Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Common Core/Research
Understand the place value system
Students extend their understanding of the base-ten system to the relationship between adjacent places, how numbers compare, and how numbers round
for decimals to thousandths. New at Grade 5 is the use of whole number exponents to denote powers of 10. 5.NBT.2 Students understand why multiplying by a
power of 10 shifts the digits of a whole number or decimal that many places to the left. For example, multiplying by 104 is multiplying by 10 four times.
Multiplying by 10 once shifts every digit of the multiplicand one place to the left in the product (the product is ten times as large) because in the base-ten
system the value of each place is 10 times the value of the place to its right. So multiplying by 10 four times shifts every digit 4 places to the left. Patterns in
the number of 0s in products of whole numbers and powers of 10, and the location of the decimal point in products of decimals with powers of 10 can be
explained in terms of place value. Because students have developed their understandings of and computations with decimals in terms of multiples
(consistent with 4.OA.4) rather than powers, connecting the terminology of multiples with that of powers affords connections between understanding of
multiplication and exponentiation.
Perform operations with multi-digit whole numbers and with decimals to hundredths
Because of the uniformity of the structure of the base-ten system, students use the same place value understanding for adding and subtracting decimals
that they used for adding and subtracting whole numbers.5.NBT.7 Like base-ten units must be added and subtracted, so students need to attend to aligning the
corresponding places correctly (this also aligns the decimal points). It can help to put 0s in places so that all numbers show the same number of places to the
right of the decimal point. Although whole numbers are not usually written with a decimal point that a decimal point with 0s on its right can be inserted (e.g.,
16 can also be written as 16.0 or 16.00). The process of composing and decomposing a base-ten unit is the same for decimals as for whole numbers and the
same methods of recording numerical work can be used with decimals as with whole numbers.
Resources:
See pages in IG
pages 183-185, 197-201
pages 51-61
pages 33-44
pages 1-52, 205-212
1
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
The chart below highlights the key understandings of this cluster along with important questions that teachers should pose
to promote these understandings. The chart also includes key vocabulary that should be modeled by teachers and used
by students to show precision of language when communicating mathematically.
Enduring Understandings
Essential Questions
Key Vocabulary
A digit in one place represents 10 times as much How do the digits in each place value column Decimal
as it represents in the place to its right and 1/10 relate to the digits in the column to the right Digit
of what it represents in the place to its left.
and to the left?
Place value
Powers of Ten
Measurements in the metric system can be
How can understanding place value help to
Exponent
converted using an understanding of place
convert in the metric system?
Round
value.
Equivalent Fractions
What strategies can we use to compare
Tenth
decimals?
Hundredth
The exponent above the 10 indicates how many
Thousandth
places the numerals shift.
How is adding and subtracting with decimals Convert
the same as computing with whole numbers Kilometer
and fractions?
Meter
Decimals can be compared by using models,
Centimeter
benchmark numbers or by finding equivalent
Kilogram
fractions.
Gram
Liter
Decimals can be added and subtracted using the
Milliliter
same properties of computing with whole
Inverse Operation
numbers and fractions.
2
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Convert like measurement units within a given measurement
system.
Maryland College and Career-Ready
Standards
5.MD.1. Convert among
different-sized standard
measurement units within
a given measurement
system (e.g., convert 5 cm
to 0.05 m), and use these
conversions in solving
multi-step, real world
problems.
* This standard provides
measurement conversion
as a context for not only
working with decimals but
a deeper understanding for
place value and the
connection to the metric
system.
SMP.2 Reason abstractly
and quantitatively
SMP.3 Construct viable
arguments and critique the
reasoning of others
SMP.5 Use appropriate
tools strategically
Instructional Strategies and Resource Support
Sizing Up Measurement- pp. 1-52, 205-212
Students worked with both metric and customary units of length in second grade. In third grade, students work with metric
units of mass and liquid volume. In fourth grade, students work with both systems and begin conversions within systems in
th
length, mass and volume using whole numbers. This is extended to metric conversions involving decimals in 5 grade.
