Defining Rigorin regards to the CCSS-MKaren McPhersonBuncombe County Schools

Toward greater focus and coherence

For over a decade, research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards must address the problem of a curriculum that is “a mile wide and an inch deep.” These Standards are a substantial answer to that challenge.

Shift 1: Focus

Focus where the standards focus Fewer things at any given time Giving students time they need to master

those things so they have a foundation from which to build

Greater mastery of fewer things

Shift 2: Coherence

Following the progressions of ideas and skills as they unfold across the grades

Mathematics comes together and becomes unified

Things that seem different merges into a central idea

Rigor

Conceptual UnderstandingProcedural S

kill & Fluency

Application

Conceptual Understanding

Deep conceptual understanding of core content at each grade is critical for student success in subsequent years.

Students with conceptual understanding know more than isolated facts and methods - they understand why a mathematical idea is important and the contexts which it is useful.

Teachers take time to understand the Standards for Mathematical Practice that describe the student expertise needed to develop a deep conceptual understanding of mathematics.

(NRC, 2001, p. 118; CCSSM, 2010, p. 4, 6-8)

Application

Identify opportunities for students to apply math concepts in “real world” situations.

The process of modeling, that includes choosing and using appropriate mathematics and statistics to analyze and understand situations, is key in improving decisions as well as linking classroom mathematics and statistics to everyday life, work, and decision-making.

Students are expected to use math and choose the appropriate mathematical models even when they are not prompted to do so.

(NRC, 2001, p. 124; CCSSM, 2010, p. 72-73; NMAP, 2008, p.49-50)

Procedural Fluency

Students are efficient and accurate in performing foundational computational procedures without always having to refer to tables and other aids.

Teachers should help students to study algorithms as “general procedures” so they can gain insights to the structure of mathematics (e.g. organization, patterns, predictability).

Students should also be able to apply a variety of appropriate procedures flexibly as they solve problems.

Helping students master key procedures will help them understand and manipulate more complex concepts in later grades.

(NRC, 2001, p. 121; CCSSM, 2010, p. 6)

The Language of the Standards Consider key verbs to determine if a

standard emphasizes conceptual understanding, procedural skill and fluency, or applicational use in problem solving.

Conceptual Understanding

Procedural Skill & Fluency

Application

A Look at the Language

F-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of  f  is the graph of the equation y = f(x).

A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions

F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational

A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Conceptual Understanding

Procedural Skill & Fluency

Application

The importance of the clusters Explain volume formulas and use them to

solve problems GMD.A.1 Give an informal argument for the

formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

GMD.A.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

Think About the Situation

A popular game held at fairs or parties is the jelly bean guessing contest. Someone fills a jar or other large transparent container with a known quantity of jelly beans and gives a prize to the person who guesses closest to the actual number.  Participant estimates often vary and rarely does the winner guess the actual number of jelly beans.

Suppose you and your friend are going to enter the “Jelly Bean Counting Contest” at the county fair.• Consider the bucket of jelly beans. How many

jelly beans do you think are in the bucket?• What strategies did you use to arrive at your

estimate?• How might your strategy change if you could

touch the bucket but not take the jelly beans out?

Finding Volume of Composite Shapes

If you would like resources

www.ncctm2013.wikispaces.com/Defining+Rigor

karen.mcpherson@bcsemail.org