Mathematics
Washington State Grade-Level Expectations
EALR 1: THE STUDENT UNDERSTANDS AND APPLIES THE CONCEPTS AND PROCEDURES OF MATHEMATICS.
Component 1.1: Understand and apply concepts and procedures from number sense.
Number and numeration
1.1.1 Understand the concept of number.
• Count to at least 31.
• Represent a number to at least 10 in different ways (e.g., numerals, spoken words, pictures, physical models). [CU]
• Show that the last count word names the quantity of the set (cardinality) (i.e., when counting fingers on a hand “one, two, three, four, five,” the “five” says how many fingers there are). [CU, MC]
• Identify the base ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Explain how numbers are used and give examples (e.g., to count, to order). [CU]1.1.2 Understand sequential relationships among whole numbers.
• Tell what number comes before or after a given number.
• Use comparative language (e.g., less than, more than, equal to) to compare numbers to at least 20. [CU]
• Use a known quantity to at least 10 (benchmark) to compare sets (e.g., sets of counters).
• Identify the ordinal position of objects at least through tenth (e.g., first, second …).Computation
1.1.5 Understand the meaning of addition.
• Express stories involving addition (e.g., join) with models, pictures, and symbols. [CU, MC]
• Use addition in the classroom environment (e.g., tables and chairs in the classroom). [MC]Component 1.2: Understand and apply concepts and procedures from measurement.
Attributes, units, and systems
1.2.1 Understand and apply appropriate terminology to compare attributes.
• Use comparative vocabulary to describe objects (e.g., longer/shorter, heavier/lighter, nearer/further, thicker/thinner, shorter/taller). [CU]
• Use terms to describe the duration of events (e.g., long time or short time). [CU]
• Identify and sort objects based on an attribute (e.g., color, shape, texture). [RL]
Procedures, precision, and estimation
1.2.4 Understand and apply procedures to measure with non-standard units.
• Use non-standard units to measure (e.g., paper strips, cubes, beans, hand widths).
• Explain how to use a non-standard unit to measure a given length (e.g., length of a table, width of a desk). [CU]Component 1.3: Understand and apply concepts and procedures from geometric sense.
Properties and relationships
1.3.2 Know the characteristics of familiar objects.
• Describe familiar objects based on characteristics (e.g., big, small, like a box). [CU, MC]
• Sort objects in their environment by characteristics (e.g., cans, balls, boxes, red, blue). [MC]
• Describe objects using comparative language (e.g., bigger, taller, shorter, smaller). [CU]Locations and transformations
1.3.3 Understand the relative position of objects in the environment.
• Describe the location of an object relative to another (e.g., in, out, over, under, behind, above, below, next to, etc.). [CU]
• Identify where a three-dimensional object is located relative to another given object (e.g., where the eraser is relative to the desk).Component 1.4: Understand and apply concepts and procedures from probability and statistics.
Statistics
1.4.3 Understand how data can be collected and organized.
• Use physical objects or pictures to build bar graphs. [CU]
• Organize objects into groups before counting them. [RL]1.4.5 Understand how a display provides information.
• Answer questions about graphs (e.g., how many cats? How many dogs?). [CU]Component 1.5: Understand and apply concepts and procedures from algebraic sense.
Patterns, functions, and other relations
1.5.1 Know how to recognize patterns.
• Identify and extend patterns (e.g., ABAB, green-green-blue, counting). [RL]
• Create an AB pattern.
Symbols and representations
1.5.3 Understand the concepts of equality and inequality.
• Use physical objects to model language (e.g., same, different, equal, not equal, more, less). [CU]
• Model/act out story problems to solve whole number equations and inequalities (e.g., there are three kids and two have three crayons, one has two crayons. How can you make it so all kids have the same number of crayons?). [CU, MC]
EALR 2: THE STUDENT USES MATHEMATICS TO DEFINE AND SOLVE PROBLEMS.
Component 2.1: Understand problems
Example: A classroom needs a playground ball for each student in the class. The class has fewer playground balls than are needed.Understand problems
2.1.1 Understand how to define a problem in a familiar situation with teacher guidance.
• State information presented in teacher-led discussion to determine if there is a problem that needs an answer (e.g., a classroom activity requires a playground ball for each student. There are some balls available in the classroom).
• State the problem in own words (e.g., are there enough playground balls? If not, how do we get enough for the class?).
• Generate questions that would need to be answered in order to solve the problem (e.g., how many balls are in the classroom? How many more do we need?).
• Identify known and unknown information with teacher guidance (e.g., known - the number of students in the class, and the number of balls needed; unknown - the number of additional playground balls needed). [1.1.5]Component 2.2: Apply strategies to construct solutions.
2.2.1 Understand how to create a plan to solve a problem with teacher guidance.
• Gather and organize categorical data (e.g., in a teacher-led activity, create a two-column chart - one column for student names and tally marks in the other to represent which students are assigned a ball). [1.4.3]2.2.2 Apply mathematical tools to solve the problem with teacher guidance. W
• Use appropriate tools to find a solution (e.g., draw pictures, use chart to count how many empty spaces there are for the playground balls). [1.1.1, 1.1.5]
• Recognize when an approach is unproductive and try a new approach.
EALR 3: THE STUDENT USES MATHEMATICAL REASONING.
Component 3.1: Analyze information.
Example: A classroom needs a playground ball for each student in the class. The class has fewer playground balls than are needed.3.1.1 Understand how to compare information presented in familiar situations with teacher guidance.
• Restate understanding of the situation (e.g., each student requires a playground ball; there are not enough in the classroom).Component 3.2: Make predictions, inferences, conjectures, and draw conclusions.
3.2.1 Understand how to make a reasonable prediction based on the information given in a familiar situation.
• Predict a numerical solution for a problem (e.g., guess how many more playground balls are needed).Component 3.3: Verify results
3.3.1 Understand how to justify results using evidence.
• Use tools (e.g., tally marks, physical models, words) to check for reasonableness of an answer (e.g., line up students; pass out the playground balls to students to see how many students do not receive one).
• Check reasonableness of an estimation by acting it out, using pictures, or physical models.
EALR 4: THE STUDENT COMMUNICATES KNOWLEDGE AND UNDERSTANDING IN BOTH EVERYDAY AND MATHEMATICAL LANGUAGE.
Component 4.2: Organize, represent, and share information
4.2.1 Understand how to organize information to communicate to a given audience with teacher guidance.
• Use a two-column chart to organize data (e.g., one column for student names and tally marks in the other to represent which students are assigned a ball) for the classroom with teacher guidance.
• Use physical objects or pictures to build bar graphs to answer a question generated by the class (e.g., how many of each kind of pet do we own?).
