EALR 1: The student understands and applies the concepts and procedures of mathematics.

Number and numeration

1.1.1 Understand the concept of decimals (money) and fractions.  W

·   Interpret fractions as parts of a whole object, number, or set (e.g., half of a medium pizza and half 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]

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

·   Explain or show how a fraction can be decomposed into smaller fractions (e.g., ¾ = ¼ + ¼ + ¼).

1.1.2 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. [RL]

·   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]

·   Demonstrate or show the order of like denominator fractions using pictures or objects. [CU]

1.1.3 Understand and apply the associative property of addition and multiplication and the commutative, 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, pictures, or physical models.  [CU]

·   Determine whether equations are 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). [CU]

·   Determine whether equations are true or false and explain, based on any of the properties (e.g., 14 + (62 + 38) = (14 + 62) + 38). [CU]

·   Demonstrate commutative, associative, or identity properties of addition or multiplication using pictures or objects. [CU]

Computation

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

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

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

·   Represent addition or subtraction of like-denominator fractions that represent sets of objects (e.g., ¼ of 24 marbles plus ¼ of 24 marbles = 2/4 of 24 marbles or 12).

·   Demonstrate the meaning of addition or subtraction of like denominators with multiple examples. [CU]

1.1.6 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.

·   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. [CU, MC]

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

·   Use calculators to compute with large numbers (e.g., multiplying tow digits times three digits; dividing three or four digits by two digits without remainders).

1.1.7 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.

Estimation

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

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

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

Attributes, units, and systems

1.2.1 Understand the concept of area.  W

·   Demonstrate and explain how area covers a shape and perimeter encloses a shape. [CU, MC]

·   Describe situations where area is the needed measurable attribute (e.g., buying carpet to cover a floor, painting a wall, building fishnets based on fishing ground, calculating needed area for teepees and lodges, amount of area needed for a pow-wow, describing the amount of floor space in a room).  [CU, MC]

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

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

1.2.2 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 centimeters produces an area in square centimeters).

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

·   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. [CU]

·   Demonstrate or explain how seconds are organized into minutes, minutes into hours, hours into days, days into weeks, and weeks into years. [CU]

·   Demonstrate or explain how months are organized into years. [CU]

·   Demonstrate or explain how ounces are organized into pounds. [CU]

Procedures, precision, and estimation

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

·   Select and use appropriate units (e.g., square units).

·   Select and use tools that match the unit (e.g., grid paper, squares).

·   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]

·   Solve problems involving area measurement. [SP]

·   Analyze a measurement situation and determine whether measurement has been done correctly. [RL]

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

·   Identify situations in which estimate measurements are sufficient.

·   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 leaves, ponds). [SP, RL, CU]

·   Compare areas of irregular shapes with different perimeters (e.g., leaves, ponds). [RL, MC]

·   Explain whether estimation or precision is needed in a given situation. [CU]

·   Determine whether a given measurement is exact or an estimate.

Properties and relationships

1.3.1 Understand concepts of parallel and perpendicular lines and line symmetry in two-dimensional shapes and figures.  W

·   Identify symmetrical two-dimensional figures and shapes (e.g., quilt blocks, textiles). [CU]

·   Complete a picture or design from a variety of cultures that incorporate a line of symmetry (e.g., basket design, beadwork, quilts, pyramids, nature).

·   Identify and draw a line of symmetry (e.g., folding or using a mirror). [CU]

·   Identify parallel and perpendicular lines in two-dimensional figures and shapes and in the environment. [MC]

·   Describe characteristics of two-dimensional geometric figures using appropriate vocabulary of parallel, perpendicular, symmetric (e.g., the U.S. flag, a stop sign, a yield sign, a race track, a football field). [CU, MC]

·   Explain parallel and perpendicular and give examples to demonstrate them. [CU]

1.3.2 Apply understanding of congruence to two-dimensional shapes and figures.  W

·   Identify, describe, and compare attributes of congruent figures in multiple orientations. [CU, SP, RL]

·   Build and draw congruent figures. [CU]

·   Identify, name, compare, and sort congruent two-dimensional figures and shapes in multiple orientations. [RL]

·   Solve problems involving congruence (e.g., create a design made out of congruent shapes). [SP]

Locations and transformations

1.3.3 Apply understanding of the location of points on a coordinate grid in the first quadrant.  W

·   Describe the location in the first quadrant on a coordinate grid in terms of horizontal and vertical position (e.g., to the right and up, longitude and latitude). [CU, MC]

