## Math is Tough!

When consulting with a high school in Southern California recently, I taught a lesson on graphing linear equations to Algebra I students. While I was teaching the lesson, I randomly selected non-volunteers to Check for Understanding, and through the student responses I found some very interesting math misconceptions!

The lesson I am sharing with you is a DataWORKS EDI Lesson. This page is referred to as Concept Development. In this lesson component, the teacher defines the concept with a bulletproof definition and then explains some examples and non-examples. In this case, I gave the definition of a linear equation to the students. Then, I gave them examples of a linear equation and a few non-examples. Next, I asked the students to write on their whiteboards which of the following is a linear equation (A, B, or C above). Most of the students selected B, a few students had A and B, and one student had C.

As per the process of Checking for Understanding, I found that the students had many misconceptions that I needed to address during the lesson. When teaching an Explicit Direct Instruction (EDI) lesson, we don’t tell the students the answer, or tell the student that they have the incorrect answer, we guide the students to the correct answer, or redirect their thinking, by having them analyze the attributes in the definition.

## Overgeneralizations

Here are some of the misconceptions that we cleared up during this lesson:

Student Misconceptions During the Lesson |
Effective Feedback |

B is a linear equation because the equation matches the equation in the example. |
T: I want you to look at the definition of a linear equation. Tell me if it says anything about matching to the example.
S: No, it says that all the variables are raised to the 1st power only. T: What are the variables for equation B? S: The variables are x and y. T: Are these variables raised to the 1st power. S: Yeah. Oh I see. Equation A is also a linear equation. |

A cannot be a linear equation because the equation does not match the example. |
T: I want you to look at the definition of a linear equation. Tell me if it says anything about matching the example.
S: No, it says that the variables are raised to the 1st power only. I want to change my answer. T: Which is a linear equation? S: Now, I know that both A and B are linear equations because the variables of x and y are raised to the 1st power only. |

A is a linear equation because this is the one that matches the direction of the line in the graph. |
T: I want you to read the first bullet about graphing a linear equation. What does it say?
S: It says that when a linear equation is graphed it makes a straight line. T: Does it say anything about the direction of the straight line? S: No, the answers are A and B because they both graph as a straight line. |

C is the linear equation because if we remove half of the graph, we have a straight line. |
T: I want you to read the first bullet about graphing a linear equation. What does it say?
S: It says that when a linear equation is graphed it makes a straight line. T: Does it say anything about the direction of the straight line? S: No, the answers are A and B because they both graph as a straight line. |

A cannot be a linear equation because y and x are raised to the 0 power, |
T: Help me understand why you think x and y are raised to the 0 power?
S: X and Y are raised to the 0 power because they do not have anything written on the top of x and y. T: When variables or quantities are raised to the 1st power, we do not put a one over the letter or number. It is understood that it is raised to the power of 1. If it is raised to the power of 0, then we would place a 0 over the letter or number. S: I did not know that. Now I see that both A and B are linear equations because both variables are raised to the 1st power. |

The point outside the line is NOT a solution because it is close to the corner. | T: I want you to go back to the criteria that must be met in order for the point to be a solution?
S: It says that the point must lie on the line. T: Does it say that the point that is NOT a solution must be in the corner? S: No, it doesn’t. It says that the solutions are ALL on the line. The points that are NOT solutions are outside the line. |

The point outside the line is NOT a solution because it is not close enough to the line. | Teacher sees the student quickly changing his answer. Teacher asks the students if he wants to change his answer.
S: Yes, I want to change my answer because I now know that points that are NOT solutions cannot be on the line, but it does not matter if they are close or far away from the line. |

## Debriefing

It was definitely an eye opener for all the teachers when they saw how many misconceptions the students had from this one concept (which had already been taught earlier in the year). Teachers definitely recognized the importance of strategic Checking for Understanding throughout the lesson.

Some of the teachers confided that because of the Pacing Calendar they feel that they cannot spend enough time doing CFU and providing feedback, but after hearing the students’ misconceptions, they stated that they were convinced that it needed to be done.

Great lesson with a tip. Life is a mathematical problem. To pick up the most, you need to realize how to change over negatives into positives.