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Dear Family,
Content
Overview
The goal of this unit of Math Expressions is for your child to
become fluent with dividing whole numbers and with comparing,
adding, subtracting, multiplying, and dividing fractions and
decimals. Below is a summary of the topics in this unit.
Addition and Subtraction of Fractions and Decimals
3 10
1.40
\\
- 0.25
1.15
4 = ___
10 + ___
12 = ___
22 = 1___
7
2 + __
__
5
3
15
15
15
15
Equivalent Fractions
Multiply the numerator and
denominator by the same
number.
Divide the numerator and
denominator by the same
number.
5 • 2 = ___
10
5 = _____
__
10 = 10
÷ 2 = __
5
___
_______
6
6•2
12
12
12 ÷ 2
6
Multiplication of Fractions and Decimals
Multiply numerators and
denominators.
4= 2
8
2 • __
• 4 = ___
__
_____
3
5
3• 5
15
Count decimal places in the
factors to place the decimal
point in the product.
0.12 • 0.4 = 0.048
2
places
1
place
3
places
Division of Fractions and Decimals
Multiply by the reciprocal.
3 = __
14
2 • __
7 = ___
2 ÷ __
__
7
5
5 3
15
Sometimes you can divide
numerators and denominators.
2 = _______
8 ÷ 2 = __
8 ÷ __
4
___
15
3
15 ÷ 3
5
Make the divisor a whole
number by multiplying it and
the dividend by the same
number.
Here we multiply both by 10.
0.4
_____
0.2 0.08
If you have any questions or comments, please call or write to me.
Sincerely,
Your child’s teacher
Unit 3 addresses the following standards from the Common Core State Standards for Mathematics with
California Additions: 6.NS.1, 6.NS.2, 6.NS.3, 6.NS.4, and all Mathematical Practices.
UNIT 3 LESSON 1
Place Value and Whole Number Division
57
Estimada familia,
Un vistazo
general al
contenido
El objetivo de esta unidad de Expresiones en matemáticas es que
su hijo domine la división de números enteros y que compare,
sume, reste, multiplique y divida correctamente fracciones y
decimales. Debajo hay un resumen de algunos de los temas de
esta unidad.
Suma y resta de fracciones y decimales
3 10
1.40
\\
- 0.25
1.15
4 = ___
10 + ___
12 = ___
22 = 1___
7
2 + __
__
5
15
15
15
15
3
Fracciones equivalentes
Multiplicar el numerador y el
denominador por el mismo
número.
Dividir el numerador y el
denominador entre el mismo
número.
5 • 2 = ___
10
5 = _____
__
10 = 10
÷ 2 = __
5
___
_______
6
6•2
12
12
12 ÷ 2
6
Multiplicación de fracciones y decimales
Multiplicar los numeradores y
denominadores.
4= 2
8
2 • __
• 4 = ___
__
_____
3
5
3• 5
15
Contar los lugares decimales
en los factores para colocar el
punto decimal en la respuesta.
0.12 • 0.4 = 0.048
2
lugares
1
lugar
3
lugares
División de fracciones y decimales
Multiplicar por el recíproco.
3 = __
2 ÷ __
2 • __
7 = ___
14
__
7
5
5 3
15
Algunas veces se pueden
dividir los numeradores y
denominadores.
2 = _______
8 ÷ 2 = __
8 ÷ __
4
___
3
15
5
15 ÷ 3
Dividir el divisor y el dividendo
entre el mismo número para
convertir el divisor en un
número entero.
Aquí se multiplican ambos
por 10.
0.4
_____
0.2 0.08
Si tiene preguntas, por favor comuníquese conmigo.
Atentamente,
El maestro de su hijo
En la Unidad 3 se aplican los siguientes estándares auxiliares, contenidos en los Estándares estatales comunes de
matemáticas con adiciones para California: 6.NS.1, 6.NS.2, 6.NS.3, 6.NS.4, y todos los de prácticas matemáticas.
58
UNIT 3 LESSON 1
Place Value and Whole Number Division
3–1
► Decimal and Whole Number Secret Code Cards
2, 0 0 0 0 .1
3 0 0 0 .0 3
3 0 1 0 .0 1
2 9 0 .4 .0 0
6 0 0 .0 0 2
7 0 0 .6
2,000
0 .1
300
0 .03
1
30
2
9
0 .01
0 .4
60
70
UNIT 3 LESSON 1
0 .002
0 .6
Place Value and Whole Number Division
59
3–1
► Decimal and Whole Number Secret Code Cards
$1,000
$1,000
$100
$100
$100
$1
$10
$10
$10
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$10
$10
$10
$10
$10
$10
$10
$10
$10
$10
$10
$10
$10
60
UNIT 3 LESSON 1
Place Value and Whole Number Division
3–1
Content Standards 6.NS.2
Mathematical Practices MP.2, MP.3, MP.6, MP.7, MP.8
► Discuss and Summarize
Fill in the blanks and discuss how the parts of each problem are related.
