Download Fifth Grade Handbook

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Mat
2015 - 2016
Grade 5
Getting the Facts about Mathematics Florida Standards
Fifth Grade Table of Contents
I.
Planning
 Introduction to Pacing and Sequencing
 Pacing and Sequencing Chart
 Test Item Specifications
 Operations and Properties Tables
II.
Standards for Mathematical Practice








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III.
What Do Good Problem Solvers Do?
What Constitutes a Cognitively Demanding Task?
Key Ideas in Mathematics
Standards for Mathematical Practice Descriptions
Standards for Mathematical Practice – Student Behaviors
Standards for Mathematical Practice – Student Friendly Language
Standards for Mathematical Practice – Sample Questions for Teachers to Ask
Standards for Mathematical Practice in Action
Standards for Mathematical Practice in 5th Grade
Standards for Mathematical Practice Posters
Getting to know the Mathematics Florida Standards (MAFS)






Breaking the Code
MAFS by Grade Level at a Glance
Mathematics Florida Standards Changes
CCSS Domains, Clusters, and Critical Areas of Focus
Domain Progression
Fourth Grade Domain/Cluster Descriptors and Clarifications
NOTE: While some of the documents in this section were written based on Common Core
Standards, they still contain information that can be used with Mathematics Florida Standards
(MAFS). The changes as listed on the chart titled Mathematics Florida Standards Changes must be
considered when using these documents.
IV.
Additional Resources




Addition and Subtraction Strategies
Basic Multiplication and Division Strategies
Four Corners and Rhombus Math Graphic Organizer
Depth of Knowledge Levels/ Cognitive Complexity of Mathematics Items
Planning
INTRODUCTION TO PACING AND SEQUENCING- GRADE 5
INSTRUCTION:
ALGORITHMS AND FORMULAS:
 All instruction must be standards-based.
 The textbook is a resource and textbook
lessons must be carefully chosen and
aligned with the standards targeted for
instruction.
 It is critical that the Pacing and Sequencing
Chart and the FSA Test Item Specifications
are used for planning and implementing
lessons.
 The entire Pacing and Sequencing Chart
should be previewed in order to begin with
the end in mind and understand how the
mathematical concepts grow throughout the
year.
 The standard algorithm for multiplication of
whole numbers is introduced. The intent is
that students will build on the knowledge
about multiplication acquired in previous
grades, especially 4th grade. This is not just
a “rule” to be followed. Students must be
encouraged to understand why it works.
 The standard algorithm for division is NOT
to be introduced. Students are expected to
solve division problems using strategies
based on place value and properties. This
algorithm is in the 6th grade standards.
 Operations with decimals are to be
performed using strategies based on place
value and properties with the ability to
relate these strategies to written form. The
standard algorithms for operations with
decimals are introduced in 6th grade.
 The formula for volume of rectangular
prisms is introduced in 5th grade. The intent
is that students understand why this
formula works, not just memorize and use
it. It is not listed on the FSA Mathematics
Reference Sheet nor will it be given with
problems on FSA since it is a part of the
standards.
CONNECTIONS BETWEEN THE DOMAINS:
 Standards are not meant to be taught in
isolation.
 Each standard supports other standards
and will continue to be developed
throughout the year.
PROBLEM-SOLVING:
 Emphasis should be on engaging students
in deeper levels of thinking and analyzing.
 Students must have many opportunities to
explore the content of the standards
through real-world problem-solving tasks.
 Mathematical discourse must be an integral
part of instruction.
MEASUREMENT:
 Hands-on opportunities for students to be
engaged in measurement are critical.
 Hands-on measurement tasks may be
taught within the science and social studies
curricula.
VOCABULARY:
 Correct mathematical vocabulary MUST be
used. For example, students are expected
to use terms such as addend, sum, factor,
product, and so on.
POST-FSA IDEAS:
 Students should continue to work on critical
areas within the grade level standards.
 Project-based lessons and activities are
encouraged.
Possible resources to use are:
 AIMS Solve It!
 Navigating Through Numbers and
Operations in Grades 3-5, NCTM
 EnVision Math Worldscapes Literature
Library
 The Super Source Series, ETA/Cuisenaire
 Teaching Student-Centered Mathematics,
Vol.1, J.A.Van de Walle and L.H. Lovin
 Good Questions for Math Teaching, by
Peter Sullivan and Pat Lilburn
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
Standards for Mathematical Practice
Make sense of
problems and
persevere in
solving them.
Reason
abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate
tools
strategically.
Attend to
precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
FIRST NINE WEEKS
Mathematics Florida Standards (MAFS)
MAFS.5.NBT.2.5:
Fluently multiply multi-digit whole numbers using the
standard algorithm.
MAFS.5.NBT.2.6:
Find whole-number quotients of whole numbers with
up to four-digit dividends and two-digit divisors, using
strategies based on place value, the properties of
operations, and/or the relationship between
multiplication and division. Illustrate and explain the
calculation by using equations, rectangular arrays,
and/or area models.
Explanation and Examples*
Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps), and flexibility (using strategies that
demonstrate number sense).
This standard builds upon students’ work with multiplying in third and fourth grades where they used various strategies to multiply.
Students can continue to use these strategies as long as they are efficient, but must also understand and be able to use
the standard algorithm.
This standard extends students’ prior experiences with using strategies based on place value, the properties of operations, and/or
the relationship between multiplication and division to solve division problems. It also extends division to include 2-digit divisors
from 4th grade. The standard algorithm is not taught until sixth grade.
MAFS.5.NBT.1.1:
In this standard, students must reason about the magnitude of numbers. Teachers should provide opportunities for students to
Recognize that in a multi-digit number, a digit in one practice the concept that the value of the place is ten times as much as the place to its right and 1 the value of the place to the
10
place represents 10 times as much as it represents in
left. This standard extends the understanding to the relationship of decimal fractions. Students must be able to explain the
1
the place to its right and
of what it represents in
1
10
relationship between the two fives in 455.721 (5 is
of 50).
10
the place to its left.
Students may say, “I can show that the 5 to the left is 10 times larger than the five to the right.” Student’s response should reflect
an understanding of how a digit’s position affects its value.
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 1 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
This standard includes multiplying by multiples of 10 and powers of 10, including 102 which is10 x 10 =100, and 103 which is
Explain patterns in the number of zeros of the product 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.
when multiplying a number by powers of 10, and
explain patterns in the placement of the decimal point
Example:
when a decimal is multiplied or divided by a power of What is the value of the missing exponent in the expression 523 ÷10□ = 52.3?
10. Use whole-number exponents to denote powers of Solution: 1
10.
MAFS.5.NBT.1.3:
Examples:
1
1
1
Read, write, and compare decimals to thousandths.
a) A number in expanded form is shown. 3 × 1 + 2 × ( ) + 6 × (
)+5×(
)
10
1000
100
a. Read and write decimals to thousandths using
What is the number in decimal form? 3.265
base-ten numerals, number names, and expanded
form, e.g., 347.392 = 3  100 + 4  10 + 7  1 + 3 b) Compare 0.207 to 0.26:
MAFS.5.NBT.1.2:
(
1
1
1
)+9(
)+2(
).
10
100
1000
b. Compare two decimals to thousandths based on
meanings of the digits in each place, using >, =,
and < symbols to record the results of comparisons.
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 0.207 is 207 thousandths (and may write
hundredths (and may write
207
). 0.26 is 26
1000
26
260
) but I can also think of it as 260 thousandths (
). So, 260 thousandths is greater than
1000
100
207 thousandths.”
MAFS.5.NBT.1.4:
Use place value understanding to round decimals to
any place.
MAFS.5.NBT.2.7:
Add, subtract, multiply, and divide 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.
Students should go beyond simply applying an algorithm or procedure for rounding.
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).
14.2
14.3
This standard builds on the work from fourth grade where students are introduced to decimals and compare them. In fifth grade,
students begin adding, subtracting, multiplying and dividing decimals. This work should focus on concrete models and pictorial
representations, rather than relying solely on the algorithm. The use of symbolic notations involves having students record the
answers to computations (2.25 x 3 = 6.75), but this work should not be done without models or pictures. This standard includes
students’ reasoning and explanations of how they use models, pictures, and strategies.
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 2 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.NBT.2.7:
(Continued)
Add, subtract, multiply, and divide 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.
Students should be able to express that when they add decimals they add tenths to tenths and hundredths to hundredths. So,
when they are adding in a vertical format (numbers beneath each other), it is important that they write numbers with the same place
value beneath each other. This understanding can be reinforced by connecting addition of decimals to their understanding of
addition of fractions. Adding fractions with denominators of 10 and 100 is a standard in fourth grade.
Examples:
a)
4 - 0.3 = _____
3 tenths subtracted from 4 wholes. The wholes must be divided into tenths.
7
or 3.7.
10
b) An area model can be useful for illustrating products.
The answer is 3
c) Find the number in each group or share.
MAFS.5.NBT.2.7:
Students
be encouraged to apply a fair sharing model separating decimal values into equal parts such as
Add, subtract, multiply, and divide decimals to hundredths,
usingshould
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.
(Continued)
Find the number of groups.
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 3 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.NBT.2.7:
(Continued)
Add, subtract, multiply, and divide 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.
d) Joe has 1.6 meters of rope. He has to cut pieces of rope that are 0.2 meters long. How many can he cut?
To divide to find the number of groups, a student might:
 draw a segment to represent 1.6 meters. In doing so, he would count in tenths to identify the 6 tenths, and be able to
identify the number of 2 tenths within the 6 tenths. The student can then extend the idea of counting by tenths to divide
the one meter into tenths and determine that there are 5 more groups of 2 tenths.
10
, a student
10
might think of 1.6 as 16 tenths. Counting 2 tenths, 4 tenths, 6 tenths, . . .16 tenths, a student can count 8 groups of 2
tenths.
 count groups of 2 tenths without the use of models or diagrams. Knowing that 1 can be thought of as
 use their understanding of multiplication and think, “8 groups of 2 is 16, so 8 groups of
2
16
6
is
or 1 .”
10 10
10
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 4 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
Standards for Mathematical Practice
Make sense of
problems and
persevere in
solving them.
Reason
abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate
tools
strategically.
Attend to
precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
Second Nine Weeks
Mathematics Florida Standards (MAFS)
MAFS.5.NF.1.1:
Add and subtract fractions with unlike denominators
(including mixed numbers) by replacing given
fractions with equivalent fractions in such a way as to
produce an equivalent sum or difference of fractions
2 5
with like denominators. For example,
+ =
3 4
a
c (ad  bc )
8
15
23
+
=
. (In general, + =
.)
12
b
d
12
12
bd
Explanation and Examples*
Students should apply their understanding of equivalent fractions developed in fourth grade to find common denominators. They
should know that multiplying the denominators will always give a common denominator but may not result in the least common
denominator.
Examples:
2 7
+
5 8
16 35
+
40 40
51
=
40
1
4
3
3
12
1
=3
12
3
1
6
2
12
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 5 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.NF.1.2:
This standard refers to number sense, which means students’ understanding of fractions as numbers that lie between whole
numbers on a number line. Number sense in fractions also includes moving between decimals and fractions to find
Solve word problems involving addition and
7
7
1
3
3
subtraction of fractions referring to the same whole,
equivalents, also being able to use reasoning such as
is greater than
because
is missing only
and
is missing
including cases of unlike denominators, e.g., by using
4
4
8
8
8
visual fraction models or equations to represent the
7
1
so
is closer to a whole. Also, students should use benchmark fractions to estimate and examine the reasonableness
problem. Use benchmark fractions and number
4
8
sense of fractions to estimate mentally and assess the
5
5
1
1 4
6
6
1
reasonableness of answers. For example, recognize of their answers. For example,
is greater than
because
is larger than
( ) and
is only
larger
10
10
10
8
8
8
2 8
2 1 3
an incorrect result + = , by observing that
1 5
5 2 7
than
( ).
10
2
3 1
< .
3
2
7 2
Example: Jerry was making two different types of cookies. One recipe needed cup of sugar and the other needed cup of
4
3
sugar. How much sugar did he need to make both recipes?
Mental estimation: A student may say that Jerry needs more than 1 cup of sugar but less than 2 cups. An explanation may
1
1
compare both fractions to and state that both are larger than so the total must be more than 1. In addition, both fractions are
2
2
slightly less than 1 so the sum cannot be more than 2.
Area model
3
9
=
4 12
Linear model
2
8
=
3 12
3 2
+
4 3
9
8
17
+
=
12 12 12
17 12
5
=
+
12 12 12
5
1
12
Solution:
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 6 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.NF.2.3:
This standard calls for students to extend their work of partitioning a number line from third and fourth grade. Students need
Interpret a fraction as division of the numerator by the ample experiences to explore the concept that a fraction is a way to represent the division of two quantities.
a
denominator = a ÷ b. Solve word problems
Students are expected to demonstrate their understanding using concrete materials, drawing models, and explaining their
b
3
involving division of whole numbers leading to
thinking when working with fractions in multiple contexts. They read as “three fifths” and after many experiences with
5
answers in the form of fractions or mixed numbers,
3
e.g., by using visual fraction models or equations to
sharing problems, learn that can also be interpreted as “3 divided by 5.”
5
3
represent the problem. For example, interpret as
4
Examples:
3
multiplied
a) Ten team members are sharing 3 boxes of cookies. How much of a box will each student get?
4
When working this problem a student should recognize that the 3 boxes are being divided into 10 groups, so s/he is seeing the
by 4 equals 3, and that when 3 wholes are shared
solution to the following equation, 10 x n = 3 (10 groups of some amount is 3 boxes) which can also be written as 3 ÷ 10 = n.
equally among 4 people each person has a share of
3
Using models or diagram, they divide each box into 10 groups, resulting in each team member getting
of a box.
3
size . If 9 people want to share a 50-pound sack of
10
4
rice equally by weight, how many pounds of rice
b) Enter the two consecutive whole numbers that the quotient 78 ÷14 is between. Solution: Between 5 and 6.
should each person get? Between what two whole
numbers does your answer lie?
c) The six fifth grade classrooms have a total of 27 boxes of pencils. How many boxes will each classroom receive?
27
Students may recognize this as a whole number division problem but should also express this equal sharing problem as
.
6
3
1
27
They explain that each classroom gets
boxes of pencils and can further determine that each classroom get 4 or 4
6
6
2
boxes of pencils.
the result of dividing 3 by 4, noting that
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 7 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
Students need to develop a fundamental understanding that the multiplication of a fraction by a whole number could be
1
1 1
Apply and extend previous understandings of
represented as repeated addition of a unit fraction (e.g., 2 x ( ) = + ).
multiplication to multiply a fraction or whole number by
4 4
4
a fraction.
a
Students are expected to multiply fractions including proper fractions, fractions greater than one, and mixed numbers. They
a. Interpret the product  q as a parts of a
b
multiply fractions efficiently and accurately as well as solve problems in both contextual and non-contextual situations.
partition of q into b equal parts; equivalently, as
the result of a sequence of operations a  q ÷
3
As they multiply fractions such as x 6, they can think of the operation in more than one way.
b. For example, use a visual fraction model to
5
2
8
18
3

