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DйMATH 60 TEACHING IDEAS FOR CHAPTERS 2, 3, 4, 5, 6, 7
TopicsSectionsWeeksCh 2 Equations, Inequalities, and Problem Solving
2.2-2.7
3Ch 3 Introduction to GraphingALL3Ch 4 Polynomials4.1-4.6, and 4.83Ch 5 Polynomials and Factoring
ALL
3Ch 6 Solving Equations with Rational Expressions6.6 and some problems in 6.7
ONLY
1Ch 7 Systems of Equations7.1-7.42
Day 1 On the first day I am planning to randomly form groups of four. The first assignment will be for students to introduce themselves to their group. The day s work will be to find and correct the errors in a version of a Pre-algebra final exam on which I have shown typical solutions containing typical errors. The goals of this activity are:A. To inform students of what I expect them to know.B. To allow me to assess what they know as I move among the groups answering questions.C. To introduce students to the fact that they will be expected to work together on problems during the class.The assignment for the first day will be to complete work on the Final Exam and to write a math autobiography.Complete your math autobiography. Be sure to include the following:
A list of the math courses you took in high school (If you know the program or text please give that information. )
What was hard and what was easy.
Events that stand out in your memory (positive or negative)
Current feelings about math
Major and/or career goals.
Preliminary (Not included in text)Student skills from preliminary section:
Gain confidence with fractions
Review order of operations and the distributive law
Basic arithmetic (in particular, signed arithmetic)
Past GTAs have noticed that many students need a review of signed arithmetic, factions, and distribution (along with order of operations). Since Chapter One is not in the text the following activities may be used to review some of these concepts. Past GTA s have noticed that somewhere between 1 2 classes, up to one week, tended to be a good amount of review. Rock, paper, scissorsTargets adding and subtracting fractions with unlike denominators.~ 10 minutes to play and keep score, ~ 5 minutes to add it up.Students work in pairs, have them keep track of who won each round with which object. Play 10 rounds then give them fractional scores. Each player computes their own score.Assign fractional scores and have them score. For example:Rock beats scissors W:+1/2, L:-1/2Scissors beats paper W: +1/3, L: -1/3Paper beats rock W: +1/5, L: -1/5Rewriting equivalent expressions:This lesson can be used in a number of contexts. Here it is used for Distribution, whole number multiplication as repeated addition, and can be used for fractions as division.Start with an example; 3(2-7)+4 can be written as 3(-5)+4, -5-5-5+4, (2-7)3 + 4, etc.Students work in groups of 3. For 10 minutes each group comes ups with as many different ways of writing the expression as they can. Class compares expressions on the board. The group with the greatest number of correct equivalent expressions not duplicated by other groups wins the game. (Kind of like Boggle)This could be modified to include signed arithmetic (i.e. 3-(-2)), and to cover fractions as division (i.e. 3*(1/4) is also 3*(1?4))Hot and cold cubesSigned arithmetic, subtracting negative numbers.Area of Rectangles Distributive propertySee Solve It! day 15, pp. 113 for lesson.Grouping activity: Distributive property Use several containers with, for example, 6 of one thing, 3 of another, and 4 of a third item in each. Find out how many of each thing and relate the physical container (bag, box, etc) to the parenthesis. Have students create expressions to represent what is in the containers, the totals of each thing, etc..
