# The Elusive Relationship of x & y

“Ali has \$10 dollars and spends \$5 dollars every 2 days.” That was the simple scenario offered for the linear relationship that my students were working with at the time. I was having them write equations from various contexts in order to emphasize the concepts of slope and y-intercept. Boy, did I unravel a serious issue.

Typically, what is asked when dealing with slope-intercept form of lines is somemthing like: “What is the slope and y-intercept of the following equation:  y = -(5/2)x + 10?” Then those values are to be used to graph the equation. The hope commonly is that the students have been paying enough attention to know that the “slope is the number in front of the x.” The frustration commonly is that students still can’t identify the slope from the equation. From a context, the slope is even more difficult to identify, like with the Ali Scenario above. What I found out through a very unique question is why it is so difficult for them.

In the previous contexts that I had offered them, the students would see only two numbers. “Sally has 1 friend and makes 2 new friends every week,” as an example. I was asking them to identify the constant and the rate of change in examples like this. That was a struggle, but they eventually started to get it. When they ran across the Ali Scenaro, I was expecting the three numbers that were representing only two values to cause issues. I was correct. But it was their response to my next question that opened my eyes: “What two quantities are related here?”

I got blank stares. So I asked them to write down the two words that represent the quantities being related in the given context. When I looked at their answers, I was shocked to find the two words that over half the class had written down: spend & has!

Really?! I wrote those two words on the board, and asked them, “Does it make sense that the number of spends that I have determines the number of ‘hasses’ that I have?” I noticed everyone’s attention in class was riveted on the two words on the board. They knew what I was saying didn’t make sense, but they were struggling to reconcile the issue, so I asked another question. “What determines what in this scenario?’ A student finally offered, “days determines dollars.” Yes!

We had to wrap up the day, since I had spent so much of the period mining this one context for mathematical understanding. So the next day, I started the lesson by posting the same Ali Scenario and the two sets of words, “spends & has” and “days & dollars.” We revisited the idea that even though the sentence claims that “Ali spends and Ali has,” we are looking for words that represent numbers. Then we can use those numbers and a few symbols (variables, operations and equal sign), to write an equation (an abstract generalization) to represent the relationship. That equation (like the scenario and eventually the graph) represents an infinite number of combinations of “days & dollars.”

The students now easily identified 10 as the constant and wrote it properly as the y-intercept of the equation. The trouble was in dealing with a slope that was represented by a negative fraction, which was the issue that I originally intended the scenario to pose. However, my adventure into the minds of my Algebra students helped remind me that we too often have them leap from the concrete to the abstract, or that we skip the concrete altogether. How many of the millions of Algebra students in this country are graphing linear equations, yet have no idea that x and y actually represent a quantitative relationship? Focusing on getting the answer correct (“Graph this by starting at this number and counting up and over this number.”) often times bypasses the ultimate goal: mathematical understanding.

# Number Tricks (student sample)

Number Tricks is a lesson that involves writing and simplifying expressions. It demands the higher order thinking skills called for in the Common Core in several ways. 1) The students are to write a mathematical model for a trick given to them. 2) They are to create their own trick and offer the algebraic expression that represents it. 3) It presses the students to understand the concept of a variable; in this case, the variable represents the number originally chosen. 4) The students are asked to compare their simplified expression to the pattern generated by the various numbers tested. The lesson offers a great opportunity for a high level of critical thinking with a rather low level piece of content.

Here is an erroneous submission from my Algebra class. I want to analyze the mistake and discuss why this lesson was so very good for this student even though the “answer was wrong.”

This was Dewey’s response to creating his own Number Trick, including 3 numbers to generate the pattern, and the algebraic expression it represents:

 Pick a number 3 10 -7 x Add 4 7 14 -3 x + 4 Multiply by 2 14 28 -6 2x + 4 Subtract 3 11 25 -9 2x + 4 – 3 Subtract the Original Number 8 15 -2 2x + 4 – 3 – x Simplified: x + 1 Common Result: one more than the number picked

Now of course we can see that the student should have included the parenthesis when multiplying by 2. The final expression should have been:

2(x + 4) – 3 – x, simplified: x + 5

So the positives? The student is showing that he is solid in his operations with negative integers, that he can simplify correctly and that he is interpreting the final expression properly (x + 1 means 1 more than the original number). The Big Negative? The pattern of numbers does not support the students simplified expression. The resultant numbers are NOT one more than the original number; they are 5 more.

My judgement call here was to ask Dewey if his expression matches the pattern. He couldn’t answer right away. There was disconnect between generating the expression and actually knowing what the expression represented. Once I pointed out that the last numbers in each column where not 1 more than the first, I asked him to find his own mistake, which he did. Dewey was then able to correctly simplify and without assistance verify that his new expressions supported the pattern of numbers.

Dewey did a great deal of complex thinking with a topic as simple as simplifying expressions.