9. Georgia Department of Education
longstanding importance in mathematics education. The first of these are the NCTM process standards of problem
solving, reasoning and proof, communication, representation, and connections. The second are the strands of
mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning,
strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and
relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately),
and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled
with a belief in diligence and one’s own efficacy).
Students are expected to:
1. Make sense of problems and persevere in solving them.
In Kindergarten, students begin to build the understanding that doing mathematics involves solving problems and
discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to
solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems.
They may check their thinking by asking themselves, “Does this make sense?” or they may try another strategy.
2. Reason abstractly and quantitatively.
Younger students begin to recognize that a number represents a specific quantity. Then, they connect the quantity
to written symbols. Quantitative reasoning entails creating a representation of a problem while attending to the
meanings of the quantities.
3. Construct viable arguments and critique the reasoning of others.
Younger students construct arguments using concrete referents, such as objects, pictures, drawings, and actions.
They also begin to develop 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.
In early grades, students experiment with representing problem situations in multiple ways including numbers,
words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating
equations, etc. Students need opportunities to connect the different representations and explain the connections.
They should be able to use all of these representations as needed.
5. Use appropriate tools strategically.
Younger students begin to consider the available tools (including estimation) when solving a mathematical
problem and decide when certain tools might be helpful. For instance, kindergarteners may decide that it might be
advantageous to use linking cubes to represent two quantities and then compare the two representations side‐by‐
side.
6. Attend to precision.
As kindergarteners begin to develop their mathematical communication skills, they try to use clear and precise
language in their discussions with others and in their own reasoning.
7. Look for and make use of structure.
Younger students begin to discern a pattern or structure. For instance, students recognize the pattern that exists in
the teen numbers; every teen number is written with a 1 (representing one ten) and ends with the digit that is first
stated. They also recognize that 3 + 2 = 5 and 2 + 3 = 5.
8. Look for and express regularity in repeated reasoning.
In the early grades, students notice repetitive actions in counting and computation, etc. For example, they may
notice that the next number in a counting sequence is one more. When counting by tens, the next number in the
sequence is “ten more” (or one more group of ten). In addition, students continually check their work by asking
themselves, “Does this make sense?”
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 9 of 51
All Rights Reserved
10. Georgia Department of Education
Counting and Cardinality K.CC
Know number names and the count sequence.
MCCK.CC.1 Count to 100 by ones and by tens.
MCCK.CC.2 Count forward beginning from a given number within the known sequence (instead of
having to begin at 1).
MCCK.CC.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0‐20
(with 0 representing a count of no objects).
Count to tell the number of objects.
MCCK.CC.4 Understand the relationship between numbers and quantities; connect counting to
cardinality.
a. When counting objects, say the number names in the standard order, pairing each object with
one and only one number name and each number name with one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number
of objects is the same regardless of their arrangement or the order in which they were
counted.
c. Understand that each successive number name refers to a quantity that is one larger.
MCCK.CC.5 Count to answer “how many?” questions about as many as 20 things arranged in a line, a
rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number
from 1–20, count out that many objects.
Compare numbers.
MCCK.CC.6 Identify whether the number of objects in one group is greater than, less than, or equal to
the number of objects in another group, e.g., by using matching and counting strategies.1
MCCK.CC.7 Compare two numbers between 1 and 10 presented as written numerals.
Operations and Algebraic Thinking K.OA
Understand addition as putting together and adding to, and understand subtraction as taking apart
and taking from.
MCCK.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings2,
sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
MCCK.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by
using objects or drawings to represent the problem.
1
Include groups with up to ten objects.
2
Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are
mentioned in the standards.)
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 10 of 51
All Rights Reserved
13. Georgia Department of Education
Mathematics | Grade 1
In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of
addition, subtraction, and strategies for addition and subtraction within 20; (2) developing
understanding of whole number relationships and place value, including grouping in tens and ones; (3)
developing understanding of linear measurement and measuring lengths as iterating length units; and
(4) reasoning about attributes of, and composing and decomposing geometric shapes.
(1) Students develop strategies for adding and subtracting whole numbers based on their prior
work with small numbers. They use a variety of models, including discrete objects and length‐
based models (e.g., cubes connected to form lengths), to model add‐to, take‐from, put‐
together, take‐apart, and compare situations to develop meaning for the operations of addition
and subtraction, and to develop strategies to solve arithmetic problems with these operations.