CHAPTER 2 - Equations, Inequalities, and Problem Solving
Student skills from chapter 2:
Translation between problems stated in English and mathematical expressions
Solve equations and inequalities
The focus of Chapter 2 is writing and solving equations. Starting with word problems allows you to introduce something different to a group of students who have already taken an algebra course. It is also an approach that most can succeed with. Most books present word problems as words to be translated into algebra. People who have studied and succeeded in mathematics can translate from relationships stated in words to algebraic expressions because they are fluent in both mathematics and English. Our students are not fluent in mathematical language, and most of them have difficulty seeing the relationships described in words in the problems. One way to help them is to give them a way to work through the words using a specific number. After doing this with several numbers they can then use a variable and work through the same steps.Day 2 Assign students randomly to groups of four. Their first job is to introduce themselves and to talk about their best and worst experiences in mathematics. Following their introductions we, as a whole class, brainstormed the do s and don t s of what a math class should be like.Demonstrate a guess-and-check approach to two problems. Choose examples that are not too easy to set up. Then assign students to work together on some from Set 2.5 Page 106-107: 10, 14, 17, 22, 27. They should use guess-and-check in order to get a feel for the method even though some may think they can do the problems without it.Possible Assignment:For each of the following problems show a Guess and Check Table and after the table an Equation. See which equations you can solve, and make sure your equation results agree with your table results. Suggested problems: Page 106-107: 11, 19. 26, 28, 34, 35 (Depends on how far they got in class.)Day 3 Start by having students work in pairs to solve the equations in problem set 2.2. Students should work on no more than three problems and then stop trade papers and check each other s work. When they have used different methods or made errors they should discuss these. Circulate and respond to questions. . This allows you to assess what they already know and can do before you start explaining everything. When you start to get the same question repeated several times, stop and hold a full class discussion and show some examples.Possible problems to assign each pair A and B.Page 83-84: Either set A or set B A:3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 59, 63, 67, 71 B: 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 57, 61, 65, 69Possible Assignment:Synthesis Problems: 93 through 103, with 104, 105 extra
Try the activity on p. 84 to give students practice simplifying expressions in groups of three.Day 4Section 2.3 Students are thrown off by variables as constants. A few examples should do the trick.Possible lesson: Modify the lesson on p. 84 using formulas and having students solve for a letter.Day 5Section 2.4 Percents. There are a few sections in Math 70 that require students to change from decimal to percent and vice versa, however, it s straightforward and doesn t require much time. Students often think there are many different types of percent problems. It is worth showing that setting up a proportion can be used to in solving most percent problems. The rest of the section is similar to section 2.5.Day 6Sections 2.6 and 2.7 Inequalities. See whether anyone can tell you what is different about solving inequalities from solving equations. I usually work through an example where the inequality relation has to be reversed, without reversing it. If no one notices I substitute some numbers to check. And then discuss why you have to flip the inequality symbol. Assign problems to do from both 2.6 and 2.7. It would be a good idea to have them work on 2.7 problems (word problems) during class in groups. However, you want students to interpret solutions in the context of the original problems (the model equation may allow for some unreasonable solutions i.e. we don t have 8.25 people)
CHAPTER 3 Introduction to Graphing
Student skills from chapter 3:
Learn to make a graph for data using the right scale factor for the axes.
Learn to graph using slope and y-intercept.
Recognize slope and y-intercept from a graph/equation.
Recognize slope as a rate of change, and that lines have constant slope.
Graphing is a weak point in most traditional high school curriculums and for many students. I like the following activity because it makes a lot of the important basic details about a Cartesian Coordinate system memorable.Students should be encouraged to get graph paper and to work on graph paper from this point on.Day 1The following activity can be used instead of Section 3.1. Introduce the appropriate language as you are leading the activity. Hope for a dry day as this is best done outside. The equipment, two ropes marked off with black tape and colored index cards with numbers from 4 to +4, is in a canvas bag in the workroom. The shorter rope is the x-axis and the longer is the y-axis. Directions for the Human Graphing activity are attached. Students will not need to do the practice exercises in Set 3.1. Possible Assignment: pg 140-141 problems 55, 59-64.Day 2For section 3.2, arrange with Audio Visual to the Burning candle video or put the data from it on the board. Have students, working in pairs, record the data and draw an accurate graph to estimate when the candle will go out. If you use the tape, you may need to have it run twice, because they don t get enough data the first time. Assign the problems from 3.2 and 3.4 that involve graphs using points and slope.The point of the video is to relate the x-intercept of the graph of a line to the solution of a one variable equation in x, when y = 0.Day 3
Section 3.3 Draw graphs based on the x- and y- intercepts.Use the graphing calculator worksheets (attached) to develop the relationship between the equation and the slope and y-intercept of the graph then assign problems from Sec 3.4 and 3.5Days 4, 5, 6In sections 3.5, 3.6, 3.7 emphasize the value of starting with a sketch of the graph as a basis for determining the equation and as a check for whether the equation makes sense. Students and their instructors are often too quick to move to the subtraction formula for calculating the slope. This is a formula that is not understood, often forgotten, and usually mixed up. Try not introducing subtraction or the slope formula at all. Instead insist that students draw a graph for each problem and use the points on the graph to figure out the slop and approximate the y-intercept before getting it exactly. Some students will want you to tell them how to find the slope without the graph. Explain that they should be able to come up with the formula themselves, and if they can do that and justify it they are welcome to use it as a shortcut. Students will like it better if you just tell them the formula, but they will not understand it well enough to apply it later on.
