1.1 Understand and apply concepts and procedures from number sense.

Grade 3

Grade 4

Grade 5

Number and Numeration

1.1.1

Understand the concept of whole numbers. W

·          Represent a number to at least 100,000 in different ways (e.g., words, numerals, pictures, physical models). [CU]

·          Translate from one representation of a whole number to another in standard, expanded, and word forms. [SP, RL, CU, MC]

·          Generate equivalent representations for a given number by decomposing and composing. [SP, RL]

·          Explain the difference between the natural numbers and the whole numbers

·          Explain what makes a number odd or even.

Understand the concept of decimals (money) and fraction. W

·          Interpret fractions as parts of a whole object, number, or set (e.g., ½ of a medium pizza and ½ of a large pizza are not equal amounts).

·          Symbolically represent parts of a whole or parts of a set with common denominators. [CU]

·          Explain how fractions (denominators of 2, 3, 4, 6, and 8) represent information across the curriculum (e.g., interpreting circle graphs, fraction of states that border an ocean). [CU, MC, SP, RL]

·           Represent decimals (money) in multiple ways (e.g., symbols, physical models)

Understand the concepts of fractions and decimals. W

·          Demonstrate understanding of the concepts and symbolic representations of mixed numbers and improper fractions and decimals

·          Create a model when given a symbolic representation or write the fraction when given a model (e.g., number line). [CU]

·          Explain the value of a given digit in a decimal to at least the thousandths place. [CU]

·          Explain how the value of a fraction changes in relationship to the size of the whole (e.g., half a pizza vs. half a cookie). [CU]

·          Use factors and multiples to rename equivalent fractions. [SP, RL]

·          Read and write decimals to at least the thousandth place. [CU]

1.1.2

Understand the relative values of whole numbers. W

·          Compare whole number values to at least 100,000 using the symbols for "greater than", "less than", and “equal to".

·          Order three or more numbers to at least 100,000 from smallest to largest. [CU]

·          Compare combined quantities (e.g. 50 + 3 is greater than 40 + 9) [SP, RL]

Understand the relative values of fractions and decimals (money). W

·          Model and describe equivalent fractions (e.g., paper folding, geoboards, parallel number lines). [CU]

·          Use a number line to approximate and label halves, thirds, and fourths in relationship to whole units. [CU, MC]

·          Order fractions with like denominators.. [CU, MC]

·          Demonstrate and explain equivalent relationships between decimals and fractions (e.g., $.50 is equal to ½ a dollar and 50/100 of a dollar) using models. [CU, MC]

Understand the relative values of non-negative fractions or decimals. W

·          Demonstrate understanding of the relative values of non-negative fractions (denominators of 2, 3, 4, 5, 6, or 10) or decimals.

·          Compare, order, or illustrate whole numbers, decimals, and fractions using concrete models (e.g., number line or shaded grid) or implementing strategies (e.g., like denominators, benchmarks, conversions). [SP, RL, CU, MC]

·          Determine equivalence among fractions. [SP, RL]

·          Explain the relative values of non-negative fractions or decimals. [CU]

1.1 Understand and apply concepts and procedures from number sense.

Grade 3

Grade 4

Grade 5

Number and Numeration

1.1.3

Understand and apply the commutative and identity properties of addition on whole numbers. W

·          Explain how the commutative property works with addition and not subtraction, using words, numbers, or physical models. [CU]

·          Describe how the identity property works with addition. [CU]

·          Evaluate addition equations as true or false and explain based on the commutative and identity properties for addition. (e.g., 14 + (0 + 38) = 38 + (14 + 0))

Understand and apply the associative property of addition, and commutative, associative, identity, and zero properties of multiplication on whole numbers. W

·          Describe how the commutative property works with multiplication and not division using words, numbers, or physical models. [CU]

·          Describe how the identity property for addition is different from the identity property for multiplication using words, numbers, or physical models.  [CU]

·          Evaluate equations as true or false and explain based on any of the properties for multiplication

(e.g., 4 x (5 x 6) = (4 x 5) x 6). [SP, RL]

·          Evaluate equations as true or false and explain based on any of the properties

(e.g., 14 + (62 + 38) = (14 + 62) + 38). [SP, RL]

Understand and apply concepts of divisibility. W

·        Apply the concepts of odd and even numbers to check for divisibility, finding factors and multiples.

