Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12
Standards for Mathematical Practice
Critical Area(s): Equivalent Fractions and Operations with Fractions
Major Work
70% of time
4.OA.A.1-2-3
4.NBT.A.1-2-3
4.NBT.B.4-5-6
4.NF.A.1-2
4.NF.B.3-4
4.NF.C.6-7
FOCUS for Grade 4
Supporting Work
20% of time
4.OA.B.4
4.MD.A.1-2-3
4.MD.B.4
Additional Work
10% of time
4.OA.C.5
4.MD.C.5-6-7
4.G.A.1-2-3
Required fluency: 4.NBT.B.4
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.
Standards in bold are specifically targeted within instructional materials.
Domains:
Number and Operations—Fractions
Measurement and Data
Clusters:
Clusters outlined in bold should drive the learning for this period of instruction.
4.NF.C Understand decimal notation for fractions, and compare decimal fractions.
4.MD.A Solve problems involving measurement and
conversion of measurements from a larger unit to a
smaller unit.
Standards:
4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and
use this technique to add two fractions with respective denominators 10 and 100.2 For example,
express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62
as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that
comparisons are valid only when the two decimals refer to the same whole. Record the results of
comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
1
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
4.MD.A.2 Use the four operations to solve word
problems involving distances, intervals of time, liquid
volumes, masses of objects, and money, including
problems involving simple fractions or decimals, and
problems that require expressing measurements
given in a larger unit in terms of a smaller unit.
Represent measurement quantities using diagrams
such as number line diagrams that feature a
measurement scale.
2
Students who can generate equivalent fractions can develop strategies for adding fractions with unlike
denominators in general. But addition and subtraction with unlike denominators in general is not a requirement
at this grade.
Revised 2/2017
Property of MPS
Page 1 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12
Foundational Learning
3.NF.A
4.NF.A-B
Key Student Understandings
 Students understand that numbers can be written in various equivalent forms; students write
equivalent fractions and decimals.
 Students deepen understanding of equivalence as they generate equivalent fractions with
denominators of 10 and 100.
 Students use place value understanding to compare and order decimals.
 Students explore how their understanding of fractions and decimals can be used to make sense of realworld contexts.
Future Learning
5.NBT.A.3
5.NBT.B.7
Assessments

Formative Assessment Strategies

Evidence for Standards-Based Grading
Common Misconceptions/Challenges
4.NF.C Understand decimal notation for fractions, and compare decimal fractions.
 Often students incorrectly represent equality of fractions and decimals, perhaps thinking 1/2 is .2 or 5/8 is .5; they are not considering equivalence and
place value. Provide extensive opportunities for students to explore and create concrete and pictorial models of fractions and decimals, including the
language used to describe these models, to develop conceptual understanding and address these kinds of misconceptions.
 Students have a difficult time providing the word form for fractions and decimals, e.g. six-tenths, thirty-seven hundredths. Teachers should model, expect,
and reinforce precision in language (SMP.6) when working with decimal numbers—3.45 should be read as “three and forty-five hundredths”; 0.4 is zero
and four tenths (or simply four tenths). Students should understand that numbers can be read flexibly—0.15 can be read as “1 tenth and 5 hundredths”
or “15 hundredths” or zero and 15 hundredths”.
 Students treat decimals as whole numbers when making comparison of two decimals. They may think the number with the most digits has the greater the
value, regardless of the placement of the decimal. For example, they think that 7.03 is greater than 8.3.
 Students think of the decimal point as the middle of the number system and think that there is a “oneths” place. Spend time representing numbers on
place value charts, to help students recognize that symmetry occurs with the respect to the ones place. (Progressions for the CCSSM; K-5, Number and Operations in Base
Ten, CCSS Writing Team, March 2015, page 13)
4.MD.A Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
 Students believe that larger units will give a larger measure. Students should be given multiple opportunities to measure the same object with different
measuring units. For example, have students measure the length of a room with one-inch tiles, with one-foot rulers, and with yard sticks. Students should
notice that it takes fewer yard sticks to measure the room than rulers or tiles and explain their reasoning.
Revised 2/2017
Property of MPS
Page 2 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12
Instructional Practices
Domain: 4.NF
Cluster: 4.NF.C Understand decimal notation for fractions, and compare decimal fractions.
4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with
respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

This standard continues the work of equivalent fractions by having students change fractions with a 10 in the denominator into equivalent fractions that
have a 100 in the denominator. In order to prepare for work with decimals (4.NF.6 and 4.NF.7), experiences that allow students to shade decimal grids
(10x10 grids) to represent numbers can support this work. This work in Grade 4 also lays the foundation for performing operations with decimal
numbers in fifth grade. Student experiences in this grade should focus on working at the conceptual level with models (see images below), rather than
the procedural/abstract level of algorithms.

Students in Grade 4 work with fractions having denominators 10 and 100. Because it involves partitioning into 10 equal parts and treating the parts as
numbers called one tenth and one hundredth, work with these fractions can be used as preparation to extend the base-ten system to non-whole
numbers.