Place Value
3 10
1
10
1
10
1
10
1
10
1
10
1
TNBMMFS
10
5IPVTBOET
)VOESFET
5FOT
0/&4
5FOUIT
)VOESFEUIT
5IPVTBOEUIT
1,000.
100.
10.
1.
0. 1
0.01
0.001
1000
1
100
1
10
1
1
1
1
10
1
100
1
1000
$1,000.00
$100.00
$10.00
$1.00
$0.10
$0.01
$0.001
2,000
300
60
$1 000
$1,000
$1,000
$100
$100
$100
$10
$10
$10
$10
$10
1
0 .6
0 .0 3
0.002
2, 03 06 001000..60.030 2
1 a.
r(SBEF1MBDF7BMVF1PTUFSr¥CZ)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH$PNQBOZr*4#/
CJMM
5JN.BJOJFSP"MBNZ
b.
c.
4 10
MBSHFS
2,361.632 = 2,000 +
+
$10
$1
+
2 a.
+
0
.6
+ 0.002
3
2
+ ____ + ______
10
100
1,000
+ ______ + ______ + ______
1,000
1,000
1,000
+
b.
c.
d.
UNIT 3 LESSON 1
+
___
0
.6
0
0
+
0
.0
3
0
+
0
.0
0
2
0
.6
3
2
Place Value and Whole Number Division
61
3–1
Vocabulary
dividend
divisor
quotient
remainder
► Discuss Division Meanings
The 49 sixth graders raised $2,361 toward their class trip.
How much is that for each student?
When $2,361 is
split 49 ways, each
student gets $40.
Write 4 in the tens
place.
Take away the
$1,960 that has
been shared out,
leaving $401 to be
shared next.
divisor
4
__________
4
__________
49 $2,361.00
49 $2,361.00
- 1 96
401
dividend
Take away the
49 dimes shared
out from the
90 dimes, leaving
41 dimes.
Change the
41 dimes to
410 pennies.
48.1
__________
49 $2,361.00
1 96
______
401
- 392
90
-49
41
3. Each student gets $
62
UNIT 3 LESSON 1
48.1
__________
49 $2,361.00
- 1 96
401
- 392
90
-49
4 10
Each student gets
8 more dollars.
Take away the
$392 shared out.
48
__________
49 $2,361.00
- 1 96
401
- 392
9
The multiplier 8,
which was used
before, works
again. Each
student gets
8 pennies.
48.18
__________
49 $2,361.00
1 96
401
- 392
90
-49
4 10
and there are
There are 9 dollars
left, which is
90 dimes. Each
student gets
1 dime, so write 1
in the tenths place.
48.1
__________
49 $2,361.00
- 1 96
401
- 392
90
That makes
392 pennies
shared out of
the 410 pennies,
leaving 18 pennies.
quotient__________
48.18
49 $2,361.00
- 1 96
401
- 392
90
-49
4 10
- 3 92
18
remainder
cents left over.
Place Value and Whole Number Division
3–3
► Rule About Multiplying Decimals
Example
0.4 • 0.02
Ignore the decimal point
and multiply: 4 • 2 = 8
Place decimal point so there
are three decimal places:
0.4 • 0.02 = 0.008
We can state a rule about multiplying with
decimal numbers.
Rule: Ignore the decimal points and multiply.
Then place the decimal point so the number of
decimal places in the product is equal to the total
number of decimal places in the factors.
1
place
2
places
3
places
3. Find each product in two ways: 1) Use the rule
above; 2) Use the method of dividing into 10 or 100 parts
and then multiplying by the number of tenths or hundredths.
Show or explain your work. The first one is done for you.
Use the Rule
a. 0.2 • 0.3
Divide into Parts and Multiply
(0.3 ÷ 10) • 2 = 0.03 • 2 = 0.06
0.06; two decimal places
b. 0.02 • 0.3
c. 0.2 • 3
d. 0.02 • 3
e. 2 • 0.3
f. 2 • 0.03
4. Explain why the two methods give the same result.
Use either method to solve these problems.
