RATIONALS (1 week)
Simplifying Rational Expressions
Overview of Objectives, students should be able to:
•
Find the value of a rational expression given a
replacement value
•
Identify values for which a rational expression is
undefined
•
Simplify, or write rational expression in lowest terms.
•
Write equivalent forms of rational expressions
Objectives:
•
•
Find the value of a rational expression given a
replacement value
Identify values for which a rational expression is
undefined
Main Underlying Questions:
1. What is a rational expression?
2. How can you tell when a rational expression is undefined?
3. How can you tell when two fractions are equivalent?
Activities and Questions to ask students:
•
•
What is the difference between a rational/irrational number?
What is a rational expression?
•
Evaluate a simple rational expression:
•
In what order did you need to simplify the expression?
•
Evaluate a more complicated rational expression:
•
•
In what order do you need to simplify?
What previous rules did you need to correctly simplify the expressions after evaluation?
[order of operations, exponent rules]
•
What is
•
•
What number cannot be divided by? What part of the fraction would that translate to?
What is another way to say a fraction is undefined? (divide by 0, 0 in the denominator, etc.)
1
for x = 6.
x−4
x−6
for x = 2.
x − 2x + 4
2
4
? If students believe it is 0, ask “how many times can 0 go into 4?”
0
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.
•
Simplify, or write rational expression in lowest terms.
•
Do you need to consider the numerator? Why not?
•
Give a simple example: for what value of x is
•
•
How did you find when the expression was undefined?
How can you check that x = 6 is when it is undefined?
•
Give a harder example: for what value of x is
•
•
•
If we can’t just “look at” where the denominator is 0, what process could we use to find it?
If we want 4x - 7 to be equal to 0, what equation could we write to find this?
Write down the process you used to find the answer.
•
Give another harder example: for what values of x is
•
•
What technique did you need to solve your equation?
Write down the process you used to find your answer and compare to the other examples.
•
How do you reduce the fraction
reduction?
•
Write equivalent forms of rational expressions
1
undefined?
x−6
x −1
undefined?
4x − 7
x−2
undefined?
x + 3x + 2
2
16
? What process or operations did you use in your
30
2x 3 2 ⋅ x ⋅ x ⋅ x 2
=
= ? What process or operations did you use in your
x⋅x⋅x⋅x x
x4
•
What about
•
reduction?
How do you know when it is reduced to lowest terms?
•
Give another example: how could you simplify
•
the last example.
What processes do you need in order to simplify a rational expression?
2x 2 + 4x
? Remember how you simplified
6 x 2 + 12 x
•
x2 − x − 2
?
How could you simplify 2
x + 5x + 4
•
What happens to the numerator when simplifying:
•
Do you see a pattern in the last two examples? (i.e. when the numerator “disappears” we
must write a 1)
How can we tell if two fractions are equivalent?
•
x5
x−5
? What about 2
?
8
x
x − 7 x + 10
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.
8
?
16
•
What are some equivalent fractions of
•
Compare the two fractions
•
Give students some other examples like the last one, does a pattern exist?
•
•
•
−3
3
and
.
6
−6
−a
a
and
? Are they the same or different?
b
−b
x+ y
−x− y
How about
and
?
−x− y
x+ y
How about
Is it correct to say there are many ways to represent equivalent fractions?
Multiplying and Dividing Rational Expressions with monomials
Overview of Objectives, students should be able to:
1.
Multiply rational expressions
2.
Divide rational expressions
3.
Multiply and divide rational expressions
Main Underlying Questions:
1. How do you multiply and divide rational expressions?
2. How do you convert between units of measure?
4. Convert between units of measure
Objectives:
•
Multiply rational expressions
Activities and Questions to ask students:
4 7
⋅
5 8
•
How do you multiply fractions? i.e.
•
Give students several numeric examples. Do you see a pattern or process to multiply
fractions?
•
•
P R P⋅R
. Then, simplify.
⋅ =
Q S Q⋅S
2 x−5
Have students work the example:
⋅
x−6 3
Have students draw out the process that
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.
•
•
•
•
Divide rational expressions
•
•
•
Multiply and divide rational expressions
•
Convert between units of measure
4 7 4/ 7 1 7 7
⋅ = ⋅ = ⋅ =
5 8 5 8/ 5 2 10
x−6 4
How can we use the shortcut to multiply:
?
⋅
3 x−6
6x + 6
y2
Have students work the example:
⋅
10 x + 10
y
3 7
How do you divide fractions? ? i.e.
