Numerical & Proportional Reasoning: Place Value Patterns & Equivalent Forms Kindergarten Grade 2 Grades 3-5

Transcription

1 Numerical & Proportional Reasoning: Place Value Patterns & Equivalent Forms 2.1 Numbers and words can be used to count and represent quantities or position. The position of a digit in any number determines its value. The number system is represented with base ten place value models. Any number or numerical expression can be represented in different ways but have the same value. Identify and order numerals 1-10 and match sets of objects to the numerals. Form and write numerals 0-9. Identify, read and write numerals to 100 Locate and label two- and three-digit whole numbers using place value models, pictures and number lines Locate and label whole numbers to 10,000 using place value models, pictures, and the number line. Locate and label whole numbers up to 100,000 using place value models, diagrams, number patterns and number lines. Locate and label whole numbers to 1,000,000 using place value models, diagrams and number lines. Determine the value and magnitude of a given digit within a two-digit whole number. Determine the value and magnitude of a given digit within a three-digit whole number. Determine the value and magnitude of a given digit within a four-digit whole number. Determine the value and magnitude of a given digit within a five-digit whole number. Determine the value and magnitude of a given digit within a six-digit whole number. Compare sets using the terms more, less or the same and order sets from least to greatest. Explore ordering two-digit whole numbers using visual representations such as the hundreds chart or a number line. Order two- and three-digit whole numbers using visual representations such as the hundreds chart or a number line and place value models. Compare and order whole numbers to 10,000 using place value models, number patterns, and number lines Compare and order whole numbers to 100,000 using place value models, number patterns, and number lines Compare and order whole numbers to 1,000,000 using place value models, number patterns, and number lines. Represent two-digit numbers using a variety of concrete and visual models, including number lines. Represent two- and three-digit numbers in expanded and regrouped forms, using place value models, pictures, and written form. Show equivalent representations for two- and three-digit numbers in expanded and regrouped forms Show equivalent representations for two-, three-, four-, and fivedigit numbers in expanded and regrouped forms Express five-digit numbers in expanded and regrouped forms and use the numbers to support computational strategies. Identify ordinal position of objects, first through fifth, and last. Identify ordinal position of objects, first through tenth 1

2 Sort pennies, dimes, nickels, quarters using physical attributes. Identify, and name pennies, dimes, and nickels. Grades K-5 Mathematics Scope and Sequence Identify and name quarters, half dollars, and a dollar. Identify, name, and count and trade pennies for objects Determine and compare values and trade with sets of pennies and dimes. Determine the value of mixed sets of dimes and pennies. Find the value of like sets of pennies, nickels, and dimes Determine and compare values and trade with sets of pennies, dimes and dollars, connecting to place value. Determine and compare given values up to a dollar using mixed sets of coins (pennies, nickels, dimes, and quarters) to show given values up to one dollar. Find the value of like sets of pennies, nickels, dimes, quarters, and half dollars and dollars. Demonstrate a variety of ways to show a given numerical amount, including whole numbers and monetary amounts. Determine and compare the value of a mixed set of coins, including pennies, nickels, dimes, quarters, and half-dollars, using decimal notation. Estimate change from whole number amounts. Use mixed money amounts to connect money notation to decimals and place value.. Explore the equivalent relationship between the fractional form and the decimal form for tenths using visual and concrete models (1/10 =.1). Build models and describe tenths and hundredths using equivalent ratio, fraction, and decimal notation. Use models to extend whole number place value concepts and patterns to decimals. Explore numbers less than zero and extend the number line to introduce the concept of integers within practical applications. 2

