Stone 1
Robert Stone
Dr. Hodges
EDSE 550
June 29, 2014
Evidence of Effective Mathematics Instruction
My lesson plan on inverse variation serves as evidence of my knowledge of effective mathematics instruction: what students know, and what and how students need to learn mathematics. In terms of the actual content of the lesson, it aligns with the following Common
Core State Standard:
“8.F.3 Interpret the equation y = mx+ b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points
(1,1), (2,4) and (3,9), which are not on a straight line.” (Common Core State Standards)
In the time before the lesson starts, the entire class reviewed what the students knew about linear equations. After this, I acknowledged that they would be learning about another type of equation that isn’t linear through an activity. I presented this to the students in a way that showed the students there was something new to learn, but that I wasn’t going to feed them the information; they would be tasked to discover it on their own. This aligns with what Principles to Actions says about goals and how “goals or essential questions motivate learning when students perceive the goal as challenging but attainable” (13).

Stone 2
In my lesson students were given sets of cut out rectangles and were told that they needed to place them in order from shortest width to longest width. As the task continued, students eventually realized the activity was a physical and concrete model of inverse variation.
According to the Leitze and Kitt article, “one way to teach for understanding is to use concrete models to introduce concepts rather than concentrate only on the abstract or symbolic”
(462). These rectangle manipulatives that were given to students were just one representation of inverse variation that students got to see in the activity. Students got to see the concept in a table they filled out with the width and lengths of the rectangles. Using the table, students had to create a graph of the data. From here students had to write a narrative describing what was occurring in the data and graph. Then using all the information they had discovered, students were tasked to develop an equation. This gave the students multiple representations of the concept and according to Principles to Actions “students’ understanding is deepened through discussion of similarities among representations that reveal underlying mathematical structures”
(26). These multiple representations also built on each other to show the students the concept of inverse variation.
This building on each other didn’t happen by accident. These representations were scaffolded for the purpose of students working up to the goal of the lesson. This is effective method for teaching a concept because according to Holiday and Duff, “an established strategy for reaching [self-reliant students] is teacher scaffolding of student learning” (328). When scaffolding, teachers must also provide similar problems. Although the first version I created of this activity didn’t have similar problems about inverse equations, I would add similar problems in the revised version. An example problem that I may add would be an equation where the students must use the information they learned about inverse variation to determine if the

Stone 3 equation is an example of inverse variation or not. The activity sheet also asks students to take their own example to create their own generalized rule for creating inverse variation equations.
This question is good as it makes students apply what they learned to create a concept.
Teachers also create good questions that help promote reasoning and problem solving by
“selecting tasks that provide multiple entry points through use of varied tools and representations” (24), according to
Principles to Actions . Again, in the revised version of my lesson plan I would add these types of questions. For example, I could have students choose if they wanted to do work that started hard and declined inversely or started medium and declined linearly, both doing the same amount of work overall in the same amount of time, just spread out at different increments of time. This question could potentially lead into or be referenced back to when talking about mean, median, and mode.
In the Five Practices For Orchestrating Productive Mathematics Discussions, it states
“the five practices were designed to help teachers to use students’ responses to advance the mathematical understanding of the class as a whole” (Smith and Stein 7). These five practices are anticipating, monitoring, selecting, sequencing, and connecting. The lesson plan I created used these five practices, but could be used more in a revised version of the plan. In the lesson plan there are anticipated answers and questions of students listed. This gives me the opportunity to be able to have responses prepared for when students produce the answers or questions. The lesson I created also leaves time where the students work on the activity in groups without direct instruction, which lets me, as a teacher, monitor and observe the various student answers. Some of the prepared questions that are on the activity sheet ask the students to explain and describe what they see visually, as well as connections to other representations. Since there is time for me to walk around and observe what students are putting for their responses, I can go

Stone 4 ahead and silently pre-select who should present their answers in an order so that the responses build on each other. This covers the last three practices with me selecting answers, sequencing them in a certain order and connecting responses to other students’ responses. I would also utilize the connecting practice by connecting similarities and differences between their answers and elements of linear equations.
This chosen artifact was also created in the middle of a course, thus leaving plenty of room for improvement. The fact that I was able use my now improved experience to make revised additions to improve this lesson also serves as evidence of my knowledge of effective teaching. I feel it serves as evidence because students change from year to year, so my lessons need to adapt and change to fit my ever-changing students.

Works Cited
CCSSTOOLBOX.COM." CCSSTOOLBOX.COM
. N.p., n.d. Web. 29 June 2014.
Holliday, Berchie, and Lauren Duff. "Using Graphing Calculators to Model Real-World
Data." Mathematics Teacher : n. page. Print.
Stone 5
Leitze, Annette, and Nancy Kitt. "Using Homemade Algebra Tiles to Develop Algebra and
PreAlgebra Concepts." Mathematics Teacher 96: n. page. Print.
Principles to Actions: Ensuring Mathematical Success for All . Reston, VA: The National
Council of Teachers of Mathematics , 2014. Print.
Smith, Margaret Schwan, and Mary Kay Stein. Five Practices for Orchestrating Productive
Mathematics Discussions . Reston, VA: National Council of Teachers of Mathematics;
2011. Print.