·          Illustrate prime or composite numbers by creating a physical model (e.g., arrays, area models). [CU]

·          Identify the prime numbers between 1 and 100

·          Explain why a whole number between 1 and 100 is prime or composite. [CU]

·          Explain a method to find the least common multiple (LCM) and greatest common factor (GCF) of two numbers.

·          Solve problems related to primes, factors, multiples, and composites in a variety of situations (e.g., find a mystery number, find unit pricing, increase or decrease a recipe, find the portions for a group) .[SP, RL, MC]

1.1.4

1.1 Understand and apply concepts and procedures from number sense.

Grade 3

Grade 4

Grade 5

Computation

1.1.5

Understand the meaning of multiplication and division on whole numbers. W

·          Illustrate multiplication and division using models and diagrams. [CU]

·          Illustrate and explain the inverse relationship between multiplication and division using physical diagrams, words, and symbols (e.g., arrays, fact families). [CU]

·          Describe and compare strategies to solve problems involving multiplication and division (e.g., alternative algorithms different strategies, decomposition, properties of multiplication). [CU]

Understand the meaning of addition and subtraction on like-denominator fractions. W

·          Represent addition and subtraction of decimals through hundredths using models (e.g., with money). [CU]

·          Represent addition and subtraction of fractions with like denominators using models (e.g., fraction circles, number lines, geoboards). [CU]

·          Explain the meaning of addition and subtraction of like denominator fractions. [CU]

Understand the meaning of addition and subtraction on non-negative decimals and fractions. W

·          Demonstrate understanding of the meaning of addition and subtraction of non-negative decimals and fractions.(e.g., fractions with denominators of 2, 4, 8 or 2, 3, 6, 12 or 5, 10 – highest LCM of 12).

·          Explain the meaning of adding and subtracting fractions and decimals using words, symbols, or other models. [CU]

·          Create a problem situation involving addition or subtraction of non-negative decimals or fractions. [SP, RL, CU, MC]

1.1.6

Apply procedures of addition and subtraction on whole numbers with fluency. W

·          Describe and compare strategies to solve three-digit addition and subtraction problems (e.g., child developed algorithms, decomposition). [SP, RL, CU, MC]

·          Use joining, separating, adding-on, and finding the difference to solve problems.

·          Write and solve multi-step problem situations that involve addition and subtraction. [SP, RL, CU, MC]

Apply procedures of multiplication and division on whole numbers with fluency. W

·          Use a variety of strategies to mentally access multiplication and division facts through 12's. [SP, RL]

·          Recall multiplication and division facts through 12’s.

·          Record, share, and evaluate algorithms used in computational situations. [CU]

·          Write and solve problem situations with whole numbers using a combination of any two operations. [SP, RL, CU, MC]

·          Interpret remainders of a division problem in a given situation. [SP, RL, MC]

Apply procedures of addition and subtraction on non-negative decimals and like-denominator fractions. W

·          Fractions with denominators of: 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 20, and 100)

·          Explain a strategy for adding fractions. [CU]

·          Write and solve problem situations to find sums or differences of decimals or like-denominator fractions. [SP, RL, CU, MC]

1.1 Understand and apply concepts and procedures from number sense.

Grade 3

Grade 4

Grade 5

Computation

1.1.7

Understand and apply strategies and tools as appropriate to tasks involving addition and subtraction on whole numbers.

