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

Number and numeration

1.1.1 Understand the concepts of fractions and decimals.  W

·   Represent mixed numbers, 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. [RL]

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

·   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]

·   Convert between improper fractions and mixed numbers. [MC]

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

·   Compare, order, or illustrate whole numbers, decimals, and fractions (denominators of 2, 3, 4, 5, 6, or 10) using concrete models (e.g., number line or shaded grid) or implementing strategies (e.g., like denominators, benchmarks, conversions). [RL, CU]

·   Determine equivalence among fractions. [RL]

·   Explain why one fraction is greater than, equal to, or less than another fraction. [CU]

·   Explain why one decimal number is greater than, equal to, or less than another decimal number. [CU]

1.1.3 Understand and apply the concept 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. [CU]

·   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]

·   Factor a number into its prime factors.

·   Determine whether one number is a factor of another number.

Computation

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

·   Explain the meaning of adding and subtracting fractions and decimals using words, symbols, or other models (e.g., fractions with denominators of 2, 4, 8 or 2, 3, 6, 12 or 5, 10 ─ highest LCM of 12). [CU]

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

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

·   Create or identify a representation of addition or subtraction of non-negative decimals or fractions.

·   Demonstrate the effect of multiplying a whole number by a decimal number. [CU]

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

·   Add and subtract like-denominator fractions (denominators of 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 20, and 100) and non-negative decimals.

·   Explain a strategy for adding fractions. [CU]

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

·   Use calculators to multiply or divide with two decimal numbers in the hundredths and/or thousandths place.

1.1.7 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 non-negative decimals and like-denominator fractions.  W

·   Identify when an approximation is appropriate.

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

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

·   Determine reasonableness of estimated answers for a given situation. [RL]

·   Demonstrate or explain various strategies used during estimation. [CU]

Attributes, units, and systems

1.2.1 Understand the concept of angle measurement.  W

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

·   Identify angles in the environment. [MC]

·   Classify or sort angles as right, acute, or obtuse. [RL, 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 degrees (30°, 45°, 60°, 90°, and 180°) as units of measurement for angles.  W

·   Describe an angle in relation to a right angle. [RL]

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

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

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

·   Identify, draw, or demonstrate angles that match or approximate 30°, 45°, 60°, 90°, and 180°. [CU]

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

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

·   Demonstrate or explain how grams are organized into kilograms. [CU]

·   Demonstrate or explain how millimeters are organized into centimeters and how centimeters are organized into meters. [CU]

·   Demonstrate or explain how milliliters are organized into liters. [CU]

Procedures, precision, and estimation

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

·   Select and use appropriate units for measuring area (e.g., square units) or dimensions.

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

·   Explain a method for measuring the area of a rectangle or right triangle (e.g., use the formula for the area of a rectangle or triangle, select grid paper). [CU]

·   Use measurements of area to describe and compare rectangles or triangles.

·   Solve problems involving measurement of area in rectangle and triangle (e.g., create a design using triangles and rectangles and determine how much paint is needed to cover the area of each of the shapes). [SP]

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

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

·   Explain how to find the perimeter or area of any rectangle using a rule. [CU]

·   Explain and use formulas to find the perimeter or 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 perimeter or area (e.g., a rectangle with an area of 24 square feet could be 1’x24’, 2’x12’,3’x8’, or 4’x6’). [RL, CU]

·   Explain why formulas are used to find area and/or perimeter. [CU]

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

·   Identify situations in which estimated measurements are sufficient.

·   Estimate measures of angles and areas in rectangles and triangles.

·   Estimate a measurement using standard or non-standard units (e.g., tiles, square feet, note cards).

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

·   Determine whether an angle is closest to 30° 45°, 60°, 90°, or 180°.

