Manipulating Radicals - Kenton County MDC
Lesson 40
Mathematics Assessment Project Formative Assessment Lesson Materials
Manipulating Radicals
MARS Shell Center University of Nottingham & UC Berkeley Alpha Version
Please Note: These materials are still at the “alpha” stage and are not expected to be perfect. The revision process concentrated on addressing any major issues that came to light during the first round of school trials of these early attempts to introduce this style of lesson to US classrooms. In many cases, there have been very substantial changes from the first drafts and new, untried, material has been added. We suggest that you check with the Nottingham team before releasing any of this material outside of the core project team.
If you encounter errors or other issues in this version, please send details to the MAP team c/o [email protected].
© 2012 MARS University of Nottingham
Manipulating Radicals
Teacher Guide
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Manipulating Radicals
2
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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•
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Alpha Version January 2012
Use the properties of exponents, including rational exponents, and manipulate algebraic statements involving radicals. Discriminate between equations that are identities and those that are not.
8
In this lesson there is also an opportunity to consider the role of the complex number i = optional.
9
Standards addressed
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10 11 12 13 14 15
This lesson relates to the following Standards for Mathematical Content in!the Common Core State Standards for Mathematics: N-RN: Extend the properties of exponents to rational exponents. A-SSE: Interpret the structure of expressions. N-CN Perform arithmetic operations with complex numbers. This lesson also relates to the following Standards for Mathematical Practice in the CCSS:
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3. 7.
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Introduction
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The unit is structured in the following way:
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Construct viable arguments and critique the reasoning of others. Look for and make use of structure.
•
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"1 , but this is
• • • • •
Before the lesson, students work individually on an assessment task that is designed to reveal their current levels of understanding. You review their work and create questions to help students improve their solutions. During the lesson, students first work in small groups on a collaborative discussion task. After sharing their solutions with another group, students extend and generalize the math. In an optional collaborative task focuses on complex numbers. In a plenary whole-class discussion, students review the main mathematical concepts of the lesson. Finally students return to the original task, to try to improve their individual responses.
Materials required Each student will need a copy of the assessment tasks, Operations With Radicals, and Operations With Radicals (revisited), a mini-whiteboard, a pen, and an eraser. • Each small group of students will need the cards Always, Sometimes, or Never True? (cut up before the lesson), a glue stick, and a large sheet of paper for making a poster. • There are some projector resources to support the whole-class discussions. You may want to copy the Card Sets onto transparencies to be used on an overhead projector to support whole-class discussions. •
Time needed Approximately fifteen minutes before the lesson, an eighty-minute lesson (more if complex numbers are introduced), and fifteen minutes in a follow-up lesson (or for homework). Precise timings will depend on the needs of your class.
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© 2012 MARS University of Nottingham
1
Manipulating Radicals
Teacher Guide
Alpha Version January 2012
41
Before the lesson
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Assessment task: Operations With Radicals (15 minutes)
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Set this task in class or for homework a few days before the lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. This should help you to target your help more effectively in the lesson.
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Give each student a copy of the assessment task Operations With Radicals.
Manipulating Radicals
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1.
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9 = +3.
x = 7
Always True
x 7
Sometimes True
Never True
It is true for !
Show your reasoning:
If you change 7 to another number, is your answer still correct? Explain.
Throughout this task, the radical sign denotes a positive square root.
Explain to students that they need not worry too much if they cannot understand or do everything in the task. In the next lesson they will engage in similar work, which should help them to progress.
means the positive square root. So
For each of the following statements, indicate whether it is true for all values of x, true for some values of x or there are no values of x for which it is true. Circle the correct answer. ! ! true”, state all values of x that make If you choose “sometimes it true.
Read through the questions and try to answer them as carefully as you can.
It is important that students are allowed to answer the questions without your help, as far as possible.
Alpha Version January 2012
Operations With Radicals In these questions, the symbol
2.
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Student Materials
x " y =
x "
y
Always True
Sometimes True
Never True
It is true for
!
Show your reasoning:
3.
!
(1" 2x )(1+ 2x ) = "5
Always True
Sometimes True
Never True
It is true for Show your reasoning:
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Assessing students’ responses
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Collect students’ responses to the task, and think about what the students’ work reveals about their current levels of understanding.
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© 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 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 the trials of this unit. Write a selection of these questions on each piece of student work. If you do not have time, 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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© 2012 MARS University of Nottingham
2
Manipulating Radicals
Teacher Guide
Common issues:
Alpha Version January 2012
Suggested questions and prompts:
Student cannot get started
• Try some values to decide whether you think the equation can ever be true. • How can you get rid of the square root?
Student uses only guess and check
• You have checked a few values of x. How can you manipulate the equation to check your answer is correct in all cases? • How can you be sure there are no other values of x for which this equation is true/false?
Student provides no, or poor explanation for conclusions
• How can you use math to explain your answer?
Student makes mistakes with manipulation of the equations
• You have changed the equation. How can you be sure that the two sides of the equation are in fact equal?
For example: The student equates expressions when she has only squared one side of the equation. Or: The student subtracts a value from one side of the equation but not the other. Student does not choose appropriate operations when manipulating the equations For example: The student does not square to remove square roots. Or: The student does not simplify the expressions on each side.
Student misinterprets the meaning of algebraic equations and expressions For example: The student assumes that because the expressions on each side of an equation are different, the equation can never be true.
• How could you manipulate the algebra to simplify the equation? • What other operation could you use to change the equation? Why would that be a helpful move? • What is the positive square root of an expression that is squared? • How could squaring both sides of the equation help? • Try some values for x. Are there any values for x that would make the equation true? • When the square root of 3 is multiplied by itself, what is the result?
Or: The student assumes that squaring an expression in x means that every x in the expression just becomes x2. Student is confused by roots of negative values For example: The student says that the statement is sometimes true because you can’t use negative values of x (Q1. or Q2.)
• Is the statement sometimes, always, or never true for positive x? • Can you take the square root of a negative number?
Student correctly answers the questions
•
The student needs an extension task. •
© 2012 MARS University of Nottingham
Look at this inequality: 2 x > 3x . Is it always, sometimes, or never true? Write an equation that is only true when x is a complex number. !
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Manipulating Radicals
Teacher Guide
Alpha Version January 2012
83
Suggested lesson outline
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Whole-class introduction (10 minutes)
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Throughout this lesson, the radical sign,
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Give each student a calculator, a mini-whiteboard, a pen, and an eraser.
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Throughout the introduction, encourage students to first tackle a problem individually, and only then discuss it with a neighbor. In that way students will have something to talk about. Maximize participation in the whole ! class discussion by asking all students to show you solutions on their mini-whiteboards. Select a few students with interesting or contrasting answers to justify them to the class. Encourage the rest of the class to challenge these explanations.
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Write this on the board:
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, denotes the positive square root.
Always, Sometimes, Or Never True? x "5 = 93
x
"
5
Ask students: !
Is this statement always true, sometimes true, or never true?
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It is important that they understand the problem, and spend some time thinking about what they need to include in their solutions. If students are struggling to get started, encourage them to choose a value to substitute into the equation.
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Bring the class together and ask the following questions in turn:
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What does always true mean? Sometimes true? Never true?
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Show me a value for x that makes the statement incorrect. Now show me another…
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Can you show me a value for x that makes the statement correct? [x = 5] Can you show me another? How do you know for certain there are no more?
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We can find many values for x that make the statement false, and one value that makes it true. So is this statement “true,” “sometimes true, ” or never true? [Sometimes true.]
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You may find students choose values of x that are less than 5, thus making the left hand side of the equation imaginary. You may choose to address this issue in detail later in the lesson (Collaborative Activity 3). For now, ask students to just use numbers that give positive values to square root. What is important is that they justify their classification, with reference to mathematically available criteria. Fully justifying the conjecture that the statement is sometimes true will involve the use of algebra. Prompt students to manipulate the equation. In this case, it is useful to square both sides of the equation.
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What operations do you know for manipulating algebraic expressions? (E.g. add, multiply, take a power.)