3 Numerical & Proportional Reasoning: Rational Numbers Fractions involve sharing equal parts of a set of objects or a whole object. A fraction with the same numerator and denominator represents the whole object or an entire set. Proportions can be compared, measured, and represented in a variety of ways. Proportional relationships express how quantities change in relationship to each other. Identify a whole and a half of an object using a variety of models. Recognize that two halves can be put together to make a whole. Share a set of objects considered to be a whole by forming two smaller sets that have equal amounts. Determine if a whole is divided into equal or unequal parts. Identify 1/2, 1/4 and 1/3 as a part of a whole. Use pictures and models to represent one half of a whole and label using symbolic notation (½.) Compare parts of a whole object and estimate whether they are closer to very little (0), one half or one whole. Make a whole of equal-sized parts of familiar objects. Identify half of a small set of objects. Use pictures and models to represent equal parts of a whole (½, ¼, 1/3) and label using symbolic notation. Estimate, compare and order unit fractions ( ½, 1/3, ¼ ) of a whole using models and familiar objects. Find equal parts of a small set of objects, (such as ½, 1/3 or ¼ of 12 cookies) by estimating, counting and grouping Recognize equivalent fractions using models. Identify and build models of various fractional parts of a whole (such as 2/3, ¾) Explore and describe addition with like denominators (halves, thirds and fourths) Use models and pictures to represent fractions and label the parts with words and fraction symbols. Identify a whole as a fraction with the same numerator and denominator. Use counting and grouping of objects to find equal parts of a set of objects. Use models and number patterns to identify amounts such as 2/3 of 12 is 8. Use models and pictures to represent fractions and label the parts with words and fraction symbols. Identify a whole as a fraction with the same numerator and denominator. Estimate and locate points representing 2 and 3 digit numbers; fractions (½, 1/3, & ¼) and decimals (tenths) on number lines and rulers. Build and label a variety of models to represent, compare and order fractional parts of a whole and mixed numbers and to identify ratios and equivalent fractions (through division and multiplication). Use counting, number patterns and grouping to identify parts of a set. Express a division problem as a fraction. Describe the relationship between the divisor and the remainder written as a fraction Represent a rational number in its equivalent fraction with models, number patterns, and common factors. Explore decimal, ratio, and fraction relationships. Represent percent forms with models, number patterns and common factors. Estimate and use counting, grouping of objects, number patterns, equivalent ratios and division to find fractional parts of a set of objects including rulers. 3

4 and write matching fraction sentences using models up to one whole. Use counting and grouping of objects to find equal parts of a set of objects. Use models and number patterns to identify amounts such as 2/3 of 12 is 8. Identify amounts such as 2/3 of 12 is 8 using picture models. Estimate fractional values and measure to the nearest half unit with the aid of number lines and rulers.. Write division problems in fraction form and round the fraction form to estimate an answer to a division problem. Describe patterns with simple ratios using familiar contexts, such as 1 cat has 4 legs, 2 cats have 8 legs. Describe simple ratios in patterns using models and pictures (e.g., in a pattern of green, green, red blocks, there are always two green blocks for each red block). Use models, pictures, and number patterns to represent ratios. Use models, pictures and number patterns to solve simple problems involving ratios and proportions. Identify patterns with equivalent ratios such as 3 out of 6 crayons are red or 4 out of 8 crayons are red are the same as 1 out of 2 crayons is red. Use ratios and proportions to solve practical problems such as interpreting maps and scale drawings or identifying probability. 4

5 Numerical & Proportional Reasoning: Operations 2.2 Operations create relationships between numbers. The relationships among the operations and their properties promote computational fluency. Computation involves taking apart and combining numbers using a variety of strategies. Flexible methods of computation involve grouping of numbers in a variety of ways. Whether we estimate or compute the exact answer depends on the needs of the situation. Count to and past 10 to 20, then to 30. Group and count up to 10 objects. Count whole numbers to 100. Identify 1 more and 1 less and explore 10 more and 10 less than a number. Count and write whole numbers to Explore and describe number patterns including odd and even numbers, counting by 2s, 5s, 10s, and 100s and counting on by 10. Count numbers beyond 1,000. Identify 10 and 100 more and less than a number. Solve practical problems and extend patterns involving 10 and 100 more and less than a number. Estimate products and missing factors using multiples of 10 and 100 Solve problems involving finding 10, 100 and 1000 more and less than a number. Estimate products and missing factors using multiples of 10, 100 and Estimate the amount of objects in a set using 10 as a benchmark, and then count to determine if the amount is more or less than 10. Identify sets and numbers, which are equal and one more. Group and skip count by 2s, 5s and 10s. Count on from a given amount, orally and with models, and count back from 10. Estimate and explain quantity with benchmark amounts such as 0, 10 and 100. Use objects and pictures to model and solve addition and subtraction story problems. Identify 10 more and 10 less and 100 more and 100 less than a number using place value models, pictures and number lines. Compare amounts in sets using the terms of greater than, less than and equal to. Use a variety of mental computation strategies to recall addition and subtraction facts and fact families to 18. Compare and round numbers to the nearest 10 and 100. Round monetary amounts to the nearest dollar. Use place value concepts, number patterns, the number line and the commutative and associative properties to develop estimation and computation strategies. Choose and use benchmarks to approximate locations on number lines and coordinate grids. 5

