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



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


(1st grade example)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


Definition

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Characteristics

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Example

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Non-Example

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