4.2.2 Understand how to communicate or represent ideas or information using mathematical language or notation.
• Explain or represent ideas using mathematical language from:
o Number sense (e.g., numbers 1 to 10) [1.1.1];
o Measurement (e.g., compare objects to describe relative size) [1.2.1];
o Geometric sense (e.g., name objects based on their characteristics - I have four equal sides, what am I?) [1.3.1];
o Algebraic sense (e.g., create a pattern such as AB). [1.5.1]
EALR 5: THE STUDENT UNDERSTANDS HOW MATHEMATICAL IDEAS CONNECT WITHIN MATHEMATICS, TO OTHER SUBJECT AREAS, AND TO REAL LIFE SITUATIONS.
Component 5.1: Relate concepts and procedures within mathematics.
5.1.1 Understand how to use concepts and procedures from any two of the content components from EALR 1 in a given problem or situation.
• Organize data collections (e.g., bar graph, sorted groups) and compare data using comparative language. [1.1.2, 1.4.3]
• Sort objects based on chosen attribute and create a simple AB pattern using the sorted objects. [1.3.2, 1.5.1]5.1.2 Understand how to recognize and create equivalent mathematical models and representations in familiar situations.
• Identify different representations of a number to 20 (e.g., numerals, pictures, physical models). [1.1.1]
• Express stories involving addition (e.g., join) with models, pictures, and symbols. [1.1.5]Component 5.2: Relate mathematical concepts and procedures to other disciplines.
5.2.1 Apply and analyze the use of mathematical patterns and ideas in familiar situations in other disciplines.
• Describe how math is used in science when a number of objects are needed for an experiment or measurement is used to illustrate change.
• Identify patterns in a piece of artwork.
Component 5.3: Relate mathematical concepts and procedures to real-world situations.5.3.1 Understand how mathematics is used in everyday life.
• Generate examples of mathematics in everyday life:
o counting (e.g., the number of people ahead of us in a line);
o sorting things (e.g., grouping socks by color in order to match them up);
o comparing things (e.g., who has the biggest piece of cake for dessert, or who is tallest/shortest in the family);
o pointing out patterns (e.g., in clothing, fence posts, designs on buildings).
• Identify objects based on a description of their geometric attributes (e.g., buildings have sides; some windows are shaped like a rectangle).
• Describe the location of objects relative to each other (e.g., in, out, over, under, school bus stops next to each other).
EALR 1: THE STUDENT UNDERSTANDS AND APPLIES THE CONCEPTS AND PROCEDURES OF MATHEMATICS.
Component 1.1: Understand and apply concepts and procedures from number sense.
Number and numeration
1.1.1 Understand different representations of whole numbers.
• Represent a number to at least 100 in different ways (e.g., numerals, pictures, words, physical models) and translate from one representation to another. [CU]
• Group and regroup objects into 1's and 10's.
• Count sets of objects less than 100 using a variety of grouping strategies.1.1.2 Understand sequential relationships among whole numbers.
• Order three or more numbers to at least 100 from smallest to largest. [RL]
• Use comparative language (e.g., less than, more than, equal to) to compare numbers to at least 100. [CU]
• Skip count by 2, 5, and 10.
• Count forward and backward, from a given number that is less than 100.Computation
1.1.5 Understand the meaning of subtraction.
• Express stories involving subtraction (e.g., separate) with models, pictures, and symbols. [CU, MC]
• Show relationships between addition and subtraction using physical models, diagrams, and acting out problems. [CU]1.1.6 Understand and apply procedures for addition of whole numbers with fluency.
• Use strategies (e.g., count on, count back, doubles) for addition to at least sums to 12. [SP, RL]
• Recall addition facts through at least sums to 12.
• Solve problems involving addition using procedures and explaining those procedures. [SP, RL, CU]1.1.7 Understand and apply strategies and appropriate tools for adding with whole numbers.
• Use strategies and appropriate tools from among mental math, paper and pencil, manipulatives, or calculator to compute in a problem situation. [SP, RL]
• Use counting strategies to combine whole numbers with sums under 12. [SP, RL]
Estimation
1.1.8 Understand and apply estimation strategies to determine the reasonableness of answers.
• Use a known quantity (e.g., chunking) to make reasonable estimates. [RL]
• Use numbers that are easy to add or subtract to make a reasonable estimate of a sum (e.g., 9 + 8 should be about 20, since 9 is about 10, 8 is about 10, and 10 + 10 is 20). [RL]Component 1.2: Understand and apply concepts and procedures from measurement.
Attributes, units, and systems
1.2.1 Understand and apply attributes to describe and compare objects.
• Order three or more objects according to an attribute (e.g., pencil lengths, students’ hand span, and thickness of books). [RL]
• Read a clock with only the hour hand and use approximate language (e.g., almost 7, a little after 7). [CU]
• Identify coins (penny, nickel, dime, quarter) and state their value. [CU]Procedures, precision, and estimation
1.2.4 Understand and apply procedures to measure with non-standard or standard units.
• Select units appropriate to the object being measured (e.g., measure length of classroom with footprints, not beans) and explain why it was selected. [CU]
• Use a uniform unit to measure an object (e.g., cubes, paper strips, ruler).
• Measure a variety of objects using appropriate non-standard tools (e.g., arm length, hand width, lengths of rope).
• Use a variety of records of time (e.g., calendar, seasonal plants, animal migrations, moon phases, tides, shadows).
• Use physical models of measuring units to fill, cover, match, or make the desired comparison of the attribute with the unit. [SP, RL]
• Explain the need for appropriate tools for measurement. [CU]Component 1.3: Understand and apply concepts and procedures from geometric sense.
Properties and relationships
1.3.2 Understand how to compare figures based on their characteristics.
• Describe two-dimensional figures based on their characteristics (e.g., number of sides, number of equal sides). [CU]
• Identify, compare, and sort two-dimensional figures in their surroundings (e.g., by lengths of sides, general shape). [RL, MC]
• Describe figures using accurate terminology (e.g., square, rectangle, triangle).
Locations and transformations
1.3.3 Understand the locations of numbers on a positive number line.
• Indicate whether a number is above or below a benchmark number (e.g., greater than or less than 100).
• Describe the location of a given number between 1 and 100 on a number line. [CU]
• Identify a point up to 100 on a positive number line.Component 1.4: Understand and apply concepts and procedures from probability and statistics.
Statistics
1.4.3 Understand how data can be organized and displayed.
• Display results of data collection by making student-invented and conventional displays. [CU]
• Construct bar graphs with physical materials and record pictorially (e.g., shoes, cats, crops, egg rolls, tacos). [CU]
• Collect data related to questions and organize the data into useful categories in familiar situations (e.g., how many students like apples? How many students do NOT like apples?).1.4.5 Understand how a display provides information.
• Answer questions about bar graphs or pictographs (e.g., how many dancers, plants, canoes, pets?). [CU]Component 1.5: Understand and apply concepts and procedures from algebraic sense.
