Standards for Mathematics
Standards for Mathematical Practice
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Apply across all grade levels
Describe habits of mind of a mathematically proficient student
Standards for Mathematical Content
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K-8 standards presented by grade level
High school standards presented by topic (Number & Quantity,
Algebra, Functions, Modeling, Geometry, Statistics & Probability)
Organized into domains that progress over several grades
Two to four “critical areas” at each grade level
Mathematical Practices
1. Make sense of problems
5. Use appropriate tools
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique
the reasoning of others.
4. Model with
mathematics.
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
Design and Organization
Critical areas at each grade level
Design and Organization
Grade Level Overviews
Domain, Cluster and Standards
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve
problems.
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.”
2. Understand the concept of a unit rate a/b associated with a ratio a:b
with 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 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15
hamburgers, which is a rate of $5 per hamburger.”
Reading the Standards
Standards define what students should understand
and be able to do.
• Specific advice is often given about instructional approaches.
• The actual language can be dense, and probably requires
“unpacking.” K-6 “I Can” statements are available on our
website.

I understand what it means to divide by a fraction, and I
can solve word problems that involve division with
fractions.
 I can create equivalent fractions by scaling up, and I can show this process
using visual fraction models.
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4
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2
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2
8
High School
The high school standards specify the mathematics that all
students should study in order to be college and career ready.
Additional mathematics that students should learn in order to
take advanced courses such as calculus, advanced statistics, or
discrete mathematics is indicated by (+), as in this example:
(+) Represent complex numbers on the complex plane in
rectangular and polar form (including real and imaginary
numbers).
All standards without a (+) symbol should be in the common
mathematics curriculum for all college and career ready
students. Standards with a (+) symbol may also appear in
courses intended for all students.
Modeling Standards
Modeling is best interpreted not as a collection of isolated
topics but rather in relation to other standards. Making
mathematical models is a Standard for Mathematical
Practice, and specific modeling standards appear
throughout the high school standards indicated by a star
symbol (★).
Course topics
• Appendix A shows which standards fit into Algebra I,
Algebra II, and Geometry.
Major Shifts in Mathematics
Working like a mathematician:
• Problem solving
• Reasoning (proportional reasoning, geometric
reasoning, etc.
• Creating viable arguments
• Modeling with mathematics
Reasoning and Sense-making
In square ABCE shown below, D is the midpoint of
CE. Which of the following is the ratio of the area of
triangle ADE to the area of triangle ADB?
F. 1:1
G. 1:2
H. 1:3
J. 1:4
K. 1:8
Videos of classrooms
From the Inside Mathematics web resource
http://InsideMathematics.org
Properties of Quadrilaterals, Grades 9-10,
Tuesday Group Work, Pt. A
Major Shifts in Mathematics
Focus and coherence
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Focus on key topics at each grade level.
Coherent progressions across grade levels.
Balance of concepts, skills and problem-solving
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Content standards require a blend of deep understanding,
procedural fluency and application.
Mathematical practices
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Foster reasoning and sense-making in mathematics.
Focus: Critical Areas
Mathematics Grade 3
In Grade 3, instructional time should focus on four critical areas:
(1) developing understanding of multiplication and division and
strategies for multiplication and division within 100;
(2) developing understanding of fractions, especially unit fractions
(fractions with numerator 1);
(3) developing understanding of the structure of rectangular arrays
and of area; and
(4) describing and analyzing two-dimensional shapes.
Coherence: Learning progressions
2nd grade
Students connect arrays and the area of
rectangles with repeated addition
3rd grade
Students solve word problems involving equal
groups, arrays and area; relate division to
multiplication; use strategies for multiplying and
dividing; and become fluent with one-digit
multiplication and division.
4th grade
Students solve word problems involving
multiplication as a comparison; solve multi-step
multiplication and division word problems;
continue to use equations to represent situations;
and multiply multi-digit numbers (4x1 and 2x2)
Coherence: Learning Progressions
Fractions, Grades 3-6
3rd grade: Develop an understanding of fractions as numbers.
4th grade: Extend understanding of fraction equivalence and ordering.
5th grade: Use equivalent fractions as a strategy to add and subtract
fractions.
5th grade: Apply and extend previous understandings of multiplication
and division to multiply and divide fractions.
6th grade: Apply and extend previous understandings of multiplication
and division to divide fractions by fractions.
On-line Learning Trajectories
http://turnonccmath.net/
Balance of concept, skill & application
Multiplication, single and multi-digit
Fluently multiply and divide within 100, using strategies… By the end
of Grade 3, know from memory all products of one-digit numbers.3.OA.7
Solve two-step word problems using the four operations. 3.OA.8
Multiply a whole number of up to four digits by a one-digit whole
number, and multiply two two-digit numbers, using strategies…
Illustrate and explain the calculation by using equations, rectangular
arrays, and/or area models. 4.NBT.5
Solve multistep word problems posed with whole numbers, including
problems in which remainders must be interpreted. 4.OA.3
Fluently multiply multi-digit whole numbers using the standard
algorithm. 5.NBT.5
Mathematical Practices
• Make sense of problems and
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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.
Marty made two types of cookies.
He used 2/3 cup of sugar for one
recipe and 1/4 cup of sugar for the
other. He only has 1 cup of sugar.
Is that enough? Explain your
reasoning.
Some people might say that
A) Why might they say this?
B) Do you agree?
C) How would you solve this
problem?