6 Write corresponding number sentences (equations) to match addition and subtraction story problems. Develop and use a variety of strategies to add and subtract onedigit numbers. Demonstrate fluency for addition and subtraction facts with sums through 10. Explore finding the sum of two 2- digit numbers using models and counting strategies without regrouping. Grades K-5 Mathematics Scope and Sequence Know basic addition and subtraction facts to 18 with Use commutative and associative automaticity. properties to solve problems. Solve addition and subtraction problems involving real-world experiences. Create story problems to match addition and subtraction number sentences (equations). Explore alternative strategies for estimating, adding and subtracting any two 2-digit numbers with and without regrouping using place value models. Estimate, add and subtract with two- and three- digit numbers using a variety of strategies. Write and solve multi-step word problems, including problems with extraneous information. Create and solve multi-step problems and explore order of operations in the context of practical situations. Develop strategies, using place value relationships, inverse operations and commutative, associative and distributive properties, to simplify computations with two-, three-, and four-digit numbers and money amounts. Explore the connection between addition and multiplication by extending number patterns, skip counting, combining repeated addends, building models of groups the same size and using arrays and pictures. Use skip counting, combining repeated addends, building models of groups the same size, arrays and pictures to represent multiplication. Develop fluency with multiplication and division fact families for factors 1 through 5 and 10. Explore the connection between multiplication and division using models and pictures of groups and arrays. Develop fluency with multiplication and division fact families for all factors 1 through 12. Relate multiplication and division to models with groups and rectangular arrays and begin to identify prime and composite numbers. Explore the property of zero in multiplication and its implication in division. Use rectangular arrays to identify factor pairs and to classify numbers as prime, composite and perfect squares. Solve problems using divisibility rules and patterns with remainders. Solve problems with the property of zero in multiplication and division. Explore divisibility rules and patterns with remainders. Identify the appropriate operation and write a word problem to match 6

7 a given addition, subtraction, multiplication or division number sentence and write the matching number sentence to solve a word problem. Compare and round numbers to the nearest 10 using place value models and number lines. Identify reasonable answers to addition and subtraction problems. Use estimation strategies to determine and justify the reasonableness of a computational answer. Explore when an estimate is appropriate and determine whether an estimation strategy will result in an over-or underestimate. Use arrays and explore using the distributive property 10 x (5 + 4) = (10 x 5) + (10 x 4) to estimate, multiply and divide two- and threedigit numbers by one-digit factors. Apply and explain a variety of estimation strategies in problemsolving situations to add and subtract money amounts less than $10.00 and two- and three-digit numbers with and without regrouping. Identify a reasonable estimate and/or exact amount to a problem, including making change. Use estimation to predict results and to recognize when an answer is or is not reasonable. Explain when an estimation strategy will result in an over- or underestimate. Determine and discuss the reasonableness of an answer and explain why a particular estimation strategy will result in an over- or underestimate. 7