Patterns, functions, and other relations
1.5.1 Understand the concept of patterns.
• Create and describe a variety of repeating patterns using sounds, objects, and symbols. [CU]
• Describe and extend a repeating pattern (e.g., ABAC, ABAC; snap, clap, snap, stomp). [CU]
• Identify the unit in a repeating pattern (e.g., in A-A-B-A-A-B the unit is A-A-B). [RL]
• Identify and describe numerical patterns in the 100’s chart. [CU, RL]
• Identify geometric patterns in art, textiles, and ceramics.Symbols and representations
1.5.3 Understand the meaning of symbols and labels used to represent equality in situations.
• Demonstrate equality by recording number sentences with balance using the “=” symbol (e.g., 9 = 4 + 5, 4 + 5 = 2 + 7, 9 = 9). [CU]
• Complete open sentences showing equalities (e.g., 5 = ____).
• Explain, using pictures or words, the meaning of equality. [CU]
• Give an example of equality in real life (e.g., on the first turn, Juan scored 4 points, on the second turn, he scored 5 points. On the first turn, Ivana scored 2 points, on the second turn, she scored 7 points. After two turns, they are tied with the same number of points). [MC]
EALR 2: THE STUDENT USES MATHEMATICS TO DEFINE AND SOLVE
PROBLEMS.
Component 2.1: Understand problems
Example: A classroom is presenting a play and everyone has invited two guests. Enough chairs are needed to seat all the guests. There are some chairs in the classroom.2.1.1 Understand how to define a problem in a familiar situation with teacher guidance.
• State information presented in a teacher-led discussion to determine if there is a problem (e.g., a classroom is having a play and each student invited two guests. Chairs are needed for the guests. There are some chairs available in the classroom).
• State the problem in own words (e.g., there aren’t enough chairs for the guests. How many more chairs do we need?).
• Generate questions that would need to be answered in order to solve the problem (e.g., how many guests are attending? How many more chairs do we need?).
• Identify known and unknown information with teacher guidance (e.g., known - number of students, number of guests invited, number of chairs in classroom; unknown - number of guests attending, number of chairs needed). [1.1.5]Component 2.2: Apply strategies to construct solutions
2.2.1 Understand how to create a plan to solve a problem with teacher guidance.
• Gather and organize categorical data (e.g., in a teacher-guided activity, create a two-column chart - one column for student names and the other to record the number of guests attending the play). [1.4.3]2.2.2 Apply mathematical tools to solve the problem with teacher guidance.
• Use strategies (chart to count, skip count, cluster, or physical models). [1.1.1, 1.1.5]
• Use appropriate tools from among mental math, paper and pencil, manipulatives, or calculator (e.g., to determine the total number of guests attending and the total number of chairs needed for the class play). [1.1.7]
• Recognize when an approach is unproductive and try a new approach.
EALR 3: THE STUDENT USES MATHEMATICAL REASONING.
Component 3.1: Analyze information
Example: A classroom is presenting a play and everyone has invited two guests. Enough chairs are needed to seat all the guests. There are some chairs in the classroom.3.1.1 Understand how to compare information presented in familiar situations.
• Restate understanding of the situation (e.g., each guest attending the play will require a chair; there are not enough in the classroom).Component 3.2: Make predictions, inferences, conjectures, and draw conclusions.
3.2.1 Understand how to make a reasonable prediction based on prior knowledge and the information given in a familiar situation.
• Predict a numerical solution for a problem (e.g., predict how many more chairs will be needed).
• Use known information to make a reasonable prediction (e.g., if two numbers are each less than 10, the sum will be less than 20).
• Make an inference based on information provided (e.g., the boys in class did a better job convincing their guests to attend because there are more guests coming for the boys than the girls).3.2.2 Understand how to draw conclusions based on prior knowledge and the information given in a familiar situation.
• Draw conclusions from displays using comparative language (e.g., more students have two guests coming, or fewer students have only one guest coming) and provide examples from displays to support conclusions.3.2.3 Analyze procedures used to solve problems in familiar situations with teacher guidance.
• Justify the importance of counting in a situation rather than making a guess at a number of items for a specific purpose (e.g., counting the number of chairs needed for the play rather than guessing).Component 3.3: Verify results.
3.3.1 Understand how to justify results using evidence.
• Check reasonableness of results by using pictures, physical models, or acting it out (e.g., students raise one hand for one guest attending and two hands if two guests are attending).3.3.2 Understand how to validate thinking about numerical, measurement, geometric, or statistical ideas by using models, known facts, patterns, or relationships.
• Explain why a strategy or tool was used in solving a problem (e.g., why a two-column chart was helpful to gather the information needed about the number of guests attending the play).
EALR 4: THE STUDENT COMMUNICATES KNOWLEDGE AND UNDERSTANDING
IN BOTH EVERYDAY AND MATHEMATICAL LANGUAGE.
Component 4.1: Gather information.
4.1.1 Understand how to develop and follow a simple plan for collecting information for a given purpose.
• Determine what information is needed and how to collect it for a given purpose (e.g., to help explain something, to find out if something is needed) and who the information is for (e.g., for the classroom, for the adults at home, for the librarian).
• Develop and follow a plan to gather data about an event (e.g., how many students will attend the Saturday Movie Afternoon at school?).4.1.2 Understand how to extract information for a given purpose from one or two different sources.
• Follow simple written directions for creating an art project using a model (e.g., requiring cutting and folding geometric shapes).
• Generate questions that could be answered using informational text (e.g., TV ads, books, menus, cereal boxes).Component 4.2: Organize, represent, and share information.
4.2.1 Understand how to organize information to communicate to a given audience with teacher guidance.
• Organize and display data on a chart to communicate solution for the given audience (e.g., use a two- or three-column chart to display the number of guests per student attending a class play and, if there is a chair for each guest, inform the custodian as to how many more chairs are needed).
• Display results of data collection by making student-invented and conventional displays (e.g., hair color, eye color, teeth missing).4.2.2 Understand how to communicate or represent ideas or information using mathematical language or notation.
• Explain or represent ideas using mathematical language from:
o Number sense (e.g., numbers to at least 100) [1.1.1];
o Measurement (e.g., order three or more objects according to an attribute and identify the chosen attribute) [1.2.1];
o Geometric sense (e.g., name and describe two-dimensional figures based on their characteristics) [1.3.1];
o Statistics (e.g., construct bar graphs with physical materials) [1.4.3];
o Algebraic sense (e.g., explain the meaning of equality). [1.5.3]
EALR 5: THE STUDENT UNDERSTANDS HOW MATHEMATICAL IDEAS
CONNECT WITHIN MATHEMATICS, TO OTHER SUBJECT AREAS, AND TO REAL LIFE SITUATIONS.
Component 5.1: Relate concepts and procedures within mathematics.