·   Plot a given set of ordered pairs in the first quadrant of a coordinate grid. [CU]

·   Give directions from one location to another using ordered pairs in the first quadrant of a coordinate grid (e.g., given a state map, specify location of landmarks). [CU, MC]

1.3.4 Understand and apply single transformations using a translation (slide) or reflection (flip).  W

·   Simulate translations and reflections using objects (e.g., pattern blocks, geo blocks). [MC]

·   Record results of a translation or a reflection (e.g., given a polygon on a grid, translate or reflect it and list the new ordered pairs of the vertices). [CU]

·   Identify and draw a single translation (slide) or a single reflection (flip). [CU]

·   Create designs using translations and/or reflections. [SP]

Probability

1.4.1 Understand when events are certain or impossible and more likely, less likely, or equally likely.  W

·   Identify the likelihood of events and use the vocabulary of probability (e.g., weather, if homework will be assigned, simple games). [CU, MC]

·   Place events in order of likelihood of occurrence (e.g., use a number line marked from 0 to 1). [SP, RL, MC]

·   Distinguish between events that are certain or uncertain. [RL]

·   Place events in order based on their likelihood of occurrence. [RL]

·   Identify or describe possible and impossible events.

·   Determine what events are more likely, less likely, or equally likely to happen given an area model (e.g., a spinner with different sized sections).

Statistics

1.4.3 Understand and apply data collection methods to obtain the desired information.  W

·   Identify appropriate questions and populations to obtain the desired kind of information.

·   Formulate questions for surveys and collect data. [CU]

·   Decide whether to conduct a survey, use observations, or measure for a given question. [RL]

·   Make a plan to answer a question including how to record and organize data. [RL, CU, MC]

·   Determine which of several questions is most likely to give the desired information. [RL]

1.4.4 Understand and apply median and range to describe a set of data.  W

·   Use a variety of strategies to determine median and range from a set of data (e.g., use a graph, pictures, or objects).

·   Calculate the range of a data set.

·   Compare the mode and median from a set of data and determine which measure better describes the average. [RL]

·   Explain what the median represents and how to find it in a set of data. [CU]

·   Explain what the range represents and how to find it in a set of data. [CU]

·   Determine data points that would result in a given median. [RL, SP]

1.4.5 Understand representations of data from line plots and pictographs.  W

·   Read data from line plots and pictographs.

·   Describe a trend from a given line plot. [CU, MC]

·   Interpret a pictograph where the scale is other than one unit. [RL]

·   Create two different graphic displays using a set of data. [CU, MC]

·   Read and interpret data from line plots and pictographs. [RL, CU]

·   Use technology to create pictographs.

·   Explain the data in a given table, chart, or graph. [CU]

·   Analyze the completeness and accuracy of data in a graph given a set of data. [RL]

Patterns, functions, and other relations

1.5.1 Understand patterns of objects including number patterns using addition, subtraction, or multiplication based on a single arithmetic operation.  W

·   Extend or create patterns of numbers, shapes, or objects using addition, subtraction, or multiplication based on a single operation between terms.

·   Extend and represent patterns using words, tables, numbers, and pictures. [RL, CU]

·   Create a number pattern and explain what makes it a pattern. [CU]

1.5.2 Understand a pattern to develop a rule describing the pattern which may include a single arithmetic operation.  W

·   Use the rule for a pattern which may include a single arithmetic operation to extend or fill in parts of a pattern.

·   Solve a problem that uses a pattern with a single operation. [SP]

·   Model growing patterns using objects and pictures (e.g., a stair step sequence, or a “growing” L shape in which a unit is added to each leg to show 3, 5, 7, 9, . . .). [RL, CU]

·   Describe the rule for a pattern based on one operation (e.g., add 4, multiply by 2). [CU]

·   Analyze a pattern to determine a rule. [RL]

·   Use a rule to generate a pattern.

Symbols and representations

1.5.3 Apply understanding of the concept of mathematical inequality.  W

·   Compare multiplication or division expressions using the symbols >, <, and = (e.g., 5 x 3 > 3 x 2). [RL]

·   Select operational and relational symbols to make a multiplication or division number sentence true (e.g., 4 _ 3 = 12; 5 x 12 _ 64).