Investigating Linear Equations and their Graphs
On the graphing calculator enter and graph the following equations. And then sketch the graphs on graph paper. What do you notice? Why should this be so?
1. y = 2x + 1
2. y = 5x + 1
3. y = x + 1
4. y = 0.5x + 1
5. y = -4x + 1
6. y = -0.4x + 1
Make up equations for two more equations whose graphs will belong the to the family of functions above.
Graph the following equations on the same set of axes. And then sketch the graphs on graph paper.
1. y = 2x 3
2. y = 2x 5
3. y = 2x
4. y = 2x + 1
5. y = 2x + 4
Graph the following equations on the same set of axes. And then sketch the graphs on graph paper.
1. y = 0.5x 3
2. y = 0.5x 5
3. y = 0.5x
4. y = 0.5x + 1
5. y = 0.5x + 4
Graph the following equations on the same set of axes. And then sketch the graphs on graph paper.
1. y = (-3/4)x 3
2. y = (-3/4)x 5
3. y = (-3/4)x
4. y = (-3/4)x + 1
5. y = (-3/4)x + 4
Make up two more equations whose graphs will belong the to the family of functions above.
What do you notice about the last three families of equations? Why do you think this is true?CHAPTER 4: Polynomials
Student skills from chapter 4:
Know rules of exponents, be able to apply them, and be able to explain them using examples and the basic definition.
Know how to add, subtract, multiply, and divide polynomials (note that students tend to need more practice on multiplication).
4.1 Properties of exponents:The class should be able to develop and articulate the rules of exponents listed on page 210 from the basic definition, through whole class discussion or by working in groups on lists of examples that lead to each one. Then most of the exercises in 4.1 are just puzzles in which to apply the various laws.
It might be worthwhile for students to make tables of powers of 2 up to the 10th, 3 up to the 6th, 5 up to§џџџ
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Good problems to assign in addition to the routine practice: 89-92, 99, 101, 102-107. Also 93-96 and require an explanation along with a numerical example.
4.2 Polynomials:
To get to the definition start with two lists one a variety of polynomials and the other a variety of algebraic expressions which are not polynomials. Then give students a third list, mixed, and ask them (in groups) to identify the polynomials and non-polynomials and come up with a description of polynomials that will exclude all the nons. Groups can report and you can refine the definition as a class.
An interesting way to compare polynomials is through graphing polynomial functions. The attached worksheet reviews what they know about linear expressions and provides some exercises to graph with the graphing calculator. As students work together on this you can move from group to group introducing and using language such as coefficient, term, degree, etc.
In addition to some practice problems, problems 83-88 and 89-90 are good ones. Plan on some class time following this assignment to discuss 83-88 in groups and as a class.
4.3 Addition and subtraction of polynomials:
This is pretty straightforward. Possibly have groups work on 51-66 in class and then share and discuss their results. Then assign the practice problems for homework.
4.4 Multiplication of polynomials:Show distributive property examples and introduce a rectangular model for multiplication based on area (Introduced on p. 222, ex 10).
In addition to some practice, problems worth doing include: 77-83, 86 90 and for extra credit (with a clear explanation) 84, 85. Note that 77-83 and 86-90 should be done in class in groups and followed by presentations and discussion.