5.1.1 Understand how to use concepts and procedures from any two of the content components from EALR 1 in a given problem or situation.
• Interpret results and draw conclusions from student-made displays using comparative language (e.g., more, fewer). [1.4.4, 3.2.2]
• Measure objects using non-standard tools and place resulting numbers in order from shortest (smallest) to longest (largest). [1.2.3, 1.1.2]5.1.2 Understand how to recognize and create equivalent mathematical models and representations in familiar situations.
• Identify different representations of a number to at least 100 (e.g., numerals, pictures, physical models). [1.1.1]
• Express stories involving subtraction (e.g., separate) with models, pictures, and symbols. [1.1.5]Component 5.2: Relate mathematical concepts and procedures to other disciplines
5.2.1 Apply and analyze the use of mathematical patterns and ideas in familiar situations in other disciplines.
• Use the characteristics of two-dimensional shapes in art projects and recognize the use of geometric shapes in artwork.
• Use a clock to determine when it is time for recess or lunch time.
• Explain how math is used whenever we use money for a purchase.5.2.2 Know the contributions of individuals and cultures to the development of mathematics.
• Recognize the contributions of women, men,, and people from different cultures (e.g., look at symbols used for numbering in the Mayan culture).
Component 5.3: Relate mathematical concepts and procedures to real-world situations.5.3.1 Understand how mathematics is used in everyday life.
• Generate examples of mathematics in everyday life:
o counting (e.g., the pennies in the penny jar);
o comparing measurements (e.g., standing up against the mark on the wall to check for growth);
o building things (e.g., a snowman with three spheres, a dog house made of a box with a triangular roof);
o playing games (e.g., when counting spaces on a board or knowing money is needed)
• Describe familiar two-dimensional shapes based on their geometric characteristics (e.g., sharp corners, sides of different lengths).
• Identify and sort two-dimensional shapes in their surroundings.
• Skip count by 5s or 10s (e.g., with nickels or dimes).
EALR 1: THE STUDENT UNDERSTANDS AND APPLIES THE CONCEPTS AND PROCEDURES OF MATHEMATICS.
Component 1.1: Understand and apply concepts and procedures from number sense.
Number and numeration
1.1.1 Understand place value in whole numbers.
• Group and regroup objects into 1's, 10's, and 100's and explain relationships. [CU]
• Determine the value of a digit based on its position in a number.
• Read and write numbers to at least 1,000. [CU]1.1.2 Understand sequential relationships among whole numbers.
• Order three or more numbers to at least 1,000 from smallest to largest. [RL]
• Use comparative language (e.g., less than, more than, equal to) to compare numbers to at least 1,000. [CU]Computation
1.1.5 Understand the meaning of addition and subtraction and how they relate to one another.
• Show relationships between addition and subtraction using physical models, diagrams, and acting out problems. [CU, MC]
• Model real life situations involving addition (e.g., Peter has 7 peanut butter cookies and 4 chocolate chip. How many cookies does he have?) and subtraction (e.g., Peter has 11 cookies which is 4 more than Teresa. How many cookies does Teresa have?) using physical models and diagrams from various cultures and acting out problems. [CU]1.1.6 Understand and apply procedures for addition and subtraction of whole numbers with fluency.
• Use strategies for addition and subtraction combinations through at least 18.
• Recall addition and subtraction facts through at least 18.
• Solve problems involving addition and subtraction with two or three digit numbers using a calculator and explaining procedures used. [SP, CU]
• Make combinations and name total value of coins.1.1.7 Understand and apply strategies and appropriate tools for adding and subtracting with whole numbers.
• Use mental math strategies to compute (e.g., composing and decomposing numbers, finding combinations that are easy to add or subtract) through 100. [RL]
• Use calculator, manipulatives, or paper and pencil to solve addition or subtraction problems.
• Explain methods to mentally group numbers efficiently (e.g., when adding 52 and 59, add the 50’s together to get 100, then add eleven more). [CU]
Estimation
1.1.8 Understand and apply estimation strategies to predict computation results and to determine the reasonableness of answers.
• Use estimation strategies (e.g., front-end estimation, clustering) to predict computation results and to determine the reasonableness of answers. [RL]
• Justify reasonableness of an estimate in addition and subtraction. [CU]
• Decide whether a given estimate for a sum or difference is reasonable. [RL]Component 1.2: Understand and apply concepts and procedures from measurement.
Attributes, units, and systems
1.2.1 Understand and apply attributes to measure objects and time.
• Identify attributes of an object that are measurable (e.g., time, length, distance around, or weight of objects).
• Compare lengths or distances where direct comparison is not possible (e.g., use a string, paper strip, arm length, or hand span to compare the height and width of a table). [RL, MC]
• Read a clock to tell time to the half hour.Procedures, precision, and estimation
1.2.4 Understand and apply procedures to measure with non-standard or standard units.
• Select the most appropriate unit to measure the time of a given situation (e.g., would you use minutes or hours to measure brushing your teeth, eating dinner, sleeping?). [MC]
• Select a tool that can measure the given attribute (e.g., analogue clock - time, string - length, balance - weight).
• Demonstrate measurement procedure (e.g., start at a beginning point, place units end-to-end, not overlapping, and straight line). [CU]
• Justify the use of one tool over another (e.g., the length of a hand is a better measurement tool for this situation than the length of a small cube). [CU, RL]
• Explain why, when the unit is smaller it takes more to measure an item than when the unit is larger (e.g., it takes more small paper clips than large paper clips to measure the same length). [CU]1.2.6 Understand how to estimate in measurement situations.
• Estimate length and weight using non-standard units. [RL]
• Use important benchmarks (referents) (e.g., 5 or 10) to make initial and revised estimates.
• Explain how a benchmark (referent) helps to make a reasonable estimate. [CU]
Component 1.3: Understand and apply concepts and procedures from geometric sense.Properties and relationships
1.3.2 Understand characteristics of two-dimensional geometric figures.
• Sort and describe characteristics of two-dimensional geometric figures (e.g., various polygons). [RL, CU]
• Draw a two-dimensional shape that matches a set of characteristics (e.g., draw a four-sided shape that has all sides the same length).Locations and transformations
1.3.3 Understand the locations of numbers on a positive number line.
• Indicate whether a number is above or below a benchmark number (e.g., greater than or less than 1000).
• Describe the location of a given number between 1 and 1000 on a number line. [CU]
• Identify a point up to 1000 on a positive number line.Component 1.4: Understand and apply concepts and procedures from probability and statistics.
Statistics
1.4.3 Understand the organization of a graph.
• Identify title, horizontal and vertical axes, and key.
• Construct a bar graph that includes a title, key, and single unit increment. [CU]
• Name an appropriate title for a display of data. [CU]1.4.5 Understand how a display provides information about a question.