·          Use appropriate strategies and tools from among mental computation, estimation, calculators, and paper and pencil to compute in a problem situation. [SP, RL]

·          Defend situations in which estimation is sufficient (e.g., grocery shopping or party supplies). [CU]

·          Use mental arithmetic, pencil and paper, or calculator as appropriate to the task involving addition and subtraction of whole numbers.

Understand and apply strategies and tools as appropriate to tasks involving multiplication and division on whole numbers.

·          Select and justify appropriate strategies and tools from among mental computation, estimation, calculators, and paper and pencil to compute in a problem situation. [SP, RL]

§          Use estimation strategies appropriately when the exact answer is not necessary. [SP, RL]

§          Identify and justify situations when estimation is not appropriate. [SP, RL, CU, MC]

§          Use mathematical tools as appropriate to the task involving multiplication and division of whole numbers

Understand and apply strategies and tools as appropriate to tasks involving addition and subtraction of non-negative, like denominator fractions or decimals.

·          Select and justify strategies and appropriate tools from among mental computation, estimation, calculators, manipulatives, and paper and pencil to compute a problem situation. [SP, RL]

·          Use mental arithmetic to add and subtract non-negative decimals and like-denominator fractions.

Estimation

1.1.8

Understand and apply estimation strategies to determine the reasonableness of answers in situations involving addition and subtraction on whole numbers. W

·          Identify when an approximation is appropriate;

·          Use estimation to determine the reasonableness of answers in situations

·          Describe and justify reasonableness of an estimate in computation. [SP, RL, CU]

·          Use a variety of estimation strategies (e.g., multiples of 10 and 100, rounding, front-end estimation, compatible numbers, clustering). [SP, RL]

·          Describe and justify whether an approximation is or is not appropriate. [SP, RL, CU]

Understand and apply estimation strategies to determine the reasonableness of answers in situations involving multiplication and division on whole numbers. W

·          Identify when an approximation is appropriate

·          Use estimation to determine the reasonableness of answers in situations.

·          Use a variety of strategies to approximate sums, differences, products, and quotients. [SP, RL]

·          Make and explain an appropriate adjustment when an estimate and a solution don't agree. [SP, RL, CU]

Understand and apply estimation strategies to determine the reasonableness of answers in situations involving addition and subtraction on non-negative decimals and like-denominator fractions. W

·          Identify when an approximation is appropriate;

·          Use estimation to determine the reasonableness of answers in situations

·          Use estimation strategies prior to computation of addition and subtraction of decimals and like-denominator fractions to determine reasonableness of answers. [SP, RL]

·          Identify reasonableness of estimated answers for a given situation. [SP, RL]

·          Articulate various strategies used during estimation. [CU]

1.2 Understand and apply concepts and procedures from measurement.

Grade 3

Grade 4

Grade 5

Attributes, Units and Systems

1.2.1

Understand how different attributes (of length, perimeter,  time, money value, weight/mass, and temperature) are used to describe objects. W

·          Describe the different measures (hour, minutes, and seconds) of time displayed on a clock. [CU]

·          Given an object, name the attributes that can be measured. [CU, MC]

·          Identify temperature on thermometers with different scales (e.g., increments of 1, 2, 5, or 10 degrees).

·          Use measurements of length, perimeter, time, money, weight, and temperature to describe and compare objects. [CU]

Understand the concept of area. W

·          Demonstrate and explain that area is covering a shape and perimeter is enclosing a shape. [CU, MC]

·          Describe situations where area is the needed measurable attribute (e.g., buying carpet to cover a floor, painting a wall, describing the amount of floor space in a room).  [CU, MC]

·          Compare areas of different shapes and sizes. [SP, RL]

·          Use measurements of area to describe and compare objects. [CU]

Understand the concept of angle measurement. W

·          Describe and compare angles in a variety of objects. [CU, MC]

·          Identify angles in the environment. [MC]

·          Classify angles as right, acute or obtuse. [CU]

·          Identify types of angles in polygons (e.g., right, acute, obtuse). [MC]

·          Explain and provide examples of how angles are formed.