Conversions among metric units such as kilometers, meters, and centimeters give an opportunity to apply these extended
th
place value relationships and exponents in a meaningful context by exploring word problems. (5 graders will convert time
and customary units of measure in Unit 4 and apply throughout the year.)
Students present solutions to multi-step problems in the form of a valid chain of reasoning, using symbols such as equal
signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as
1+4=5+7=12, even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and
present corrected solutions. (SMP.3, 5,6)
The metric system of measurement is based on 10 and powers of 10. The prefixes used for length, capacity, and mass tell
what part of the basic unit is being considered. The symbols for each unit of measure are given in parentheses. The most
th
commonly used units are in bold and are the units students should be working with in 5 grade.
Sample Formatives
Write <, >, = to
compare
measurements.
Shawn measured a
piece of wood at 2
meters in length. He
needs to know how
many centimeters
long it is in length.
Explain how he will
find the answer
without using a ruler.
If the wood
measures 2.5 meters
in length, how many
centimeters is it in
length?
Prefix
Meaning
Measure of Length
Measure of Capacity
Measure of Mass
Kilo1,000
Kilometer
Kiloliter
Kilogram
Hecto100
Hectometer
Hectoliter
Hectogram
Deka10
Dekameter
Dekaliter
Dekagram
Base unit
1
Meter
Liter
Gram
Deci0.1
Decimeter
Deciliter
Decigram
Centi0.01
Centimeter
Centiliter
Centigram
Milli0.001
Millimeter
Milliliter
Milligram
To change from a larger unit to a smaller unit, multiply by the appropriate power of 10.
To change from a smaller unit to a larger unit, divide by the appropriate power of 10.
SMP.6 Attend to precision
The standard should be taught throughout the decimal unit connected to the understanding of the base 10 system.
3
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Understand the place value system.
Maryland College and CareerReady Standards
5.NBT.1 Recognize that
in a multi-digit number,
a digit in one place
represents 10 times as
much as it represents
in the place to its right
and 1/10 of what it
represents in the place
to its left.
SMP.2 Reason
abstractly and
quantitatively
Instructional Strategies and Resource Support
Sample Formatives
TSCM pages 183-185
NCTM Focus in Grade 5 pages 51-55
Digi-Block pages 108-119
Determine if the
statement is true or
false. Explain.
Begin with a thousand Digi-block. Ask the students what will be inside when you open the block. (10
hundreds).What will be inside when I open a hundred? (10 tens) One ten? (10 ones) What if I could open a 1, what
would be inside? (10 tenths) Show students a tenth Digi. What if I could open the tenth (10 hundredths) Show
students the hundredth Digi. What if I could open a hundredth Digi? (10 thousandths) What would that look like?
(It would be of the size of the hundredth Digi). What pattern do we see? (A digit in one place represents 10 times
Write an equivalent
number sentence
using the inverse
operation of the given
number sentence and
show one solution.
as much as it represents in the place to its right and
of what it represents in the place to its left.) What can you
say about the value of 2 compared to 0.2 (2 is 10 times the value of 0.2 and 0.2 is
than 0.2. Compare the other values as well.
) Prove that 2 is 10 times more
Complete the equation
below and explain the
pattern.
Explain the difference
between the digit 7 in
the number 75,712.
SMP.6 Attend to
precision
SMP.7 Look for and
make use of structure
Use the metric system and money (penny, dime, dollar, ten dollars) to apply (5.NBT.1)
Students use their understanding of structure of whole numbers to generalize this understanding of decimals
(MP.7) and explain the relationship between the numerals. (SMP.2,SMP.6)
Students apply their understanding of the structure within the base-ten system and fraction-decimal equivalencies
to precisely communicate their understanding of relative sizes of decimal numbers (SMP.2, SMP.6, SMP.7)
4
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Understand the place value system.
