Download It`s Hip To Be Square CCSS.Math.Content.8.NS.A.2

 Know that there are numbers that are not rational, and approximate
them by rational numbers.
• CCSS.Math.Content.8.NS.A.1 Know that numbers that are not
rational are called irrational. Understand informally that every
number has a decimal expansion; for rational numbers show that
the decimal expansion repeats eventually, and convert a decimal
expansion which repeats eventually into a rational number.
CCSS.Math.Content.8.NS.A.2 Use rational approximations of irrational
numbers to compare the size of irrational numbers, locate them
approximately on a number line diagram, and estimate the value of
expressions (e.g., π2). For example, by truncating the decimal expansion
of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and
explain how to continue on to get better approximations.
Standards for Mathematical Practices:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Key Vocabulary:
approximate, convert, decimal expansion, irrational number, rational number,
compare, estimate, expression, number line, rational approximations
Content Standard Mapping
Exploration 1
Exploration 2
Exploration 3
Exploration 4
Exploration 5
Exploration 6
Exploration 7
8.NS.A.1 8.NS.A.2
x
x
x
x
x
x
x
Mathematical Practices Mapping
MP.1
Exploration 1
Exploration 2
Exploration 3
Exploration 4
Exploration 5
Exploration 6
Exploration 7
x
MP.2
MP.3
MP.4
MP.5
MP.6
MP.7
x x x x x
x
x
x
x
x
x
x
x
x
x
x
x x MP.8
x x Are You Rational?
All real numbers are either rational or irrational. They can be classified
using the following diagram.
Early in our mathematical endeavors we began working with integers, whole
numbers, and natural numbers. Then we spent effort learning to work with
fractions. Any number that can be expressed as a fraction is a rational
number. Any decimal that terminates or repeats can be expressed as a
fraction. Any number that does not repeat or terminate is irrational.
Example A:
Express 0.85 as a fraction.
0.85 is 85 hundredths or
85
100
=
17
20
Therefore 0.85 is rational.
85
100
Example B:
Express 0.85 as a fraction.
Let x = 0.85
x = 0.8585858585858585..........
Multiply both sides by 100
100x = 85.85858585858585.........
Subtract the two equations
100x = 85.85858585858585.........
-x = 0.85858585858585..........
99x = 85
Solve the equation.
85
x=
99
Therefore 0.85 is rational.
Example C:
0.212112111....... does not terminate or repeat.
It is not the ratio of two rational numbers.
Therefore 0.212112111........ is irrational.
Example D:
π ! 3.14
22
π!
7
These are only approximations!
π = 3.1415926535........
π does not repeat or terminate.
Therefore π is irrational.
Exploration #1
Determine if the following are rational or irrational. If the number is
rational determine if it is repeating or terminating.
1. 13
2.
10. 0.3555
4
11. 0.3555....
9
3. 0.675
4. -
2
7
12. 4 +
13.
7
4
7
5π
5.
36
14.
6.
24
15. 0.33333333......
7.
50
2
2π
16. (3 +
7 )(3 -
7)
8. π
17. ( 7 +3)( 7 - 3)
9. 0.315315
18. 0.315
Exploration #2
As we already know we can express repeating and terminating decimals as
fractions. Many years ago mathematicians would express fractions as a sum
of unit fractions. This is a challenging activity. It is your task to express all
the 12ths as a sum of unit fractions. The 6ths are completed as an example
to go by.
1
6
4
6
1
=
=
10
1
2
+
+
1
2
15
6
1
5
6
6
=
=
1
4
1
2
+
+
1
3
12
6
1
6
3
6
=
=
1
3
1
2
+
+
1
6
1
3
+
1
6
The 12ths Challenge
1
12
5
12
9
12
=
=
=
2
12
6
12
10
12
=
=
=
3
12
7
12
11
12
=
=
=
4
12
8
12
12
12
=
=
=
Exploration #3
1. Show that 3.999999......... = 4
2. Show that 0.2499999999...... = 1/4
3. Show that it only takes 0.9999..... mathematicians to screw in a light bulb.
4. Find the sum of 2.99999...... and 3.99999...........
Exploration #4
15
3
2 3
1. Mario states the perimeter and area are both irrational.
2. Isabella the perimeter is rational, but the area is irrational.
3. Necie states the perimeter is irrational, but the area is rational.
4. Deshawn states the perimeter and area are both rational.
Decide who is correct and defend your reasoning.
Exploration #5
1. Charlie has determined that
7 and
3 are both irrational.
If that is true he makes the statement that the following are also irrational.
7x3 ,
7÷3 ,
7+3 ,
7-3
Is Charlie correct? Defend your reasoning!
2. Charlene's teacher showed her today that even though
2 x
2 resulted in an rational number.
She remembered that π was considered irrational.
She concludes that π x π must also be a rational number.
Is Charlene correct? Defend your reasoning!
2 was irrational
It's Hip To Be Square
CCSS.Math.Content.8.NS.A.2
The square root of 4 is 2.
You can make a square of
length 2 with 4 items.
4 =2
The square root of 9 is 3.
