More of What Zalman Usiskin Taught Me

equation-transform-pic1In my previous blog, I wrote about how Zalman Usiskin, Director of the University of Chicago School Mathematics Project, showed me a better way to teach slope of a line and linear relationships. Usiskin also demonstrated a better way to teach the equation transformation rules. Over the years, many of my math teachers have given excellent presentations on a variety of math concepts. However, Usiskin’s presentation was sensational. If you have ever attended one of his presentations, you know what I mean.

Many students find the equation transformation rules to be counterintuitive. When we replace every instance of the variable x in an equation with x + 5, most students will guess that the graph is slid 5 units to the right in the positive direction, not 5 units left in the negative direction. When we replace every instance of the variable x in an equation with 2x, most students think that the graph will be stretched horizontally by a factor of 2, not shrunk horizontally by a factor of ½.

Usiskin showed how the solutions of two x-y variable relations compared. The two examples shown below are similar to his. The first shows how we can slide the graph of a circle, and the second shows how we can shrink or stretch the graph of a circle. After he generated a solution of the equation x2 + y2 = 25, Usiskin asked the audience how he generated a solution of the transformed equation. After comparing and checking the solutions of pairs of equations, it quickly became clear why the transformation rules work the way they do. What a cool way to explain the equation transformation rules to my students.

Slide the graph 5 units to the left in the negative direction and up 3 units in the positive direction.

x2 + y2 = 25            (x + 5)2 + (y – 3)2 = 25
(3,4) —————>    (-2, 7)
(-4, 3) ————->    (-9, 6)
(0, -5) ————->    (-5, -2)
(5, 0) ————–>    (0, 3)
(-3, -4) ————>    (-8, -1)

Shrink the graph horizontally by a factor of ½ and stretch the graph vertically by a factor of 3.

x2 + y2 = 25           (2x)2 + (y/3)2 = 25
(3, 4) ————–>    (3/2, 12)
(-4, 3) ————->    (-2, 9)
(0, -5) ————->    (0, -15)
(5, 0) ————–>    (5/2, 0)
(-3, -4) ————>    (-3/2, -12)

Last week I used the above examples to teach the equation transformation rules to my college algebra students. I had a student enter the equations in my Basic Trig Functions program. This allowed me to write on the board and ask/answer questions. While students did not find my presentation sensational, most of them gained a good understanding of how the equation transformation rules work.

You can download my free summary of the Equation Transformation Rules by going to Check out the Basic Trig Functions program by viewing a wide variety of screen shots that demonstrate many features of the program. Equations can be entered as an explicitly defined function of x, an explicitly defined function of y or an implicitly defined relation in the variables x and y. Just enter the equations. The program will figure out the type of equation entered.

What moments of math epiphany have you experienced? What methods have you found effective in teaching math to kids?

Yours in math,
George Johnson

Teaching the Slope of a Line and Linear Relationships: What I Learned from Zalman Usiskin

Blog4PicWhenever I attended conferences where the brilliant math educator, Zalman Usiskin, Director of the University of Chicago School Mathematics Project, was giving a presentation, I always made sure I was near the front of the line to get in. Usiskin had a profound influence on how I viewed the world and approached teaching math.

In one of Usiskin’s presentations, he told about a time when he gave his babysitter a ride home. She explained that she did not understand slope of a line. She just didn’t get it. He asked, “ How much per hour did I pay you?” After a few micro-seconds, she came up with the correct answer. To her surprise, he told her that she had just calculated a slope. In the presentation, as Usiskin went on to explain that all slopes are unit rates, I wondered why I hadn’t thought of that before. The amount that the y-variable increases or decreases when the x-variable increases one unit is the slope of the line. After his talk, I placed much more emphasis on teaching slope as rate, such as cost per part, profit per item sold, weight loss per week, etc.

To help teachers with the concept of a slope of a line and linear relationships, you can download my free Linear Growth and Decay handout by going to The first example in this handout is about predicting gross pay given the number of parts produced in 8 hours. Students seem to have no problem understanding how I derive the equation P(n) = 0.35n + 60. When I ask them why 0.35 in the equation should be no surprise, I’m still amazed that many of them don’t make the connection that the piece work rate stated in the problem was 35 cents per part. But after more coaching, almost all students understand the concept.

Two short examples:

  • The first four digits of the square root of 3 = 1.732. What is special about the year 1732?

Answer: George Washington was born.

  • The first ten digits of the irrational number e = 2.718281828. My son’s significant other remembers e as 2.7 Andrew Jackson Andrew Jackson. I googled Andrew Jackson and found out that he was elected President in 1828 and served from 1829 – 1837. It worked for her.

I’ll have more moments of epiphany for you in my next blog. Feel free to share your stories about how teachers have added to your understanding of math concepts.

All the best,

George Johnson

photo courtesy of Used by permission.