DATE
NAME
Lowest Common Multiples in Fractions
1. 2.
How many pieces are in pie A?
How many pieces are in pie A?
How many pieces are in pie B?
How many pieces are in pie B?
Find the LCM of the number of pieces in pies A and B: LCM =
Find the LCM of the number of
pieces in pies A and B: LCM =
Cut pie A and pie B into this many pieces.
Cut pie A and pie B into this many pieces.
How many pieces did you cut each piece of pie A into?
How many pieces did you cut each piece
of pie A into?
How many pieces did you cut each piece of pie B into?
How many pieces did you cut each piece
of pie B into?
3. A
B
A
B
4.
Write the name of each fraction.
Write the name of each fraction.
A
A
Find the LCM of the number of pieces in pies A and B: LCM =
Find the LCM of the number of
pieces in pies A and B: LCM =
Cut each pie into the number of pieces given by the LCM.
Cut each pie into the number of pieces
given by the LCM.
Now write a new name for each fraction.
Now write a new name for each fraction.
A
A
B-172
B
B
B
A
B
B
B
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A
DATE
NAME
5. 6.
Write the name of each fraction.
Write the name of each fraction.
A
A
Find the LCM of the number of pieces in pies A and B: LCM =
Find the LCM of the number of
pieces in pies A and B: LCM =
Cut each pie into the number of pieces given by the LCM.
Cut each pie into the number of pieces
given by the LCM.
Now write a new name for each fraction.
Now write a new name for each fraction.
A
A
7. A
B
B
B
A
B
B
B
8.
A
B
Write the name of each fraction.
Write the name of each fraction.
A
A
Find the LCM of the number of pieces in pies A and B: LCM =
Find the LCM of the number of
pieces in pies A and B: LCM =
Cut each pie into the number of pieces given by the LCM.
Cut each pie into the number of pieces
given by the LCM.
Now write a new name for each fraction.
Now write a new name for each fraction.
A
A
B
A
B
B
B
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B
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B-173
DATE
NAME
Adding Reciprocals
If a fraction
INVESTIGATION 1
How does the sum
a
b
is less than 1, its reciprocal
is more than 1.
b
a
a b
+ compare to 2?
b a
A.Start with a simpler problem: Without adding the whole numbers, decide whether
the sum is more than 100 or less than 100.
a) 38 + 61
38 is
61 is
So 38 + 61 is
less
than 40 by
than 60 by
b) 53 + 48
2
53 is
than 50 by
48 is
than 50 by
than 40 + 60 = 100 So 53 + 48 is
than 50 + 50 = 100
c) 76 + 19 d) 69 + 33
76 is
than 80 by
69 is
than
by
19 is
than 20 by
33 is
than
by
So 76 + 19 is
than 80 + 20 = 100
So 69 + 33 is
than
B.Describe what worked for the simpler problems: How did you decide whether
the sum is greater or less than 100 without calculating the sum?
C.Use the method that worked for the simpler problem for the problem we
are investigating.
a) Is
5
6
6
5
5 6
+ more or less than 2?
b) Is
6 5
1
5
is
less
than 1 by 6
7
1
7
is more than 1 by 5
5
The amount more than 1 is
than the amount less than 1,
so
B-174
5 6
+ is
6 5
more
more
than 1 + 1 = 2. 5 7
+ more or less than 2?
7 5
is
than 1 by
is
than 1 by
The amount more than 1 is
than the amount less than 1,
so
5 7
+ is
7 5
than 1 + 1 = 2.
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NAME
c) Is
5 8
+ more or less than 2? 8 5
5
is
8
8
is
5
d) Is
than 1 by
than 1 by
The amount more than 1 is
than the amount less than 1,
so
5 8
+ is
8 5
than 1 + 1 = 2.
D. Check your answers by calculating the sums.
5 6
a)
+ 6 5
c) 5 8
+ 8 5
8 11
+
more or less than 2?
11 8
8
is
11
11
is
8
than 1 by
than 1 by
The amount more than 1 is
than the amount less than 1,
so
8 11
+
is
11 8
than 1 + 1 = 2.
5 7
b) + 7 5
d)
8 11
+
11 8
Generalize the result for any a < b: What expression for x works for
both equations?
BONUS
a x b
a x b
+ = and + = then x =
b b b
a a a
a
b
How much more than 1 is ?
How much less than 1 is ?
b
a
If
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The amount more than 1 is
Is
than the amount less than 1. How do you know?
a b
+ more or less than 1 + 1 = 2? Explain.
b a
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B-175
NAME
DATE
Dividing Decimals by Decimals
1. Estimate the quotients.
a) 9.7 ÷ 1.3 ≈
b) 8.24 ÷ 2.16 ≈
÷
≈
÷
≈
c) 19.6 ÷ 5.2 ≈
d) 420.8 ÷ 6.95 ≈
÷
≈
÷
≈
e) 16.34 ÷ 3.87 ≈
f)
÷
99.5 ÷ 4.07 ≈
≈
÷
≈
2. Estimate the quotients.
a) 0.462 ÷ 0.208
≈ 0.4
≈
4
b) 0.629 ÷ 0.346
÷ 0.2
÷
2
≈
c) 0.084 ÷ 0.0426
≈ 0.6
÷
≈
÷ 0.04
≈
÷
≈
÷
≈
d) 0.0965 ÷ 0.0316
≈
e) 1.1548 ÷ 0.1863
f) 12.3956 ÷ 0.2015
≈
÷ 0.03
≈
÷
≈
÷ 0.02
≈
÷
≈
÷
≈
÷
≈
≈
≈
3. Use estimation to place the decimal point correctly in each quotient.
a) 50.46 ÷ 2.14 ≈ 2 3 5 7 9 4 3 9 2 5
b) 0.684 ÷ 0.27 ≈ 2 5 3 3 3 3 3 3 3
c) 562.6 ÷ 0.047 ≈ 1 1 9 7 0 2 1 2 7
d) 16.9 ÷ 0.1287 ≈ 1 3 1 3 1 3 1 3 1 3 1
e) 35.04 ÷ 0.0381 ≈ 9 1 9 6 8 5 0 3 9
f)
1.75 ÷ 2.515 ≈ 6 9 5 8 2 5 0 4 9 7
4. The decimal point is in the wrong place. Estimate the quotient to correct the answer.
a) 5.43 ÷ 0.96 ≈ 56.5625
b) 252 ÷ 5.5 ≈ 4581.8181
c) 15.78 ÷ 3.75 ≈ 0.4208
a) 18.72 ÷ 1.2
b) 2.921 ÷ 2.3
c) 37.26 ÷ 6.9
d) 1.264 ÷ 0.016
e) 3.192 ÷ 0.042
f)
12.194 ÷ 5.2
6. Calculate using a calculator. Estimate to check your answer. Round answers to
two decimal places.
a) 2.174 ÷ 0.649
B-176
b) 1 000 ÷ 0.068
c) 2 ÷ 54.873
d) 987.54 ÷ 3.13
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5. Multiply both the dividend and divisor by 10, 100, or 1000 to change them to whole
numbers. Then divide using a calculator. Estimate the quotient to check your answer.