l35_equations-of-cir.. - Kenton County MDC
Lesson 35
Mathematics Assessment Project Formative Assessment Lesson Materials
Equations of Circles 2 MARS Shell Center University of Nottingham & UC Berkeley Alpha Version If you encounter errors or other issues in this version, please send details to the MAP team c/o [email protected].
© 2011 MARS University of Nottingham
Equation of Circles 2
Teacher Guide
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Equations of Circles 2
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Mathematical goals
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This lesson unit is intended to help you assess how well students are able to: • •
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Alpha Version August 2011
Translate between the equations of circles and their geometric features. Sketch a circle from its equation.
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Standards addressed
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This lesson involves mathematical content in the standards from across the grades, with emphasis on:
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G-GPE: Expressing geometric properties with equations. A-CED: Create equations that describe numbers or relationships. N-RN: Extend the properties of exponents to rational exponents. This lesson involves a range of mathematical practices, with emphasis on:
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1. 7.
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Introduction
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The unit is structured in the following way:
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Make sense of problems and persevere in solving them. Look for and make use of structure.
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Materials required • •
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Before the lesson, students work individually on an assessment task that is designed to reveal their current understanding and difficulties. You then review their work and create questions for students to answer in order to improve their solutions. During the lesson, small groups work on a collaborative task, categorizing equations and geometric descriptions of circles. After a plenary discussion, students revise their solutions to the assessment task.
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Each student will need two copies of the assessment task, Going Round in Circles Again. Each small group of students will need a mini-whiteboard, a pen and an eraser, the cut up cards Equations 1 and Equations 2, a large sheet of paper for making a poster, and a glue stick. You may also need a set of transparencies of the cards Equations 1 and Equations 2 to support whole class discussions. There are some projector resources to support whole class discussions.
Time needed Approximately 15 minutes before the lesson, and a single 90-minute lesson (or two 45-minute lessons). Exact timings will depend on the needs of the class.
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© 2011 MARS University of Nottingham
1
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
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Before the lesson
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Assessment task: Going Round in Circles Again (15 minutes)
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Set this task, in class or for homework, a few days before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it You will then be able to target your help more effectively in the follow-up lesson.
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Read through the questions and try to answer them as carefully as you can.
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Assessing students’ responses
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Collect students’ responses to the task, and note down what their work reveals about their current levels of understanding and their different approaches.
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Alpha Version July 2011
Going Round in Circles Again 1. Complete the drawing of the graph of the equation (x + 1)2 + (y ! 3)2 = 9 to show the x- axis and y-axis. Add numbers to each of these axes.
Figure out the co-ordinates of any x-intercepts and y-intercepts. Explain your answer(s).
It is important that students are allowed to answer the questions without your assistance, as far as possible.
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Student Materials
Give each student a copy of the assessment task Going Round in Circles Again.
Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to be able to answer questions such as these confidently. This is their goal.
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Equations of Circles 2
2. Write an equation of a second circle that has two x-intercepts, but just one y-intercept and a radius of 6. Explain your answer.
© 2011 MARS University of Nottingham S-1
We suggest that you do not score students’ work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit. We suggest that you write a list of your own questions, based on your students’ work, using the ideas that follow. You may choose to write questions on each student’s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson.
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© 2011 MARS University of Nottingham
2
Equation of Circles 2
Teacher Guide
Common issues:
Alpha Version August 2011
Suggested questions and prompts:
Student cannot get started (Q1)
• What does the equation tell you about its graph? • Try sketching a graph on your own axes.
Student adds the coordinates of the center of the circle, but does not draw any axes (Q1)
• Now draw in axes to fit your center
Student assumes the origin is the center of the circle (Q1)
• How can you check your center is correct?
Student incorrectly identifies the position of the axes (Q1)
• Substitute y = 0 into the equation. Does this agree with your drawing? • Substitute x = 0 into the equation. Does this agree with your drawing?
Student uses the Pythagorean Theorem incorrectly when figuring out the coordinates of the intercept (Q1)
• Check your answer for the coordinates of the y-intercept. You may want to substitute the value for x at this point into the equation.
Student uses substitution incorrectly when figuring out the coordinates of the intercept (Q1)
• Check your answer for the intercept by using the Pythagorean rule.
Student cannot get started (Q2)
• Try sketching a circle with these features. Write what you know on your sketch. • Try out an equation. Check to see if it fits the conditions.
Student correctly sketches a graph, but provides an incorrect equation (Q2)
• Check your equation is correct by figuring out the coordinates of the x-intercepts and y-intercepts.
For example: (x − 2)² + (y −6)² = 36 Student provides little or no explanation (Q2)
• Sketch your circle and write the coordinates of its center and its radius.
Student correctly answers the questions
• In Question 2, suppose the coordinates of the intercepts of the circle are always integers. How many circles could you draw? Explain your answer. [8.]
The student needs an extension task.