8 ALGEBRAIC REASONING: Patterns & Functions 1.1 Patterns and relationships can be represented numerically, graphically, symbolically, and verbally Patterns provide insights into potential relationships. Sort and classify objects by a single attribute (size, shape, color, texture, use, position and orientation). Identify and copy patterns of sounds, colors, shapes, textures and numbers in a variety of contexts. Create simple patterns using s variety of materials. Describe the rule for a given pattern. Sort and classify objects based on one and two attributes and describe the rule used. Identify and extend a variety of concrete, spatial, auditory, and kinesthetic patterns. Describe counting patterns and number patterns. Describe the rule used to create a pattern. Develop and test simple conjectures (generalizations) based on Sort and classify data and objects based on more than one attribute and describe the rule used. Explore the use of Venn diagrams to illustrate the common attributes between two sets of objects or data. Identify missing object(s) with common or different attributes, and the complement (those that do not belong) of a set of objects using patterns and the rules that describe them. Create and extend a variety of concrete, spatial, auditory, and kinesthetic patterns. Translate the same pattern from one representation (such as color) to another representation (such as shape). Explore a variety of ways to describe and write rules for patterns. Develop and test simple Sort and classify the same set of objects and data in more than one way and explain the reason for each sort. Use Venn diagrams to sort and classify data, including the intersection and complement of sets. Construct, reproduce, describe and extend numerical and spatial patterns. Explore and describe patterns and sequences using tables, graphs and charts. Develop and test conjectures (generalizations) based on observations of number relationships such as using t- charts to find the rule. Create Venn diagrams to sort and classify data, including the intersection and complement of sets, as well as to determine missing member(s). Extend and describe a variety of patterns and trends including repeating and growing patterns. Explore extending and comparing arithmetic and geometric sequences using tables, graphs, and charts. Develop and test generalizations of patterns and relationships. Use Venn diagrams to solve problems involving the organization of data and information. Extend and compare arithmetic and geometric sequences. Represent geometric and numeric patterns using words, tables, graphs and equations. Analyze patterns and data to make generalizations and predictions. 8

9 observations of number relationships, such as commutativity, adding zero and adding one. Grades K-5 Mathematics Scope and Sequence conjectures (generalizations) based on observations of number relationships such as using t- charts to find the rule. ALGEBRAIC REASONING: Equivalence and Variables Real world situations can be represented symbolically and graphically. Algebraic expressions and equations generalize relationships from specific cases. Identify sets of objects (10 and less) as equivalent quantities or non-equivalent quantities. Demonstrate understanding of the = sign as an equality symbol. Demonstrate equivalence using models. Model real-life situations that involve addition and subtraction of whole numbers using objects, pictures and open number sentences. (2 + = 5) Apply the = sign as an equality symbol in simple addition and subtraction number sentences and equations. (4 + 2 = 3 + 3) Model situations involving addition and subtraction of whole numbers using objects, pictures, symbols and open number sentences. (6 = 10-2) Demonstrate understanding of the = sign as an equality symbol and explore inequalities and the symbol. ( , 10 6 = 3 + 1, = + 3) Demonstrate equivalence using the commutative and associative properties of whole numbers. Represent mathematical relationships involving addition, subtraction, multiplication and division as open number sentences by modeling with story problems. Use variables to represent quantities in expressions and number sentences. Demonstrate equivalence with the commutative, associative and distributive properties of whole numbers. Use equations to describe the rules for number patterns and to model story problems. Represent mathematical relationships using variables in expressions, equations and inequalities. Model and solve one-step equations using materials that model equivalence. Use the properties (commutative, associative and distributive) to demonstrate equivalence in order to solve equations. Describe how a change in one variable relates to a change in a second variable in a given situation. 9

10 GEOMETRY AND MEASUREMENT: Spatial Relationships 3.1 Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems. Geometry and spatial sense offer ways to interpret and reflect on our physical environment. Shapes and solids may be identified and classified by physical characteristics. Sort small sets of objects and sequence by size or length. Identify, sort, and compare two- dimensional shapes and solids in the environment, such as triangles, squares, rectangles, circles, cubes, cones, spheres, and cylinders. Sort, build and name twodimensional shapes such as triangles, squares, rectangles, circles. Sort and name threedimensional objects such as cubes, spheres, cylinders and cones. Create two- dimensional designs using a variety of materials. Identify and describe polygons, solids and other two- and threedimensional objects found in the environment. Build polygons, solids and other two- and three-dimensional objects. Identify shapes that have one line of reflective symmetry or that can be divided into two congruent parts. Create and draw 2 dimensional polygons. Sort polygons and solids, and other 3 dimensional objects by a variety of characteristics such as number of sides, relationship of sides (parallel, perpendicular) symmetry, and congruence. Identify and draw a variety of geometric shapes (pentagon, hexagon, trapezoid, and octagon). Identify shapes with right angles and explore acute and obtuse angles. Draw, describe and classify two- and three-dimensional figures using physical features such as number of sides, number of angles, lengths of sides and straight and curved parts. Identify two-dimensional shapes and determine lines of symmetry and congruence. Identify polygons and solids through using characteristics such as the relationship of sides (parallel, perpendicular), types of angles (acute, right and obtuse). Use geometric relationships such as parallel, perpendicular and congruent to describe the attributes of sets and subsets of shapes and solids. Make and test conjectures about geometric relationships. Measure, draw, and name acute, right, and obtuse angles. Name and classify triangles based on measurement of their sides and/or angles. 10

