Exploring Linear Functions

Name:          Son Thieu

Title of lesson:  Exploring Linear Functions

Date of lesson:  1st Week of 5th Sixth-Weeks- 2nd Lesson.

Length of lesson:  2 Days

Description of the class: 9/10 Grade –Regular/Honor

TEKS addressed:

(b) Foundations for functions: knowledge and skills and performance descriptions.

(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.

(B) The student gathers and records data, or uses data sets, to determine functional (systematic) relationships between quantities.

(D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

(c) Linear functions: knowledge and skills and performance descriptions.

(1) The student understands that linear functions can be represented in different ways and translates among their various representations.

(A) The student determines whether or not given situations can be represented by linear functions.

(C) The student translates among and uses algebraic, tabular, graphical, or verbal descriptions of linear functions.

(2) The student understands the meaning of the slope and intercepts of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations.

(A) The student develops the concept of slope as rate of change and determines slopes from graphs, tables, and algebraic representations.

(B) The student interprets the meaning of slope and intercepts in situations using data, symbolic representations, or graphs.

            I.  Overview

In this lesson, students model linear data in a variety of settings that range from car repair costs to sports to medicine. Students can work alone or in small groups to construct scatter plots, interpret data points and trends, and investigate the notion of line of best fit. Students use linear function to represent real world applications. Students recognize how linear functions are useful in everyday events and generate linear data for these events.

II. Performance or learner outcomes

            Students will be able to:

  • construct scatter plots of two-variable data
  • interpret individual data points and make conclusions about trends in data, especially linear relationships
  • estimate and write equations of lines of best fit

    III. Resources, materials and supplies needed

 

  • Paper and pencil.

 

           IV. Supplementary materials, handouts.

Lesson Resources:

            http://illuminations.nctm.org/index_d.aspx?id=298

Five-E Organization

 Teacher Does                                                                           Student Does

Day 1

Engage:

Learning Experience

Today we are going to do something pretty interesting and fun. We are going to explore real world applications using linear function.

  1. Can someone tell me what does a linear equation looks like?
  2. Who can tell me how a linear graph looks like?
  3. What is a rate of change?
  4. Why do you think linear graphs is a straight line, or what makes it a straight line?

But first of all, we are using the computer to find out how each data is represented linearly and what happen when the values are changed. You’ll have 15 minutes to do the applets.

http://illuminations.nctm.org/tools/tool_detail.aspx?id=82

This applet allows you to investigate a regression line, sometimes known as a "line of best fit."

 

4. Who knows what a line of best fit is?

(Write instructions on board also)

Ø      Click anywhere on the grid to plot points.

 

Ø      To delete a point, move the mouse over the point, hold down the Ctrl key, and press the Delete key.

 

Ø      To move a point, highlight the point with the mouse, hold down the Shift key, and drag the point to a new location.

 

Ø      The x- and y‑axes are set initially from ‑10 to 10. To change the scale of values of x‑min, x‑max, y‑min, and y‑max, enter the number and hit the Set Scale button.

Ø      Click Show Line to display the linear regression line.

 

In the upper left corner, the following values are displayed after Show Line is clicked.

n - The number of points on the graph.

 

r - The correlation coefficient. This measure indicates the association between the x‑variable and the y‑variable. Its absolute value roughly indicates how well the line of best fit approximates the data.

 

y =An equation describing the line of best fit.

 

The Clear button will reset the graph.

 

Hoped for student response

 Answer questions.

1. y = mx + b,

    y = m(x - x 1 ) + y 1 or y - y 1 =   

    m(x - x 1 ), Ax + By + C = 0 or y = (-A/B)x + (-C/B)

 

  1. A straight line.
  2. The slope of the graph, the change in y divided by the corresponding change in x.
  3. Constant of rate change, the slope stays constantly.
  4. A straight line that best represents the data on a scatter plot. 

 

 

 

 

                                                      Evaluate 

Teacher would walk around and monitor students if they are doing the activity correctly.

1. How well does the line approximate the scatter plot?

2. How does the regression line if the points are changed?

 

Explore:

Learning Experience(s)

Pass out activity 1 and activity 2 for students to complete (activities attached).

You will complete these activities individually. Make sure your answers are clearly explained with reasons, not just the answers alone. For each activity, you’ll have 15 minutes to complete. Then we’ll wrap up for the day. For tomorrow, make sure you have your activities and be ready to discuss your answers.

