Understanding the New York State Standards in Mathematics Frayer Model...
Transcript of Understanding the New York State Standards in Mathematics Frayer Model...
Understanding the New York State
Standards in Mathematics
Frayer Model Answer Key
Definition
• Understanding numbers and their relationships as well as computing and estimating
Characteristics
• Number Systems • Number Theory • Operations • Estimation
Example
• Scientific Notation • Place Value • Multiplying / Dividing • Fractions / Decimals • Commutative Property • Odd / Even Numbers • Comparing Numbers • Estimating
Non-Example
• Identifying Shapes • Using Mathematical tools (protractor,
compass) • Graphing • Classifying Angles
Number Sense & Operations (1)
Ulster County Math Initiative 2005
Definition
• Students will understand and manipulate numbers within our number system to make reasonable estimates and accurate computations
Characteristics
• Symbols that represent a numerical value
• All operations have predictable outcomes.
Example
• Number lines • Place value • Operations / # Sentences • Family of facts • Comparing numbers • Order of operations • Number properties
Non-Example
• Incorrect place value • Unreasonable estimates • Inaccurate computations
Number Sense & Operations (2)
Ulster County Math Initiative 2005
Definition
• Adding and subtracting accurately using single digits, whole numbers and positive numbers
• Estimation • Multiplication as a form of repeated
addition
Characteristics
• Ordering • Place value • Skip counting by 10’s, 5’s, 2’s • Counting forwards and backwards • Recognizing numerals in written,
verbal, and with objects
Example
• 7 + 7 = 14 • 2 + 2 + 2 + 2 + 2 = 10 • 9 – 2 = 7
Non-Example
• 14 - = 7 • -2 + 5 = 3
Number Sense & Operations (3)
(1st grade example)
Ulster County Math Initiative 2005
Definition
• Solving to find an unknown
Characteristics
• Variables • Equations • <, >, =, ≠ • patterns – geo / num • balancing sides • function
Example
• Input / output boxes • 2x = 6 • 0 0 __ __
•
• 6 > 2 + • 5 + 5 + 5 = 5x
Non-Example
• 2 + 1 = 3 • graphing data • square, circle
Algebra (1)
Ulster County Math Initiative 2005
Definition
• Solve for an unknown quantity
Characteristics
• Students will have the ability to perform algebraic operations correctly
• Isolate the variable • Set up a variable equation
Example
• 3x + 4 = 10 • 10 - = 4 • .
3 2 P = 10
x
Non-Example
• 3 X 4 = • 2 + 3 = • .
3 2 Find the perimeter
5
Algebra (2)
Ulster County Math Initiative 2005
Definition
• Analyze the characteristics of geometric shapes and their properties/ relationships (formal & informal)
Including coordinate geometry in problem solving situations
Characteristics
• Shapes • Geometric relationships • Transformational geometry • Constructions • Coordinate geometry • Locus • Informal proofs • Formal proofs
Example
• Polygons • Line segments • Angles • Rays • Point • Plane • Plotting Points • Transformations (reflection, rotation, dialation,
symmetry, translations) • 2 dimensional shapes • 3 dimensional shapes • Parallel lines • Intersecting lines • Perpendicular lines • Circles • Area • Perimeter • Volume • Locus • Proofs • Constructions
Non-Example
• Mean • Median • $ • Time • Negative integers
Geometry (1)
Ulster County Math Initiative 2005
Definition
• Understanding characteristics, properties, and relationships of geometric shapes
Characteristics
• Transformation • Symmetry • Coordinate Geometry • Analyze Problem Solving
Example
• K-1: Identify basic shapes • 3-4:
o Use correct shapes o Congruent and similar o Describe/ Compare 3-D
• 5: Calculate perimeter of regular • 6: Circumference/ Diameter • 7: Calculate radius & diameter with
given circumference
Non-Example
• Solving Equations • Number Systems • Algebra
Geometry (2)
Ulster County Math Initiative 2005
Definition
• Using a standard unit to give a comparable value to a real life object, concept, etc.
Characteristics
• Distance • Weight • Estimation
Example
• Perimeter • Ruler • Inches • Feet • Diameter • Distance to space • Hours • Area • Centimeters
Non-Example
• Personal Association • Negative Numbers • Reading
Measurement (1)
Ulster County Math Initiative 2005
Definition
• Using appropriate measurement tools to determine length, mass, or capacity.
Characteristics
• Select appropriate measurement tools and units
• Develop personal references for relative size
• Recognize error and significance of error
Example
• Area, volume, length, perimeter, mass • Elapsed time (hours / days) • Temperature • Conversions
Non-Example
• Algebraic Equation (2X + 3 = X – 6) • Tree diagram • Find the value of 5
Measurement (2)
Ulster County Math Initiative 2005
Definition
• Collecting data and showing data in some formal and organized manner and then explaining what the data means and predicting likely outcomes based upon the data presented.