4.5 Multiplication of binomials and special products:Show three methods: Distributive property, FOIL, and generic rectangle and how they are related. I generally require students to show a couple of problems done three ways, and then tell them to use the way they prefer.After practice, be sure to assign 85-96, some from 97-102, and 113-122.
4.6 and 4.7 Polynomials in more than one variable and division by a monomial.
You can probably do enough of both of these in one day. All that is required is division by a monomial.
DO NOT SPEND TIME ON DIVISION BEYOND DIVINDING BY A MONOMIAL.
4.8 Negative exponents:
As in 4.1, start with a multiple choice test with explanations of choices or set up sequences of examples that lead to the general properties, and have students write the general rules. Pull this together with a class discussion. The problems set is just exercises to practice using the rules, more puzzles to simplify.
CHAPTER 5: Factoring Polynomials
Student skills from chapter 5:
Factoring common factors
Factoring quadratic polynomials with leading coefficient 1
Recognizing perfect square trinomials and difference of squares
Use factoring and Zero Product principle to solve
5.1 Factoring using the distributive property.
Straight forward use of what they know about exponents and the distributive property. Students will need examples of factoring by grouping.
In addition to practice, assign problems 64 and 71
5.2 Factoring x2 + bx + c
Start with Diamond Puzzles.
Show factoring by grouping, backwards FOIL, and rectangles as ways of organizing.After practice, assign 69, 70, some of 71-76, 85 and 86. 87 and 88 make good extra credit problems. (for 87 require work be shown, since the answer is in the back.)
5.4 Factoring Perfect Squares and the Difference of Two Squares:
Recognition of perfect squares is the most difficult part. After practice factoring both forms students should give a thorough explanation for 87 and 88.
5.5 You can skip this.
5.6 General
This is where students have difficulty deciding what approach to use after factoring out common factors. Select problems carefully to avoid a x2 + bx + c trinomials that are not perfect squares.
5.7 Solving Polynomial Equations by FactoringStart with graphs of quadratic polynomials (See worksheet). Use the graphing calculator to sketch the graph and the trace and zoom buttons to locate the x-intercepts. Students will only be able to approximate intercepts with the grapher. Ask them how to find them exactly.
In addition to practice, be sure to assign problems 51-58.
5.8 Applications, the Pythagorean TheoremHave students do a hands on proof of the Pythagorean Theorem. They will all know that a2 + b2 = c2 , but probably don t really believe it., or believe it only because an expert said it was true.
Have groups work on the applications problems in class. When they have trouble setting up the equations, remind them that they can use guess and check tables to get started. But also remind them that even if they get the answer by guessing, you want to see the equation and its solution. The table just lets them know they are right.
Use a selection of problems from 1-16 , NOT # 17. Other problems to include are 23, 24, 29, 30, and 31-34. The others are just plugging numbers into formulas; 31-34 covers that idea using a more important formula. Other recommended problems that require a little thinking are 47, 48, 50, 52, 53.
CHAPTER 6 (Sections 6.6 and 6.7 ONLY)
Student skills from chapter 6:
Understanding how to use guess and check to solve word problems, and to develop equations for word problems.
Be able to solve rational equations
Day 1
Start with 6.7 and work problems. The following is a good example to work through with the class:
An Abel Paper Shredder can shred the week s accumulation in 12 hours, the Blaster Shredder takes only 8 hours. How long would it take to do the job using both machines?
Develop a guess and check table with the whole class similar to the following:
Guess # hoursAbelBlasterTogetherCheck10
6
4
5
.
.
x5/6
1/2
1/3
5/12
x/1210/8???
3/4
1/2
5/8
x/8
5/4
5/6
25/24
1
Too much
Not enough
Close
At this point students will know that they want x/12 + x/8 to equal 1, and they should be able to come up with the equation:
x/12 + x/8 = 1
Then you can ask them for ideas on how to solve this.