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What operation is it useful to perform here? [Squaring both sides of the equation.]
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Why square both sides? [To keep them equal.]
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Square the left side of the equation:
(
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x " 5
2
)
= x " 5
© 2012 MARS University of Nottingham
!
(1)
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Manipulating Radicals 118 119 120
( x " 5)2 = x " 5 Is this correct? [No.] Explain. [Squaring an expression in the brackets does not just mean squaring each component within the brackets separately.]
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!Ask students to help expand the brackets. Write this on the board:
(
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x "
2
) (
5 =
x "
5
)(
x "
)
5 = x " 2 x 5 + 5
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The two expressions (1) and (2) are equal if:
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x " 5 = x " 2 x 5 + 5 ! Prompt students to manipulate and simplify the resulting equation.
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!
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Help students to show that the expressions are equal, when x = 5:
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x "5 = x "2 x 5 +5
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" 2 x 5 = 10
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! ! !
(2)
Is there any way of simplifying this? Remember to perform the same operation on both sides when manipulating the equation.
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Alpha Version January 2012
Now square the right hand side. You may wish to make a deliberate mistake to address the common misconception that squaring an expression involving square roots involves only removing the square root signs. Write this on the board:
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Teacher Guide
" x 5 =5 ! x= 5 ! x=5
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Collaborative Activity 1 (20 minutes) Organize the class into groups of two or three students. Give each group a large sheet of paper and Cards A - F from the sheet Always, Sometimes, or Never True? On the board, show students how to divide their large sheet of paper into three columns, with headings Always True, Sometimes True, Never True.
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Explain the activity to students.
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You have a set of cards with statements on them. Your task is to decide whether the statements are always, sometimes, or never true. In your group, take turns to place a card in a column. Then explain your choice to your partner or partners. If you think the statements is sometimes true, you will need to find the values of x for which it is true and values of x for which it is not true. You will need to explain why the equation is true for just those values of x.
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If you think the statement is always true, or never true, you will need to explain why you’re sure of that. Remember, if you’re showing it is always true or never true, substituting just a few values of x is not enough.
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Your partner then has to decide whether the card is in the correct place.
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When you agree, write down your explanations on paper. You may want to use the slide, Always, Sometimes, or Never True? to display the instructions for group work. © 2012 MARS University of Nottingham
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Manipulating Radicals 156 157 158 159 160 161 162 163
While students work in small groups, you have two tasks: to note students’ work on the task, and to use what you notice to support students’ reasoning. Make a note of student approaches to the task Listen to and watch students carefully. In particular, be aware of whether students are addressing the difficulties they experienced in the assessment. • •
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Alpha Version January 2012
It does not matter if students do not manage to place all of the cards. It is more important that everyone in the group understands the categorization of each card.
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Teacher Guide
•
Which values do students choose to substitute into equations? Is their choice of values purposive, or does it seem random? Which operations do students choose to perform on the equations? Do they simplify expressions and manipulate the equations without errors? Do they make errors in the manipulation of exponents? What kinds of explanations do students give for their choices? Are they satisfied with a “guess and check” approach? Or do they move to algebraic manipulation of expressions to find a general solution? Are they able to articulate the consequences of their algebraic work? Are the others in the group convinced by the arguments?
You can use the information to support students’ reasoning, and also to focus the whole-class discussion towards the end of the lesson. Support student reasoning Try not to correct students’ errors for them, but instead, direct their attention to the error, and ask them to explain what they have done. Use questions like those in the Common issues table to help students address their errors and misconceptions. If students are having difficulties in moving toward an algebraic solution, use prompts to remind them of what they already know:
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Your equation is complicated in its current form. Is there any way to simplify it? Does that help you find the values of x for which the equation is true?
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What do you know about manipulating algebraic expressions that might be useful?
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What do you know about operations on exponents? Might that be useful?
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Try not to make suggestions that move students towards a particular categorization of statements. Instead, prompt students to reason together. Once one student has explained the placement of a card, encourage another member of the group to participate either by explaining that reasoning again in his or her own words, or by challenging the reasons the first student gave. Alice, Gabriel placed that card in the “always true” column. Do you agree with her? OK, then explain her reasons to me in your own words/why you don’t agree. The purpose of this structured work is to encourage students to engage with each other’s explanations, and to take responsibility for each other’s understanding.
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Most students will substitute into each equation positive integers for x. If you want the lesson to involve consideration of complex numbers, you may want to encourage them also substitute negative values.
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Encourage your students to work algebraically, to develop convincing reasoning:
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You have shown the statement is true for this specific value of x. Now convince me it is always true for all values.
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Can you use algebra to justify your decision for this card?
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© 2012 MARS University of Nottingham
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Manipulating Radicals 200 201 202 203 204
Teacher Guide
Alpha Version January 2012
You may need to explain the reasoning required for statements that are sometimes true. Some students will leave their responses at “guess and check”, having found a few values for which the equation is true, and a few for which it is not true. In that case, emphasize that the task is to find exactly which values of x make the equation and which don’t. The student also needs to explain why that result holds. Sharing posters (10 minutes)
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As students finish matching the cards, ask one student from each group to make a quick copy of how cards were placed on his or her poster on scrap paper. This only involves writing the letters used to identify each card (A-F.)
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Ask one student from each group to visit another group's poster.
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If you are staying at your desk, be ready to explain the reasons for your group's placement of equations on your poster.
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If you are visiting another group, take the scrap paper with you.
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Go to another group's desk and look for differences. Check to see which equations are in a different category 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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When you return to your own desk, you need to consider as a group whether to make any changes to your own poster.
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You may want to use the slide Sharing Posters to display these instructions.
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When students are satisfied that they have identified and discussed differences, they are to return to their own posters and negotiate any changes they think are needed with their partner(s).
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Give each small group a glue stick, to glue the cards onto the poster.
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Whole Class Discussion (10 minutes)
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This activity provides students with an opportunity to generalize their work in the previous task.
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Write the equation on the board again:
x "5 = x " 5
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You decided before that the statement is sometimes true. It’s true just when x = 5.
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The equation has x's in it, and fives. Now change the fives to another number.
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!
Allow students a few minutes to think about this on their own. Then give the students a couple of minutes to discuss their initial ideas in pairs. Students may argue informally that changing the five to, for example, a four, 'just has to' make the solution four. Prompt them to provide a stronger justification, by manipulating the expression algebraically. We have looked at replacing five with lots of other numbers. Can anyone now think of a quicker way to check all numbers? [Replace the particular number with a variable.]
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Write this statement on the board:
x"y = x " y
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You can check for five, four, or any other number all at the same time if you can figure out when this algebraic equation is true. How might you do that?
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Is the equation now always, sometimes or never true?
!
Which operations did you use before, with y = 5?
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© 2012 MARS University of Nottingham
7
Manipulating Radicals
Teacher Guide
Alpha Version January 2012
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Ask students to work first individually, and then in pairs to show you how to manipulate the equation, to show that:
x" y =
x "
y
Squaring the left side of the equation :
(
x" y
2
)
= x " y
Squaring the right side of the equation :
(
x "
y
2
) =(
x "
y
)(
x "
y
)
= x + y - 2 x y 242
The two expressions are equal when : x " y = x + y - 2 x y 2 x y = 2y x y = y Squaring both sides : xy = y 2
(
)
y y " x = 0 y = 0 and y = x.
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As they work on the problem, support the students as in the first collaborative activity. If necessary, discuss the solution as a class. ! Collaborative Activity 2 (10 minutes)
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Once you've established how to replace the number with a variable:
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Choose one or two of the statements from your poster.
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Replace the numbers in the equations with a variable, say, y.
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Check to see if the statement should still remain in the same category.
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For the statements you place in the “sometimes true” category, you should figure out exactly which values of x and y make the equation true.
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They can write their equations and answers on the posters. Support the students as in the first collaborative activity.
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Collaborative Activity 3 (10 minutes) (optional)
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If you want to work on complex numbers in the lesson, give each group Cards G and H. These two cards introduce the students to complex numbers.
