Apr

Algebra: Investigating Positive and Negative Slopes

Learning algebra is difficult for most students. Ask any student what they are learning in algebra and you will probably receive an answer similar to this (after - Why do I need to know this?).

“Using Xs and Ys a lot, a bunch of numbers and symbols, and memorizing a lot of rules.”

This often comes with learning algebra without connection to anything students can relate to. When students learn basic math, they can make a lot of connections.

For example - If the sales tax is 10%. How much tax will you pay when you buy an new MP3 player that costs $135.00?

Slope: A Definition

Lets take a look at another concept students study in algebra. The properties of lines such as slope, y-intercept, and their interactions.

This is a definition they typically find in an algebra textbook:

“The slope is defined as the ratio of the rise divided by the run between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line.”

Once they are able to get beyond the definition and understand how rise and run are connected to the term slope, then we start throwing positive and negative slope at them.

Equations: Negative and Positive Slopes

Start with the following graph to help students develop and understanding of negative slopes.

Now ask the students to find the slope for each line.

For the Green Line: y = 6 and x =6

For the Purple Line: y = 2 and x = 10

Now ask the students to find an equation for each slope.

y = rise/run times axis crossing point

y = -6/6 x 6

y = -2/10 x 2

Use the following graph to help students develop and understanding of positive slopes.

Now ask the students to find the slope for each line in the second graph.

For the Green Line: y = 0 and x = 10

For the Purple Line: y = 0 and x = 8

Now ask the students to find an equation for each slope.

y = 10/4 x 0

y = 8/6 x 0

This helps students with a fundamental concept:

“When lines go down from left to right this is a negative slope and when lines climb from left this a positive slope.”

Extension: Opportunity for Students to Demonstrate True Understanding

Now lets provide students with an opportunity to demonstrate true understanding with the following graph.

Now ask the students to find the slope for each line on this third graph.

From Quadrant I to Quadrant II: The “y” intercepts actually don’t change, and neither do the ‘run’ values. In fact the only thing different is that now the lines go up (positive slope). These five equations are the same as the first quadrant with the sign of the slope changed.

From Quadrant I to Quadrant III: The signs of the slope now stay the same (both negative), they can form the 3rd quadrant equations by simply changing the sign of the “y” intercept.

From Quadrant I to Quadrant IV: They need to change both the sign of the slopes and the “y” intercepts.

After completing these exercises the students should be more comfortable with translating equations of lines into corresponding lines on graph paper and vice-a-versa, along with using the slope-intercept form of linear equations.

Real World Applications: Additional Extension Activities

Additional extensions for finding the slope: students apply these concepts to a slide on a playground, schools stairs, non-flat drive ways, football field stands, any area of the school grounds which has a rolling or hilly terrain, etc.