Activity 1: Oil Changes and Engine Repair. The table gives data relating the number of oil changes per year to the cost of car repairs. Plot the data on the grid provided, with the number of oil changes on the horizontal axis. 

 

Activity 2: Bike Weights and Jump Heights. In BMX dirt-bike racing, jumping high or "getting air" depends on many factors: the rider's skill, the angle of the jump, and the weight of the bike. Here are data about the maximum height for various bike weights. Plot (weight, height). If the data are linear, draw a trend or best-fit line.

But first of all, let make sure that everyone’s on the same page.

Ok take a look at both activities,

  1. For activity 1, how would you represent the table using x and y axis?
  2. How about for activity 2?
  3. What are the different types of slope? How do they look like?
  4. How do you find slope of a line?

When working on your activities, make sure you read the directions carefully, which are at the top of the page and answer each questions with reasoning. If anyone of you didn’t finish your activities at the end of class, then your homework for tonight is to complete them. I expect that all of you completed your activities before entering class tomorrow with no exceptions. I will check them as each of you enters my class. Points will be deducted from your grades for incompleteness.

 

 

Hoped for student response

Students will use the laptops to use the java applets.

Work on activity handouts.

      Expected Student Response

  1. x-axis = Oil Changes Per Year, y axis= Cost of Repairs

 

  1. x-axis = Weight, y-axis = height

 

  1. Negative slope-downward from left to right, positive slope-upward from left to right, zero slope-horizontal line, undefined slope-vertical line.

 

  1. Rise/run, the ratio of change in y over change in x, m=(y2-y1)/(x2-x1)

 

      Evaluate 

The teachers will walk around to help students with any question that might arise and to make sure that they are doing their works.

Day 2: 

Explain:

Learning Experience(s)

(Check for students’ completed activities as they enter the class and take proper actions if any activity is incomplete.)

Ask the students to explain any strategy that they found to make linear equations, such as slope, difference forms. Ask students to explain their activity handouts and the significant of each data.

       Questions

Let’s go over activity 1.

1. Are the data linear? Why?

2. What does the slope represents?

3. Find the x- and y-intercepts. Explain in terms of oil changes and engine repairs what each represents.

4. What is the equation of the line?  

5. If you change your oil four times a year, how much can you expect to pay in engine repairs? Explain.

Let’s go over activity 2:

6. What’s the relationship between the bike weight and the jump height?

7. Find the slope or rate of change. What does this mean in words?

8. Predict the maximum height for a bike that weighs 21.5 pounds if all other factors
are held constant.

 

 

 

Hoped for student response

 Answer questions and explain solutions.

      Expected Student Response

1. Answers will be varied.

 

2. A rate of change of the cost of engine repair.

 

3. The y-intercept is about 750. This means that if there are zero oil changes, engine repairs will cost about $750. The x-intercept is about 7.5, which means that a car owner would expect to spend nothing on engine repairs if she changed the oil 7.5 times a year.

 

4. y = -100x +750

 

5. Let x = 4, then y = -100(4) + 750, or y = $350. You can expect to pay about $350 in engine repairs.

 

6. As the weight increases, the height decreases.

 

7. -.150; for every 1-pound increase in weight, the height decreases slightly less than 2/10 of an inch.

 

8. A 21.5-lb bike would be able to jump about 10 inches.

 

 

 

      Evaluate

Extend / Elaborate:

Learning Experience(s)

1. Ask students to identify other real life situations that use linear functions and how the functions are useful in each situation. (Pass out activity 3 as an extension of this lesson.)

Ok, let’s move on to our last activity. Use the rest of the class period to finish this activity.  

The dosage chart was prepared by a drug company for doctors who prescribed Tobramycin, a drug that combats serious bacterial infections such as those in the central nervous system, for life-threatening situations. Plot the data in the graph in the appropriate axis; make sure you label the axis. Draw the line of best fit and answer the questions follow. Same deal as before, answers must accompany with reasons and explanations with no exceptions. Grade will be deducted otherwise.

 

        

 

Hoped for student response

 Answer questions.

1. Answers will be varied.

Students working on the third activity.

      Evaluate 

The teachers will walk around to help students with any question that might arise and to make sure that they are doing their works.