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Now consider whether these new statements are always, sometimes, or never true.
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What values of x make this statement true? Support student reasoning:
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What is the result of substituting x = -1 into the statement?
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How can you check the statement is correct for just this value / these values of x? © 2012 MARS University of Nottingham
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Manipulating Radicals 262 263
Teacher Guide
Alpha Version January 2012
You may also want to ask students to check that their categorization of the cards is correct for negative as well as positive values.
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Whole-class discussion (10 minutes) Organize a whole-class discussion about what students have learned. Ask each group to write on a mini-whiteboard an equation from their poster that meets some chosen criterion. For example, you might ask: Show me an equation that is always true. Is your equation still always true when the equation contains two variables?
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Show me an equation that is sometimes true. For what values is it true for the equation with two variables?
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Encourage students to explain their answers using examples and justifications.
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Does anyone disagree with this classification? Why? Which explanation do you prefer? Why?
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Draw out any issues you have noticed as students worked on the activity.
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(3 + x )(3 " x )
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Is this statement ever true? If so, for which x values is it true?
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Ask students to write the equation for Card G with x = -1 on their whiteboards.
!
What do you notice when you substitute this value of x into the equation? What do you think
"1 means?
Ask students to put this complex numbers into context, by thinking about their background knowledge about types of numbers. Tell me ! some of the different types of numbers you know. Push for a wide range of numbers, including rationals, decimals, negative numbers, irrational numbers such as
2 and transcendental numbers such as
3" . 4
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These are all called “real numbers”. The square root of a negative number is not a real number. A new kind of number is needed, which we call a complex number. The new symbol i is used to define the complex number √(-1)
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You can use i to represent lots of other complex numbers.
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How can
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What about "2 ? [ 2 " #1 = 2 "
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Optional Extension: Discussion of complex numbers using cards G and H If your students worked on Cards G and H, you may want to extend the discussion to include complex numbers. Write the statement from Card G on the board.
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!
"9 be represented as a complex number? [ 9 " #1 = 9 " #1 = 3i .]
A similar discussion can be made about Card H.
!
!
#1 = 2i .] !
!
© 2012 MARS University of Nottingham
9
Manipulating Radicals
Teacher Guide
Alpha Version January 2012
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Follow-up lesson: Revisiting the initial assessment task and reflecting on learning (15 minutes)
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Return to the students their original assessment task, and a copy of Operations With Radicals (revisited).
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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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Look at your original responses and think about what you have learned this lesson.
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Carefully read through the questions I have written.
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Spend a few minutes thinking about how you could improve your work.
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You may want to make notes on your mini-whiteboard.
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Using what you have learned, try to answer the questions on the new task: Operations With Radicals (revisited).
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If you find you are running out of time, you could set this task in the next lesson or for homework.
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© 2012 MARS University of Nottingham
10
Manipulating Radicals
Teacher Guide
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Solutions
312
Assessment Task: Operations With Radicals
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1.
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Alpha Version January 2012
x x = The statement is always true, for any value of x. 7 7 Whatever the value of x substituted into the equation, it remains true. Statements that are always true are identities. Some students might justify this claim only by substituting a few values of x and showing that the equation is true for those values. This is not sufficient justification. Others may just appeal directly to the laws for the x
" a %x a . manipulation of indices: $ ' = x #b& b
() ()
Whilst this shows that the statement is true, it does not help explain why the statement is always true. Squaring and taking the positive square root of the right hand side of the equation uses the laws for the manipulation of indices, but shows how this does not change the value of the fraction: !
" x %2 $$ '' = 7 # &
x( x 7( 7
=
x 7
This is clearly true for any value of x.
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Replacing 7 by any other numbers does not affect the truth of the statement. This can be shown by the same
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!methods for a particular value, but that does not establish the generality. A full solution will involve replacing
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7 with a variable, y, say, and manipulating the algebra to show equality.
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© 2012 MARS University of Nottingham
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Manipulating Radicals 328
2.
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x " y =
x "
Alpha Version January 2012
y.
This statement is sometimes true, that is, true when either x = y or when y = 0.
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Teacher Guide
!
For a full solution, the student is asked not only to identify some values of x for which the statement is true and some for which it is false, but to state exactly which values of x make the statement true.
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Students may just notice that if x = y, both sides are zero. This is not a complete solution, as it does not provide any justification for the general statement and it misses the solution x = y.
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Consider the left hand side of the equation. Squaring:
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(
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(
!
)
2
2
!
!
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Squaring both sides:
y 2 = xy Therefore x = y or y = 0 . A common error is to divide
! 3.
!
x y = y by
y , which hides the solution y = 0.
! (1" 2x )(1+ 2x ) = "5 . This statement is sometimes true, when x = 3.
!
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2
= x "2 x y + y = x "2 x y + y
"y= x y
!
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x" y
" 2y = 2 x y
!
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= x"y # x"y =x"y
x " y = x "2 x y + y
!
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2
If the two sides of the equation are equal, then
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)
Squaring the right hand side of the equation:
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x"y
(1! 2x )(1+ 2x ) = !5 " 1! 2x = !5 " x=3
!
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© 2012 MARS University of Nottingham
12
Manipulating Radicals
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Card A
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x +2 = x + 2
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(
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!
!
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2 x 2 =0 " x 2 =0
!
This is true only if x =0. For Collaborative Activity 2, we replace 2 with y, so the equation becomes x + y =
!
x + y.
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Following the same method as before, find that the statement is true only if x = 0 or y = 0 . Logically, this includes the possibility that both values are zero.
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Card B
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3x = 3 " x
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! !
This equation is always true.
!
Squaring the right hand side of the equation, then taking the positive square root:
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(
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3" x
)
2
=
(
) (
3" x "
)
3" x =
3 " 3 " x " x = 3 " x = 3x
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Therefore this equation is an identity, true for any value of x.
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Card C
!
x 2 + 22 = x + 2
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This equation is sometimes true.
Some students may simply remove the square root and square notation, and assume that this equation is always true. Other students may try a few values of x and assume it is never true.
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!
2
Subtracting x + 2 from each side of the equation:
! !
x +2
" x +2 = x +2+2 x 2
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2
) = ( x + 2) " x + 2 = ( x + 2 )( x + 2 )
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This equation is sometimes true, when x = 0.
Squaring both sides of the equation:
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!
Alpha Version January 2012
Lesson Task For each card, students should identify values for x for which the equation is true, and explain why the equation is true (or is not true) for just those values.
350
!
Teacher Guide
x 2 + 22 = x + 2
375
Suppose that
376
2 2 Then $ x + 2 ' = x + 2
" #
%2 &
(
)2
! " x 2 + 22 = ( x + 2)( x + 2) = x 2 + 2 # 2x + 22 = x 2 + 4x + 4
377
Thus ! the equation is true just when 4x = 0 which occurs only when x = 0 .
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For Collaborative Activity 2, we replace 2 with y, say.
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! !
© 2012 MARS University of Nottingham
!
13
Manipulating Radicals
Teacher Guide
x2 + y2 = x + y
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If we suppose that
381
" % 2 Then $ x 2 + y 2 ' = x + y
2
#
(
&
)
! " x + y 2 = ( x + y )( x + y ) = x 2 + 2xy + y 2 = x 2 + 2xy + y 2 2
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Thus ! the equation is true just when 2xy = 0 in which case either x = 0 or y = 0 .
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Card D
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! 385
!
Alpha Version January 2012
x 2 " 22 = x " 2
This equation is sometimes true, ! when x = 2 .
!
!
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As with Card C, an error that students sometimes make is to assume that you can simply remove the squares and
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root notation from
x 2 " 22 and that the equation is thus always true. 2
x " 2 = x " 2.
Suppose that
388
2
!
2
2 ! # 2 2 & Then % x " 2 ( = x " 2 $ ' ! So x 2 " 22 = x " 2 x " 2 = x 2 " 2 # 2x + 22 = x 2 " 4x + 4
389
(
390
)
)(
)
Thus!the equation is true just when 4x = 8 .