1.2.2

Understand the differences between non-standard and standard units of measurement for length and weight/mass in either U.S. or metric systems. W

·          Given an attribute of length or weight/mass, identify an appropriate unit of measurement.

·          Measure the length or weight/mass of objects and compare measurements using standard units.

·          Explain when standard units of measurement are more appropriate than nonstandard units.

·          Identify when two measurements are not necessarily equal (e.g., 1 pace long can represent). [CU, MC]).

Understand the differences between length units and area (square) units in U.S. or metric systems. W

·          Measure perimeter and area for regular and irregular shapes (e.g., use tiles, inches, or grid paper to find perimeter or area of mats, CDs, or skateboards). [SP, RL, MC]

·          Compare and describe area measurements made using different units (e.g., square inches vs. square centimeters). [SP, RL]

·          Describe how the unit chosen to measure linear dimensions can determine the unit used to measure area (e.g., measuring perimeter in cm produces an area in square cm) [CU].

Understand degrees (30°, 45°, 60°, and 90°) as units of measurement for angles. W

·          Demonstrate understanding of the concept of degrees in angles as units of measurement - specifically 30°, 45°, 60°, 90°, and 180° angle measurements and their relation to right angle.

·          Measure angles to the nearest 5 degrees using a protractor, angle ruler, or other appropriate tool. [SP, RL]

·          Measure angles in assorted polygons and determine the total number of degrees in the polygon. [SP, RL]

·          Describe relationships between angle measures (e.g., two 30° angles have the same total measure as one 60° angle) [CU]

·          Explain how degrees are used as measures of angles (e.g., a circle can be divided into 360°).

1.2 Understand and apply concepts and procedures from measurement.

Grade 3

Grade 4

Grade 5

Attributes, Units and Systems

1.2.3

Understand how measurement units of length (U.S.) and capacity (U.S.) are organized into systems.

·          Describe the various units of measurement for length and capacity and explain how they are organized.

·          Explain the benefits and appropriate uses of standard units of measurement for length and capacity using  our customary (U.S.) system. [CU]

Understand how measurement units of time and weight (U.S.), are organized into systems. W

·          Demonstrate understanding of how units of measurement are organized into time and weight in U.S. systems.

·          Know and correctly label the basic units of measurement for time and weight measure in the metric and customary system. [CU]

·          Explain the benefits and appropriate uses of standard units of measurement for area using both customary and metric systems.

Understand how measurement units of capacity, weight, and length are organized in the metric system. W

·          Demonstrate understanding of how units of measurement are organized into capacity, weight and length in the metric systems.

·          Explain and give examples of the metric system standard units for capacity, weight, and length..

Procedures, Precision, and Estimation

1.2.4

Understand and apply systematic procedures to measure length, perimeter, time, weight, money value, and temperature. W

·          Use systematic procedures to measure length, perimeter, time, weight, money value, and temperature to describe and compare objects.

·   Identify attribute to measure

·   Select and use appropriate units

·   Select and use tools that match the unit

·   Count, or compute, and label measures

·          Determine the attribute to be measured.

·          Explain and use a method for making change with coins. [CU].

Understand and apply systematic procedures to determine the area of figures composed of rectangles. W

·          Use systematic procedures to measure, describe and compare the areas of figures composed of rectangles..

·   Identify attribute to measure

·   Select and use appropriate units

·   Select and use tools that match the unit

·   Count, or compute, and label area measures

·          Explain and use a method for measuring the area of an irregular shape (e.g., Describe an irregular shape in terms of the composition of regular figures). [CU]

Understand and apply systematic procedures to determine the areas of rectangles and right triangles. W

·          Use systematic procedures to measure, describe and compare the areas of rectangles (including squares) and right triangles.

