## Sample Lessons

### Lesson Plan 1: Use Your Noodle

- Grade Level
- 5-8
- Central Topic/Theme:
- Scatter plots and introduction to regression lines
- Estimated Time
- Full version

#### Overview

This lesson is to introduce the idea of Best Fit Line (regression lines) to student. Students learn about positive and negative correlations and how a line can help make estimations/ predictions about missing data. Students also get practice graphing and interpreting their work by using both graphic and symbolic representations to represent the relationship between two variables. Students can work in groups or individually.

#### Prerequisite Concepts

- Familiarity with coordinate plane
- Graphing
- Slope
- Plotting points
- Ratios
- Proportions
- Measures of central tendency
- (Equation of a line)

#### Standards / Benchmarks Addressed

- Data Analysis and Probability: Select and use appropriate structures using symbols
- Representations: Use representations to model and interpret physical social and mathematical phenomena

#### Connections to Other Topics / Disciplines

- Lesson is preliminary step to regression lines, and interpreting data.
- Related to science data (volcanoes, earthquake predictions)
- An
*actuary*is a statistical expert who determines the financial impact of randomly occurring events like earthquakes, injuries from an accident, even death. Actuaries work primarily in the insurance industry and for state and federal government agencies. Actuaries use math and statistics to determine the probability of major events in a geographic area (like an earthquake in central California). T

#### Objectives

- Graph, interpret and analyze data
- Investigate the relationship between two variables in a problem
- Use graphic and symbolic representation to represent the relationship between two variables.
- Use the concept of best fit line to describe the relationship between two variables
- Estimate missing data/ variable based on line of best fit.

#### Materials

Describe the materials needed to present and conduct the lesson, including any special materials (overhead projector, hand outs, pencils, laptop, spaghetti noodles, tape measurer/rulers, graphing paper, a basketball.)

#### Engagement Activity

- Teacher hold up a drawing or replica of a 41 cm shoe in real size. And ask students questions.
- What does this look like? Yes, it is a size 20 men’s shoe. I WONDER how big this person is? Who would wear a shoe this size? Shaquille O’Neal!
- Teacher shows another picture of his/her own hand and foot.
- This is my hand and foot. So you think that Shaq has big or small hands? How much could it wrap around this basketball?
- We aren’t sure of the size of his hands, so we are going to investigate this by using a graph.

#### Instructional Plan

- Good Afternoon , please get started on the do now you have 2 mins. Done? Prepare notebook for lesson.
- Can I have a volunteer to read the answers to number 2 and a second volunteer for number 3.
- ANY QUESTIONS?
- The reason I wanted you to think about what this graph looks is because we will be working with this idea more today.
- We are also going to be working with some concepts from our previous units when we worked with ratios and proportions & talking about something new Best Fit Line is related to them.

#### ***Anticipatory Set aka Engagement Activity said here.

Lesson:

- Teacher writes on board
**scatter plot**- a graph that relates data from two different sets.**line of best fit**(trend line) - A line on a scatter plot which can be drawn near the points to more clearly show the trend between two sets of data.

- Let’s look at some examples on the over head: Keep in mind these examples are graphed and the dataset is given to you. Sometimes you will collect your own data
- What do you notice about these graphs? Teacher and students discuss graphs. HOW ARE THESE GRAPHS DIFFERENT?
- Teacher writes more notes on board:
- The line of best that rises quickly from left to right is called a
**positive correlation**. - The line of best that falls down quickly from left to the right is called a
**negative correlation** **Strong**positive and negative correlations have data points very close to the line of best fit.**Weak**positive and negative correlations have data points that are not clustered near or on the line of best fit.- Data points that are not close to the line of best fit are called
**outliers**. - (optional) Notice a scatter plot with a
**strong correlation****Weak correlations**have data points that are further from the line of best fit.