• Conduct a survey for a predetermined question and collect data using tallies, charts, lists, or pictures (e.g., who has animals at home, how many, what type?). [SP, RL]
• Identify a question that could be answered from a display.
• Interpret results and draw conclusions from displays (e.g., pictographs, bar graphs) using comparative language (e.g., more, fewer). [CU, MC]
• Read the labels from each axis of a graph. [CU[Component 1.5: Understand and apply concepts and procedures from algebraic sense.
Patterns, functions, and other relations
1.5.1 Understand how patterns are generated.
• Translate a pattern from one representation to another (e.g., snap-clap-stomp translates to ABC). [CU, MC]
• Identify, extend, create, and explain patterns of addition and subtraction represented in charts and tables. [CU, RL, MC]
Symbols and representations
1.5.3 Understand the meaning of symbols and labels used to represent situations.
• Use number sentences with symbols and labels to represent real-world problems involving addition and subtraction. [MC]
• Give an example of inequality in real life (e.g., on the first turn, Juan scored 6 points, on the second turn, he scored 8 points. On the first turn, Ivana scored 9 points, on the second turn, she scored 7 points. After two turns, Juan’s points are less than Ivana’s points). [CU, MC]Evaluating and solving
1.5.6 Understand and apply strategies to solve for the unknown using addition and subtraction.
• Solve equations with an “unknown” (e.g., 6 + ? = 11; 11=?+6). [RL]
• Justify the selection of a particular value for an unknown quantity in a real world situation (e.g., Two girls had 10 cookies. If Kwame had 6, how many did Ellie have? Explain). [RL, MC]
EALR 2: THE STUDENT USES MATHEMATICS TO DEFINE AND SOLVE PROBLEMS.
Component 2.1: Understand problems
Example: A classroom is planning an all-day skating party on Thursday. Each student must pay for admission ($2); a box lunch ($3); and skate rental ($2). The teacher needs a total amount to reserve the rink.2.1.1 Understand how to define a problem in a familiar situation.
• State or record information presented in situation (e.g., the classroom is planning a skating party on Thursday. Each student must pay for admission, lunch, and skates. The teacher needs to know the total cost in order to reserve the rink).
• Explain the problem, verbally or in writing, in own words (e.g., how much will the skating party cost?).
• Generate questions that would need to be answered in order to solve problem (e.g., what is the cost of a ticket and skate rental for the skating rink? What is the cost of food? What is the cost for each student? What will a skating party cost?). [1.4.4]
• Identify known and unknown information (e.g., known - the cost of admission, skates, lunch, and the number of students going; unknown - cost for each student and total cost).
• Identify extraneous information (e.g., the party is planned for Thursday).Component 2.2: Apply strategies to construct solutions
2.2.1 Understand how to create a plan to solve a problem.
• Gather and organize relevant information (e.g., create a four-column chart with student names in one column and the other three for costs related to the party - admission, skates, lunch; draw a seating chart and write in costs by each student).2.2.2 Apply mathematical tools to solve the problem.
• Use estimation strategies (e.g., front-end estimation, clustering) to predict computation results. [1.1.8]
• Use appropriate tools from among mental math, paper and pencil, manipulative, or calculator (e.g., to determine the total cost of the skating party). [1.1.7]
• Recognize when an approach is unproductive and try a new approach.
EALR 3: THE STUDENT USES MATHEMATICAL REASONING.
Component 3.1: Analyze information.
Example: A classroom is planning an all-day skating party on Thursday. Each student must pay for admission ($2); a box lunch ($3); and skate rental ($2). The teacher needs a total amount to reserve the rink.3.1.1 Understand how to compare information presented in familiar situations.
• Explain understanding of a situation, verbally or in writing (e.g., there are costs for admission, skates, lunch for the party; we need to know what it will cost for all of us so our teacher can reserve the rink).
• Estimate how much money will be needed for all 25 students to attend.Component 3.2: Make predictions, inferences, conjectures, and draw conclusions.
3.2.1 Understand how to make a reasonable prediction based on prior knowledge and the information given in a familiar situation.
• Predict a numerical solution for a problem (e.g., predict how much it will cost for the class to attend the skating party).
• Use known information to make a reasonable prediction (e.g., if most students in one class like red apples, then most students in another class will like red apples).
• Make an inference based on information provided (e.g., when you skate at the rink with a big group it costs less for each person than when you go with a friend).3.2.2 Understand how to draw conclusions based on prior knowledge and the information given in a familiar situation.
• Draw conclusions from displays using comparative language (e.g., greater than, less than).
• Provide data to justify conclusions.
• Provide examples from displays to support conclusions.3.2.3 Analyze procedures used to solve problems in familiar situations.
• Justify the use of a chart or table to collect and organize information used to solve a problem (e.g., the two- or four-column chart helped to keep track of the information).
• Justify the use of one mathematical tool over another (e.g., is a calculator or 100’s chart a better tool in this situation?).Component 3.3: Verify results.
3.3.1 Understand how to justify results using evidence.
• Check for reasonableness of results by using a calculator for repeated addition (e.g., to determine the total cost of the skating party).3.3.2 Understand how to validate thinking about numerical, measurement, geometric, or statistical ideas by using models, known facts, patterns, or relationships.
• Explain why a strategy or tool used in solving a problem (e.g., why a seating chart was helpful to help determine total cost of skating).
EALR 4: THE STUDENT COMMUNICATES KNOWLEDGE AND UNDERSTANDING IN BOTH EVERYDAY AND MATHEMATICAL LANGUAGE.
Component 4.1: Gather information.
4.1.1 Understand how to develop and follow a simple plan for collecting information for a given purpose.
• Determine what information is needed and how to collect it for a given purpose (e.g., to help explain something, to find out if something is needed) and who the information is for (e.g., for the classroom, for the adults at home, for the cafeteria, for the principal).
• Develop and follow a plan to gather information about supplies needed for a project (e.g., how many pieces of paper will be needed to create a pattern design for each of the kindergarten windows?).4.1.2 Understand how to extract information for a given purpose from one or two different sources.
• Decide what information would be important to learn about the students in the second grade after reading an informational text (e.g., health article) in class (e.g., how many students eat a nutritious breakfast). Determine what questions to ask in a survey. Graph the results.Component 4.2: Organize, represent, and share information.
4.2.1 Understand how to organize information to communicate to a given audience.
• Organize and display data on a chart to communicate a solution to a specific audience (e.g., use a chart to display individual costs and total cost for the skating party for parents and PTA).
• Construct a bar graph with a title, key, and single unit increment to display survey results (e.g., the number of brothers and sisters of students in the class).
