Area of circle discovery ppt shortened
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Discovering Area of a Circle
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Area of circle discovery ppt shortened
- 1. 1 Warm Up OBJECTIVE: Students will be able to describe the shortcut to calculate the area of any circle in the world. Find the area of each shaded region: #1 #2 Explain how you found your answer to problem 1.#3
- 2. 2 A = 36 square units Describe how you found the area of this shape. WHY? Warm Up Review 1 Agenda
- 3. 3 Warm Up Review A = 9 square units Describe how you found the area of this shape. WHY? The shaded part is called a “radius square”. Why do you think it is called this? 1 Agenda
- 4. 4 Launch – Finding the Area of a Circle Think-Pair-Share 1) How would you find the area of this circle? 2) Why is it hard to find the area of a circle? Agenda wait..
- 5. 5 • 1) Not all of the squares are completely in the circle.Hey, is this a square? What about this? 1 2 3 4 5 6 7 8 ??? ? • 2) There are just TOO many to count! Nice Job! I found the same two problems. • PROBLEM 1 With these 2 problems, do you think there is a way to find the area of a circle more quickly ? • PROBLEM 2 Agenda 1wait.. Launch
- 6. 6 People have always run into these problems, when finding the area of a circle years ago, the Babylonian mathematicians discovered that the radius square was the key to quickly finding the area of a circle. Radius Square = r2 A certain number of radius squares fits inside of every circle. Agenda Launch
- 7. 7 Explore: Getting Ready How many radius squares fit inside one circle? This is a radius. This is a radius square. • To find out the shortcut for the area of a circle you will first answer this question. What is your prediction? 1 Agenda wait..
- 8. Explore: The Question… How many radius squares fit inside a circle? More than 1 square? More than 2 squares? How many radius squares do YOU think fit inside one circle? 1wait.. 8
- 9. 9 Explore: The Activity • You will get a paper circle and 4 radius squares. • You need to find out the answer to this question: How many radius squares fit inside a circle? by cutting and gluing the radius squares into the circle. Agenda wait..
- 10. 10 Explore: How to do the activity • To count a radius square all parts of it must be inside the lines of the circle. How many radius squares fit inside one circle? This circle does not have one whole radius square covering its surface. Agenda wait..
- 11. 11 How many radius squares fit inside one circle? You may cut the squares to make them fit inside the circle. 1 Agenda Explore: How to do the activity
- 12. 12 How many radius squares fit inside one circle? • But all parts of the radius square must be inside the circle to count. This circle does have one whole radius square covering its surface. # of RADIUS SQUARES THAT FIT INSIDE 1 wait.. Explore: How to do the activity
- 13. 13 Explore: Area of a Circle Challenge How many radius squares fit inside one circle? r r r You will have 10 minutes to try this activity. Agenda wait..
- 14. 14 Explore: Clean Up Even though you may not have finished the activity, it is time to clean up. Agenda
- 15. 15 Think-Pair • Now that you have tried the activity do you know… • Take one minute to think about the answer to this question. • Next, talk to your partner. Take turns sharing the answer to this question. How many radius squares fit inside a circle? Agenda
- 16. 16 Video Click on video to start Agenda
- 17. Debrief How many radius squares fit inside one circle? • What is a number that you recently learned that is more than 3? It’s π • Is that true for every circle no matter its size? YES! 1wait.. Agenda18
- 18. 19 Debrief: Is There a Shortcut? • We know that EXACTLY π (3.1415…) radius squares fit inside a circle. • Can we write a rule that will work for any circle? The area of a circle is a little bit more than 3 times the radius square. A shortcut makes you faster! Looking at the Rule Agenda wait..