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© 2011 MARS University of Nottingham
3
Equation of Circles 2
Teacher Guide
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Suggested lesson outline
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Whole class introduction (20 minutes)
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Give each student a mini-whiteboard, a pen and an eraser.
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Show Slide 1 of the projector resource.
Alpha Version August 2011
Odd One Out Graph A
Graph B
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Draft Version February 2011
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Each of these three circles might be considered to be the odd one out. Choose one and write on your mini-whiteboard why it is the odd one out. What properties have the other two circles got in common that the third does not have?
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© 2011 MARS, University of Nottingham
Projector resources:
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Ask students for justifications. As they do this, write the properties that they mention on the board. You may need to encourage students to think about the properties of the intercepts of each circle. There may be more than one reason for each. • • • •
Graphs B and C have the same radius. Graph A has a different radius. Graphs A and C have centre (5,-3). Graph B has center (-3, 2). Graphs A and B have two x intercepts. Graph C only has one x intercept. Graphs A and B have one y intercept. Graph C because has no y intercept.
Show Slide 2 of the projector resource: Odd One Out
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A.
(x + 4)2 + (y – 3)2 = 16
B.
(x – 6)2 + (y – 7)2 = 25
C.
(x – 5)2 + y2 = 25
Ask students to repeat the exercise with the three equations, again looking at centers, radii and intercepts. These will take a bit more thought, so allow students time to think about this. If students are struggling, then encourage them to discuss the problem with a neighbor before asking one or two students for an explanation. Encourage the rest of the class to challenge explanations. Draft Version February 2011
• • • •
© 2011 MARS, University of Nottingham
Projector resources:
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Equations B and C have radius 5. Equation A has a radius of 4. Equations A and C have two x-axis intercepts. Equation B has no x -axis intercept. Equations A and C have one y-axis intercept. Equation B has no y -axis intercept. Equation C is the odd one out as its center is the only one on the x-axis.
Students who struggle may be encouraged to either (i) substitute x = 0, or y = 0 into each equation, or (ii) to sketch a graph using the key features and the Pythagorean theorem:
© 2011 MARS University of Nottingham
4
Equation of Circles 2 104
(i)
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Teacher Guide
Equation A: When x = 0 then
(x + 4)2 + (y − 3)2 = 16. 42 + (y − 3)2 = 16 (y − 3)2 = 0 y = 3. There is one y-intercept.
When y = 0 then
(x + 4)2 + (−3)2 = 16 (x + 4)2 + 9 = 16 (x + 4)2 = 7
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There are two x-intercepts: x = −4 + 7 and x = −4 − 7 .
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Alpha Version August 2011
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© 2011 MARS University of Nottingham
5
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
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Collaborative activity 1 (15 minutes)
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Organize the class into groups of two or three students. Give each group the cards 1 to 4 of Equations 1 and a large sheet of paper.
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Show Slide 3 of the projector resource and ask students to copy the table onto their poster. The table should cover all of the poster. Categorizing Equations No x-axis intercept
One x-axis intercept
Two x-axis intercepts
No y-axis intercept
One y-axis intercept
Two y-axis intercepts Draft Version 9 Mar 2011
© 2011 MARS, University of Nottingham
Projector Resources:
3
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Explain how students are to work collaboratively. You are now going to continue to sort equations of circles according to their number of x-intercepts and y-intercepts. Take it in turns to place the equation cards in one of the categories in the table. To do this, figure out the coordinates of any intercepts. You may want to use the axes in each cell to sketch the graph.
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If you place a card, explain how you came to your decision. Your partner should check your answer using a different method.
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Both of you need to agree on, and explain the placement of every card.
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Write additional information as part of your explanation. Once you have placed all four cards, figure out what the equations have in common.
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Slide 4 of the projector resource summarizes these instructions.
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The purpose of this structured group work is to make students engage with each others’ explanations, and take responsibility for each others’ understanding.
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Encourage students not to rush into the activity, but spend some time thinking about how they can approach the task.
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What information can you get from the cards?
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How can you use this information?
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Encourage students to spend time justifying fully the position of each card.
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While students work in small groups you have two tasks, note different student approaches to the task, and support student reasoning.
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© 2011 MARS University of Nottingham
6
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
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Note different student approaches to the task Notice how students make a start on the task, any interesting ways of explaining a categorization, any attempts to generalize, where they get stuck, and how they respond if they do come to a halt. For example, do students plot the graph of an equation accurately? Do students use the Pythagorean theorem or substitution to figure out the coordinates of the intercepts on the circle? What do students do when the x or y coordinate of an intercept is imaginary? Do students sketch a graph of the equation? If so, when do they sketch a graph: at the start to figure out the number of intercepts or after they have figured out the coordinates of the intercepts? Do students notice that the equations in a column or row have a common feature?
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You can use this information to focus a whole-class discussion.