Students can use base ten blocks, graph paper, and other place value models to explore the relationship between fractions with denominators of 10 and
denominators of 100.
o Base-ten blocks: students may represent 3/10 with 3 longs and may also write the fraction as 30/100 with the whole in this case being the flat
(the flat represents one hundred units with each unit equal to one hundredth). Students begin to make connections to the place value chart as
shown in 4.NF.6.

In decimal numbers, the value of each place is 10 times the value of the place to its immediate right. Students need an understanding of decimal
notations before they try to do conversions in the metric system. Understanding of the decimal place value system is important prior to the
generalization of moving the decimal point when performing operations involving decimals.
Revised 2/2017
Property of MPS
Page 3 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12

Fractions with denominator 10 and 100, called decimal fractions, arise naturally when student convert between dollars
and cents, and have a more fundamental importance, developed in Grade 5, in the base 10 system. For example,
because there are 10 dimes in a dollar, 3 dimes is 3/10 of a dollar; and it is also 30/100 of a dollar because it is 30 cents,
and there are 100 cents in a dollar. Such reasoning provides a concrete context for the fraction equivalence
(Progressions for the CCSSM; Number and Operations—Fractions, 3-5, CCSS Writing Team, September 2013)
4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate
0.62 on a number line diagram.

Decimals are introduced for the first time in Grade 4. Students should have ample opportunities to explore and reason about the idea that a number can
be represented as both a fraction and a decimal.
o
Students can make connections between fractions with denominators of 10 and 100 and the place value chart. By reading fraction names,
students say 32/100 as thirty-two hundredths and rewrite this as 0.32 or represent it on a place value model as shown below.
Revised 2/2017
Property of MPS
Page 4 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12
o
o
Students can use the representations explored in 4.NF.5 to understand 32/100 can be expanded to 3/10 and 2/100.
Students can represent values such as 0.32 or 32/100 on a number line. 32/100 is more than 30/100 (or 3/10) and less than 40/100 (or 4/10). It
is closer to 30/100 so it would be placed on the number line near that value.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to
the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

The decimal point is used to signify the location of the ones place, but its location may suggest there should be a “oneths" place to its right in order to
create symmetry with respect to the decimal point. However, because one is the basic unit from which the other base ten units are derived, the
symmetry occurs instead with respect to the ones place. (Progressions for the CCSSM; Number and Operations in Base Ten, CCSS Writing Team, March
2015, p 13-14)

Ways of reading decimals aloud vary. Mathematicians and scientists often read 0.15 aloud as “zero point one five" or “point one five." (Decimals smaller
than one may be written with or without a zero before the decimal point.) Decimals with many non-zero digits are more easily read aloud in this
manner. (For example, the number π, which has infinitely many non-zero digits, begins 3.1415...) Other ways to read 0.15 aloud are “1 tenth and 5
hundredths” and “15 hundredths,” just as 1,500 is sometimes read “15 hundred” or “1 thousand, 5 hundred.” Similarly, 150 is read “one hundred and
fifty” or “a hundred fifty” and understood as 15 tens, as 10 tens and 5 tens, and/or as 100 + 50.
Revised 2/2017
Property of MPS
Page 5 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12

Just as 15 is understood as 15 ones and also as 1 ten and 5 ones in computations with whole numbers, 0.15 is viewed as 15 hundredths and as 1 tenth
and 5 hundredths in computations with decimals. It takes time to develop understanding and fluency with the different forms. Layered cards for
decimals can help students become fluent with decimal equivalencies such as “three tenths is the same as thirty hundredths”.

Students should reason that comparisons are only valid when they refer to the same whole. Visual models should include area models, decimal grids,
decimal circles, number lines, and meter sticks.

Students build area and other models to compare decimals. Through these experiences and their work with fraction models, they build the
understanding that comparisons between decimals or fractions are only valid when the whole is the same for both cases.
o Example: Draw a model to show that 0.3 < 0.5.
Each of the models below shows 3/10, but the whole on the right is much bigger than the whole on the left. So although both models represent
the same part of a whole, the 3/10 on the right is a much larger quantity than the model on the left.
Students should sketch two models of approximately the same size to show that the area that represents three-tenths is smaller than the area
that represents five-tenths.

When comparing two decimals, remind students that as in comparing two fractions, the decimals need to refer to the same whole. Allow students to use
visual models to compare two decimals. They can shade in a representation of each decimal
on a 10 x 10 grid. The 10 x 10 grid is defined as one whole. The decimal must relate to the
whole.
Revised 2/2017
Property of MPS
Page 6 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12
(Progressions for the CCSSM; Number and Operations—Fractions, CCSS Writing Team, September 2013, p 10)
Domain: 4.MD
Cluster: 4.MD.A Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
4.MD.A.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including
problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.
Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

