# 1A: Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

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1 NCTM STANDARD 1: Numbers and Operations Kindergarten Grade 2 1A: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Kindergarten Grade One Grade Two 1. Count orally to Count objects to Write numerals from 0 to Write numbers from 0 to Given two sets of less than 10 objects, recognize which set has more or less. 6. Match and write the numeral for a set of less than 10 objects. 7. Identify the next number less and the next number more than any named number. 8. Recognize a fractional part of a whole (i.e. one half of the class, one fourth cup of milk, one third of the paper). 1. Count objects to Count orally to 100 by 1 s, 5 s and 10 s. 3. Count orally to 20 by odd or even numbers. 4. Given any two numbers, read and write all the whole numbers between them. 5. Write numbers to Use manipulatives to model place value of ones, tens and hundreds. 7. Identify ordinals 1 st through 10 th. 8. Be able to identify the numbers before and after a given number. 9. Use the hundreds chart to identify number patterns (i.e. odd and even numbers, counting by 5 s and 10 s, etc.). 10. Represent a whole number as a combination of two other whole numbers by composing and decomposing numbers (i.e = 3, 4-1 = 3, etc.). 11. Identify and write the numerical words zero to ten. 12. Break a whole into halves, thirds and quarters. 13. Compare fraction parts which is more/less, equal to. 1. Read and write numbers to 1, Given any two numbers, read and write all the whole numbers between them. 3. Demonstrate place value using manipulatives to represent numbers to1, Use models to represent the relative position of cardinal and ordinal numbers and their connections (i.e. hundreds chart use, base 10 blocks). 5. Read and write cardinal and ordinal words and symbols. 6. Represent a whole number as a combination of two or more other whole numbers by composing and decomposing numbers (i.e = 10, = 10, = 10, = 10, 12-2 = 10, etc.). 7. Break a whole into halves, thirds, quarters, sixths and eighths. 8. Compare fraction parts which is more/less, equal to

7 NCTM STANDARD 2: Algebra Kindergarten Grade 2 2A: Understand patterns, relations, and functions. Grade K Grade 1 Grade 2 1. Sort and classify by size, shape and color. 2. Identify patterns using people, objects and numbers. 3. Observe and identify patterns in the environment. 1. Sort, classify and order objects by size, shape and color. 2. Recognize, describe and extend patterns using objects and numbers (i.e. geometric patterns, skip counting). 3. Recognize that patterns can repeat and/or grow. 1. Sort, classify and order by size, weight and number and explain their reasoning. 2. Sort and classify objects by use and other attributes. 3. Describe and extend numeric and geometric patterns. 4. Use different symbols to translate a given pattern (i.e. ABC, 123). 5. Extend both repeating and/or growing patterns

8 NCTM STANDARD 2: Algebra Kindergarten Grade 2 2B: Represent and analyze math situations and structures using algebraic symbols. Grade K Grade 1 Grade 2 1. Recognize commutativity using concrete objects and number symbols. 2. Explore the concept of unknown numbers using manipulatives and games. 1. Recognize the property of commutativity (i.e = 3 + 5). 2. Represent various ways to group numbers to find a given solution (i.e. use unifix cubes to show the different ways to represent 9). 3. Use concrete, pictorial and verbal representations to develop an understanding of conventional symbols for addition, subtraction and equals. 1. Apply commutativity to addition. 2. Write number sentences using the conventional symbols for addition, subtraction and equals

9 NCTM STANDARD 2: Algebra Kindergarten Grade 2 2C: Use mathematical models to represent and understand quantitative relationships. Grade K Grade 1 Grade 2 1. Listen to and tell number stories. 2. Use manipulatives to model addition and subtraction. 1. Model and explain situations that involve addition/subtraction of whole numbers using objects, pictures and symbols. 1. Write simple word problems involving addition and subtraction

10 NCTM STANDARD 2: Algebra Kindergarten Grade 2 2D: Analyze change in various contexts. Grade K Grade 1 Grade 2 1. Learn language to describe qualitative change (i.e. taller, smaller). 2. Learn language to describe quantitative change (i.e. 2 inches more, 3 feet less). 1. Compare and describe qualitative change over a period of time (i.e. taller, smaller). 2. Describe quantitative change that involves measuring length, weight and capacity. 1. Describe qualitative change, such as a student growing taller. 2. Measure quantitative change (i.e. distance, height, weight and capacity)