391
This occurs when x = 2 . ! For Collaborative Activity 2, replace 2 with y.
392 393
If
394
!
!
x2 " y2 = x " y .
2 # 2 2 & % x " y ( = x " y Then $ '
(
395
! 396
2
)
So x 2 " y 2 = x " y x " y = x 2 " 2xy + y 2
(
)(
)
397
2 2 2 Thus!the general equation is true just when " y = " 2xy + y , that is, when 2 y = 2xy
398
2 This ! occurs when y = xy
399
A possible student error is to divide ! through by y at this point, which!hides the solution y=0.
400
Instead, factorize: ! y 2 = xy
401
" y 2 # xy = 0
402 403 404 405
" y( y # x) = 0 ! The equation is true just when y=x or y=0. ! Card E !
406 407
!
(
x 2 " 32 = 3x
This equation is always true.
Using the laws for the manipulation of indices,
x 2 = x and
32 = 3 . So
© 2012 MARS University of Nottingham
x2 "
32 = 3x 14
!
!
!
Manipulating Radicals 408
Alpha Version January 2012
For Collaborative Activity 2, replace 3 with y.
x2 "
409
Since
410 411
Teacher Guide
y 2 = xy
x 2 = x for any value,
x2 "
y 2 = xy
Card F !
x2 ! x = 3 32
412
This equation is ! always true.
x 2 = x and
32 = 3 . Thus
414
As with Card G, this is also true if 3 is replaced with y, except that we must then exclude the possibility that y=0. ! ! Card G ! 3 + x 3 " x = 10 . This equation is sometimes true if x may take negative values (and the square root of
415 416
(
)(
32
=
)
418
a negative number is permitted). In this case, x = -1. If x is only considered to be zero or positive, then the statement is never true.
419
To find the values of x for which it is true, first clear the brackets.
417
420
(3 + x )(3 " x ) = 10
421
" 9 # 3 x + 3 x # x = 10
422 423 424
" 9 # x = 10 ! Therefore the equation is true if x = "1 ! Substituting these values of x back into the equation uses i, the square root of -1. !
425
(3 + "1)(3!" "1) = 10
426
(3 + i)(3 " i) = 10
427
!Thus if students look only at values of x that are positive or zero, this statement is never true.
429
Since Card G and H are introduced after Collaborative Activity 2, the general case (substituting y for 3) is not ! considered by students. This could, though, be used as an extension activity.
430
Card H
431 432
x + 1+ x " 1 = allowed, when x = -1.
433
Consider
434
Squaring this expression:
435
!
428
!
x . 3
Using the laws for the manipulation of indices,
!
!
x2
413
x + 1+
( x + 1+
2x
This equation is sometimes true, when x = 1, and also if complex numbers are
x " 1.
x " 1)2 = ( x + 1 +
x " 1)( x + 1 +
= x + 1 + 2 x +1 x " 1 + x " 1 = 2( x +
436
x " 1)
x + 1 x " 1) = 2( x +
x 2 " 1)
! ! © 2012 MARS University of Nottingham
15
Manipulating Radicals
Teacher Guide
( ) 2x
437
Squaring the right hand expression gives
438
Thus the two sides are equal if 2x = 2( x +
2
Alpha Version January 2012
= 2x .
x 2 " 1)
442
2 This occurs if x 2 " 1 = 0 , that ! is, if x = 1. Thus x = 1 or x = -1. In the latter case, both sides of the expression are complex numbers. ! Operations With Radicals (revisited) ! ! These solutions are analogous to those in the initial assessment, so we will not repeat all the reasoning.
443
1.
439 440 441
x ! x = 2 x is sometimes true when x = 0 or x = 4. If the two expressions are equal x = 2 x " x 2 = 4x " x(x # 4) = 0
444
" x = 0 or x = 4. If 2 were replaced by y then the equation is sometimes true when x = 0 and when x = y 2 445
2.
x = y
446
x y
is always true.
447 448
3.
(1! 4x )(1+ 4x ) = !15 is sometimes true when x = 4
449
© 2012 MARS University of Nottingham
16
Manipulating Radicals
Student Materials
Alpha Version January 2012
Operations With Radicals In these questions, the symbol
means the positive square root. So
9 = +3.
For each of the following statements, indicate whether it is true for all values of x, true for some values of x or there are no values of x for which it is true. Circle the correct answer. ! ! If you choose “sometimes true”, state all values of x that make it true.
1.
x = 7
Always True
x 7
Sometimes True
Never True
It is true for !
Show your reasoning:
If you change 7 to another number, is your answer still correct? Explain.
2.
x " y =
x "
y
Always True
Sometimes True
Never True
It is true for
!
Show your reasoning:
3.
!
(1" 2x )(1+ 2x ) = "5
Always True
Sometimes True
Never True
It is true for Show your reasoning:
© 2012 MARS University of Nottingham S-1
Manipulating Radicals
Student Materials
Alpha Version January 2012
Cards: Always, Sometimes, or Never True? A.
B.
x +2 =
!
x +
2
3x =
!!
!
C.
D.
x 2 + 22
!
3• x
x 2 " 22
= x +2
= x " 2
!
E.
F.
2
2
x • 3
x2
= 3x
32
! G.
H.
(3 +
)(
x 3 "
x
)
= 10
=
x 3
!
x + 1 +
x " 1 =
2x
!
!
© 2012 MARS University of Nottingham S-2
Manipulating Radicals
Student Materials
Alpha Version January 2012
Operations With Radicals (revisited) In these questions, the symbol
means the positive square root. So
9 = +3.
For each of the following statements, indicate whether it is true for all values of x, true for some values of x or there are no values of x for which it is true. Circle the correct answer. ! ! If you choose “sometimes true”, state all values of x that make it true.
1.
x "
x = 2 x
Always True
Sometimes True
Never True
It is true for
!
Show your reasoning:
If you change 2 to another number, is your answer still correct? Explain.
2.
x = y
x
Always True
Sometimes True
Never True
y
It is true for !
Show your reasoning:
3.
(1! 4x )(1+ 4x ) = !15
Always True
Sometimes True
Never True
It is true for Show your reasoning:
© 2012 MARS University of Nottingham S-3
Always, Sometimes, or Never True? 1. Take turns to place a card in a column. 2. If you think the equation is sometimes true: • Find the values of x for which the equation is true, and the values of x for which it is not true. • Explain why the equation is true for just those values of x. If you think the equation is always true or never true: • Explain how you can be sure that this is the case. 3. Other members of the group then decide if the card is in the correct place. • If you think it is correctly placed, explain why. • If you think it is not correctly placed, move the card to a new position and explain your reasoning. 4. When you all agree, write your reasoning on the poster next to the card. Alpha Version January 2012
© 2012 MARS, University of Nottingham
Projector Resources:
1
Sharing posters 1. If you are staying at your desk, be ready to explain the reasons for the placement of cards on your poster. 2. If you are visiting another group: • Copy the table from your poster onto a piece of paper. • Go to another group and read their poster. • Check: Do they place any cards in categories that differ from your group s poster? • If there are differences, ask for an explanation. • If you still don't agree, explain your own thinking. 3. Return to your own group. • Do you need to make any changes to your own poster? Alpha Version January 2012
© 2012 MARS, University of Nottingham
Projector Resources:
2
Mathematics Assessment Project Formative Assessment Lesson Materials
Manipulating Radicals
MARS Shell Center University of Nottingham & UC Berkeley Alpha Version
Please Note: These materials are still at the “alpha” stage and are not expected to be perfect. The revision process concentrated on addressing any major issues that came to light during the first round of school trials of these early attempts to introduce this style of lesson to US classrooms. In many cases, there have been very substantial changes from the first drafts and new, untried, material has been added. We suggest that you check with the Nottingham team before releasing any of this material outside of the core project team.
If you encounter errors or other issues in this version, please send details to the MAP team c/o [email protected].
