Introduction to the Slopes of Lines in Coordinate Geometry:
A Constructivist Lesson

Paper written for Dr. Samson Nashon, ETEC 530, April 2006

Topic

Introduction to slopes of lines in coordinate geometry

Grade/level

Grade 10 Principles of Mathematics, Province of British Columbia

Rationale and Purpose

This lesson uses White and Gunstone’s (1992) Prediction – Observation – Explanation (POE) model, as well as adhering to the stages of constructivist teaching described by Driver and Oldham (1986; in Matthews, 1994). The stages are Orientation; Elicitation; Restructuring of Ideas; Application of Ideas; and Review; and these will form the framework used in the “Instructional Strategies and Activities” section below. However, it is noted that the Driver and Oldham (1986) “Review” stage involves asking the students to make comparisons between their thinking at the beginning of the lesson and the end of the lesson, but this stage is encompassed by the Explanation phase of the POE model, which will take precedence in this lesson. Instead, the Review stage of this lesson invites the students to think back on their learning in order to enhance the use of metacognitive strategies.

The Prediction – Observation – Explanation (POE) model can be easily used in activities that allow students to observe the results of an activity or experiment, and thus lends itself to studies in science and mathematics. Before the demonstration or activity, students must make a written prediction and provide reasons for the predictions based on concepts and information that they already know. Students then observe the results of the demonstration or activity, and must provide an explanation that reconciles any differences between what they predicted and what they observed. The added value in this lesson is that the students themselves create the observable situation by producing their own graphs.

This lesson also conforms to the criteria set out by the Learning Resources Unit at BCIT (2003) for “Best practices in constructivist e-learning” (see p. iv). Learners are able to construct their knowledge by relating prior conceptions to new knowledge and using information to produce and analyse graphs. The focus of the lesson is on process, not product, as students are encouraged and evaluated throughout the lesson, not just at the end, and not by producing one “correct” answer. Multiple perspectives are encouraged by providing the learners with opportunities to work as part of a team. Situated cognition is addressed by having the students apply their new knowledge of slopes to plan the design of some playground equipment. Reflexive cognition is encouraged through the use of the POE model and the inclusion of self and peer evaluations at the end of the lesson. Modelling and coaching by the instructor address the issue of cognitive apprenticeship, and finally, process-based evaluation is attended to in two ways: by assessing the students’ abilities to graph a line and name the slope, and by including multiple perspectives in the evaluation process (instructor, student and peers).

Objectives

This lesson addresses the following learning outcomes for the grade 10 British Columbia Principles of Mathematics stream:

Resources

Students need graph paper and a pre-recorded CD which has video, still picture and word processing files. The video files give demonstrations of the steps involved in working through some of the lessons. Students do not need to refer to these files unless they need guidance in completing some of the exercises. Still picture files will be used to show diagrams (such as graphs) so that students can check their answers to some of the problems and make sure they are completing questions correctly. Word processing files contain templates in Rich Text Format (RTF) for reporting results in some of the discussion forums so that student responses are complete and consistent.

Students and instructor need access to a Learning Management System (LMS) that allows both synchronous and asynchronous discussions and includes an audio chat feature. Word processing and “paint” programs are also required (the programs included with the computer’s operating system are adequate).

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Instructional strategies and activities

Orientation

Students will read a short introduction to graphing lines at the beginning of their online lesson. The introduction will stress the importance of being able to graph lines, and note that there are several characteristics that students will need to be able to identify when given an equation for a line. Students will be given the learning outcomes for the lesson, including how they are situated within the whole math program, and how other lessons build on the successful mastering of the learning outcomes in this lesson. Students will also be reminded of the POE format for the lesson (which would have been introduced earlier in the course).
(5 minutes)

Elicitation

For prior knowledge, in synchronous audio chat mode the instructor will ask the students to consider what they know about the word "slope". It has all sorts of meanings outside math. The students will also be provided with an exercise on plotting pairs of points in coordinate geometry as a review if necessary.
(5 minutes)

The instructor will then ask the students to locate a diagram from the CD of four lines drawn in different colours with four randomly given numbers that match the slopes of the lines. Students must make a prediction about which number belongs to which line, and provide reasons. They will be directed to post their predictions in a discussion forum.
(5 minutes)

Restructuring of ideas

Once all of the students have registered their predictions, they will then move on to the next page in their lesson. It will provide four to six groups of ordered pairs which form lines that the students have to graph. All lines will run through the origin; only the slopes of the lines will differ. (Note that this begins the observation phase, as the students must construct their lines in order to observe the results of the activity).