4.2.2 Understand how to communicate or represent ideas or information using mathematical language or notation.
• Explain or represent ideas using mathematical language from:
o Number sense (e.g., numbers to at least 1000) [1.1.1];
o Measurement (e.g., identify attributes of an object that are measurable - time, length, distance around, capacity, area or weight of objects) [1.2.1];
o Geometric sense (e.g., describe characteristics of two-dimensional geometric figures, various polygons) [1.3.1];
o Statistics (e.g., construct bar graph using a single increment scale) [1.4.3];
o Algebraic sense (e.g., explain and use the symbols < and > to express relationships). [1.5.3]
EALR 5: THE STUDENT UNDERSTANDS HOW MATHEMATICAL IDEAS CONNECT WITHIN MATHEMATICS, TO OTHER SUBJECT AREAS, AND TO REAL LIFE SITUATIONS.
Component 5.1: Relate concepts and procedures within mathematics.
5.1.1 Understand how to use concepts and procedures from any two of the content components from EALR 1 in a given problem or situation.
• Conduct a survey for a predetermined question, collect data, and use addition and subtraction procedures to compute the results of the survey. [1.4.4, 1.1.6]
• Interpret a bar graph for comparative information (e.g., how many more than, less than) and draw conclusions about the data. [1.4.5, 3.2.2]5.1.2 Understand how to recognize and create equivalent mathematical models and representations in familiar situations.
• Represent addition and subtraction situations with physical models, diagrams, and acting out problems. [1.1.5]
• Identify different representations of a pattern (e.g., snap-clap-stomp translates to ABC). [1.5.1]Component 5.2: Relate mathematical concepts and procedures to other disciplines
5.2.1 Apply and analyze the use of mathematical patterns and ideas in familiar situations in other disciplines.
• Collect and display data based on a science experiment (e.g., plant growth, magnetism).
• Identify patterns used in the design of common objects (e.g., skateboards, clothing).
• Describe how estimation can be used to know about how much something costs.5.2.2 Know the contributions of individuals and cultures to the development of mathematics.
• Recognize the contributions of women, men, and people from different cultures (e.g., examine design and patterns on tapestry from various African cultures).Component 5.3: Relate mathematical concepts and procedures to real-world situations
5.3.1 Understand how mathematics is used in everyday life.
• Generate examples of mathematics in everyday life:
o counting (e.g., tallies to keep score during a game);
o comparing lengths or distances where direct comparison is not possible (e.g., using a string or paper strip to compare the height and width of a desk to see if it fits in the room);
o drawing geometric shapes (e.g., using a ruler to create shapes with equal sides);
• Select the most appropriate unit to measure a given time (e.g., would you use minutes or hours to measure brushing your teeth, eating dinner, sleeping?);
• Estimate the cost of two items knowing the approximate cost of one (e.g., one game costs about $8).
EALR 1: THE STUDENT UNDERSTANDS AND APPLIES THE CONCEPTS AND PROCEDURES OF MATHEMATICS.
Component 1.1: Understand and apply concepts and procedures from number sense.
Number and numeration
1.1.1 Understand the concept of whole numbers. W
• Represent a number to at least 10,000 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. [MC]
• Generate equivalent representations for a given number by decomposing and composing. [MC]
• Explain the difference between the natural numbers and the whole numbers.
• Identify place values of digits of whole number to the hundreds or thousands place using words, pictures, or numbers.
• Write whole numbers to 999.
• Decompose whole numbers into components (e.g., 35 is made of 3 tens and 5 ones) using words, numbers, or pictures.1.1.2 Understand the relative values of whole numbers. W
• Compare whole number values to at least 10,000 using the symbols for "greater than," "less than," and “equal to".
• Order three or more numbers to at least 10,000 from smallest to largest. [CU]
• Compare combined quantities (e.g., 50 + 3 is greater than 40 + 9). [RL]1.1.3 Understand and apply the commutative and identity properties of addition on whole numbers. W
• Explain or show 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]
• Determine whether addition equations are true or false and explain, based on the commutative or identity properties for addition (e.g., 15+ 3+5 = 15+5 +3). [CU]
• Identify an equivalent expression using the commutative property.
• Show how the commutative property works using pictures or objects. [CU]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]
• Demonstrate the relationship between multiplication and repeated addition.
• Demonstrate the relationship between division and repeated subtraction.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). [RL, CU]
• Use joining, separating, adding-on, and finding the difference to add and subtract.
• Write and solve multi-step problem situations that involve addition and subtraction. [CU, MC]
• Use calculators to compute with large numbers (e.g., adding three or more 3-digit numbers; subtracting 3 digit from 4 digit numbers).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.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. [RL]
• Describe and justify reasonableness of an estimate in computation. [RL, CU]
• Use a variety of estimation strategies (e.g., multiples of 10 and 100, rounding, front-end estimation, compatible numbers, clustering).
• Describe and justify whether an approximation is or is not appropriate. {RL, CU]Component 1.2: Understand and apply concepts and procedures from measurement.
Attributes, units, and systems
1.2.1 Understand how different attributes (length, perimeter, time, money value, weight/mass, and temperature) are used to describe objects. W
• Given an object, name the attributes that can be measured. [CU, MC]
• Explain how length is used to describe objects. [CU]
• Explain or show how height and weight are different. [CU]
• Explain or show how clocks measure the passage of time. [CU]
• Explain how money is used to describe the value of purchased items. [CU]
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
• Identify when two unit measurements are not necessarily equal (e.g., one pace long can represent different lengths). [CU, MC]
• Determine whether measurement can or cannot be compared based on whether the units are the same or different.
• Show how length units are shown on rulers, tape measures, and other linear measuring tools. [MC, CU]
• Show how weight units are shown on a grocery scale. [MC]
• Explain why people created standard units for length or weight/mass. [CU]1.2.3 Understand how measurement units of length (U.S.) and capacity (U.S.) are organized into systems. W
• 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]
• Demonstrate or explain how inches are organized into feet and feet are organized into yards. [CU]
• Demonstrate or explain how cups are organized into pints, pints into quarts, and quarts into gallons. [CU]Procedures, precision, and estimation
1.2.4 Understand and apply systematic procedures to measure length, time, weight, money value, and temperature. W
• Identify attribute to measure.
• Select and use appropriate units (e.g., meters, minutes, pounds, dollars, degrees).
• Select and use tools that match the unit (e.g., ruler, clock, scales, calculator, thermometer).
• Count or compute and label measures.
• Explain and use a method for making change with coins. [CU].
• Compare measures of two or more like objects. [RL]1.2.6 Understand and apply strategies to obtain reasonable estimates of length, time, weight, and temperature measurements. W
• Identify situations in which estimated measurements are sufficient; estimate length, time, money, weight or temperature.
• Estimate a measurement using standard or non-standard units (e.g., fingers, arms, paper clips, inches, minutes, or foot lengths).
• Create and use referents to standard units (e.g., width of pinkie finger is similar to a centimeter). [RL, MC]
• Use estimation to decide whether standard or non-standard units of measurement have been used in a situation. [RL]
• Determine when estimation is useful.