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Support student reasoning
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Try not to make suggestions that move students towards particular matchings, instead draw student's attention to critical mathematical features that they might not yet understand. Try to create a conjecturing atmosphere by asking questions to help students to reason together. You may decide to use some of the questions and prompts from the Common Issues table.
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If a student struggles to get started, encourage them to ask a specific question about the task. Articulating the problem in this way can sometimes offer a direction to pursue that was previously overlooked. However, if the student needs their question answered, ask another member of the group for a response.
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Some students may try to accurately plot the graph of the equation. Encourage them to sketch the graph instead.
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What does the equation of the circle tell you about the center of the circle and the radius?
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Can you use this information to sketch a circle?
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When students substitute x = 0 or y = 0 into the equations, some may have difficulty manipulating it. For example, students may write:
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(x + 3)2 + 62 = 25
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x+3+6=5
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x = −4
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In which case you may decide to ask students to check their answers. The formative assessment lesson 'Manipulating Roots' addresses this misconception. If you find one student has placed a card in a particular category on the table, challenge another student in the group to provide an explanation.
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Chan placed this card here. Cheryl, why does Chan think this equation has this number of intercepts?
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Can you show me a different method to the one Chan used?
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How can you check that the number of intercepts for this equation is correct?
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If you find the student is unable to answer these questions, ask them to discuss the work further. Explain that you will return in a few minutes to ask a similar question.
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© 2011 MARS University of Nottingham
7
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
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Sharing Work (10 minutes)
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As students finish categorizing the first four cards, ask them to check their work with that of a neighboring group.
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Check to see which cells are different from your own.
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If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.
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Once there is agreement, look to see what these equations or their graphs have in common. What is it about these equations that give them this common graphical feature? Can you use this to think of a quick way of figuring out the number of intercepts for a particular equation?
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These instructions are summarized on Slide 5 of the projector resource.
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Encourage students to focus on the structure of the equations and start to make generalizations.
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What do all four sketches have in common? [No x-axis intercept.]
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What is it about the equations that gives them this common feature?
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What is it about the equation that means the graph has no/one or two intercepts?
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Tell me about the relationship between the coordinates of the center of a circle and its radius for each circle. [Suppose (m,n) is the center of a circle and r its radius. For no x-axis intercept |n| > radius. For no y-axis intercept, |m| > radius. For one y-axis intercept, |m| = radius. For two y-axis intercepts, |m| < radius.]
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Collaborative activity 2 (20 minutes)
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After students have checked each others work and started to think about generalizations ask them to return to their original groups.
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Give each group cards 5 - 10 of Equations 2.
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Students are now to categorize the remaining cards. They can either continue figuring out the values of the x-axis and y-axis intercepts for each equation or they can use the structure of the equation to categorize it.
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If there is no equation on the card they are to add one that fits the sketch. More than one equation fits the sketch for cards 8, 9 and 10. Card 10 should be placed in the empty cell on the poster. If students struggle to figure out the equations for these cards, you may want to ask: Look at the other cards in this row/column. What do they have in common? What is it about the graphs that give them this common feature? What is it about the equations of these graphs that give them this common feature?
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Again listen carefully to student's explanations and note any difficulties they encounter. You can use this information in the plenary.
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Support students as in the first collaborative activity.
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© 2011 MARS University of Nottingham
8
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
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Plenary discussion (15 minutes)
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During the discussion you may want to use Slide 6 of the projector resource and transparencies of Equations 1 and Equations 2.
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Categorizing Equations No x-axis intercept
One x-axis intercept
Two x-axis intercepts
No y-axis intercept
One y-axis intercept
Two y-axis intercepts Draft Version 9 Mar 2011
© 2011 MARS, University of Nottingham
Projector Resources:
3
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Organize a discussion about what has been learned. The intention is that you focus on getting students to understand the methods of working out the answers, not just checking that everyone produced the same table. You may want to focus the discussion on cards students found difficult. Michael, where did you place this card? How did you decide? Why does that card not go in this category? In addition to asking for a variety of methods, pursue the theme of listening and comprehending others’ methods by asking students to rephrase others’ reasoning. Re-explaining someone else’s thoughts may encourage students to reflect on how the explanation compared with their own mathematical knowledge. Can someone else, put that into your own words/show me a different method? You want to explore, in a more general way, when x- and y-intercepts occur, write on the board: (x − m) 2 + (y − n) 2 = radius2 And ask students: What do you notice about equations that have no y-intercepts? [|m| > radius where (x − m) 2 + (y − n) 2 = radius2] What do you notice about equations that have just one y-intercept? [|m| = radius where (x − m) 2 + (y − n) 2 = radius2] What do you notice about equations that have two y-intercepts? [|m| < radius where (x − m) 2 + (y − n) 2 = radius2]
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Revisiting the initial assessment task (10 minutes)
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Return to the students their original assessment Going Round in Circles Again, as well as a second blank copy of the task.
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Look at your original responses and think about what you have learned this lesson. Using what you have learned, try to improve your work.