Maryland College
and Career-Ready
Standards
5.NBT.2. Explain
patterns in the
number of zeros
of the product
when
multiplying a
number by
powers of 10,
and explain
patterns in the
placement of
the decimal
point when a
decimal is
multiplied or
divided by a
power of 10.
Use wholenumber
exponents to
denote powers
of 10.
SMP.6 Attend
to precision
SMP.7 Look for
and make use
of structure
Sample
Formatives
Instructional Strategies and Resource Support
Digi-Block pages 248-259 (negative exponents may be used as an extension lesson)
2
3
This standard includes multiplying by multiples of 10 and powers of 10, including 10 which is 10 x 10=100, and 10 which is 10 x 10 x 10=1,000.
Students should have experiences working with connecting the pattern of the number of zeros in the product when you multiply by powers of 10.
3
Example: 2.5 x 10 = 2.5 x (10 x 10 x 10) = 2.5 x 1,000 = 2,500 Students should reason that the exponent above the 10 indicates how many places the
numerals are shifting (not just that the numerals are shifting, but that you are multiplying or making the number 10 times greater three times) when
you multiply by a power of 10. Since we are multiplying by a power of 10 the numerals shift to the left.
3
350 ÷ 10 = 350 ÷ 1,000 = 0.350 = 0.35
350/10 = 35, 35 /10 = 3.5, 3.5 /10 =.0.35
350 x 1/10, 35 x 1/10, 3.5 x 1/10=0.35
This example shows that when we divide by powers of 10, the exponent above the 10 indicates how many places the numerals shift to the right (how
many times we are dividing by 10 , the number becomes ten times smaller). Since we are dividing by powers of 10, the numerals shift to the right.
Identify the
pattern below
and explain
using your
understanding
of exponents.
Students need to be provided with opportunities to explore this concept and come to this understanding; this should not just be taught procedurally.
Students should look for a pattern(s).
Students might write:
1

36 x 10 = 36 x 10 = 360
2

36 x 10 x 10 = 36 x 10 = 3,600
3

36 x 10 x 10 x 10 = 36 x 10 = 36,000
Students might say:

I noticed that every time I multiplied by 10, I added a zero to the end of the number. That makes sense because each digit's value became
10 times larger. To make a digit 10 times larger, I have to move it one place value to the left.

When I multiplied 36 by 10, the 30 became 300. The 6 became 60, or the 36 became 360. So I had to add a 0 at the end to have the 3
represent 3 one-hundreds (instead of 3 tens) and the 6 represents 6 tens (instead of 6 ones).
Students should be able to use the same type of reasoning as above to explain why the following multiplication and division problems by powers of
10 make sense.
3

523 x 10 =523,000 – The place value 523 is increased by 3 places.
2

5.223 x 10 =522.3 – The place value of 5.223 is increased by 2 places.
1
 52.3÷10 = 5.23 - The place value of 52.3 is decreased by 1 place.
Students use their understanding of structure of whole numbers to generalize this understanding of decimals (SMP.7) and explain the relationship
between the numerals. (SMP.6)
Students apply their understanding of the structure within the base-ten system and fraction-decimal equivalencies to precisely communicate their
understanding of relative sizes of decimal numbers (SMP.6, SMP.7)
5
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Write and interpret numerical expressions.
Maryland College and
Career-Ready Standards
Instructional Strategies and Resource Support
rd
th
Sample Formatives
th
5.OA.1. Use
parentheses,
brackets, or braces
in numerical
expressions, and
evaluate
expressions with
these symbols.
Students in 3 and 4 grade have been working with parenthesis and order of operations in an informal way. In 5 grade, students
are introduced to the order of operations through a set of reasonable rules. It is important NOT to use PEMDAS with students.
Student should be taught the order of operations through a series of steps that help them understand the order. When using
PEMDAS, students often end up thinking multiplication comes before division and addition before subtraction.