Students understand connections between counting and addition and subtraction (e.g., adding
two is the same as counting on two). They use properties of addition to add whole numbers and
to create and use increasingly sophisticated strategies based on these properties (e.g., “making
tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution
strategies, children build their understanding of the relationship between addition and
subtraction.
(2) Students develop, discuss, and use efficient, accurate, and generalizable methods to add within
100 and subtract multiples of 10. They compare whole numbers (at least to 100) to develop
understanding of and solve problems involving their relative sizes. They think of whole numbers
between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as
composed of a ten and some ones). Through activities that build number sense, they
understand the order of the counting numbers and their relative magnitudes.
(3) Students develop an understanding of the meaning and processes of measurement, including
underlying concepts such as iterating (the mental activity of building up the length of an object
with equal‐sized units) and the transitivity principle for indirect measurement.4
(4) Students compose and decompose plane or solid figures (e.g., put two triangles together to
make a quadrilateral) and build understanding of part‐whole relationships as well as the
properties of the original and composite shapes. As they combine shapes, they recognize them
from different perspectives and orientations, describe their geometric attributes, and determine
how they are alike and different, to develop the background for measurement and for initial
understandings of properties such as congruence and symmetry.
Professional Learning: 2011‐2012
Read, discuss, and understand the breadth and depth of the MCC Note areas in which what has
been taught in the past will remain the same, which concepts are taught more deeply, and which
concepts have been moved to another grade.
4
Students should apply the principle of transitivity of measurement to make indirect comparisons, but they need not use this
technical term.
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 13 of 51
All Rights Reserved
14. Georgia Department of Education
See http://public.doe.k12.ga.us/ci_services.aspx?PageReq=CIServMath for resources to support
discussion and understanding of CCGPS Mathematical Content Standards and Mathematical Practice
Standards.
Implementation/Transition: 2012‐2013
1st ‐ Teach CCGPS. Also, teach the additional CCGPS which moves from 1st GPS to K CCGPS. This
standard has been embedded in this document with CCGPS Geometry to prevent gaps in learning:
MCCK. Geometry Identify and describe shapes (squares, circles, triangles, rectangles, hexagons,
cubes, cones, cylinders, and spheres.)
MCCK.G.1 Describe objects in the environment using names of shapes, and describe the relative
positions of these objects using terms such as above, below, beside, in front of, behind, and next
to. (Not listed are the terms left and right, since CC suggests terms rather than providing a checklist
of terms. Please include the teaching of left and right as descriptors, as has been done under GPS.)
Note the following: Use money and time, including calendar time, as models and contexts for counting.
This will ensure that students are familiar with these contexts when they appear in the MCC Ordinal
numbers should also be incorporated into daily routines, for example, “Please line up your color tiles so
that red tiles make up the first group on the left, green tiles make the second group, and blue tiles make
the third group.”
Standards for Mathematical Practice
Mathematical Practices are listed with each grade’s mathematical content standards to reflect the need to connect
the mathematical practices to mathematical content in instruction.
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels
should seek to develop in their students. These practices rest on important “processes and proficiencies” with
longstanding importance in mathematics education. The first of these are the NCTM process standards of problem
solving, reasoning and proof, communication, representation, and connections. The second are the strands of
mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning,
strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and
relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately),
and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled
with a belief in diligence and one’s own efficacy).
Students are expected to:
1. Make sense of problems and persevere in solving them.
In first grade, students realize that doing mathematics involves solving problems and discussing how they solved
them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students
may use concrete objects or pictures to help them conceptualize and solve problems. They may check their
thinking by asking themselves, “Does this make sense?” They are willing to try other approaches.
2. Reason abstractly and quantitatively.
Younger students recognize that a number represents a specific quantity. They connect the quantity to written
symbols. Quantitative reasoning entails creating a representation of a problem while attending to the meanings of
the quantities.
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 14 of 51
All Rights Reserved
16. Georgia Department of Education
Understand and apply properties of operations and the relationship between addition and
subtraction.
MCC1.OA.3 Apply properties of operations as strategies to add and subtract.6
Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To
add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12.
(Associative property of addition.)
MCC1.OA.4 Understand subtraction as an unknown‐addend problem. For example, subtract 10 – 8 by
finding the number that makes 10 when added to 8.