© 2012 MARS University of Nottingham
Manipulating Radicals
Teacher Guide
1
Manipulating Radicals
2
Mathematical goals
3
This lesson unit is intended to help you assess how well students are able to: •
4 5
•
6
Alpha Version January 2012
Use the properties of exponents, including rational exponents, and manipulate algebraic statements involving radicals. Discriminate between equations that are identities and those that are not.
8
In this lesson there is also an opportunity to consider the role of the complex number i = optional.
9
Standards addressed
7
10 11 12 13 14 15
This lesson relates to the following Standards for Mathematical Content in!the Common Core State Standards for Mathematics: N-RN: Extend the properties of exponents to rational exponents. A-SSE: Interpret the structure of expressions. N-CN Perform arithmetic operations with complex numbers. This lesson also relates to the following Standards for Mathematical Practice in the CCSS:
17
3. 7.
18
Introduction
19
The unit is structured in the following way:
16
20
Construct viable arguments and critique the reasoning of others. Look for and make use of structure.
•
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
"1 , but this is
• • • • •
Before the lesson, students work individually on an assessment task that is designed to reveal their current levels of understanding. You review their work and create questions to help students improve their solutions. During the lesson, students first work in small groups on a collaborative discussion task. After sharing their solutions with another group, students extend and generalize the math. In an optional collaborative task focuses on complex numbers. In a plenary whole-class discussion, students review the main mathematical concepts of the lesson. Finally students return to the original task, to try to improve their individual responses.
Materials required Each student will need a copy of the assessment tasks, Operations With Radicals, and Operations With Radicals (revisited), a mini-whiteboard, a pen, and an eraser. • Each small group of students will need the cards Always, Sometimes, or Never True? (cut up before the lesson), a glue stick, and a large sheet of paper for making a poster. • There are some projector resources to support the whole-class discussions. You may want to copy the Card Sets onto transparencies to be used on an overhead projector to support whole-class discussions. •
Time needed Approximately fifteen minutes before the lesson, an eighty-minute lesson (more if complex numbers are introduced), and fifteen minutes in a follow-up lesson (or for homework). Precise timings will depend on the needs of your class.
40
© 2012 MARS University of Nottingham
1
Manipulating Radicals
Teacher Guide
Alpha Version January 2012
41
Before the lesson
42
Assessment task: Operations With Radicals (15 minutes)
43
Set this task in class or for homework a few days before the lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. This should help you to target your help more effectively in the lesson.
44 45 46 47 48 49 50 51 52 53 54
Give each student a copy of the assessment task Operations With Radicals.
Manipulating Radicals
56 57 58
1.
59 60 61 62
9 = +3.
x = 7
Always True
x 7
Sometimes True
Never True
It is true for !
Show your reasoning:
If you change 7 to another number, is your answer still correct? Explain.
Throughout this task, the radical sign denotes a positive square root.
Explain to students that they need not worry too much if they cannot understand or do everything in the task. In the next lesson they will engage in similar work, which should help them to progress.
means the positive square root. So
For each of the following statements, indicate whether it is true for all values of x, true for some values of x or there are no values of x for which it is true. Circle the correct answer. ! ! true”, state all values of x that make If you choose “sometimes it true.
Read through the questions and try to answer them as carefully as you can.
It is important that students are allowed to answer the questions without your help, as far as possible.
Alpha Version January 2012
Operations With Radicals In these questions, the symbol
2.
55
Student Materials
x " y =
x "
y
Always True
Sometimes True
Never True
It is true for
!
Show your reasoning:
3.
!
(1" 2x )(1+ 2x ) = "5
Always True
Sometimes True
Never True
It is true for Show your reasoning:
63 64
Assessing students’ responses
65
Collect students’ responses to the task, and think about what the students’ work reveals about their current levels of understanding.
66 67 68 69 70 71 72 73 74 75
© 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 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 the trials of this unit. Write a selection of these questions on each piece of student work. If you do not have time, 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.
76 77 78 79 80 81 82
© 2012 MARS University of Nottingham
2
Manipulating Radicals
Teacher Guide
Common issues:
Alpha Version January 2012
Suggested questions and prompts:
Student cannot get started
• Try some values to decide whether you think the equation can ever be true. • How can you get rid of the square root?
Student uses only guess and check
• You have checked a few values of x. How can you manipulate the equation to check your answer is correct in all cases? • How can you be sure there are no other values of x for which this equation is true/false?
Student provides no, or poor explanation for conclusions
• How can you use math to explain your answer?
Student makes mistakes with manipulation of the equations
• You have changed the equation. How can you be sure that the two sides of the equation are in fact equal?
For example: The student equates expressions when she has only squared one side of the equation. Or: The student subtracts a value from one side of the equation but not the other. Student does not choose appropriate operations when manipulating the equations For example: The student does not square to remove square roots. Or: The student does not simplify the expressions on each side.
Student misinterprets the meaning of algebraic equations and expressions For example: The student assumes that because the expressions on each side of an equation are different, the equation can never be true.
• How could you manipulate the algebra to simplify the equation? • What other operation could you use to change the equation? Why would that be a helpful move? • What is the positive square root of an expression that is squared? • How could squaring both sides of the equation help? • Try some values for x. Are there any values for x that would make the equation true? • When the square root of 3 is multiplied by itself, what is the result?
Or: The student assumes that squaring an expression in x means that every x in the expression just becomes x2. Student is confused by roots of negative values For example: The student says that the statement is sometimes true because you can’t use negative values of x (Q1. or Q2.)
• Is the statement sometimes, always, or never true for positive x? • Can you take the square root of a negative number?
Student correctly answers the questions
•
The student needs an extension task. •
© 2012 MARS University of Nottingham
Look at this inequality: 2 x > 3x . Is it always, sometimes, or never true? Write an equation that is only true when x is a complex number. !
3
Manipulating Radicals
Teacher Guide
Alpha Version January 2012
83
Suggested lesson outline
84
Whole-class introduction (10 minutes)
85
Throughout this lesson, the radical sign,
86
Give each student a calculator, a mini-whiteboard, a pen, and an eraser.
87
91
Throughout the introduction, encourage students to first tackle a problem individually, and only then discuss it with a neighbor. In that way students will have something to talk about. Maximize participation in the whole ! class discussion by asking all students to show you solutions on their mini-whiteboards. Select a few students with interesting or contrasting answers to justify them to the class. Encourage the rest of the class to challenge these explanations.
92
Write this on the board:
88 89 90
, denotes the positive square root.
Always, Sometimes, Or Never True? x "5 = 93
x
"
5
Ask students: !
Is this statement always true, sometimes true, or never true?
94
97
It is important that they understand the problem, and spend some time thinking about what they need to include in their solutions. If students are struggling to get started, encourage them to choose a value to substitute into the equation.
98
Bring the class together and ask the following questions in turn:
95 96
What does always true mean? Sometimes true? Never true?
99 100
Show me a value for x that makes the statement incorrect. Now show me another…
101
Can you show me a value for x that makes the statement correct? [x = 5] Can you show me another? How do you know for certain there are no more?
102 103
We can find many values for x that make the statement false, and one value that makes it true. So is this statement “true,” “sometimes true, ” or never true? [Sometimes true.]
104 105 106 107 108 109 110 111
You may find students choose values of x that are less than 5, thus making the left hand side of the equation imaginary. You may choose to address this issue in detail later in the lesson (Collaborative Activity 3). For now, ask students to just use numbers that give positive values to square root. What is important is that they justify their classification, with reference to mathematically available criteria. Fully justifying the conjecture that the statement is sometimes true will involve the use of algebra. Prompt students to manipulate the equation. In this case, it is useful to square both sides of the equation.
113
What operations do you know for manipulating algebraic expressions? (E.g. add, multiply, take a power.)
114
What operation is it useful to perform here? [Squaring both sides of the equation.]
115
Why square both sides? [To keep them equal.]
112
116
Square the left side of the equation:
(
117
x " 5
2
)
= x " 5
© 2012 MARS University of Nottingham
!
