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).
1 Make sense of problems and persevere in solving them.
In grade 6, students solve problems involving ratios and rates and discuss how they solved them. Students
solve real world problems through the application of algebraic and geometric concepts. 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.
In grade 6, students represent a wide variety of real world contexts through the use of real numbers and
variables in mathematical expressions, equations, and inequalities. Students contextualize to understand
the meaning of the number or variable as related to the problem and decontextualize to manipulate
symbolic representations by applying properties of operations.
3 Construct viable arguments and critique the reasoning of others.
In grade 6, students construct arguments using verbal or written explanations accompanied by
expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot
plots, histograms, etc.). They further refine their mathematical communication skills through
mathematical discussions in which they critically evaluate their own thinking and the thinking of other
students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always
work?” They explain their thinking to others and respond to others’ thinking.
4 Model with mathematics.
In grade 6, students model problem situations symbolically, graphically, tabularly, and contextually.
Students form expressions, equations, or inequalities from real world contexts and connect symbolic and
graphical representations. Students begin to explore covariance and represent two quantities
simultaneously. Students use number lines to compare numbers and represent inequalities. They use
measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences
about and make comparisons between data sets. Students need many opportunities to connect and explain
the connections between the different representations. They should be able to use all of these
representations as appropriate to a problem context.
5 Use appropriate tools strategically.
Students consider available tools (including estimation and technology) when solving a mathematical
problem and decide when certain tools might be helpful. For instance, students in grade 6 may decide to
represent similar data sets using dot plots with the same scale to visually compare the center and
variability of the data. Additionally, students might use physical objects or applets to construct nets and
calculate the surface area of three-dimensional figures.
6 Attend to precision.
In grade 6, students 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 rates, ratios, geometric figures, data displays, and components of
expressions, equations or inequalities.
7 Look for and make use of structure.
Students routinely seek patterns or structures to model and solve problems. For instance, students
recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties.
Students apply properties to generate equivalent expressions
(i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by
subtraction property of equality), c=6 by division property of equality). Students compose and decompose
two- and three-dimensional figures to solve real world problems involving area and volume.
8 Look for and express regularity in repeated reasoning.
In grade 6, students use repeated reasoning to understand algorithms and make generalizations about
patterns. During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d =
ad/bc and construct other examples and models that confirm their generalization. Students connect place
value and their prior work with operations to understand algorithms to fluently divide multi-digit numbers and perform all operations with multi-digit decimals. Students informally begin to make connections
between covariance, rates, and representations showing the relationships between quantities