ccss mathematics grades 6-8

CCSS MathematicsGrades 6-8

Curriculum Review Week 2012

Today’s Learning Targets

I can understand that CCSS math standards require a different type of teaching, which also requires a different type of assessing.

I can understand that this different way of teaching and assessing has an impact on the work that we will be doing this week, as we create our Baseline and Benchmark Assessments.

teachers & students:Focus- Teach less, go deeperCoherence- Build on previous learning and

connect concepts Rigor- Balance of conceptual

understanding, procedural skill and fluency, and application of skills in problem solving situations

Math Shifts

When planning, ask

“What task can I give that will build student understanding?”

rather than“How can I explain clearly so they will

understand?”

Grayson Wheatley, NCCTM, 2002

Definitions

Understanding mathematics Asking a student to understand something means asking a

teacher to assess whether the student has understood it.

But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to

justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y).

Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

http://www.corestandards.org/

Understanding the Math

The “Why” behind the math- students need to know why and be able to explain why!

Develop the concept then teach the procedure that goes along with that.

Concrete Representational Abstract

Timeline for Common Core Mathematics Implementation

Year Standards To Be Taught Standards To Be Assessed 2011 – 2012 2003 NCSCOS 2003 NCSCOS2012 – 2013 CCSS CCSS (NC)2013 – 2014 CCSS CCSS (NC)2014 – 2015 CCSS CCSS (SBAC)

Common Core State Standards Adopted June, 2010

Let’s look at a familiar problem…

Which of the following represents ?

a.

b.

c.

d.

Same problem- a new twist…

For numbers 1a – 1d, state whether or not each figure has of its whole shaded.

1a.

1b.

1c.

1d.

ο Yes ο No

ο Yes ο No

ο Yes ο No

ο Yes ο No

For numbers 1a – 1d, state whether or not each figure has of its whole shaded?

1a.

1b.

1c.

1d.

ο Yes ο No

ο Yes ο No

ο Yes ο No

ο Yes ο No

This item is worth 0 – 2 points depending on the responses. What series of the yes and no responses would give a student:

“Turn and Talk”

2 points? 1 point? 0 points?

2 points: YNYN 1 point: YNNN, YYNN, YYYN0 point: YYYY, YNNY, NNNN, NNYY, NYYN, NYNN NYYY, NYNY, NNYN, NNNY, YYNY, YNYY

For numbers 1a - 1d, state whether or not each figure has of its whole shaded.

The kind and level of thinking required of students to successfully engage with and solve a task.

Teaching for Understanding Through Cognitive Demand

Cognitive Demand Spectrum

MEMORIZATIONPROCEDURESWITHOUT

CONNECTIONS(TO

UNDERSTANDING,MEANING, OR CONCEPTS)

PROCEDURESWITH

CONNECTIONS(TO

UNDERSTANDING,MEANING,

OR CONCEPTS)

DOINGMATHEMATICS

Tasks that require memorized procedures in routine ways

Tasks that require engagement with

concepts, and stimulate students to make

connections to meaning, representation, and other

mathematical ideas

Procedures Without Connections

Convert the fraction to a decimal and a percent.

MemorizationWhat are the decimal and percent equivalents for the fractions:

and

Using a 10x10 grid, identify

the decimal and percent

equivalents of

Procedures WithConnections

Level 1: Recalling and Recognizing Student is able to recall routine facts or

knowledge and can recognize shape, symbols, attributes and other qualities.

Level 2: Using Procedures Student uses or applies procedures and

techniques to arrive at solutions or answers.

Depth of Knowledge

Depth of Knowledge

Level 3: Explaining and Concluding Student reasons and derives conclusions.

Student explains reasoning and processes. Student communicates procedures and findings.

Level 4: Making Connections, Extending and Justifying:

Student makes connections between different concepts and strands of mathematics. Extends and builds on knowledge to a situation to arrive at a conclusion. Student uses reason and logic to prove and justify conclusions.

Balance Defined

Problem Solving

Computational & Procedural

SkillsDOING MATH

Conceptual Understanding

“WHERE” THE MATHEMATICSWORK

“HOW” THE

MATHEMATICSWORK

“WHY” THE

MATHEMATICSWORK

Let’s do some Mathematics

• Shade 6 small squares in a 4 x 10 rectangle.

• Using the rectangle, explain how to determine each of the following:a. the fractional part of the area

shaded. b. the decimal part of the area

shaded. c. the percent of the area that is

shaded. Doing Mathematics

Different Objects, Same RatiosTrucks and BoxesOn which cards is the ratio of trucks to boxes the same? Given one card, students are to select a card on which the ratio of the two types of objects is the same. This task moves students to numeric approach rather than a visual one and introduces the notion of ratios as rates. A unit rate is depicted on a card that shows exactly one of either of the two types of objects. For example the card with three boxes and one truck provides one unit rate. A unit rate for the other ratio is not shown. What would it be?  