÷
4 8
Was there a shortcut to our first example?
Give students several numeric examples. Do you see a pattern or process to divide fractions?
P R P S P⋅S
. Then, simplify.
÷ = ⋅ =
Q S Q R Q⋅R
•
Have students draw out the process that
•
•
•
•
•
•
Give students an example to use this process.
If you want to use the shortcut from multiplication, at what step would you perform it?
Have students work multiplication and division problems on a worksheet in groups.
Ask students to talk through the strategy they use to solve each problem.
Is there more than one way to solve each problem? Explain.
How many feet are in a yard? How many inches are in a foot? How could write these
equivalencies mathematically?
•
What is 6 written as fraction? Have students work several examples to see that
•
Now, what is 6 ⋅ 1 ? Have students verify that in general, a ⋅1 = a
•
Have students write 6 inches as a fraction:
•
Now have students multiply by 1:
a
=a
1
6 inches
.
1
6 inches
⋅ 1 . Have students check that
1
6 inches
⋅ 1 = 6 inches .
1
1 foot
? Tell students this is called a conversion
12 inches
•
If 1 foot = 12 inches, what is the ratio:
•
ratio.
Using what we have written, what is another way to write
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.
6 inches
6 inches 1 foot
?
⋅1 =
⋅
1
1
12 inches
•
•
Now, how can you use the multiplication process to simplify your result?
Summarize the process to convert from one unit to another. How do you know which unit
fraction to use? Have students work several examples to see the pattern or process emerge.
Adding and Subtracting Rational Expressions with the Same Denominator
Overview of Objectives, students should be able to:
•
1. How do you add and subtract rational expressions that have the same denominator?
Add and subtract rational expressions with common
denominators
•
Find the LCD of a list of rational expressions
•
Write a rational expression as an equivalent expression
whose denominator is given
Objectives:
•
Main Underlying Questions:
Add and subtract rational expressions with common
denominators
2. How do you find the LCD of a list of rational expressions?
Activities and Questions to ask students:
3 1
+
8 8
•
How do you add fractions with the same denominator? i.e.
•
Give students several numeric examples. Do you see a pattern or process to add fractions?
•
How do you subtract fractions with the same denominator? i.e.
•
Give students several numeric examples. Do you see a pattern or process to subtract
fractions?
•
•
5 1
−
8 8
P R P±R
. Then, simplify.
± =
Q Q
Q
3n 8n
Have students work out the example:
+
2m 2m
Have students draw out the process that
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.
•
•
Find the LCD of a list of rational expressions
Write a rational expression as an equivalent expression
whose denominator is given
3y
6
− 2
y + 3 y − 10 y + 3 y − 10
•
Have students work out the example:
•
How do you find the least common denominator?
•
Give several numeric examples. Do you see a pattern or process?
•
What is the least common denominator of
•
find the least common denominator.
When is this process useful? For what purpose could we use this?
•
How do you write equivalent fractions? Give a numeric example:
•
•
Give several numeric examples. Do you see a pattern or process?
Ask students what a cross product is. Can we use this to solve the problem another way? Do
we get the same answer?
•
Have students work the example:
•
•
Does using cross products work in this case? When can we use cross products?
When could you use this process to add or subtract fractions that do not have the same
denominator?
2
2 1
, .
3 6
2 3
, ? Write down the process you used to
x2 y
3 ?
= .
4 8
5
?
=
3 y 12 y 2
Adding and Subtracting Rational Expressions without the same Denominator
Overview of Objectives, students should be able to:
1.
Add and subtract rational expressions containing
monomials with different denominators
Main Underlying Questions:
1. How do you add or subtract rational expressions that contain different denominators?
2. Why do you have to get common denominators before adding/subtracting rational
expressions?
3. How is adding/subtracting rational expressions similar to adding/subtracting fractions?
Objectives:
Activities and Questions to ask students:
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.
•
Add and subtract rational expressions containing
monomials with different denominators
2 1
+
3 6
•
How do you add fractions with different denominators?
•
•
Give several numeric examples. Do you see a pattern or process?
How can you use what you have already learned to add or subtract fractions with different
denominators?
•
Have students work the example:
•
•
3
4
. Write down the process you used to solve
+
2
4x
2x
the problem.
Have students work in groups on a worksheet of problems containing a mixture of addition,
subtract, multiplication, and division problems.
Is there more than one way to solve each problem? Explain.
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.