·   Explain or identify an appropriate process for estimating area or angle measurement. [CU]

Properties and relationships

1.3.1 Understand properties of angles and polygons.  W

·   Explain the difference between a regular and irregular polygon. [CU]

·   Identify, sort, classify, or explain the properties of angles, polygons, or circles based on attributes (e.g., triangles [right, equilateral, isosceles, or scalene], angles [acute, right, obtuse, or straight], or quadrilaterals [squares, rectangles, parallelograms, or trapezoids]). [RL,  CU]

·   Construct a geometric shape using geometric properties. [MC]

1.3.2 Apply understanding of the properties of parallel and perpendicular and line symmetry to two-dimensional shapes and figures.  W

·   Identify, name, compare, and sort parallel and perpendicular lines in two-dimensional figures. [SP, RL, CU]

·   Draw and label a design that includes a given set of attributes (e.g., create a design that has only two lines of symmetry; parallel and perpendicular lines). [SP, CU]

·   Sort figures based on characteristics of parallel lines, perpendicular lines, and/or lines of symmetry.

·   Draw figures or shapes that have particular characteristics (e.g., create a figure that has two parallel lines and one line of symmetry).

·   Identify parallel and perpendicular lines and/or lines of symmetry in the environment.

·   Construct a geometric shape using given geometric properties. [CU]

·   Use technology to draw figures with given characteristics. [MC]

Locations and transformations

1.3.3 Apply understanding of the location of non-negative rational numbers on a positive number line.  W

·   Use a number line to order fractions or decimals from least to greatest (e.g., not limited to a number line marked from 0 to 1). [SP, RL]

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

·   Identify the appropriate values of points on an incomplete number line involving fractional or decimal increments (e.g., using a ruler, reading a fuel gauge). [CU]

1.3.4 Apply understanding of translations (slides) or reflections (flips) to congruent figures.  W

·   Identify a specific transformation as a translation (slide) or reflection (flip). [CU]

·   Given a shape on a grid, perform and draw at least one transformation (i.e., translation or reflection). [SP, RL]

·   Draw congruent figures and shapes in multiple orientations using a transformation. [SP, RL]

·   Explain a series of transformations in art, architecture, or nature. [CU, MC]

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

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

Probability

1.4.1 Understand the likelihood (chance) of events occurring.  W

·   Predict and test how likely it is that a certain outcome will occur (e.g., regions of a spinner, flip of a coin, toss of dice). [SP, RL]

·   Represent the probability of a single event on a scale of 0 to 1. [MC]

·   Given a fair game, create an advantage for one of the players (e.g., if the game selecting marbles include more marbles of one color than the other). [SP, RL]

·   Explain the likelihood of a single event. [CU]

·   Determine whether a game for two people is fair. [RL]

·   Create a game that would make it more or less likely for an event to happen. [SP]

1.4.2 Understand and apply the Fundamental Counting Principle to situations.  W

·   Calculate the number of different combinations of different objects (e.g., three shirts and two pants could be combined in 3 x 2 = 6 ways).

·   Describe a situation that might include three different selections combined (e.g., describe a situation that could be calculated by 10 x 10 x 26 ─ two digits and a letter of the alphabet). [CU]

Statistics

1.4.3 Understand how different collection methods or different questions can affect the results.  W

·   Ask the same question using different data collection methods that result in other points of view being supported and explain why the method affected the data. [SP, RL, CU]

·   Explain how different data collection methods affect the nature of the data set with a given question (e.g., phone survey, internet search, person-to-person survey). [CU, MC]

·   Identify or describe the appropriate sample for a given question.

·   Identify or describe the appropriate population for a given sample.

1.4.4 Understand and apply the mean of a set of data.  W

·   Explain how to find the mean of a set of data and explain the significance of the mean. [CU].

·   Find the mean from a given set of data using objects, pictures, or formulas.

·   Given a problem situation, determine and defend whether mean, median, or mode is the most appropriate measure of average. [SP, RL, CU, MC]

·   Compare the mean, median, and mode for a given set of data. [RL]

·   Find and compare mean for two samples from the same population. [RL]

1.4.5 Apply strategies to organize, display, and interpret data.  W

·   Read and interpret data from text, line and bar graphs, histograms, stem-and-leaf plots, and circle graphs and determine when using each of these is appropriate.

·   Use histograms, pictographs, and stem-and-leaf plots to display data. [CU, MC]

·   Construct assorted graphs that include labels, appropriate scale, and key. [CU]

·   Determine what type of data should be represented on a bar graph, circle graph, histogram, or line graph. [RL]

·   Compare the consistency of results from two different displays that address the same question.

