Algebra: Investigating Positive and Negative Slopes

Posted by David Wetzel

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.

Teaching Algebra - Making Real World Connections

Posted by David Wetzel

Making Real World Connections with Algebra

Teaching Algebra is always a challenge with students, because it is procedural driven and typically taught without any connection to the real-world.

This is why students constantly ask - “I will never use this, so why do I need to learn it?”

This explanation - “You need to learn it, because algebra helps you develop logic thinking skills and employers expect you to have passed an algebra class in high school.” falls on students’ deaf ears.

Algebra needs to be taught in the context of real-life applications so students can develop a better understanding of why and where algebra is needed.

Real World Algebra Problems

Two sample real-world algebra problems are:

“If a batter goes into a game with a seasonal batting average of “S” after a total of “T” times at bat, and gets “K” hits in that game for “N” times at bat, the batters new batting average is determined by this expression:”

“Suppose a roller coaster ride begins by climbing to a height of 50 meters, and then falling rapidly to a height of 10 meters. If you ignore the effects of friction, then as it falls its velocity will be related to its height above the ground by this equation:”

Learning Variables, Patterns, and Functions

Variables is used in algebra for developing expressions and equations, because students need to determine what independent and dependent variables are, for example y = x.

This has real-world connections with science, since students must identify independent and dependent variables when conducting science experiments.

Lets take a look at a sample real-world problem in which students use variables to solve algebra problems.

High school algebra students are fascinated with getting their drivers license to experience the freedom of driving a car.

One thing they do not understand is center of gravity and the fact cars with a high center of gravity will turn over quickly when turning corners at a high rate of speed.

Note: Watch high school students peeling out of their school’s parking lot any day to observe this first hand.

Problem: Which car do you think is likely to roll over in a sharp turn?

Show students pictures of an SUV and sports car, when asking this question.

Then provide them with materials that represent the differences between these two types of cars (different size blocks of wood or cardboard boxes) for them to test.

In this real-world application, students are using algebra concepts related to patterns, functions, and variables.

Students should find the amount of force required to lift the opposite side of the vehicle depends on the width of the vehicle - the greater the width, the more force required.

Graphing the data should yield points which lie roughly on a straight line and pass through the origin (proportional function).

Students then determine the slope of the line to find the formula which describes the relationship - Force required = (slope) × width of vehicle.

Connections with Real-Life

When I took algebra in school, a long time ago, it was boring and I asked the same questions students ask today. All we did was work problems in class after the teacher demonstrated how to solve the problem on the board. I passed, although I hated every minute of algebra class.

Fast forward a couple of decades - Has anything changed? In most cases no! My children learned algebra the same way as I did, except this time they are completely turned off to mathematics.

When algebra is connected with real life situations, it gives students a personal connection. They can draw upon their prior knowledge and life experiences to help make these critical connections.

Everything else is evolving and changing, so should algebra continue to be taught the same way?

Solving Weaknesses in Math Education using Project Based Learning

Posted by David Wetzel

Project Based Learning and Math

Estimates are about 80 percent of Math Education at the K-12 level is strictly focused on solving math problems without any real-world context (Computational Thinking and Math Maturity: Improving Math Education in K-8 Schools, Dave Moursund, 2007). Students’ math learning strategies are narrowly confined to written and mental skills, speed, and accuracy in solving problems.

The weakness of this approach is there is not enough emphasis on placing math problems within a real world context. This is often referred to as the root of students constantly complaining – “Why do I need to learn this, I will never use it!”

Math Teaching Framework

The following is a framework for teaching math for conceptual understanding and making real world connections.

• Math Problems – these comes in form of numerical and word problems for developing a reasonable level of speed and accuracy in performing addition, subtraction, multiplication, and division on integers, decimal fractions, and fractions. This also applies to knowledge of basic algebra, geometry, statistics, probability, and other higher math topics.

• Solving Math Problems – emphasis is on learning course material through rote memorization of steps and procedures in preparation for the next course. This strategy does not prepare students in developing the skills necessary to transfer new math knowledge and skills into other subject areas or into situations requiring the use of math outside the math classroom.

• Real World Math Problems – students need to be taught math is the foundation of product purchases, engineering, culinary arts, medical fields, sciences, geography, sports, and more real world applications.

• Solving Real World Math Problems – the ideal approach for integrating real world math into math classes is through the use of project based learning. Other titles for this type of teaching are problem based learning or inquiry based learning. Whatever you want to call it – students are solving basic math problems with the context of real world math.

Project Based Learning

For teachers new to project based learning, the following are recommendations for helping you integrate this strategy and helping students make math connections to real world applications.

• Start Small – do not try to completely reinvent your teaching strategies and techniques all at once. Integrate project based learning into one chapter per unit at a time. This will help you work out the kinks and become more comfortable in later units.

• Train Your Students – you will often find students may not be used to this style of teaching and learning. Just like you are transforming your teaching strategies, your students need to learn how to transform themselves into self-directed learners, presenters, and problems solvers.

• Use Interdisplinary Projects – work with teachers in other subject areas if possible to create and begin projects. This helps you in getting feedback from colleagues and also helps students make those all important connections outside the classroom.

For teachers who are veterans with project based learning integration, try using the following strategies:

• Mentorship – become a mentor for a colleague who is striking out into the world of project based learning. This helps you by becoming stronger in using this strategy and you also have the opportunity to gain insights into new project based learning activities through this collaboration.

• Technology – try integrating new technological tools, especially Web 2.0 tools, into your students’ projects. Other technology tools include the use of interactive whiteboards, global-positioning-system (GPS) devices, digital still cameras, video cameras, and associated editing equipment.

• Collaborative Projects – if projects do not currently rely on collaboration with other classrooms or schools, try a multi-classroom approach. This can be accomplished by contacting colleagues in your within your school, colleagues in classrooms in other schools in your district or state, or other classrooms in schools in other countries.

Using these strategies help your students make connections to the world outside their classroom and most importantly build bridges to higher levels of learning and math understanding.

The road in mathematics can take many paths; however, solving real world math problems must move beyond its current estimated of only 20 percent application in math today. This approach will alleviate the inherent weaknesses in mathematics teaching and learning today.

Additional Resources

6 Steps to Creating a Project Based Learning Activity

Project Based Learning - Chemistry or Physical Science

Pythagorean Theorem - Real World Context