Math Framework Standard for Mathematical Practice #3
Kindergarten
Construct viable arguments
and critique the reasoning of
others
Grade 1
Construct viable arguments
and critique the reasoning of
others.
Grade 2
Construct viable arguments
and critique the reasoning of
others.
Grade 3
Construct viable arguments
and critique the reasoning of
others.
Grade 4
Construct viable arguments
and critique the reasoning of
others.
Younger students construct arguments using actions and concrete
materials, such as objects, pictures, and drawings. They 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. They begin to develop the ability to reason
and analyze situations as they consider questions such as, “Are you
sure...?”, “Do you think that would happen all the time...?”, and “I
wonder why...?”
First graders construct arguments using concrete referents, such as
objects, pictures, drawings, and actions. They practice mathematical
communication skills as they participate in mathematical discussions
involving questions like “How did you get that?” or “Explain your
thinking,” and “Why is that true?” They explain their own thinking and
listen to the explanations of others. For example, “There are 9 books
on the shelf. If you put some more books on the shelf and there are
now 15 books on the shelf, how many books did you put on the shelf?”
Students might use a variety of strategies to solve the task and then
share and discuss their problem solving strategies with their
classmates.
Second graders may construct arguments using concrete referents,
such as objects, pictures, math drawings, and actions. They practice
their mathematical communication skills as they participate in
mathematical discussions involving questions like “How did you get
that?” “Explain your thinking,” and “Why is that true?” They not only
explain their own thinking, but also listen to others’ explanations. They
decide if the explanations make sense and ask appropriate questions.
Students critique the strategies and reasoning of their classmates. For
example, to solve 74 – 18, students might use a variety of strategies
and discuss and critique each other’s
Students may construct arguments using concrete referents, such as
objects, pictures, and drawings. They refine their mathematical
communication skills as they participate in mathematical discussions
that the teacher facilities by asking questions such as “How did you get
that?” and “Why is that true?” Students explain their thinking to others
and respond to others’ thinking. For example, after investigating
patterns on the 100s chart, students might explain why the pattern
makes sense.
Students may construct arguments using concrete referents, such as
objects, pictures, drawings, and actions. They practice their
mathematical communication skills as they participate in mathematical
discussions involving questions like “How did you get that?”, “Explain
your thinking,” and “Why is that true?” They not only explain their own
thinking, but listen to others’ explanations and ask questions. Students
explain and defend their answers and solution strategies as they
answer question that require an explanation.
Mathematics Framework Chapters – MP.3
The State Board of Education adopted the Mathematics Framework on November 6, 2013.
Grade 5
Construct viable arguments
and critique the reasoning of
others.
Grade 6
Construct viable arguments
and critique the reasoning of
others.
Grade 7
Construct viable arguments
and critique the reasoning of
others.
Grade 8
Construct viable arguments
and critique the reasoning of
others.
In fifth grade students may construct arguments using visual models,
such as objects and drawings. They explain calculations based upon
models, 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.
Students use various strategies to solve problems and they defend
and justify their work with others. For example, two afterschool clubs
are having pizza parties. The teacher will order 3 pizzas for every 5
students in the math club; and 5 pizzas for every 8 students in the
student council. If a student is in both groups, decide which party
he/she should to attend. How much pizza will each student get at each
party? If a student wants attend the party with the most pizza (if
divided equally between the students at the party), which party should
he/she attend?
Students construct arguments using verbal or written explanations
accompanied by expressions, equations, inequalities, models, and
graphs, tables, and other data displays (e.g., box plots, dot plots,
histograms). 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?” and
“Does that always work?” They explain their thinking to others and
respond to others’ thinking.
Students construct arguments using verbal or written explanations
accompanied by expressions, equations, inequalities, models, and
graphs, and tables. They further refine their mathematical
communication skills through mathematical discussions in which they
critically evaluate their own thinking and the thinking of other students.
For example, as students notice when given geometric conditions
determine a unique triangle, more than one triangle or no triangle
(7.G.2), they have an opportunity to construct viable arguments and
critique the reasoning of others. Students should be encouraged to
answer questions, such as “How did you get that?”, “Why is that true?”
and “Does that always work?”
Students construct arguments using verbal or written explanations
accompanied by expressions, equations, inequalities, models, and
graphs, tables, and other data displays (e.g., box plots, dot plots,
histograms). 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?” and
“Does that always work? ”They explain their thinking to others and
respond to others’ thinking.
Mathematics Framework Chapters – MP.3
The State Board of Education adopted the Mathematics Framework on November 6, 2013.
Algebra 1/ Mathematics 1
Construct viable arguments
Students reason through the solving of equations, recognizing that
and critique the reasoning of
solving an equation is more than simply a matter of rote rules and
others.
steps. They use language such as “if… then...” when explaining their
Students build proofs by
solution methods and provide justification.
induction and proofs by
contradiction. CA 3.1 (for
higher mathematics only).
Geometry
Construct viable arguments
Students construct proofs of geometric theorems. They write coherent
and critique the reasoning of
logical arguments and understand that each step in a proof must follow
others. Students build
from the last, justified with a previously accepted or proven result.
proofs by induction and
proofs by contradiction. CA
3.1 (for higher mathematics
only).
Mathematics 2
Construct viable arguments
Students construct proofs of geometric theorems based on
and critique the reasoning of
congruence criteria of triangles. They understand and explain the
others. Students build
definition of radian measure.
proofs by induction and
proofs by contradiction. CA
3.1 (for higher mathematics
only).
Algebra 2/ Mathematics 3/ Precalculus
Construct viable arguments
Students continue to reason through the solution of an equation and
and critique the reasoning of
justify their reasoning to their peers. Students defend their choice of a
others. Students build
function to model a real world situation.
proofs by induction and
proofs by contradiction. CA
3.1 (for higher mathematics
only).
Statistics and Probability
Construct viable arguments
Students defend their choice of a function to model data. Students pay
and critique the reasoning of
attention to the precise definitions of concepts such as causality and
others. Students build
correlation and learn how to discern between the two, becoming aware
proofs by induction and
of potential abuses of statistics.
proofs by contradiction. CA
3.1 (for higher mathematics
only).
Assessment Chapter
Journals and Learning Logs are tools in which students do mathematical writing that serves to
illuminate their current understandings. For example, a teacher may provide each student with a journal
that is kept in the classroom, which is then used for students to solve an “exit problem” of the day. Or
students may be asked to explain what they learned that given day or what they think the major idea of
the lesson was. Such journals have a variety of uses. Teachers should not feel required to grade
everything in the math journal; in fact, this may diminish its use as students feel they need to write a
“correct response.” Instead, teachers can simply read all or some of their students’ journals to get
feedback on student understanding.
Mathematics Framework Chapters – MP.3
The State Board of Education adopted the Mathematics Framework on November 6, 2013.