Week 1: - Kent



Chapter 1 Suggestions: (Week 1)

Sec. 1.0 Algebra Toolbox

Goals for students:

• Understand subsets of real number system; be able to classify numbers, identify subsets, etc.

• Understand and use interval notation; change from inequality notation to interval and vice versa

• Simplify algebraic expressions using distributive property and solve simple linear equations

Review syllabus with students; discuss grading policies, etc. Students need a graphing calculator for class: starting tomorrow! You might email students ahead of time welcoming them to class and informing them of class times and locations (especially if you have lab on Monday)!

The following is a good “ice breaker” and takes about 10 -15 minutes of class time (including the whole class discussion following group work): Students break up into groups of 3 or 4 students. Their tasks in the groups are: introduce self, share contact info, discuss major, why taking course, attitude toward math, expectations for course, and expectations for instructor. Then we have a full class discussion with one person from each group reporting the group’s comments, especially “attitudes toward math” and “expectations from instructor.”

Continue then with subsets of real numbers: a Venn diagram is a great way to summarize this. You can view a quick lesson with checkpoints for students at:



Other topics to discuss today are interval notation and a review of distributive property and solve simple linear equations.

Suggested homework: p. 9: 7 – 18 all, 19, 21, 27 – 39 odd, 38, 40

Sec 1.1 Day 1 Introduction to Functions

Goals for students;

• Get a feel for the notion of inputs and outputs, with one variable dependent on the other

• Identify dependent and independent variable in a given scenario

• Correctly use function notation

I like to start this section with the application problems. A good rule of thumb is to begin with something with which students are familiar, i.e. something concrete and understandable, then generalize to the mathematical notation and concepts behind the scenario.

For example, you might start with a scenario like #34, on page 21 and ask such questions as:

• What two things are changing in this scenario? These are the variables

• Which one would do you think depends upon the other

• Are the outputs increasing or decreasing?

Try to get them used to the notion of inputs and outputs, with the output depending upon the input. I don’t use function notation right away, but introduce it gradually. You might say something like “Temperature depends upon time and Temp =” , then gradually introduce T(m) notation, perhaps joking that this is the “lazy way out” (less writing). E.g. after writing “Temperature(minutes)” a few times, they are happy to write T(m). Be sure to emphasize that the notation in parenthesis gives the independent variable and that this notation does NOT REFER TO MULTIPLICATION, but is taken as a whole to mean the output of the process that is the function.

Most students will be familiar with inequality notation, so that is a good place to start when teaching interval notation. You might have students come to the board and sketch graphs of inequalities, then you provide the “new and efficient” interval notation.

Quickly (class is almost over!) review distributive prop and solving equations. Of all the material we discuss today, most students are most comfortable with these topics.

Suggested homework: pp. 19 – 26: 1 – 19 odd, 20 (omit domain and range parts); 31- 37 odd, 39, 41, 45– 55 odd . (Omit all parts dealing with domain and range.)

Sec 1.1 Day 2: Domains and ranges of functions

Goals for students:

• Identify domains and ranges of functions in context

• Identify domains and ranges of functions given in symbolic form

• Identify domains and ranges from graphs of functions

The concepts of domain and range are not easy ones for students to understand. Perhaps again begin in context, in a situation with which they are familiar and asking: “What values of the input make sense in the problem?”

Be sure to also emphasize reading domain and range from a graph.

Suggested homework: pp. 19 – 26: all the domain and range stuff from yesterdays homework,; also # 42, 44, 58, 59, 61, 62; Bring handout with graph activity to class tomorrow.

Sec 1.2 Graphs of functions

Goals for students:

• Graph a function by hand, using a table of values

• Correctly interpret a given graph as a relationship between two varying quantities

• Use the graphing calculator to graph a function using an appropriate viewing window, make a stat plot, and find function values.

As a small group in-class activity, you might make copies of the “Graph Activity” from the Instructor site. This is also available on the students’ VISTA site, but your site may not yet be active. Of course, relate this activity to this week’s “Distance Match” lab. Some students mistake a graph for a picture of what’s actually happening rather than a relationship between two varying quantities.

I’ll probably sketch a few graphs by hand, then teach basics of graphing calculator, time permitting.

I’ll probably not get to scatterplots on the TI today, but if you do, you can also assign #47 and #49 for hw.

Suggested homework: p.36 - 40: 1 – 39 odd, 43, 44

(Week 2)

Sec. 1.3 Linear Functions

Goals for students:

• Given two points, find the slope of the line containing them

• Given two points in a real world context, find and interpret the slope of the linear function containing them

• Understand that slope represents a constant rate of change

• Given a function in symbolic form, determine if the function is linear

• Identify and interpret the slope and y-intercept of a linear function given in symbolic form

• Given a function in symbolic form, find the x- and y-intercepts.

Start with two points and ask students to find the slope of the line between them. Review the formula for finding slope and let students try several traditional examples. (This skill is a review for most students.) Then introduce an example in context, like #51 on page 54. Let students find the slope, then develop the notion of interpreting it in context. Here emphasize that slope is a rate of change:[pic]. For #51, we might right [pic] and say, “Each year the value of the property decreases by $61,000.”

I would then continue with other in context examples, like #52 – 54.

A real world scenario like example 2 on p. 44 is one way to intuitively develop the slope-intercept form of a linear function. Another simpler example might be the following:

For female infants at birth, the median weight is 7 pounds. If the babies grow an average of 1.5 pounds per month, write a mathematical model representing this scenario.