You can make a square of
length 3 with 9 items.
9 =3
The question is what about the square root of 11? It is clear that the square root of 11 is between 3 and 4!
The question the early mathematician faced was to approximate numbers
such as the square root of 11. We will examine a procedure for
approximating irrational numbers such as the square of 11 using rational
numbers.
Given:
The square root of 9 is 3.
You can make a square of
length 3 with 9 items.
9 =3
The square root of 16 is 4.
You can make a square of
length 4 with 16 items.
16 = 4
How many items how to be added to model of three squared to get the model
that represents four squared?
As you can see it would require 7 tokens.
If we had 11 tokens then how many of these 7 would we have?
You have 2 of the 7 needed to go from the model of
3 squared to the model representing 4 squared.
2
Thus 11 ! 3
7
Example:
Approximate the square root of 22.
Four squared is 16. Five squared is 25. The difference in 25 and 16 is 9. The
difference in 22 and 16 is 6.
Therefore,
6
2
22 ! 4 or 4
9
3
Exploration #6
Making Irrational Rational!!!!!
Approximate
the square
root of
these
numbers
This
number
is more
than
what
perfect
square
This
number
is less
than
what
perfect
square
The
difference
in the two
perfect
squares
The
difference in
the number
being
approximated
and the
perfect
square it is
more than.
The rational
approximation
of the
irrational
number
Estimate of
the square
root
(Using rational
approximation)
Actual
square
root
(calculator)
11
19
34
37
54
90
1. Discuss the "quality" of the approximation as the size of the square root
increases.
2. If this technique is used will the square root approximation ever be
greater than the actual square root?
Exploration #7
Just beyond the town of Xanadu there exists a place commonly called
"Radical Land". All numbers here are expressed in square roots.
For example if you needed what we refer to as 8 gallons of gas you would
need
64 gallons of gas.
Billy's House
School
Midpoint
Store
Billy lives on a road that is 25 miles from school and 36 miles from the store.
There is a sign that marks the point that is in the middle of the school and the store.
1. If there is a park in between the school and the store that is on this road and
from Billy's house would it be before or after the midpoint sign.
State how you reached your conclusion.
2. How far would someone that lived in "Radical Land" say that the school was
from the park?
30 miles
8th Grade Number System Key
Exploration #1
1. rational and terminating
2. rational and repeating
3. rational and terminating
4. rational and repeating
5. rational and terminating
6. irrational
7. rational and terminating
8. irrational
9. rational and terminating
10. rational and terminating
11. rational and repeating
12. irrational
13. irrational
14. rational and terminating
15. rational and repeating
16. rational and terminating
17. rational and terminating
18. rational and repeating
Exploration #2
1
12
2(
3(
4(
5(
6(
=
1
12
1
12
1
12
1
12
1
12
1
18
)=
)=
)=
)=
)=
+
1
10
1
6
1
4
1
6
1
3
1
7(
36
+
+
+
+
+
1
15
1
12
1
12
1
4
1
6
8(
1
12
1
12
12
10(
12(
)=
1
9(
11(
)=
1
12
1
12
1
12
1
3
1
1
)=
)=
Exploration #3
1.
Let x = 3.99999.....
10x = 39.99999.....
Subtract he two expressions
9x = 36
x=4
Therefore 3.99999.....= 4
1
1
2
1
2
1
4
1
+
2
1
6
+
2
)=
4
+
2
)=
1
+
+
+
3
1
3
1
3
+
+
1
12
1
6
2.
Let x = 0.2499999....
1000x = 249.99999.....
100x = 24.99999.....
Subtract he two expressions
990x =225
225 1
x=
=
990 4
1
Therefore 0.2499999.....=
4
3.
Let x = 0.99999.....
10x = 9.99999.....
Subtract he two expressions
9x = 9
x=1
Therefore 0.99999.....= 1
4.
2.99999..... = 3
3.99999..... = 4
Therefore 2.99999..... + 3.99999..... = 3 + 4 = 7
Exploration #4
Answers will vary but Necie is correct.
Exploration #5
1. sqrt (7-3) is rational so beware but answers will vary
2. itis not but answers will vary
Exploration #6
Approximate
the square
root of
these
numbers
This
number
is more
than
what
perfect
square
This
number
is less
than
what
perfect
square
The
difference
in the two
perfect
squares
11
9
16
7
The
difference in
the number
being
approximated
and the
perfect
square it is
more than.
2
The rational
approximation
of the
irrational
number
3
16
25
9
3
19
4
25
36
11
9
34
5
36
49
13
1
37
6
49
64
15
5
54
7
81
100
19
90
9
9
1. It gets better and better
2. No, that is impossible
2
Estimate of
the square
root
(Using rational
approximation)
Actual
square
root
(calculator)
3.285714
3.316625
4.333333
4.358899
5.818182
5.830952
6.076923
6.082763
7.333333
7.348469
9.473684
9.486833
7
3
9
9
11
1
13
5
15
9
19
Exploration #7
1. It is before the midpoint. Answers will vary.
2.
30 -
25