- The line of best that rises quickly from left to right is called a
- Now if I wanted to place my noodle close to all the points where would I put it? Let’s consider a line of best fit for the data below.
- Teacher can ask volunteers to help at projector. “A Line of Best Fit is a straight line that relates to data plots on a graph. Where would you put it?
- Yes, lets look at this data which is done. Just what you had.
- What does it mean? It is a prediction of missing data. So if I wanted to know how high a bike of 21.5 lbs can likely jump I look at the graph. I don’t have an exact data point but a likely guess.
- How could the line of best fit be helpful to us? It helps us to make predictions. We don’t have data for what happened here because we don’t have all these scatter plots, … but we have the line and that’s helping us predict our
- Teacher writes the following on board:
#### Predicting:

- If you are looking for values that fall within the plotted values, you are
*interpolating.* - If you are looking for values that fall outside the plotted values, you are
*extrapolating.***Be careful**when extrapolating. The further away from the plotted values you go, the less reliable is your prediction.

- If you are looking for values that fall within the plotted values, you are
- NOW let’s extrapolate data to see the likely size of Shaq’s hand. Teacher passes out noodles, graph paper, and hand out and has students do activity as a group. Each student must turn in separate sheet. Or divide task and work in group of 4 or 4.

#### Accommodations

- Students will work in pairs and may collaborate to accomplish the task in end.
- Many visual aids will be used to help ESOL students make meaning from the written and spoken parts of lesson.

#### Sustaining, Concluding, or Extending Activities

- Count down from 10 to gain attention
- Ok some students have extrapolated would you like to share your findings?
- Discuss answers and validity.
- Can also use basketball and answer UB quick finisher question.

#### Evaluation and Assessment

A formative assessment will take place during the lesson where the teacher will continually monitor the progress of the students in an informal manner. This will take place by asking them questions throughout the lesson to keep them actively engaged.

- I will know that students can graph, interpret and analyze data if they are able to complete their assignment sheet.
- I will know students can estimate missing data/ variable based on line of best fit, investigate the relationship between two variables in a problem and use the concept of best fit line to describe the relationship between two variables if they find a close answer to Shaq’s hand.
- I will know students can use graphic and symbolic representation to represent the relationship between two variables if they graph the data on their assignment sheet.

#### Reflection

I don’t know if the lesson will have the flow I want it to have because the black board is right in front of the projector. I need to be able to write on the chalkboard and walk over to the projector with out too much hassle, but that may not be possible in this class. I think this is a lesson that addresses NCTM standards and would also be interesting to a wide range on individuals students and pre-service teacher alike. I think this lesson has the right balance of engagement and instruction to make it interesting and not boring.

#### References

Constructing a best fit line. (n.d.). *SERC*.

Retrieved September 30, 2010, from http://serc.carleton.edu/mathyouneed/bestfit.html

### Lesson Plan 2: Adding It All Up

- Grade Level
- 8
- Central Topic / Theme
- Exploring interior angles of a polygon
- Estimated Time
- 45-60 mins (for longer periods extension activities should be utilized or more sharing should be encouraged)

#### Overview

In this lesson, students use various polygons to investigate their interior angles. The investigation can be done using both an applet and paper and pencil to foster an understanding of how different patterns can lead to solutions. After comparing results with a partner, students develop a formula showing the relationship between the number of sides of a polygon and the sum of the interior angles. Students are guided to use triangulation to build upon previous knowledge. This lesson heavily reinforces pre requisite concepts while introduces a new one.

#### Prerequisite Concepts

- Basic geometry (points, lines, angles, regular shapes, and non-regular shapes)
- Polygons (convex, concave)
- Complementary and supplementary angles
- Interior and exterior angles
- Vertical angles
- Sum of interior angles of triangles and squares
- Familiarity with circles & degrees if circles

#### Standards / Benchmarks Addressed

##### NY State Standards

- 8.G.1 Identify pairs of vertical angles as congruent
- 8.G.2 Identify pairs of supplementary and complementary angles
- 8.G.3 Calculate the missing angle in a supplementary or complementary pair
- 8.G.6 Calculate the missing angle measurements when given two intersecting lines and an angle
- G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts
- G.R.1 Investigate relationships between different representations and their impact on a given problem

##### NCTM Standards

- Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects
- Draw geometric objects with specified properties, such as side lengths or angle measures.

- Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.

#### Connections to Other Topics / Disciplines

- Lesson is preliminary step to formal proofs
- Various discovered formulas can be checked using cross multiplication principle [ ie. n(180) -360/n = (n-2)180/n ]
- Emphasizes geometric properties and provides experience as is recommended by Van Heilis. Students can take steps in moving up in levels by understanding properties of geometric shapes and visualizing how their manipulation effects various concepts and why.
- Has applications in engineering and architecture --used for designs, bridges, buildings and sporting facilities. Carpenters use interior and exterior angles to make things. Artists use their knowledge of interior and exterior angles to make tessellations as is the signature of much Middle Eastern art.

#### Objectives

By the end of a lesson a student should be able to:

- Investigate the pattern between the number of sides of a polygon and the sum of the interior angles using provided methods
- Explore interior angles in regular polygons
- Create a formula to find the interior angle sum given the number of sides
- Determine that the interior angle sum is always the same for polygons with the same number of sides

#### Materials

- Warm up activity sheet (see attached)
- Chalk board & chalk
- 3x5 index cards
- Compasses & Rulers
**or**Preprinted shapes - Unlined paper (optional)
- Colored pencils or markers (optional)
- Adding It All Up Activity Sheet (http://illuminations.nctm.org/Lessons/AddingItAllUp/AddingItAllUp-AS.pdf)
- Adding It All Up Answer Key (http://illuminations.nctm.org/Lessons/AddingItAllUp/AddingItAllUp-AK.pdf)

If extension activity is used:

- Computers with Internet access
- Angle Sum Applet (http://illuminations.nctm.org/ActivityDetail.aspx?ID=9)

#### Engagement Activity

The warm up activity asks students to connect vocabulary words (polygon, convex, concave, regular, triangulation) to definitions and identify a favorite character or friend.

**Teacher asks students to start the provided warm up activity****Teacher asks volunteers to read answers: “Who thinks they have #1 correct? How about #2?...”****Teacher gets to #5 and says, “Well some of you may have guessed, today we will be working with this new concept-- TRIANGULATION” (don’t elaborate on this yet continue to HOOK students).****Teacher continues “But before we get into that let’s do a quick wrap around and tell me your favorite cartoon character, movie star or friend”.****Teacher says, “Ok awesome, I like a lot of them too. Now suppose a robot spell has been cast upon your character and now he/she is trapped in a regular polygon. In order to get out he/she can only walk along the sides and turn if you give then angle measurements in degrees. How would your character get out of the polygon?”****Talk to your neighbor and discuss what information you would need to know****Teacher ask for students answers “ and says, “YES, and we are going to find out how to calculate interior angles of regular polygons when they are not given and use what we learned about supplemental angles to be able to save our characters****”.**

#### Instructional Plan

Teacher attempts to guide students through lesson and pushes for students to answer and justify all questions.

- Students are given a triangle and asked to start at one point and navigate their character back to the same point.
- Students are asked to show their work (students can work in pairs).
- After a few mins, teacher asks a student to come to the board and show what they did.
- Class discusses process.
- Students are then given a square and asked to start at one point and navigate their character back to the same point
- Students are asked to show their work (students can work in pairs).
- After a few mins, teacher asks a student to come to the board and show what they did.
- Class discusses process.
- Teacher says, “But what if we had a regular pentagon, a regular hexagon, or some other regular shape, how many degrees are each angle? We need to know so we will know how much to turn”.
- Students are asked to investigate this problem with the handouts provided.
- Shapes are also provided (students could also be asked to make their own regular shapes.
- Teacher pauses and discusses answers as a class or guides students as they work in their groups
- Upon completing this lesson teacher can have student’s read their answers for the questions if this was not done in the section above.
- Teacher then passes out the game sheet and challenges students to have their characters pass through the maze (students can work in pairs).