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If you have not added questions to individual pieces of work, then write your list of questions on the board. Students should select from this list only the questions they think are appropriate to their own work.
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If you find you are running out of time, then you could set this task in the next lesson or for homework.
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© 2011 MARS University of Nottingham
9
Equation of Circles 2
Teacher Guide
241
Solutions
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Assessment Task: Going Round in Circles Again
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Alpha Version August 2011
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The graph shows there is one x-intercept at (−1,0)
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Students may use substitution or the Pythagorean theorem to figure out the y-intercepts for the graph of the equation (x + 1)2 + (y − 3)2 = 9.
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Substitution:
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When x = 0 then
12 + (y − 3)2 = 9
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(y − 3)2 = 8
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y = 3 + 8 or y = 3 − 8
or y = 3 + 2√2 or y = 3 – 2√2.
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Therefore, the y-intercepts are at (0, 3 + 8 ) and (0,3 − 8 ).
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Using the Pythagorean theorem: !
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Therefore, the y-intercepts are at (0,3 + 8 ) and (0,3 − 8 ) or (0,3 + 2√2) and (0, 3 – 2√2)
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h = 8.
2.
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(x − 6)2 + (y − m) 2 = 36, or (x + 6)2 !+ (y − m)2 = 36!or
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(x − 6)2 + (y + m) 2 = 36, or (x + 6)2 + (y + m)2 = 36 where |m| 4 No y-axis intercept
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1. (x – 3)² + (y – 5)² = 9
y-intercept at (0,5)
10. The equation is: 2
7. The equation is:
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(x − m) + (y − n) = radius where m = n = radius.
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(x – 4)² + (y – 3)² = 16 x intercepts at (4-√7, 0) and (4+√7, 0)
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2. (x + 3)² + (y + 6)² = 25
y-intercepts at (0, -2)and (0. -10)
6. (y + 5)² + (3 + x)² = 25 x intercept (-3, 0)
9. The equation is:
y intercepts (0, -1) or (0, -9)
where r > 3.
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4. (x + 3)² + (5 – y)² = 16 y - intercepts at (0,5 + 7 ) and (0,5 " 7 ). ! !"#$%&
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© 2011 MARS University of Nottingham
11
Equations of Circles 2
Student Materials
Alpha Version August 2011
Going Round in Circles Again 1. Complete the drawing of the graph of the equation (x + 1)2 + (y − 3)2 = 9 to show the x- axis and y-axis. Add numbers to each of these axes.
Figure out the co-ordinates of any x-intercepts and y-intercepts. Explain your answer(s).
2. Write an equation of a second circle that has two x-intercepts, but just one y-intercept and a radius of 6. Explain your answer.
© 2011 MARS University of Nottingham
Equations of Circles 2
Student Materials
Alpha Version August 2011
Equations 1 1.
2.
(x – 3)² + (y – 5)² = 9
(x + 3)² + (y + 6)² = 25 !
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3.
(x – 5)² + (y + 5)² = 16 !
(x + 3)² + (5 – y)² = 16 !
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© 2011 MARS University of Nottingham
4.
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Equations of Circles 2
Student Materials
Alpha Version August 2011
Equations 2 5.
(x –6)² + (y + 4)² = 25
6.
(y + 5)² + (3 + x)² = 25 !
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© 2011 MARS University of Nottingham
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Odd One Out Graph A
Graph B
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© 2011 MARS, University of Nottingham
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Odd One Out
Alpha Version August 2011
A.
(x + 4)2 + (y – 3)2 = 16
B.
(x – 6)2 + (y – 7)2 = 25
C.
(x – 5)2 + y2 = 25
© 2011 MARS, University of Nottingham
Projector Resources:
2
Categorizing Equations No x-axis intercept
One x-axis intercept
Two x-axis intercepts
No y-axis intercept
One y-axis intercept
Two y-axis intercepts Alpha Version August 2011
© 2011 MARS, University of Nottingham
Projector Resources:
3
Categorizing equations 1. Take it in turns to place the equation cards in one of the categories in the table. • •
To do this, figure out the coordinates of any intercepts. You may want to sketch the graph of the equation.
2. If you place a card, explain how you came to your decision. 3. Your partner should check your answer using a different method. 4. You all need to be able to agree on, and explain the placement of every card. 5. Write some additional information or include a drawing as part of your explanation. 6. You are to ask each other for help before asking the teacher. Alpha Version August 2011
© 2011 MARS, University of Nottingham
Projector Resources:
4
Working with a neighboring group 1. Check to see which equations have been placed in different categories from your own. 2. If there are differences, ask for an explanation. If you still don't agree, explain your own thinking. 3. Once there is agreement, look to see what these equations or their graphs have in common. – What is it about these equations that give them this common graphical feature? – Can you use this to think of a quick way of figuring out the number of intercepts for a particular equation?
