## Giving Students Meaningful Practice with Signed Numbers The purpose of this post is to show how I have used simple equation graphing activities very early in a beginning algebra course to practice basic operations with signed numbers. Students not only practice applying the rules for adding, subtracting, multiplying and dividing signed numbers, they also learn how to plot x-y data pairs, draw graphs of equations, and get a glimpse of future course concepts which include the ideas that 1) equations and graphs of equations describe a relationship between two variables and 2) variables in linear relationships change at a constant or steady rate with respect to each other. After students have learned how to do the basic operations with signed numbers, there is no real reason to wait another two or three chapters in the book to start learning how to graph simple equations. They now have a reason to use what they have just learned.

You can download two free graphing activities by clicking the links below. These activities are examples of the approach presented in this post that you can directly use with your students or as a starting point for making your own activities.

Intro to Graphing Equations 1 (student version)

Intro to Graphing Equations 2 (student version)

🙂 The completed teacher versions of these activities are also available on our free instructional content page under the algebra and pre-calculus tab.

I will begin by describing the setup and parameters for a graphing activity.

1. Each student is provided a handout containing directions for the activity, equations to be graphed, a table of equation input values, and x-y coordinate axes. I prefer lattice point coordinate axes, but grid line coordinate axes are fine. The coordinate axes should be properly labeled and laid out so that it’s easy to plot points.
2. Each student should have a ruler to aid in drawing graphs. Sloppy hand drawn graphs are not allowed. I have found that a 6 inch or 15 cm ruler works best.
3. No ink pens or calculators allowed; strictly old school. Student may use scratch paper of course.
4. The input values for each equation are carefully selected so that most points in the relation are integer pairs, and the key features of the graph are graphed.
5. Even if all values in a table are the same, students are required to write every value in the table; no down arrows to indicate that numeric values continue.

Shown below are sample equations and table setups that I have used to introduce graphing equations to my students. My handouts usually have two or three equations with tables located at the sides and/or bottom of a blank x-y coordinate axes. The tables presented in this post show the output values in red color text. In an actual graphing activity handout for students, the column of output values are blank. This is brand new stuff for beginners, and therefore I do a lot of coaching by explaining what the equations mean, and by doing a few of each type of problem. What is obvious to more advanced students is not so obvious to beginning students.

Tables A, B and C below are similar to tables that I have used to introduce graphing equations. These equations are so simple that some students initially find these equations somewhat difficult to understand; it’s true. As shown below, I like to give intuitive descriptions of the equations.

• Table A: The value of the y-variable always equals the opposite of x.
• Table B: No matter what x equals, the value of y always equals 7.
• Table C: No matter what y equals, the value of x always equals -6. The equations in tables D, E and F give students meaningful practice adding, subtracting and multiplying signed numbers. After filling in the tables, hopefully some students will notice a common pattern in the tables in that the y-variable goes up or down a certain amount whenever the x-variable goes up or down a certain amount. Depending on the class, I might ask students if they see a way to predict the steady rate of change between the variables from the equation. Later in the course, I use similar tables and corresponding graphs to show the relationship between the slopes of parallel lines and perpendicular lines. Intuitive descriptions of the equations D, E and F are shown below.

• Table D: y always equals 2/3 of x plus 4.
• Table E: y always equals 2/3 of x minus 2.
• Table F: y always equals the opposite of 3/2 of x plus 1. The equations in tables G, H and I give students more meaningful practice adding, subtracting and multiplying signed numbers. The equation x + y = 4 requires students to think differently because the y-variable is not explicitly stated in terms of the x-variable. After filling in the table, hopefully students will notice patterns in the tables. Depending on the class, I might discuss the steady rate of change pattern in table G and the symmetry patterns in tables H and I. Intuitive descriptions of the equations G, H and I are shown below.

• Table G: The sum of x and y must always equal 4.
• Table H: y always equals x squared minus 8.
• Table I: y equals the product of the quantities (x + 2) and (x – 4). After filling in the tables of equation x-y data pairs, I decrible how equations should be graphed as follows:

1. For each x-y data pair that fits on the x-y coordinate axes provided, draw a heavy dot at location (x, y). The x-value indicates how many spaces to the left/right of the origin (0, 0) the point is. The y-value indicates how many spaces above/below the x-axis the point is.
2. The points for equations A through G should fall on a straight line. If they don’t, correct the mistake in your table and replot the point.
3. For each of the graphs A through G, use your ruler to draw a line segment through all of the equation points. At each endpoint of the line segment, draw an arrow to indicate that the graph continues forever in both directions.
4. The graphs of equations H and I are named parabolas. If the points don’t fall on a smooth U shaped curve, correct the mistake in your table and replot the points.
5. For graphs H and I, draw a smooth U shaped curve through the x-y data pairs. At each endpoint of the parabolic curve, draw an arrow to indicate that the graph continues forever in both directions.
6. Remind students that they are only plotting a small sample of the infinitely many x-y real number data pairs that satisfy the equation, not just x-y integer pairs.

After drawing the graph of an equation, I show students what the graph of some equations should look like so that they can make corrections before turning in the graphing activity for grading. Shown below are the graphs of equations A through I. I cheated by using my equation graphing program.   I will close this post by mentioning that readers can download blank x-y coordinate axes graphs in either lattice point or grid line format by clicking the links below. These graphs are in JPEG format which makes it easy to paste and resize them to create handouts, tests, presentations, etc.

Blank graph with grid lines (5×5 x-y axes scale)

Blank graph with grid lines (10×10 x-y axes scale)

Blank graph with grid lines (15×15 x-y axes scale)

Blank graph with lattice points (5×5 x-y axes scale)

Blank graph with lattice points (10×10 x-y axes scale)

Blank graph with lattice points (15×15 x-y axes scale)

🙂 These blank graphs are also available on our free instructional content page under the algebra and pre-calculus tab.