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4.4.1
Number and Numerical Operations
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4.4.1.4 A
Number Sense
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4.4.1.4 A.1
Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well).
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4.4.1.4 A.1.a
Whole numbers through millions
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4.4.1.4 A.1.b
Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 16) as part of a whole, as a subset of a set, and as a location on a number line
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4.4.1.4 A.1.c
Decimals through hundredths
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4.4.1.4 A.2
Demonstrate an understanding of place value concepts.
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4.4.1.4 A.3
Demonstrate a sense of the relative magnitudes of numbers.
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4.4.1.4 A.4
Understand the various uses of numbers.
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4.4.1.4 A.4.a
Counting, measuring, labeling (e.g., numbers on baseball uniforms), locating (e.g., Room 235 is on the second floor)
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4.4.1.4 A.5
Use concrete and pictorial models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number.
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4.4.1.4 A.6
Compare and order numbers.
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4.4.1.4 A.7
Explore settings that give rise to negative numbers.
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4.4.1.4 A.7.a
Temperatures below 0°, debts
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4.4.1.4 A.7.b
Extension of the number line
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4.4.1.4 B
Numerical Operations
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4.4.1.4 B.1
Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems.
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4.4.1.4 B.1.a
Addition and subtraction: joining, separating, comparing
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4.4.1.4 B.1.b
Multiplication: repeated addition, area/array
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4.4.1.4 B.1.c
Division: repeated subtraction, sharing
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4.4.1.4 B.2
Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as "skip counting" and "repeated subtraction") and then commit them to memory.
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4.4.1.4 B.3
Construct, use, and explain procedures for performing whole number calculations and with:
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4.4.1.4 B.3.a
Pencil-and-paper
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4.4.1.4 B.3.b
Mental math
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4.4.1.4 B.3.c
Calculator
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4.4.1.4 B.4
Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.
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4.4.1.4 B.4.a
Addition of 3-digit numbers
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4.4.1.4 B.4.b
Subtraction of 3-digit numbers
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4.4.1.4 B.4.c
Multiplication of 2-digit numbers
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4.4.1.4 B.4.d
Division of 3-digit numbers by 1-digit numbers
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4.4.1.4 B.5
Construct and use procedures for performing decimal addition and subtraction.
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4.4.1.4 B.6
Count and perform simple computations with money.
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4.4.1.4 B.6.a
Standard dollars and cents notation
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4.4.1.4 B.7
Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
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4.4.1.4 B.8
Check the reasonableness of results of computations.
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4.4.1.4 B.9
Use concrete models to explore addition and subtraction with fractions.
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4.4.1.4 B.10
Understand and use the inverse relationships between addition and subtraction and between multiplication and division.
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4.4.1.4 C
Estimation
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4.4.1.4 C.1
Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.
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4.4.1.4 C.2
Construct and use a variety of estimation strategies (e.g., rounding and mental math) for estimating both quantities and the results of computations.
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4.4.1.4 C.3
Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.
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4.4.1.4 C.4
Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.
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4.4.2
Geometry and Measurement
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4.4.2.4 A
Geometric Properties
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4.4.2.4 A.1
Identify and describe spatial relationships of two or more objects in space.
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4.4.2.4 A.1.a
Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)
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4.4.2.4 A.1.b
Relative shapes and sizes
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4.4.2.4 A.1.c
Shadows (projections) of everyday objects
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4.4.2.4 A.2
Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them.
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4.4.2.4 A.2.a
Vertex, edge, face, side, angle
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4.4.2.4 A.2.b
3D figures - cube, rectangular prism, sphere, cone, cylinder, and pyramid
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4.4.2.4 A.2.c
2D figures - square, rectangle, circle, triangle, quadrilateral, pentagon, hexagon, octagon
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4.4.2.4 A.2.d
Inclusive relationships - squares are rectangles, cubes are rectangular prisms
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4.4.2.4 A.3
Identify and describe relationships among two-dimensional shapes.
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4.4.2.4 A.3.a
Congruence
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4.4.2.4 A.3.b
Lines of symmetry
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4.4.2.4 A.4
Understand and apply concepts involving lines, angles, and circles.
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4.4.2.4 A.4.a
Point, line, line segment, endpoint
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4.4.2.4 A.4.b
Parallel, perpendicular
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4.4.2.4 A.4.c
Angles - acute, right, obtuse
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4.4.2.4 A.4.d
Circles - diameter, radius, center
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4.4.2.4 A.5
Recognize, describe, extend, and create space-filling patterns.
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4.4.2.4 B
Transforming Shapes
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4.4.2.4 B.1
Use simple shapes to cover an area (tessellations).
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4.4.2.4 B.2
Describe and use geometric transformations (slide, flip, turn).
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4.4.2.4 B.3
Investigate the occurrence of geometry in nature and art.
