x + 2 | > 4      x <  _______ or x >   __________

10. | 4x – 5 | < 3        ________   < x <   __________

Daily Lesson Plan 3

Mathematics

I.      Title:  Systems of Equations and Inequalities: The Substitution Method

II.      Organization:  Whole group / small groups

III.      Objectives:

·        The students will be able to find an exact solution to a system of linear equations by using the substitution method.

IV.      Standards Covered:

·        2.1: Numbers and Operations

·        2.2: Computation and Estimation

·        2.5: Problem Solving and Communication

·        2.6: Statistics and Data Analysis

·        2.8: Patterns, Functions, and Algebra

·        2.9: Geometry

V.      Materials:

·        Pencil

·        Paper

·        Textbook: Holt Algebra I

·        Calculator

·        Worksheet: Solving Linear Systems by Substitution

VI.      Procedure:

Intro / Motivation:

·        Assign Pre-class/Warm-up, walk around room, take roll, and monitor students.

·        Review questions from homework for the previous night.

·        Give 3 or 4 substitution practice problems for review.

Solve each equation for y:

·        5x + 3y = 8

x = 2

·        2x + 4y = 16

x = 4

Developmental Activities:  Today we are going to be solving systems of equations by using the substitution method.  If we know the value of one variable in a system of equations, we can find the solution for the system by substituting the known value of the variable into one of the equations.  This method is called the substitution method.

Let’s look at an example.

Solve by using substitution:

8x + 2y = 19

x = 3

Since x = 3, we can substitute the value of x into the first equation.

8 (3) + 2y = 19

Solve the resulting equation for y.

24 + 2y = 19

2y = -5

y = -2.5

The solution for this system of equations is ( 3 , -2.5 ).

Let’s check our work to see if this is correct!

We can check the solution by substituting the values for x and y into the first equation.

8 (3) + 2 (-2.5) = 19

24 + (-5) = 19

19 = 19

TRUE!

Let’s look at another example:

Solve by using substitution:

15x – 5y = 30

y = 2x + 3

Substitute 2x + 3 for y into the first equation, and solve for x.

15x – 5 (2x + 3) = 30

15x – 10x – 15 = 30

5x – 15 = 30

5x = 45

x = 9

Substitute 9 for x into the equation y = 2x + 3, and solve for y.

y = 2 (9) + 3

y = 18 + 3

y = 21

The solution for this system of equations is ( 9 , 21 ).

Let’s check our work to see if this is correct!

We can check the solution by substituting the values for x and y into the original equations.

15 (9) – 5 (21) = 30

135 – 105 = 30

30 = 30

TRUE!

OR,

21 = 2 (9) + 3

21 = 18 + 3

21 = 21

TRUE!

Now let’s take a look at a trickier example!

Solve by using substitution:

3x + y = 4

5x – 7y = 11

To use substitution for this example, we must solve one equation for y.  Which equation do you think would be easier to solve for y?  The first equation is right!  To solve the first equation for y, all we have to do is subtract the 3x from both sides.  To solve the second equation for y, we would have to subtract 5x from both sides and then divide both sides by -7.  So lets solve the first equation for y.

3x + y = 4

y = 4 – 3x

Now that we have found the value for y, we can substitute 4 – 3x for y in the second equation and solve for x.

5x – 7y = 11

5x – 7 (4 – 3x) = 11

5x – 28 + 21x = 11

26x – 28 = 11

26x = 39

x = 1.5

Now that we have found the value of x, substitute 1.5 for x in the first equation and solve for y.

3 (1.5) + y = 4

4.5 + y = 4

y = -0.5

The solution for this system of equations is ( 1.5 , -0.5 ).

We can check the solution by substituting the values for x and y into the original equations.

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** If there is extra time, do another example: **

6x – 2y = 11

x + 3y = 4

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After going over these example problems, the students will be given a worksheet to work on individually or in small groups.  The worksheet contains problems on solving linear systems by using the substitution method.  The students will be given the remainder of the period to work in small groups on this worksheet.

Closure:  Students who have not completed the assigned worksheet in class will be expected to finish it for homework.

·        For ESL students, allow them to work on problems with a buddy who can help with the language.

·        For special needs students, repeat key concepts.

·        Circulate around the room and answer any questions, or help students who are having trouble with the lesson.

·        For students who finish the worksheet early, give a second worksheet for them to work on.

VIII.      Evaluation:

Student:  The students will be evaluated using the worksheet given in class.

Teacher:

·        Was the class enthusiastic about the lesson?  Was there a lot of participation?

·        Did the students understand the concept of substitution?

·        Could the students identify which equation was easiest to solve for?

·        Have students complete the worksheet given in class for homework.  Review this worksheet tomorrow at the beginning of class.      "Mathematics is the alphabet with which God has written the universe." -- Galileo Galilei This site is written and maintained by Tiffany Young