- 19. Exit Ticket • What is the best estimate for the area of this circle? When I say so “go” make the gesture next to the answer you think is correct.. A) 5 square units B) 15 square units C) 25 square units D) 75 square units Agenda 21
- 20. The goal of 21st Century Lessons is simple: We want to assist teachers, particularly in urban and turnaround schools, by bringing together teams of exemplary educators to develop units of high-quality, model lessons. These lessons are intended to: •Support an increase in student achievement; •Engage teachers and students; •Align to the Common Core State Standards and the Massachusetts curriculum frameworks; •Embed best teaching practices, such as differentiated instruction; •Incorporate high-quality multi-media and design (e.g., PowerPoint); •Be delivered by exemplary teachers for videotaping to be used for professional development and other teacher training activities; •Be available, along with videos and supporting materials, to teachers free of charge via the Internet. •Serve as the basis of high-quality, teacher-led professional development, including mentoring between experienced and novice teachers. The goal… 29 21st Century Lessons
- 21. Directors: Kathy Aldred - Co-Chair of the Boston Teachers Union Professional Issues Committee Ted Chambers - Co-director of 21st Century Lessons Tracy Young - Staffing Director of 21st Century Lessons Leslie Ryan Miller - Director of the Boston Public Schools Office of Teacher Development and Advancement Kevin Qazilbash - Co-director of 21st Century Lessons 21st Century Lessons The people… Lesson Designers: Stephanie Conklin Meghan McGoldrick Lisa Schad Sarita Thomas Shane Ulrich Project Specialist: Kimberly Anderson Meideros 30
Notes de l'éditeur
- 0 - 3 In Class Notes: Students work independently ~3 min. Debrief another ~2 min Click to the next slides when ready to go over the answers. Use calling sticks for #1 and #2 Ask students how #2 and #3 are similar/different
- 4-5 In Class Notes: Although you can count the squares, the most efficient way to find the area of this square is to multiply the sides.
- 5-6 In Class Notes: Every circle has a radius square. This is the building block of finding the area of any circle. The radius square is the key element in today’s lesson. Key Questions: (If class discussion does not generate these naturally) Did you count the shaded squares? Is there a short cut where you don’t have to count?
- 8-9 In Class Notes: this is the BIG IDEA for today’s lesson. Finding the area of this circle is harder because… partial squares too many squares to count.
- 10-11 In Class Notes: emphasize how helpful a shortcut is. Tell students they DO NOT want to count every square to find area. “I’ll teach you the shortcut!”
- 12 In Class Notes: Historical connection to yesterday’s lesson. May also connect to students’ history class with Babylonian and Egyptian Empires.
- 13 In Class Notes: Key Question: How many radius squares fit inside one circle? Preparation Notes: This question should be strongly focused on and referred to throughout the student activity. The more attention you give this question, the more likely students will be able to remember a purpose to their work and have an answer when time runs out or after the video viewing.
- 14 - 15 In Class Notes: Wait for animation Show students paper circles, squares, scissors, etc so students understand what they will be using. Preparation Notes: A quick description of what they are about to do. You may even want to hold up the paper circles and squares, scissors and glue so students can see and understand what they are about to do. The next slides clarify some common student errors in this activity and it’s important that students are really paying attention with the knowledge that they will soon do this for themselves. Students should not have the materials in hand at this point.
- 16 In Class Notes: Show students paper circles, squares, scissors, etc so students understand what they will be using. Preparation Notes A quick description of what they are about to do. You may even want to hold up the paper circles and squares, scissors and glue so students can see and understand what they are about to do. The next slides clarify some common student errors in this activity and it’s important that students are really paying attention with the knowledge that they will soon do this for themselves. Students should not have the materials in hand at this point.
- 17 In Class Notes Make sure students hear common errors! 1. A square cannot be counted if part of it is outside the circle. Overhang must be cut off!
- 18-19 In Class Notes: Make sure students hear common errors! 2. A radius square may be cut without following the lines. You can even cut curves!
- 20 In Class Notes: Make sure students hear common errors! 3. Do not throw away your cut pieces. ALL PIECES MUST BE FIT INTO THE RADIUS SQUARE WITHOUT OVERLAPPING.
- 21-31 In Class Notes: Distribute materials Have students check that their squares do indeed measure the length of a radius on each side. That makes them radius squares. Students must answer question in yellow. Preparation Notes: The stopwatch image will take you to an online stopwatch. But be aware that the page will hide the powerpoint slide so you may want to have the stopwatch running in the background by clicking again on Powerpoint so that students can see the key question. Questions to ask as you circulate: How many radius squares do you think will fit? Where will you put the first radius square? Can you fit another whole radius square? Where will you put it? How will you make all of it fit in the circle? How many radius squares are inside your circle now? Do you think you can fit any more? I notice that there are still some spaces that haven’t been filled. But can you fit another whole radius square? (no) Can you say how much of a radius square would fill those spaces? Is it a little or a lot? If I asked you to estimate a number for how many radius squares fill up your circle, what would you say? {Encourage estimation and that they do not have to be exact. “a little more than __” is perfectly acceptable.} Ultimately we want students to answer this question with “A little more than 3 radius squares fit inside any size circle.”