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4.4.2.4 C
Coordinate Geometry
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4.4.2.4 C.1
Locate and name points in the first quadrant on a coordinate grid.
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4.4.2.4 C.2
Use coordinates to give or follow directions from one point to another on a map or grid.
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4.4.2.4 D
Units of Measurement
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4.4.2.4 D.1
Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.
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4.4.2.4 D.2
Select and use appropriate standard units of measure and measurement tools to solve real-life problems
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4.4.2.4 D.2.a
Length - fractions of an inch (1/8, 1/4, 1/2), mile, decimeter, kilometer
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4.4.2.4 D.2.b
Area - square inch, square centimeter
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4.4.2.4 D.2.c
Volume - cubic inch, cubic centimeter
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4.4.2.4 D.2.d
Weight - ounce
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4.4.2.4 D.2.e
Capacity - fluid ounce, cup, gallon, milliliter
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4.4.2.4 D.3
Develop and use personal referents to approximate standard units of measure (e.g., a common paper clip is about an inch long).
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4.4.2.4 D.4
Incorporate estimation in measurement activities (e.g., estimate before measuring).
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4.4.2.4 D.5
Solve problems involving elapsed time.
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4.4.2.4 E
Measuring Geometric Objects
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4.4.2.4 E.1
Determine the area of simple two-dimensional shapes on a square grid.
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4.4.2.4 E.2
Distinguish between perimeter and area and use each appropriately in problem-solving situations.
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4.4.2.4 E.3
Measure and compare the volume of three-dimensional objects using materials such as rice or cubes.
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4.4.3
Patterns and Algebra
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4.4.3.4 A
Patterns
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4.4.3.4 A.1
Recognize, describe, extend, and create patterns.
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4.4.3.4 A.1.a
Descriptions using words, number sentences/expressions, graphs, tables, variables (e.g., shape, blank, or letter)
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4.4.3.4 A.1.b
Sequences that stop or that continue infinitely
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4.4.3.4 A.1.c
Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11,... or 800, 400, 200,...)
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4.4.3.4 A.1.d
Sequences can often be extended in more than one way (e.g., the next term after 1, 2, 4,... could be 8, or 7, or ...)
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4.4.3.4 B
Functions and Relationships
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4.4.3.4 B.1
Use concrete and pictorial models to explore the basic concept of a function.
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4.4.3.4 B.1.a
Input/output tables, T-charts
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4.4.3.4 B.1.b
Combining two function machines
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4.4.3.4 B.1.c
Reversing a function machine
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4.4.3.4 C
Modeling
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4.4.3.4 C.1
Recognize and describe change in quantities.
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4.4.3.4 C.1.a
Graphs representing change over time (e.g., temperature, height)
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4.4.3.4 C.1.b
How change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration)
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4.4.3.4 C.2
Construct and solve simple open sentences involving any one operation (e.g., 3 x 6 = __, n = 15 ÷ 3, 3 x __ = 0, 16 - c = 7).
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4.4.3.4 D
Procedures
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4.4.3.4 D.1
Understand, name, and apply the properties of operations and numbers.
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4.4.3.4 D.1.a
Commutative (e.g., 3 x 7 = 7 x 3)
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4.4.3.4 D.1.b
Identity element for multiplication is 1 (e.g., 1 x 8 = 8)
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4.4.3.4 D.1.c
Associative (e.g., 2 x 4 x 25 can be found by first multiplying either 2 x 4 or 4 x 25)
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4.4.3.4 D.1.d
Division by zero is undefined
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4.4.3.4 D.1.e
Any number multiplied by zero is zero.
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4.4.3.4 D.2
Understand and use the concepts of equals, less than, and greater than in simple number sentences.
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4.4.3.4 D.2.a
Symbols (= , < , >)
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4.4.4
Data Analysis, Probability, and Discrete Mathematics
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4.4.4.4 A
Data Analysis
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4.4.4.4 A.1
Collect, generate, organize, and display data in response to questions, claims, or curiosity.
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4.4.4.4 A.1.a
Data collected from the school environment
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4.4.4.4 A.2
Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data.
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4.4.4.4 A.2.a
Pictograph, bar graph, line plot, line graph, table
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4.4.4.4 A.2.b
Average (mean), most frequent (mode), middle term (median)
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4.4.4.4 B
Probability
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4.4.4.4 B.1
Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability.
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4.4.4.4 B.1.a
Likely, unlikely, certain, impossible, improbable, fair, unfair
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4.4.4.4 B.1.b
More likely, less likely, equally likely
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4.4.4.4 B.1.c
Probability of tossing "heads" does not depend on outcomes of previous tosses
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4.4.4.4 B.2
Determine probabilities of simple events based on equally likely outcomes and express them as fractions.
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4.4.4.4 B.3
Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color).
