The Mathematics in Moving Straight Ahead

In this unit, students investigate linear relationships. A relationship is linear if there is a constant rate of change between the two variables. That is, for each unit change in x, there is a constant change in y. The mathematics embedded in this unit are illustrated through the use of many real- world contexts. Several representative examples are used below

Developing the Concept of a Constant Rate or Slope

Example 1   If a car is driven at 50 miles per hour, the distance it travels can be represented by the equation distance = 50 x time or 4 = 50t, where 4 is the distance in miles after t hours.

Such a constant rate of change can be observed as a pattern in a table. As t increases from 0 to 1 hours, d increases from 0 to 50 miles. As t increases from 1 to 2 hours, d increases another 50 miles to 100 miles.

In the symbolic representation, d = 50t, the constant rate of change shows up as the coefficient of t

If we graph the data, the resulting straight line indicates a constant rate of change.

For a rate of change of 60 miles per hour, the line would have a steeper incline, and the constant rate of change would be 60. The equation of this line is d = 60t.

This constant rate of change is called the slope of the line. Slope is the ratio of the vertical change to the horizontal change. In the Comparing and Scaling unit, all of the situations that were studied had a constant rate of change.

To help strengthen students’ understanding of linear situations, they wilt periodically took at nonlinear situations, usually represented in tables or graphs.

Example 2 Which of the following patterns represents a linear relationship?

The pattern in the first table is linear: the constant rate of change in y is 0, and the equation is y = 3. The second pattern is also linear: the constant rate of change in y is 3, and the equation is y = -3x + 1. The third pattern is not linear: as x increases 1 unit, there is not a constant rate of change in y; this pattern can be represented as y=x². Such patterns of change are studied in a later Connected Mathematics unit.

Graphing calculators are used to display graphs of equations, and graphs can be used to describe patterns in a linear relationship. For example, we can see whether the tine is increasing, decreas- ing, or neither, and we can find y given x or vice versa. The graphing calculator is also used to explore families of lines or the effect of changing the slope or the y-intercept. Students can deepen their understanding of constant rates by investigating problems such as the following on a graphing calculator.

Example 3   Investigate equations of the form y = mx. Substitute values for m, and graph each equation. What can you say about the slope m?

From this exercise, several patterns can be seen:

• All the lines pass through the origin (have a y-intercept of 0).
• If m is positive, the line increases from left to right.
• The greater the slope, the steeper the line.
• If m is negative, the line decreases from left to right.
• If m is 0, the line is horizontal.

Developing the Concept of the y-Intercept

Students also work through examples to help develop their understanding of the meaning of the y-intercept. In the next example, the concept of the slope is revisited in a context that introduces the y-intercept.

Example 4   Suppose the cost to rent bikes is $150 plus $10 per bike. Symbolically, we can represent this as C = 150 + 10n, where c is the cost in doflars and ii is the number of bikes.

As there is a constant rate being charged for the bikes, there will be a linear pattern in the rela- tionship between cost and number of bikes. The constant rate of change is $10. We can create a table and a graph for this relationship.

As n increases from 0 to 1, C increases by 10; as n increases from 1 to 2, C increases by 10 again. The graph of the data is a straight line.

The slope of the line is 10. Also, notice that the line does not pass through the origin. It crosses the y-axis (the vertical axis) at $150. This point is called the y-intercept. For 0 bikes, the cost is $150: there is a fixed charge in addition to the charge per bike. The y-intercept is the constant term in the related equation, C = 150+10n.

Students' use of graphing calculators will help them strengthen their understanding of the y-intercept, as in the following activity.

Example 5   Investigate lines of the form y = x + b. Substitute values for b, into the equation, and graph each line.

From this exercise, these things can be seen:

• The lines are all parallel to each other, and they all have a slope of l.
• The lines are parallel to the line y = x. If b is positive, the line y=x moves b units up; if b is negative, the line y = x moves b units down.

Solving an Equation

All three representations—symbolic expressions, tables, and graphs—can be helpful in solving equations. For instance, in example 4 above, we might want to know the cost to rent 75 bikes or to know how many bikes we can rent for $300. In either case, we are solving for one of the variables. We can find this information in several ways:

• • We can read the information from the table or graph.
• • We can reason about the situation verbally: “If it costs $10 per bike, 75 bikes will cost $750 plus the fixed fee of $150 for a total of $900.”
• • We can solve the equation by manipulating the symbols, as done in the next example.

Example 6   In example 4, the equation for the cost to rent bikes was C = 150 + 10n, If the cost is $750, how many bikes were rented?

Students might initially solve this by using a table or graph or by reasoning backward: “First I must subtract the $150 from the cost, which leaves $600. Next I divide by 10 to find the number of bikes, which is 600 ÷ 10 = 60.” This reasoning leads to a ‘balancing the equation” method for solving an equation:

750 = 150 + 10n
750—150=150—150+10n
600 = 10n
600 = 10n
10   10
60 = n

Finding the Equation of a Line Given the Slope and y-Intercept

In this unit, students investigate only linear equations of the form y=mx + b. The slope of the line is m, and the y-intercept is b. Linear situations are represented in words, tables, graphs, and symbols. We can find a symbolic representation in two ways:

• We can translate the verbal situation directly into symbols, as was done in example 4.
• We can find the slope and the y-intercept and substitute them into y = mx + b.

Finding the Slope of a Line Given Two Points

To find the slope of a line, we need to find the constant rate of change. This rate can be found in various ways:

• • The rate can be found directly from the verbal situation (see examples 1 and 4).
• • The rate can be read from a table by noting the constant difference (see example 2).
• • The rate can be determined by finding the ratio of vertical change to horizontal change between two points on a line (see example 7).

Example 7   The points (1, 4) and (3, 10) lie on a line

Finding the y-Intercept

The y-intercept can be found in various ways:

• • The y-intercept can be obtained from the verbal situation (see example 4).
• • The y-intercept can be read from a graph or table. It is the point (0, b).
• • The y-intercept can be found by substituting the slope and one of the points into the equation y = mx + b and solving for b; see example 8. (This method is only briefly explored in this unit. It is considered in more depth in the Thinking with Mathematical Models unit.)

Example 8   To find the y-intercept for the line in example 7, substitute the slope into the equation y = mx + b to obtain y = 3x + b. Since both points lie on the line, the x and y values of the points satisfy the equation. Substituting (1, 4) gives 4 = 3(1) + b. Solving for b gives b = 1. We now substitute for b to obtain the equation of the line going through the two points: y = 3x + 1.