*The expressions
will include the use
of parentheses and
brackets, but will
not contain braces
at grade 5.
Expressions have
depth no greater
than two, e.g.,
3×[5 + (8 ÷ 2)] is
acceptable but
3×[5 + (8 ÷ {4−2})]
is not.
Students should come to the conclusion that it doesn’t matter the order if they are only adding, but it does matter if it is only
subtraction or a mix of the two operations. At this time the teacher should explain that (in the absence of parenthesis) you perform
the operations of addition and subtraction from left to right.
SMP.6 Attend to
precision
SMP.7 Look for
and make use of
structure
Begin with what students already know: Ask- If I have 4 numbers I want to add, does it matter the order in which I add them? 12 + 3
+ 2 + 5 Prove it. If I have 4 numbers I want to subtract, does it matter the order in which I subtract them? 12-3-2-5 Prove it. When
I have a mix of adding and subtracting, does it matter the order in which I perform the operations?
12 -3+2 -5 Prove it.
Place <,>, or = in
each circle to make
the equation true.
Sort the equations
according to equal or
not equal.
Next complete the same process with multiplication and division. After this, students should realize it doesn’t matter the order if it
is multiplication alone, but does matter if multiplication and division are mixed. The teacher explains that if you are only dividing or
if multiplication and division are mixed, you compute from left to right just as you do with addition and subtraction.
So, what if all of the operations are mixed. Does the order matter? Students now learn that you perform the multiplication and
division operations before the addition and subtraction operations, calculating multiplication from left to right first and then the
addition and subtraction from left to right.
Next parenthesis can be discussed. Students have some previous experience with parenthesis, but they can be formally introduced
th
at this point as well as introducing brackets which is a new symbol for 5 graders.
Ask student to evaluate various numerical expressions that contain the same integers and operations yet have differing results due
to the placement of parentheses. It helps students see the purpose of using parentheses. Ask students if the parentheses could be
removed to point out that sometimes the parentheses are not necessary.
Students discuss the meaning of symbols and interpret numerical expressions precisely (SMP.6)
https://www.illustrativemathematics.org/content-standards/tasks/555
It is important to understand that order of operations will work, but as students begin to understand the underlying mathematics
they will need not be so ridged.
6
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Understand the place value system.
Maryland College and CareerReady Standards
5.NBT.3. Read, write,
and compare decimals
to thousandths.
 Read and write
decimals to
thousandths using
base-ten
numerals, number
names, and
expanded form,
e.g., 347.392 = 3 ×
100 + 4 × 10 + 7 ×
1 + 3 × (1/10) + 9 ×
(1/100) + 2 ×
(1/1000).
 Compare two
decimals to
thousandths based
on meanings of
the digits in each
place, using >, =,
and < symbols to
record the results
of comparisons.
SMP.6 Attend to
precision
Instructional Strategies and Resource Support
NCTM Focus in Grade 5 pages 54-59
Digi-Block Book pages 26-31
This standard references the expanded form of decimals with fractions included. Students should build on their work
from Fourth Grade, where they worked with both decimals and fractions interchangeably. Expanded form is included to
build upon work in 5.NBT.2 and deepen students’ understanding of place value.
Students build on the understanding they developed in fourth grade to read, write, and compare decimals to the
thousandths place. They connect their prior experiences with using decimal notation for fractions and addition of
fractions with denominators of 10 and 100. They use concrete models and number lines to extend this understanding to
decimals to the thousandths. Models may include base ten blocks, place value charts, grids, pictures, drawings,
manipulatives, technology-based, etc. They read decimals using fractional language and write decimals in fractional
form, as well as in expanded notation. This investigation leads them to understanding equivalence of decimals
(0.8 = 0.80 = 0.800).