·   Explain inequality and the use of “>” or “<” in inequalities. [CU]

·   Identify or write a situation that represents it given an expression or equation using < or >. [CU, MC]

1.5.4 Understand and apply operational and relational symbols and notations to write expressions and equations involving multiplication and division.  W

·   Identify and use mathematical symbols and notations in reading and writing expressions and equations.

·   Write a situation that represents it given an equation involving multiplication or division. [CU, MC]

·   Write an equation that represents it given a situation involving multiplication or division. [CU, MC]

Evaluating and solving

1.5.5 Understand and apply a variety of strategies to evaluate expressions with addition, subtraction, or multiplication.  W

·   Substitute a numeric value for a symbol in expressions or equations (e.g., if £ = 7, find £ x 3; if w= 12 and I= 36, what is w x I?).

1.5.6 Understand and apply strategies to solve equations that include multiplication.  W

·   Solve missing factor equations (e.g., £´ 3 = 12). [SP, RL]

·     Describe and compare strategies used to solve an equation with multiplication. [SP, RL, CU]

EALR 2: The student uses mathematics to define and solve problems.

Example: Jamal and his sister, Aleesha, want to buy a pet. Their mother said she will help by paying for the ongoing cost of food if they can save the money to buy the pet and all the needed equipment, bedding, and food to get started. They have $17.83 saved already and most of that money is in quarters. They are reading pet store ads to see what the costs would be if they bought a mouse, a hamster, or a guinea pig.

2.1.1 Analyze a situation to define a problem.  W

·   Use strategies/approaches to examine the situation and determine if there is a problem to solve (e.g., ask questions, make lists, or paraphrase information provided in ads: two kids want to buy a pet. They have some money but they need to find out if they can afford a mouse, hamster, or guinea pig and the equipment and food for it).

·   Determine the problem using information from investigation (e.g., do Jamal and Aleesha have enough money?).

·   Generate questions that would need to be answered in order to solve the problem (e.g., how much will each animal cost? How much is equipment and food for each animal?).

·   Identify known and unknown information (e.g., known: how much money Jamal and Aleesha have; unknown: all the costs for each animal).

·   Identify information that is needed or not needed (e.g., needed: all costs related to purchasing the animals, the amount that the kids have saved; not needed: the money is in quarters).

2.2.1 Apply strategies, concepts, and procedures to devise a plan to solve the problem.  W

·   Gather and organize data (e.g., determine how to break information into categories such as cost of animal, cost of cage, cost of food, cost of bedding, cost of equipment in order to create a table).

·   Determine what tools should be used to construct a solution (e.g., calculators, paper and pencil, calculator, mental math physical models such as play money).

2.2.2 Apply mathematical tools to solve the problem.  W

·   Use strategies to solve problems (e.g., column addition, play money to determine costs, and subtraction to determine how much money is needed if they don’t have enough).

·   Use appropriate tools to solve problems (e.g., paper and pencil, calculator, or physical models, play money).

·   Recognize when an approach is unproductive and try a new approach.

EALR 3: The student uses mathematical reasoning.

Example: Jamal and his sister, Aleesha, want to buy a pet. Their mother said she will help by paying for the ongoing cost of food if they can save the money to buy the pet and all the needed equipment, bedding, and food to get started. They have $17.83 saved already and most of that money is in quarters. They are reading pet store ads to see what the costs would be if they bought a mouse, a hamster, or a guinea pig.

3.1.1 Analyze information presented in familiar situations.  W

·   Break down the research information in order to explain or paraphrase it (e.g., each animal has costs related to cage, bedding, food which must be calculated in order to see if the kids have enough money to buy an animal).

3.2.1 Apply prediction and inference skills.  W

·   Make a reasonable prediction based on prior knowledge and investigation of situation (e.g., after reading the pet store ads, predict whether the kids will be able to buy a pet).

·   Defend prediction with evidence from the situation.

·   Make inferences (conjectures) using information from the situation or data to support the inference (e.g., guinea pig equipment/food is more expensive because the animal is larger and requires a bigger cage and pellets).

3.2.2 Apply the skill of drawing conclusions and support those conclusions using evidence.  W

·   Draw conclusions from displays, texts, or oral discussions and justify those conclusions with logical reasoning or other evidence.

3.2.3 Analyze procedures used to solve problems in familiar situations.  W

·   Describe and compare data organization methods (e.g., charts used for organizing costs for each animal). [1.4.3]

3.3.1 Understand how to justify results using evidence.  W

·   Check for reasonableness of results by using a different strategy or tool to solve the problem (e.g., use front end estimation to determine about how much each animal will cost).

