Lesson Background and Concepts for Teachers
Begin by bringing back the line from last class that passed between the points (0,3) and (6,5). Remind students that they determined this slope to be 1/3. However, it would be nice to recognize this line beyond just its slope. What would really be great is to create a relationship between the two variables x and y. Remind students that for any given linear function, every value of x has exactly one corresponding y value. For a different linear function, that x value may have a different corresponding y-value. So part of what we need to do is characterize the line such that we give a relationship between x and y, no matter what the numerical values are for that function.
One "special case" that is interesting to start with is the relationship of variables that have direct variation with one another. That is, they follow the relationship y = k x , where k is the "constant of variation." This is seen in the relationship between distance, speed, and time, where speed is held constant as y and x vary directly from one another. Because students are widely familiar with the relationship between speed, distance, and time, have them do a hands-on activity to demonstrate this relationship (see below for the Matching the Motion activity). These relationships found in the activity are important for engineers when analyzing data. For example, mechanical engineers must understand the different relationships between speed, distance and time in order to best and and safely design products such as cars and airplanes. For now, students should know how to determine the constant of variation (which is the slope of the line), given a pair of values (x, y). They also need to know how to determine one value given the other value and the constant of variation. For example, "y varies directly as x. y=4 when x=0.5. What is k? Write an equation. When y=2, what is x?" Sample work is shown below.
The direct variation relationship is really just a special case of the next form of equations students will look at, slope-intercept form. The only difference is that it always has a y-intercept of 0 (crosses at the origin).
The first widely accepted form of an equation is slope-intercept form. For this form, we need to know – you guessed it – the slope of the line and its y-intercept. (The y-intercept is where the line crosses the y-axis.) Slope intercept equations take the following form: y = m x + b , where m is the slope and b is the y-intercept. So if you know the slope of the line, as discussed in our previous lesson, and you know where it crosses the y-axis (the y-intercept), then you can write the equation of the line. For example, y=2x+3 has a slope of 2 and crosses the y-axis at positive 3. Students should be able to take an equation in any other form and convert it to slope-intercept form by re-arranging the equation using properties of equality. An example is shown below.
Students should also be able to move easily between algebraic and graphical representations of the linear relationship.
The second important form of equations is standard form, which is written as Ax + By = C. Students should know that, unlike slope-intercept form, it is more difficult to obtain this form directly from the graph because A, B and C are not the slope, intercepts or any other characteristic of the line. The best way to write this type of equation from a graphed line is to find the line in another form, like slope-intercept form, and then re-arrange the equation so that it is in standard form. he beauty of the standard form of an equation is that one can easily determine x- and y-intercepts from it and use these points to graph. To do this, encourage students to make an x-y table as shown below. Plug in 0 for x and see what y is. Then plug in 0 for y and see what x is. With these two points, students can easily make a graph of the linear function. Students should also be able to convert an equation into standard form.
The third form equation students should know about is point-slope form of a line, which iswhere m is the slope, and (x1,y1) is a point on the line. This form of equation is helpful because if one knows a point on the graph and the slope, s/he can easily graph the line. Students should be able to look at a graph and tell the point-slope form of the line's equation as well as take an equation in point-slope form and draw a graph. Additionally, they need to be able to convert between this form and other forms of equations. It is interesting to note the similarities between this formula and the formula for the slope of a line.
Wrap up the lessons by relating them back to the challenge question. Ask students to get out their sheets of paper or journals where they have been recording ideas. Tell them to write down how this information on forms of equations might help them solve the challenge question.
For homework each night, have students complete the attached worksheets.Provide them with the Forms of Lines handout to study with.
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