6
show  4 = , and create a story context for
or
3
3
5
51
2
4
18
this equation. Do the same with ( ) x ( ) =
(3 x 6) ÷ 5 or 18 ÷ 5 = ( )
3
5
5
8
a
c
ac
. (In general, x
=
).
3
b
d
bd
15
Students create a story problem for x 6 such as:
5
3
Isabel had 6 feet of wrapping paper. She used of the paper to wrap some presents. How much does she have left?
5
3
3
b. Find the area of a rectangle with fractional side
Every day Tim ran of a mile. How far did he run after 6 days? (Interpreting this as 6 x )
5
5
lengths by tiling it with unit squares of the
Example:
appropriate unit fraction side lengths, and show
2
2
4
that the area is the same as would be found by
Rectangle with dimensions of 2 and showing that 2 x
=
3
3
3
multiplying the side lengths. Multiply fractional
side lengths to find areas of rectangles, and
represent fraction products as rectangular areas.
MAFS.5.NF.2.4:
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 8 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.NF.2.5:
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of
one factor on the basis of the size of the other
factor, without performing the indicated
multiplication.
This standard calls for students to examine the magnitude of products in terms of the relationship between two types of problems.
This extends the work of 5.OA.1.
Examples:
3
a)
x 7 is less than 7 because 7 is multiplied by a factor less than 1 so the product must be less than 7.
4
7
b. Explaining why multiplying a given number by a
fraction greater than 1 results in a product greater
3
than the given number (recognizing multiplication
of 7
by whole numbers greater than 1 as a familiar
4
case); explaining why multiplying a given number
2
2
b) 2 x 8 must be more than 8 because 2 groups of 8 is 16 and 2 is almost 3 groups of 8. So the answer must be close to, but less
by a fraction less than 1 results in a product
3
3
smaller than the given number; and relating the
than 24.
a na
principle of fraction equivalence =
to the c) 3 = 5  3 because multiplying 3 by 5 is the same as multiplying by 1.
b nb
4 5 4
4
5
a
effect of multiplying by 1.
b
MAFS.5.NF.2.6:
Examples:
2
Solve real world problems involving multiplication of
a) Evan bought 6 roses for his mother.
of them were red. How many red roses were there?
3
fractions and mixed numbers, e.g., by using visual
fraction models or equations to represent the problem. Using a visual, a student divides the 6 roses into 3 groups and counts how many are in 2 of the 3 groups.
A student can use an equation to solve.
2
12
x6=
red roses
3
3
b) What is the area, in square units, of the rectangle.
Solution:
6
3 2
× =
units2
7 9 63
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 9 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.NF.2.7:
Apply and extend previous understandings of division
to divide unit fractions by whole numbers and whole
numbers by unit fractions.
This is the first time students are dividing with fractions. In fifth grade, students experience division problems with whole number divisors
and unit fraction dividends or with unit fraction divisors and whole number dividends. Students extend their understanding of the meaning
of fractions, how many unit fractions are in a whole, and their understanding of multiplication and division as involving equal groups or
shares and the number of objects in each group/share. In sixth grade, they will use this foundational understanding to divide more
complex fractions and develop abstract methods of dividing with fractions.
a. Interpret division of a unit fraction by a non-zero
whole number, and compute such quotients. For Examples:
a) Knowing the number of groups/shares and how many/much in each group/share:
example, create a story context for
1
1
Four students sitting at a table were given of a pan of brownies to share. How much of a pan will each student get if they share the
÷ 4, and use a visual fraction model to show
3
3
pan of brownies equally?
the quotient. Use the relationship between
1
1
The diagram shows the pan divided into 4 equal shares with each share equaling
of the pan.
1
3
12
multiplication and division to explain that ÷ 4 =
3
1
1
1
because
 4= .
3
12
12
b) Knowing how many in each group/share and finding how many groups/shares:
b. Interpret division of a whole number by a unit
1
1
Angelo has 4 lbs. of peanuts. He wants to give each of his friends lb. How many friends can receive lb. of peanuts?
fraction, and compute such quotients. For
5
5
example, create a story context for
1
1
A diagram for 4 ÷ = n is shown below. Students explain that since there are five-fifths in one whole, there must be 20 fifths in 4 lbs.
4 ÷ , and use a visual fraction model to show
5
5
1 lb. of peanuts.
the quotient. Use the relationship between
multiplication and division to explain that for 4 ÷
1
1
= 20 because 20  = 4.
5
5
1
c) How much rice will each person get if 3 people share
lb. of rice equally?
c. Solve real world problems involving division of
2
unit fractions by non-zero whole numbers and
1
division of whole numbers by unit fractions, e.g.,
÷3
2
by using visual fraction models and equations to
3
1
represent the problem. For example, how much
÷3=
6
6
chocolate will each person get if 3 people share
1
1
1
lb. of chocolate equally? How many cup
A student may think or draw a whole and shade half. Next s/he may divide that
into 3 equal parts and then determine that each of those
3
2
2
servings are in 2 cups of raisins?
1
parts is .
6
1
3
3
1
A student may think of as equivalent to
and
divided by 3 is .
2
6
6
6
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 10 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
Standards for Mathematical Practice
Make sense of
problems and
persevere in
solving them.
Reason
abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate
tools
strategically.
Attend to
precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
Third Nine Weeks
Mathematics Florida Standards (MAFS)
Explanation and Examples*
MAFS.5.G.1.1:
Example: Students can use a classroom size coordinate system to physically locate the coordinate point (5, 3) by starting at the
Use a pair of perpendicular number lines, called axes, to origin point (0,0), walking 5 units along the x axis to find the first number in the pair (5), and then walking up 3 units for the second
number in the pair (3). The ordered pair names a point in the plane (x, y).
define a coordinate system, with the intersection of the
lines (the origin) arranged to coincide with the 0 on each
line and a given point in the plane located by using an
ordered pair of numbers, called its coordinates.
Understand that the first number indicates how far to
travel from the origin in the direction of one axis, and the
second number indicates how far to travel in the
direction of the second axis, with the convention that the
names of the two axes and the coordinates correspond
(e.g., x-axis and x-coordinate, y-axis and y-coordinate).
MAFS.5.G.1.2:
Examples:
Represent real world and mathematical problems by Some locations in Lamar’s town are shown in the coordinate plane.
graphing points in the first quadrant of the coordinate
plane, and interpret coordinate values of points in the
context of the situation.
Lamar moved from one location to another by traveling 1
unit left and 5 units up. Which ways could he have traveled?
A.
B.
C.
D.
From home to the park
From the park to the library
From home to the library
From school to the park
Solution: C
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 11 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.OA.1.1:
Use parentheses, brackets, or braces in numerical
expressions, and evaluate expressions with these
symbols.
Students need experiences with multiple expressions that use grouping symbols throughout the year to develop understanding of when
and how to use parentheses, brackets, and braces. First, students use these symbols with whole numbers. Then the symbols can be
used as students add, subtract, multiply and divide decimals and fractions. (Items may not require division with fractions.)
Examples:
1
x {6 x 1 + 7} + 11
2
1
Step 1: x {6 x 8}+ 11
2
1
Step 2: x 48 + 11
2
Step 3: 24 + 11
a)
Step 4: 35
In which step does a mistake first appear?
A. Step 1
B. Step 2
C. Step 3
D. Step 4
Solution: A
b) To further develop students’ understanding of grouping symbols and facility with operations, students place grouping symbols in
equations to make the equations true or they compare expressions that are grouped differently.
3 + 8 – 4 × 2 – 12
Create an equivalent expression that includes a set of parentheses so that the value of the expression is 2.
Solution: (3 + 8 – 4) ×2 - 12
This standard calls for students to verbally describe the relationship between expressions without actually calculating them. Students
MAFS.5.OA.1.2:
should apply their understanding of the four operations, grouping symbols, and place value to write expressions and interpret the
Write simple expressions that record calculations with
meaning of a numerical expression.
numbers, and interpret numerical expressions without
evaluating them. For example, express the
Examples:
calculation “add 8 and 7, then multiply by 2” as
a) Write an expression for the steps “double 5 then add 25.” (2 x 5 ) + 25
2 x (8 + 7). Recognize that 3 x (18,932 + 921) is three
1
b) Recognize that 0.5 x (300 ÷ 15) is of (300 ÷ 15) without calculating the quotient.
times as large as 18,932 + 921 without having to
2
calculate the indicated sum or product.
c) Describe how the expression 5(10 x 10) relates to 10 x 10 (it is 5 times larger).
* Multiplication cross symbol (×) is the only acceptable symbol for multiplication. The multiplication dot (•)
may not be used.
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 12 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.OA.2.3:
This standard extends the work from fourth grade, where students generate numerical patterns when they are given one rule.
In fifth grade, students are given two rules and generate two numerical patterns. The graphs that are created should be line
graphs to represent the pattern.
Generate two numerical patterns using two given
rules. Identify apparent relationships between
corresponding terms. Form ordered pairs consisting of
corresponding terms from the two patterns, and graph Example:
the ordered pairs on a coordinate plane. For example, a) Use the rule “add 3” to write a sequence of numbers. Starting with a 0, students write 0, 3, 6, 9, 12, . . .
Use the rule “add 6” to write a sequence of numbers. Starting with 0, students write 0, 6, 12, 18, 24, . .
given the rule “Add 3” and the starting number 0, and
After comparing these two sequences, the students notice that each term in the second sequence is twice the corresponding
given the rule “Add 6” and the starting number 0,
terms of the first sequence. One way they justify this is by describing the patterns of the terms. Their justification may include
generate terms in the resulting sequences, and
some mathematical notation (see example below). A student may explain that both sequences start with zero and to generate
observe that the terms in one sequence are twice the
each term of the second sequence he/she added 6, which is twice as much as was added to produce the terms in the first
corresponding terms in the other sequence. Explain
sequence. Students may also use the distributive property to describe the relationship between the two numerical patterns by
informally why this is so.
reasoning that 6 + 6 + 6 = 2 (3 + 3 + 3).
0,
+6
+3
6,
3,
+6
+3
12,
6,
+3
+618,
9,
+6
+312, . . .
24,. . .
b) Once students can describe that the second sequence of numbers is twice the corresponding terms of the first sequence, the
terms can be written in ordered pairs and then graphed on a coordinate grid. They should recognize that each point on the
graph represents two quantities in which the second quantity is twice the first quantity.
Ordered pairs
MAFS.5.MD.3.3:
Students’ prior experiences with volume were limited to liquid volume. As students develop their understanding of volume they
understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. This cube has a length of 1 unit, a width of 1
Recognize volume as an attribute of solid figures and
unit and a height of 1 unit and is called a cubic unit. This cubic unit is written with an exponent of 3 (e.g., in3, m3). Students connect this
understand concepts of volume measurement.
notation to their understanding of powers of 10 in our place value system. Models of cubic inches, cubic centimeters, cubic feet, etc., are
a. A cube with side length 1 unit, called a “unit cube” is
helpful in developing an image of a cubic unit. Students estimate how many cubic yards would be needed to fill the classroom or how
said to have “one cubic unit” of volume, and can be
many cubic centimeters would be needed to fill a pencil box.
used to measure volume.
b. A solid figure which can be packed without gaps or
overlaps using n unit cube is said to have a volume
of n cubic units.
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 13 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.MD.3.4:
Students understand that same sized cubic units are used to measure volume. They select appropriate units to measure volume. For
example, they make a distinction between which units are more appropriate for measuring the volume of a gym and the volume of a box
Measure volumes by counting unit cubes, using cubic
of books. They can also improvise a cubic unit using any unit as a length (e.g., the length of their pencil). Students can apply these ideas
cm, cubic in., cubic ft., and improvised units.
by filling containers with cubic units (wooden cubes) to find the volume.
Students need multiple opportunities to measure volume by filling rectangular prisms with cubes. By looking at the relationship between
MAFS.5.MD.3.5:
the length, the width, and the height, they develop understanding of both formulas and explore how these ideas would apply to other
Relate volume to the operations of multiplication and
prisms. Students use the associative property of multiplication and decomposition of numbers using factors to investigate rectangular
addition and solve real world and mathematical
prisms with a given number of cubic units.
problems involving volume.
Examples:
a) When given 24 cubes, students make as many rectangular prisms as possible with a volume of 24 cubic units. Students build the
a. Find the volume of a right rectangular prism with
prisms and record possible dimensions.
whole-number side lengths by packing it with unit
Length
Width
Height
cubes, and show that the volume is the same as
1
2
12
would be found by multiplying the edge lengths,
equivalently by multiplying the height by the area of
2
2
6
the base. Represent threefold whole-number
4
2
3
products as volumes, e.g., to represent the
8
3
1
associative property of multiplication.
b. Apply the formulas V = l  w  h and V = B  h for
rectangular prisms to find volumes of right
rectangular prisms with whole-number edge lengths
in the context of solving real world and
mathematical problems.
b) Select all the options that could be the dimensions of a rectangular prism with a volume of 384 cubic feet (ft).
□
□
□
□
□
Length: 6 ft., width: 8 ft., height: 8 ft.
Length: 4 ft., width: 12 ft., height: 24 ft.
Length: 4 ft., width: 6 ft., height: 16 ft.
Length: 4 ft., width: 8 ft., height: 12 ft.
Length: 3 ft., width: 10 ft., height: 20 ft.
Solution:
Length: 6 ft., width: 8ft., height: 8 ft.
Length: 4 ft., width: 6 ft., height: 16 ft.
Length: 4 ft., width: 8 ft., height: 12 ft.
c. Recognize volume as additive. Find volumes of
solid figures composed of two non-overlapping right c) A homeowner is building a swimming pool and needs to calculate the volume of water needed to fill the pool. The design of the
pool is shown in the illustration below.
rectangular prisms by adding the volumes of the
non-overlapping parts, applying this technique to
solve real world problems.
Prism 1: 20 ft. x 5 ft. x 5 ft.
Prism 2: 14 ft. x 5 ft. x 5 ft.
MAFS.5.MD.1.1:
Convert among different-sized standard measurement
units (i.e., km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec.)
within a given measurement system (e.g., convert 5 cm
to 0.05 m), and use these conversions in solving multistep, real world problems.
In fifth grade, students build on their prior knowledge of related measurement units to determine equivalent measurements. Prior to
making actual conversions, they examine the units to be converted, determine if the converted amount will be more or less units than the
original unit, and explain their reasoning. They use several strategies to convert measurements. When converting metric measurement,
students apply their understanding of place value and decimals.
(This standard may be carried over into the fourth nine weeks, if FSA testing schedule allows.)
Fifth Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 14 of 17, Brevard Public Schools, 2015 – 2016
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction and FSA Test Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
Standards for Mathematical Practice
Make sense of
problems and
persevere in
solving them.
Reason
abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate
tools
strategically.
Attend to
precision.
Look for and
make use of
structure.
Look for and
express
regularity in
repeated
reasoning.
Fourth Nine Weeks
Mathematics Florida Standards (MAFS)
Explanation and Examples*
MAFS.5.MD.1.1:
In fifth grade, students build on their prior knowledge of related measurement units to determine equivalent
measurements. Prior to making actual conversions, they examine the units to be converted, determine if the
Convert among different-sized standard measurement units (i.e.,
converted amount will be more or less units than the original unit, and explain their reasoning. They use several
km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec.) within a given
measurement system (e.g., convert 5 cm to 0.05 m), and use these strategies to convert measurements. When converting metric measurement, students apply their understanding
of place value and decimals.
conversions in solving multi-step, real world problems.
MAFS.5.MD.2.2:
Make a line plot to display a data set of measurements in fractions
1 1 1
of a unit ( , , ). Use operations on fractions for this grade to
2 4 8
solve problems involving information presented in line plots. For
example, given different measurements of liquid in identical
beakers, find the amount of liquid each beaker would contain if the
total amount in all the beakers were redistributed equally.
This standard provides a context for students to work with fractions by measuring objects to one-eighth of a
unit. This includes length, mass, and liquid volume. Students are making a line plot of this data and then
adding and subtracting fractions based on data in the line plot.
Example:
Ten beakers, measured in liters, are filled with a liquid.
Liquid in Beakers
Amount of Liquid (in Liters)
The line plot above shows the amount of liquid in liters in 10 beakers. If the liquid is redistributed equally, how
5
much liquid would each beaker have?
16
Students apply their understanding of operations with fractions. They use either addition and/or multiplication to
determine the total number of liters in the beakers. Then the sum of the liters is shared evenly among the ten
beakers.
Fifth Grade Mathematics Florida Standards, Explanations and Examples, page 15 of 17, Brevard Public Schools, 2014 – 2015
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction, and FSA Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
MAFS.5.G.2.3:
Understand that attributes belonging to a category of twodimensional figures also belong to all subcategories of that
category. For example, all rectangles have four right angles and
squares are rectangles, so all squares have four right angles.
MAFS.5.G.2.4:
Classify and organize two-dimensional figures into Venn diagrams
based on the attributes of the figures.
This standard calls for students to reason about the attributes (properties) of shapes. Students should have many
opportunities to discuss the properties of shapes.
Examples:
a) Do all quadrilaterals have right angles? No
b) A parallelogram has 4 sides with both sets of opposite sides parallel. What types of quadrilaterals are
parallelograms? square, rectangle, and rhombus
c) Regular polygons have all of their sides and angles congruent. Name or draw some regular polygons.
d) All rectangles have 4 right angles. Squares have 4 right angles so they are also rectangles. True or False?
e) A trapezoid has 2 sides parallel so it must be a parallelogram. True or False?
This standard builds on what was done in fourth grade. Figures from previous grades include polygon,
rhombus/rhombi, rectangle, square, triangle, quadrilateral, pentagon, hexagon, cube, trapezoid,
half/quarter circle, and circle.
Properties of a figure may include:
Properties of sides—parallel, perpendicular, congruent, number of sides
Properties of angles—types of angles, congruent
Lines of symmetry
Examples:
A right triangle can be both scalene and isosceles, but not equilateral.
A scalene triangle can be right, acute or obtuse.
Triangles can be classified by angles and sides:
Examples:
Right: The triangle has one angle that measures 90º.
Acute: The triangle has exactly three angles that measure between 0º and 90º.
Obtuse: The triangle has exactly one angle that measures greater than 90º and less than 180º.
Equilateral: All sides of the triangle are the same length.
Isosceles: At least two sides of the triangle are the same length.
Scalene: No sides of the triangle are the same length.
Fifth Grade Mathematics Florida Standards, Explanations and Examples, page 16 of 17, Brevard Public Schools, 2014 – 2015
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction, and FSA Item Specifications.
Pacing and Sequencing Chart
Fifth Grade Mathematics Florida Standards
2015 - 2016
FIFTH GRADE – CRITICAL AREAS OF FOCUS
In Grade 5, instructional time should focus on three critical areas:
(1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions
and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit
fractions);
(2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing
understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal
operations; and
(3) developing understanding of volume.
(1) Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike
denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions,
and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship
between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense.
(Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)
(2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of
operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of
models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency
in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions,
as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is
a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They
compute products and quotients of decimals to hundredths efficiently and accurately.
(3) Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the
total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by
1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that
involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by
viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine
volumes to solve real world and mathematical problems.
Fifth Grade Mathematics Florida Standards, Explanations and Examples, page 17 of 17, Brevard Public Schools, 2014 – 2015
*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction, and FSA Item Specifications.
DRAFT
Grade5Mathematics ItemSpecifications
Grade 5 Mathematics Item Specifications Florida Standards Assessments The draft Florida Standards Assessments (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as provided in CPALMs. The Specifications are a resource that defines the content and format of the test and test items for item writers and reviewers. Each grade‐level and course Specifications document indicates the alignment of items with the Florida Standards. It also serves to provide all stakeholders with information about the scope and function of the FSA. Item Specifications Definitions Also assesses refers to standard(s) closely related to the primary standard statement. Clarification statements explain what students are expected to do when responding to the question. Assessment limits define the range of content knowledge and degree of difficulty that should be assessed in the assessment items for the standard. Item types describe the characteristics of the question. Context defines types of stimulus materials that can be used in the assessment items.  Context – Allowable refers to items that may but are not required to have context.  Context – No context refers to items that should not have context.  Context – Required refers to items that must have context. 2 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Technology‐EnhancedItemDescriptions:
TheFloridaStandardsAssessments(FSA)arecomposedoftestitemsthatinclude
traditionalmultiple‐choiceitems,itemsthatrequirestudentstotypeorwritearesponse,
andtechnology‐enhanceditems(TEI).Technology‐enhanceditemsarecomputer‐delivered
itemsthatrequirestudentstointeractwithtestcontenttoselect,construct,and/orsupport
theiranswers.
Currently,thereareninetypesofTEIsthatmayappearoncomputer‐basedassessmentsfor
FSAMathematics.ForstudentswithanIEPor504planthatspecifiesapaper‐based
accommodation,TEIswillbemodifiedorreplacedwithtestitemsthatcanbescannedand
scoredelectronically.
Forsamplesofeachoftheitemtypesdescribedbelow,seetheFSATrainingTests.
Technology‐EnhancedItemTypes–Mathematics
1. EditingTaskChoice–Thestudentclicksahighlightedwordorphrase,which
revealsadrop‐downmenucontainingoptionsforcorrectinganerroraswell
asthehighlightedwordorphraseasitisshowninthesentencetoindicate
thatnocorrectionisneeded.Thestudentthenselectsthecorrectwordor
phrasefromthedrop‐downmenu.Forpaper‐basedassessments,theitemis
modifiedsothatitcanbescannedandscoredelectronically.Thestudentfills
inacircletoindicatethecorrectwordorphrase.
2. EditingTask–Thestudentclicksonahighlightedwordorphrasethatmaybe
incorrect,whichrevealsatextbox.Thedirectionsinthetextboxdirectthe
studenttoreplacethehighlightedwordorphrasewiththecorrectwordor
phrase.Forpaper‐basedassessments,thisitemtypemaybereplacedwith
anotheritemtypethatassessesthesamestandardandcanbescannedand
scoredelectronically.
3. HotText–
a. SelectableHotText–Excerptedsentencesfromthetextarepresented
inthisitemtype.Whenthestudenthoversovercertainwords,phrases,
orsentences,theoptionshighlight.Thisindicatesthatthetextis
selectable(“hot”).Thestudentcanthenclickonanoptiontoselectit.
Forpaper‐basedassessments,a“selectable”hottextitemismodified
sothatitcanbescannedandscoredelectronically.Inthisversion,the
studentfillsinacircletoindicateaselection.
3 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments b. Drag‐and‐DropHotText–Certainnumbers,words,phrases,or
sentencesmaybedesignated“draggable”inthisitemtype.Whenthe
studenthoversovertheseareas,thetexthighlights.Thestudentcan
thenclickontheoption,holddownthemousebutton,anddragittoa
graphicorotherformat.Forpaper‐basedassessments,drag‐and‐drop
hottextitemswillbereplacedwithanotheritemtypethatassessesthe
samestandardandcanbescannedandscoredelectronically.
4. OpenResponse–Thestudentusesthekeyboardtoenteraresponseintoatext
field.Theseitemscanusuallybeansweredinasentenceortwo.Forpaper‐based
assessments,thisitemtypemaybereplacedwithanotheritemtypethatassesses
thesamestandardandcanbescannedandscoredelectronically.
5. Multiselect–Thestudentisdirectedtoselectallofthecorrectanswersfrom
amonganumberofoptions.Theseitemsaredifferentfrommultiple‐choiceitems,
whichallowthestudenttoselectonlyonecorrectanswer.Theseitemsappearin
theonlineandpaper‐basedassessments.
6. GraphicResponseItemDisplay(GRID)–Thestudentselectsnumbers,words,
phrases,orimagesandusesthedrag‐and‐dropfeaturetoplacethemintoagraphic.
Thisitemtypemayalsorequirethestudenttousethepoint,line,orarrowtoolsto
createaresponseonagraph.Forpaper‐basedassessments,thisitemtypemaybe
replacedwithanotheritemtypethatassessesthesamestandardandcanbescanned
andscoredelectronically.
7. EquationEditor–Thestudentispresentedwithatoolbarthatincludesavarietyof
mathematicalsymbolsthatcanbeusedtocreatearesponse.Responsesmaybein
theformofanumber,variable,expression,orequation,asappropriatetothetest
item.Forpaper‐basedassessments,thisitemtypemaybereplacedwithamodified
versionoftheitemthatcanbescannedandscoredelectronicallyorreplacedwith
anotheritemtypethatassessesthesamestandardandcanbescannedandscored
electronically.
8. MatchingItem–Thestudentchecksaboxtoindicateifinformationfromacolumn
headermatchesinformationfromarow.Forpaper‐basedassessments,thisitem
typemaybereplacedwithanotheritemtypethatassessesthesamestandardand
canbescannedandscoredelectronically.
9. TableItem–Thestudenttypesnumericvaluesintoagiventable.Thestudentmay
completetheentiretableorportionsofthetabledependingonwhatisbeing
asked.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanother
itemtypethatassessesthesamestandardandcanbescannedandscored
electronically.
4 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments MathematicalPractices:
TheMathematicalPracticesareapartofeachcoursedescriptionforGrades3‐8,Algebra1,
Geometry,andAlgebra2.Thesepracticesareanimportantpartofthecurriculum.The
MathematicalPracticeswillbeassessedthroughout.
Makesenseofproblemsandpersevereinsolvingthem.
MAFS.K12.MP.1.1:
MAFS.K12.MP.2.1:
5 | P a g e Mathematicallyproficientstudentsstartbyexplainingto
themselvesthemeaningofaproblemandlookingforentrypoints
toitssolution.Theyanalyzegivens,constraints,relationships,and
goals.Theymakeconjecturesabouttheformandmeaningofthe
solutionandplanasolutionpathwayratherthansimplyjumping
intoasolutionattempt.Theyconsideranalogousproblems,andtry
specialcasesandsimplerformsoftheoriginalprobleminorderto
gaininsightintoitssolution.Theymonitorandevaluatetheir
progressandchangecourseifnecessary.Olderstudentsmight,
dependingonthecontextoftheproblem,transformalgebraic
expressionsorchangetheviewingwindowontheirgraphing
calculatortogettheinformationtheyneed.Mathematically
proficientstudentscanexplaincorrespondencesbetween
equations,verbaldescriptions,tables,andgraphsordrawdiagrams
ofimportantfeaturesandrelationships,graphdata,andsearchfor
regularityortrends.Youngerstudentsmightrelyonusingconcrete
objectsorpicturestohelpconceptualizeandsolveaproblem.
Mathematicallyproficientstudentschecktheiranswersto
problemsusingadifferentmethod,andtheycontinuallyask
themselves,“Doesthismakesense?”Theycanunderstandthe
approachesofotherstosolvingcomplexproblemsandidentify
correspondencesbetweendifferentapproaches.
Reasonabstractlyandquantitatively.
Mathematicallyproficientstudentsmakesenseofquantitiesand
theirrelationshipsinproblemsituations.Theybringtwo
complementaryabilitiestobearonproblemsinvolvingquantitative
relationships:theabilitytodecontextualize—toabstractagiven
situationandrepresentitsymbolicallyandmanipulatethe
representingsymbolsasiftheyhavealifeoftheirown,without
necessarilyattendingtotheirreferents—andtheabilityto
contextualize,topauseasneededduringthemanipulationprocess
inordertoprobeintothereferentsforthesymbolsinvolved.
May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Quantitativereasoningentailshabitsofcreatingacoherent
representationoftheproblemathand;consideringtheunits
involved;attendingtothemeaningofquantities,notjusthowto
computethem;andknowingandflexiblyusingdifferentproperties
ofoperationsandobjects.
Constructviableargumentsandcritiquethereasoningof
others.
MAFS.K12.MP.3.1:
MAFS.K12.MP.4.1:
6 | P a g e Mathematicallyproficientstudentsunderstandandusestated
assumptions,definitions,andpreviouslyestablishedresultsin
constructingarguments.Theymakeconjecturesandbuildalogical
progressionofstatementstoexplorethetruthoftheirconjectures.
Theyareabletoanalyzesituationsbybreakingthemintocases,and
canrecognizeandusecounterexamples.Theyjustifytheir
conclusions,communicatethemtoothers,andrespondtothe
argumentsofothers.Theyreasoninductivelyaboutdata,making
plausibleargumentsthattakeintoaccountthecontextfromwhich
thedataarose.Mathematicallyproficientstudentsarealsoableto
comparetheeffectivenessoftwoplausiblearguments,distinguish
correctlogicorreasoningfromthatwhichisflawed,and—ifthere
isaflawinanargument—explainwhatitis.Elementarystudents
canconstructargumentsusingconcretereferentssuchasobjects,
drawings,diagrams,andactions.Suchargumentscanmakesense
andbecorrect,eventhoughtheyarenotgeneralizedormade
formaluntillatergrades.Later,studentslearntodetermine
domainstowhichanargumentapplies.Studentsatallgradescan
listenorreadtheargumentsofothers,decidewhethertheymake
sense,andaskusefulquestionstoclarifyorimprovethearguments.
Modelwithmathematics.
Mathematicallyproficientstudentscanapplythemathematicsthey
knowtosolveproblemsarisingineverydaylife,society,andthe
workplace.Inearlygrades,thismightbeassimpleaswritingan
additionequationtodescribeasituation.Inmiddlegrades,a
studentmightapplyproportionalreasoningtoplanaschoolevent
oranalyzeaprobleminthecommunity.Byhighschool,astudent
mightusegeometrytosolveadesignproblemoruseafunctionto
describehowonequantityofinterestdependsonanother.
Mathematicallyproficientstudentswhocanapplywhattheyknow
arecomfortablemakingassumptionsandapproximationsto
simplifyacomplicatedsituation,realizingthatthesemayneed
revisionlater.Theyareabletoidentifyimportantquantitiesina
practicalsituationandmaptheirrelationshipsusingsuchtoolsas
diagrams,two‐waytables,graphs,flowchartsandformulas.They
cananalyzethoserelationshipsmathematicallytodraw
May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments MAFS.K12.MP.5.1:
conclusions.Theyroutinelyinterprettheirmathematicalresultsin
thecontextofthesituationandreflectonwhethertheresultsmake
sense,possiblyimprovingthemodelifithasnotserveditspurpose.
Useappropriatetoolsstrategically.
Mathematicallyproficientstudentsconsidertheavailabletools
whensolvingamathematicalproblem.Thesetoolsmightinclude
pencilandpaper,concretemodels,aruler,aprotractor,a
calculator,aspreadsheet,acomputeralgebrasystem,astatistical
package,ordynamicgeometrysoftware.Proficientstudentsare
sufficientlyfamiliarwithtoolsappropriatefortheirgradeorcourse
tomakesounddecisionsaboutwheneachofthesetoolsmightbe
helpful,recognizingboththeinsighttobegainedandtheir
limitations.Forexample,mathematicallyproficienthighschool
studentsanalyzegraphsoffunctionsandsolutionsgeneratedusing
agraphingcalculator.Theydetectpossibleerrorsbystrategically
usingestimationandothermathematicalknowledge.Whenmaking
mathematicalmodels,theyknowthattechnologycanenablethem
tovisualizetheresultsofvaryingassumptions,explore
consequences,andcomparepredictionswithdata.Mathematically
proficientstudentsatvariousgradelevelsareabletoidentify
relevantexternalmathematicalresources,suchasdigitalcontent
locatedonawebsite,andusethemtoposeorsolveproblems.They
areabletousetechnologicaltoolstoexploreanddeepentheir
understandingofconcepts.
Attendtoprecision.
MAFS.K12.MP.6.1:
7 | P a g e Mathematicallyproficientstudentstrytocommunicatepreciselyto
others.Theytrytousecleardefinitionsindiscussionwithothers
andintheirownreasoning.Theystatethemeaningofthesymbols
theychoose,includingusingtheequalsignconsistentlyand
appropriately.Theyarecarefulaboutspecifyingunitsofmeasure,
andlabelingaxestoclarifythecorrespondencewithquantitiesina
problem.Theycalculateaccuratelyandefficiently,express
numericalanswerswithadegreeofprecisionappropriateforthe
problemcontext.Intheelementarygrades,studentsgivecarefully
formulatedexplanationstoeachother.Bythetimetheyreachhigh
schooltheyhavelearnedtoexamineclaimsandmakeexplicituseof
definitions.
May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Lookforandmakeuseofstructure.
MAFS.K12.MP.7.1:
MAFS.K12.MP.8.1:
Mathematicallyproficientstudentslookcloselytodiscernapattern
orstructure.Youngstudents,forexample,mightnoticethatthree
andsevenmoreisthesameamountassevenandthreemore,or
theymaysortacollectionofshapesaccordingtohowmanysides
theshapeshave.Later,studentswillsee7×8equalsthewell
remembered7×5+7×3,inpreparationforlearningaboutthe
distributiveproperty.Intheexpressionx²+9x+14,olderstudents
canseethe14as2×7andthe9as2+7.Theyrecognizethe
significanceofanexistinglineinageometricfigureandcanusethe
strategyofdrawinganauxiliarylineforsolvingproblems.Theyalso
canstepbackforanoverviewandshiftperspective.Theycansee
complicatedthings,suchassomealgebraicexpressions,assingle
objectsorasbeingcomposedofseveralobjects.Forexample,they
cansee5–3(x–y)²as5minusapositivenumbertimesasquare
andusethattorealizethatitsvaluecannotbemorethan5forany
realnumbersxandy.
Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsare
repeated,andlookbothforgeneralmethodsandforshortcuts.
Upperelementarystudentsmightnoticewhendividing25by11
thattheyarerepeatingthesamecalculationsoverandoveragain,
andconcludetheyhavearepeatingdecimal.Bypayingattentionto
thecalculationofslopeastheyrepeatedlycheckwhetherpointsare
onthelinethrough(1,2)withslope3,middleschoolstudents
mightabstracttheequation(y–2)/(x–1)=3.Noticingthe
regularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x
–1)(x²+x+1),and(x–1)(x³+x²+x+1)mightleadthemtothe
generalformulaforthesumofageometricseries.Astheyworkto
solveaproblem,mathematicallyproficientstudentsmaintain
oversightoftheprocess,whileattendingtothedetails.They
continuallyevaluatethereasonablenessoftheirintermediate
results.
8 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments ReferenceSheets:
•Referencesheetsandz‐tableswillbeavailableasonlinereferences(inapop‐upwindow).A
paperversionwillbeavailableforpaper‐basedtests.
•ReferencesheetswithconversionswillbeprovidedforFSAMathematicsassessmentsin
Grades4–8andEOCMathematicsassessments.
•ThereisnoreferencesheetforGrade3.
•ForGrades4,6,and7,Geometry,andAlgebra2,someformulaswillbeprovidedonthe
referencesheet.
•ForGrade5andAlgebra1,someformulasmaybeincludedwiththetestitemifneededto
meettheintentofthestandardbeingassessed.
•ForGrade8,noformulaswillbeprovided;however,conversionswillbeavailableona
referencesheet.
•ForAlgebra2,az‐tablewillbeavailable.
Grade
Conversions
SomeFormulas
z‐table
3
No
No
No
4
OnReferenceSheet
OnReferenceSheet
No
5
OnReferenceSheet
WithItem
No
6
OnReferenceSheet
OnReferenceSheet
No
7
OnReferenceSheet
OnReferenceSheet
No
8
OnReferenceSheet
No
No
Algebra1
OnReferenceSheet
WithItem
No
Algebra2
OnReferenceSheet
OnReferenceSheet
Yes
Geometry
OnReferenceSheet
OnReferenceSheet
No
9 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.OA Operations and Algebraic Thinking MAFS.5.OA.1 Write and interpret numerical expressions. MAFS.5.OA.1.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Assessment Limits Expressions may contain whole numbers and up to one fraction with a denominator of 10 or less. Items may not require division with fractions. Items may not contain nested grouping symbols. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Context No context Sample Item Item Type Equation Editor An expression is shown. 3 + 8 – 4 x 2 – 12 Create an equivalent expression that includes a set of parentheses so that the value of the expression is 2. Equation Editor What is the value of the expression x [4 + 6] – 9? A numerical expression is evaluated as shown. x {6 x 1 + 7} + 11 Step 1: x {6 x 8} + 11 Step 2: x 48 + 11 Step 3: 24 + 11 Step 4: 35 In which step does a mistake first appear? A. Step 1 B. Step 2 C. Step 3 D. Step 4 See Appendix for the practice test item aligned to this standard. 10 | P a g e May 2016 Multiple Choice Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.OA Operations and Algebraic Thinking MAFS.5.OA.1 Write and interpret numerical expressions. MAFS.5.OA.1.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Assessment Limits Expressions may contain whole numbers or fractions with a denominator of 10 or less. Expressions may not include nested parentheses. Multiplication cross symbol is the only acceptable symbol for multiplication. The multiplication dot ( ) may not be used. When grouping symbols are part of the expression, the associative property or distributive property must be found in the expression. Calculator No Item Types Equation Editor Multiple Choice Multiselect Open Response Context No context Sample Item Item Type Multiple Choice Which expression could represent the following phrase? Divide 10 by 2, then subtract 3. A.
B.
C.
D.
2 ÷ 10 – 3 2 ÷ (10 – 3) 10 ÷ 2 – 3 10 ÷ (2 – 3) Which statement describes the expression 18 + x (9 – 4) ? A.
B.
C.
D.
Half the difference of 4 from 9 added to 18 Subtract half the quantity of 9 and 4 from 18 The sum of 18 and half the product of 9 and 4 Half of 9 added to 18 minus 4 See Appendix for the practice test item aligned to this standard. 11 | P a g e May 2016 Multiple Choice Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.OA Operations and Algebraic Thinking MAFS.5.OA.2 Analyze patterns and relationships. MAFS.5.OA.2.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Assessment Limits Expressions may contain whole numbers or fractions with a denominator of 10 or less. Ordered pairs many only be located within Quadrant I of the coordinate plane. Operations in rules limited to: addition, subtraction, multiplication, and division. Patterns that require division may not lead to fractional terms. Items may not contain rules that exceed two procedural operations. Items must provide the rule. Expressions may not include nested parentheses. Calculator No Item Types Editing Task Choice Equation Editor GRID Hot Text Multiple Choice Multiselect Open Response Table Item Context Allowable Sample Item Item Type Equation Editor Michael and John are creating patterns.  Michael uses the rule “multiply by 2” and starts at 5.  John uses the rule “add 8” and starts at 16. What is the first number in Michael’s pattern that also appears in John’s pattern? 12 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Michael and John are creating patterns. Each pattern starts at 1.  Michael uses the rule “multiply by 2, then add 1.”  John uses the rule “multiply by 2, then add 2.” A. Drag numbers into the table to show the next 2 terms for Michael’s pattern and John’s pattern. B. Use the Add Point tool to plot the ordered pairs that are created from the first three terms of their patterns. Michael's pattern provides the x values and John's pattern provides the y values. See Appendix for the practice test item aligned to this standard. 13 | P a g e May 2016 Item Type GRID Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NBT Number and Operations in Base Ten MAFS.5.NBT.1 Understand the place value system. MAFS.5.NBT.1.1 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 of what it represents in the place to its left. Assessment Limit Items may require a comparison of the values of digits across multiple place values, including whole numbers and decimals from millions to thousandths. Calculator No Item Types Editing Task Choice Equation Editor Hot Text Multiple Choice Multiselect Open Response Context Allowable Sample Item Item Type Equation Editor What is the missing value in the equation shown? x = 0.034 What is the value of the missing number in the following equation? 0.34 x = 3.4 A. 10 B. 100 C.
D.
Multiple Choice How many times greater is the value 0.34 than the value 0.0034? Equation Editor See Appendix for the practice test item aligned to this standard. 14 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NBT Number and Operations in Base Ten MAFS.5.NBT.1 Understand the place value system. MAFS.5.NBT.1.2 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 whole‐number exponents to denote powers of 10. Assessment Limits Items may contain whole number and decimal place values from millions to thousandths. Items may contain whole number exponents with bases of 10. Calculator No Item Types Editing Task Choice Equation Editor GRID Hot Text Multiple Choice Multiselect Open Response Context No context Sample Item Item Type Equation Editor What is 0.523 x 10²? What is the value of the missing exponent in the equation ? Equation Editor Which statement is equivalent to multiplying a number by 103? A. adding 10 three times B. adding 3 ten times C. multiplying by 10 three times D. multiplying by 3 ten times Multiple Choice When dividing a number by 103, how is the decimal point moved? A. 3 places to the right B. 3 places to the left C. 4 places to the right D. 4 places to the left Multiple Choice See Appendix for the practice test item aligned to this standard. 15 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NBT Number and Operations in Base Ten MAFS.5.NBT.1 Understand the place value system. MAFS.5.NBT.1.3 Read, write, and compare decimals to thousandths. MAFS.5.NBT.1.3a 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 × + 9 × + 2 × ,
. MAFS.5.NBT.1.3b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Assessment Limit Items may contain decimals to the thousandths with the greatest place value to the millions. Calculator No Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Context Allowable Sample Item Item Type Multiple Choice What is “two hundred sixty‐five thousandths” in decimal form? A. 260.005 B. 265.0 C. 0.265 D. 2.65 Select the decimal form for each number name. Matching Item 0.650 0.605 0.065 6.050 Sixty‐five thousandths Six hundred five thousandths □ □ □ □ □ □ □ □ Equation Editor A number in expanded form is shown. 3 x 1 + 2 x + 6 x + 5 x ,
What is the number in decimal form? 16 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Select all the expressions that show 2.059 written in expanded form. □
2 x 1 + 0 x + 5 x + 9 x □
2 x 1 + 5 x + 9 x □
2 x 1 + 0 x + 59 x □
20 x + 59 x □
20 x + 5 x ,
,
+ 9 x ,
See Appendix for the practice test item aligned to a standard in this group. 17 | P a g e May 2016 Item Type Multiselect Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NBT Number and Operations in Base Ten MAFS.5.NBT.1 Understand the place value system. MAFS.5.NBT.1.4 Use place value understanding to round decimals to any place.
Assessment Limits Items may contain decimals to the thousandths with the greatest place value to the millions. The least place value a decimal may be rounded to is the hundredths place. Calculator No Item Types Equation Editor Matching Item Multiple Choice Multiselect Table Item Context Allowable Sample Item Item Type Multiselect Select all the numbers that round to 4.3 when rounded to the nearest tenth. □
□
□
□
□
□
4.25 4.24 4.31 4.352 4.219 4.305 What is 3.149 rounded to the nearest hundredth? Equation Editor Numbers are rounded to the nearest tenth and hundredth, as shown in the table. Complete the table to show the numbers that could be rounded. Number Rounded to Nearest Tenth Rounded to Nearest Hundredth 1.5 3.2 9.4 1.55 3.18 9.35 See Appendix for the practice test item aligned to this standard.
18 | P a g e May 2016 Table Item Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NBT Number and Operations in Base Ten MAFS.5.NBT.2 Perform operations with multi‐digit whole numbers and with decimals to hundredths. MAFS.5.NBT.2.5 Fluently multiply multi‐digit whole numbers using the standard algorithm.
Assessment Limit Calculator Item Types Multiplication may not exceed five digits by two digits. No Equation Editor GRID Multiple Choice Multiselect Allowable Context Sample Item Multiply: Item Type Equation Editor 423 x 79 See Appendix for the practice test item aligned to this standard. 19 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NBT Number and Operations in Base Ten MAFS.5.NBT.2 Perform operations with multi‐digit whole numbers and with decimals to hundredths. MAFS.5.NBT.2.6 Find whole‐number quotients of whole numbers with up to four‐
digit dividends and two‐digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Assessment Limit Calculator Item Types Division may not exceed four digits by two digits. No Equation Editor GRID Multiple Choice Multiselect Open Response Allowable Context Sample Item Select all the expressions that have a value of 34. □ 340 ÷ 16 □ 380 ÷ 13 □ 408 ÷ 12 □ 510 ÷ 15 □ 680 ÷ 24 See Appendix for the practice test item aligned to this standard. 20 | P a g e May 2016 Item Type Multiselect Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NBT Number and Operations in Base Ten MAFS.5.NBT.2 Perform operations with multi‐digit whole numbers and with decimals to hundredths. MAFS.5.NBT.2.7 Add, subtract, multiply, and divide 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.
Assessment Limits Items may only use factors that result in decimal solutions to the thousandths place (e.g., multiplying tenths by hundredths). Items may not include multiple different operations within the same expression (e.g., 21 + 0.34 x 8.55). Expressions may have up to two procedural steps of the same operation. Calculator No Item Types Editing Task Choice Equation Editor GRID Hot Text Multiple Choice Multiselect Open Response Context Allowable Sample Item Item Type Equation Editor What is the value of the expression? 5.2 x 10.38 Equation Editor An expression is shown. 12.25 + 3.05 + 0.6 What is the value of the expression? See Appendix for the practice test item aligned to this standard. 21 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NF Numbers and Operations – Fractions MAFS.5.NF.1 Use equivalent fractions as a strategy to add and subtract fractions. MAFS.5.NF.1.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, . (In general, .) Assessment Limits Fractions greater than 1 and mixed numbers may be included. Expressions may have up to three addends. Least common denominator is not necessary to calculate sums or differences of fractions. Items may not use the terms “simplify” or “lowest terms.” For given fractions in items, denominators are limited to 1‐20. Items may require the use of equivalent fractions to find a missing addend or part of an addend. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Context No context Sample Item Item Type Multiple Choice What is the value of the expression? A.
B.
C.
D.
What is the value of the expression 6
4 ? See Appendix for the practice test item aligned to this standard. 22 | P a g e May 2016 Equation Editor Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NF Number and Operations ‐ Fractions MAFS.5.NF.1 Use equivalent fractions as a strategy to add and subtract fractions. MAFS.5.NF.1.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result , by observing that . Assessment Limits Fractions greater than 1 and mixed numbers may be included. Expressions may have up to three addends. Least common denominator is not necessary to calculate sums or differences of fractions. Items may not use the terms “simplify” or “lowest terms.” For given fractions in items, denominators are limited to 1‐20. Items may require the use of equivalent fractions to find a missing addend or part of an addend. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Open Response Context Required Sample Item Item Type Equation Editor John and Sue are baking cookies. The recipe lists cup of flour. They only have cup of flour left. How many more cups of flour do they need to bake the cookies? Javon, Sam, and Antoine are baking cookies. Javon has cup of flour, Sam has 1 cups of flour, and Antoine has 1 cups of flour. How many cups of flour do they have altogether? 23 | P a g e May 2016 Equation Editor Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Richard and Gianni each bought a pizza. The pizzas are the same size.  Richard cut his pizza into 12 slices.  Gianni cut his pizza into 6 slices, and ate 2 slices.  Together, Richard and Gianni ate of one pizza. How many slices of his pizza did Richard eat? A. 3 B. 5 C. 6 D. 7 See Appendix for the practice test item aligned to this standard. 24 | P a g e May 2016 Item Type Multiple Choice Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NF Numbers and Operations – Fractions MAFS.5.NF.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. MAFS.5.NF.2.3 Interpret a fraction as division of the numerator by the . Solve word problems involving division of whole denominator numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, 3
3
interpret as the result of dividing 3 by 4, noting that multiplied by 4 equals 3, 4
4
and that when 3 wholes are shared equally among 4 people each person has a 3
share of size . If 9 people want to share a 50‐pound sack of rice equally by 4
weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
Assessment Limits Quotients in division items may not be equivalent to a whole number. Items may contain fractions greater than 1. Items may not use the terms “simplify” or “lowest terms.” Only use whole numbers for the divisor and dividend of a fraction. For given fractions in items, denominators are limited to 1‐20. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Open Response Table Item Context Allowable Sample Item Item Type Multiple Choice Which expression is equivalent to ? A. 8 – 15 B. 15 – 8 C. 8 ÷ 15 D. 15 ÷ 8 Joe has a board that is 6 feet long. He needs to cut the board into 15 equal‐length Equation Editor pieces. How many feet long should each piece of the board be? See Appendix for the practice test item aligned to this standard. 25 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NF Number and Operations – Fractions MAFS.5.NF.2 Apply and extend previous understanding of multiplication and division to multiply and divide fractions. MAFS.5.NF.2.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
MAFS.5.NF.2.4a Interpret the product as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show context for this equation. Do the same with 4
, and create a story . (In general,
).
MAFS.5.NF.2.4b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. Also Assesses: MAFS.5.NF.2.6 Solve real‐world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
Assessment Limits Visual models may include:  Any appropriate fraction model (e.g., circles, tape, polygons, etc.)  Rectangle models tiled with unit squares For tiling, the dimensions of the tile must be unit fractions with the same denominator as the given rectangular shape. Items may not use the terms “simplify” or “lowest terms.” Items may require students to interpret the context to determine operations. Fractions may be greater than 1. For given fractions in items, denominators are limited to 1‐20. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Context Allowable for MAFS.5.NF.2.4; Required for MAFS.5.NF.2.6 26 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Which expression is equivalent to Item Type Multiple Choice ? A.
B.
C.
D.
Roger has 2 gallons of water in a jug. He pours of the water into a new container. How many gallons of water does Roger have left in the jug? Courtney has 4 gallons of milk. She uses of the milk to make hot chocolate. Then, she uses of the remaining milk to make cookies. How many gallons of milk does Courtney have left after making hot chocolate and cookies? See Appendix for the practice test item aligned to a standard in this group. 27 | P a g e May 2016 Equation Editor Equation Editor Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NF Number and Operations — Fractions MAFS.5.NF.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. MAFS.5.NF.2.5 Interpret multiplication as scaling (resizing), by: MAFS.5.NF.2.5a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. MAFS.5.NF.2.5b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence to the effect of multiplying by 1. Assessment Limits For given fractions in items, denominators are limited to 1‐20. Non‐fraction factors in items must be greater than 1,000. Scaling geometric figures may not be assessed. Scaling quantities of any kind in two dimensions is beyond the scope of this standard. Calculator No Item Types Editing Task Choice Hot Text Matching Item Multiple Choice Multiselect Open Response Context Allowable Sample Item Item Type Multiple Choice Two newspapers are comparing sales from last year.  The Post sold 34,859 copies.  The Tribune sold 34,859 x copies. Which statement compares the numbers of newspapers sold? A. The Post sold half the number of newspapers that the Tribune sold. B. The Tribune sold half the number of newspapers that the Post sold. C. The Tribune sold twice the number of newspapers that the Post sold. D. The Post sold the same number of newspapers that the Tribune sold. 28 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Two newspapers are comparing sales from last year.  The Post sold 34,859 copies.  The Tribune sold one‐and‐a‐half times as many copies as the Post. Which expression describes the number of newspapers the Tribune sold? A. 34,859 × 1 Item Type Multiple Choice B. 34,859 ÷ 1 C. 34,859 × D. 34,859 ÷ Select all the expressions that have a value greater than 1,653. □ 1,653 × □
1,653 × 4 □
1,653 × 12 □
1,653 × □
1,653 × 1 See Appendix for the practice test item aligned to a standard in this group. 29 | P a g e May 2016 Multiselect Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.NF Number and Operations – Fractions MAFS.5.NF.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. MAFS.5.NF.2.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. MAFS.5.NF.2.7a Interpret division of a unit fraction by a non‐zero whole number, 4, and compute such quotients. For example, create a story context for and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4
4
because . MAFS.5.NF.2.7b Interpret division of a whole number by a unit fraction, and ,and use compute such quotients. For example, create a story context for 4
a visual fraction model to show the quotient. Use the relationship between 20 because 20
4. multiplication and division to explain that 4
MAFS.5.NF.2.7c Solve real world problems involving division of unit fractions by non‐zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share lb. of chocolate equally? How many cup servings are in 2 cups of raisins? Assessment Limit Calculator Item Types For given fractions in items, denominators are limited to 1‐20. No Equation Editor GRID Multiple Choice Multiselect Open Response Allowable Context Sample Item An expression is shown. Item Type Equation Editor ÷ 12 What is the value of the expression? 30 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Item Type Julio has 8 pounds of candy. He wants to put the candy into bags so that each bag Multiple Choice has pound. Which equation shows how to calculate the number of bags of candy Julio can make? A. 16 × = 8 B. 16 × 2 = 32 C. 16 × 8 = D. 16 × 8 = 128 Julio has 12 pounds of candy. He wants to put the candy into bags so that each bag Equation Editor has pound of candy. How many total bags of candy can Julio make? See Appendix for the practice test item aligned to a standard in this group. 31 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.MD Measurement and Data MAFS.5.MD.1 Convert like measurement units within a given measurement system. MAFS.5.MD.1.1 Convert among different‐sized standard measurement units (i.e., km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec) 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. Assessment Limits Measurement values may be whole, decimal, or fractional values. Conversions must be within the same system. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Open Response Table Item Context Allowable Sample Item Item Type Equation Editor Michael is measuring fabric for the costumes of a school play. He needs 11.5 meters of fabric. He has 28.5 centimeters of fabric. How many more centimeters of fabric does he need? See Appendix for the practice test item aligned to this standard. 32 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.MD Measurement and Data MAFS.5.MD.2 Represent and interpret data. MAFS.5.MD.2.2 Make a line plot to display a data set of measurements in fractions of a unit , , . Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
Assessment Limit Items requiring operations on fractions must adhere to the Assessment Limits for that operation’s corresponding standard. Calculator No Item Types Equation Editor GRID Multiple Choice Multiselect Table Item Context Allowable Sample Item Item Type Equation Editor A line plot with Kelly’s lengths of ribbons is shown. What is the total length, in inches, of the longest piece and shortest piece of ribbon? A line plot with Kelly’s lengths of ribbons is shown. She adds another ribbon so that the difference between the longest ribbon and shortest ribbon is 1 inches. What length of ribbon, in inches, could Kelly have added? 33 | P a g e May 2016 Equation Editor Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item A line plot with Kelly’s ribbon lengths is shown. She adds two more ribbons so that the total length of ribbon is 200 inches. What are two possible lengths of ribbon, in inches, that Kelly could have added? See Appendix for the practice test item aligned to this standard. 34 | P a g e May 2016 Item Type Equation Editor Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.MD Measurement and Data MAFS.5.MD.3 Geometric measurement: understand concepts of volume and relate volume to multiplication and division. MAFS.5.MD.3.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. MAFS.5.MD.3.3a A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. MAFS.5.MD.3.3b A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. Also Assesses: MAFS.5.MD.3.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
Assessment Limits Items may contain right rectangular prisms with whole‐number side lengths. Figures may only be shown with unit cubes. Labels may include cubic units (i.e. cubic centimeters, cubic feet, etc.) or exponential units (i.e., cm3, ft3, etc.). Items requiring measurement of volume by counting unit cubes must provide a key of the cubic unit. Calculator No Item Types Equation Editor Matching Item Multiple Choice Multiselect Context Allowable Sample Item Item Type Ellen is shopping for boxes. Which measurement should she use to determine the Multiple Choice amount the box will hold? A. Area B. Perimeter C. Length D. Volume 35 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item A rectangular prism is shown. Item Type Equation Editor What is the volume, in cubic inches (in.), of the rectangular prism? Which prisms have a volume between 20 and 40 cubic units? See Appendix for the practice test item aligned to a standard in this group. 36 | P a g e May 2016 Multiselect Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.MD: Measurement and Data MAFS.5.MD.3 Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. MAFS.5.MD.3.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. MAFS.5.MD.3.5a Find the volume of a right rectangular prism with whole‐number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole‐number products as volumes, e.g., to represent the associative property of multiplication. MAFS.5.MD.3.5b Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole‐number edge lengths in the context of solving real world and mathematical problems. MAFS.5.MD.3.5c Recognize volume as additive. Find volumes of solid figures composed of two non‐overlapping right rectangular prisms by adding the volumes of the non‐overlapping parts, applying this technique to solve real world problems.
Assessment Limits Items may not contain fraction or decimal dimensions or volumes. Items may contain no more than two non‐overlapping prisms – non‐overlapping means that two prisms may share a face, but they do not share the same volume. Items assessing MAFS.5.MD.3.5b may not contain the use or graphic of unit cubes. Items assessing MAFS.5.MD.3.5c must contain a graphic of the figures. Calculator No Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Context Allowable 37 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item A shipping box in the shape of a rectangular prism has the dimensions shown. What is the volume, in cubic feet, of the box? Select all the options that could be the dimensions of a rectangular prism with a volume of 384 cubic feet (ft). □ length: 6 ft, width: 8 ft, height: 8 ft □ length: 4 ft, width: 12 ft, height: 24 ft □ length: 4 ft, width: 6 ft, height: 16 ft □ length: 4 ft, width: 8 ft, height: 12 ft □ length: 3 ft, width: 10 ft, height: 20 ft See Appendix for the practice test item aligned to a standard in this group. 38 | P a g e May 2016 Item Type Equation Editor Multiselect Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.G Geometry MAFS.5.G.1 Graph points on the coordinate plane to solve real‐world and mathematical problems. MAFS.5.G.1.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x‐axis and x‐coordinate, y‐axis and y‐coordinate). Also Assesses: MAFS.5.G.1.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. Assessment Limits Items assessing MAFS.5.G.1.1 may not require directions between two given points. Points must rely on the origin. Items assessing MAFS.5.G.1.1 may require identifying the point (e.g., Point A) on a coordinate grid that represents a given ordered pair. Items assessing MAFS.5.G.1.1 may require determining the ordered pair that represents a given point on the coordinate plane. Items assessing MAFS.5.G.1.1 may not require graphing/plotting a point given an ordered pair. Points may only contain positive, whole number ordered pairs. Mathematical and real‐world problems must have axes scaled to whole numbers (not letters). Calculator No Item Types Editing Task Choice Equation Editor GRID Hot Text Matching Item Multiple Choice Multiselect Open Response Context No context for MAFS 5.G.1.1; Allowable for MAFS.5.G.1.2 39 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Point Z is 3 units away from the origin on the x‐axis. What could be the coordinates of point Z? A. (0, 3) B. (3, 0) C. (3, 3) D. (3, 6) Item Type Multiple Choice Point M is 3 units away from the origin along the x‐axis, and 5 units away along the Multiple Choice y‐axis. What could be the coordinates of point M? A. (3, 5) B. (5, 3) C. (3, 8) D. (5, 8) Multiple Choice Which point is located at (5, 1) on the coordinate grid? A.
B.
C.
D.
Point A Point B Point C Point D 40 | P a g e May 2016 Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Use the Add Point tool to plot the point (3, 4). Item Type GRID Point A has the coordinates (3, 5). Point B is located 5 units above point A. Drag points A and B to show their locations in the coordinate plane. 41 | P a g e May 2016 GRID Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Some locations in Lamar’s town are shown in the coordinate plane. Lamar moved from one location to another by traveling 1 unit left and 5 units up. Which ways could he have traveled? A. from home to the park B. from the park to the library C. from home to the library D. from school to the park See Appendix for the practice test items aligned to these standards. 42 | P a g e May 2016 Item Type Multiple Choice Grade 5 Mathematics Item Specifications Florida Standards Assessments Content Standard MAFS.5.G Geometry MAFS.5.G.2 Classify two‐dimensional figures into categories based on their properties. MAFS.5.G.2.3 Understand that attributes belonging to a category of two‐
dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. Also Assesses: MAFS.5.G.2.4 Classify and organize two‐dimensional figures into Venn diagrams based on the attributes of the figures.
Assessment Limit Attributes of figures may be given or presented within given graphics. Items that include trapezoids must consider both the inclusive and exclusive definitions. Items may not use the term "kite" but may include the figure. Calculator No Item Types Editing Task Choice GRID Hot Text Matching Item Multiple Choice Multiselect Open Response Context No context Sample Item Item Type Multiselect Select all the properties that both rectangles and parallelograms always share. □ 4 right angles □ 4 sides of equal length □ 2 pairs of parallel sides □ 2 pairs of sides with equal length □ 2 acute angles and 2 obtuse angles Which kinds of shapes are always rectangles? A.
B.
C.
D.
Parallelograms Quadrilaterals Rhombuses Squares 43 | P a g e May 2016 Multiple Choice Grade 5 Mathematics Item Specifications Florida Standards Assessments Sample Item Select all the shapes that are also always parallelograms. Select all the names of figures that could also be classified as a rhombus. □ Parallelogram □ Square □ Rectangle □ Quadrilateral □ Triangle See Appendix for the practice test item aligned to a standard in this group. 44 | P a g e May 2016 Item Type Multiselect Multiselect Grade 5 Mathematics Item Specifications Florida Standards Assessments Appendix A The chart below contains information about the standard alignment for the items in the Grade 5 Mathematics FSA Computer‐Based Practice Test at http://fsassessments.org/students‐and‐
families/practice‐tests/. Content Standard Item Type MAFS.5.OA.1.1 MAFS.5.OA.1.2 MAFS.5.OA.2.3 MAFS.5.NBT.1.1 MAFS.5.NBT.1.2 MAFS.5.NBT.1.3 MAFS.5.NBT.1.4 MAFS.5.NBT.2.5 MAFS.5.NBT.2.6 MAFS.5.NBT.2.7 MAFS.5.NF.1.1 MAFS.5.NF.1.2 MAFS.5.NF.2.3 MAFS.5.NF.2.4b MAFS.5.NF.2.5 MAFS.5.NF.2.7 MAFS.5.MD.1.1 MAFS.5.MD.2.2 MAFS.5.MD.3.3 MAFS.5.MD.3.5 MAFS.5.G.1.1 MAFS.5.G.1.2 MAFS.5.G.2.3 Equation Editor Equation Editor Table Item Multiselect Multiselect Multiselect Matching Item Multiple Choice Multiple Choice Equation Editor Equation Editor Multiple Choice Table Item Equation Editor Multiselect GRID Equation Editor Multiple Choice Multiple Choice GRID Open Response GRID GRID 45 | P a g e May 2016 Computer‐Based Practice Test Item Number 4 8 20 19 13 22 10 1 12 2 14 11 7 21 5 18 17 3 23 16 15 9 6 Grade 5 Mathematics Item Specifications Florida Standards Assessments Appendix B: Revisions Page(s) 10 11 12‐13 14 15 16‐17 19 20 21 22 23‐24 25 26‐27 28‐29 32 33‐34 35‐36 37‐38 39‐42 43‐44 45 Revision Assessment limits and sample items revised. Item types revised. Assessment limits, item types, and sample items revised. Item types revised. Item types and sample items revised. Assessment limits revised. Item types revised. Sample items revised. Item types revised. Sample items revised. Item types revised. Item types revised. Assessment limits revised. Item types revised. Assessment limits and item types revised. Item types revised. Assessment limits revised. Corrected standard language for MAFS.5.MD.3.5b. Item types and sample items revised. Assessment limits, item types, and sample items revised. Appendix A added to show Practice Test information. 46 | P a g e May 2016 Date May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 May 2016 Table 1. Common addition and subtraction situations.6
Add to
Result Unknown
Change Unknown
Start Unknown
Two bunnies sat on the grass.
Three more bunnies hopped
there. How many bunnies are on
the grass now?
Two bunnies were sitting on the
grass. Some more bunnies
hopped there. Then there were
five bunnies. How many
bunnies hopped over to the
first two?
Some bunnies were sitting on
the grass. Three more bunnies
hopped there. Then there were
five bunnies. How many
bunnies were on the grass
before?
2+?=5
?+3=5
Five apples were on the table.
I ate some apples. Then
there were three apples. How
many apples did I eat?
Some apples were on the
table. I ate two apples. Then
there were three apples. How
many apples were on the
table before?
2+3=?
Take from
Five apples were on the table.
I ate two apples. How many
apples are on the table now?
5–2=?
Put together/take
apart2
Total Unknown
Addend Unknown
Three red apples and two
green apples are on the table.
How many apples are on the
table?
Five apples are on the table.
Three are red and the rest are
green. How many apples are
green?
3+2=?
3 + ? = 5, 5 – 3 = ?
difference Unknown
Difference Unknown
Compare3
5–?=3
Bigger Unknown
Bigger Unknown
?–2=3
Both Addends
Unknown1
Grandma has five flowers.
How many can she put in her
red vase and how many in
her blue vase?
5=0+5
5=1+4
5=2+3
5=5+0
5=4+1
5=3+2
Smaller Unknown
Smaller Unknown
(“How many more?” version):
Lucy has two apples. Julie
has five apples. How many
more apples does Julie have
than Lucy?
(Version with “more”):
Julie has three more apples
than Lucy. Lucy has two
apples. How many apples
does Julie have?
(Version with “more”):
Julie has three more apples
than Lucy. Julie has five
apples. How many apples
does Lucy have?
(“How man fewer?” version):
Lucy has two apples. Julie
has five apples. How many
fewer apples does Lucy have
than Julie?
(Version
(Version with “fewer”):
Lucy has 3 fewer apples than
Julie. Julie has five apples.
How many apples does Lucy
have?
2+?=5
5–2=?
with “fewer”):
Lucy has 3 fewer apples than
Julie. Lucy has two apples.
How many apples does Julie
have?
2+3=?
3+2=?
5–3=?
?+3=5
1These take apart situations can be used to show all the decompositions of a given number. The associated equations,
which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or
results in but always does mean is the same number as.
2Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a
productive extension of this basic situation, especially for small numbers less than or equal to 10.
3For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using
more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult.
6Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).
Common Operation Situations and Properties, page 1 of 3, Brevard Public Schools, 2015 – 2016
Table 2. Common multiplication and division situations.7
Unknown Product
3×6 =?
Group Size Unknown
(“How many in each
group?” Division)
3 × ? = 18, and 18 ÷ 3 = ?
Equal Groups
There are 3 bags with 6
plums in each bag. How
many plums are there in all?
Compare3
General
equally into 3 bags, then how
many plums will be in each
bag?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed
6 to a bag, then how many
bags are needed?
Measurement example: You
have 18 inches of string,
which you will cut into 3 equal
pieces. How long will each
piece of string be?
Measurement example: You
have 18 inches of string,
which you will cut into pieces
that are 6 inches long. How
many pieces of string will you
have?
If 18 apples are arranged into
3 equal rows, how many
apples will be in each row?
If 18 apples are arranged into
equal rows of 6 apples, how
many rows will there be?
Area example: A rectangle
has area 18 square
centimeters. If one side is 3
cm long, how long is a side
next to it?
Area example: A rectangle
has area 18 square
centimeters. If one side is 6
cm long, how long is a side
next to it?
A red hat costs $18 and that
is 3 times as much as a blue
hat costs. How much does a
blue hat cost?
A red hat costs $18 and a
blue hat costs $6. How many
times as much does the red
hat cost as the blue hat?
Measurement example: A
rubber band is 6 cm long.
How long will the rubber band
be when it is stretched to be 3
times as long?
Measurement example: A
rubber band is stretched to
be18 cm long and that is 3
times as long as it was at first.
How long was the rubber
band at first?
Measurement example: A
rubber band was 6 cm long at
first. Now it is stretched to be
18 cm long. How many times
as long is the rubber band
now as it was at first?
a×b=?
a × ? = p, and p ÷ a = ?
? × b = p, and p ÷ b = ?
Measurement example: You
need 3 lengths of string, each
6 inches long. How much
string will you need
altogether?
Arrays4,
Area5
If 18 plums are shared
Number of Groups
Unknown
(“How many groups?”
Division)
There are 3 rows of apples
with 6 apples in each row.
How many apples are there?
Area example: What is the
area of a 3 cm by 6 cm
rectangle?
A blue hat costs $6. A red
hat costs 3 times as much as
the blue hat. How much does
the red hat cost?
4The
language in the array examples shows the easiest form of array problems. A harder form is to use the terms
rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there?
Both forms are valuable.
5Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array
problems include these especially important measurement situations.
7The
first examples in each cell are examples of discrete things. These are easier for students and should be given
before the measurement examples.
Common Operation Situations and Properties, page 2 of 3, Brevard Public Schools, 2015 – 2016
TABLE 3. THE PROPERTIES OF OPERATIONS. Here a, b and c stand for arbitrary numbers in a given
number system. The properties of operations apply to the rational number system, the real number system, a nd
the complex number system.
Associative property of addition
Commutative property of addition
Additive identity property of 0
Existence of additive inverses
Associative property of multiplication
Commutative property of multiplication
(a + b) + c = a + (b + c)
a+b=b+a
a+0=0+a=a
For every a there exists –a so that a + (–a) = (–a) + a = 0
(a × b) × c = a × (b × c)
a×b=b×a
Multiplicative identity property of 1
a×1=1×a=a
Existence of multiplicative inverses
For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1
Distributive property of multiplication over addition
a × (b + c) = a × b + a × c
TABLE 4. THE PROPERTIES OF EQUALITY. Here a, b and c stand for arbitrary numbers in the rational, real,
or complex number systems.
Reflexive property of equality
Symmetric property of equality
Transitive property of equality
a=a
If a = b, then b = a.
If a = b and b = c, then a = c.
Addition property of equality
If a = b, then a + c = b + c.
Subtraction property of equality
If a = b, then a – c = b – c.
Multiplication property of equality
If a = b, then a × c = b × c.
Division property of equality
Substitution property of equality
If a = b and c ≠ 0, then a ÷ c = b ÷ c.
If a = b, then b may be substituted for a in any
expression containing a.
TABLE 5. THE PROPERTIES OF INEQUALITY. Here a, b and c stand for arbitrary numbers in the rational or
real number systems.
Exactly one of the following is true: a < b, a = b, a > b.
If a > b and b > c then a > c.
If a > b, then b < a.
If a > b, then–a < –b.
If a > b, then a ± c > b ± c.
If a > b and c > 0, then a × c > b × c.
If a > b and c < 0, then a × c < b × c.
If a > b and c > 0, then a ÷ c > b ÷ c.
If a > b and c < 0, then a ÷ c < b ÷ c
Common Operation Situations and Properties, page 3 of 3, Brevard Public Schools, 2015 – 2016
Standards
for
Mathematical
Practice
Do what makes
sense and be
persistent
Look for
and use
patterns and
connections
Use math to
describe a
real situation
or problem
Use number sense
when representing
a problem
What do good
problem
solvers do?
Make
conjectures
and prove or
disprove them
Look for
and create
efficient
strategies
Be precise
with words,
numbers, and
symbols
Use tools
and
technology
strategically
What Constitutes a Cognitively Demanding Task?
Lower-level demands (memorization)
• Involve either reproducing previously learned facts, rules, formulas, or definitions or committing facts,
rules, formulas or definitions to memory.
• Cannot be solved using procedures because a procedure does not exist or because the time frame in
which the task is being completed is too short to use a procedure
• Are not ambiguous. Such tasks involve the exact reproduction of previously seen material, and what is to
be reproduced is clearly and directly stated.
• Have no connection to the concepts or meaning that underlie the facts, rules, formulas, or definitions
being learned or reproduced.
Lower-level demands (procedures without connections to meaning)
• Are algorithmic. Use of the procedure either is specifically called for or is evident from prior instruction,
experience, or placement of the task.
• Require limited cognitive demand for successful completion. Little ambiguity exists about what needs to
be done and how to do it.
• Have no connection to concepts or meaning that underlie the procedure being used.
• Are focused on producing correct answers instead of on developing mathematical understanding.
• Require no explanation or explanations that focus solely on describing the procedure that was used.
Higher-level demands (procedures with connections to meaning)
• Focus students’ attention on the use of procedures for the purpose of developing deeper levels of
understanding of mathematical concepts and ideas.
• Suggest explicitly or implicitly pathways to follow that are broad general procedures that have close
connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with
respect to underlying concepts.
• Usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols, and problem
situations. Making connections among multiple representations helps develop meaning.
• Require some degree of cognitive effort. Although general procedures may be followed, they cannot be
followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to
complete the task successfully and that develop understanding.
Higher-level demands (doing mathematics)
• Require complex and non-algorithmic thinking - a predictable, well-rehearsed approach or pathway is
not explicitly suggested by the task, task instructions, or a worked-out example.
• Require students to explore and understand the nature of mathematical concepts, processes, or
relationships.
• Demand self-monitoring or self-regulation of one’s own cognitive processes.
• Require students to access relevant knowledge and experiences and make appropriate use of them in
working through the task.
• Require considerable cognitive effort and may involve some level of anxiety for the student because of
the unpredictable nature of the solution process required.
Arbaugh, F., & Brown, C.A. (2005). Analyzing mathematical tasks: a catalyst for change? Journal of Mathematics Teacher
Education, 8, p. 530.
Key Ideas in the Mathematics Florida Standards (MAFS)
 Focus: Greater focus on fewer topics
 Focus deeply on the standards for mastery and the ability to transfer skills.
 Focus deeply on the major work of each grade as follows:
 In grades K-2: Concepts, skills, problem solving related to addition and subtraction.
 In grades 3-5: Concepts, skills, and problem solving related to multiplication and division of
whole numbers and fractions.
 In grade 6: Ratios and proportional relationships, and early algebraic expressions and
equations.
 This focus will enable students to gain strong foundations, including a solid understanding of
concepts, and the ability to apply the math they know to solve problems inside and outside the
classroom.
 Coherence: Linking topics and thinking across grades