This standard includes multi-step word problems related to expressing measurements from a larger unit in terms of a smaller unit (e.g., feet to inches,
meters to centimeter, dollars to cents).
o Example: Charlie and 10 friends are planning for a pizza party. They purchased 3 quarts of milk. If each glass holds 8 oz. will everyone get at least
one glass of milk?
Possible solution: Charlie plus 10 friends = 11 total people
11 people x 8 ounces (glass of milk) = 88 total ounces
1 quart = 2 pints = 4 cups = 32 ounces
2 quarts = 4 pints = 8 cups = 64 ounces
3 quarts = 6 pints = 12 cups = 96 ounces
If Charlie purchased 3 quarts (6 pints) of milk there would be enough for everyone at his party to have at least one glass of
milk. If each person drank 1 glass then he would have 1- 8 oz. glass or 1 cup of milk left over.
Revised 2/2017
Property of MPS
Page 7 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12

Students should have ample opportunities to use number line diagrams and tape diagrams to solve word problems.
(Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page 20)
Revised 2/2017
Property of MPS
Page 8 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12

Additional Examples with various operations:
o
Division/fractions: Susan has 2 feet of ribbon. She wants to give her ribbon to her 3 best friends so each friend gets the same amount. How much
ribbon will each friend get?
Students may record their solutions using fractions or inches. (The answer would be 2/3 of a foot or 8 inches. Students are able to express the
answer in inches because they understand that 1/3 of a foot is 4 inches and 2/3 of a foot is 2 groups of 1/3.)
o
Addition: Mason ran for an hour and 15 minutes on Monday, 25 minutes on Tuesday, and 40 minutes on Wednesday. What was the total number
of minutes Mason ran?
o
Subtraction: A pound of apples costs $1.20. Rachel bought a pound and a half of apples. If she gave the clerk a $5.00 bill, how much change will
she get back?
o
Multiplication: Mario and his 2 brothers are selling lemonade. Mario brought one and a half liters, Javier brought 2 liters, and Ernesto brought
450 milliliters. How many total milliliters of lemonade did the boys have?
o
Elapsed Time: At 7:00 a.m. Melisa wakes up to go to school. It takes her 8 minutes to shower, 9 minutes to get dressed and 17 minutes to eat
breakfast. How many minutes does she have until the bus comes at 8:00 a.m.? Use the number line below to help solve the problem.
Differentiation
4.NF.C Understand decimal notation for fractions, and compare decimal fractions.
 Utilize multiple concrete and pictorial models to represent decimal numbers: base-ten blocks, place value disks, 100bead strings, number lines, decimal grids, and decimal circles. Provide students with blank models to help aid with
precision in representations and thinking.
 Decimals used in computation should connect to real life contexts beyond money, including speed, measures
(distance, liquid, linear) and the radio dial.
 Students should work with a variety of representations (see image at right) to
reinforce their understanding of decimal numbers and the relationship to
fractions. Students can represent and describe these relationships with concrete,
pictorial, contextual, symbolic, and verbal/written representations. “Students’
understanding is deepened through discussion of similarities among
representations” and connections between representations (Principles to Actions,
2014).
 English language learners may confuse the terms decompose and compose.
Revised 2/2017
Property of MPS
Literacy Connections

Academic Vocabulary Terms

Vocabulary Strategies

Literacy Strategies
Page 9 of 10
Mathematics 2016-17—Grade 4
Weeks 28-29—March/April
enVisionmath2.0—Topic 12

Demonstrate that the prefix de can be placed before some words to add an opposite meaning. Use gestures to clarify
the meanings: Decompose is to take apart, and compose is to put together. Refresh students’ memory of
decomposition and composition in the context of the operations with whole numbers.
Practice counting by decimals with students: one tenth, two tenths, three tenths, four tenths, five tenths/one half, …
10 tenths/1 whole. Count forward and backward in both tenths and hundredths. Count flexibly—sometimes name
the numbers as hundredths (27 hundredths, 28 hundredths, 29 hundredths,…) and sometimes as tenths and
hundredths (2 tenths 7 hundredths, 2 tenths 8 hundredths, 2 tenths 9 hundredths,…).
4.MD.A Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
 Vary the questions posed when analyzing and interpreting data. Consider using simpler addition and subtraction
situations (CCSSM Glossary, Table 1, at right) when solving problems involving measurement, and add increasingly
complex situations as students gain confidence with line plot analysis and interpretation.
 Give students measuring tools and manipulatives so that they can visually see unit conversions. For example, allow
students to use a ruler that has inches and feet marked. This way students can count to convert from one to the
other, developing their understanding of conversion.
 Support English language learners as they explain their thinking. Provide sentence starters and a word bank.
Word bank:
- decimal, tenth, hundredth, equivalent
Sentence Starters:
- When converting from ___ to ___, the units get smaller so I will need more of them.
- When converting from ___ to ___, the units get larger so I will need less of them.
 Explain and use measurement conversions in a real world context; have students write and share word problems in
which conversion is necessary.
The Common Core Approach to Differentiating Instruction (engageny How to Implement a Story of Units, p. 14-20)
Linked document includes scaffolds for English Language Learners, Students with Disabilities, Below Level Students, and
Above Level Students.
Resources
enVisionmath2.0
Developing Fluency
Multiplication Fact Thinking Strategies
Topic 12 Pacing Guide
Grade 4 Games to Build Fluency
Multi-Digit Addition & Subtraction Resources
Revised 2/2017
Property of MPS
Page 10 of 10