Download RADICALS (2 weeks)

RADICALS (2 weeks)
Introduction to Radicals
Overview of Objectives, students should be able to:
1.
Find square roots
2.
Approximate square roots
3.
Find cube roots
4.
Simplify radical containing variables
Objectives:
•
Find square roots
Main Underlying Questions:
1. How do you find square and cube roots?
2. How do you simplify radical expressions containing variables?
3. What is the relationship between exponents and roots?
Activities and Questions to ask students:
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What is 5 2 ? What is 6 2 ?
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Tell students the process of “undoing” a square is called the square root, symbolized by
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What is another way to explain the process of taking the square root?
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What is
25 ? What is 36 ?
50 ? Is there a way to approximate the number?
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Approximate square roots
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What is
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Find cube root
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Tell students the cube root is symbolized by
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•
•
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Discuss the parts of the radical using terminology: index radicand
Give examples of higher order radicals and ask students what relationship they see between
the index and the type of the root.
If the square root is the process of “undoing” a square root, what does the cube root do?
What is another way to explain the process of taking the cube root?
•
What is
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What is
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Have students draw the conclusion that
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Simplify radical containing variables
3
3
8 ? What is 3 64 ?
9 ? What is 3 2 ? Have students work several examples like this.
a2 = a
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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Remind students to think about what the process of the square root is.
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What if we have numbers and variables
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What is
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Have students work several examples to show that
9a 2 ?
3 4 = 81 = 9 ? Have students then write 9 as a power of 3: 32
an = a
n
2
Simplifying Radicals
Overview of Objectives, students should be able to:
1.
Use the product rule to simplify radicals
2.
Use the quotient rule to simplify radicals
3.
Use both rules to simplify radicals containing variables
Objectives:
•
•
Use the product rule to simplify radicals
Use the quotient rule to simplify radicals
Main Underlying Questions:
1.
How do you remember the rules to simplify radical expressions?
2. How do you decide which rule to use when simplifying radical expressions?
Activities and Questions to ask students:
4⋅ 9.
4 ⋅ 9 . Do you see a relationship?
•
Ask students to evaluate:
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Ask students to evaluate:
Have students work several pairs of examples of this type. Does a pattern exist?
•
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What is a 2 ⋅ b 2 ? What about
square root process is.
Do you notice a pattern?
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Have students draw the conclusion that:
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How would you simplify
•
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How about 2 12 ? How does the “2” on the outside change the process?
Have students work several numeric examples of this type.
•
Ask students to evaluate:
a 2 ⋅ b 2 ? Remind students to think about what the
a ⋅b = a ⋅ b
18 = 9 ⋅ 2 using the product rule?
16
4
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
Use both rules to simplify radicals containing variables
16
. Do you see a relationship?
4
•
Ask students to evaluate:
•
Have students work several pairs of examples of this type. Does a pattern exist?
a2
a2
?
b2
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What is
•
Do you notice a pattern?
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Have students draw the conclusion that:
•
•
b2
? What about
a
=
b
a
b
Start with an example like x 6 . Ask students to simplify the radical. Give several examples
with even exponents for students to work.
How is this like what we have done before?
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What about x 7 = x 6 ⋅ x 1 ? Do you see a pattern or process to if the exponent is odd?
Give students several more examples to work.
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Now continue with examples like
•
•
•
•
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How do you know which rule to apply?
Give a worksheet with several problems of varying difficulty.
Ask students to talk through their strategy on how to solve.
Is there more than one way to perform the examples?
Is it correct to say there are many ways to simplify? Why?
8x 3 ,
12
.
x4
Adding and Subtracting Radicals
Overview of Objectives, students should be able to:
1.
Add or subtract like radicals
2.
Simplify square root radical expressions, and then add or
Main Underlying Questions:
1. How do you know when to add or subtract radicals?
2. How do you know when a radical expression is fully simplified?
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
subtract any like radical
Objectives:
•
Add or subtract like radicals
Activities and Questions to ask students:
•
•
•
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Ask students how they would simplify 2 x + 3 x = 5 x .
Where did the “5” come from?
Why didn’t x become x2?
If students have difficulty, ask “What is 2 apples plus 3 apples?” Is it 5 apples (x) , or 5
tangerines (x2)?
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Now give simple radical example: What is 2 5 + 3 5 . Ask students to consider 5 is “x”
or the apple as in the previous examples.
Write down the process you used to add the radicals. Give several more simple examples
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•
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Simplify square root radical expressions, and then add or
subtract any like radical
9 + 16 ? Can you simplify like you did before? Why or why not?
If students have trouble and guess 9 + 16 = 25 = 5 ask them if there was another way
to find the answer. 9 + 16 = 3 + 4 = 7 . Why did the first way not work?
What about
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How can you tell that square roots cannot be added or subtracted together? Write down
several examples where they can and where they cannot be added or subtracted.
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If 4 + 9 cannot be combined together, is there another way to simplify?
Have students work several examples of this type. What process can you use to simplify?
(simplify and then add or subtract)
•
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What about 12 + 3 ? How can you simplify each radical first? Have you done this
before? What rule or process can you use to help you simplify? How does the “2” on the
outside of the radical change the process?