(1)
4
Manipulating Radicals 118 119 120
( x " 5)2 = x " 5 Is this correct? [No.] Explain. [Squaring an expression in the brackets does not just mean squaring each component within the brackets separately.]
122 123
!Ask students to help expand the brackets. Write this on the board:
(
125
x "
2
) (
5 =
x "
5
)(
x "
)
5 = x " 2 x 5 + 5
126
The two expressions (1) and (2) are equal if:
127
x " 5 = x " 2 x 5 + 5 ! Prompt students to manipulate and simplify the resulting equation.
128 129
!
131
Help students to show that the expressions are equal, when x = 5:
132
x "5 = x "2 x 5 +5
133
" 2 x 5 = 10
135
136
! ! !
(2)
Is there any way of simplifying this? Remember to perform the same operation on both sides when manipulating the equation.
130
134
Alpha Version January 2012
Now square the right hand side. You may wish to make a deliberate mistake to address the common misconception that squaring an expression involving square roots involves only removing the square root signs. Write this on the board:
121
124
Teacher Guide
" x 5 =5 ! x= 5 ! x=5
140
Collaborative Activity 1 (20 minutes) Organize the class into groups of two or three students. Give each group a large sheet of paper and Cards A - F from the sheet Always, Sometimes, or Never True? On the board, show students how to divide their large sheet of paper into three columns, with headings Always True, Sometimes True, Never True.
141
Explain the activity to students.
137 138 139
142 143 144 145 146 147 148
You have a set of cards with statements on them. Your task is to decide whether the statements are always, sometimes, or never true. In your group, take turns to place a card in a column. Then explain your choice to your partner or partners. If you think the statements is sometimes true, you will need to find the values of x for which it is true and values of x for which it is not true. You will need to explain why the equation is true for just those values of x.
151
If you think the statement is always true, or never true, you will need to explain why you’re sure of that. Remember, if you’re showing it is always true or never true, substituting just a few values of x is not enough.
152
Your partner then has to decide whether the card is in the correct place.
149 150
153 154 155
When you agree, write down your explanations on paper. You may want to use the slide, Always, Sometimes, or Never True? to display the instructions for group work. © 2012 MARS University of Nottingham
5
Manipulating Radicals 156 157 158 159 160 161 162 163
While students work in small groups, you have two tasks: to note students’ work on the task, and to use what you notice to support students’ reasoning. Make a note of student approaches to the task Listen to and watch students carefully. In particular, be aware of whether students are addressing the difficulties they experienced in the assessment. • •
166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
Alpha Version January 2012
It does not matter if students do not manage to place all of the cards. It is more important that everyone in the group understands the categorization of each card.
164 165
Teacher Guide
•
Which values do students choose to substitute into equations? Is their choice of values purposive, or does it seem random? Which operations do students choose to perform on the equations? Do they simplify expressions and manipulate the equations without errors? Do they make errors in the manipulation of exponents? What kinds of explanations do students give for their choices? Are they satisfied with a “guess and check” approach? Or do they move to algebraic manipulation of expressions to find a general solution? Are they able to articulate the consequences of their algebraic work? Are the others in the group convinced by the arguments?
You can use the information to support students’ reasoning, and also to focus the whole-class discussion towards the end of the lesson. Support student reasoning Try not to correct students’ errors for them, but instead, direct their attention to the error, and ask them to explain what they have done. Use questions like those in the Common issues table to help students address their errors and misconceptions. If students are having difficulties in moving toward an algebraic solution, use prompts to remind them of what they already know:
182
Your equation is complicated in its current form. Is there any way to simplify it? Does that help you find the values of x for which the equation is true?
183
What do you know about manipulating algebraic expressions that might be useful?
184
What do you know about operations on exponents? Might that be useful?
181
185 186 187 188 189 190 191 192 193
Try not to make suggestions that move students towards a particular categorization of statements. Instead, prompt students to reason together. Once one student has explained the placement of a card, encourage another member of the group to participate either by explaining that reasoning again in his or her own words, or by challenging the reasons the first student gave. Alice, Gabriel placed that card in the “always true” column. Do you agree with her? OK, then explain her reasons to me in your own words/why you don’t agree. The purpose of this structured work is to encourage students to engage with each other’s explanations, and to take responsibility for each other’s understanding.
195
Most students will substitute into each equation positive integers for x. If you want the lesson to involve consideration of complex numbers, you may want to encourage them also substitute negative values.
196
Encourage your students to work algebraically, to develop convincing reasoning:
194
198
You have shown the statement is true for this specific value of x. Now convince me it is always true for all values.
199
Can you use algebra to justify your decision for this card?
197
© 2012 MARS University of Nottingham
6
Manipulating Radicals 200 201 202 203 204
Teacher Guide
Alpha Version January 2012
You may need to explain the reasoning required for statements that are sometimes true. Some students will leave their responses at “guess and check”, having found a few values for which the equation is true, and a few for which it is not true. In that case, emphasize that the task is to find exactly which values of x make the equation and which don’t. The student also needs to explain why that result holds. Sharing posters (10 minutes)
206
As students finish matching the cards, ask one student from each group to make a quick copy of how cards were placed on his or her poster on scrap paper. This only involves writing the letters used to identify each card (A-F.)
207
Ask one student from each group to visit another group's poster.
205
209
If you are staying at your desk, be ready to explain the reasons for your group's placement of equations on your poster.
210
If you are visiting another group, take the scrap paper with you.
211 212
Go to another group's desk and look for differences. Check to see which equations are in a different category from your own.
213
If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.
214
When you return to your own desk, you need to consider as a group whether to make any changes to your own poster.
208
215 216
You may want to use the slide Sharing Posters to display these instructions.
217 218
When students are satisfied that they have identified and discussed differences, they are to return to their own posters and negotiate any changes they think are needed with their partner(s).
219
Give each small group a glue stick, to glue the cards onto the poster.
220
Whole Class Discussion (10 minutes)
221
This activity provides students with an opportunity to generalize their work in the previous task.
222
Write the equation on the board again:
x "5 = x " 5
223
You decided before that the statement is sometimes true. It’s true just when x = 5.
224
The equation has x's in it, and fives. Now change the fives to another number.
225 226 227 228 229 230
!
Allow students a few minutes to think about this on their own. Then give the students a couple of minutes to discuss their initial ideas in pairs. Students may argue informally that changing the five to, for example, a four, 'just has to' make the solution four. Prompt them to provide a stronger justification, by manipulating the expression algebraically. We have looked at replacing five with lots of other numbers. Can anyone now think of a quicker way to check all numbers? [Replace the particular number with a variable.]
231 232 233
Write this statement on the board:
x"y = x " y
234
You can check for five, four, or any other number all at the same time if you can figure out when this algebraic equation is true. How might you do that?
235 236 237
Is the equation now always, sometimes or never true?
!
Which operations did you use before, with y = 5?
238
© 2012 MARS University of Nottingham
7
Manipulating Radicals
Teacher Guide
Alpha Version January 2012
239 240 241
Ask students to work first individually, and then in pairs to show you how to manipulate the equation, to show that:
x" y =
x "
y
Squaring the left side of the equation :
(
x" y
2
)
= x " y
Squaring the right side of the equation :
(
x "
y
2
) =(
x "
y
)(
x "
y
)
= x + y - 2 x y 242
The two expressions are equal when : x " y = x + y - 2 x y 2 x y = 2y x y = y Squaring both sides : xy = y 2
(
)
y y " x = 0 y = 0 and y = x.
245
As they work on the problem, support the students as in the first collaborative activity. If necessary, discuss the solution as a class. ! Collaborative Activity 2 (10 minutes)
246
Once you've established how to replace the number with a variable:
243 244
247
Choose one or two of the statements from your poster.
248
Replace the numbers in the equations with a variable, say, y.
249
Check to see if the statement should still remain in the same category.
250
For the statements you place in the “sometimes true” category, you should figure out exactly which values of x and y make the equation true.