- 32 - 35 In Class Notes: Students must clean up even if they are not done with the activity. Preparation Notes It is very important that students stop the activity after 10 minutes and begin clean up. This activity can easily take 1 hour for some students and 10 minutes for others. In order for all students to be actively engaged in meaning making, we need to limit the activity time. This is why the video was created. The video will help students see, to completion, ideas that were forming as they worked. You may add in the text box and specific directions to students on how to clean up, such as where to put scissors, glue, pieces of scrap paper and the finished or unfinished circles.
- 36- 40 Preparation Notes: Students need some time to process what they did in the activity and what they discovered. It’s likely they did not finish but they probably had enough time to develop some ideas about how many radius squares would fit. They need to take a moment to consider what they would say at this point. The video (up next) will make clear how many fit so they don’t have to be right, they just have to know what they think at this point. Give students a timed 1 minute to think silently with no talking. Telling someone else pushes the accountability for this thinking. Give students 1 minute timed to tell each other what they think.
- 41 – 46 In-Class Notes: The video is titled “Day 3 How many radius squares fit in a circle? Movie” Video length: 3 min 31 sec The video file is located on your 21st Century flash drive. If there are any problems playing the video through PowerPoint then you can exit PowerPoint and play the video directly by double clicking on the video file itself. opening the window showing the video and then Preparation Notes: You will need speakers to make sure all students can hear the video. The video is titled “Day 3 How many radius squares fit in a circle? Movie” Video length: 3 min 31 sec The video file is located on your 21st Century flash drive. If there are any problems playing the video through PowerPoint then you can exit PowerPoint and play the video directly by opening the window showing the video and then double clicking on the video file itself.
- 45/60 If you run out of time you may choose to share out verbally answers to these questions, but writing an answer on an index card accomplishes two things: 1.) Writing pushes students to be accountable for their thinking. 2.) Collected writing will give the teacher data on who has accomplished the objective today. Do not rush the thinking part. Make sure they are silent and give them space to reflect on what they have just done and seen. Next, they share for just 1 minute taking turns. Encourage them to talk to one another to answer the question, even if it seems clear. After writing, these cards should be collected at the end of 3 minutes.
- 50 - 52 In Class Notes: Use calling sticks for 3 or more students Students should go from “3 and a little bit” to “Pi” Preparation Notes: The purpose of this slide is to bring students from saying “there are 3 and a little bit radius squares in every circle” to the knowledge that the “3+” is really π. Remind students what they know about π, most importantly that it is approximately 3.14 or 3 and 1/7.
- 53/60 In Class Notes: If you think your students are ready to see this formula as a more official preview, you can click on the “Looking at the Rule” button at the bottom of this slide. Otherwise, wait until tomorrow’s lesson to see the formula. Preparation Notes: The purpose of this slide is for students to use the ideas of the day’s lesson to encourage students to verbalize a rule that they could use, knowing what they know, to find the area of ANY circle. This is a preview of tomorrow’s lesson about the formula for the area of a circle. Tricky aspects of the formula: r squared means r times r - this represents the area of the radius square. You can prompt students to remember this from the warm up. π r^2 means π times the radius squared. Students may be able to visualize 2 radius squares and 3 radius squares but π number of radius squares is a stretch for the mind.
- Optional The purpose of this slide is for students to use the ideas of the day’s lesson to better understand the formula. Once you click the red underline the rest of the formula will appear in a delayed timing. Once all the letters and symbols appear you can click again to finish the slide. Tricky aspects of the formula: r squared means r times r - this represents the area of the radius square. You can prompt students to remember this from the warm up. πr^2 means π times the radius squared. Students may be able to visualize 2 radius squares and 3 radius squares but π number of radius squares is a stretch for the mind.
- 56 Preparation Notes: The exit ticket is a quick assessment to see if students grasp the essential meaning of all that they did today. The big understanding is that if you know the radius you can find the area of the radius square and multiply that by 3. That will give you a number a little less than the actual area of the circle. Use the time after this slide to pass out and preview the homework in order to answer any clarifying questions.