5. 0.04 • 2 =
6. 0.4 • 2 =
7. 0.3 • 0.3 =
8. 3 • 0.3 =
9. 0.04 • 6 =
10. 0.5 • 7 =
11. 0.3 • 0.8 =
12. 7 • 0.6 =
UNIT 3 LESSON 3
Multiplying by a Decimal
63
64
UNIT 3 LESSON 3
Multiplying by a Decimal
3–5
► How Many Decimal Places?
Use the fact that 39 • 74 = 2,886 to solve each problem.
20. 0.39 • 7,400 =
_______
21. 0.39 • 740 =
_______
22. 0.39 • 74 =
_______
23. 39 • 7.4 =
_______
24. 0.74 2,886
25. 0.74 288.6
26. 0.74 28.86
27. 74 288.6
28. 3.9 • 740 =
29. 3.9 • 74 =
30. 3.9 • 7.4 =
31. 39 • 0.74 =
_______
32. 7.4 2,886
PATH to
FLUENCY
_______
33. 7.4 288.6
_______
34. 7.4 28.86
_______
35. 74 28.86
Solve.
36. 0.7 • 0.3 =
___
37. 0.06 • 7 =
____
38. 8 • 0.4 =
___
_____
39. 0.12 42
40. 3.2 2.4
41. 0.06 54
42. 0.27 1.35
0.09
43. × 52
8.5
44. × 4.2
7.2
45. × 0.25
18
46. × 0.6
_____
47. 3.02 90.6
UNIT 3 LESSON 5
_____
48. 7.5 0.06
____
49. 0.8 6.8
_____
50. 0.2 0.95
Multiplication or Division
65
3–9
► Strategies for Comparing
Look for patterns in the
fractions and decimals below.
Below are general and special cases for comparing
fractions and decimals.
General Cases
1= 2 = 4 = 8
2 4 8
7
8
3=6
4 8
5
8
1=2=4
2 4 8
1.000 = 1.0
0.9
0.875
0.83 ≈ 5
6
0.8 = 4
5
0.750 = 0.75
0.7
0.67 ≈ 4 = 2
0.625 3
0.6 =
6
3
5
0.500 = 0.50
= 0.5 = 3
6
3
8
1=2
4 8
1
8
0.4 = 2
0.375 5
0.33 ≈ 2 = 1
6 3
0.3
0.250 1 = 0.25
0.2 =
5
0.17 ≈ 1
0.125
0.1
6
Case 1: Same denominator or number of places
Fraction with greater
3
4
__
__
numerator is greater.
5
5
Ignore decimal point.
0.7 > 0.5
Greater number is greater.
Case 2: Same numerator or digits
Fraction with lesser
6
__
denominator is greater.
9
Decimal with leftmost non-zero
0.3
digit is greater.
Case 3: Different denominators
Find equivalent fractions with
a common denominator to
change fractions to Case 1.
3
__
6
__
8
0.03
2
__
4
3
Case 4: Mixed numbers
Number with greater whole
9
1
2___
5__
8
10
number is greater. If whole
numbers are the same, compare 4.7 > 4.07
fractions using Cases 1–3.
Special Cases for Fractions
Case 5: Denominators are factors of 10 or 100.
Compare the decimal
3
4
__
___
equivalents.
5
10
1
1
. One fraction < __
.
Case 6: One fraction > __
1
Fraction > __
is greater.
2
2
5
__
2
3
__
7
8
Use strategies to compare.
0= 0=0
2 4 8
0.000
28. 0.76
5
31. __
8
66
UNIT 3 LESSON 9
0.67
5
__
6
2
29. __
3
32. 0.09
7
___
1
30. 1__
12
2
1__
5
2
0.1
4
33. __
9
3
__
5
Mixed Problem Solving
Unit 3
Select all that apply.
_____
______
2. 25 2,515
1. 2 629
A
213
A
55
B
314 R1
B
95.6
C
314.5
C
100.6
D
345
D
100 R15
3. Is the problem below a multiplication or a division problem?
Explain your answer.
The rectangular floor of a playpen has an area of 14 square feet.
1
feet. What is the other measure?
One measure of the floor is 3__
2
4. Select True or False for each statement.
4a.
0.9 ÷ 7.5 = 0.12
True
False
4b.
63.2 ÷ 8 = 7.8
True
False
4c.
72.8 ÷ 5.2 = 1.4
True
False
4d.