Alpha Version August 2011
© 2011 MARS, University of Nottingham
Projector Resources:
5
Categorizing Equations No x-axis intercept
One x-axis intercept
Two x-axis intercepts
No y-axis intercept
One y-axis intercept
Two y-axis intercepts Alpha Version August 2011
© 2011 MARS, University of Nottingham
Projector Resources:
6
Mathematics Assessment Project Formative Assessment Lesson Materials
Equations of Circles 2 MARS Shell Center University of Nottingham & UC Berkeley Alpha Version If you encounter errors or other issues in this version, please send details to the MAP team c/o [email protected].
© 2011 MARS University of Nottingham
Equation of Circles 2
Teacher Guide
1
Equations of Circles 2
2
Mathematical goals
3
This lesson unit is intended to help you assess how well students are able to: • •
4 5
Alpha Version August 2011
Translate between the equations of circles and their geometric features. Sketch a circle from its equation.
6
Standards addressed
7
This lesson involves mathematical content in the standards from across the grades, with emphasis on:
8
G-GPE: Expressing geometric properties with equations. A-CED: Create equations that describe numbers or relationships. N-RN: Extend the properties of exponents to rational exponents. This lesson involves a range of mathematical practices, with emphasis on:
9 10 11
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1. 7.
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Introduction
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The unit is structured in the following way:
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Make sense of problems and persevere in solving them. Look for and make use of structure.
•
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•
Materials required • •
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•
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Before the lesson, students work individually on an assessment task that is designed to reveal their current understanding and difficulties. You then review their work and create questions for students to answer in order to improve their solutions. During the lesson, small groups work on a collaborative task, categorizing equations and geometric descriptions of circles. After a plenary discussion, students revise their solutions to the assessment task.
•
Each student will need two copies of the assessment task, Going Round in Circles Again. Each small group of students will need a mini-whiteboard, a pen and an eraser, the cut up cards Equations 1 and Equations 2, a large sheet of paper for making a poster, and a glue stick. You may also need a set of transparencies of the cards Equations 1 and Equations 2 to support whole class discussions. There are some projector resources to support whole class discussions.
Time needed Approximately 15 minutes before the lesson, and a single 90-minute lesson (or two 45-minute lessons). Exact timings will depend on the needs of the class.
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© 2011 MARS University of Nottingham
1
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
34
Before the lesson
35
Assessment task: Going Round in Circles Again (15 minutes)
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Set this task, in class or for homework, a few days before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it You will then be able to target your help more effectively in the follow-up lesson.
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Read through the questions and try to answer them as carefully as you can.
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Assessing students’ responses
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Collect students’ responses to the task, and note down what their work reveals about their current levels of understanding and their different approaches.
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Alpha Version July 2011
Going Round in Circles Again 1. Complete the drawing of the graph of the equation (x + 1)2 + (y ! 3)2 = 9 to show the x- axis and y-axis. Add numbers to each of these axes.
Figure out the co-ordinates of any x-intercepts and y-intercepts. Explain your answer(s).
It is important that students are allowed to answer the questions without your assistance, as far as possible.
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Student Materials
Give each student a copy of the assessment task Going Round in Circles Again.
Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to be able to answer questions such as these confidently. This is their goal.
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Equations of Circles 2
2. Write an equation of a second circle that has two x-intercepts, but just one y-intercept and a radius of 6. Explain your answer.
© 2011 MARS University of Nottingham S-1
We suggest that you do not score students’ work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit. We suggest that you write a list of your own questions, based on your students’ work, using the ideas that follow. You may choose to write questions on each student’s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson.
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© 2011 MARS University of Nottingham
2
Equation of Circles 2
Teacher Guide
Common issues:
Alpha Version August 2011
Suggested questions and prompts:
Student cannot get started (Q1)
• What does the equation tell you about its graph? • Try sketching a graph on your own axes.
Student adds the coordinates of the center of the circle, but does not draw any axes (Q1)
• Now draw in axes to fit your center
Student assumes the origin is the center of the circle (Q1)
• How can you check your center is correct?
Student incorrectly identifies the position of the axes (Q1)
• Substitute y = 0 into the equation. Does this agree with your drawing? • Substitute x = 0 into the equation. Does this agree with your drawing?
Student uses the Pythagorean Theorem incorrectly when figuring out the coordinates of the intercept (Q1)
• Check your answer for the coordinates of the y-intercept. You may want to substitute the value for x at this point into the equation.
Student uses substitution incorrectly when figuring out the coordinates of the intercept (Q1)
• Check your answer for the intercept by using the Pythagorean rule.
Student cannot get started (Q2)
• Try sketching a circle with these features. Write what you know on your sketch. • Try out an equation. Check to see if it fits the conditions.
Student correctly sketches a graph, but provides an incorrect equation (Q2)
• Check your equation is correct by figuring out the coordinates of the x-intercepts and y-intercepts.