Coherence is about making math make sense.
Mathematics is a coherent body of knowledge made up of interconnected concepts.
The standards are designed around coherent progressions from grade to grade.
Learning is carefully connected across grades so that students can build new understanding
onto foundations built in previous years.
 Each standard is not a new event, but an extension of previous learning.
 It is critical to think across grade levels and examine the progressions to see how major
content is developed across grades.
 Rigor: Calls for a balance of tasks that require conceptual understanding,
procedural skills and fluency, and application of mathematics to solve problems
 Rigor refers to deep, authentic command of mathematical concepts.
 The following three aspects of rigor must be pursued with equal intensity to help students
meet the standards:
 Conceptual understanding: The standards call for conceptual understanding of key
concepts. Students must be able to access concepts from a number of perspectives. This
will allow them to see math as more than a set of mnemonics or discrete procedures.
 Procedural skills and fluency: The standards call for speed and accuracy in calculation with
a balance of practice and understanding. Students must practice simple calculations such
as single-digit multiplication with meaning, in order to have access to more complex
concepts and procedures.
 Application: The standards call for students to have solid conceptual understanding and
procedural fluency. They are expected to apply their understanding and procedural skills in
mathematics to problem solving situations.
-Adapted from www.corestandards.org
Standards for Mathematical Practice
The Standards for Mathematical Practice describe behaviors that all students will develop in the Common Core
Standards. These practices rest on important “processes and proficiencies” including problem solving, reasoning and
proof, communication, representation, and making connections. These practices will allow students to understand and
apply mathematics with confidence.
When given a problem, I can
make a plan to solve it and check
my answer.
1. Make sense of problems and persevere in solving them.
• Find meaning in problems
• Analyze, predict, and plan solution pathways
• Verify answers
• Ask them the question: “Does this make sense?”
2.
Reason abstractly and quantitatively.
• Make sense of quantities and their relationships in problems
• Create coherent representations of problems
I can explain my thinking and
consider the mathematical
thinking of others.
I can use numbers and words
to help me make sense of
problems.
3. Construct viable arguments and critique the reasoning of others.
• Understand and use information to construct arguments
• Make and explore the truth of conjectures
• Justify conclusions and respond to arguments of others
4. Model with mathematics.
• Apply mathematics to problems in everyday life
• Identify quantities in a practical situation
• Interpret results in the context of the situation and reflect on
whether the results make sense
I can recognize math in everyday
life and use math I know to solve
problems.
I can use math tools to
help me explore and
understand math in my
world.
5. Use appropriate tools strategically.
• Consider the available tools when solving problems
• Be familiar with tools appropriate for their grade or course (pencil and paper, concrete models, ruler,
protractor, calculator, spreadsheet, computer programs, digital content located on a website, and other
technological tools)
6. Be precise.
• Communicate precisely to others
• Use clear definitions, state the meaning of symbols and be careful
about specifying units of measure and labeling axes
• Calculate accurately and efficiently
I can see and understand how
numbers and shapes are put
together as parts and wholes.
I can be careful when I
use math and clear
when I share my ideas.
7. Look for and make use of structure.
• Recognize patterns and structures
• Step back for an overview and shift perspective
• See complicated things as single objects or as being composed of several objects
8. Look for and identify ways to create shortcuts when doing problems.
• When calculations are repeated, look for general methods, patterns and
shortcuts
• Be able to evaluate whether an answer makes sense
I can notice when
calculations are
repeated.
Standard for Mathematical
Practice
Student Friendly
Language
1. Make sense of problems and
persevere in solving them.
• I can try many times to
understand and solve a
math problem.
2. Reason abstractly
and quantitatively.
• I can think about the math
problem in my head, first.
3. Construct viable arguments
and critique the reasoning
of others.
• I can make a plan, called a
strategy, to solve the
problem and discuss other
students’ strategies too.
4. Model with mathematics.
5. Use appropriate tools
strategically.
• I can use math symbols and
numbers to solve the
problem.
• I can use math tools,
pictures, drawings, and
objects to solve the problem.
6. Attend to precision.
• I can check to see if my
strategy and calculations
are correct.
7. Look for and make use
of structure
• I can use what I already
know about math to solve
the problem.
8. Look for and express regularity
in repeated reasoning.
• I can use a strategy that I
used to solve another math
problem.
Carroll County Public Schools, http://www.carrollk12.org/instruction/instruction/elementary/math/curriculum/common/default.asp
Florida State Standards Standards for
Mathematical Practice Sample
Questions for Teachers to Ask
Make sense of problems and
persevere in solving them
Reason abstractly and
quantitatively
Teachers ask:
• What is this problem asking?
• How could you start this
problem?
• How could you make this
problem easier to solve?
• How is ___’s way of solving
the problem like/different from
yours?
• Does your plan make sense?
Why or why not?
• What tools/manipulatives
might help you?
• What are you having trouble
with?
• How can you check this?
Teachers ask:
• What does the number ____
represent in the problem?
• How can you represent the
problem with symbols and
numbers?
• Create a representation of the
problem.
Use appropriate tools
strategically
Attend to precision
Teachers ask:
• How could you use
manipulatives or a drawing to
show your thinking?
• Which tool/manipulative would
be best for this problem?
• What other resources could
help you solve this problem?
Teachers ask:
• What does the word ____
mean?
• Explain what you did to solve
the problem.
• Compare your answer to
_____’s answer
• What labels could you use?
• How do you know your answer
is accurate?
• Did you use the most efficient
way to solve the problem?
Construct viable arguments and
critique the reasoning of others
Teachers ask:
• How is your answer different
than _____’s?
• How can you prove that your
answer is correct?
• What math language will help
you prove your answer?
• What examples could prove or
disprove your argument?
• What do you think about
_____’s argument
• What is wrong with ____’s
thinking?
• What questions do you have
for ____?
*it is important that the teacher
implements tasks that involve
discourse and critiquing of
reasoning
Model with mathematics
Teachers ask:
• Write a number sentence to
describe this situation
• What do you already know
about solving this problem?
• What connections do you see?
• Why do the results make
sense?
• Is this working or do you need
to change your model?
*It is important that the teacher
poses tasks that involve real world
situations
Look for and make use of
structure
Look for and express regularity
in repeated reasoning
Teachers ask:
• Why does this happen?
• How is ____ related to ____?
• Why is this important to the
problem?
• What do you know about ____
that you can apply to this
situation?
• How can you use what you
know to explain why this
works?
• What patterns do you see?
Teachers ask:
• What generalizations can you
make?
• Can you find a shortcut to
solve the problem? How would
your shortcut make the
problem easier?
• How could this problem help
you solve another problem?
*deductive reasoning (moving
from general to specific)
*inductive reasoning (moving from
specific to general)
Standards for Mathematical Practice in Action
Practice
Sample Student Evidence
1. Make sense
of problems
and
persevere in
solving them







