At the conclusion of this lesson, the student will be able to:
- Choose between point-slope and slope-intercept forms of linear equations as a starting point, given resources such as slope, one or more points, and intercepts.
- Substitute and simplify chosen form and then convert to standard form.
Subject and ELL Standards
- CA State Algebra Standard 7.0: Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.
- Strategy 5b: Use multiple media to support concepts.
- LCD projector, with remote
- OpenOffice presentation software
- OpenOffice presentation files
- Overhead projector
- Pull-down eraseable coordinate plane
Introduction (Anticipatory Set)
- Post three homework problems on the board (from PDFs supplied by textbook publisher) and ask students to write down, in complete sentences, what the given info is for each problem and what information the problem is asking for. Provide examples of “given info,” recalling earlier lessons’ examples such as y-intercept, slope, a point (or two points).
- Randomly call students (using deck of cards or dice) to read answers. List answers on board, without commentary, under headings of “given info” and “type of answer.”
Teacher Activities (Instruction)
- Quickly review prior lesson’s activity: building a table used to choose between two linear-equation forms and converting to slope-intercept (if in point-slope) and then graphing the result.
- Announce that today’s activity will be very similar, but instead of trying to get all information into slope-intercept, we’ll be converting to standard form. Announce that there will be a short quiz at the end of the activity, and students will be turning in their work.
- Display sample homemade student worksheet on LCD projector. (See illustration.)
- Demonstrate populating first row, using first set of given information: m=-2, b=3. Complete row will look like this:
- Demonstrate second row, using given info: m = -2, (1,3). This time, point-slope will be the initial form of linear equation, but the process of conversion to standard form (after the distribution step) will be reminiscent of row 1.
- Give students two more rows to work on, supplying a slope and intercept for one and a slope and non-intercept point for the other. Guide them through this practice.
- Survey students to recall ideas from previous lessons: What if there is no explicit m (students are given two points)? What if there is no explicit b, but the point contains an x-coordinate of zero? Conversely, what if there is an x-intercept given? How do they choose between initial forms? (Students should say that S-I is preferred, for its ease of graphing.) During these questions, call on students randomly (using dice or deck of cards), giving the class time to consider and discuss answers before rolling, and allowing no “I don’t know” answers. (Students who say “I don’t know” should be given time to discuss the question with neighbors.)
- Give students two more rows to work on independently, supplying a slope and the point (0,1) for the first (students should recognize the number 1 as the y-intercept) and two points for the second (students should realize that they will need to compute m using the slope formula, which they have previously studied).
- Administer quiz: three more questions, on new paper. Students should work alone.
Accommodations for ELL/SDAIE/Exceptional Learners
- Use of LCD projector.
- Use of white board.
- Use of multiple colors of text on LCD presentation; use of multiple colors of marker on white board.
- Speak slowly, clearly.
- Use of gestures/expressions.
- Students build semantic concepts table which leads them through the steps of choosing an initial form of linear equation, substituting the given info, simplifying, and converting to standard form.
- Students use new paper and build table which they use for quiz. After the quiz, students hand in work for grading by teacher.
- Students hand in quizzes for grading by teacher.
The work itself represents a lot of steps for some students, manageable by few. (This is a “foundations” class; I believe many of these students do poorly in algebra because they cannot focus on a problem longer than two steps. This was demonstrated in this exercise, and it’s demonstrated dramatically when solving linear equations: they can do a two-step equation, but when it comes to equations requiring distribution or collecting terms on one side, they seem to run out of steam almost before they even begin.)
Students occasionally needed reminders of tracking negatives and distributing. I also ran into the classic questions about how to handle more than two factors at once. (When point-slope equation has a fractional m, students have been taught to “clear the fraction” by multiplying both sides of the equation by the denominator. This confuses students who are drawn to next-door factors in a construction like this: ½ (x-3) (2). The students don’t see the cancellation of the 2 and the ½ but instead want to distribute one or the other – or both! – of these. Teacher should take care to write the denominator-canceling integer next to the fraction.)
A student who is conscientious failed to plan and ended up squeezing a lot of work into a single line on her lined paper. The teacher should anticipate this and tell students to allow five rows for each problem.
I have trained the class to recognize b (given either as “b=2” or as the point “(0,2)”) and to use this as their criterion for deciding whether to begin with point-slope or slope-intercept. However, some of the students in this exercise failed to recognize the “hidden” b and chose point-slope when they could have worked with slope-intercept instead.
The formatting of these papers represents the typical poor quality I see each day. I’ve spent a lot of time explaining how the papers should look. I may devote a whole day to this in the future. It’s hard to resist telling the students that this work is insulting to me. Since the students can’t all be counted on to actually draw the table, next time I would stop at this point and have students hold up working showing table before proceeding.
Illustration: Projector Slide demonstrating table students are expected to draw and populate.