251
253
They can write their equations and answers on the posters. Support the students as in the first collaborative activity.
254
Collaborative Activity 3 (10 minutes) (optional)
255 256
If you want to work on complex numbers in the lesson, give each group Cards G and H. These two cards introduce the students to complex numbers.
257
Now consider whether these new statements are always, sometimes, or never true.
252
258 259
What values of x make this statement true? Support student reasoning:
260
What is the result of substituting x = -1 into the statement?
261
How can you check the statement is correct for just this value / these values of x? © 2012 MARS University of Nottingham
8
Manipulating Radicals 262 263
Teacher Guide
Alpha Version January 2012
You may also want to ask students to check that their categorization of the cards is correct for negative as well as positive values.
264 265 266 267 268
Whole-class discussion (10 minutes) Organize a whole-class discussion about what students have learned. Ask each group to write on a mini-whiteboard an equation from their poster that meets some chosen criterion. For example, you might ask: Show me an equation that is always true. Is your equation still always true when the equation contains two variables?
269 270
Show me an equation that is sometimes true. For what values is it true for the equation with two variables?
271 272 273
Encourage students to explain their answers using examples and justifications.
275
Does anyone disagree with this classification? Why? Which explanation do you prefer? Why?
276
Draw out any issues you have noticed as students worked on the activity.
274
277 278 279 280 281
(3 + x )(3 " x )
282
Is this statement ever true? If so, for which x values is it true?
283 284 285 286 287 288 289 290
Ask students to write the equation for Card G with x = -1 on their whiteboards.
!
What do you notice when you substitute this value of x into the equation? What do you think
"1 means?
Ask students to put this complex numbers into context, by thinking about their background knowledge about types of numbers. Tell me ! some of the different types of numbers you know. Push for a wide range of numbers, including rationals, decimals, negative numbers, irrational numbers such as
2 and transcendental numbers such as
3" . 4
293
These are all called “real numbers”. The square root of a negative number is not a real number. A new kind of number is needed, which we call a complex number. The new symbol i is used to define the complex number √(-1)
294
You can use i to represent lots of other complex numbers.
295
How can
296
What about "2 ? [ 2 " #1 = 2 "
291 292
!
Optional Extension: Discussion of complex numbers using cards G and H If your students worked on Cards G and H, you may want to extend the discussion to include complex numbers. Write the statement from Card G on the board.
297 298
!
"9 be represented as a complex number? [ 9 " #1 = 9 " #1 = 3i .]
A similar discussion can be made about Card H.
!
!
#1 = 2i .] !
!
© 2012 MARS University of Nottingham
9
Manipulating Radicals
Teacher Guide
Alpha Version January 2012
299
Follow-up lesson: Revisiting the initial assessment task and reflecting on learning (15 minutes)
300
Return to the students their original assessment task, and a copy of Operations With Radicals (revisited).
301
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.
302 303
Look at your original responses and think about what you have learned this lesson.
304
Carefully read through the questions I have written.
305
Spend a few minutes thinking about how you could improve your work.
306
You may want to make notes on your mini-whiteboard.
307
Using what you have learned, try to answer the questions on the new task: Operations With Radicals (revisited).
308 309
If you find you are running out of time, you could set this task in the next lesson or for homework.
310
© 2012 MARS University of Nottingham
10
Manipulating Radicals
Teacher Guide
311
Solutions
312
Assessment Task: Operations With Radicals
313
1.
314 315 316
! 317 318
319 320 321
322
323
Alpha Version January 2012
x x = The statement is always true, for any value of x. 7 7 Whatever the value of x substituted into the equation, it remains true. Statements that are always true are identities. Some students might justify this claim only by substituting a few values of x and showing that the equation is true for those values. This is not sufficient justification. Others may just appeal directly to the laws for the x
" a %x a . manipulation of indices: $ ' = x #b& b
() ()
Whilst this shows that the statement is true, it does not help explain why the statement is always true. Squaring and taking the positive square root of the right hand side of the equation uses the laws for the manipulation of indices, but shows how this does not change the value of the fraction: !
" x %2 $$ '' = 7 # &
x( x 7( 7
=
x 7
This is clearly true for any value of x.
324
Replacing 7 by any other numbers does not affect the truth of the statement. This can be shown by the same
325
!methods for a particular value, but that does not establish the generality. A full solution will involve replacing
326
7 with a variable, y, say, and manipulating the algebra to show equality.
327
© 2012 MARS University of Nottingham
11
Manipulating Radicals 328
2.
331
x " y =
x "
Alpha Version January 2012
y.
This statement is sometimes true, that is, true when either x = y or when y = 0.
329 330
Teacher Guide
!
For a full solution, the student is asked not only to identify some values of x for which the statement is true and some for which it is false, but to state exactly which values of x make the statement true.
333
Students may just notice that if x = y, both sides are zero. This is not a complete solution, as it does not provide any justification for the general statement and it misses the solution x = y.
334
Consider the left hand side of the equation. Squaring:
332
(
335 336
337
(
!
)
2
2
!
!
346
Squaring both sides:
y 2 = xy Therefore x = y or y = 0 . A common error is to divide
! 3.
!
x y = y by
y , which hides the solution y = 0.
! (1" 2x )(1+ 2x ) = "5 . This statement is sometimes true, when x = 3.
!
348
2
= x "2 x y + y = x "2 x y + y
"y= x y
!
344
347
x" y
" 2y = 2 x y
!
345
= x"y # x"y =x"y
x " y = x "2 x y + y
!
342
343
2
If the two sides of the equation are equal, then
340 341
)
Squaring the right hand side of the equation:
338 339
x"y
(1! 2x )(1+ 2x ) = !5 " 1! 2x = !5 " x=3
!
349
© 2012 MARS University of Nottingham
12
Manipulating Radicals
352 353
Card A
351
x +2 = x + 2
354
(
357
359 360
!
!
363
2 x 2 =0 " x 2 =0
!
This is true only if x =0. For Collaborative Activity 2, we replace 2 with y, so the equation becomes x + y =
!
x + y.
365
Following the same method as before, find that the statement is true only if x = 0 or y = 0 . Logically, this includes the possibility that both values are zero.
366
Card B
364
3x = 3 " x
367
! !
This equation is always true.
!
Squaring the right hand side of the equation, then taking the positive square root:
368
(
369
3" x
)
2
=
(
) (
3" x "
)
3" x =
3 " 3 " x " x = 3 " x = 3x
370
Therefore this equation is an identity, true for any value of x.
371
Card C
!
x 2 + 22 = x + 2
372
This equation is sometimes true.
Some students may simply remove the square root and square notation, and assume that this equation is always true. Other students may try a few values of x and assume it is never true.
373 374
!
2
Subtracting x + 2 from each side of the equation:
! !
x +2
" x +2 = x +2+2 x 2
361 362
2
) = ( x + 2) " x + 2 = ( x + 2 )( x + 2 )
356
358
This equation is sometimes true, when x = 0.
Squaring both sides of the equation:
355
!
Alpha Version January 2012
Lesson Task For each card, students should identify values for x for which the equation is true, and explain why the equation is true (or is not true) for just those values.
350
!
Teacher Guide
x 2 + 22 = x + 2
375
Suppose that
376
2 2 Then $ x + 2 ' = x + 2
" #
%2 &
(
)2
! " x 2 + 22 = ( x + 2)( x + 2) = x 2 + 2 # 2x + 22 = x 2 + 4x + 4
377
Thus ! the equation is true just when 4x = 0 which occurs only when x = 0 .
378
For Collaborative Activity 2, we replace 2 with y, say.
379
! !
© 2012 MARS University of Nottingham
!
13
Manipulating Radicals
Teacher Guide
x2 + y2 = x + y
380
If we suppose that
381
" % 2 Then $ x 2 + y 2 ' = x + y
2
#
(
&
)
! " x + y 2 = ( x + y )( x + y ) = x 2 + 2xy + y 2 = x 2 + 2xy + y 2 2
382
Thus ! the equation is true just when 2xy = 0 in which case either x = 0 or y = 0 .