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4.4.4.4 B.3.a
What students think will happen (intuitive)
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4.4.4.4 B.3.b
Collect data and use that data to predict the probability (experimental)
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4.4.4.4 B.3.c
Analyze all possible outcomes to find the probability (theoretical)
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4.4.4.4 C
Discrete Mathematics-Systematic Listing and Counting
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4.4.4.4 C.1
Represent and classify data according to attributes, such as shape or color, and relationships.
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4.4.4.4 C.1.a
Venn diagrams
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4.4.4.4 C.1.b
Numerical and alphabetical order
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4.4.4.4 C.2
Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation.
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4.4.4.4 C.2.a
Organized lists, charts, tree diagrams
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4.4.4.4 C.2.b
Dividing into categories (e.g., to find the total number of rectangles in a grid, find the number of rectangles of each size and add the results)
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4.4.4.4 D
Discrete Mathematics-Vertex-Edge Graphs and Algorithms
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4.4.4.4 D.1
Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers).
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4.4.4.4 D.2
Play two-person games and devise strategies for winning the games (e.g., "make 5" where players alternately add 1 or 2 and the person who reaches 5, or another designated number, is the winner).
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4.4.4.4 D.3
Explore vertex-edge graphs and tree diagrams.
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4.4.4.4 D.3.a
Vertex, edge, neighboring/adjacent, number of neighbors
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4.4.4.4 D.3.b
Path, circuit (i.e., path that ends at its starting point)
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4.4.4.4 D.4
Find the smallest number of colors needed to color a map or a graph.
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4.4.5
Mathematical Processes
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4.4.5 A
Problem Solving
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4.4.5 A.1
Learn mathematics through problem solving, inquiry, and discovery.
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4.4.5 A.2
Solve problems that arise in mathematics and in other contexts.
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4.4.5 A.2.a
Open-ended problems
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4.4.5 A.2.b
Non-routine problems
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4.4.5 A.2.c
Problems with multiple solutions
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4.4.5 A.2.d
Problems that can be solved in several ways
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4.4.5 A.3
Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.
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4.4.5 A.4
Pose problems of various types and levels of difficulty.
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4.4.5 A.5
Monitor their progress and reflect on the process of their problem solving activity.
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4.4.5 A.6
Distinguish relevant from irrelevant information, and identify missing information.
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4.4.5 B
Communication
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4.4.5 B.1
Use communication to organize and clarify their mathematical thinking.
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4.4.5 B.1.a
Reading and writing
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4.4.5 B.1.b
Discussion, listening, and questioning
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4.4.5 B.2
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.
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4.4.5 B.3
Analyze and evaluate the mathematical thinking and strategies of others.
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4.4.5 B.4
Use the language of mathematics to express mathematical ideas precisely.
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4.4.5 C
Connections
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4.4.5 C.1
Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).
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4.4.5 C.2
Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point).
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4.4.5 C.3
Recognize that mathematics is used in a variety of contexts outside of mathematics.
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4.4.5 C.4
Apply mathematics in practical situations and in other disciplines.
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4.4.5 C.5
Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).
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4.4.5 C.6
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
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4.4.5 D
Reasoning
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4.4.5 D.1
Recognize that mathematical facts, procedures, and claims must be justified.
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4.4.5 D.2
Use reasoning to support their mathematical conclusions and problem solutions.
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4.4.5 D.3
Select and use various types of reasoning and methods of proof.
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4.4.5 D.4
Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.
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4.4.5 D.5
Make and investigate mathematical conjectures.
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4.4.5 D.5.a
Counterexamples as a means of disproving conjectures
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4.4.5 D.5.b
Verifying conjectures using informal reasoning or proofs.
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4.4.5 D.6
Evaluate examples of mathematical reasoning and determine whether they are valid.
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4.4.5 E
Representations
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4.4.5 E.1
Create and use representations to organize, record, and communicate mathematical ideas.
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4.4.5 E.1.a
Concrete representations (e.g., base-ten blocks or algebra tiles)
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4.4.5 E.1.b
Pictorial representations (e.g., diagrams, charts, or tables)
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4.4.5 E.1.c
Symbolic representations (e.g., a formula)
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4.4.5 E.1.d
Graphical representations (e.g., a line graph)
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4.4.5 E.2
Select, apply, and translate among mathematical representations to solve problems.
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4.4.5 E.3
Use representations to model and interpret physical, social, and mathematical phenomena.
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4.4.5 F
Technology
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4.4.5 F.1
Use technology to gather, analyze, and communicate mathematical information.
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4.4.5 F.2
Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.
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4.4.5 F.3
Use graphing calculators and computer software to investigate properties of functions and their graphs.
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4.4.5 F.4
Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).
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4.4.5 F.5
Use computer software to make and verify conjectures about geometric objects.
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4.4.5 F.6
Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).
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