• Problems should be within 20.
Add and subtract within 20
MCC1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
MCC1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a
number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition
and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but
easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Work with addition and subtraction equations
MCC1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition
and subtraction are true or false. For example, which of the following equations are true and which
are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
• The equal sign describes a special relationship between two quantities. In the case of a true
equation, the quantities are the same.
MCC1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating to
three whole numbers. For example, determine the unknown number that makes the equation true in
each of the equations 8 + ? = 11, 5 = □ – 3, 6 + 6 = ∆.
Number and Operations in Base Ten 1.NBT
Extend the counting sequence
MCC1.NBT.1 Count to 120, starting at any number less than 120. In this range, read and write
numerals and represent a number of objects with a written numeral.
• Correctly count and represent the number of objects in a set using numerals.
6
Students need not use formal terms for these properties. Problems should be within 20.
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 16 of 51
All Rights Reserved
17. Georgia Department of Education
Understand place value
MCC1.NBT.2 Understand that the two digits of a two‐digit number represent amounts of tens and
ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven,
eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven,
eight, or nine tens (and 0 ones).
MCC1.NBT.3 Compare two two‐digit numbers based on meanings of the tens and ones digits,
recording the results of comparisons with the symbols >, =, and <.
• Use tools such as a sequential number line, number chart, manipulatives, pictorial
representation, etc.
• No use of ≠.
Use place value understanding and properties of operations to add and subtract.
MCC1.NBT.4 Add within 100, including adding a two‐digit number and a one‐digit number, and adding
a two‐digit number and a multiple of 10, 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. Understand that in adding
two‐digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose
a ten.
• See Glossary, Computation strategy.
• Use real‐life situations and/or story problems to provide context.
• See Glossary, Addition and subtraction within 5, 10, 20, 100, or 1000.
MCC1.NBT.5 Given a two‐digit number, mentally find 10 more or 10 less than the number, without
having to count; explain the reasoning used.
MCC1.NBT.6 Subtract multiples of 10 in the range 10‐90 from multiples of 10 in the range 10‐90
(positive or zero differences), 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.
• Use real‐life situations and/or story problems to provide context.
Measurement and Data 1.MD
Measure lengths indirectly and by iterating length units
MCC1.MD.1 Order three objects by length; compare the lengths of two objects indirectly by using a
third object.
MCC1.MD.2 Express the length of an object as a whole number of length units, by laying multiple
copies of a shorter object (the length unit) end to end; understand that the length measurement of an
object is the number of same‐size length units that span it with no gaps or overlaps. Limit to contexts
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 17 of 51
All Rights Reserved
18. Georgia Department of Education
where the object being measured is spanned by a whole number of length units with no gaps or
overlaps.
Tell and write time.
MCC1.MD.3 Tell and write time in hours and half‐hours using analog and digital clocks.
• It is important for students to focus on understanding the hour hand. The use of a clock with
only an hour hand is helpful.
• Once students understand the hour hand, develop understanding of the minute hand and how it
relates to the hour hand.
• Conversation should include telling time with descriptions such as about, halfway, almost, and
the connection between an hour‐handed clock and a single number line to 12.
Represent and interpret data.
MCC1.MD.4 Organize, represent, and interpret data with up to three categories; ask and answer
questions about the total number of data points, how many in each category, and how many more or
less are in one category than in another.
Geometry 1.G
Reason with shapes and their attributes.
Transition Standard to be taught 2012‐2013
MCCK.G.1 Describe objects in the environment using names of shapes, and describe the relative
positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
(include left and right of)
MCC1.G.1 Distinguish between defining attributes (e.g., triangles are closed and three‐sided) versus
non‐defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess
defining attributes.
• Manipulatives may be used to build shapes.
MCC1.G.2 Compose two‐dimensional shapes (rectangles, squares, trapezoids, triangles, half‐circles,
and quarter‐circles) or three‐dimensional shapes (cubes, right rectangular prisms, right circular cones,
and right circular cylinders) to create a composite shape, and compose new shapes from the
composite shape.7
MCC1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using
the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
Describe the whole as two of, or four of the shares. Understand for these examples that decomposing
into more equal shares creates smaller shares.
7
Students do not need to learn formal names such as “right rectangular prism.”