Display sense-making behaviors
Show patience and listen to others
Turn and talk for first steps and/or generate solution plan
Analyze information in problems
Use and recall multiple strategies
Self-evaluate and redirect
Assess reasonableness of process and answer
2. Reason
abstractly
and
quantitatively







Represent abstract and contextual situations symbolically
Interpret problems logically in context
Estimate for reasonableness
Make connections including real life situations
Create and use multiple representations
Visualize problems
Put symbolic problems into context












Questions others
Use examples and non-examples
Support beliefs and challenges with mathematical evidence
Forms logical arguments with conjectures and counterexamples
Use multiple representations for evidence
Listen and respond to others well
Uses precise mathematical vocabulary
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





















Connect math (numbers and symbols) to real-life situations
Symbolize real-world problems with math
Make sense of mathematics
Apply prior knowledge to solve problems
Choose and apply representations, manipulatives and other
models to solve problems
Use strategies to make problems simpler
Use estimation and logic to check reasonableness of an answer
Choose appropriate tool(s) for a given problem
Use technology to deepen understanding
Identify and locate resources
Defend mathematically choice of tool
Communicate (oral and written) with precise vocabulary
Carefully formulate questions and explanations (not retelling
steps)
Decode and interpret meaning of symbols
Pay attention to units, labeling, scale, etc.
Calculate accurately and effectively
Express answers within context when appropriate
Look for, identify, and interpret patterns and structures
Make connections to skills and strategies previously learned to
solve new problems and tasks
Breakdown complex problems into simpler and more
manageable chunks
Use multiple representations for quantities
View complicated quantities as both a single object or a
composition of objects
Design and state “shortcuts”
Generate “rules” from repeated reasoning or practice (e.g.
integer operations)
Evaluate the reasonableness of intermediate steps
Make generalizations
Sample Teacher Actions






Provide open-ended problems
Ask probing questions
Probe student responses
Promote and value discourse
Promote collaboration
Model and accept multiple approaches


Model context to symbol and symbol to context
Create problems such as “what word problem
will this equation solve?”
Give real world situations
Offer authentic performance tasks
Place less emphasis on the answer
Value invented strategies
Think Aloud































