Standards for Mathematical Practice

December 1, 2011

NCTM Principles• Representation• Connections• Reasoning• Problem Solving• Communication

Make Sense of Problems and Persevere in Solving Them

• A problem is a situation where you do not know what to do.

• Teachers seldom are able to model problem solving and when they do, it is an accident.

• Problems for students are usually exercises for teachers.

Make Sense of Problems and Persevere in Solving Them

• Teachers identify rich mathematical tasks that will lead to the understanding of the content they are teaching and which they believe will be “problems” for their students.

• Teachers present these tasks to the students.• Teachers use on-going formative assessment to modify

tasks so that all students are engaged and challenged.

Make Sense of Problems and Persevere in Solving Them

• Students should be given rich mathematical tasks presented as a “problem” for them to engage in and make sense of (related to the task).

• Students should make and implement plans to try to accomplish the goal of the task.

• Students should monitor progress and revise and refine their plans based on the intermediate findings.

Make Sense of Problems and Persevere in Solving Them

• Teachers should encourage students to compare and evaluate results and processes.

• Teachers should provide students with information about mathematical vocabulary, notations, and conventions to enhance their ability to communicate effectively with others.

Make Sense of Problems and Persevere in Solving Them

• Problem-solving often results in the creation of factual or procedural knowledge that can be used to accomplish future tasks, which are no longer problems but merely exercises.

Reason Abstractly and Quantitatively• Use less than, greater than, and equal to as you

compare the following:6 812 47 7

Reason Abstractly and Quantitatively• Use less than, greater than, and equal to as you

compare the following:6 pounds 8 ounces12 nickels 4 dollars7 meters 7 centimeters

Reason Abstractly and Quantitatively• What is the sum of 4 and 3?• What is the sum of 4 nickels and 3 dimes?• What is the sum of 4 third-cups and 3 half-cups?• What is 4x + 3y?

Reason Abstractly and Quantitatively• Compare the following rates:

2/3 meters per second

1.5 seconds per meter

Reason Abstractly and Quantitatively• Why is the slope of a line change in y over change in

x?

Reason Abstractly and Quantitatively

• The school has 782 students that need to be transported by bus. Each bus can transport 48 students. How many buses are needed?

Reason Abstractly and Quantitatively

• Karen has 12 yellow pencils and 8 red pencils. Does she have more yellow pencils or red pencils? How many more?

Construct a Viable Argument and Critique the Reasoning of Others

• Often more is learned by being wrong than by being right.

• Competition for grades and awards often make it unwise to risk being wrong at school.

Construct a Viable Argument and Critique the Reasoning of Others

• What is 7 times 8?I don’t know, but I do know 5 times 8 is 40. So 6 times 8 would be 40 plus 8, 48. To get 7 times 8, I have to add 8 more to 48. I know 48 is 2 away from 50, if you take 2 from 8 that leaves 6, so 7 times 8 must be 56.

Construct a Viable Argument and Critique the Reasoning of Others

• I asked a class of third graders to find 203-78?

The class got 35, 125, 135, and 205. The most popular answer was 205.

• Which answers are unreasonable? Why?

Construct a Viable Argument and Critique the Reasoning of Others.

• Students measured the perimeter of a table and got the data below (all measures were in cm):

168, 209, 241.5 271, 400, 432, 436, 438, 440, 446, 450, 450, 450, 458, 460, 460, 460, 462, 464, 468, 470, 480, 494, 530

• Why is the range of this data so large?

Construct a Viable Argument and Critique the Reasoning of Others

• If two line segments intersect and are perpendicular, what kind of quadrilateral will be formed by connecting the end points of the two intersecting segments?

Model with Mathematics• Everyday Life• Society• Workplace

Model with Mathematics• What time will I get home today?• How much money do I need to budget to buy

Christmas gifts?• Can I get a table top that is 7.5 feet wide through a

door that is 3 feet by 7 feet?• What long distance calling plan should I buy?

Model with Mathematics• Should I cut a cedar tree?• How do I calibrate a crop sprayer?• How much should I charge for a candy bar?• How do surveyors calculate some distances?• Where is the center of a room that is an isosceles

trapezoid?

Model with Mathematics• If two candidates have very different plans for

collecting taxes, for which one should I cast my vote?

• Should children have to be immunized to attend school?

• What does it mean if a child or a school is at the 48th percentile?

Model with Mathematics• Write a word problem that can be solved by

simplifying the following expression.

Attend to Precision

• Ten refers to exactly ten and no more or less.• -teen means one ten and some toward a second

group of ten.• -ty means more than one group of ten.

Attend to Precision• What is a circle?• What is an angle?• What do we mean when we say a segment is 15 cm

long?