Patterns, functions, and other relations

1.5.1 Understand patterns of objects including relationships between two sets of numbers based on a single arithmetic operation.  W

·   Extend or create patterns of numbers, shapes, or objects based on a single arithmetic operation between the terms.

·   Determine the operation that changes the elements of one set of numbers into the elements of another set of numbers (e.g., if one set is 1,2,3,… and another set is 5,10, 15 …, one rule is to multiply each number in the first set by 5 to get the corresponding number in the second set). [RL]

·   Explain why a given rule fits a pattern based on a single arithmetic operation in the rule. [RL, CU]

1.5.2 Apply understanding of a pattern to develop a rule describing the pattern including combinations of two arithmetic operations.  W

·   Use the rule for a pattern which may include a combination of two arithmetic operations to extend a pattern. [SP, RL]

·   Solve a problem that uses a pattern of alternating operations (e.g., a frog climbed up 3 feet each day and then slipped down 1 foot each night, how long did it take the frog to reach the top of the building that is 15 feet high?). [SP]

·   Analyze a pattern to determine a rule with two operations between terms. [RL]

·   Use a rule to generate a pattern.

Symbols and representations

1.5.3 Apply understanding of the concept of mathematical inequality.  W

·   Express relationships between quantities using “≠, £, or ³”.

·   Given a number sentence using ≠, £, or ³, identify or write a situation that represents it. [CU, MC]

·   Given a real-world situation, use =, ≠, £, or ³ to describe quantities. [RL, , MC]

·   Explain inequality and the use of “≠”, “£”, or “³”. [CU]

1.5.4 Understand how to represent situations involving one operation or two alternating arithmetic operations.  W

·   Translate a situation involving one arithmetic operation into algebraic form using equations, tables, and graphs. [CU, MC]

·   Translate a situation involving two alternating arithmetic operations into algebraic form using equations, tables, and graphs (e.g., a snail crawls up 3 feet each day and slides back 2 feet each night). [CU, MC]

·   Identify or describe a situation involving one arithmetic operation that may be modeled by a graph. [CU]

·   Identify or describe a situation involving two alternating arithmetic operations that may be modeled by a graph (e.g., a snail crawls up 3 feet each day and slides back 2 feet each night). [CU]

Evaluating and solving

1.5.5 Understand and apply a variety of strategies to evaluate expressions with division.  W

·   Evaluate expressions with division using manipulatives, pictures, and symbols.

·   Substitute a symbol for a numeric value in an expression (e.g., X = 4, find 20 ¸ X; if H= 12 and t= 36, what is t¸H?). [SP, RL]

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

·   Solve for a missing value in an equation involving division (e.g., 12 ¸£ = 3). [SP, RL]

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

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

Example: Mrs. Allen’s class won a pizza party sponsored by the PTA for best school attendance. There are 30 students in the class. Ten pizzas arrived but they were cut in three different ways. Three pizzas were cut in eighths, three were cut in fourths, and four were cut in halves. Mrs. Allen wouldn’t let the students start eating until she was sure everyone received equal shares.

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., draw pictures, ask questions, or paraphrase information provided: 30 students in a class have ten pizzas to divide fairly. Three are sliced in eighths, three are sliced in fourths and four are sliced in halves).

·   Generate questions that would need to be answered in order to solve the problem (e.g., how should the pizzas be sliced? Can we use the slices that have already been made? How many pieces is each student’s fair share?).

·   Identify known and unknown information (known: number of students, number of pizzas to share; the ways in which the pizzas have been sliced; unknown: size of each slice, number of equal slices, number of pieces per student).

·   Identify information that is needed or not needed (e.g., needed: number of students, number of pizzas, how pieces have already been cut; not needed: reason for the pizza party).

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

·   Gather and organize the necessary information or data from the problem (e.g., draw pictures, create a chart or table, or use models to organize information).

·   Determine what tools should be used to construct a solution (e.g., paper and pencil, pictures, physical models).

2.2.2 Apply mathematical tools to solve the problem.  W

·   Use strategies to solve problems (e.g., draw pictures, use physical models).

·   Use appropriate tools to solve problems (e.g., paper and pencil, mental math, manipulatives).