Create a safe and collaborative environment
Model respectful discourse behaviors
“Find the error” problems
Promote student to student discourse (do not
mediate discussion)
Plan effective questions or Socratic formats
Provide time and value discourse
Model reasoning skills
Provide meaningful, real world, authentic
performance-based tasks
Make appropriate tools available
Model various modeling techniques
Accept and value multiple approaches and
representations
Provide a “toolbox” at all times with all available
tools – students then choose as needed
Model tool use, especially technology for
understanding
Model problem solving strategies
Give explicit and precise instruction
Ask probing questions
Use ELA strategies of decoding,
comprehending, and text-to-self connections for
interpretation of symbolic and contextual math
problems
Guided inquiry
Let students explore and explain patterns
Use open-ended questioning
Prompt students to make connections and
choose problems that foster connections
Ask for multiple interpretations of quantities
Provide tasks that allow students to generalize
Don’t teach steps or rules, but allow students to
explore and generalize in order to discover and
formalize
Ask deliberate questions
Create strategic and purposeful check-in points
STANDARDS FOR MATHEMATICAL PRACTICE IN FIFTH GRADE
The Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students Grades
K-12. Below are a few examples of how these Practices may be integrated into tasks that students complete.
Practice
Explanation and Example
1. Make sense of problems
and persevere in solving
them.
Mathematically proficient students in fifth grade solve problems by applying their understanding of
operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems
related to volume and measurement conversions. Students seek the meaning of a problem and look for
efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is
the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in
a different way?”.
2. Reason abstractly and
quantitatively.
Mathematically proficient students in fifth grade should recognize that a number represents a specific
quantity. They connect quantities to written symbols and create a logical representation of the problem
at hand, considering both the appropriate units involved and the meaning of quantities. They extend this
understanding from whole numbers to their work with fractions and decimals. Students write simple
expressions that record calculations with numbers and represent or round numbers using place value
concepts.
3. Construct viable
arguments and critique
the reasoning of others.
Mathematically proficient students in fifth grade may construct arguments using concrete referents, such
as objects, pictures, and drawings. They explain calculations based upon models and properties of
operations and rules that generate patterns. They demonstrate and explain the relationship between
volume and multiplication. They refine their mathematical communication skills as they participate in
mathematical discussions involving questions like “How did you get that?” and “Why is that true?”. They
explain their thinking to others and respond to others’ thinking.
4. Model with mathematics.
Mathematically proficient students in fifth grade experiment with representing problem situations in
multiple ways including numbers, words (mathematical language), drawing pictures, using objects,
making a chart, list, or graph, creating equations, etc. Students need opportunities to connect these
different representations and explain the connections. They should be able to use all of these
representations as needed. Fifth graders should evaluate their results in the context of the situation and
whether the results make sense. They also evaluate the utility of models to determine which models are
most useful and efficient to solve problems.
Standards for Mathematical Practice, (from North Carolina Department of Education, http://www.ncpublicschools.org/), Fifth Grade, page 1 of 2, 2013 - 2014
Practice
Explanation and Example
5. Use appropriate tools
strategically.
Mathematically proficient students in fifth grade consider the available tools (including estimation) when
solving a mathematical problem and decide when certain tools might be helpful. For instance, they may
use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions. They use
graph paper to accurately create graphs and solve problems or make predictions from real- world data.
6. Attend to precision.
Mathematically proficient students in fifth grade continue to refine their mathematical communication
skills by using clear and precise language in their discussions with others and in their own reasoning.
Students use appropriate terminology when referring to expressions, fractions, geometric figures, and
coordinate grids. They are careful about specifying units of measure and state the meaning of the
symbols they choose. For instance, when figuring out the volume of a rectangular prism, they record
their answers in cubic units.
7. Look for and make use
of structure.
Mathematically proficient students in fifth grade look closely to discover a pattern or structure. For
instance, students use properties of operations as strategies to add, subtract, multiply, and divide with
whole numbers, fractions, and decimals. They examine numerical patterns and relate them to a rule or a
graphical representation.
8. Look for and express
regularity in repeated
reasoning.
Mathematically proficient students in fifth grade use repeated reasoning to understand algorithms and
make generalizations about patterns. Students connect place value and their prior work with operations
to understand algorithms to fluently multiply multi-digit numbers and perform all operations with decimals
to hundredths. Students explore operations with fractions with visual models and begin to formulate
generalizations.
Standards for Mathematical Practice, (from North Carolina Department of Education, http://www.ncpublicschools.org/), Fifth Grade, page 2 of 2, 2013 - 2014
BEFORE…
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What is the question?
What do I know?
What do I need to find out?
What tools/strategies will I use?
MAKE A PLAN to
solve the problem
myself.
EXPLAIN the problem to
it isn’t working out
CHANGE my plan if
make sense?”
Jordan School District 2012, Grades 4-5 What worked/didn’t work?
How was my solution similar or
different from my classmates’?
EVALUATE
ASK myself, “Does this
CHECK
MONITOR my work
(Stick to it!)
AFTER…
Is my answer correct?
How do my representations connect to
my solution?
PERSEVERE
DURING…
Mathematical Practice 1
Make sense of problems and persevere in
solving them.
When presented with a problem, I
can make a plan, carry out my plan,
and check its success.
0
1
2
Day 2
1
Day 3
1
2
Day 4
2
Day 5
1
2
Day 6
3
Reasoning Habits
Hours
1
Day 3
1
2
Day 4
2
Day 5
1
2
Day 6
1
2
Day 2
0
Day 1
3
Mary practices the piano hour a day for 6 days.
How many total hours does she practice?
2) Think about the units involved.
Jordan School District 2012, Grades 4-5
4) Use the properties of operations or objects.
1) Make an understandable representation of the problem. 3) Pay attention to the meaning of the numbers.
Hours
Day 1
Mary practices the piano hour a day for 6 days.
How many total hours does she practice?
Decontextualize (Words to Numbers)
Mathematical Practice 2
Contextualize (Numbers to Words)
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Reason abstractly and quantitatively.
I can use numbers, words, and
reasoning habits to help me make
sense of problems.
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• relating to contexts
• using examples and non-examples
• listening
• using objects, drawings, diagrams
and actions
Jordan School District 2012, Grades 4-5
• comparing strategies and
arguments
• asking and answering questions
I can analyze the reasoning
of others by…
I can make and present
arguments by…
Mathematical Practice 3
Construct viable arguments and critique the
reasoning of others.
I can make logical arguments and
respond to the mathematical
thinking of others.
Think about the relationship
to find an answer.
The tank (30") is 5 times
bigger than the turtle
length (6").
I thought about the
problem again and a 30"
side length on the tank
makes sense!
Turtle: About 6" long
Tank: 5 times the length
of the turtle
Find important
numbers.
20
25
30
35
40
Jordan School District 2012, Grades 4-5 …to solve everyday problems.
4
5
6
7
8
Turtle
Tank Use tools to show
Length Length
relationships.
(inches) (inches)
I will round 5 1/2" to 6".
Use estimates to
make the problem
simpler.
Consider my answer -Does it make sense?
My box turtle is getting a new tank. He is 5
1/2" long and 3" tall. One side length of the
tank needs to be 5 times his length. How long
will the length of the tank need to be?
I can…
Mathematical Practice 4
I can recognize math in everyday
life and use math I know to solve
problems.
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Model with mathematics.
Mathematical Practice 5
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• I can reason: “Did the tool I used
give me an answer that makes
sense?”
-1
0
1
2
3
4
Cº
5
6
8
9
Jordan School District 2012, Grades 4-5 7
axb=bxa
Aº
I can use certain tools to help me
explore and deepen my math
understanding.
• I know HOW and WHEN to use
math tools.
Use appropriate tools strategically.
I can be precise when solving
problems and clear when
communicating my ideas.
Mathematical practice 6
units of
measure
48 inches = 4 feet
equal
(the same as)
symbol:
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Jordan School District 2012, Grades 4-5
• calculations that are accurate
and efficient
• units of measure
• context labels
• symbols that have meaning
• math vocabulary with clear
definitions
Mathematicians communicate with others using…
Attend to precision.
3
10
4
+ 100 =
34
.
100
So,
3
30
is equal to
.
10
100
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Mathematical Practice 7
Symmetry
Lines and Angles
For Example:
A
Jordan School District 2012, Grades 4-5
Location
0 1 2 3 4 5 6 7 8 9
9
8
7
6
5
4
3
2
1
I can see and understand how
numbers and spaces are organized
and put together as parts and
wholes.
Numbers
Spaces
I know that
For Example:
Look for and make use of structure.
…I evaluate if my
results are
reasonable.
…I think about
what I’m trying to
figure out while I
pay attention to
the details
As I work…
….jump three size jumps
on a number line.
0
0
8
1
8
+ +
2
8
….add eighths.
I CAN…..
3
8
4
8
s.
5
8
6
8
7
8
,
1
8
8
,
Jordan School District 2012, Grades 4-5
….count by eighths.
(one-eighth, two eighths, three eighths)
There are many ways to decompose because it is composed of repeated
….draw a whole and shade
in three s parts.
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Mathematical Practice 8
Look for and express regularity in
repeated reasoning.
I can notice when calculations are
repeated. Then, I can find more
general methods and short cuts.
Getting
to Know
MAFS
Breaking the Code
Mathematics Florida Standards
Subject/Standards
Domain
Standard
MAFS.5.OA.1.1
Grade Level
Cluster
MAFS = Mathematics Florida Standards
5 = Fifth Grade
OA = Operations and Algebraic Thinking
1 = Cluster – Write and interpret numerical
expressions.
1 = Use parentheses, brackets, or braces in
numerical expressions, and evaluate
expressions with these symbols.
K – 5 Domains
CC = Counting and Cardinality
OA = Operations and Algebraic Thinking
NBT = Number and Operations in Base Ten
MD = Measurement and Data
G = Geometry
Fifth Grade Mathematics Florida Standards
2015 – 2016
Domain: OPERATIONS AND ALGEBRAIC THINKING
Cluster 1: Write and interpret numerical expressions.
MAFS.5.OA.1.1:
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
MAFS.5.OA.1.2:
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For
example, express the calculation “add 8 and 7, then multiply by 2” as 2 x (8 + 7). Recognize that 3 x (18,932 + 921) is three times as
large as 18,932 + 921 without having to calculate the indicated sum or product.
Cluster 2: Analyze patterns and relationships.
MAFS.5.OA.2.3:
Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs
consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule
“Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and
observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Domain: NUMBER AND OPERATIONS IN BASE TEN
Cluster 1: Understand the place value system.
MAFS.5.NBT.1.1:
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
of
10
what it represents in the place to its left.
MAFS.5.NBT.1.2:
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 whole-number exponents to denote powers of 10.
MAFS.5.NBT.1.3:
Read, write, and compare decimals to thousandths.
a. 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
1
1
)+9(
)+ 2  (
).
10
100
1000
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results
of comparisons.
MAFS.5.NBT.1.4:
Use place value understanding to round decimals to any place.
Fifth Grade Mathematics Florida Standards, page 1 of 5, Brevard Public Schools, 2015 - 2016
Fifth Grade Mathematics Florida Standards
2015 – 2016
Cluster 2: Perform operations with multi-digit whole numbers and with decimals to hundredths.
MAFS.5.NBT.2.5:
Fluently multiply multi-digit whole numbers using the standard algorithm.
MAFS.5.NBT.2.6:
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place
value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by
using equations, rectangular arrays, and/or area models.
MAFS.5.NBT.2.7:
Add, subtract, multiply, and divide 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.
Domain: NUMBER AND OPERATIONS - FRACTIONS
Cluster 1: Use equivalent fractions as a strategy to add and subtract fractions.
MAFS.5.NF.1.1:
Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in
2 5 8
15
23
such a way as to produce an equivalent sum or difference of fractions with like denominators. For example,
+ =
+
=
.
3 4 12 12 12
(In general, a + c = (ad  bc ) ).
b
d
bd
MAFS.5.NF.1.2:
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators,
e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to
2 1 3
estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result + = , by observing that
5 2 7
3 1
< .
7 2
Cluster 2: Apply and extend previous understandings of multiplication and division to multiply and
divide fractions.
MAFS.5.NF.2.3:
Interpret a fraction as division of the numerator by the denominator (
a
= a ÷ b). Solve word problems involving division of whole
b
numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the
3
3
problem. For example, interpret as the result of dividing 3 by 4, noting that multiplied by 4 equals 3, and that when 3 wholes are
4
4
3
shared equally among 4 people each person has a share of size . If 9 people want to share a 50-pound sack of rice equally by weight,
4
how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
Fifth Grade Mathematics Florida Standards, page 2 of 5, Brevard Public Schools, 2015 - 2016
Fifth Grade Mathematics Florida Standards
2015 – 2016
MAFS.5.NF.2.4:
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product
a
 q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations
b
a  q ÷ b. For example, use a visual fraction model to show
2
8
 4 = , and create a story context for this equation. Do the same
3
3
2
4
8
c
a
ac
with ( )  ( ) =
. (In general, 
=
).
d
bd
b
15
3
5
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and
show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of
rectangles, and represent fraction products as rectangular areas.
MAFS.5.NF.2.5:
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated
multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing
multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1
results in a product smaller than the given number; and relating the principle of fraction equivalence
multiplying
a n ×a
=
to the effect of
b n×b
a
by 1.
b
MAFS.5.NF.2.6:
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to
represent the problem.
MAFS.5.NF.2.7:
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for
1
÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that
3
1
1
1
1
÷4=
because
 4= .
3
3
12
12
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for
1
4 ÷ , and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that
5
1
1
for 4 ÷ = 20 because 20  = 4.
5
5
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit
fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each
1
1
person get if 3 people share lb. of chocolate equally? How many cup servings are in 2 cups of raisins?
3
2
Fifth Grade Mathematics Florida Standards, page 3 of 5, Brevard Public Schools, 2015 - 2016
Fifth Grade Mathematics Florida Standards
2015 – 2016
Domain: MEASUREMENT AND DATA
Cluster 1: Convert like measurement units within a given measurement system.
MAFS.5.MD.1.1:
Convert among different-sized standard measurement units (i.e., km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec.) 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.
Cluster 2: Represent and interpret data.
MAFS.5.MD.2.2:
1 1 1
Make a line plot to display a data set of measurements in fractions of a unit ( , , ). Use operations on fractions for this grade to solve
2 4 8
problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the
amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
Cluster 3: Geometric measurement: understand concepts of volume and relate volume to
multiplication and to addition.
MAFS.5.MD.3.3:
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube” is said to have “one cubic unit” of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cube is said to have a volume of n cubic units.
MAFS.5.MD.3.4:
Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft., and improvised units.
MAFS.5.MD.3.5:
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is
the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent
threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
b. Apply the formulas V = l  w  h and V = B  h for rectangular prisms to find volumes of right rectangular prisms with whole-number
edge lengths in the context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the
volumes of the non-overlapping parts, applying this technique to solve real world problems.
Fifth Grade Mathematics Florida Standards, page 4 of 5, Brevard Public Schools, 2015 - 2016
Fifth Grade Mathematics Florida Standards
2015 – 2016
Domain: GEOMETRY
Cluster 1: Graph points on the coordinate plane to solve real-world and mathematical problems.
MAFS.5.G.1.1:
Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged
to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates.
Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates
how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond
(e.g., x-axis and x-coordinate, y-axis and y-coordinate).
MAFS.5.G.1.2:
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate
values of points in the context of the situation.
Cluster 2: Classify two-dimensional figures into categories based on their properties.
MAFS.5.G.2.3:
Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For
example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
MAFS.5.G.2.4:
Classify and organize two-dimensional figures into Venn diagrams based on the attributes of the figures.
Fifth Grade Mathematics Florida Standards, page 5 of 5, Brevard Public Schools, 2015 - 2016
3rd Grade
Mathematics Florida Standards Changes
STANDARD
CODE
REVISED/
DELETED/NEW
STANDARD
MACC.3.MD.1.2
PREVIOUS
Measure and estimate liquid volumes and masses of objects using standard units of grams
(g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same units, e.g., by using
drawings (such as a beaker with a measurement scale) to represent the problem.
MAFS.3.MD.1.2
REVISED
Measure and estimate liquid volumes and masses of objects using standard units of grams
(g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same units.
4th Grade
Mathematics Florida Standards Changes
STANDARD
CODE
MACC.4.MD.1.2
MAFS.4.MD.1.2
REVISED/
DELETED/NEW
STANDARD
PREVIOUS
Use the four operations to solve word problems1 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.
REVISED
Use the four operations to solve word problems1 involving distances, intervals of time, and
money, including problems involving simple fractions or decimals.2 Represent fractional
quantities of distance and intervals of time using linear models. (1See Table 2 Common
Multiplication and Division Situations) (2Computational fluency with fractions and decimals is
not the goal for students at this grade level.)
MAFS.4.OA.1.a
MAFS.4.OA.1.b
MACC.4.OA.2.4
MAFS.4.OA.2.4
NEW
Determine whether an equation is true or false by using comparative relational thinking. For
example, without adding 60 and 24, determine whether the equation 60 + 24 = 57 + 27 is true
or false.
NEW
Determine the unknown whole number in an equation relating four whole numbers using
comparative relational thinking. For example, solve 76 + 9 = n + 5 for n by arguing that nine is
four more than five, so the unknown number must be four greater than 76.
PREVIOUS
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is
a multiple of each of its factors. Determine whether a given whole number in the range 1–100
is a multiple of a given one-digit number. Determine whether a given whole number in the
range 1–100 is prime or composite.
REVISED
Investigate factors and multiples.
A. Find all factor pairs for a whole number in the range 1–100.
B. Recognize that a whole number is a multiple of each of its factors. Determine whether a
given whole number in the range 1–100 is a multiple of a given one-digit number.
C. Determine whether a given whole number in the range 1–100 is prime or composite.
5th Grade
Mathematics Florida Standards Changes
STANDARD
CODE
REVISED/
DELETED/NEW
MACC.5.G.2.4
PREVIOUS
MAFS.5.G.2.4
REVISED
Classify and organize two-dimensional figures into Venn diagrams based on the attributes of
the figures.
MACC.5.MD.1.1
PREVIOUS
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.
MAFS.5.MD.1.1
REVISED
STANDARD
Classify two-dimensional figures in a hierarchy based on properties.
Convert among different-sized standard measurement units (i.e., km, m, cm; kg, g; lb., oz.; l,
ml; hr., min, sec) 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.
MACC.5.MD.3.5
MAFS.5.MD.3.5
PREVIOUS
Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it
with unit cubes, and show that the volume is the same as would be found by multiplying the
edge lengths, equivalently by multiplying the height by the area of the base. Represent
threefold whole-number products as volumes, e.g., to represent the associative property of
multiplication.
b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right
rectangular prisms with whole- number edge lengths in the context of solving real world and
mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts,
applying this technique to solve real world problems.
REVISED
Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it
with unit cubes, and show that the volume is the same as would be found by multiplying the
edge lengths, equivalently by multiplying the height by the area of the base. Represent
threefold whole-number products as volumes, e.g., to represent the associative property of
multiplication.
b. Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of
right rectangular prisms with whole- number edge lengths in the context of solving real world
and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts,
applying this technique to solve real world problems.
6th Grade
Mathematics Florida Standards Changes
STANDARD
CODE
REVISED/
DELETED/NEW
STANDARD
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning
about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole- number measurements, find
missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables
to compare ratios.
MACC.6.RP.1.3
PREVIOUS
b. Solve unit rate problems including those involving unit pricing and constant speed. For
example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be
mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times
the quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
Use ratio and rate reasoning to solve real-world and mathematical problems1, e.g., by
reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or
equations.
a. Make tables of equivalent ratios relating quantities with whole- number measurements, find
missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables
to compare ratios.
MAFS.6.RP.1.3
REVISED
b. Solve unit rate problems including those involving unit pricing and constant speed. For
example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be
mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times
the quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
e. Understand the concept of Pi as the ratio of the circumference of a circle to its diameter.
(1See Table 2 Common Multiplication and Division Situations)
DOMAIN PROGRESSION
OPERATIONS AND ALGEBRAIC THINKING
Third Grade
Represent and solve problems involving
multiplication and division.
3.OA.1.1: Interpret products of whole numbers (e.g.,
interpret 5 × 7 as the total number of objects
in 5 groups of 7 objects each).
For example, describe a context in which a total
number of objects can be expressed as 5 × 7.
3.OA.1.2: Interpret whole number quotients of whole
numbers (e.g., interpret 56 ÷ 8 as the
number of objects in each share when 56
objects are partitioned equally into 8 shares,
or as a number of shares when 56 objects
are partitioned into equal shares of 8 objects
each).
For example, describe a context in which a number of
shares or a number of groups can be expressed as
56 ÷ 8.
3.OA.1.3: Use multiplication and division within 100 to
solve word problems in situations involving
equal groups, arrays, and measurement
quantities (e.g., by using drawings and
equations with a symbol for the unknown
number to represent the problem).
Domain Progression, Brevard Public Schools, 2013-2014
Fourth Grade
Use the four operations with whole numbers to
solve problems.
Fifth Grade
Write and interpret numerical expressions.
5.OA.1.1: Use parenthesis, brackets, or braces in
numerical expressions, and evaluate
expressions with these symbols.
4.OA.1.1: Interpret a multiplication equation as a
comparison (e.g., interpret 35 = 5 × 7 as a
5.OA.1.2: Write simple expressions that record
statement that 35 is 5 times as many as 7
calculations with numbers, and interpret
and 7 times as many as 5). Represent verbal
numerical expressions without evaluating
statements of multiplicative comparisons as
them.
multiplication equations.
For example, express the calculation “add 8 and 7,
then multiply by 2” as 2 × (8 + 7). Recognize that
4.OA.1.2: Multiply or divide to solve word problems
3 × (18,932 + 921) is three times as large as
involving multiplicative comparison (e.g., by
using drawings and equations with a symbol 18,932 + 921, without having to calculate the indicated
sum or product.
for the unknown number to represent the
problem, distinguishing multiplicative
Analyze patterns and relationships.
comparison from additive comparison).
5.OA.2.3: Generate two numerical patterns using two
given rules. Identify apparent relationships
4.OA.1.3: Solve multi-step word problems posed with
between corresponding terms. Form ordered
whole numbers and having whole number
pairs consisting of corresponding terms from
answers using the four operations, including
the two patterns, and graph the ordered pairs
problems in which remainders must be
on a coordinate plane.
interpreted. Represent these problems using
For example, given the rule “Add 3” and the starting
equations with a letter standing for the
number 0, and given the rule “Add 6” and the starting
unknown quantity. Assess the
number 0, generate terms in the resulting sequences
reasonableness of answers using mental
and observe that the terms in one sequence are twice
computation and estimation strategies
the corresponding terms in the other sequence.
including rounding.
Explain informally why this is so.
Page 1 of 22
Third Grade
3.OA.1.4: Determine the unknown whole number in a
multiplication or division equation relating three whole
numbers.
For example, determine the unknown number that
makes the equation true in each of the equations
8 × ? = 48, 5 = ☐ ÷ 3, 6 × 6 = ?.
Understand properties of multiplication and the
relationship between multiplication and division.
3.OA.2.5: Apply properties of operations as strategies
to multiply and divide.
Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is
also known. (commutative property of multiplication)
3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30,
or by 5 × 2 = 10, then 3 × 10 = 30. (associative
property of multiplication)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find
8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.
(distributive property)
3.OA.2.6: Understand division as an unknown-factor
problem.
For example, find 32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Domain Progression, Brevard Public Schools, 2013-2014
Fourth Grade
Fifth Grade
Gain familiarity with factors and multiples.
4.OA.2.4: Find all factor pairs for a whole number in
the range 1-100. Recognize that a whole
number is a multiple of each of its factors.
Determine whether a given whole number in
the range 1-100 is a multiple of a given onedigit number. Determine whether a given
whole number in the range 1-100 is prime or
composite.
Generate and analyze problems.
4.OA.2.5: Generate a number or shape pattern that
follows a given rule. Identify apparent
features of the pattern that were not explicit
in the rule itself.
For example, given the rule “Add 3” and the starting
number 1, generate terms in the resulting sequence
and observe that the terms appear to alternate
between odd and even numbers. Explain informally
why the numbers will continue to alternate this way.
Page 2 of 22
Third Grade
Fourth Grade
Fifth Grade
Multiply and divide within 100.
3.OA.3.7: Fluently multiply and divide within 100, using
strategies such as the relationship between
multiplication and division (e.g., knowing that
8 × 5 = 40, one knows 40 ÷ 5 = 8) or
properties of operations.
By the end of Grade 3, know from memory all products
of two one-digit numbers.
Solve problems involving the four operations, and
identify and explain patterns in arithmetic.
3.OA.4.8: Solve two-step word problems using the four
operations. Represent these problems using
equations with a letter standing for the
unknown quantity. Assess the
reasonableness of answers using mental
computation and estimation strategies
including rounding.
3.OA.4.9: Identify arithmetic patterns (including
patterns in the addition table or multiplication
table), and explain them using properties of
operations.
For example, observe that 4 times a number is always
even, and explain why 4 times a number can be
decomposed into two equal addends.
Domain Progression, Brevard Public Schools, 2013-2014
Page 3 of 22
NUMBERS AND OPERATIONS BASE IN TEN
Third Grade
Use place value understanding and properties of
operations to perform multi-digit arithmetic.
3.NBT.1.1: Use place value understanding to round
whole numbers to the nearest 10 or 100.
Fourth Grade
Generalize place value understanding for multidigit whole numbers.
4.NBT.1.1: Recognize that in a multi-digit whole
number, a digit in one place represents ten
times what it represents in the place to its
right.
For example, recognize that 700 ÷ 7 = 10 by applying
concepts of place value and division.
3.NBT.1.2: Fluently add and subtract within 1,000
using strategies and algorithms based on
place value, properties of operations, and/or
the relationship between addition and
4.NBT.1.2: Read and write multi-digit whole numbers
subtraction.
using base ten numerals, number names,
and expanded form. Compare two multi3.NBT.1.3: Multiply one-digit whole numbers by
digit numbers based on meanings of the
multiples of 10 in the range 10-90
digits in each place, using >, =, and <
(e.g., 9 × 80, 5 × 60) using strategies based
symbols to record the results of
on place value and properties of operations.
comparisons.
4.NBT.1.3: Use place value understanding to round
multi-digit whole numbers to any place.
Use place value understanding and properties of
operations to perform multi-digit arithmetic.
4.NBT.2.4: Fluently add and subtract multi-digit whole
numbers using the standard algorithm.
4.NBT.2.5: Multiply a whole number of up to four digits
by a one-digit whole number, and multiply
two two-digit numbers, using strategies
based on place value and the properties of
operations. Illustrate and explain the
calculation by using equations, rectangular
arrays, and/or area models.
Domain Progression, Brevard Public Schools, 2013-2014
Fifth Grade
Understand the place value system.
5.NBT.1.1: 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.
5.NBT.1.2: Explain patterns in the number of zeros of
the product when multiplying a number of
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 whole number exponents to
denote powers of 10.
5.NBT.1.3: Read, write, and compare decimals to
thousandths.
a. 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
1
1
+9×
+2×
).
1,000
100
10
b. Compare two decimals to thousandths
based on meanings of the digits in
each place, using >, =, and < symbols
to record the results of comparisons.
5.NBT.1.4: Use place value understanding to round
decimals to any place.
Page 4 of 22
Third Grade
Fourth Grade
Fifth Grade
Perform operations with multi-digit whole numbers
and with decimals to hundredths.
5.NBT.2.5: Fluently multiply multi-digit whole numbers
using the standard algorithm.
4.NBT.2.6: Find whole number quotients and
remainders with up to four-digit dividends
and one-digit divisors, using strategies
based on place value, the properties of
operations, and/or the relationship between
multiplication and division. Illustrate and
explain the calculation by using equations,
rectangular arrays, and/or area models.
Domain Progression, Brevard Public Schools, 2013-2014
5.NBT.2.6: Find whole number quotients of whole
numbers with up to four-digit dividends and
two-digit divisors, using strategies based on
place value, the properties of operations,
and/or the relationship between
multiplication and division. Illustrate and
explain the calculation by using equations,
rectangular arrays, and/or area models.
5.NBT.2.7: Add, subtract, multiply, and divide 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.
Page 5 of 22
NUMBER AND OPERATIONS - FRACTIONS
Third Grade
Develop understanding of fractions as numbers.
3.NF.1.1: Understand a fraction
1
as the quantity
b
formed by 1 part when a whole is partitioned
a
b
1
as the quantity formed by a parts of size .
b
into b equal parts; understand a fraction
3.NF.1.2: Understand a fraction as a number on the
number line; represent fractions on a
number line diagram.
1
a. Represent a fraction on a number line
b
diagram by defining the interval from 0 to
1 as the whole and partitioning it into b
equal parts. Recognize that each part has
1
and the endpoint of the part
b
1
based at 0 locates the number on the
b
size
number line.
a
on a number line
b
1
diagram by marking off a lengths from
b
b. Represent a fraction
0. Recognize that the resulting interval
has size
a
on the number line.
b
Domain Progression, Brevard Public Schools, 2013-2014
Fourth Grade
Fifth Grade
Extend understanding of fractional equivalence
and ordering.
4.NF.1.1: Explain why a fraction
fraction
a
is equivalent to a
b
n×a
by using visual fraction
n×b
models, with attention to how the number
and size of the parts differ even though the
two fractions themselves are the same size.
Use this principle to recognize and generate
equivalent fractions.
4.NF.1.2: Compare two fractions with different
numerators and different denominators (e.g.,
by creating common numerators and
denominators, or by comparing to a
Use equivalent fractions as a strategy to add and
subtract fractions.
5.NF.1.1: Add and subtract fractions with unlike
denominators (including mixed numbers) by
replacing given fractions with equivalent
fractions in such a way as to produce an
equivalent sum or difference of fractions
with like denominators.
5
8
15 23
2
+ =
+
=
. (In general,
4
12
12 12
3
(ad + bc)
a c
+ =
).
b d
bd
For example:
5.NF.1.2: Solve word problems involving addition and
subtraction of fractions referring to the same
whole, including cases of unlike
that comparisons are valid only when the two
denominators (e.g., by using visual fraction
fractions refer to the same whole. Record the
models or equations to represent the
results of comparisons with symbols >, =, or
problem). Use benchmark fractions and
<, and justify the conclusions (e.g., by using
number sense of fractions to estimate
the visual fraction model).
mentally and assess the reasonableness of
Extend understanding of fractional equivalence
answers.
and ordering.
For example, recognize an incorrect result
benchmark fraction such as
4.NF.2.3: Understand a fraction
of fractions
1
.
b
1
). Recognize
2
a
with 𝑎 > 1 as a sum
b
2
1
3
3 1
+ = , by observing that < .
5
2 7
7 2
Page 6 of 22
Third Grade
Fourth Grade
3.NF.1.3: Explain equivalence of fractions in special
cases, and compare fractions by reasoning
about their size.
a. Understand two fractions as equivalent (equal)
if they are the same size, or the same
point on a number line.
b. Recognize and generate simple equivalent
fractions (e.g.,
1
2 4 2
= , = ). Explain
2
4 6 3
why the fractions are equivalent (e.g., by
using a visual fraction model).
c. Express whole numbers as fractions, and
recognize fractions that are equivalent to
whole numbers.
For example: Express 3 in the form 3 =
that
3
; recognize
1
6
4
= 6; locate and 1 at the same point on a
1
4
number line diagram.
d. Compare two fractions with the same
numerator or the same denominator by
reasoning about their size. Recognize
that comparisons are valid only when the
two fractions refer to the same whole.
Record the results of comparisons with
the symbols >, =, or <, and justify the
conclusions (e.g., by using a visual
fraction model).
Domain Progression, Brevard Public Schools, 2013-2014
a. Understand addition and subtraction of
fractions as joining and separating parts
referring to the same whole.
b. Decompose a fraction into a sum of fractions
with the same denominator in more than
one way, recording each decomposition
by an equation. Justify decompositions
(e.g., by using a visual fraction model).
1 1 1 3
1 2
3
= + + ;
= + ;
8 8 8 8
8 8
8
1
8
1
1
8
2 =1+1+ = + +
8
8
8
8
8
For example:
Fifth Grade
Use equivalent fractions as a strategy to add and
subtract fractions.
5.NF.2.3: Interpret a fraction as division of the numerator by
a
the denominator ( = 𝑎 ÷ b). Solve word
b
problems involving division of whole numbers
leading to answers in the form of fractions or
mixed numbers (e.g., by using visual fraction
models or equations to represent the problem).
3
For example, interpret as the result of dividing
4
3
3 by 4, noticing that multiplied by 4 equals 3, and that
4
when 3 wholes are shared equally among 4 people, each
3
person has a share of size . If 9 people want to share a
4
50-pound sack of rice equally by weight, how many pounds
of rice should each person get? Between what two whole
numbers does your answer lie?
c. Add and subtract mixed numbers with like
denominators (e.g., by replacing each
mixed number with an equivalent fraction,
and/or by using properties of operations
and the relationship between addition and
subtraction).
d. Solve word problems involving addition and
5.NF.2.4: Apply and extend previous understandings of
subtraction of fractions referring to the
multiplication to multiply a fraction or whole
same whole and having like denominators
number by a fraction.
(e.g., by using visual fraction models and
a
a. Interpret the product × q as a parts of a partition
equations to represent the problem).
4.NF.2.4: Apply and extend previous understandings of
multiplication to multiply a fraction by a whole
number.
a. Understand a fraction
a
1
as a multiple of .
b
b
b
of q into b equal parts; equivalently, as the result of
a sequence of operations 𝑎 × q ÷ b.
For example, use a visual fraction model to show
2
8
× 4 = , and create a story context for this equation. Do
3
3
2 4 8
a c
ac
the same with × =
. (In general, ×
=
.)
3 5 15
b d
db
Page 7 of 22
Third Grade
Fourth Grade
Fifth Grade
For example: use a visual fraction model to represent
5
as the product 5 ×
4
5
the equation = 5 ×
4
1
, recording the conclusion by
4
1
.
4
1
a
b. Understand a multiple of as a multiple of ,
b
b
and use this understanding to multiply a
fraction by a whole number.
For example: use a visual fraction model to express
2
1
6
as 6 × , recognizing this product as . (In
5
5
5
a n×a
general, n × =
).
b
b
3×
c. Solve word problems involving multiplication of
a fraction by a whole number (e.g., by
using visual fraction models and
equations to represent the problem).
For example: if each person at a party will eat
3
of a
8
pound of roast beef, and there will be 5 people at the
party, how many pounds of roast beef will be needed?
Between what two whole numbers does your answer
lie?
b. Find the area of a rectangle with fractional side
lengths by tiling it with unit squares of the
appropriate unit fraction side lengths, and
show that the area is the same as would
be found by multiplying the side lengths.
Multiply fractional side lengths to find
areas of rectangles, and represent
fraction products as rectangular areas.
5.NF.2.5: Interpret multiplication as scaling (resizing),
by:
a. Comparing the size of a product to the size of
one factor on the basis of the size of the
other factor, without performing the
indicated multiplication.
b. Explaining why multiplying a given number by a
fraction greater than 1 results in a product
greater than the given number
(recognizing multiplication by whole
numbers greater than 1 as a familiar
case); explaining why multiplying a given
number by a fraction less than 1 results in
a product smaller than the given number;
and relating the principle of fraction
equivalence
multiplying
Domain Progression, Brevard Public Schools, 2013-2014
a n×a
=
to the effect of
b n×b
a
by 1.
b
Page 8 of 22
Third Grade
Fourth Grade
Fifth Grade
Understand decimal notation for fractions, and
compare decimal fractions.
5.NF.2.6: Solve real world problems involving
multiplication of fractions and mixed numbers
4.NF.3.5: Express a fraction with denominator 10 as an
(e.g., by using visual fraction models or
equivalent fraction with denominator 100,
equations to represent the problem).
and use this technique to add two fractions
with respective denominators 10 and 100.
5.NF.2.7: Apply and extend previous understandings of
3
30
For example: express
as
, and add
division to divide unit fractions by whole
10
100
numbers and whole numbers by unit
3
4
34
+
=
.
fractions.
10
100
100
4.NF.3.6: Use decimal notation for fractions with
denominators 10 or 100.
For example: rewrite 0.62 as
62
; describe a length
100
as 0.62 meters; locate 0.62 on a number line diagram.
4.NF.3.7: 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).
a. Interpret division of a unit fraction by a non-zero
whole number, and compute such
quotients.
For example, create a story context for
1
÷ 4, and use
3
a visual fraction model to show the quotient. Use the
relationship between multiplication and division to
explain that
1
1
÷4=
because
3
12
1
1
×4= .
12
3
b. Solve real-world problems involving division of
unit fractions by non-zero whole numbers
and division of whole numbers by unit
fractions (e.g., by using visual fraction
models and equations to represent the
problem).
For example, how much chocolate will each person get
if 3 people share
many
Domain Progression, Brevard Public Schools, 2013-2014
1
lb. of chocolate equally? How
2
1
cup servings are in 2 cups of raisins?
3
Page 9 of 22
MEASUREMENT AND DATA
Third Grade
Solve problems involving measurement and
estimation of intervals of time, liquid volumes, and
masses of objects.
3.MD.1.1: Tell and write time to the nearest minute and
measure time intervals in minutes. Solve
word problems involving addition and
subtraction of time intervals in minutes (e.g.,
by representing the problem on a number
line diagram).
3.MD.1.2: Measure and estimate liquid volumes and
masses of objects using standard units of
grams (g), kilograms (kg), and liters (L). Add,
subtract, multiply, or divide to solve one-step
word problems involving masses or volumes
that are given in the same units (e.g., by
using drawings such as a beaker with a
measurement scale to represent the
problem).
Represent and interpret data.
3.MD.2.3: Draw a scaled picture graph and a scaled
bar graph to represent a data set with
several categories. Solve one- and two-step
“how many more” and “how many less”
problems using information presented in
scaled bar graphs.
For example, draw a bar graph in which each square in
the bar graph might represent 5 pets.
Domain Progression, Brevard Public Schools, 2013-2014
Fourth Grade
Fifth Grade
Solve problems involving measurement and
Convert like measurement units within a given
conversion of measurements from a larger unit to a measurement system.
smaller unit.
5.MD.1.1: Convert among different sized standard
4.MD.1.1: Know relative sizes of measurement units
measurement units within a given
within one system of units including km, m,
measurement system (e.g., convert 5 cm to
cm; kg, g; lb., oz.; L, mL; hr., min., and sec.
0.05 m) and use these conversions in solving
Within a single system of measurement,
multi-step, real-world problems.
express measurements in a larger unit.
Record measurement equivalents in a twoRepresent and interpret data.
column table.
For example, know that 1 ft. is 12 times as long as 1 in. 5.MD.2.2: Make a line plot to display a data set of
Express the length of a 4 ft. snake as 48 in. Generate a
measurements in fractions of a unit
1 1 1
conversion table for feet and inches listing the number
( , , ). Use operations on fractions for
pairs (1,12), (2,24), (3,36),…
2 4 8
this grade to solve problems involving
4.MD.1.2: Use the four operations to solve word
information presented in line plots.
problems involving distances, intervals of
For example, given different measurements of liquid in
time, liquid volumes, masses of objects, and identical beakers, find the amount of liquid each beaker
money, including problems involving simple
would contain if the total amount in all the beakers
fractions or decimals, and problems that
were redistributed equally.
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.
Page 10 of 22
Third Grade
Fourth Grade
3.MD.2.4: Generate measurement data by measuring
lengths using rulers marked with halves and
fourths of an inch. Show the data by making
a line plot, where the horizontal scale is
marked off in appropriate units - whole
numbers, halves, or quarters.
4.MD.1.3: Apply the area and perimeter formulas for
rectangles in real world and mathematical
problems.
For example, find the width of a rectangular room given
the area of the flooring and the length, by viewing the
area formula as a multiplication equation with an
unknown factor.
Geometric measurement: understand concepts of
area and relate area to multiplication and to
addition.
Fifth Grade
Represent and interpret data.
4.MD.2.4: Make a line plot to display a data set of
measurements in fractions of a unit
1 1 1
2 4 8
( , , ). Solve problems involving addition
3.MD.3.5: Recognize area as an attribute of plane
and subtraction of fractions by using
figures and understand concepts of area
information presented in line plots.
management.
For
example,
from a line plot, find and interpret the
a. A square with side length 1 unit, called “a unit
square,” is said to have “one square unit” of area, difference in length between the longest and shortest
specimens in an insect collection.
and can be used to measure area.
b. A plane figure that can be covered without gaps
or overlaps by n unit squares is said to have an
area of n square units.
3.MD.3.6: Measure areas by counting unit squares
(square cm, square m, square in., square ft.,
and improvised units).
3.MD.3.7: Relate area to the operations of
multiplication and addition.
a. Find the area of a rectangle with whole number
side lengths by tiling it, and show that the area is
the same as would be found by multiplying the
side lengths.
Domain Progression, Brevard Public Schools, 2013-2014
Geometric measurement: understand concepts of
angles and measure angles.
4.MD.3.5: Recognize angles as geometric shapes that
are formed wherever two rays share a
common endpoint, and understand concepts
of angle measurement.
a An angle is measured with reference to a circle
with its center at the common endpoint of the
rays, by considering the fraction of the circular arc
between the points where the two rays intersect
the circle. An angle that turns through
1
of a
360
circle is called a “one-degree angle,” and can be
used to measure angles.
Geometric measurement: understand concepts of
volume and relate volume to multiplication and to
addition.
5.MD.3.3: Recognize volume as an attribute of solid
figures and understand concepts of volume
measurement.
a. A cube with side length 1 unit, called a “unit
cube,” is said to have “one cubic unit” of volume,
and can be used to measure volume.
b. A solid figure which can be packed without gaps
or overlaps using n unit cubes is said to have a
volume of n cubic units.
5.MD.3.4: Measure volumes by counting unit cubes,
using cubic cm, cubic in., cubic ft., and
improvised units.
5.MD.3.5: Relate volume to the operations of
multiplication and addition and solve realworld and mathematical problems involving