Attend to Precision• 3 red tiles + 4 blue tiles = 4 blue tiles + 3 red tiles

The above statement is an example of the commutative property for addition.

• 3 red tiles + 4 blue tiles = 4 red tiles + 3 blue tilesIf the statement above is true, it is because of the transitive property of equality not the commutative property for addition.

Attend to PrecisionWhat property do you see?• 2 packages x 6 cookies/package =

6 packages x 2 cookies/package• 2 packages x 6 cookies/package =

6 cookies/package x 2 packages• 2 feet x 6 feet = 6 feet x 2 feet

Attend to PrecisionWhat does “=“ mean?

• 1/5 gallon = 2/10 gallon• 1 gallon/5 miles = 2 gallons/10 miles• 1 woman/ 5 people = 2 women/10 people• 2x + 3 = 15• 2(x+3) = 2x + 6

Attend to Precision

• Why is 2(x + 5) = 2x + 10?

• Why is 2x + 5x = 7x?

Attend to Precision

• What do we mean in mathematics class by “cancel”? Is cancel a mathematical term?

• What understanding could we promote by avoiding the use of pronouns without antecedents?

Attend to Precision

What are:• Solutions?• X-intercepts?• Zeros?• Roots?

Attend to Precision• What connections do we need to make with ELA that

will promote vocabulary development in mathematics?

• Many of our terms are compound words. When and how do students best learn about compound words in general? Are we using the same strategies in mathematics vocabulary development that are used in ELA and other disciplines?

Attend to PrecisionTriangle indicates 3 angles.Quadrilateral indicates 4 sides.• Why not trilateral or quadrangle?

Pentagon is 5 sides. Hexagon is 6 sides.• Why not trigon and tetragon?

Attend to Precision• Are there ways we can activate students’ prior

knowledge by using terminology and examples they would recognize from earlier grades?

• What are the appropriate mathematics vocabulary words for each grade?

• Are all teachers using the same mathematical symbols and conventions?

Select Appropriate Tools and Use Them Strategically.

• With and without manipulatives.• With and without technology.

Select Appropriate Tools and Use Them Strategically

• Technology is not evil and has not caused students to lack automaticity with facts or fail to develop procedural fluency.

Select Appropriate Tools and Use Them Strategically

• Computer software – graphing utilities, spread sheets, dynamic geometry packages, fluency development programs, etc.

• Measuring tools – rulers, meter sticks, measuring tapes, scales, balances, graduated cylinders, protractors, clocks, thermometers, etc.

• Constructions tools- compass and straightedge• Other – manipulatives, scissors, gridded paper, etc.

Select Appropriate Tools and Use Them Strategically

• Teachers will not select the tool and direct the students on the strategy for using the tool in each problem or task.

• Students engaged in a problem or task will realize a need for a tool, think about the purpose for using the tool, find the tool or go to the teacher for help with finding a tool that allows them to collect the information they need or create the product they need to accomplish their goal.

Look For and Make Use of Structure• Properties of equality• Field Properties• Properties of inequality• Number naming conventions have structure• Classification of numbers, 2-D figures, and 3-D objects

imposes structure• Logic has structure• Algebraic notation imposes structure

Look For and Make Use of Structure• Structure does not force everyone to think the same way.

If structure is understood, then students can be flexible in their thoughts and make decisions based on context.

• The structure of the base ten number system and the distributive property allow us to generalize an algorithm for multiplying multi-digit numbers. In some cases, we can modify that algorithm to do the work more efficiently.

Look For and Make Use of Structure• Consider 8 x 24:

8(20 + 4) = 160 + 32 = 1928(25 – 1) = 200 – 8 = 192

Look For and Make Use of Structure• The Distance Formula, the Pythagorean Theorem, and

the one of the trigonometric identities are not three different things they are the same thing.

• Think of the difference in the x-coordinates as “a”, the difference in the y-coordinates as “b”, and the distance itself as “c”.

• Think of sin(θ) as “a”, cos(θ) as “b”, and since the unit circle is being used we know “c” is 1 unit.

Look For and Make Use of Regularity in Repeated Reasoning

• Perhaps the writers of the CCSS for mathematics intentionally avoided saying “look for patterns” because educators have (over time) developed some very limited ideas about what patterning is and why it might be important.

• Maybe because patterning was being done with very young children, most curriculum material on patterning is somewhat elementary and isn’t transferring to more rigorous mathematics situations.

Look For and Make Use of Regularity in Repeated Reasoning

Activity: Fat Is

CCSS Standards for Mathematical Practice

Http://www.youtube.com/watch?v=m1rxkW8ucAI