·   Identify attribute to measure

·   Select and use appropriate units

·   Select and use tools that match the unit

·   Count, or compute, and label measures

·          Determine the appropriate unit to measure the area of objects (e.g., square cm, sq. feet, and sq. miles). [SP, RL]

·          Use measurements of area to describe and compare rectangles

·          Select an appropriate tool according to the unit chosen. [MC]

·          Compare the object being measured with the units on the tool being used and record and label the units. [SP, RL, CU]

·          Explain and use a method for measuring the area of a rectangle. [CU]

1.2 Understand and apply concepts and procedures from measurement.

Grade 3

Grade 4

Grade 5

Procedures, Precision, and Estimation

1.2.5

Understand and apply formulas to measure perimeter of rectangles. W

·          Demonstrate understanding of or use formulas to find the perimeter of any rectangle.

·          Explain how to find the perimeter of any rectangle using a rule

·          Explain why linear units are used for perimeter. [CU]

·          Explain how to use a formula to find the perimeter of a rectangle. [CU]

·          Find and compare all possible rectangles with whole number dimensions and a whole number perimeter. (e.g. Find all possible rectangles with a perimeter of 28 and whole number measures as lengths of sides). [SP, RL, CU]

Understand and apply formulas to measure area of rectangles and right triangles. W

·          Demonstrate understanding of or use formulas to find the area of any rectangle or right triangle

·          Explain how to find the area of any rectangle using a rule

·          Explain why square units are used for area. [CU]

·          Explain and use formulas to find the area of a rectangle. [CU]

·          Explain and use a formula to find the area of a right triangle. [CU]

·          Find and compare all possible rectangles or right triangles with whole number dimensions with a given area (e.g., a rectangle with an area of 24 square feet could be 1’x24’, 2’x12’,3’x8’, or 4’x6’). [SP, RL, CU]

·           

1.2.6

Understand and apply strategies to obtain reasonable estimates of linear, time, weight, and temperature measurements. W

·          Identify situations in which estimated measurements are sufficient; estimate length, perimeter, time, money, weight or temperature.

·          Estimate a measurement using standard or nonstandard units (e.g., paper clips, inches, minutes, or foot lengths). [SP, RL]

·          Use referents to standard units (e.g., width of pinkie finger is similar to a centimeter). [MC]

Understand and apply strategies to obtain reasonable estimates of area measurements for irregular figures. W

·          Identify situation in which estimate measurements are sufficient; estimate areas of irregular figures.

·          Compare areas of irregular shapes with different perimeters {e.g., leafs, ponds) [SP, RL, MC]

·          Apply a process that can be used to find a reasonable estimate of the area measurement of an irregular shape (e.g., use tiles or pieces of paper to measure leafs, ponds) [SP, RL, CU]

·          Describe a procedure to estimate the area of an irregularly shaped room. [SP, RL, CU

Understand and apply strategies to obtain reasonable estimates of angles, and area measurements for rectangles and triangles. W

·          Identify situation in which estimated measurements are sufficient; estimate measures of angles  and areas in rectangles and triangles

·          Estimate a measurement using standard or nonstandard units (e.g., tiles, square feet, note cards). [SP, RL]

·          Use estimation to justify reasonableness of a measurement (e.g., estimate the area of the classroom by using carpet squares). [SP, RL]

·          Determine whether an angle is closest to 30° 45°,60°, 90°

1.3 Understand and apply concepts and procedures from geometric sense.

Grade 3

Grade 4

Grade 5

Properties and Relationships

1.3.1

Understand the concept of congruence. W

·          Identify, describe and compare congruent 2-dimensional geometric figures. [SP, RL, CU, MC]

·          Given a variety of figures, determine which figures are congruent. [MC]

·          Draw a shape that is congruent to a given2-dimensional shape. [CU]

·          Explain congruence and use an example to demonstrate it.

Understand and illustrate concepts of parallel and perpendicular lines and line symmetry in 2-dimensional shapes and  figures.