For example: (x − 2)² + (y −6)² = 36 Student provides little or no explanation (Q2)
• Sketch your circle and write the coordinates of its center and its radius.
Student correctly answers the questions
• In Question 2, suppose the coordinates of the intercepts of the circle are always integers. How many circles could you draw? Explain your answer. [8.]
The student needs an extension task.
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© 2011 MARS University of Nottingham
3
Equation of Circles 2
Teacher Guide
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Suggested lesson outline
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Whole class introduction (20 minutes)
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Give each student a mini-whiteboard, a pen and an eraser.
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Show Slide 1 of the projector resource.
Alpha Version August 2011
Odd One Out Graph A
Graph B
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Draft Version February 2011
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Each of these three circles might be considered to be the odd one out. Choose one and write on your mini-whiteboard why it is the odd one out. What properties have the other two circles got in common that the third does not have?
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© 2011 MARS, University of Nottingham
Projector resources:
1
Ask students for justifications. As they do this, write the properties that they mention on the board. You may need to encourage students to think about the properties of the intercepts of each circle. There may be more than one reason for each. • • • •
Graphs B and C have the same radius. Graph A has a different radius. Graphs A and C have centre (5,-3). Graph B has center (-3, 2). Graphs A and B have two x intercepts. Graph C only has one x intercept. Graphs A and B have one y intercept. Graph C because has no y intercept.
Show Slide 2 of the projector resource: Odd One Out
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A.
(x + 4)2 + (y – 3)2 = 16
B.
(x – 6)2 + (y – 7)2 = 25
C.
(x – 5)2 + y2 = 25
Ask students to repeat the exercise with the three equations, again looking at centers, radii and intercepts. These will take a bit more thought, so allow students time to think about this. If students are struggling, then encourage them to discuss the problem with a neighbor before asking one or two students for an explanation. Encourage the rest of the class to challenge explanations. Draft Version February 2011
• • • •
© 2011 MARS, University of Nottingham
Projector resources:
2
Equations B and C have radius 5. Equation A has a radius of 4. Equations A and C have two x-axis intercepts. Equation B has no x -axis intercept. Equations A and C have one y-axis intercept. Equation B has no y -axis intercept. Equation C is the odd one out as its center is the only one on the x-axis.
Students who struggle may be encouraged to either (i) substitute x = 0, or y = 0 into each equation, or (ii) to sketch a graph using the key features and the Pythagorean theorem:
© 2011 MARS University of Nottingham
4
Equation of Circles 2 104
(i)
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Teacher Guide
Equation A: When x = 0 then
(x + 4)2 + (y − 3)2 = 16. 42 + (y − 3)2 = 16 (y − 3)2 = 0 y = 3. There is one y-intercept.
When y = 0 then
(x + 4)2 + (−3)2 = 16 (x + 4)2 + 9 = 16 (x + 4)2 = 7
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There are two x-intercepts: x = −4 + 7 and x = −4 − 7 .
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Alpha Version August 2011
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© 2011 MARS University of Nottingham
5
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
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Collaborative activity 1 (15 minutes)
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Organize the class into groups of two or three students. Give each group the cards 1 to 4 of Equations 1 and a large sheet of paper.
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Show Slide 3 of the projector resource and ask students to copy the table onto their poster. The table should cover all of the poster. Categorizing Equations No x-axis intercept
One x-axis intercept
Two x-axis intercepts
No y-axis intercept
One y-axis intercept
Two y-axis intercepts Draft Version 9 Mar 2011
© 2011 MARS, University of Nottingham
Projector Resources:
3
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Explain how students are to work collaboratively. You are now going to continue to sort equations of circles according to their number of x-intercepts and y-intercepts. Take it in turns to place the equation cards in one of the categories in the table. To do this, figure out the coordinates of any intercepts. You may want to use the axes in each cell to sketch the graph.
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If you place a card, explain how you came to your decision. Your partner should check your answer using a different method.
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Both of you need to agree on, and explain the placement of every card.
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Write additional information as part of your explanation. Once you have placed all four cards, figure out what the equations have in common.
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Slide 4 of the projector resource summarizes these instructions.
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The purpose of this structured group work is to make students engage with each others’ explanations, and take responsibility for each others’ understanding.
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Encourage students not to rush into the activity, but spend some time thinking about how they can approach the task.
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What information can you get from the cards?
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How can you use this information?
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Encourage students to spend time justifying fully the position of each card.
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While students work in small groups you have two tasks, note different student approaches to the task, and support student reasoning.
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© 2011 MARS University of Nottingham
6
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
149
Note different student approaches to the task Notice how students make a start on the task, any interesting ways of explaining a categorization, any attempts to generalize, where they get stuck, and how they respond if they do come to a halt. For example, do students plot the graph of an equation accurately? Do students use the Pythagorean theorem or substitution to figure out the coordinates of the intercepts on the circle? What do students do when the x or y coordinate of an intercept is imaginary? Do students sketch a graph of the equation? If so, when do they sketch a graph: at the start to figure out the number of intercepts or after they have figured out the coordinates of the intercepts? Do students notice that the equations in a column or row have a common feature?