Another good example to work through with the class (because of the opportunity it provides to tweak their memories about fractions) is #9 from page 382.
Carpenter Juanita can build a fence in 12 hrs.
Anton can build the same fence in 16 hours. How long would it take for them to complete the job working together.
Start a table again and go through at least the guesses 10, 6, 8, and 7 before setting up the equation.
Assign: From 6.7 starting on page 382 selected problems from 1-4 (direct set ups), 5-14 (all work problems), and 23-47 (all are proportions). Start them in class in groups with one problem from each of these sections so they can ask questions if they need to.
Note: You do not need to assign problems from the other groups (#16-22); in fact, it could be counter productive to deal with the distance, rate, time problems here because of the convoluted method they show in order to come up with a rational equation. These are much more easily understood using two variables.
Day 2
On day you ve already dealt with how to handle the easy denominators in rational equations. Today you can focus on the more complicated ones like the ones they show in Example 2 in the text at the bottom of page 370. Two examples with audience participation should be enough to get started.
Allow some group time for students to work on a selection from page 274 #13-26.
Possibly assign each group a particular problem to prepare to put up on the board and explain.
Assign: A selection of about 5 beyond what they finish in class from 13-26 or 47-54. Give them all the answers ahead of time so they can see these as puzzles to solve. Be sure you have actually worked the ones you assign (some are just ugly). Also assign one chosen from 60, 61, 62 or 63 , and (maybe) # 72, page 386 of section6.7.
CHAPTER 7
Student skills from chapter 7:
Be able to solve systems of linear equations graphically and algebraically (using either substitution or elimination).
Understand what it means to be a solution of a system (graphically and algebraically).
Understand what it means to be a solution set (i.e. feasible region) of a system of linear inequalities.
Be able to graph a system of linear inequalities (by shading in the feasible region).
Throughout this chapter, students should be allowed to use whatever algebraic solution method they choose. They should be encouraged to look for efficient methods.
Day 1:
Sections 7.1, 7.2, and 7.3 can be consolidated into two days by starting (again) with a couple of word problems. A couple of starters are attached.
The Amusement Park Problem:
Have students work in pairs (or groups of four) to make and compare tables, draw the graphs on the same set of axes, and write and solve the equations. They should be able to do all of these things without much guidance from you. It s important that they use three methods (table, graphs, equations) to solve the problem and then discuss the relationship of the methods, and the advantages and disadvantages of each.
Perimeter Problem:
Next lead a class solution of a problem such as the perimeter problem posed on page 384:
A rectangular garden in 4 feet longer than it is wide. If the perimeter is 92 feet, find the length and width.
Start with a guess and check table (even though this is an easy one that they may be able to set up, the point of this exercise is to see two ways to set up the equation(s))
So here s a possible scenario for the guess and check table:
Guess WidthLengthPerimeterCheck 92
50
20
22
21.
.
.
x
x
54
24
26
25
.
.
x + 4
.
y
208
88
96
92
2x + 2(x + 4)
2x + 2y
Too long
Too short
Too long
"
92
92
They are used to solving problems like this with one variable- so go through that possibility with the equation:
2x + 2(x + 4) = 92
Then ask if they could have used two variables what if they just called the length y instead of x + 4. What equations would they write? They will usually come up with
2x + 2y = 92
But they may have difficulty figuring out the other one (which is, ironically, the easy one ). Make a big deal out of the fact that they need two and wait for someone to find it! The whole point of working through this guess and check chart was to see how to get two equations. This will become the key to their dealing with the problems in section 7.4 without using the authors formulaic method.
Once they have 2x + 2y = 92 and y = x + 4 give them a few minutes to work with a neighbor to solve the pair of equations. Then have students explain their methods. This should be enough to remind them of how to solve systems of equations.
Put the class in groups to work on the following three problems. Generally you will find that students need just a little nudging to remember how to use the Substitution and Elimination methods. Questions like can you solve the first equation for x? and then use that result with the second equation. OR Could you multiply the second equation by 3 and then combine them? OR for the third, could you multiply one equation by one number and the other by another?