volume.
a. Find the volume of a right-rectangular prism with
whole number side lengths by packing it with
unit cubes, and show that the volume is the
same as would be found by multiplying the edge
lengths, equivalently by multiplying the height
by the area of the base. Represent threefold
whole number products as volumes (e.g., to
represent the associative property of
multiplication).
Page 11 of 22
Third Grade
Fourth Grade
Fifth Grade
b. Multiply side lengths to find areas of rectangles
with whole number side lengths in the context of
solving real world and mathematical problems,
and represent whole number products as
rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the
area of a rectangle with whole number side
lengths a and b + c is the sum of a × b and a × c.
Use area models to represent the distributive
property in mathematical reasoning.
b. An angle that turns through n one-degree angles
is said to have an angle measurement of n
degrees.
4.MD.3.6: Measure angles in whole number degrees
using a protractor. Sketch angles of a
d. Recognize area as additive. Find areas of
specified measure.
rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of
4.MD.3.7: Recognize angle measure as additive. When
the non-overlapping parts, applying this
an angle is decomposed into nontechnique to solve real world problems.
overlapping parts, the angle measure of the
whole is the sum of the angle measures of
Geometric measurement: recognize perimeter as
the parts. Solve addition and subtraction
an attribute of plane figures and distinguish
problems to find unknown angles on a
between linear and area measures.
diagram in real world and mathematical
problems (e.g., by using an equation with a
3.MD.4.8: Solve real world and mathematical problems
symbol for the unknown angle measure).
involving perimeters of polygons, including
b. Apply the formulas V = l × w × h and
V = B × h for rectangular prisms to find
volumes of right-rectangular prisms with
whole number edge lengths in the context of
solving real world and mathematical
problems.
c. Recognize volume as additive. Find volumes
of solid figures composed of two nonoverlapping right-rectangular prisms by
adding the volumes of the non-overlapping
parts, applying this technique to solve real
world problems.
finding the perimeter given the side lengths,
finding an unknown side length, and
exhibiting rectangles with the same
perimeter and different areas or with the
same area and different perimeters.
Domain Progression, Brevard Public Schools, 2013-2014
Page 12 of 22
GEOMETRY
Third Grade
Reason with shapes and their
attributes.
3.G.1.1: Understand that shapes in
different categories (e.g.,
rhombuses, rectangles, and
others) may share attributes
(e.g., having four sides), and
that the shared attributes can
define a larger category (e.g.,
quadrilaterals). Recognize
rhombuses, rectangles, and
squares as examples of
quadrilaterals, and draw
examples of quadrilaterals that
do not belong to any of these
subcategories.
3.G.1.2: Partition shapes into parts with
equal areas. Express the area
of each part as a unit fraction of
the whole.
For example, partition a shape into 4
parts with equal area, and describe the
area of each part as
the shape.
1
of the area of
4
Fourth Grade
Draw and identify lines and angles,
and classify shapes by properties of
their lines and angles.
4.G.1.1: Draw points, lines, line
segments, rays, angles (right,
acute, obtuse), and
perpendicular and parallel
lines. Identify these in twodimensional figures.
4.G.1.2: Classify two-dimensional
figures based on the presence
or absence of parallel or
perpendicular lines, or
presence or absence of angles
of a specified size. Recognize
right triangles as a category,
and identify right triangles.
4.G.1.3: Recognize a line of symmetry
for a two-dimensional figure as
a line across the figure such
that the figure can be folded
along the line into matching
parts. Identify line-symmetric
figures and draw lines of
symmetry.
Domain Progression, Brevard Public Schools, 2013-2014
Fifth Grade
Graph points on the coordinate plane
to solve real-world and mathematical
problems.
Sixth Grade
Solve real-world and mathematical
problems involving area, surface
area, and volume.
5.G.1.1: Use a pair of perpendicular
number lines, called axes, to
6.G.1.1: Find the area of right triangles,
define a coordinate system,
other triangles, special
with the intersection of the
quadrilaterals, and polygons by
lines (the origin) arranged to
composing into rectangles or
coincide with the 0 on each line
decomposing into triangles and
and a given point in the plane
other shapes; apply these
located by using an ordered
techniques in the context of
pair of numbers, called its
solving real-world and
coordinates. Understand that
mathematical problems.
the first number indicates how
far to travel from the origin in
6.G.1.2: Find the volume of a rightthe direction of one axis, and
rectangular prism with fractional
the second number indicates
edge lengths by packing it with
how far to travel in the direction
of the second axis, with the
unit cubes of the appropriate
convention that the names of
unit fraction edge lengths, and
the two axes and the
show that the volume is the
coordinates correspond (e.g.,
same as would be found by
x-axis and x-coordinate, y-axis
multiplying the edge lengths of
and y-coordinate).
the prism. Apply the formulas
V = l× w× h and V = B× h to
5.G.1.2: Represent real-world and
find volumes of rightmathematical problems by
rectangular prisms with
graphing points in the first
fractional edge lengths in the
quadrant of the coordinate
context of solving real-world
plane, and interpret coordinate
and mathematical problems.
values of points in the context
of the situation.
Page 13 of 22
Third Grade
Fourth Grade
Fifth Grade
Classify two-dimensional figures into
categories based on their properties.
Sixth Grade
6.G.1.3: Draw polygons in the
coordinate plane given
coordinates for the vertices; use
coordinates to find the length of
a side joining points with the
same first coordinate or the
same second coordinate. Apply
these techniques in the context
of solving real-world and
mathematical problems.
5.G.2.3: Understand that attributes
belonging to a category of twodimensional figures also
belong to all subcategories of
that category.
For example, all rectangles have four
right angles and squares are rectangles,
6.G.1.4: Represent three-dimensional
so all squares have four right angles.
figures using nets made up of
rectangles and triangles, and
5.G.2.4: Classify two-dimensional
use the nets to find the surface
figures in a hierarchy based on
area of these figures. Apply
properties.
these techniques in the context
of solving real-world and
mathematical problems.
Domain Progression, Brevard Public Schools, 2013-2014
Page 14 of 22
RATIOS AND PROPORTIONAL RELATIONSHIPS
Sixth Grade
Seventh Grade
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.1.1: Understand the concept of a ratio and use ratio language to describe a
ratio relationship between two quantities.
For example, the ratio of wings to beaks in the bird house at the zoo was 2:1
because for every 2 wings there was 1 beak. For every vote candidate A received,
candidate C received nearly three votes.
Analyze proportional relationships and use them to solve real-world and
mathematical problems.
7.RP.1.1: Compute unit rates associated with ratios of fractions, including ratios of
lengths, areas, and other quantities measured in like or different units.
6.RP.1.2: Understand the concept of a unit rate
a
associated with a ratio 𝑎:b with
b
b ≠ 0, and use rate language in the context of a ratio relationship.
For example, this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is
For example, if a person walks
the complex fraction
1
2
1
4
1
1
mile in each hour, compute the unit rate as
2
4
miles per hour, equivalently 2 miles per hour.
7.RP.1.2: Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship (e.g., by
testing for equivalent ratios in a table or graphing on a coordinate
rate of $5 per hamburger.
plane and observing whether the graph is a straight line through
the origin).
6.RP.1.3: Use ratio and rate reasoning to solve real-world and mathematical
b. Identify the constraint of proportionality (unit rate) in tables, graphs,
problems (e.g., by reasoning about tables of equivalent ratios, tape
equations, diagrams, and verbal descriptions of proportional
diagrams, double number line diagrams, or equations).
relationships.
a. Make tables of equivalent ratios relating quantities with whole number
c.
Represent
proportional relationships by equations.
measurements, finding missing values in the tables, and plot the
For
example,
if
total
cost t is proportional to the number n of items purchased at a
pairs of values on the coordinate plane. Use tables to compare
constant price p, the relationship between the total cost and the number of items
ratios.
b. Solve unit rate problems including those involving unit pricing and constant can be expressed t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship
speed.
means in terms of the situation, with special attention to the points
For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns
could be mowed in 35 hours? At what rate were lawns being mowed?
(0, 0) and (1, r) where r is the unit rate
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity
30
7.RP.1.3: Use proportional relationships to solve multi-step ratio and percent
means
times the quantity); solve problems involving finding the
problems.
100
whole, given a part and the percent.
Examples: simple interest, tax, markups and markdowns, gratuities and
d. Use ratio reasoning to convert measurement units; manipulate and
commissions, fees, percent increase and decrease, percent error
transform units appropriately when multiplying or dividing quantities.
3
cup of flour for each cup of sugar. We paid $75 for 15 hamburgers, which is a
4
Domain Progression, Brevard Public Schools, 2013-2014
Page 15 of 22
THE NUMBER SYSTEM
Sixth Grade
Apply and extend previous understandings of multiplication and division to
divide fractions by fractions.
6.NS.1.1: Interpret and compute quotients of fractions, and solve world problems
involving division of fractions by fractions (e.g., by using visual fraction
models and equations to represent the problem).
For example, create a story context for
2
3
÷ and use a visual fraction model to
3
4
show the quotient; use the relationship between multiplication and division to
3
8
2
a
2
c ad
3 8
÷ = because of is (In general ÷ =
.) How
4 9
3
9
3
b d
4
bc
1
much chocolate will each person get if 3 people share lb. of chocolate equally?
2
2
3
How many cup servings are in cup of yogurt? How wide is a rectangular strip
3
4
3
1
of land with a length of mile and an area of square mile?
4
2
explain that
Compute fluently with multi-digit numbers and find common factors and
multiples.
6.NS.2.2: Fluently divide multi-digit numbers using the standard algorithm.
Seventh Grade
Apply and extend previous understandings of operations with fractions to
add, subtract, multiply, and divide rational numbers.
7.NS.1.1: Apply and extend previous understandings of addition and subtraction to
add and subtract rational numbers; represent addition and subtraction
on a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0.
For example, a hydrogen atom has 0 charge because its two constituents are
oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the
positive or negative direction depending on whether q is positive or
negative. Show that a number and its opposite have a sum of 0 (are
additive inverses). Interpret sums of rational numbers by describing
real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse,
p - q = p + (-q). Show that the distance between two rational numbers
on the number line is the absolute value of their difference, and apply
this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational
numbers.
7.NS.1.2. Apply and extend previous understandings of multiplication and division
and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational
numbers by requiring that operations continue to satisfy the
6.NS.2.4: Find the greatest common factor of two whole numbers less than or equal
properties of operations, particularly the distributive property, leading
to 100 and the least common multiple of two whole numbers less than or
to products such as (-1)(-1)=1 and the rules for multiplying signed
equal to 12. Use the distributive property to express a sum of two whole
numbers. Interpret products of rational numbers by describing realnumbers 1-100 with a common factor as a multiple of a sum of two whole
world contexts.
numbers with no common factor.
For example, express 36 + 8 as 4(9 + 2).
6. NS.2.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the
standard algorithm for each operation.
Domain Progression, Brevard Public Schools, 2013-2014
Page 16 of 22
Sixth Grade
Seventh Grade
Apply and extend previous understandings of numbers to the system of
rational numbers.
6.NS.3.5: Understand that positive and negative numbers are used together to
describe quantities having opposite directions or values (e.g.,
temperature above/below zero, elevation above/below sea level,
credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real-world contexts,
explaining the meaning of 0 in each situation.
6.NS.3.6: Understand a rational number as a point on the number line. Extend
number line diagrams and coordinate axes familiar from previous grades
to represent points on the line and in the plane with negative number
coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite
sides of 0 on the number line; recognize that the opposite of the
opposite of a number is the number itself (e.g., -(-3) = 3) and that 0 is
its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in
quadrants of the coordinate plane; recognize that when two ordered
pairs differ only by signs, the locations of the points are related by
reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or
vertical number line diagram; find and position pairs of integers and
other rational numbers on a coordinate plane.
b. Understand that integers can be divided, provided that the divisor is not
zero, and every quotient of integers is a rational number. If p and q
are integers, then –
p
−p
p
=
=
. Interpret quotients of rational
−q
q
q
numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide rational
numbers.
d. Convert a rational number to a decimal using long division; know that the
decimal form of a rational number terminates to 0s or eventually
repeats.
7.NS.1.3. Solve real-world and mathematical problems involving the four operations
with rational numbers.
6.NS.3.7: Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position
of two numbers on a number line diagram.
Domain Progression, Brevard Public Schools, 2013-2014
Page 17 of 22
Sixth Grade
Seventh Grade
For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a
number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in realworld contexts.
For example, write -3°C > -7°C to express the fact that -3°C is warmer than -7°C.
c. Understand the absolute value of a rational number as its distance from 0
on the number line; interpret absolute value as magnitude for a
positive or negative quantity in real-world situation.
For example, for an account balance of -30 dollars, write |-30| = 30 to describe the
size of debt in dollars.
d. Distinguish comparisons of absolute value from statements about order.
For example, recognize that an account balance less than -30 dollars represents a
debt greater than 30 dollars.
6.NS.3.8: Solve real-world and mathematical problems by graphing points in all four
quadrants of the coordinate plane. Include use of coordinates and
absolute value to find distances between points with the same first
coordinate or the same second coordinate.
Domain Progression, Brevard Public Schools, 2013-2014
Page 18 of 22
EXPRESSIONS AND EQUATIONS
Sixth Grade
Apply and extend previous understandings of arithmetic to algebraic
expressions.
6.EE.1.1: Write and evaluate numerical expressions involving whole number
exponents.
6.EE.1.2: Write, read, and evaluate expressions in which letters stand for
numbers.
a. Write expressions that record operations with numbers and with letters
standing for numbers.
For example, express the calculation “Subtract y from 5” as 5 – y.
b. Identify parts of an expression using mathematical terms (sum, term,
product, factor, quotient, coefficient); view one or more parts of
an expression as a single entity.
For example, describe the expression 2(8 + 7) as a product of two factors; view
(8 + 7) as both a single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include
expressions that arise from formulas used in real-world problems.
Perform arithmetic operations, including those involving whole
number exponents, in the conventional order when there are no
parentheses to specify a particular order (Order of Operations).
For example, use the formulas V = s3 and A = 6s2 to find the volume and surface
area of a cube with sides of length s =
1
.
2
6.EE.1.3: Apply the properties of operations to generate equivalent expressions.
For example, apply the distributive property to the expression 3(2 + x) to produce
the equivalent expression 6 + 3x; apply the distributive property to the expression
24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of
operations to y + y + y to produce the equivalent expression 3y.
Domain Progression, Brevard Public Schools, 2013-2014
Seventh Grade
Use properties of operations to generate equivalent expressions.
7.EE.1.1: Apply properties of operations as strategies to add, subtract, factor, and
expand linear expressions with rational coefficients.
7.EE.1.2: Understand that rewriting an expression in different forms in a problem
context can shed light on the problem and how the quantities in it are
related.
For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as
“multiply by 1.05”.
Solve real-life and mathematical problems using numerical and algebraic
expressions and equations.
7.EE.2.3: Solve multi-step real-life and mathematical problems posed with positive
and negative rational numbers in any form (whole numbers, fractions, and
decimals) using tools strategically. Apply properties of operations to
calculate with numbers in any form; convert between forms as appropriate;
and assess the reasonableness of answers using mental computation and
estimation strategies.
For example: If a woman making $25 an hour gets a 10% raise, she will make an
1
of her salary an hour, or $2.50, for a new salary of $27.50. If you want
10
3
1
to place a towel bar 9 inches long in the center of a door that is 27 inches
2
4
additional
wide, you will need to place the bar about 9 inches from each edge; this estimate
can be used as a check on the exact computation.
7.EE.2.4: Use variables to represent quantities in a real-world or mathematical
problem, and construct simple equations and inequalities to solve
problems by reasoning about the quantities.
Page 19 of 22
Sixth Grade
Seventh Grade
6.EE.1.4: Identify when two expressions are equivalent (e.g., when the two
expressions name the same number regardless of which value is
substituted into them).
For example, the expressions y + y + y and 3y are equivalent because they name
the same number regardless of which number y stands for.
Reason about and solve one-variable equations and inequalities.
6.EE.2.5: Understand solving an equation or inequality as a process of answering a
question: which values from a specified set, if any, make the equation or
inequality true? Use substitution to determine whether a given number in a
specified set makes an equation or inequality true.
a. Solve word problems leading to equations of the form
px + q = r and p(x + q) = r, where p, q, and r are specific rational
6.EE.2.6: Use variables to represent numbers and write expressions when solving a
numbers. Solve equations of these forms fluently. Compare an
real-world or mathematical problem; understand that a variable can
algebraic solution to an arithmetic solution, identifying the sequence
represent an unknown number, or, depending on the purpose at hand, any
of the operations used in each approach.
number in a specified set.
For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its
6.EE.2.7: Solve real-world and mathematical problems by writing and solving
width?
equations of the form x + p = q and px = q for cases in which p, q and x are
b. Solve word problems leading to inequalities of the form
all non-negative rational numbers.
px + x > r or px + q < r, where p, q, and r are specific rational
6.EE.2.8: Write an inequality of the form x > c or x < c to represent a constraint or
numbers. Graph the solution set of the inequality and interpret it in the
condition in a real-world or mathematical problem. Recognize that
context of the problem.
inequalities of the form x > c or x < c have infinitely many solutions;
For example, as a salesperson, you are paid $50 per week plus $3 per sale. This
represent solutions of such inequalities on number line diagrams.
week you want your pay to be a least $100. Write an inequality for the number of
sales you need to make and describe the solutions.
Represent and analyze quantitative relationships between dependent and
independent variables.
6.EE.3.9: Use variables to represent two quantities in a real-world problem that change
in relationship to one another; write an equation to express one quantity,
thought of as the dependent variable, in terms of the other quantity, thought of
as the independent variable. Analyze the relationship between the dependent
and the independent variables using graphs and tables, and relate these to
the equation.
For example, in a problem involving motion at constant speed, list and graph ordered
pairs of distances and times, and write the equation d = 65t to represent the relationship
between distance and time.
Domain Progression, Brevard Public Schools, 2013-2014
Page 20 of 22
STATISTICS AND PROBABILITY
Sixth Grade
Seventh Grade
Develop understanding of statistical variability.
Use random sampling to draw inferences about a population.
6.SP.1.1: Recognize a statistical question as one that anticipates variability in the
data related to the question and accounts for it in the answers.
For example, “How old am I?” is not a statistical question, but “How old are the
students in my school?” is a statistical question because one anticipates variability
in students’ ages.
7.SP.1.1: Understand that statistics can be used to gain information about a
population by examining a sample of the population; generalizations
about a population from a sample are valid only if the sample is
representative of that population. Understand that random sampling
tends to produce representative samples and support valid inferences.
6.SP.1.2: Understand that a set of data collected to answer a statistical question
has a distribution which can be described by its center, spread, and
overall shape.
7.SP.1.2: Use data from a random sample to draw inferences about a population
with an unknown characteristic of interest. Generate multiple samples (or
simulated samples) of the same size to gauge the variation in estimates
or predictions.
6.SP.1.3: Recognize that a measure of center for a numerical data set summarizes For example, estimate the mean word length in a book by randomly sampling
all of its values with a single number, while a measure of variation
words from the book; predict the winner of a school election based on randomly
describes how its values vary with a single number.
sampled survey data. Gauge how far off the estimate or prediction might be.
Summarize and describe distributions.
6.SP.2.4: Display numerical data in plots on a number line, including dot plots,
histograms, and box plots.
6.SP.2.5: Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it
was measured and its units of measurement.
c. Giving quantitative measures of center (median and/or mean) and
variability (interquartile range and/or mean absolute deviation), as
well as describing any overall pattern and any striking deviations
from the overall pattern with reference to the context in which the
data were gathered.
d. Relating the choice of measures of center and variability to the shape of
the data distribution and the context in which the data were
gathered.
Domain Progression, Brevard Public Schools, 2013-2014
Draw informal comparative inferences about two populations.
7.SP.2.3: Informally assess the degree of visual overlap of two numerical data
distributions with similar variabilities, measuring the difference between
the centers by expressing it as a multiple of a measure of variability.
For example, the mean height of players on the basketball team is 10 cm greater
than the mean height of players on the soccer team, about twice the variability
(mean absolute deviation) on either team; on a dot plot, the separation between the
two distributions of heights is noticeable.
7.SP.2.4: Use measures of center and measures of variability for numerical data
from random samples to draw informal comparative inferences about two
populations.
For example, decide whether the words in a chapter of a seventh grade science
book are generally longer than the words in a chapter of a fourth grade science
book.
Page 21 of 22
Sixth Grade
Seventh Grade
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.3.5: Understand that the probability of a chance event is a number between 0 and 1
that expresses the likelihood of the event occurring. Larger numbers indicate
greater likelihood. A probability near 0 indicates an unlikely event, a probability
1
around indicates an event that is neither unlikely nor likely, and a probability
2
near 1 indicates a likely event.
7.SP.3.6: Approximate the probability of a chance event a probability around by collecting
data on the chance process that produces it and observing its long-run relative
frequency, and predict the approximate relative frequency given the probability.
For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled
roughly 200 times, but probably not exactly 200 times.
7.SP.3.7: Develop a probability model and use it to find probabilities of events. Compare
probabilities from a model to observed frequencies; if the agreement is not
good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes,
and use the model to determine probabilities of events.
For example, if a student is selected at random from a class, find the probability that Jane
will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in
data generated from a chance process.
For example, find the approximate probability that a spinning penny will land heads up or
that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny
appear to be equally likely passed on the observed frequencies?
7.SP.3.8: Find probabilities of compound events using organized lists, tables, tree
diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the
fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists,
tables, and tree diagrams. For an event described in everyday language (e.g., “rolling
double sixes”) identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events.
For example, use random digits as a simulation tool to approximate the answer to the
question: If 40% of donors have type A blood, what is the probability that it will take at least
4 donors to find one with type A blood?
Domain Progression, Brevard Public Schools, 2013-2014
Page 22 of 22
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Operations and Algebraic Thinking
Cluster: Write and interpret numerical expressions.
Students should be given ample opportunities to explore numerical expressions with mixed
operations. This is the foundation for evaluating numerical and algebraic expressions that will
include whole number exponents in Grade 6. There are conventions (rules) determined by
mathematicians that must be learned with no conceptual basis. For example, multiplication
and division are always done before addition and subtraction.
Begin with expressions that have two operations without any grouping symbols (multiplication
or division combined with addition or subtraction) before introducing expressions with multiple
operations. Using the same digits, with the operations in a different order, have students
evaluate the expressions and discuss why the value of the expression is different. For
example, have students evaluate 5 × 3 + 6 and 5 + 3 × 6. Discuss the rules that must be
followed. Have students insert parentheses around the multiplication or division part in an
expression. A discussion should focus on the similarities and differences in the problems and
the results. This leads to students being able to solve problem situations which require that
they know the order in which operations should take place.
After students have evaluated expressions without grouping symbols, present problems with
one grouping symbol, beginning with parentheses, then in combination with brackets and/or
braces.
Have students write numerical expressions in words without calculating the value. This is the
foundation for writing algebraic expressions. Then, have students write numerical
expressions from phrases without calculating them.
Common Misconceptions
Students may believe the order in which a problem with mixed operations is written is the
order to solve the problem. Allow students to use calculators to determine the value of the
expression, and then discuss the order the calculator used to evaluate the expression.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 1 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Operations and Algebraic Thinking
Cluster: Analyze patterns and relationships.
The graph of both sequences of numbers is a visual representation that will show the
relationship between the two sequences of numbers.
Encourage students to represent the sequences in T-charts so that they can see a
connection between the graph and the sequences.
Common Misconceptions
Students reverse the points when plotting them on a coordinate plane. They count up first on
the y-axis and then count over on the x-axis. The location of every point in the plane has a
specific place. Have students plot points where the numbers are reversed such as (4, 5) and
(5, 4). Begin with students providing a verbal description of how to plot each point. Then,
have them follow the verbal description and plot each point.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 2 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Number and Operations in Base Ten
Cluster: Understand the place value system.
In Grade 5, the concept of place value is extended to include decimal values to thousandths.
The strategies for Grades 3 and 4 should be drawn upon and extended for whole numbers
and decimal numbers. For example, students need to continue to represent, write and state
the value of numbers including decimal numbers. For students who are not able to read,
write, and represent multi-digit numbers, working with decimals will be challenging.
Money is a good medium to compare decimals. Present contextual situations that require the
comparison of the cost of two items to determine the lower or higher priced item. Students
should also be able to identify how many pennies, dimes, dollars, and ten dollars, etc., are in
a given value. Help students make connections between the number of each type of coin and
the value of each coin, and the expanded form of the number. Build on the understanding
that it always takes ten of the number to the right to make the number to the left.
Number cards, number cubes, spinners, and other manipulatives can be used to generate
decimal numbers. For example, have students roll three number cubes, then create the
largest and smallest number to the thousandths place. Ask students to represent the number
with numerals and words.
Common Misconceptions
A misconception that is directly related to comparing whole numbers is the idea that the more
digits the number has the greater the number. With whole numbers, a 5-digit number is
always greater than a 1-, 2-, 3-, or 4-digit number. However, with decimals, a number with
one decimal place may be greater than a number with two or three decimal places. For
example, 0.5 is greater than 0.12, 0.009 or 0.499. One method for comparing decimals it to
make all numbers have the same number of digits to the right of the decimal point by adding
zeros to the number, such as 0.500, 0.120, 0.009 and 0.499. A second method is to use a
place-value chart to place the numerals for comparison.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 3 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Number and Operations in Base Ten
Cluster: Perform operations with multi-digit whole numbers and with
decimals to hundredths.
Because students have used various models and strategies to solve problems involving
multiplication with whole numbers, they should be able to transition to using standard
algorithms effectively. With guidance from the teacher, they should understand the
connection between the standard algorithm and their strategies. Connections between the
algorithm for multiplying multi-digit whole numbers and strategies such as partial products or
lattice multiplication are necessary for students’ understanding.
The multiplication can also be done without listing the partial products by multiplying the
value of each digit from one factor by the value of each digit from the other factor.
Understanding of place value is vital in using the standard algorithm. As students develop
efficient strategies to do whole number operations, they should also develop efficient
strategies with decimal operations.
Students should learn to estimate decimal computations before they compute with pencil and
paper. The focus on estimation should be on the meaning of the numbers and the operations,
not on how many decimal places are involved. Have students use estimation to find the
product by using exactly the same digits in one of the factors with the decimal point in a
different position each time. For example, have students estimate the product of 275 × 3.8,
27.5 × 3.8, and 2.75 × 3.8, and discuss why the estimates should or should not be the same.
Common Misconceptions
Students might compute the sum or difference of decimals by lining up the right-hand digits
as they would whole number. For example, in computing the sum of 15.34 + 12.9, students
will write the problem in this manner:
15.34
+ 12.9
16.63
To help students add and subtract decimals correctly, have them first estimate the sum or
difference. Providing students with a decimal-place value chart will enable them to place the
digits in the proper place. Students should understand that the decimal points should be
lined up when writing an addition or subtraction problem vertically.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 4 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Number and Operations – Fractions
Cluster: Use equivalent fractions as a strategy to add and subtract
fractions.
To add or subtract fractions with unlike denominators, students use their understanding of
equivalent fractions to create fractions with the same denominators. Start with problems that
require the changing of one of the fractions and progress to changing both fractions. Allow
students to add and subtract fractions using different strategies such as number lines, area
models, and fraction bars or strips. Have students share their strategies and discuss
commonalities in them.
Students need to develop the understanding that when adding or subtracting fractions, the
fractions must refer to the same whole. Any models used must refer to the same whole.
Students may find that a circular model might not be the best model when adding or
subtracting fractions.
The concept of adding or subtracting fractions with unlike denominators will develop through
solving problems. Mental computations and estimation strategies should be used to
determine the reasonableness of answers. Students need to prove or disprove whether an
answer provided for a problem is reasonable. Estimation is about getting useful answers, it is
not about getting the right answer. It is important for students to learn which strategy to use
for estimation. Students need to think about what might be a close answer.
Common Misconceptions
Students often mix models when adding, subtracting, or comparing fractions. Students will
use a circle for thirds and a rectangle for fourths when comparing fractions with thirds and
fourths. Remind students that the representations need to be from the same whole models
with the same shape and size.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 5 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Number and Operations – Fractions
Cluster: Apply and extend previous understandings of multiplication and
division to multiply and divide fractions.
Connect the meaning of multiplication and division of fractions with whole number
multiplication and division. Consider area models of multiplication and both sharing and
measuring models for division.
Ask questions such as, “What does 2 × 3 mean?” and “What does 12 ÷ 3 mean?”. Then,
follow with questions for multiplication with fractions, such as, “What does
3
3
3
3
4
×
3
1
3
“What does 4 × 7 mean?” (4 of a set of 7), and “What does 7 × 4 mean?” (7 sets of 4).
The meaning of 4 ÷
1
2
1
(How many 2’s are in 4?) and
1
2
should also be illustrated with models or drawings like:
mean?”,
1
÷ 4 (How many groups of 4 are in 2)
Encourage students to use models or drawings to multiply or divide with fractions. Begin with
students modeling multiplication and division with whole numbers. Have them explain how
they used the model or drawing to arrive at the solution. Models to consider when multiplying
or dividing fractions include, but are not limited to: area models using rectangles or squares,
fraction strips/bars and sets of counters.
Present problem situations and have students use models and equations to solve the
problem. It is important for students to develop understanding of multiplication and division of
fractions through contextual situations.
Common Misconceptions
Students may believe that multiplication always results in a larger number. Additionally,
students may believe that division always results in a smaller number. Using models when
multiplying and dividing with fractions will enable students to see that the results are not
always so.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 6 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Measurement and Data
Cluster: Convert like measurement units.
Students should gain ease in converting units of measures in equivalent forms within the
same system. To convert from one unit to another unit, the relationship between the units
must be known. In order for students to have a better understanding of the relationships
between units, they need to use measuring tools in class. The number of units must relate to
the size of the unit. For example, students have discovered that there are 12 inches in 1 foot
and 3 feet in 1 yard. This understanding is needed to convert inches to yards. Using 12-inch
rulers and yardsticks, students can see that three of the 12-inch rulers are equivalent to one
yardstick (3 × 12 inches = 36 inches; 36 inches = 1 yard). Using this knowledge, students can
decide whether to multiply or divide when making conversions.
Once students have an understanding of the relationships between units and how to do
conversions, they are ready to solve multi-step problems that require conversions within the
same system. Allow students to discuss methods used in solving the problems. Begin with
problems that allow for renaming the units to represent the solution before using problems
that require renaming to find the solution.
Common Misconceptions
When solving problems that require renaming units, students use their knowledge of
renaming the numbers as with whole numbers. Students need to pay attention to the unit of
measurement which dictates the renaming and the number to use. The same procedures
used in renaming whole numbers should not be taught when solving problems involving
measurement conversions. For example, when subtracting 5 inches from 2 feet, students
may take 1 foot from the 2 feet and use it as 10 inches. Since there were no inches with the 2
feet, they put 1 with 0 inches and make it 10 inches, when it should be 1 foot 12 inches.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 7 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Measurement and Data
Cluster: Represent and interpret data.
Using a line plot to solve problems involving operations with unit fractions now includes
multiplication and division. Revisit using a number line to solve multiplication and division
problems with whole numbers. In addition to knowing how to use a number line to solve
problems, students also need to know which operation to use to solve problems.
Use the tables for common addition, subtraction, multiplication, and division situations (Table
1 and Table 2 in the Common Core State Standards for Mathematics) as a guide to the types
of problems students need to solve without specifying the type of problem. Allow students to
share methods used to solve the problems. Also have students create problems to show their
understanding of the meaning of each operation.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 8 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Measurement and Data
Cluster: Geometric measurement: understand concepts of volume and
relate volume to multiplication and to addition.
Volume refers to the amount of space that an object takes up and is measured in cubic units,
such as cubic inches or cubic centimeters.
Students need to experience finding the volume of rectangular prisms by counting unit cubes,
in metric and standard units of measure, before the formula is presented. Provide multiple
opportunities for students to develop the formula for the volume of a rectangular prism with
activities similar to the one described below.
Give students one block (a cubic-centimeter or cubic-inch cube), a ruler with the appropriate
measure based on the type of cube, and a small rectangular box. Ask students to determine
the number of cubes needed to fill the box. Have students share their strategies with the
class using words, drawings or numbers. Allow them to confirm the volume of the box by
filling the box with cubes of the same size.
A right-rectangular prism has three pairs of parallel faces that are all rectangles. Have
students build a prism in layers. Then, have students determine the number of cubes in the
bottom layer and share their strategies. By stacking geometric solids with cubic units in
layers, students can begin understanding the concept of how addition plays a part in finding
volume. Students should use multiplication based on their knowledge of arrays and its use in
multiplying two whole numbers. This will lead to an understanding of the formula for the
volume of a right-rectangular prism, 𝐵 × ℎ, where 𝐵 is the area of the base.
Ask what strategies can be used to determine the volume of the prism based on the number
of cubes in the bottom layer. Expect responses such as “adding the same number of cubes in
each layer as were on the bottom layer” or multiply the number of cubes in one layer times
the number of layers.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 9 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Geometry
Cluster: Graph points on the coordinate plane to solve real-world and
mathematical problems.
Students need to understand the underlying structure of the coordinate system and see how axes
make it possible to locate points anywhere on a coordinate plane. This is the first time students are
working with coordinate planes, and only in the first quadrant. It is important that students create the
coordinate grid themselves. This can be related to two number lines and reliance on previous
experiences with moving along a number line.
Multiple experiences with plotting points are needed. Provide points plotted on a grid and have
students name and write the ordered pair. Have students describe how to get to the location.
Encourage students to articulate directions as they plot points.
Present real-world and mathematical problems and have students graph points in the first quadrant of
the coordinate plane. Gathering and graphing data is a valuable experience for students. It helps them
to develop an understanding of coordinates and what the overall graph represents. Students also
need to analyze the graph by interpreting the coordinate values in the context of the situation.
Common Misconceptions
When playing games with coordinates or looking at maps, students may think the order in plotting a
coordinate point is not important. Have students plot points so that the position of the coordinates is
switched. For example, have students plot (3, 4) and (4, 3) and discuss the order used to plot the
points. Have students create directions for others to follow so that they become aware of the
importance of direction and distance.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 10 of 11, 2013-2014
FIFTH GRADE
DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION
Domain: Geometry
Cluster: Classify two-dimensional figures into categories based on their
properties.
This cluster builds from Grade 3 when students described, analyzed, and compared properties of twodimensional shapes. They compared and classified shapes by their sides and angles, and connected
these with definitions of shapes. In Grade 4, students built, drew, and analyzed two-dimensional
shapes to deepen their understanding of the properties of two-dimensional shapes. They looked at
the presence or absence of parallel and perpendicular lines or the presence or absence of angles of a
specified size to classify two-dimensional shapes. Now, students classify two-dimensional shapes in a
hierarchy based on properties. Details learned in earlier grades need to be used in the descriptions of
the attributes of shapes. The more ways that students can classify and discriminate shapes, the better
they can understand them. The shapes are not limited to quadrilaterals.
Students can use graphic organizers such as flow charts or t-charts to compare and contrast the
attributes of geometric figures. Have students create a t-chart with a shape on each side. Have them
list attributes of the shapes, such as number of sides, number of angles, types of lines, etc. They need
to determine what is alike or different about the two shapes to get a larger classification for the
shapes.
Pose questions such as, “Why is a square always a rectangle?” and “Why is a rectangle not always a
square?”.
Fifth Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education,
www.ode.state.oh.us/), page 11 of 11, 2013-2014
Additional
Resources
ADDITION AND SUBTRACTION STRATEGIES
The development of strategies for addition and subtraction is a critical area in the Common Core State
Standards. By using and comparing a variety of solution strategies students build their understanding
of the relationship between addition and subtraction.
*The following information regarding addition and subtraction strategies has been adapted from:
Van de Walle, J.A., & Lovin, L.H. (2006). Teaching Student-Centered Mathematics, Volume I. Boston:
Pearson. See chapters three and four of this book for further clarification of addition and subtraction
strategies.
Addition Strategies
Subtraction Strategies
Zero
One More/Two More
Doubles
Near Doubles
Sums of Ten
Make Ten
Ten Plus
Invented
Commutative Property
Associative Property
Think-Addition
Build Up Through Ten
Back Down Through Ten
Invented
SPECIAL NOTES:
 Basic facts for addition are combinations of numbers where both addends are less than 10.
Subtraction facts correspond to the addition facts.
 Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps that does
not include counting), and flexibility (using strategies that demonstrate number sense).
 Every child, including ESE children, can master the basic facts with efficient mental tools.
 Steps to Mastery:
1) Children must develop an understanding of number relationships and the operations.
2) Children need to develop efficient strategies for fact retrieval.
3) Teachers need to provide practice of selection of strategies once they have been developed.
 Children who do not learn mental strategies will continue to count on their fingers since they
have no other strategies to solve basic addition and subtraction problems.
 AVOID PREMATURE DRILL: if a child does not know a fact and is given a timed test; the
child will revert to counting.
 Downplay counting on as a strategy because children often get confused as to why they can
count for some problems but not others. It is used as a crutch where other strategies would be
more efficient.
 Many of the strategies apply to more than one fact. Therefore, students need to choose the one
that works best for them through discussion and justification.
 Encourage discussion so students can justify and defend their method. This allows the students
to hear other methods that might lead to the development of a more effective strategy.
Addition and Subtraction Strategies, page 1, 2015 - 2016
NOTE: Counting on is not a sophisticated strategy. Children coming from Kindergarten are expected to
recognize small sets of numbers but may count. Children in first and second grade are expected to take the
next step by creating and using more sophisticated strategies such as the ones listed below.
Addition Strategies (Continued)