Some equivalent forms of 0.72 are:
70/100 + 2/100
0.720
72/100
720/1000
7/10 + 2/100
Sample
Formatives
Compare
using the
symbols <,>, or
=. Show the
value of each
number being
compared.
7 x (1/10) + 2 x (1/100)
0.70 + 0.02
7 x (1/10) + 2 x (1/100) + 0 x (1/1000)
Students need to understand the size of decimal numbers and relate them to common benchmarks such as 0, 0.5(0.50
and 0.500), and 1. Comparing tenths to tenths, hundredths to hundredths, and thousandths to thousandths is simplified
if students use their understanding of fractions to compare decimals.
Example: Comparing 0.25 and 0.17, a student might think, “25 hundredths is more than 17 hundredths”. They may also
think that it is 8 hundredths more. They may write this comparison as 0.25 > 0.17 and recognize that 0.17 < 0.25 is
another way to express this comparison. Comparing 0.207 to 0.26, a student might think, “Both numbers have 2 tenths,
so I need to compare the hundredths. The second number has 6 hundredths and the first number has no hundredths so
the second number must be larger. Another student might think while writing fractions, “I know that 0.207 is 207
thousandths (and may write 207/1000). 0.26 is 26 hundredths (and may write 26/100) but I can also think of it as 260
thousandths (260/1000). So, 260 thousandths is more than 207 thousandths.”
Students use their understanding of structure of whole numbers to generalize this understanding of decimals (SMP.7)
and explain the relationship between the numerals. (SMP.6)
SMP.7 Look for and
make use of structure
Students apply their understanding of the structure within the base-ten system and fraction-decimal equivalencies to
precisely communicate their understanding of relative sizes of decimal numbers (SMP.6, SMP.7)
7
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Maryland College and CareerReady Standards
Understand the place value system.
5.NBT.4. Use place
value understanding to
round decimals to any
place.
SMP.2 Reason
abstractly and
quantitatively
Instructional Strategies and Resource Support
Digi-Block Book pages 32-43
This standard refers to rounding. Students should go beyond simply applying an algorithm or procedure for rounding.
The expectation is that students have a deep understanding of place value and number sense and can explain and
reason about the answers they get when they round. Students should have numerous experiences using a number line
to support their work with rounding. This standard should be taught throughout the unit, connected with the
understanding of the base ten system and applied in the computation portion of the unit when students round answers
to the nearest tenth, hundredth, etc.
Example: Round 14.235 to the nearest tenth. Students recognize that the possible answer must be in tenths thus, it is
either 14.2 or 14.3. They then identify that 14.235 is closer to 14.2 (14.20) than to 14.3 (14.30).
Sample
Formatives
Which three
numbers round
to 9.5?
Write two
decimal
numbers that
round to 15.
Solve and round
to the nearest
tenth.
Rounding occurs all the time in everyday life. For example, cash registers are programmed to round off automatically to
the nearest hundredth. Since one cent is one hundredth of a dollar, what we are charged must be rounded off to the
hundredths place. For example, if the sales tax is 8.25%, or .0825, the amount charged would sometimes need to be
rounded.
All measurements of real objects are approximate. The number of digits in your answer indicates the accuracy with
which the measurement was made. Understanding rounding and significant figures is important for everyday life and
work. The major obstacle to learning these topics is lack of understanding the decimal notation system.
See the link below for a complete explanation.
https://extranet.education.unimelb.edu.au/DSME/decimals/SLIMversion/backinfo/rounding.shtml
Students should be able to round answers when calculating with decimals. For example: Solve and round to the nearest
tenth or solve and round to the nearest hundredth.
8
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Perform operations with decimals to hundredths.
Maryland College and Career-Ready
Standards
5.NBT.7. Add and subtract decimals
to hundredths, using concrete
models or drawings and strategies
based on place value, properties of
operations, and/or the relationship
between addition and subtraction;
relate the strategy to a written
method and explain the reasoning
used.
