LESSON STUDY TEMPLATE

DEVELOPED FOR UTS 180

SPRING, 2011

I.BACKGROUND INFORMATION

This being my first year teaching a Pre-AP course, I have always heard how the upper-level math teachers complain about how their students are not prepared for pre-calculus and calculus. In particular, they say the kids are not prepared for the Calculus AP exam. This lesson has been developed in an effort to bridge that gap by exposing the students to integration early enough (in Geometry Pre-AP) in their high school career. This is an effort to be able to raise our AP scores, as well as the percentage of students passing the AP Calculus exam as a school.

II.Unit Information

  1. Goal(s) of Lesson: The goal of the lesson is to expose students to integration on a very basic level. The long-term goal is to raise the percentage of students passing the AP Calculus exam.
  2. How this Lesson is related to the curriculum: This is an extension of our unit on areas of polygons, in particular, composite and irregular figures.
  3. Instructional sequence for the Lesson: We have just finished teaching areas of polygons, as well as composite figures, so this should be prior knowledge to the students. I have created a stations activity that will be the exploration. It has students find the areas under a quadratic function (spiraled from algebra 1) using a triangle, rectangles of different widths (different number of partitions), and trapezoids. As an engagement, they will do similar tasks using the graph of a linear parent function (spiraled from algebra 1).
  1. Lesson Information
  2. Name of the Lesson: Finding the Area Under a Curve
  3. Goal(s) of the Lesson
  4. How Lesson relates to broad goals of lesson study

This lesson has been designed to introduce the concept of an integral using what the students know about areas of polygons. It is supposed to be a basic exposure to some of the vocabulary and idea of Riemann Sums. The idea is that this way, they will be more receptive to those concepts in Calculus AP.

  1. Learner outcomes that are measurable / observable for this research-based lesson

Students should be able to identify polygons and find the area of polygons used per station. They should be able to use scaling to create rectangles, triangles or trapezoids to find the area under a given curve, as well as find the sum of those polygons.

  1. Length of lesson 80-90 minutes
  2. Description of the class The students participating in this research lesson are all freshmen in Geometry Pre-AP.
  3. Source of the lesson created by the teacher; adapted from worksheets from Kuta software, and (for the animation), as well as a Holt Geometry worksheet
  4. TEKS (or other state /national standards) addressed

G.8A find areas of regular polygons, circles, and composite figures

G.9B formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models;

IV. Resources, materials and supplies needed

  • Computer with access to the internet
  • Projector
  • TI-83+ calculator or higher
  • Plastic sheet covers
  • Whiteboard markers and erasers (1 pair per station)

V. Supplementary materials, handouts.

  • Warm-Up: Areas of Polygons worksheet
  • Stations for Area Under the Curve
  • Stations Answer Sheet
  • 9-4 Challenge Holt worksheet
  1. Lesson Plan write-up: This only needs to be an OUTLINE of intended lesson flow and activities over the duration of the lesson.

1)Warm-up – Students find the areas of a square, rectangle, trapezoid, and a triangle. The additional questions are meant to start a discussion about finding the sums of multiples of the same polygon. (8-10 minutes)

2)Engagement/Explanation – Teacher models and facilitates a class discussion using y=x to find the area using different polygons. Use to show the students the graph and use the drop-down menu to change the polygons and rectangle sums (upper sum, lower sum, etc.) and discuss the difference between each one. Have students reflect on which one could be closer to the area the true area, as well as what happens when the number of partitions increases. (10 minutes)

3)Exploration – Make only 1 copy of each of the stations and place inside the plastic sheet covers, in color if possible. Groups of 2-3 students rotate through each of the 5 stations. This is the same basic process teacher modeled with the function y=x except now it is a quadratic curve (8-10 minutes per station). Have students record their work on the “Stations Answer Sheet.” Provide 1 pair of whiteboard markers and erasers so students can “write” on the station plastic sheet cover. Remind students that they have to erase it before next group comes to that station. Project in case students want to go up to the computer and try it out as they work each station using the quadratic curve given on the station.

4)Evaluation: Give students about 5 minutes to answer the reflection questions on the “Stations Answer Sheet.” Teacher facilitates quick and brief discussion to end the lesson about the questions.

5)Extension/Homework: Assign Challenge 9-4 from Holt for homework.

Name: ______Date: ______

Period: ______Travis HS Geometry Pre-AP

Warm-Up: Areas of Polygons

Identify each figure and find the area of each.

1.) 2.)

Area: ______Area: ______

Area of 10 squares: ______Area of 25 triangles: ______

3.) 4.)

Area: ______Area: ______

Area of 100 trapezoids: ______Area of 250 rectangles: ______

Area Under a Curve (Guided Practice)

Below you will find the graph corresponding to . Find the area above the x-axis of that line by counting the square units from 0 to 5.

Area: ______

Now create a right triangle with a base of 5 units (from 0 to 5) and find the area once more.

Area: ______

Now create 4 rectangles each 1 unit in width where the vertices lie under the curve. Find the area by finding the sum of all the rectangles.

Area: ______

Split each unit into halves and create more rectangles. Find the area. ______

Now use a trapezoid to find the area from 1 to 5.

Area: ______

You will now get into groups. You will do a similar task to the one you just did, but with the curve .

After the stations, take 10 minutes to reflect on the following:

  1. Visually, which method of summation seemed to most closely approximate the true value of the function? Why do you think that is?
  1. How did the error of the various sum types change as the number of partitions increased? Why is that?
  1. What do you think would happen as the number of partitions increased towards infinity?

Station #1

Use a triangle to estimate the area under the curve using a triangle from 0 to 6.

Station #2

Use a triangle to estimate the area under the curve using a sum of rectangles 1 unit in width from 0 to 6. Make sure all rectangles lie under the curve.

Station #3

Use a triangle to estimate the area under the curve using a sum of rectangles ½ unit in width from 0 to 6. Make sure all rectangles lie under the curve.

Station #4

Use a triangle to estimate the area under the curve using one single trapezoid fitting in the space from 0 to 6.

Station #5

Use a triangle to estimate the area under the curve using a sum of trapezoids ½ unit in height from 0 to 6.

Name:______Date:___Per.___

Stations Answer Sheet

You must copy all problems and show work for full credit. Just answers will not be accepted. Do not spend a lot of time copying…you only have 10 minutes per station!

Station #1 Station #2

Station #3 Station #4

Station #5 Station #6

Based on the results, does it seem like the sums of the areas is approximating some specific number? If so, what is that number?

______

After watching the animation, what was the actual area under the curve? ______

Which station came closest to the area? What was the task for that specific station? ______

How can we incorporate other types of curves into this same task? ______

What if we wanted areas of curves that had function values below 0? What could be done then? Draw an graph of what this situation would look like. ______