Solve this problem without using any numeric algorithms such as cross-products. This could include pictures or counters, but there is no prescribed method. 

Two camps of scouts are having pizza parties. The Bear Camp ordered enough so that every 3 campers will have 2 pizzas. The leader of the Raccoons ordered enough so that there would be 3 pizzas for every 5 campers. Did the Bear or the Raccoon campers have more pizza to eat? 

A bird flew 20 miles in 100 minutes at a constant speed. At that speed:(a) How long would it take the bird to fly 6 miles? Explain.(b) How far would the bird fly in 15 minutes? Explain.(c) How fast is the bird flying in miles per hour?(d) What is the bird’s pace in minutes per mile?(e) Explain the meaning of the answers for parts c and d?

How would using the table help? 

Assessments

Assessment Specifications Summary is in your CRW folders and on our ISS Common Core wiki to guide you as you create your Baseline and Benchmarks.

Thank You!!

Let’s put our brains to work, creating quality baselines and benchmarks that reflect the teaching and understanding of the CCSS!!

Given: x2 – 9Which is true?I. x = 3II. x = -3III. x2 = 9

a. I only b. I and II onlyc. I and III only d. I, II, and IIIe. none

Given: (a – b)(a2 – b2) = 0,which is true?I. a = bII. a = -bIII. a2 = b2

a. I only b. I and II onlyc. I and III only d. III onlyhttp://mathnotations.blogspot.com/2007/07/understanding-algebra-conceptually.html

A high level task…Has no predictable pathwayRequires student to access and use

relevant knowledgeRequires student to explainHas the potential to engage student in

higher level thinking

Soup and Beans

Consider these two Measurement Problems

Carpeting TaskMartha is re-carpeting

her bedroom, which is 15 feet wide. How many square feet of carpet will she need to purchase, and how much baseboard will she need to run around the edge of the carpet? Explain your thinking.

The Fencing TaskMs Brown’s class will raise

rabbits for their science fair. They have 24 feet of fencing with which to build a rectangular pen. If Ms. Brown’s students want their rabbits to have as much room as possible, how long should each of the sides of the pen be? Explain your thinking.

For each problem, what kind of thinking is required?How are they alike? How are they different?

Carpeting Task vs. Fencing Task

Alike

Both require Area and perimeter calculations

Both require students to “explain your thinking”

Both are word problems, set in a “real world” context

Not Alike

Fencing requires a systematic approach

Fencing leads to generalization and justification

The “thinking in Fencing is complex – requires more than applying a memorized formula

Technology and Testing Content of the North Carolina

assessments is aligned to the CCSS-M; however, the technology will not be as sophisticated as in assessments created by the Smarter Balanced Assessment Consortium (SBAC).

www.smarterbalanced.org

www.ncdpi.wikispaces.net

DPI Contact Information

Kitty RutherfordElementary Mathematics Consultant919-807-3934kitty.rutherford@dpi.nc.gov

Amy ScrinziElementary Mathematics Consultant919-807-3839amy.scrinzi@dpi.nc.gov

Robin BarbourMiddle Grades Mathematics Consultant919-807-3841robin.barbour@dpi.nc.gov

Johannah MaynorSecondary Mathematics Consultant919-807-3842johannah.maynor@dpi.nc.gov

Barbara BissellK – 12 Mathematics Section Chief919-807-3838barbara.bissell@dpi.nc.gov

Susan HartProgram Assistant919-807-3846susan.hart@dpi.nc.gov

Let’s do some MORE math!a) Pick any integer greater than 10: __________b) Square the number you picked: __________c) Find the product of the integer that is one more than your

original number and the integer that is one less than your original number: ___________

d) Compare your result in part a to your result in part b. What do you notice?

e) Repeat steps a – c three or four times, using a different positive integer as your starting number each time.

f) Create an algebraic equation to represent what you see happening, but for ANY starting number.

g) Turn to someone sitting near you and explain as clearly as you can what your equation represents.

Place ValueTraditional worksheet vs Common Core

worksheet

What do you notice?

Mathematical LanguageGrowing StaircaseBuilding staircaseBlocksPatternTotal blocks for nth stepTotal blocks for a staircase with n number of stepsBase of a staircase

Top step of staircaseStep vs. stageGeneralizeConjectureExplicit or general/closed formula or functionRecursive formula or functionJustify/prove

Growing Staircases