·   Provide examples to support results.

3.3.2 Understand how to validate thinking about numerical, measurement, geometric, or statistical ideas by using models, known facts, patterns, or relationships.  W

·   Explain the meaning of decimal using physical models. [1.1.5]

·   Explain what the relative position of numbers on a positive number line means (e.g., to the right means greater than).  [1.3.3]

EALR 4: The student communicates knowledge and understanding in both everyday and mathematical language.

4.1.1 Understand how to develop and follow a plan for collecting information for a given purpose.  W

·   Determine how to collect information for a specific purpose or audience (e.g., to convince a parent or other adult, to demonstrate a need for change, to provide information).

·   Develop and follow a plan based on the kind of information needed, the purpose, and the audience (e.g., survey, gather data from a chart or graph, read in a text to gather information).

4.1.2 Understand how to extract information for a given purpose from one or two different sources using reading, listening, and observation.  W

·   Listen and observe to simulate translations and reflections using objects (e.g., pattern blocks, geo blocks). [1.3.4]

·   Read and follow directions using a coordinate grid (e.g., on a city street map). [1.3.3]

4.2.1 Understand how to organize information for a given purpose.  W

·   Organize information on a chart and create a summary of the results to inform a specific audience (e.g., chart all related costs for the purchase of each pet; write a summary explaining the results and the kids possible decisions based on the results).

·   Construct assorted line and pictographs that include labels, a scale that is not one, and a key. [1.4.5]

·   Create a chart or display to represent equivalent fractions. [1.1.2]

4.2.2 Understand how to communicate or represent ideas using mathematical language or notation.  W

·   Symbolically represent parts of a whole or parts of a set with common denominators. [1.1.1]

·   Use measurements of area to describe and compare objects. [1.2.1]

·   Describe a location in the first quadrant on a coordinate grid in terms of horizontal and vertical position (e.g., to the right and up, longitude and latitude). [1.3.3]

·   Describe a trend from a given line plot. [1.4.5]

·   Describe the rule for a pattern with a single arithmetic operation in the rule. [1.5.2]

EALR 5: The student understands how mathematical ideas connect within mathematics, to other subject areas, and to real-life situations.

5.1.1 Understand how to use concepts and procedures from any two of the content components in a given problem or situation.  W

·   Conduct a survey for a question; collect data, and use multiplication and/or division to compute the results of the survey. [1.1.6, 1.4.4]

·   Identify, describe, and compare attributes of congruent shapes in multiple orientations. [1.3.2]

5.1.2 Understand how to recognize equivalent mathematical models and representations in familiar situations.  W

·   Demonstrate and explain equivalent relationships between decimals and fractions (e.g., $.50 is equal to ½ a dollar and 50/100 dollar) using models. [1.1.2]

·   Interpret remainders of a division problem in a given situation (e.g., remainder 3 or 3/5). [1.1.6]

·   Represent addition and subtraction of decimals through hundredths using models (e.g., base ten blocks, fraction circles with decimal ring, money). [1.1.]

5.2.1 Apply mathematical patterns and ideas in familiar situations in other disciplines.

·   Read and interpret data from line plots and pictographs. [1.4.5]

·   Make a plan to answer a question including how to record and organize data. [1.4.3]

·   Use estimation strategies appropriately when the exact answer is not necessary. [1.1.7]

5.2.2 Know the contributions of individuals and cultures to the development of mathematics.

·   Recognize the contributions to the development of mathematics by women, men, and various cultures (e.g., what is the history of fractions?).

5.3.1 Understand that mathematics is used in daily life and extensively outside the classroom.

·   Describe situations where area is the needed measurable attribute (e.g., the pricing of buying carpet, painting a wall, picking largest bedroom). [1.2.1]

·   Measure perimeter and area for regular and irregular shapes (e.g., use tiles, inches, or grid paper to find perimeter or area of blankets, CDs, skateboards). [1.2.2]

·   Identify situations in which estimated measurements are sufficient and use estimation to obtain reasonable measurements. [1.2.6]

·   Identify parallel and perpendicular lines in two-dimensional shapes and figures and in the environment. [1.3.1]

·   Identify the likelihood of events and use the vocabulary of probability (e.g., weather, simple games, if homework will be assigned). [1.4.1]