To develop students’ understanding of the slope-intercept form of a line , you might try making a table of values for the babies’ weight (W) as a function of age (m for months).

|Age, m, in months |Weight, W, in pounds |

|0 |[pic] |

|1 |[pic] |

|2 |[pic] |

|3 |[pic] |

|… |… |

|m |[pic] |

Students will (hopefully) see that the coefficient of the independent variable, m, represents the average rate of change and the constant represents the initial value.

Then discuss the y-intercept as initial value and time permitting, develop the procedures for finding intercepts.

Suggested homework: p. 51-52: 1- 23 odd, 33, 35, 38, 39, 43, 45, 47, 51, 52, 53, 54

ASSIGN LINEAR PROJECT

Sec 1.4 Day 1 Equations of Lines (Skill day)

Goals for students:

• Given the slope and y-intercept of a linear function, write its equation

• Given two points, write the equation of the line passing through them

• Write the equation of a line parallel or perpendicular to a given line

• Determine if a given data set is linear (IMPORTANT FOR STUDENT PROJECTS)

You might use the file on the Instructor site entitled “Interpret slope” to strengthen interpretation skills.

Building on what we did in class yesterday, develop students’ skill in writing equations for lines.

You might want to use the file “Data tables” on the instructor site to develop skill in identifying linear data.

Suggested homework: pp. 63-64: 1, 3, 5, 7, 9, 13-23 (all),25, 27, 29, 35, 36.

p. 52: 25, 26

Sec 1.4 Day 2: Equations of Lines

Goals for students:

• Given an initial value and average rate of change in context, write a linear model.

• Given two data points in context, write a linear model.

Continue with the concepts developed over the last two days, except now we are back in context.

Suggested homework: pp. 65-69: 37-47 odd, 48, 55, 57, 63

(Week 3)

Lab Day: Walking Student Lab: Students will be using the CBR2s for this week’s lab.   We currently have (only) 3 working CBRs.  (Michelle is talking to the TI people to repair the 4th one).   Students will need to get in 3 large groups and share the data they collect.  You might consider asking one person from each group to write their data on the whiteboard so that everyone in the group can copy the data.  Each student should write up his/her own lab, though you might want to consider allowing them to work in pairs.  

Mary Lou (in 156 MSB) and Doug (in 108), the two of you will have to share the 3 CBR2s during your Monday 1:10 lab.  One suggestion is to do the main activity together, but let students do the write up individually.  All students will have the same data and their equations should be very similar, if not identical.  Easier grading for you!

 

  I plan to order more CBR2s for next semester.  If you haven’t already done so, please send me a copy of your syllabus so I can expedite the order.  Thanks!

Day 1: Sec. 1.5 Day 1: Algebraic solutions of Linear Equations

Goals for students:

• Solve linear equations (including those with fractions) algebraically

• Understand that the solution to [pic] is the same as the x-intercept of the graph of [pic], which is the same as the zero of the function, f.

• Solve linear equations in context.

Students actually don’t have too much trouble solving equations like those in the skill check, so a quick review would suffice. You will need to carefully develop the connections between solutions, x-intercepts, and zero of functions, however. Try to take a natural approach, perhaps asking such questions as:

If we set [pic] equal to 0, which coordinates are we setting to 0, the x- or the y- ? How does this relate to the graph of the function? (When the y- coordinate is 0, where is the point on the graph?)

In this way, you will review the meaning of function notation (f(x) is the output). You will need, then, to define the term, zero of a function, since it will be new to most students.

Since equation solving is a review for students, you might spend the most time solving the problems in context, like #39 – 66, perhaps asking volunteer students to come to the board.

Suggested homework: pp. 78-81: 1- 19 odd, 8, 12, 39, 41, 43, 45, 47, 49, 51, 53

Day 2: Sec 1.5 Day 2 Equations with only variables; Graphical solutions of Linear Equations

Goals for students:

• Solve a literal equation, i.e. one containing only variables

• Solve a linear equation graphically

Though solving equations including only variables seems natural to us, it is a big step for students. They generally have much trouble with these. I’d do example 7, p. 77 and a few more from other sources. Students need to try ALL of 29 – 34 I the text.

You can build on yesterday’s lesson connecting x-intercepts to solutions of equations, as you introduce the graphical method of solving equations. You have a couple options here. You can use the ZOOM BOX

Feature of the graphing calculator, or use the 2nd TRACE – ZERO method. Please stop by my office if you are uncertain of these calculator techniques. Do use BOTH the x-intercept method and the intersection method, time permitting.

Suggested homework: pp. 79-80: 21 – 34 all; 55, 57, 61, 65

Day 3: Sec 1.6 Fitting Lines to Data Points

Goals for students:

• Use a scatterplot to determine if a linear model is appropriate for a set of data

• Recognize when data is linear by using first differences

• Use the linear regression feature on the graphing calculator (optional)

Be sure students can make a scatterplot on their graphing calculator. Also, reinforce the idea that a linear function has a constant rate of change. These are the two main skills in this section. If you have time, you may want to show them the linear regression feature, but remind them that on the exam, they will be asked to write the equation for a line algebraically. I plan NOT to show them this feature now, but use the time to review for the exam.

There is a review sheet on the instructor website and also on the VISTA site for the exam.

Suggested problems: pp. 94ff: Together (orally): 1 – 7, 19; then 23, 27, 29, 29, 34, 35, 36

Day 4: Exam on Chapter 1. An updated sample is on the Instructor site.

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