383
Card D
384
! 385
!
Alpha Version January 2012
x 2 " 22 = x " 2
This equation is sometimes true, ! when x = 2 .
!
!
386
As with Card C, an error that students sometimes make is to assume that you can simply remove the squares and
387
root notation from
x 2 " 22 and that the equation is thus always true. 2
x " 2 = x " 2.
Suppose that
388
2
!
2
2 ! # 2 2 & Then % x " 2 ( = x " 2 $ ' ! So x 2 " 22 = x " 2 x " 2 = x 2 " 2 # 2x + 22 = x 2 " 4x + 4
389
(
390
)
)(
)
Thus!the equation is true just when 4x = 8 .
391
This occurs when x = 2 . ! For Collaborative Activity 2, replace 2 with y.
392 393
If
394
!
!
x2 " y2 = x " y .
2 # 2 2 & % x " y ( = x " y Then $ '
(
395
! 396
2
)
So x 2 " y 2 = x " y x " y = x 2 " 2xy + y 2
(
)(
)
397
2 2 2 Thus!the general equation is true just when " y = " 2xy + y , that is, when 2 y = 2xy
398
2 This ! occurs when y = xy
399
A possible student error is to divide ! through by y at this point, which!hides the solution y=0.
400
Instead, factorize: ! y 2 = xy
401
" y 2 # xy = 0
402 403 404 405
" y( y # x) = 0 ! The equation is true just when y=x or y=0. ! Card E !
406 407
!
(
x 2 " 32 = 3x
This equation is always true.
Using the laws for the manipulation of indices,
x 2 = x and
32 = 3 . So
© 2012 MARS University of Nottingham
x2 "
32 = 3x 14
!
!
!
Manipulating Radicals 408
Alpha Version January 2012
For Collaborative Activity 2, replace 3 with y.
x2 "
409
Since
410 411
Teacher Guide
y 2 = xy
x 2 = x for any value,
x2 "
y 2 = xy
Card F !
x2 ! x = 3 32
412
This equation is ! always true.
x 2 = x and
32 = 3 . Thus
414
As with Card G, this is also true if 3 is replaced with y, except that we must then exclude the possibility that y=0. ! ! Card G ! 3 + x 3 " x = 10 . This equation is sometimes true if x may take negative values (and the square root of
415 416
(
)(
32
=
)
418
a negative number is permitted). In this case, x = -1. If x is only considered to be zero or positive, then the statement is never true.
419
To find the values of x for which it is true, first clear the brackets.
417
420
(3 + x )(3 " x ) = 10
421
" 9 # 3 x + 3 x # x = 10
422 423 424
" 9 # x = 10 ! Therefore the equation is true if x = "1 ! Substituting these values of x back into the equation uses i, the square root of -1. !
425
(3 + "1)(3!" "1) = 10
426
(3 + i)(3 " i) = 10
427
!Thus if students look only at values of x that are positive or zero, this statement is never true.
429
Since Card G and H are introduced after Collaborative Activity 2, the general case (substituting y for 3) is not ! considered by students. This could, though, be used as an extension activity.
430
Card H
431 432
x + 1+ x " 1 = allowed, when x = -1.
433
Consider
434
Squaring this expression:
435
!
428
!
x . 3
Using the laws for the manipulation of indices,
!
!
x2
413
x + 1+
( x + 1+
2x
This equation is sometimes true, when x = 1, and also if complex numbers are
x " 1.
x " 1)2 = ( x + 1 +
x " 1)( x + 1 +
= x + 1 + 2 x +1 x " 1 + x " 1 = 2( x +
436
x " 1)
x + 1 x " 1) = 2( x +
x 2 " 1)
! ! © 2012 MARS University of Nottingham
15
Manipulating Radicals
Teacher Guide
( ) 2x
437
Squaring the right hand expression gives
438
Thus the two sides are equal if 2x = 2( x +
2
Alpha Version January 2012
= 2x .
x 2 " 1)
442
2 This occurs if x 2 " 1 = 0 , that ! is, if x = 1. Thus x = 1 or x = -1. In the latter case, both sides of the expression are complex numbers. ! Operations With Radicals (revisited) ! ! These solutions are analogous to those in the initial assessment, so we will not repeat all the reasoning.
443
1.
439 440 441
x ! x = 2 x is sometimes true when x = 0 or x = 4. If the two expressions are equal x = 2 x " x 2 = 4x " x(x # 4) = 0
444
" x = 0 or x = 4. If 2 were replaced by y then the equation is sometimes true when x = 0 and when x = y 2 445
2.
x = y
446
x y
is always true.
447 448
3.
(1! 4x )(1+ 4x ) = !15 is sometimes true when x = 4
449
© 2012 MARS University of Nottingham
16
Manipulating Radicals
Student Materials
Alpha Version January 2012
Operations With Radicals In these questions, the symbol
means the positive square root. So
9 = +3.
For each of the following statements, indicate whether it is true for all values of x, true for some values of x or there are no values of x for which it is true. Circle the correct answer. ! ! If you choose “sometimes true”, state all values of x that make it true.
1.
x = 7
Always True
x 7
Sometimes True
Never True
It is true for !
Show your reasoning:
If you change 7 to another number, is your answer still correct? Explain.
2.
x " y =
x "
y
Always True
Sometimes True
Never True
It is true for
!
Show your reasoning:
3.
!
(1" 2x )(1+ 2x ) = "5
Always True
Sometimes True
Never True
It is true for Show your reasoning:
© 2012 MARS University of Nottingham S-1
Manipulating Radicals
Student Materials
Alpha Version January 2012
Cards: Always, Sometimes, or Never True? A.
B.
x +2 =
!
x +
2
3x =
!!
!
C.
D.
x 2 + 22
!
3• x
x 2 " 22
= x +2
= x " 2
!
E.
F.
2
2
x • 3
x2
= 3x
32
! G.
H.
(3 +
)(
x 3 "
x
)
= 10
=
x 3
!
x + 1 +
x " 1 =
2x
!
!
© 2012 MARS University of Nottingham S-2
Manipulating Radicals
Student Materials
Alpha Version January 2012
Operations With Radicals (revisited) In these questions, the symbol
means the positive square root. So
9 = +3.
For each of the following statements, indicate whether it is true for all values of x, true for some values of x or there are no values of x for which it is true. Circle the correct answer. ! ! If you choose “sometimes true”, state all values of x that make it true.
1.
x "
x = 2 x
Always True
Sometimes True
Never True
It is true for
!
Show your reasoning:
If you change 2 to another number, is your answer still correct? Explain.
2.
x = y
x
Always True
Sometimes True
Never True
y
It is true for !
Show your reasoning:
3.
(1! 4x )(1+ 4x ) = !15
Always True
Sometimes True
Never True
It is true for Show your reasoning:
© 2012 MARS University of Nottingham S-3
Always, Sometimes, or Never True? 1. Take turns to place a card in a column. 2. If you think the equation is sometimes true: • Find the values of x for which the equation is true, and the values of x for which it is not true. • Explain why the equation is true for just those values of x. If you think the equation is always true or never true: • Explain how you can be sure that this is the case. 3. Other members of the group then decide if the card is in the correct place. • If you think it is correctly placed, explain why. • If you think it is not correctly placed, move the card to a new position and explain your reasoning. 4. When you all agree, write your reasoning on the poster next to the card. Alpha Version January 2012
© 2012 MARS, University of Nottingham
Projector Resources:
1
Sharing posters 1. If you are staying at your desk, be ready to explain the reasons for the placement of cards on your poster. 2. If you are visiting another group: • Copy the table from your poster onto a piece of paper. • Go to another group and read their poster. • Check: Do they place any cards in categories that differ from your group s poster? • If there are differences, ask for an explanation. • If you still don't agree, explain your own thinking. 3. Return to your own group. • Do you need to make any changes to your own poster? Alpha Version January 2012
© 2012 MARS, University of Nottingham
Projector Resources:
2