11 GEOMETRY AND MEASUREMENT: Spatial Relationships Analyzing geometric relationships develops reasoning and problem solving skills. 3.2 Use spatial reasoning, location and geometric relationships to solve problems. Describe the position, location and direction of objects, or parts of objects, using terms such as inside, outside, top, bottom, over, under, near, far, close, closer, etc. Describe the relative position, direction and location of objects with terms such as above, below, middle, left and right. Create and explore shapes and designs with a line of symmetry. Explore translations (slides), reflections (flips), and rotations (turns) of simple polygons using manipulative materials. Explore filling a two-dimensional region with different shapes. Investigate a line of symmetry when a simple shape is transformed through translations (slides), reflections (flips) and rotations (turns) Read and interpret simple maps using coordinate systems and shapes or pictures. Explore translations (slide), reflections (flip) and rotations (turn) in geometric designs. Use coordinate systems to specify locations. Identify and use translations (slide), reflections (flip) and rotations (turn) in geometric designs. Represent the surface of threedimensional objects through the use of two-dimensional nets. Investigate and develop strategies to determine the volume of rectangular solids. Plot points on the rectangular coordinate system and estimate and determine the distance between points. 11

12 GEOMETRY AND MEASUREMENT: Measurement The units of time are used to reflect duration and order of events. Tools, techniques and formulas exist for determining measurements. Opportunities exist for reasonable approximations. 3.3 Develop and apply units, systems, formulas and appropriate tools to estimate and measure. Locate a date on the calendar (yesterday, today and tomorrow). Sequence events using terms such as before and after. Use the calendar to identify dates, days, weeks and months and to plan and sequence events. Estimate and compare the length of time needed to complete tasks using terms such as longer or shorter. Use the calendar to write and solve problems involving time. Tell time to the half hour and quarter hour, and explore time to five minute intervals. Classify time periods as AM and PM. Use calendars and clocks to plan and sequence events. Tell time to the minute, using analog and digital clocks. Investigate problems involving elapsed time (hour, ½ hour, and 15 minute intervals) Solve problems involving time, elapsed time (minutes, hours, A.M., P.M.), and calendars. Solve problems involving the conversion of measures of time (minutes, hours, and days). Solve problems involving the conversion of time. Use estimation and counting to compare quantities that are the same, more, or less. Explore strategies to estimate and compare length, and weight using nonstandard units. Compare the weight of two objects using a balance scale and identify which is heavier. Tell time to the hour with analog and digital clocks. Explore half-hour with analog and digital clocks. Estimate the amount of objects in a set using benchmarks of 10, and count to determine if the estimate is more or less Determine the reasonableness of a measurement or estimate using nonstandard units. (i.e. length, weight and temperature) Explore elapsed time to the hour and ½ hour. Identify reasonable estimates and describe the strategies used to determine the estimates. Develop and use nonstandard referents and standard benchmarks to estimate and measure length, area, weight, capacity and volume. Use customary and metric units to estimate and measure length. (i.e. in., ft., yd., cm, m). Develop and explain strategies for using nonstandard and standard referents to estimate customary and metric units of length, area, perimeter, weight and temperature. Draw line segments using standard linear measurement tools (to the nearest inch and centimeter.) Identify and use the appropriate customary and metric units and tools for measuring length (to the nearest 1/4 in. or ½ cm), area, perimeter, weight, temperature, volume and capacity. Solve practical problems that involve estimation and measurement of length, perimeter, area, volume, capacity, mass, weight, and temperature. Estimate and choose appropriate units (using metric and customary systems) and tools to measure and solve a variety of problems involving length, perimeter, area, and volume of (parallelograms, rectangular prisms, and triangles). 12