Note:
Zero
One addend is always zero
The sum of any addend and zero is the original addend.
There are 19 facts where zero is one of the addends
Be sure to show 0 + 6 and 6 + 0
Children assume that addition sentences result in a larger number
This may seem easy; however, students over generalize that an addition sentence always equals a larger
sum.
One/Two More
 One addend is 1 or 2
 36 facts
 Students are ready for these activities when they can identify 1 or 2 more without counting
Doubles
 The two addends are the same 0 + 0, 1 + 1, 2 + 2, etc.
 There are 10 doubles facts
 These facts will be anchors for other facts (such as 4 + 4 = 8 so 4 + 5 = 9 , see Near-Doubles)
Near-Doubles
 All combinations where one addend is more than the other
Note: Some children will double the smaller fact and add up 6 + 6 = 12 so 6 + 7 = 13. Others will double the
greater fact and subtract one 7 + 7 = 14 so 7 + 6 = 13
*Be sure students are exposed to both so they can decide which is better for them.
Sums of Ten
 The two addends equal the sum of ten
 These facts will be anchors for other facts (such as 9 + 1=10, so 9 + 4 becomes 10 + 3)
Ten Plus
 One addend is 10, 10 + 4, 4 + 10
 Children need to recognize that a set of ten and a set of 4 total 14 without counting.
* This is not an appropriate place for the term 1 ten as regrouping for first graders. The term 1 set of ten not a 1 in
the tens place should be used to meet the needs of the early first grade student.
Make-Ten
 These facts all have 8 or 9 as one of the addends
 Children use 10 as a way to “bridge” to get the sum 6 + 8. Start with 8; decompose the 6 into 4 + 2 add the
2 to 8 and get a sum of 10. 10 and the remaining 4 equals 14 so 6 + 8 = 14.
Commutative Property
The order of the addends does not change the sum
2+5=5+2
Associative Property
The sum is the same regardless of the grouping of the addends.
2 + (6 + 4) = 2 + 10 = 12
Addition and Subtraction Strategies, page 2, 2015 - 2016
Remaining 4 Facts
3+5
3+6
4+7
5+7
The children have learned or discovered strategies to solve the 4 strategies above. Now encourage the students to
apply and choose a strategy that will work for them.
7+4
7+5
decompose the 4 into 3 + 1 to make ten, add 1 more
decompose the 5 to make 3 + 2, therefore making a ten creating a fact they know (7 + 3 = 10), then
add 2 more