Component 1.3: Understand and apply concepts and procedures from geometric sense.Properties and relationships
1.3.1 Understand the concept of congruence. W
• Identify, describe, and compare congruent two-dimensional geometric figures. [RL, CU]
• Given a variety of figures, determine which figures are congruent.
• Draw a shape that is congruent to a given two-dimensional shape. [CU]
• Explain congruence and use an example to demonstrate it. [CU]1.3.2 Understand and apply attributes and properties to two-dimensional shapes and figures. W
• Use attributes and properties to identify, name, draw, compare, and/or sort two-dimensional shapes and figures. [RL, CU]
• Draw and label two-dimensional figures given particular attributes (e.g., triangle, rectangle with all sides the same length). [CU]
• Identify, name, and describe the attributes and properties of polygons. [CU]
• Given two polygons, explain how they are alike and different in terms of their attributes and properties (e.g., using a Venn diagram). [CU]
• Give directions so that someone else can duplicate a design involving polygons (e.g., a friend who can’t see the design). [CU]Locations and transformations
1.3.3 Understand relative locations including intervals of numbers on a positive number line. W
• Given directions for movement on a positive number line, identify the point of final destination using real-world examples (e.g., travel back and forth on a street, temperature variation at different times of the day, dance steps from diverse cultures). [SP, RL, MC]
• Identify the interval on a given number line (e.g., describe the scale on a graph). [CU]
• Describe the relative locations of points on a number line with positive coordinates. [CU]
• Use unit values to describe the location of objects on a number line.
• Draw points or objects on a number line based on unit values given.Component 1.4: Understand and apply concepts and procedures from probability and statistics.
Statistics
1.4.3 Understand how to use data collection and display methods to obtain desired information. W
• Interpret graphs for comparative information (e.g., find the difference in selected data). [RL, CU, MC]
• Pose questions and gather data relevant to the questions posed.
• Design a survey; collect, and record data in easy-to-use formats (e.g., use tally marks, make a table). [CU]
• Organize category data into bar graphs with unit scales for ease of interpretation. [RL]
• Organize data into picture graphs with unit scales for ease of interpretation. [RL]
• Determine questions needed to gather data about themselves and their classmates.1.4.4 Understand and apply mode to describe a set of data.
• Create and solve a problem situation where mode is meaningful for a set of data. [RL, CU, MC]
• Explain what the mode represents and how to find it in a given set of data. [CU]
• Identify the mode for a given set of data.1.4.5 Understand representations of data from tables, charts, and bar graphs. W
• Pose questions that can be answered from a given graph. [CU, MC]
• Make inferences based on the data or determine if the data can support inferences made. [CU, MC]
• Read and report on data from tables, charts, and bar graphs. [CU]
• Explain how types of graphs or the graph construction can support different points of view (e.g., starting the axis numbers at 50 rather than 0). [CU, SP, RL]
• Create bar graphs including labels for title, both axes, scale units (e.g., 2’s, 5’s, 10’s), and key if needed. [SP, RL, CU, MC]
• Interpret graphs for comparative information (e.g., find the difference in selected data). [RL, CU, MC]Component 1.5: Understand and apply concepts and procedures from algebraic sense.
Patterns, functions, and other relations
1.5.1 Understand patterns of objects including number patterns with a single addition or subtraction operation. W
• Recognize and extend patterns of numbers, figures, and objects using addition and subtraction based on a single arithmetic operation between the terms (e.g., stacking cans in a pyramid, observing textile patterns).
• Identify, extend, and describe numerical patterns (e.g., skip counting, 100 chart, multiplication table). [RL, CU]
• Describe the pattern in a number sequence (e.g., Guess My Rule, Function Machine). [CU]
• Identify the rule for a pattern based on a single operation (e.g., add 3). [RL]
• Explain what makes a given pattern a pattern. [CU]
• Complete a pattern by supplying missing elements in the pattern.
• Compare two patterns to determine whether they are alike or different and explain the decision. [RL, CU]
Symbols and representations
1.5.3 Apply understanding of the concept of mathematical equality. W
• Write an equation or expression for a given situation (e.g., there are 23 dogs at a kennel; if 15 are present, how many are absent?). [SP, RL, CU]
• Explain equality and the use of “=” in equations. [CU]
• Compare expressions to determine whether they are equal (e.g., 3+4 and 2+5). [RL]
• Write a situation that represents it given an equation involving addition or subtraction. [CU, MC]
• Identify a situation that represents it given an equation involving addition or subtraction. [CU, MC]1.5.4 Understand and apply operational and relational symbols and notations to write equations involving addition and subtraction. W
• Write and explain mathematical statements (e.g., 7 + ? = 8 or ? +8 = 10). [CU]
• Identify and use mathematical symbols and notations in reading and writing expressions and equations involving addition and subtraction.
• Write an equation for a given situation (e.g., there are 23 children in class; if 15 are present, how many are absent?). [CU]Evaluating and solving
1.5.6 Understand and apply strategies to solve equations that include addition or subtraction. W
• Solve problems involving equality (e.g., 5 + 3 = ? + 2). [SP, RL]
• Solve equations with addition and subtraction using manipulatives, pictures, and symbols. [SP, RL, CU]
• Describe a strategy used to solve an equation with addition or subtraction. [CU]
EALR 2: THE STUDENT USES MATHEMATICS TO DEFINE AND SOLVE PROBLEMS.
Component 2.1: Understand problems.
Example: Miguel’s reading class has set a goal to increase nightly reading to at least 30 minutes. He is taking a survey of his nine classmates to determine about how many minutes they read each night to see if they have met the goal. Miguel likes to read books by Matt Christopher.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, or paraphrase information provided: Miguel is taking a survey to determine about how many minutes students read on school nights. The class goal is at least 30 minutes each night).
• Determine the problem using information from investigation (e.g., has the class met its reading goal for the week?).
• Generate questions that would need to be answered in order to solve the problem (e.g., about how many minutes did each person read? Can we estimate or do we need an exact number? What is the difference between the goal and the minutes read?).
• Identify known and unknown information (e.g., known: who the students are, the class goal [30 minutes x 5 nights x 10 students is 1500 total minutes]; unknown: the number of minutes each student read, if the class reached the goal).
• Identify information that is needed and not needed to solve the problem (e.g., needed: the class goal; not needed: Miguel likes Matt Christopher books).Component 2.2: Apply strategies to construct solutions.
2.2.1 Apply strategies, concepts, and procedures to devise a plan to solve the problem. W
• Gather and organize data and information (e.g., create a survey to find out about how many minutes students are watching TV; organize data on a two-column chart).
