The Common Core State Standards and Critical Thinking in Math F-E-H BOCES 15 October 2010 Dr. Tim McNamara www.mathdrtim.us
Agenda Warm- Up (KenKen) Introductions and E- Share
www.kenken.com
Issues with Basics CCSS and Its Effects
www.corestandards.org/the-
standards/mathematics
Content and Process Learning Styles Quiz Tangrams & “Mushrooms”
www.nctm.org
www.eric.ed.gov
(ED281725)
Heuristics (P- S Strategies) Some Critical- Thinking Problems
www.exemplars.com
Finger Multiplication A Taste of Differentiation Web Resources www.cesa6.k12.wi.us/products_services/curriculuminstructionasse/mathcenter.cfm
Wrap- Up (Ten for Teachers)
Multiple- Choice Question:
1264 soldiers of the United States Armed Forces are being transported to field exercises by bus, and each bus holds 50 soldiers, how many buses would be needed to transport all of the soldiers? (calculator usage allowed) (a) 25 buses (b) 25.28 buses (c) 26 buses
NYS Math Performance Indicator CCSS (1.G.5) Recognize geometric shapes and structures in the environment (2.M.9) Tell time to the half-hour...using both digital and analog clocks
K 1
(3.N.2) Read and write whole numbers to 1,000
2
(4.N.6) Understand, use, and explain the associative property of multiplication
3
(6.N.3) Define and identify the distributive property of multiplication over addition
3
(4.N.26) Round numbers less than 1,000 to the nearest 10’s and 100’s
3
(5.G.11) Identify and draw lines of symmetry of basic geometric shapes
4
(6.N.16) Add and subtract fractions with unlike denominators
5
(7.A.5)
6
Solve one-step inequalities
(9.S.15) Identify and describe sources of bias and its effect, drawing conclusions from data
7
NCTM Standards www.nctm.org
Content
Process
Numbers & Operations
Problem Solving
Algebra
Reasoning & Proof
Geometry Communication Measurement
Connections
Data Anal. & Probability
Representation
An Example of Polyá’s “Mushrooms”: 1) Can you reconstruct the original tangram square with all 7 shapes? 2) Can a square be constructed with less than 7 shapes? 3) Can a square be constructed with the 2 small triangles? 4) Can the parallelogram be constructed in a similar fashion? 5) What is the ratio of the area of 1 small triangle to the square?
6) Express that same ratio as a fraction, as a decimal, and as a percent. 7) Suppose the original tangram square has an area = 1. Express each of the 7 shapes that make up the tangram as a fraction. 8) Suppose the original tangram square has an area = 32. Express each of the 7 shapes that make up the tangram as an area. 9) If the area of the original tangram = 32, what is its perimeter? 10) What is the length of the diagonal of the original tangram square?
Some Problem- Solving Strategies: (a) Solve a Simpler Problem (b) Look for a Pattern (c) Guess & Check (d) Draw a Graph (e) Make a Chart (f) Draw a Picture (g) Use a Model
“The Homecoming Float”: Styrofoam cubes are glued together in strings of 7 before being placed lengthwise on a workbench to be spraypainted. After the paint dries, the cubes are then glued to a float in preparation for the Homecoming parade. a) How many faces in a string of 7 cubes get painted? b) How many faces would get painted for any number of cubes in a string? Solve this problem in as many different ways as you can.
“The Mystery Store”: At a new furniture store in a nearby shopping only three-legged stools and four-legged chairs are If there are 8 complete pieces of furniture in the made up of 27 total legs, how many stools and many chairs are on sale?
mall, sold. store how
Solve this problem in as many different ways as you can.
Ten for Teachers (Pólya) 1) Be interested in your subject(s). 2) Know your subject matter- inside- out, forwards and backwards. 3) Know about the ways of learning. (The best way to learn anything is to discover it yourself.) 4) Try to read the faces of your students, try to see their expectations and their difficulties- put yourself in their places. 5) Give your students not only information but also “know-how” - the attitudes of mind, the habit of methodical work. 6) Let them learn guessing. 7) And let them learn proving. 8) Look out for some features of the problem at hand that might be helpful to know for the problems to come- patterns are huge! 9) Don’t give away your whole secret! (Let students guess and find out as much as they can for themselves before you tell them.) 10) Suggest, beckon, invite- don’t force and never exclude. And one more: “Leave ‘em laughing”- plant the seeds for tomorrow.