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You can use this information to focus a whole-class discussion.
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Support student reasoning
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Try not to make suggestions that move students towards particular matchings, instead draw student's attention to critical mathematical features that they might not yet understand. Try to create a conjecturing atmosphere by asking questions to help students to reason together. You may decide to use some of the questions and prompts from the Common Issues table.
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If a student struggles to get started, encourage them to ask a specific question about the task. Articulating the problem in this way can sometimes offer a direction to pursue that was previously overlooked. However, if the student needs their question answered, ask another member of the group for a response.
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Some students may try to accurately plot the graph of the equation. Encourage them to sketch the graph instead.
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What does the equation of the circle tell you about the center of the circle and the radius?
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Can you use this information to sketch a circle?
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When students substitute x = 0 or y = 0 into the equations, some may have difficulty manipulating it. For example, students may write:
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(x + 3)2 + 62 = 25
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x+3+6=5
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x = −4
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In which case you may decide to ask students to check their answers. The formative assessment lesson 'Manipulating Roots' addresses this misconception. If you find one student has placed a card in a particular category on the table, challenge another student in the group to provide an explanation.
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Chan placed this card here. Cheryl, why does Chan think this equation has this number of intercepts?
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Can you show me a different method to the one Chan used?
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How can you check that the number of intercepts for this equation is correct?
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If you find the student is unable to answer these questions, ask them to discuss the work further. Explain that you will return in a few minutes to ask a similar question.
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© 2011 MARS University of Nottingham
7
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
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Sharing Work (10 minutes)
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As students finish categorizing the first four cards, ask them to check their work with that of a neighboring group.
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Check to see which cells are different from your own.
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If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.
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Once there is agreement, look to see what these equations or their graphs have in common. What is it about these equations that give them this common graphical feature? Can you use this to think of a quick way of figuring out the number of intercepts for a particular equation?
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These instructions are summarized on Slide 5 of the projector resource.
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Encourage students to focus on the structure of the equations and start to make generalizations.
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What do all four sketches have in common? [No x-axis intercept.]
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What is it about the equations that gives them this common feature?
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What is it about the equation that means the graph has no/one or two intercepts?
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Tell me about the relationship between the coordinates of the center of a circle and its radius for each circle. [Suppose (m,n) is the center of a circle and r its radius. For no x-axis intercept |n| > radius. For no y-axis intercept, |m| > radius. For one y-axis intercept, |m| = radius. For two y-axis intercepts, |m| < radius.]
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Collaborative activity 2 (20 minutes)
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After students have checked each others work and started to think about generalizations ask them to return to their original groups.
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Give each group cards 5 - 10 of Equations 2.
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Students are now to categorize the remaining cards. They can either continue figuring out the values of the x-axis and y-axis intercepts for each equation or they can use the structure of the equation to categorize it.
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If there is no equation on the card they are to add one that fits the sketch. More than one equation fits the sketch for cards 8, 9 and 10. Card 10 should be placed in the empty cell on the poster. If students struggle to figure out the equations for these cards, you may want to ask: Look at the other cards in this row/column. What do they have in common? What is it about the graphs that give them this common feature? What is it about the equations of these graphs that give them this common feature?
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Again listen carefully to student's explanations and note any difficulties they encounter. You can use this information in the plenary.
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Support students as in the first collaborative activity.
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© 2011 MARS University of Nottingham
8
Equation of Circles 2
Teacher Guide
Alpha Version August 2011
210
Plenary discussion (15 minutes)
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During the discussion you may want to use Slide 6 of the projector resource and transparencies of Equations 1 and Equations 2.
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Categorizing Equations No x-axis intercept
One x-axis intercept
Two x-axis intercepts
No y-axis intercept
One y-axis intercept
Two y-axis intercepts Draft Version 9 Mar 2011
© 2011 MARS, University of Nottingham
Projector Resources:
3
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Organize a discussion about what has been learned. The intention is that you focus on getting students to understand the methods of working out the answers, not just checking that everyone produced the same table. You may want to focus the discussion on cards students found difficult. Michael, where did you place this card? How did you decide? Why does that card not go in this category? In addition to asking for a variety of methods, pursue the theme of listening and comprehending others’ methods by asking students to rephrase others’ reasoning. Re-explaining someone else’s thoughts may encourage students to reflect on how the explanation compared with their own mathematical knowledge. Can someone else, put that into your own words/show me a different method? You want to explore, in a more general way, when x- and y-intercepts occur, write on the board: (x − m) 2 + (y − n) 2 = radius2 And ask students: What do you notice about equations that have no y-intercepts? [|m| > radius where (x − m) 2 + (y − n) 2 = radius2] What do you notice about equations that have just one y-intercept? [|m| = radius where (x − m) 2 + (y − n) 2 = radius2] What do you notice about equations that have two y-intercepts? [|m| < radius where (x − m) 2 + (y − n) 2 = radius2]
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Revisiting the initial assessment task (10 minutes)
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Return to the students their original assessment Going Round in Circles Again, as well as a second blank copy of the task.