• Determine what strategy will be used to solve the problem (e.g., estimate minutes read per night per week from data gathered).2.2.2 Apply mathematical tools to solve the problem. W
• Use strategies to solve problems (e.g., use number estimation - if one student reads 45 minutes [around 50] one night and if the same student reads 18 [around 20] minutes the next night, that is about 70 minutes).
• Use appropriate tools to estimate solution (e.g., mental math or paper and pencil).
• Recognize when an approach is unproductive and try a new approach.
EALR 3: THE STUDENT USES MATHEMATICAL REASONING.
Component 3.1: Analyze information.
Example: Miguel’s reading class has set a goal to increase nightly reading to at least 30 minutes. He is taking a survey of his nine classmates to determine about how many minutes they read each night to see if they have met the goal. Miguel likes to read books by Matt Christopher.3.1.1 Analyze information presented in familiar situations. W
• Break down results from data to determine about how many minutes per night students are reading in order to estimate whether the class has met 30 minutes each night goal.Component 3.2: Make predictions, inferences, conjectures, and draw conclusions.
3.2.1 Apply prediction and inference skills. W
• Make a reasonable prediction based on prior knowledge and investigation of situation (e.g., after collecting survey data and before estimation, predict whether the class will meet its goal).
• Defend prediction with evidence from the situation.
• Make inferences (conjectures) using information from the situation to support the inference (e.g., the class probably did not make the reading goal because the community softball league has started up and most kids are involved in the evenings).3.2.2 Apply the skills of drawing conclusions and support the 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 estimation strategies used (e.g., front end estimation vs. using compatible numbers). [1.1.8]Component 3.3: Verify results.
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 many minutes students were reading each night).
• Justify whether estimation is appropriate for the situation.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 comparisons can be used to draw a conclusion (e.g., the class won’t have met the reading goal because fewer students read less than more this week and didn’t make the goal last week).
EALR 4: THE STUDENT COMMUNICATES KNOWLEDGE AND UNDERSTANDING IN BOTH EVERYDAY AND MATHEMATICAL LANGUAGE.
Component 4.1: Gather information.
4.1.1 Understand how to 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
• Read and report on data from tables, charts, and bar graphs. [1.4.5]
• Read directions for movement on a positive number line, identify the point of final destination using real–world examples (e.g., travel back and forth on a street, temperature variations during the day). [1.3.3]Component 4.2: Organize, represent, and share information.
4.2.1 Understand how to organize information for a given purpose. W
• Create a display to represent information from survey results (e.g., the approximate number of minutes read and whether or not the goal was met).
• Create bar graphs including labels for title, both axes, scale units (e.g., 2’s, 5’s, 10’s), and key if needed. [1.4.2]
• Create and solve a problem situation where mode is meaningful for a set of data. [1.4.4]
• Display information to be shared.4.2.2 Understand how to communicate or represent ideas using mathematical language or notation. W
• Translate from one representation of a whole number to another in standard, expanded, and word forms. [1.1.1]
• Name attributes of an object that can be measured. [1.2.4]
• Identify, describe, and compare congruent two-dimensional geometric shapes. [1.3.1]
• Make a survey and collect data (e.g., use tally marks, make a table). [1.4.3]
• Identify and use appropriate symbols and notation in reading and writing simple expressions and equations involving addition and subtraction. [1.5.4]
EALR 5: THE STUDENT UNDERSTANDS HOW MATHEMATICAL IDEAS CONNECT WITHIN MATHEMATICS, TO OTHER SUBJECT AREAS, AND TO REAL LIFE SITUATIONS.
Component 5.1: Relate concepts and procedures within mathematics.
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 three-digit addition and subtraction to compute the results of the survey. [1.1.6, 1.4.4]
• Explain and use a method for making change with coins. [1.1.1, 1.2.4]
5.1.2 Understand how to recognize equivalent mathematical models and representations in familiar situations. W
• Translate from one representation of a whole number to another in standard, expanded, and word forms. [1.1.1]
• Compare strategies to solve problems involving multiplication and division (e.g., alternative algorithms, use of properties of multiplication). [1.1.5]
• Use the inverse relationship between multiplication and division using physical diagrams, words, and symbols (e.g., arrays, fact families). [1.1.5]Component 5.2: Relate mathematical concepts and procedures to other disciplines.
5.2.1 Apply mathematical patterns and ideas in familiar situations in other disciplines.
• Given an object, identify geometric attributes that can be measured.
• Interpret graphs for comparative information. [1.4.3]
• Pose questions and gather data about self and surroundings. [1.4.3]
• Make inferences based on data or determine if the data can support inferences made. [1.4.5]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., complete a mathematically based project that researches the history of 0?).Component 5.3: Relate mathematical concepts and procedures to real-world situations.
5.3.1 Understand that mathematics is used in daily life and extensively outside the classroom.
• Write and solve multi-step situations that involve addition and subtraction. [1.1.6]
• Use referents to standard units (e.g., width of pinkie finger is similar to a centimeter). [1.2.6]
• Identify the point of final destination using real-world examples given directions for movement on a positive number line (e.g., travel back and forth on a street, temperature variation at different times of the day, climbing up and down stairs). [1.3.3]
• Pose questions and gather data about self and surroundings. [1.4.2]
• Create and solve a problem situation where mode is meaningful for a set of data. [1.4.4]
• Make inferences on data from a real-world context and then use the context to determine if the inference is valid. [1.4.5]
EALR 1: THE STUDENT UNDERSTANDS AND APPLIES THE CONCEPTS AND PROCEDURES OF MATHEMATICS.
Component 1.1: Understand and apply concepts and procedures from number sense
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]
Component 1.2: Understand and apply concepts and procedures from measurement.
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.Component 1.3: Understand and apply concepts and procedures from geometric sense.
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]Component 1.4: Understand and apply concepts and procedures from probability and statistics.
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]Component 1.5: Understand and apply concepts and procedures of algebraic sense.
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.
Component 2.1: Understand 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).
Component 2.2: Apply strategies to construct solutions.
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.
Component 3.1: Analyze information.
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).Component 3.2: Make predictions, inferences, conjectures, and draw conclusions.
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]Component 3.3: Verify results.
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.
Component 4.1: Gather information.
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]
Component 4.2: Organize, represent, and share information.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.
Component 5.1: Relate concepts and procedures within mathematics.
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.]
Component 5.2: Relate mathematical concepts and procedures to other disciplines.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?).Component 5.3: Relate mathematical concepts and procedures to real-world situations.
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]
EALR 1: THE STUDENT UNDERSTANDS AND APPLIES THE CONCEPTS AND PROCEDURES OF MATHEMATICS.
Component 1.1: Understand and apply concepts and procedures from number sense
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]Component 1.2: Understand and apply concepts and procedures from measurement.
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]Component 1.3: Understand and apply concepts and procedures from geometric sense.
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]Component 1.4: Understand and apply concepts and procedures from probability and statistics.
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 th