Working in pairs or groups of three, students will divide up the groups of ordered pairs among (or between) them so that no one student has to graph all of the lines. Each group will be assigned to a thread in a discussion forum, and one student will be made responsible for summarizing the group’s conclusions.

After they graph the lines, students will be asked to look at the ordered pairs and describe in words the relationship between the x and y coordinates (for example, the relationship “the y-coordinate is twice the x-coordinate” leads to y = 2x). From there they will develop the standard form of the equation of a line: y = mx + b (however, b will be zero because the lines run through the origin). Students who are struggling with this exercise can discuss it with group mates in synchronous format or view a short video from the CD that will take them through the steps needed to complete the activity. From the resulting y = mx, students should be able to see that the value of m represents the slope (this is the end of the observation phase).
(15 minutes)

At this point, if time permits, students could also use an online graphing calculator such as GCalc (Kim, 2005). With this calculator, students must enter the equation of the line (e.g., y = 2x is entered as 2x), and the program uses a Java applet to draw the graph. Students can take a screenshot of the graphs that they have drawn, paste them into a Paint program, save them as jpeg files and them attach them to their discussion messages. If the instructor chooses to work with a program such as Whiteboard within WebCT, the picture files can also be pasted there.
(10 minutes – optional)

Once they have drawn their graphs, students must post in their small group discussion thread a description of each graph that they have drawn. The following format will be available on the CD as a word processing file so that the student can complete the record and then cut and paste the information to the discussion forum.

Observation and Explanation Record

Graph #______

Equation of the graph: y = ___________

The slope of this graph is ____________ (a number)

Describe the steepness of the slope of the graph (you can use words like steep, flat, very, almost medium, etc.) ___________________________________

This graph ____________ (rises/falls) to the right. Rising to the right means it goes up (the y-value gets higher) as you look from left to right, or as you go from -x to +x. Falling to the right means that it goes down (the y-value gets smaller) as you look from left to right.

This graph appears to go through the following points: (give two points other than the ones that you have already graphed.) ______________

Answer the following questions: (Explanation phase)

What can you conclude about the slope of the line and the number that multiplies the x?

Look back at the first exercise where you predicted the slopes of the lines. Were your predictions correct? If so, why do you think you were able to predict correctly, and if not, why do you think you were wrong?

Application of ideas

Students can work in pairs or individually to complete the Application Task, as follows: On graph paper and using a scale of 1 cm = 1 m, design a slide (playground equipment) for a 3-year old, a 9-year-old or a 15-year-old. Use a line segment running through the origin of the graph as the run of the slide. Give the equation of the line, the slope and the endpoints of the line segment (this will tell us how long your slide is). Use Gcalc to plot your graph, do a screenshot, transfer to a Paint program and save as a jpeg image. Justify your choice of slope and length of slide by referring to the age of the rider. Post your results (including your jpeg graph image) in the “Slide” discussion forum.
(10 minutes)

Review

As the final part of the lesson, students will fill out a short questionnaire in order to facilitate their use of metacognitive strategies and provide a self and peer evaluation. The document template will be available on the CD, and students will email it to their instructor immediately after the lesson is finished.

Samples questions for the lesson review:

Assessment strategies

This lesson will be marked out of 60 marks. Students will earn marks for posting (participation), completeness and correctness of answers or strategies, and for peer and self-evaluation at the end of the lesson. Students will be given marks for the following:

References

Learning Resources Unit at BCIT (2003). Contructivist e-learning methodologies: A Module development guide. Pan-Canadian Health Informatics Collaboratory.

Kim, J. (2005). Online Graphing Calculator. Retrieved March 28, 2006, from http://gcalc.net/

Matthews, M. R. (1994). Constructivism and science education (Chapter 7). Science Teaching: The Role of History and Philosophy of Science. New York: Routledge.

White, R. & Gunstone, R. (1992). Prediction – Observation – Explanation (Chapter 3). Probing Understanding. Philadelphia, PA: The Falmer Press.

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