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

EALR 3: The student uses mathematical reasoning.

Example: Mrs. Allen’s class won a pizza party sponsored by the PTA for best school attendance. There are 30 students in the class. Ten pizzas arrived but they were cut in three different ways. Three pizzas were cut in eighths, three were cut in fourths, and four were cut in halves. Mrs. Allen wouldn’t let the students start eating until she was sure everyone received equal shares.

3.1.1 Analyze information in familiar situations.  W

·   Break down the research information in order to explain or paraphrase it (e.g., 26 students need to share ten pizzas equally. The pizzas are already sliced, but not evenly.  Using eighths, determine how the pizza can be cut and shared equally).

3.2.1 Apply prediction and inference skills.  W

·   Make a reasonable prediction based on prior knowledge and investigation of situation (e.g., using mental math, predict how many pieces each student will receive).

·   Defend prediction with evidence from the situation.

·   Make inferences (conjectures) using information from the situation or data to support the inference (e.g., all the pizzas were the same size when whole).

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 strategies and tools used (e.g., drawing pizzas, fraction wheels or strips, paper and pencil calculations).

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., compare the results from students who used physical models vs. those who used computation).

·   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 how the value of a fraction changes in relationship to the size of the whole (e.g., half a pizza vs. half a cookie). [1.1.1]

·   Create three-dimensional shapes from two-dimensional figures (e.g., cylinder from two circles and a rectangle) and explain the relationship. [1.3.2]

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).

·   Ask the same question using different data collection methods that result in other points of view being supported. [1.4.3]

·   Explain how different data collection methods affect the nature of the data set with a given question (e.g., phone survey, person-to-person survey, internet search). [1.4.3]

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

·   After reading a text, generate questions and develop a survey (e.g., to determine how many students agree or disagree with the author).

·   Identify and use data from text passages, histograms, stem-and-leaf plots, and circle graphs. [1.4.5]

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

·   Determine the best method for organizing and representing information for a specific purpose (e.g., a physical model or a calculation to inform the teacher how many pieces of pizza each student should receive).

·   Represent and interpret all possible outcomes of experiments (e.g., an organized list, a table, a tree diagram, or a sample space). [1.4.2]

·   Construct assorted graphs including histograms, pictographs, and stem-and leaf-plots that include labels, appropriate scale, and key. [1.4.5]

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

·   Explain the value of a given digit in a decimal to at least the thousandths place. [1.1.1]

·   Describe a procedure for measuring an angle.

·   Describe relationships between angle measures (e.g., two 30° angles have the same total measure as one 60° angle). [1.2.2]

·   Draw and label a design that includes a given set of attributes. [1.3.2]

·   Explain how to find the mean of a set of data and explain the significance of the mean. [1.4.4]

·   Given an expression or equation, identify or write a situation that represents it. [1.5.3]

EALR 5: The student understands how mathematical ideas connect within mathematics, to other subjects, 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

·   Explain why angle measure does not change when the size of the circle or length of the sides of the angle change. [1.2.3]

·   Interpret skew, clusters, and gaps in given one-variable data displays. [1.4.5]

·   Translate a situation involving one arithmetic operation into algebraic form using equations, tables, and graphs.

·   Judge the appropriateness of inferences made from a set of data and support the judgment. [1.4.6]

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

·   Use factors and multiples to rename equivalent fractions. [1.1.1]

·   Determine equivalence among fractions. [1.1.2]

·   Graphically represent the same data in two different ways.

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

·   Find the mean from a given set of data using objects, pictures, or formulas.

·   Interpret skew, clusters, and gaps in given one-variable data displays.

·   Use estimation strategies and identify the reasonableness of answers. [1.1.8]

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 probability theory?).

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

·   Identify angles in the environment (e.g., in architecture, furniture, nature). [1.2.1]

·   Identify types of angles in polygons on a plane and in the environment. [1.2.1]

·   Solve problems involving angle measurements in real life situations (e.g., determine if a piece of tile will fit in a corner by measuring the angle). [1.2.3]

·   Determine whether a situation needs a precise measurement or an estimated measurement. [1.2.6]

·   Explain a series of transformations in art, architecture, or nature. [1.3.4]