Note:

- Teacher is to serve as a guide and facilitator
- Teacher is expected to acknowledge all answers with praise to promote a safe learning environment.
- Students should provide answers & teacher should challenge student with “why” and “how do you know” questions and reiterate correct answers.

#### Differentiated Instructional Strategies

- Many visual aids will be used to help ESOL students make meaning from the written and spoken parts of lesson.
- Students, who need to refer back to previous lessons, should have their notebooks available to them to refer to prior lessons if needed.
- Students can work in pairs and may collaborate to accomplish the task in end. Teacher can chose to chose seating arrangements based on ability if determined to be helpful to students.
- Game activity is distributed based on ability; each child can be further challenged by being asked to “go another route in their game. Games can also be traded so students can gain exposure to different and more challenging material.
- Extension activity can be designated for advanced students and quick finishers or completed as a group based on time and computers availability.
- Additional differentiation strategies available in assessment section

#### Sustaining, Concluding, or Extending Activities

Teacher will close by reintegrating the main point learned—how to find interior angles of a regular polygon. Then students will write in their journals about the experience. Extension activities should be done as a class if entire class gets through required materials in a timely fashion. Discovery is important so material should not be rushed.

- Concluding:
- Teacher emphasizes triangulation, how to find interior angles of regular polygons, mentions formula, and the fact that the sum of angles in convex or concave polygons don’t change.
- Teacher has students write in their journals about the topics above.

- Extension
- Have students explore exterior angles. The sum of the exterior angles for any polygon is 360°, and therefore the measure of one exterior angle in a regular polygon is 360/
*n* - In the bottom right-hand corner of the Angle Sum applet, there is an animation for the triangle and square showing how the sum of the interior angles relates to tiling. Have students watch the animations and write a journal entry on what they demonstrate.

- Have students explore exterior angles. The sum of the exterior angles for any polygon is 360°, and therefore the measure of one exterior angle in a regular polygon is 360/
- The ending game allows students the opportunity to apply concepts to new situation.
- Additional geometrical games can easily be created to extend upon this game/concept as well as incorporate new one.

#### Evaluation and Assessment

Multiple forms of assessment are appropriate one or all can be utilized.

- Students will write a journal entry describing how to find the interior angle sum of any polygon. Students should include details such as patterns they discovered and other questions on the topic they would like to explore.
- Students will randomly be given one 3x5 index card to each student from a created set of cards numbered from 50 to 100. All students will find the sum of the interior angles of a polygon with the number of sides shown on their card and the measure of 1 interior angle of a regular polygon with that same number of sides which will sever a the ticket out the door. NOTE: preselecting index cards can be used as differentiation strategy.
- A formative assessment will take place during the lesson where the teacher will continually monitor the progress of the students in an informal manner. This will take place by asking them questions throughout the lesson to keep them actively engaged.
- I will know students can calculate the exterior angles of a polygon if they are able to navigate their robot about the triangle and square provided and later the provided game.
- I will know students can relate their investigation in the table to algebraic formula if they create a acceptable version of the formula [ ie. n(180) -360/n , (n-2)180/n ]
- I will know students can determine the interior sum of polygons if they relate triangulation to interior sums in their journal writings as well as find answer to ticket out the door question.

#### Reflection

In an effort to merge my RIP topic (constructivist teaching methods) with a lesson I had to perform I adapted this lesson. This lesson is a combination of an NCTM lesson and some original ideas that I wanted to incorporate. I have not preformed this lesson before, so it is hard to predict how it would go. I feel like it is a more student centered approach which has its merits, but also can be difficult depending on the setting. In conducting discussions in classrooms I have previously been in, there have been some factors that may be relevant in a real classroom setting. My experience has been that middle school students are more likely to be interested in classroom discussion and also more likely to get out of hand where as high school students are less interested in discussion and less interested in presenting their work on the board.