Direct them to choose one of the three to do in two ways. Have students write up solutions. Take a poll to see which they chose to do two ways.
Pg. 404 # 29
x 3y = -1
5y 2x = 4 (-7,-2)Pg. 413 # 15
x + 3y = 19
x y = -1 (4,5)Pg. 413 # 19
2w 3z = -1
3w + 4z = 24 (4,3)
Assign: A few problems from Section 7.2 page 404-405. Include 43, 44, 49, and 50.
Day 2: (Dependent and Inconsistent systems & practice 7.3)
Start off with four problems for them to do in pairs or groups.
From pg. 413:
#24
5z = 2b
2a +11 = 3b
#12
8x + 3y =4
-8x 3y = -4
#30
2x + y = 13
4x + 2y = 23
#32
x (3/2)y = 13
(3/2)x y = 17
One pair is dependent and another in inconsistent. When student put these on the board, use that opportunity to put the equations into y-form and compare graphs and their relation to what happens when you try to solve algebraically.
Respond to questions on the previous assignment.
Then have students work in groups on Section 7.3, page 413-414
Problems: 18, 22, 36, 39, 47, 48, 61 and maybe 65. ( extra: 63, 64)
Assign: Finish what they have left to do on the above.
Day 3: (Section 7.4)
After responding to a few questions on the previous assignment, put class into groups of four and assign the following word problems with no further introduction. They can start with guess and check and that should lead them to writing two equations in two variables, which they should show how to solve, even if they already have a solution in their guess and check table.
Recommended problems:
Page 421-423: Problems 4, 6, 8, 10, 12 14, 15, 18, and possibly 20. Each group should be assigned three problems with responsibility for putting the complete solution to one of them up on the board and presenting it. For example, if you have 8 groups:
Group 1: Do 18, 4, 6, Present #4
Group 2: Do 4, 6, 8, Present # 6
Group 3: Do 6, 8, 10, Present # 8
etc.
Group 8: Do 15, 18, 4, Present # 18
Assign: Groups choose four additional problems (other than the ones presented) from Pages 421-423 # 1-24, that they will do for homework.
Day 4: (Sections 7.5 and 7.6)
Since different groups have done different problems there is not much point in going over particular problems unless one turns up to be a stickler for a large number of students,
Remind students of graphing inequalities: Give them two examples to graph on separate axes. Have them do these before you start. Wander around to see who remembers what.
2x + 3y e" 6 3x y < 5
Have the class tell you how to graph these and use the two graphs to discuss shading, solid vs. dotted lines etc.
Then give each group a different project to graph. Their job is
---Put each inequality into graphing form,
---Graph each inequality and shade the intersection,
---Identify the closed figure formed by the intersection,
---Label each vertex of the figure with the exact coordinates,
Each group gets one of the following systems to graph these can be done directly on the blackboards (colored chalk helps) or they can be done as posters on large easel graph paper with colored pens:
A.
2x + 3y d" 6
x y > -1
3x + y e" -3
x 3y < 9
B.
5x 4y d" 12
x + y > -3
2y x d" 10
2x + y d" 10
C.
x + 2y d" 14
2y x d" 10
2x + y > -5
2x y d" 8D.
x + y d" 9
7y 2x > -28
x y e" -1
x d" 7
E.
2y x d" 10
x + 2y d" 14
x d" 8
x e" -8
x 8y < 40
F.
3x + 2y > -10
3x + 2y < 20
x y d" 0
x y e" -5G.
2x + 3y d" 12
3y 2x > -18
2x + 3y e" -30
-2x + 3y d" 24
x > -9
x < 3H.
-3x + 2y e"12
2x + 5y d" 30
-2x + 5y d" 50
3x + 2y e" -18
y e" 3
Assign: Maybe two, at most three systems of inequalities to graph from 433. Be careful which you assign.
Day 5----If there is time:
A linear programming problem, a group assignment that pulls all the ideas from chapter 7 together. For example, try the Toy Factory lesson.
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