# 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), 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.

back to top

## 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.

back to top

## 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.

back to top

## 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/.

back to top

## 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.

back to top

## 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.

back to top

## 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.

back to top

## 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.

back to top

## 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.

back to top

## 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.

back to top

## Watertown Middle School, Fall 2002

Area models for probability

Our group tried to cooperatively plan this lesson to teach students
how to successfully use the area model to find probability. The lesson
is based upon investigation 3.2 from the “What Do You Expect”
unit of CMP (Connected Mathematics.) In this investigation, students
are trying to determine where the treasure might be hidden based upon
the floor plan. As a group, we spent time thinking about the different
ways students could use the floor plan and this led to many ideas about
how students understand or misunderstand the area model of probability.

back to top