It is noteworthy to mention another way of doing the warm would be to have students draw a picture to represent the vocabulary as according to Marzzano teaching tips (Stewart, 2007).

#### References

- Hendrickson, K. (n.d.). Illuminations: Adding It All Up.
*Illuminations: Welcome to Illuminations*. Retrieved November 08, 2010, from http://illuminations.nctm.org/LessonDetail.aspx - Illuminations: Tessellation Creator. (n.d.).
*Illuminations: Welcome to Illuminations*. Retrieved November 08, 2010, from http://illuminations.nctm.org/ActivityDetail.aspx - Stewart, K. (2007).
*Traditional vocabulary strategies versus Robert Marzano's vocabulary builder*.

"YouTube - Interior and Exterior Angles of a Polygon."*YouTube - Broadcast Yourself.*Web. 12 Nov. 2010.

http://www.youtube.com/watch?v=lHmZlKby1dE

Preparation

On chalkboard

-participation chart

-draw triangle and square

Assign seating

Put do now on each table

Lesson Script

Pre Lesson Talk

This lesson was adapted from NCTM’s illuminations site. The original lesson can be found there I had some other ideas that I wanted to run with so I decided to go in a different direction than the original so I don’t typically do anything fast but for this I’m going to take a page from Christine’s book and talk a bit faster to push through this so you can see the full picture. I’m interested in your feed as far as how you felt the pieces connected.

I assume that students have a basic geometric background aw well as an understanding of enter having had experience with concave/convex polygons, complementary and supplementary angles, interior and exterior angles, vertical angles, sum of interior angles of triangles and squares as well as familiarity with circles & degrees if circles

Ok so now lesson starts.

Good afternoon, Its great to see you all today. Please begin your warm up activity you have 2 mins to complete it then we will share our answers.

Teacher walks around the room

Ok I see most of you have finished. “Who thinks they have #1 correct? How about #2?...”

Give points

#5 and says, “Well some of you may have guessed, today we will be working with this new concept-- TRIANGULATION” and we will work with that a little later don’t worry if u didn’t quite understand it yet.

“But before we get into that let’s do a quick wrap around and tell me your favorite cartoon character, movie star or friend”.

Give points

“Ok awesome, I like a lot of those same characters too. NOW SUPPOSE a robot spell has been cast upon your character and now he/she is trapped in a polygon like a regular pentagon, or regular heptagon. In order to get out they have to start at a vertex and ends at the same vertex he/she can only walk along the sides and turn if you give then angle measurements in degrees. How would your character get out of the polygon?” Talk to your neighbor and discuss what information you would need to know

Give points

Ask for students answers “ and says, “YES, and we are going to find out how to calculate interior angles of regular polygons when they are not given to us and use what we learned last week about supplemental angles to be able to save our characters”.

“Lets start by trying these”

Hand out TRIANGLES AND SQUARE sheet/packet

Walk around the room. If you have any questions please ask me. You have 3 mins to try this

“I see most of you are finished can we have one person come to the board and explain what they did for the triangle.” (for the sake of time we do not do both at the board)

Give points

Ok great now I want you to look at the next page and begin to work on this read carefully and you can ask me questions if u need to. You can use your warm up or notes to help you also.

Walk around the room.

Great I see most of you have finished. So lets share our answers

Give point

Ask WHY!!! HOW DO YOU KNOW THAT?

Ok now turn to the last page and play the game in pairs.

If they finished I’d give another game or tell them to go another route

Note Games are by differentiated ability

-I could further differentiate by intensifying game

-quick finisher would go to applet and use computer

-do extension activity.

Begin journal work