Characteristics
• Organized information in chars and tables
• Representation in a variey
Example
• Data – surveying growth and change over time
• Event occurance – theoretical vs. experimental
• Representation – graphs & tables • Numerical expression of data (raw or
percent) • Analysis – trends – predictions • Certainty – likeliness of an even
occurring • Dependent vs independent events
Non-Example
• Random interference
Statistics &
Probability (1)
Ulster County Math Initiative 2005
Definition
• Collection and organization of numerical data
• Predicting outcomes and drawing conclusions
Characteristics
• Collections of data • Using tables/ charts • Graphs (histograms) • Measures of central tendancies • Survey - P
Example
• Monthly temperatures • Test scores • Stem and leaf plot • Box and whisker plot • Histogram • Mean/ median/ mode/ range • Tree diagrams • Ordered pairs of outcomes • Graphs of outcomes
Non-Example
• Small samples of data • Biased survey • Graphing a line • Balancing a checkbook • Exponents • Scientific notation
Statistics &
Probability (2)
Ulster County Math Initiative 2005
Definition
• Understand math through problem solving
• Solve problems that arise in math, science, social sciences and daily life
• Use and invent a repertoire of strategies to solve problems
• Assess, monitor, and reflect on the strategies and processes you used to solve a problem
Characteristics
• Guess and check • Write an algebraic expression and
solve for the unknown • Draw a diagram • Find relationships • Use a formula • Evaluate the reasonableness of you
answer • Explain how you got you answer • Use manipulative to solve a problem • Explain or teach a classmate
Example
• Find the area of a rectangle • Find the percent of a number • Write a proportion to describe a
relationship between similar figures
Non-Example
• 5 +4 = • 7 x 8 = • 54/9 =
Problem Solving (1)
Ulster County Math Initiative 2005
Definition
• Applying prior knowledge skills to perform mathematical tasks in a variety of ways
Characteristics
• Knowledge of a variety of strategies to solve problems and when to use them
Example
• Modeling • Guess and check • Work backwards • Using manipulatives • Drawing pictures • Eliminate extraneous information
Non-Example
• Incorrect equations • Fairies dancing with bears in the woods
while eating donuts
Problem Solving (2)
Ulster County Math Initiative 2005
Definition
• Being able to understand and apply real world problems using strategies to arrive at an answer
Characteristics
• One step / multi-step problems • Application / process problems • Strategies
Example
• Working backwards • Drawing diagrams • Understanding directed questions • Choosing correct operation
Non-Example
• Use incorrect operation • Can’t read for data • Tunnel vision
Problem Solving (3)
Ulster County Math Initiative 2005
Definition
• Justify an answer • Prove/ argue your point • Support
Characteristics
• Prove • Argue with you point • Solve an equation • Check you answer • Try something else/ various methods
Example
• Show your work • Explain • True/ false • Trial and error / guess and check • Create an equation
Non-Example
• Jus the answer/ # • No labels • Answer doesn’t make sense
(reasonableness) • Inappropriate strategies
Reasoning & Proof (1)
Ulster County Math Initiative 2005
Definition
• Explain or justify answer
Characteristics
• Explaining you answer • Prove through the use of representation • Looking for patterns
Example
• Commutative Property 3 + 4 = 4 + 3 6 x 10 = 10 x 6
• Manipulatives 1, 3, 5, 7, ….
Non-Example
• Answer = 5 • 3 + 3 + 4 ≠ 5 + 2 + 1 • 6 + 8 = 12 + 2
Reasoning & Proof (2)
Ulster County Math Initiative 2005
Definition
• Analyze and discuss mathematical thinking and strategies through words and pictures
Characteristics
• Show understanding • Use mathematical language to express
mathematical ideas precisely
Example
• Speaking • Writing • Pictures • Charts • tables
Non-Example
• “I just knew it”
Communication (1)
Ulster County Math Initiative 2005
Definition
• Expressing understanding in different forms
Characteristics
• Organized math thinking • Explaining strategies • Showing steps to achieve answers/
solutions
Example
• Written response • Journal writing • Sharing orally in groups • Showing numerical reasoning in
sequence
Non-Example
• Giving answers w/o justification • Multiple choice response • Mental math
Communication (2)
Ulster County Math Initiative 2005
Definition
• Relate mathematical knowledge to real life, between strands, and across curriculum
Characteristics
• Recognizing similarities for application
Example
• Money management, traveling, cooking
• Algebraic word problems • Timeliness and science experiments
Non-Example
• Isolated skills 8 + 8 = 16 28 ÷ 4 = 7 A = L x W
Connection (1)
Ulster County Math Initiative 2005
Definition
• Ability to link mathematical concept with prior knowledge
Characteristics
• Real world application • Inter-disciplinary units
(science, chemistry, physics, geography)
Example
• Cooking • Building a house • Mechanics • Science labs • Geography
Non-Example
• Worksheets • Repetitive practice
Connection (2)
Ulster County Math Initiative 2005
Definition
• Use of representation to solve problems, communicate and organize mathematical ideas.
Characteristics
• Physical object (model) • Drawing or diagram (graphic
representation) • Words and/ or numbers
Example
• Unifix cubes • Base-ten blocks • Number lines • Hundreds charts • Diagrams • Scaled drawings • Graphs • Narrative • Equations
Non-Example
• Blank paper • Just an answer without an explanation
Representation (1)
Ulster County Math Initiative 2005
Definition
• Using & applying representations to organize, record and communicate math
Characteristics
• Organize • Apply • Record • Illustrate • Diagram • Graphs and charts
Example
• Graphs • Draw a picture • Make a chart • Act out • Number sentences • Equations • Using manipulatives
Non-Example
• Direct explanations • Calculations
Representation (2)
Ulster County Math Initiative 2005
Definition
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Characteristics
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Example
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Non-Example
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Template
Ulster County Math Initiative 2005