Write down the process you used to solve the problem.
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•
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How about 5 x + 4 x ? Write down the process you used to solve.
Give a worksheet with several problems of varying difficulty (some with variables).
Ask students to talk through their strategy on how to solve.
Is there more than one way to perform the examples?
Is it correct to say there are many ways to simplify? Why?
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
Multiplying and Dividing Radicals
Overview of Objectives, students should be able to:
1.
Multiply radicals
2.
Divide radicals
3.
Rationalize denominators
4.
Rationalize using conjugates
Objectives:
•
Multiply radicals
Main Underlying Questions:
1. How do you multiply or divide radicals?
2. How do you know when a radical expression is fully simplified?
Activities and Questions to ask students:
•
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Ask students to rewrite the product rule for radicals: a ⋅ b = a ⋅ b
How could you use the rule to multiply two radicals? Write the process you would use.
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Ask students to multiply: 6 ⋅ 3 = 18 . Can we simplify this further? Why? What
processes have you used before to solve.
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Ask students to multiply: 6 x x − 3 . Which previously learned multiplication
technique did you need to solve the problem?
Give a worksheet with several multiplication problems (FOIL, square of a binomial). Ask
students to write down the previously learned multiplication technique they needed to solve.
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Divide radicals
•
(
)
Ask students to rewrite the quotient rule for radicals:
a
=
b
a
b
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How could you use the rule to divide two radicals? Write the process you would use.
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Ask students to divide:
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Ask students to divide:
40
10
. Write the process you used to simplify.
120 x 3
3x
. Which previously learned technique did you need to solve
the problem?
Give a worksheet with several division problems (FOIL, square of a binomial). Is there more
than one way to simplify each problem? Compare with your classmates. Which one works
the best?
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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Rationalize denominators
•
Ask students to simplify:
examples of this type.
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Ask students to divide:
Rationalize using conjugates
4
2
. What is different about this problem vs. other division
=
5
5
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problems?
How could you write an equivalent fraction that has no radical in the denominator? Ask
students to think about how they can write equivalent fractions when adding and subtraction
fractions. Write the process and operation you would use.
What could you multiply by to achieve your goal? Write down your process.
Tell students this process is called “rationalizing the denominator”
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Give an example: rationalize the denominator of
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5 ⋅ 5 = 5 . Do you notice a pattern? Give several more
2x 2
3x
. Write down your steps.
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How do you know when you need to rationalize the denominator?
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Ask students to simplify: (2 + 5 )(2 − 5 ) . Do you notice a pattern? Give several more
examples of this type. What happens to the radical terms?
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How could you use this pattern to rationalize the denominator of:
•
multiply by to remove the radical term? Write the process you would use.
Tell students that if an expression is a + b then its conjugate is a − b or vice versa.
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Give an example: rationalize the denominator of
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What is the conjugate of the denominator?
Write down your steps to solve.
4
x −1
2
2+ 5
. What could you
.
Solving Equations Containing Radicals
Overview of Objectives, students should be able to:
1.
2.
Solve radical equations by using the squaring property of
equality
Main Underlying Questions:
1. How do you solve radical equations?
2. How do you check your solutions are correct?
Solve radical equations by using the squaring property of
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.
equality twice.
Objectives:
•
Solve radical equations by using the squaring property of
equality
3. What is an extraneous solution?
Activities and Questions to ask students:
•
•
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Give students a simple example: if x = 2. What is x2 = ? What about if x = 3, x2 = ?
Do you see a pattern? What “operation” are we performing on both sides of the equation?
Have students draw the conclusion that if a = b then a 2 = b 2 (squaring principle)
•
Now ask students how they would solve: x = 4 . If students just observe the answer is 16,
ask them how they would solve the equation using the squaring principle. Does it match the
solution you observed?
How can you check your solution is correct? Write down the process you used to solve and
check your answer.
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•
•
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Solve radical equations by using the squaring property of
equality twice.
How do you solve x = −4 ? What happens if you use the squaring principle? How could
you check that 16 is not the solution?
Give another example similar to this one. Do you see a pattern? How could you predict
there would be no solution?
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How would you solve: x + 4 − 6 = 2 . How is this example different than the last one?
How would you need to modify your process to solve it?
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Ask students how they would find:
x + 4 − 2 . Have you done this before? Which
previously learned processes or rules are you using?
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How would you solve: x = x + 4 − 2 ? What is different about this example than the last
ones? How can you use the squaring process at the beginning to help you solve?
After using the squaring property you still have a radical in the equation, now what do you
do?
How many radicals do you have now? Does it look similar to the first type of radical
equations you solved? Write the process you would use to continue.
Ask students to summarize the process of solving radical equations of the types studied.
Give students a worksheet with several radical equations (1 and 2 radicals, some with real
solutions, and some with no real solution) to complete. Have them use the process they
wrote down.
Is there more than one way to solve? Compare with your classmates.
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(
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2
The contents of this website were developed under Congressionally-directed grants (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the
U.S. Department of Education, and you should not assume endorsement by the Federal Government.