7 ÷ 0.35 = 20
True
False
UNIT 3 TEST
67
Unit 3
Name
Date
5. Without doing any computation, predict if the quotient 73 ÷ 0.42
will be greater than or less than the product 0.42 • 73. Explain your
reasoning in terms of the size of 0.42.
Use the numbers and the decimal point (if needed) to write the
product. Each tile may be used more than once.
0
1
2
4
6
7
8
.
0.06
7. ×12
0.4
6. ×0.3
Fraction Pair
2 __
__
,1
5 __
___
,3
7 3
12 4
LCM
Equivalent Fraction Pair
Multiply. Write the product.
9.
5.8
× 2.5
Product:
68
UNIT 3 TEST
10.
2.9
× 0.81
Product:
7 ___
___
, 4
10 15
© Houghton Mifflin Harcourt Publishing Company
8. The fraction pairs in the table are not equivalent. Find the LCM
for each fraction pair. Rewrite each pair as an equivalent
fraction pair using the LCM as the denominator.
Unit 3
Find the sum or difference. Select one number from each
column to make the equation true.
3
7
+ __
=
11. __
8
5
7
12. __
+ __
=
8
whole
number
0
1
2
3
9
fraction
5
__
8
4
__
5
1
__
4
3
__
4
6
whole
number
fraction
0
3
__
1
7
__
2
11
___
3
11
___
7
9
13
18
Write a number in the box to make the equation true.
2
13. 3__
5
1
= 1__
2
- 1.78 = 0.82
14.
15
.
15. George calculated a sum of ___
24
15
? Choose all that apply.
Which fractions are equivalent to ___
24
A
10
___
D
5
__
B
5
__
E
3
__
C
7
___
F
30
___
16
6
12
UNIT 3 TEST
8
8
48
69
Unit 3
16. Read the division problem below. Then select True or False
for each statement.
_3_ ÷ 7 =
4
16a. To find the quotient, first find a common denominator.
True
False
16b. The quotient will be less than _34_.
True
False
16c. You will need to simplify the quotient.
True
False
Draw a line to match each missing numerator or denominator
to its value. You will not use all of the choices.
•
•
•
•
3
6
• ________ = ___
17. __
5
3
1
18. __
• ________ = ___
4
35
•
•
•
•
•
•
2
3
4
5
6
7
Choose the expressions that make the equation true.
Select all that apply.
8
4
19. ___
÷ __
=
15
5
1
A __
3
70
1
5
20. 1__
÷ 1__
=
8
A
2
7
2___
16
B
2
__
B
39
__
16
C
20
___
C
26
__
24
D
40
___
D
1
1__
3
60
60
UNIT 3 TEST
12
20
Unit 3
Name
Date
21. Seth wrote an expression that simplifies to 0.36.
Choose Yes or No to indicate if the expression
has a value of 0.36.
21a. 36 • 0.01
Yes
No
21b. 3.6 • 0.1
Yes
No
21c. 3.6 ÷ 0.1
Yes
No
21d. 0.036 ÷ 0.01
Yes
No
22. Darnell has two scraps of cloth. The first scrap is 3.75 cm
1
as long as the length of the first.
in length. The second scrap is __
5
Part A
What is the total length of the scraps? Show your work.
cm
Part B
Answer the following question without making a calculation.
Explain how you know the answer.
1
as long as the first,
If Darnell’s second scrap was __
3
would the total length of the scraps be greater than or
less than the answer you found in Part A?
UNIT 3 TEST
71
Unit 3
Name
Date
1
23. Lydia has 8__
cups of juice to serve.
2
Part A
3
-cup, how many full servings can she make?
If each serving is __
4
What amount of juice, if any, will be left over? Show your work.
servings
leftover
Part B
If she needs 15 servings, how much more juice does she need?
A
1
2__
cups
4
B
3 cups
C
3
2__
cups
4
D
4 cups
Part C
5
-cup. Without calculating,
Lydia decides to make each serving of juice __
8
predict if she can make more full servings or fewer full servings using
1
cups of juice. Explain your choice. Then confirm your prediction.
the 8__
2
Predict:
© Houghton Mifflin Harcourt Publishing Company
Confirm:
24. Dana’s dog Skippy weighs 20.4 pounds. In comparison, her cat Bruiser
weighs 0.75 as much as Skippy.
24a. Does Bruiser weigh more or less than Skippy? Explain.
24b.How much does Bruiser weigh?
pounds
72
UNIT 3 TEST