Invented
Students create and/or apply any of the above strategies to other equations.
Students will create ways to solve problems that are not noted above.
Encourage students to create other ways to solve problems other than counting.
Invented strategies are number-oriented, flexible, and constructed by students.
Addition Strategies
Circle Map
Make Ten
7 + 3 = 10
+2
12
Invented
(using what I
know)
7+5
7 + 7 = 14
7 + 5 = 12
Invented
(applying a
near double)
7 + 6 = 13
7 + 5 = 12
Addition and Subtraction Strategies, page 3, 2015 - 2016
Subtraction Strategies
Think-Addition
The student understands subtraction as an unknown addend problem.


This strategy works best for sums less than 10 because 64 % of the 100 subtraction facts
fall into this category, for example: 9 – 4 (think 4 + 5 = 9)
Such facts as 7 – 2 would go along well with 2 more, now think 2 less along with
2 + 5 = 7, so 7 – 2 = 5
Build Up Through Ten


This group includes all the facts where the part is either 8 or 9
Start with the 8 or 9 and ask how much to ten and then build up
Back Down Through 10



It is most useful for facts where one digit is close to the number it is being subtracted
from
14 - 6, remove six from a ten frame and then two more to get the eight
Known as decomposing a number leading to a ten in Common Core
Invented


Students will create ways to solve problems that are not noted above
Encourage students to create other ways to solve problems other than counting
Circle Map
Think
addition
Build Up
Through
10
8+2=10
+4
14
8+6
Back
Down
Through
10
14
- 8 (4+4)
14-8
14 - 4=10-4=6
8+7=15
8+6=14
8+8=16
-2
14
8-2=6
Invented
Invented (using
a fact I know)
Addition and Subtraction Strategies, page 4, 2015 - 2016
MACC.2.NBT.2



Use place value understanding and properties of operations to add and subtract.
The standard algorithm is introduced and taught in fourth grade: 4.NBT.4.
Second and third grade students are encouraged to invent strategies when solving
multi digit addition and subtraction problems for the following reasons.
 Place value concepts are enhanced.
 Students make fewer errors as they are focused on the number and number
relationships.
 Less reteaching is necessary as they are inventing for themselves what makes
sense.
 Mental computation and estimation are enhanced.
 Flexible thinking of number leads to strategies and this thinking is often faster
than standard algorithms.
 Strategies serve students just as well as traditional algorithms on tests (including
FCAT 2.0).
 Students who look at the meaning of numbers and use what they know to solve
problems know and use more mathematics than those that follow a procedure.
Samples of Invented Strategies for Addition
Place Value:
352 + 675
300 + 50 + 2
600 + 70 + 5
900 + 120 + 7
1,027 = 1,000 + 20 + 7
Friendly Tens:
352 + 675
327 + 700 = 1,027
Compensate:
352 + 675
350 is easier to add to
650
1,000
Now I pick up my 25 + 2 = 27
1,000 + 27 = 1,027
Adding Hundreds or Ones first:
352 + 675
300 + 600 = 900
50 + 70 = 120
2+
5=
7
1,027
Addition and Subtraction Strategies, page 5, 2015 - 2016
Samples of Invented Strategies for Subtraction
Place Value:
675 – 352
600 + 70 + 5
300 + 50 + 2
323 = 300 + 20 + 3
Add up:
675 – 352
352 + 8 = 360
360 + 40 = 400
400 + 275 = 675
323
 Students may extend 275 + 5 (5+ 3 = 8) to get 280 + 20 (20 + 20 = 40) to get 300 +
23 = 323
See Chapter 6 strategies for whole-number computation in Van de Walle, J.A., & Lovin, L.H.
(2006). Teaching Student-Centered Mathematics, Volume I, Boston: Pearson
.
Addition and Subtraction Strategies, page 6, 2015 - 2016
Strategies for BASIC Multiplication and Division Facts
The development of strategies for multiplication and division is a critical area in the Mathematics
Florida Standards (MAFS). By using and comparing a variety of solution strategies students build
their understanding of the relationship between multiplication and division.
The following information regarding multiplication and division strategies has been adapted from: Van
de Walle, J.A., Lovin, L.H, Karp, K.S, & Bay-Williams, J.M. (2014). Teaching Student-Centered
Mathematics, Volume II. Boston; Pearson. See chapters eight, nine, and eleven of this book for
further clarification of multiplication and division strategies and.
Multiplication Strategies
Division Strategies
Doubles
Think Multiplication and then
Apply a Known Multiplication Fact
Fives
Zeros and Ones
Nifty Nines
Using Known Facts to Derive Other Facts
SPECIAL NOTES:
 The use of a problem-based approach and a focus on reasoning strategies are critical to
developing mastery of the multiplication and related division facts. Thus, story problems should be
used to develop reasoning strategies for basic fact mastery.
BASIC Multiplication Fact Strategies
Doubles
 These are facts with 2 as a factor and are equivalent to the addition doubles, so students should
already know these.
 Students need experiences to help them realize that 2 x 8 is the same as double 8 (8 + 8).
Fives
 These are facts that have 5 as a first or second factor.
 Mastery development ideas:
 Skip count by fives: 0, 5, 10, 15, 20 . . .
 Connect counting by fives with arrays that have 5 dots
For example three rows is 3 x 5
 Connect to counting minutes on the clock.
Basic Multiplication and Division Fact Strategies, page 1 of 3, adapted from Van de Walle, J. A., Lovin, L. H, Karp, K. S, &
Bay-Williams, J. M. (2014). Teaching Student-Centered Mathematics, Volume II. Boston; Pearson.
Zeros and Ones
 These are facts that have at least one factor that is either 0 or 1.
 While these facts seem easy, they can be confusing to students because of rules for addition. For
example, when zero is added to a number, it does not change the number (8 + 0 = 8). However,
8 x 0 = 0. Adding 1 to a number results in the next number, or one more (8 + 1 = 9), but a number
multiplied by one does not change the number (8 x 1 = 8).
 The use of rules that are strictly procedural, such as “anything times zero is zero” should be avoided.
Nifty Nines
 Facts with factors of 9 may be among the easiest to learn because of reasoning strategies and
patterns.
 9 x 8 is the same as 10 x 8 less one set of 8, or 80 – 8 = 72
 The tens digit is always one less than the other factor (the factor other than 9) and the sum of the
digits in the product is always 9. Therefore, for the fact 9 x 8, the tens digit is 7 and since the two
digits in the product must add to 9 the ones digit is 2 and the product is 72.
 Patterns are not rules without reasons. Students should be challenged to understand why they work.
Using Known Facts to Derive Other Facts
Reasoning Strategies:
 Double and Double Again
 This applies to all facts with a factor of 4.
 For example, 4 x 6 is the same as 2 x 6 doubled. Note that for some facts such as 4 x 8, doubling
the product may result in a difficult addition problem. For 4 x 8, a student knows 2 x 8 is 16, and
then doubles 16. Doubling 16 is a difficult addition and simply adding 16 + 16 defeats the purpose
of efficient reasoning. Students should use effective and efficient addition strategies such as, “I
know 15 + 15 is 30 and 16 + 16 is 2 more, or 32.”
 Double and One More
 This works with facts that have 3 as one factor. For example, 3 x 6 is 2 x 6 and 6 more
(12 + 6 = 18).
 Note that 3 x 8 and 3 x 9 result in challenging mental additions.
 Half then Double
 This applies to all facts with one even factor. For example, 6 x 8; half of 6 eights is 3 eights,
3 times 8 is 24, double 24 is 48.
 Close Fact strategy
 This involves adding one more set to a known fact. For example, think of 6 x 8 as 6 eights. Five
eights is close and results in 40. Six eights is one more eight, or 48.
 Using 5 x 8 to figure out 6 x 8, the language “6 groups of eight” or “6 eights” can help students
remember to add 8 more not 6 more.
 The Close Fact strategy can be used with any multiplication fact. It reinforces students’ number
sense and relationships between numbers.
Basic Multiplication and Division Fact Strategies, page 2 of 3, adapted from Van de Walle, J. A., Lovin, L. H, Karp, K. S, &
Bay-Williams, J. M. (2014). Teaching Student-Centered Mathematics, Volume II. Boston; Pearson.
BASIC Division Fact Strategies
Reasoning
 Mastery of basic division facts is dependent on the inverse relationship of multiplication and
division. For example, to solve 48 ÷ 6, we might naturally ask ourselves, “Six times what is 48?”
The reasoning strategy is to (1) think multiplication, and then (2) apply a known multiplication
fact.
 Near facts: 60 ÷ 8; mentally review a short sequence of multiplication facts comparing each
product to 60: 6 x 8 = 48 (too low), 7 x 8 = 56 (close), 8 x 8 = 64 (too high). It must be 7, so that
is 56 with 4 left over.
NOTE: Division with remainders if much more prevalent in the real world than basic division facts
that have no remainders. Students should be able to solve these near fact problems with
reasonable speed.
Basic Multiplication and Division Fact Strategies, page 3 of 3, adapted from Van de Walle, J. A., Lovin, L. H, Karp, K. S, &
Bay-Williams, J. M. (2014). Teaching Student-Centered Mathematics, Volume II. Boston; Pearson.
Four-Corners-and-a-Rhombus Math Graphic Organizer
What do you already know?
Brainstorm ways to solve this problem.
What do
you need to
find out?
Try two ways to solve the problem here.
List words and phrases you need to include in
your communication write up.
Cognitive Complexity of Mathematics Items
Low Complexity
This category relies heavily on the recall and recognition of previously learned concepts and principles. Items typically
specify what the student is to do, which is often to carry out come procedure that can be performed mechanically. It is not
left to the student to come up with an original method or solution. The list below illustrates some, but not all, of the
demands that items in the low complexity category might make:
• Recall or recognize a fact, term, or property.
• Identify appropriate units or tools for common measurements.
• Compute a sum, difference, product, or quotient.
• Recognize or construct an equivalent representation.
• Perform a specified operation or procedure.
• Evaluate a variable expression, given specific values for the variables.
• Solve a one-step problem.
• Retrieve information from a graph, table, or figure.
• Perform a single-unit conversion.
Moderate Complexity
Items in the moderate complexity category involve more flexibility of thinking and choice among alternatives than do
those in the low complexity category. They require a response that goes beyond the habitual, is not specified, and
ordinarily has more than a single step. The student is expected to decide what to do, using informal methods of reasoning
and problem solving strategies, and to bring together skill and knowledge from various domains. The list below illustrates
some, but not all, of the demands that items of moderate complexity might make.
• Solve a problem requiring multiple operations.
• Solve a problem involving spatial visualization and/or reasoning.
• Retrieve information from a graph, table, or figure and use it to solve a problem.
• Compare figures or statements.
• Determine a reasonable estimate.
• Extend an algebraic or geometric pattern.
• Provide a justification for steps in a solution process.
• Formulate a routine problem, given data and conditions.
• Represent a situation mathematically in more than one way.
• Select and/or use different representations, depending on situation and purpose.
High Complexity
High complexity items make heavy demands on student thinking. Students must engage in more abstract reasoning,
planning, analysis, judgment, and creative thought. The item requires that the student think in an abstract and
sophisticated way. The list below illustrates some, but not all, of the demands that items in the high complexity category
might make:
• Perform a procedure having multiple steps and multiple decision points.
• Describe how different representations can be used for different purposes.
• Solve a non-routine problem (as determined by grade-level appropriateness).
• Analyze similarities and differences between procedures and concepts.
• Generalize an algebraic or geometric pattern.
• Formulate an original problem, given a situation.
• Solve a problem in more than one way.
• Explain and justify a solution to a problem.
• Describe, compare, and contrast solution methods.
• Formulate a mathematical model for a complex situation.
• Analyze or produce a deductive argument.
• Provide a mathematical justification.
NOTE: The complexity of an item is generally NOT dependent on the multiple-choice distractors. The options may
affect the difficulty of the item, not the complexity of the item.