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 18 of 51
All Rights Reserved
23. Georgia Department of Education
MCC2.NBT.2 Count within 1000; skip‐count by 5s, 10s, and 100s.
• Skip counting forward and backwards within 1000. Avoid sing‐song memorization. Skip counting
is a useful strategy for multiplication, but only if students understand what is happening when
they skip count. Develop understanding by using a number line or number chart when skip
counting.
MCC2.NBT.3 Read and write numbers to 1000 using base‐ten numerals, number names, and expanded
form.
• For example: 473 represented as 400 + 70 + 3, and units, 47 tens + 3, or 470 + 3. (in any
arrangement)
MCC2.NBT.4 Compare two three‐digit numbers based on meanings of the hundreds, tens, and ones
digits, using >, =, and < symbols to record the results of comparisons.
• No use of ≠.
Use place value understanding and properties of operations to add and subtract.
MCC2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction.
MCC2.NBT.6 Add up to four two‐digit numbers using strategies based on place value and properties of
operations.
MCC2.NBT.7 Add and subtract within 1000, 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. Understand that in adding or subtracting three‐digit numbers,
one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is
necessary to compose or decompose tens or hundreds.
MCC2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100
from a given number 100–900.
MCC2.NBT.9 Explain why addition and subtraction strategies work, using place value and the
properties of operations.10
Measurement and Data 2.MD
Measure and estimate lengths in standard units.
MCC2.MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers,
yardsticks, meter sticks, and measuring tapes.
10
Explanations may be supported by drawings or objects.
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 23 of 51
All Rights Reserved
24. Georgia Department of Education
MCC2.MD.2 Measure the length of an object twice, using length units of different lengths for the two
measurements;
• Students should develop and use personal benchmarks for frequently used units of measure.
(i.e. A centimeter is about the width of a pinky fingernail or pencil eraser.)
MCC2.MD.3 Estimate lengths using units of inches, feet, centimeters, and meters.
MCC2.MD.4 Measure to determine how much longer one object is than another, expressing the
length difference in terms of a standard length unit.
Relate addition and subtraction to length.
MCC2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that
are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a
symbol for the unknown number to represent the problem.
MCC2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally
spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole‐number sums and
differences within 100 on a number line diagram.
MCC2.MD.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m.
and p.m.
• Develop understanding by representing the relationship between the minute hand and the hour
hand using a double number line.
MCC2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $
and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you
have?
• Relate to whole‐number place value and base‐ten understandings. For example, 23¢ = 2 dimes
and 3 pennies.
• Understand the relationship between quantity and value. For example 1 dime = 10¢. Help
students to understand that the relationship between coin size and value is inconsistent.
• Limit problems to the use of just dollar and cents symbols. There should be no decimal notation
for money at this point.
Represent and interpret data
MCC2.MD.9 Generate measurement data by measuring lengths of several objects to the nearest
whole unit, or by making repeated measurements of the same object. Show the measurements by
making a line plot, where the horizontal scale is marked off in whole‐number units.
• See Glossary, Line plot.
MCC2.MD.10 Draw a picture graph and a bar graph (with single‐unit scale) to represent a data set
with up to four categories. Solve simple put‐together, take‐apart, and compare problems11 using
information presented in a bar graph.
11
See Glossary, Table 1
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 24 of 51
All Rights Reserved
25. Georgia Department of Education
Geometry 2.G
Reason with shapes and their attributes.
MCC2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or
a given number of equal faces.12 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
MCC2.G.2 Partition a rectangle into rows and columns of same‐size squares and count to find the total
number of them.
• Grid paper or color tiles may be used.
MCC2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares
using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three
thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape
Second Grade Glossary
Addition and subtraction within 5, 10, 20, 100, or 1000. Addition or subtraction of two whole numbers
with whole number answers, and with sum or minuend in the range 0‐5, 0‐10, 0‐20, or 0‐100,
respectively. Example: 8 + 2 = 10 is an addition within 10, 14 – 5 = 9 is a subtraction within 20, and 55 –
18 = 37 is a subtraction within 100.
Line plot. A method of visually displaying a distribution of data values where each data value is shown as
a dot or mark above a number line. Also known as a dot plot.
Strategies for addition. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14);
decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between
addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or
known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
12
Sizes are compared directly or visually, not compared by measuring.
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
August 19, 2011 • Page 25 of 51
All Rights Reserved