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Look at your original responses and think about what you have learned this lesson. Using what you have learned, try to improve your work.
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If you have not added questions to individual pieces of work, then write your list of questions on the board. Students should select from this list only the questions they think are appropriate to their own work.
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If you find you are running out of time, then you could set this task in the next lesson or for homework.
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© 2011 MARS University of Nottingham
9
Equation of Circles 2
Teacher Guide
241
Solutions
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Assessment Task: Going Round in Circles Again
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Alpha Version August 2011
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The graph shows there is one x-intercept at (−1,0)
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Students may use substitution or the Pythagorean theorem to figure out the y-intercepts for the graph of the equation (x + 1)2 + (y − 3)2 = 9.
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Substitution:
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When x = 0 then
12 + (y − 3)2 = 9
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(y − 3)2 = 8
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y = 3 + 8 or y = 3 − 8
or y = 3 + 2√2 or y = 3 – 2√2.
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Therefore, the y-intercepts are at (0, 3 + 8 ) and (0,3 − 8 ).
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Using the Pythagorean theorem: !
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Therefore, the y-intercepts are at (0,3 + 8 ) and (0,3 − 8 ) or (0,3 + 2√2) and (0, 3 – 2√2)
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h = 8.
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(x − 6)2 + (y − m) 2 = 36, or (x + 6)2 !+ (y − m)2 = 36!or
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(x − 6)2 + (y + m) 2 = 36, or (x + 6)2 + (y + m)2 = 36 where |m| 4 No y-axis intercept
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y-intercepts at (0, -2)and (0. -10)
6. (y + 5)² + (3 + x)² = 25 x intercept (-3, 0)
9. The equation is:
y intercepts (0, -1) or (0, -9)
where r > 3.
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© 2011 MARS University of Nottingham
11
Equations of Circles 2
Student Materials
Alpha Version August 2011
Going Round in Circles Again 1. Complete the drawing of the graph of the equation (x + 1)2 + (y − 3)2 = 9 to show the x- axis and y-axis. Add numbers to each of these axes.
Figure out the co-ordinates of any x-intercepts and y-intercepts. Explain your answer(s).
2. Write an equation of a second circle that has two x-intercepts, but just one y-intercept and a radius of 6. Explain your answer.
© 2011 MARS University of Nottingham
Equations of Circles 2
Student Materials
Alpha Version August 2011
Equations 1 1.
2.
(x – 3)² + (y – 5)² = 9
(x + 3)² + (y + 6)² = 25 !
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(x – 5)² + (y + 5)² = 16 !
(x + 3)² + (5 – y)² = 16 !
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© 2011 MARS University of Nottingham
4.
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Equations of Circles 2
Student Materials
Alpha Version August 2011
Equations 2 5.
(x –6)² + (y + 4)² = 25
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© 2011 MARS University of Nottingham
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Alpha Version August 2011
A.
(x + 4)2 + (y – 3)2 = 16
B.
(x – 6)2 + (y – 7)2 = 25
C.
(x – 5)2 + y2 = 25
© 2011 MARS, University of Nottingham
Projector Resources:
2
Categorizing Equations No x-axis intercept
One x-axis intercept
Two x-axis intercepts
No y-axis intercept
One y-axis intercept
Two y-axis intercepts Alpha Version August 2011
© 2011 MARS, University of Nottingham
Projector Resources:
3
Categorizing equations 1. Take it in turns to place the equation cards in one of the categories in the table. • •
To do this, figure out the coordinates of any intercepts. You may want to sketch the graph of the equation.
2. If you place a card, explain how you came to your decision. 3. Your partner should check your answer using a different method. 4. You all need to be able to agree on, and explain the placement of every card. 5. Write some additional information or include a drawing as part of your explanation. 6. You are to ask each other for help before asking the teacher. Alpha Version August 2011
© 2011 MARS, University of Nottingham
Projector Resources:
4
Working with a neighboring group 1. Check to see which equations have been placed in different categories from your own. 2. If there are differences, ask for an explanation. If you still don't agree, explain your own thinking. 3. Once there is agreement, look to see what these equations or their graphs have in common. – What is it about these equations that give them this common graphical feature? – Can you use this to think of a quick way of figuring out the number of intercepts for a particular equation?
Alpha Version August 2011
© 2011 MARS, University of Nottingham
Projector Resources:
5
Categorizing Equations No x-axis intercept
One x-axis intercept
Two x-axis intercepts
No y-axis intercept
One y-axis intercept
Two y-axis intercepts Alpha Version August 2011
© 2011 MARS, University of Nottingham
Projector Resources:
6
