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# Lesson Study Resources

## Sample lessons from our participants

Here are brief descriptions of and reflections on some of the research lessons created by teams from the Lesson Study Communities in Secondary Mathematics project. These brief descriptions will give you an image of the content and pedagogical issues that our lesson study teams focused on for their work. Full research lesson reports form our participants may be posted at a later date.

 Brookline High School (Team 1) "What makes stairs steep?" Brookline High School (Team 2) "Mystery Triangle Game" Danvers High School Slope as a rate of change Lexington High School "Exploring Position and Velocity: Bicycle Rides." Lexington High School "Hot Peppers: Making an Exponent Ruler." Newton South High School Introduction to Functions Wareham High School (team 1) Solving multi-step linear equations Wareham High School (team 2) Angles formed by parallel lines and a transversal Wareham High School, (team 3) Culminating activity for Exponential Functions Watertown High School Introduction to Slope with Applications Watertown Middle School Area models for probability

## Brookline High School (Team 1), Fall 2002 “What makes stairs steep?”

Our lesson was designed to have students “discover” slope by measuring the height and tread of stairs. Using graph paper, we had students draw their own stairs and then discuss which staircases were steeper. We used Geometer’s Sketchpad to demonstrate their conjectures. We then brainstormed with students how we could measure steepness looking at the ratios of height:tread versus tread:height. We then applied our newly defined way of measuring steepness to other situations.

## Brookline High School (Team 2), Fall 2002 “Mystery Triangle Game”

Our lesson was designed to have students discover what is “sufficient information” to make a triangle that is congruent to a given “mystery” triangle. The teacher would select a mystery triangle, and student teams asked questions about the triangle. After each question, they would try to guess the triangle. We informally introduced students to the terms SSS, SAS, ASA and AAS as they used them.

## Danvers High School, Fall 2002 Slope as a rate of change

In our lesson, we wanted students to be able to make predictions from a graph, to be able to find a rate of change, and to be able to use the term “per” correctly. Our lesson was focused around planning for a long trip. Students had to think about how much of a particular supply they needed to order so that they would not run out too soon.

## Lexington High School, Fall 2002 "Exploring Position and Velocity: Bicycle Rides"

Designed to preface the concept of the derivative in calculus, this 1-2 class period computer-based lab explores the relationship between the position and velocity of a moving object (or, more generally stated, between any function and its rate of change). The lab materials include spreadsheets containing either position or velocity data. Students are asked to draw upon their intuition about velocity, position, and time to supply missing information. Through completion of various spreadsheets and analysis of corresponding Excel graphs, students begin to deduce the story behind the concept of an instantaneous rate of change. Other early calculus issues such as the Intermediate Value Theorem can be explored and the lab can be easily modified to suit a wide range of student abilities. For a look at the student materials and spreadsheets, visit the Lexington High School website at http://lhs.lexingtonma.org/Dept/Math/ldt/.

## Lexington High School, Fall 2002 "Hot Peppers: Making an Exponent Ruler".

An activity that introduces the concept of logarithms by having students construct their own "exponent rulers." Heat values of chili peppers are used as a motivating example. Follow-up questions ask the student to analyze the ruler's properties, making observations that eventually lead to properties of logarithms. The optional sequel lesson "Hot Tamale" introduces 2-dimensional graphing on semi-logarithmic "exponent paper." You can find a brief description at http://lhs.lexingtonma.org/Dept/Math/ldt/ and if you want to see the student materials for the lesson, they are just two clicks away from that page.

## Newton South High School, Fall 2002 Introduction to Functions

Our objective was to create a lesson that would serve as a first introduction to functions. We wanted students to become comfortable with the building blocks of functions including input and output values and rules associating the two. We also wanted to present these ideas as far from the normal mathematical formalism as possible. We presented the students with an interactive game that involved the entire class. For input values we used a deck of playing cards and we used a variety of outputs such as other cards, numbers, and a variety of body motions. Student groups were then asked to design their own game. The student presentations and display of their game rules demonstrated their understanding of our lesson’s objectives as well as forming the groundwork for the more formal introduction to functions, which was to be presented in the next class.

## Wareham High School (team 1), Fall 2002 Solving multi-step linear equations

Our school-wide goals for lesson study were to get students to communicate mathematically, to become independent thinkers and to appreciate alternative learning methods. In this research lesson, we introduced the concept of solving multi-step linear equations using cards and envelopes. The first problem told students there were 3 envelopes with an equal number of cards in each, and one extra card. Altogether, there were seven cards. Students had to figure out how many cards were in each envelope. Discussion of this problem led into a discussion of solving multi-step equations.

## Wareham High School (team 2), Fall 2002 Angles formed by parallel lines and a transversal

Our school-wide goals for lesson study were to get students to communicate mathematically, to become independent thinkers and to appreciate alternative learning methods. We used our research lesson to introduce students to the unique kinds of angles formed by parallel lines when they are crossed by a transversal. We wanted students to discover these relationships on their own, so we had them use a ruler and a protractor to individually discover the congruent and supplementary angles formed by the parallel lines and the transversal. We then summarized their discoveries by introducing the mathematical terms used to describe these angles.

## Wareham High School, (team 3), Fall 2002 Culminating activity for Exponential Functions

Our school-wide goals for lesson study were to get students to communicate mathematically, to become independent thinkers and to appreciate alternative learning methods. In this lesson, we wanted students to demonstrate their understanding of exponential functions by working with real world data and to demonstrate their understanding of the mathematics algebraically, graphically, numerically and verbally.

## Watertown High School, Fall 2002 Introduction to Slope with Applications

This unit will introduce students to the concept of slope and related terminology. Using "real-life" situations, students will engage in predicting future results by drawing conclusions from the data they graph and linear relationships they identify. Students will spend one day experiencing slope in various locations throughout the school, including
the wheelchair ramp, sit-up bench, different staircases, and a treadmill.