(This unit focuses on adding and
subtracting 2 decimals to
hundredths. Multiplication and
division of decimals will occur in
unit 7.)
SMP.2 Reason abstractly and
quantitatively
SMP.3 Construct viable arguments
and critique the reasoning of
others
SMP.5 Use appropriate tools
strategically
Instructional Strategies and Resource Support
Digi-Block Book pages 52-82 TSCM pages pg. 197 – 198
NCTM Focus in Grade 5 pages 60-61
pg. 198 - *Exact Sums & Differences - Activity 7.11
As with whole numbers, when computing with decimals, it is important to ask the students to
estimate the answers. Estimating helps students focus on the meaning of the numbers and
operations and not on counting decimal places or lining up the decimal point. Students will
need a solid understanding of decimal place value prior to computing with decimals.
When adding and subtracting decimals, begin with whole number understandings. We add
ones to ones, tens to tens, hundreds to hundreds, so when adding and subtracting decimals, we
add tenths to tenths and hundredths to hundredths, etc.
Number Talks will provide students to think flexibly about decimals in the same fashion they
have become accustomed to with whole numbers and will deepen their understanding of
number.
Sample Formatives
Two decimal
numbers have a sum
of 50.15. What two
addends could have
been used to get this
sum? Show your
thinking.
Below are the
wrapper lengths of
five popular candy
bars. Use the data to
answer the
questions.
Instead of just computing answers, students reason about both the relationship between
fraction and decimal operations and the relationship between whole number computation and
fractional/ decimal computation (SMP.2., SMP.3)
Example of a model for subtracting decimals:
4 - 0.3 means 3 tenths subtracted from 4 wholes. The wholes must be divided into tenths.
The answer is 3
or 3.7.
Digi-blocks provide a concrete model for adding and subtracting decimals to hundredths.
Money can also be used as an example of when tenths and hundredths are added and
subtracted, but is a limited model since it does not extend to thousandths and beyond.
9
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
Convert like measurement units within a given
measurement system.
Maryland College and Career-Ready Standards
5.MD.1. Convert among differentsized standard measurement units
within a given measurement system
(e.g., convert 5 cm to 0.05 m), and use
these conversions in solving multistep, real world problems.
* This standard provides measurement
conversion as a context for not only
working with decimals but a deeper
understanding for place value and the
connection to the metric system.
SMP.2 Reason abstractly and
quantitatively
SMP.3 Construct viable arguments and
critique the reasoning of others
SMP.5 Use appropriate tools
strategically
Instructional Strategies and Resource Support
Sizing Up Measurement- pp. 1-52, 205-212
This standard calls for students to convert measurements within the same system of
measurement in the context of multi-step, real-world problems. Both customary and standard
measurement systems are included; students worked with both metric and customary units of
length in second grade. In third grade, students work with metric units of mass and liquid
volume. In fourth grade, students work with both systems and begin conversions within systems
in length, mass and volume. In this unit, student work specifically with the metric system
because of the connection with place value and powers of ten.
Students use their understanding of place value and powers of ten (5.NBT.1 and 5.NBT.2) as a
method for expressing equivalent measures. For example, they multiply by powers of ten to
convert between meters and centimeters or ounces and cups with measurements in both whole
number and decimal form. These conversions offer opportunity for students to not only apply
their new found knowledge of the place value system and powers of 10 with both whole and
decimal numbers, but to also reason deeply about the relationships between unit size and
quantity—how the choice of one affects the other.
Sample Formatives
It is recommended
that teenagers
drink 2 liters of
water a day. A
water bottle is 500
ml. How many
water bottles
should a teenager
drink each day?
Show your
thinking.
Students will be allowed to use a reference chart on CCPS assessments and PARCC assessments.
See the reference chart below.
SMP.6 Attend to precision
10
Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
Sept-Oct. (23 days)
Unit #2: Place Value, Addition and Subtraction of Decimals and Measurement Conversions
11