13 Explore using the standard units of inch and centimeters to estimate and measure length. Explore using measurement tools such as thermometers, basic rulers and balance scales to measure temperature, length and weight. Explore converting customary units of measurement. (yd., ft., in.) Solve problems involving the conversions of measures of length (yd., ft., inches, cm, and m). Explore relationships between the lengths of sides of rectangles and their areas and perimeters and generalize the patterns as simple formulas. Estimate and choose appropriate units and tools to measure and solve a variety of problems involving, capacity, mass, time, angle and temperature Explore the relationship between area and perimeter when the dimensions of a polygon change. Develop formulas to find the perimeter and area of squares, rectangles and triangles. 13

14 WORKING WITH DATA: Graphs and Statistics 4.1 Collect, Data sets can be organized, displayed and analyzed. The way that data is collected, organized and displayed influences interpretation. Pose questions about personal information, experiences and environment. Pose questions and collect data about personal information, experiences, and the environment. Pose questions and systematically collect and sort data. Explore a variety of ways to collect, organize, record, analyze and interpret data. Design and conduct surveys and samplings to collect data that represent a general population. Design, conduct and analyze methods to collect, organize and display data that represent a general population. Explore ways to record and organize data using real and picture graphs. Organize and record data using tallies and tables. Construct real graphs, picture graphs and bar graphs to display data. Organize, record and analyze data using tallies, tables, and organized lists. Construct and label the parts of real graphs, pictographs and bar graphs from information collected or given. Identify basic patterns and explore trends in data. Construct and interpret line graphs, line plots, bar graphs, pictographs, and read circle graphs. Identify patterns and trends in data. Construct and interpret line graphs, line plots, bar graphs, pictographs, circle graphs, and stem and leaf plots and evaluate how ell each kind of display represents the features of the data. Identify and explain patterns and trends in data Interpret circle graphs and recognize that they represent data proportionally. Use systematic listing and counting strategies to solve problems. Data can be used to make predictions. 4.2 Analyze data sets to form hypotheses and make predictions Explore the use of comparative language to describe data with terms such as more, less, the same as, and equal. Use comparative language to describe the data in tables and graphs (how many more, how many less, same as, equal, fewer, more, etc) Analyze data displayed in graphs and tables using comparative terms. Explore the use of range, mode, and median to describe features of a data set. Make predictions and defend conclusions based on data. Use the range, mode, median and mean to describe features of a data set. Explore how a change in an outlier can change the measures of central tendency (mode, median and mean). Describe the shape of data sets using measures of spread (range and outliers) and central tendency (mode, median and mean). Explain how changes in a data set can affect the mode, median, mean and range. 14

15 WORKING WITH DATA: Probability 4.3 Understand and apply basic concepts of probability The probability of an event s occurrence can be predicted with varying degrees of confidence. Describe the likelihood of events related to students experiences. Engage in simple probability activities and discuss the results. Observe, record, and describe the results of simple probability activities and games in order to make predictions. Describe and explain the likelihood of various events in the real world using terms such as possible, impossible, likely, unlikely, certain, definite, probably, maybe. Discuss the likelihood of various events, state possibilities, make predictions and test them in practical situations using the terms fair and unfair. Conduct simple probability experiments and record the results in tables and charts to make predictions. Conduct multiple probability experiments and express the probability based on possible outcomes. Recognize the fairness of tests and determine likely/unlikely outcomes. Identify possible outcomes using combinations (where order does not matter) and explore situations resulting in permutations (where order does matter). Identify possible outcomes and express the likelihood of events as a fraction. Analyze the fairness of tests and determine likely/unlikely outcomes. Design and conduct probability experiments and games of chance. Make and test predictions of probability and fairness. Describe possible outcomes of events using combinations (where order does not matter) and explore situations resulting in permutations (where order does matter). Explore the relationship between the number of trials in an experiment and the predicted outcomes. Design and conduct probability experiments and make predictions about outcomes that are equally likely or not equally likely. Express probabilities as fractions, ratios, decimals and percents. Evaluate possible outcomes of events using combinations (where order does not matter) and explore situations resulting in permutations (where order does matter). 15