11 NCTM STANDARD 2: Algebra Grade 3 Grade 5 2A: Understand patterns, relations, and functions. Grade 3 Grade 4 Grade 5 1. Describe, extend and make generalizations about simple geometric and numeric patterns. 2. Write and explain the rule of a given geometric or numeric pattern. 1. Recognize, describe, extend and create a wide variety of numeric and geometric patterns. 2. Describe and represent a pattern in a table or a graph. 3. Use variables and open sentences to express relationships. 4. Use graphical representations to discover patterns and relationships. 5. Translate data from a table to a graph. 1. Describe, extend and make generalizations about geometric and numeric patterns. 2. Represent and explain numeric and geometric patterns with concrete materials, tables and graphs and translate among them

12 NCTM STANDARD 2: Algebra Grade 3 Grade 5 2B: Represent and analyze math situations and structures using algebraic symbols. Grade 3 Grade 4 Grade 5 1. Use models to demonstrate the commutative, associative and distributive properties. 2. Apply the commutative and associative properties to the addition of whole numbers. 3. Represent an unknown quantity using a variable. 4. Write simple equations using the symbols +, -, x, =, n,?,. 1. Use models to demonstrate the commutative, associative and distributive properties. 2. Apply the commutative and associative properties to whole numbers. 3. Apply the distributive property to basic math facts (i.e. 8x9 =8x6 + 8x3). 4. Represent an unknown quantity using a shape or variable (i.e. 2 x triangle = square + circle or 4 x 6 = n). 5. Represent a number sentence with appropriate algebraic symbols. 1. Using appropriate algebraic symbols, write open sentences that describe everyday situations. 2. Recognize and apply the commutative, associative and distributive properties to compute with whole numbers. 3. Represent the idea of a variable as an unknown quantity using a letter or?. 4. Express mathematical relationships using equations

13 NCTM STANDARD 2: Algebra Grade 3 Grade 5 2C: Use mathematical models to represent and understand quantitative relationships. Grade 3 Grade 4 Grade 5 1. Discuss how and why different models can be used to represent the same relationship. 2. Draw conclusions from representations involving graphs, tables and equations. 1. Create graphs and tables that can be used to represent the same relationship. 2. Draw conclusions from representations involving graphs, tables and equations. 1. Analyze tables and graphs to identify algebraic relationships. 2. Interpret tables and graphs for their practical implications

14 NCTM STANDARD 2: Algebra Grade 3 Grade 5 2D: Analyze change in various contexts. Grade 3 Grade 4 Grade 5 1. Recognize constant and variable changes that occur over a period of time (i.e. plant growth, height). 1. Recognize constant and variable changes that occur over a period of time (i.e. temperature, weight). 1. Recognize, describe and compare situations with constant or varying rates of change. 2. Investigate how a change in one variable relates to change in a second variable

15 NCTM STANDARD 3: Geometry Kindergarten Grade 2 3A: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Grade K Grade 1 Grade 2 1. Name, trace, build and sort 2-dimensional shapes (i.e. triangle, rectangle, square, circle and hexagon). 2. Play with a wide variety of two- and threedimensional shapes. 1. Identify, compare and sort 2-dimensional shapes (i.e. circle, square, triangle, rectangle, trapezoid, hexagon and diamond). 2. Identify, compare and sort 3-dimensional shapes (i.e. sphere, cube, cone, cylinder, box and pyramid). 3. Describe and compare corners, sides and angles of 2-dimensional shapes. 4. Describe and compare edges, bases, faces and corners of 3-dimensional shapes. 1. Build and/or draw two- and threedimensional shapes (i.e. circle, square, triangle, rectangle, trapezoid, diamond, hexagon, sphere, cube, cone, cylinder, box and pyramid). 2. Identify, describe, compare and sort twoand three-dimensional shapes. 3. Describe and compare corners, sides and angles of 2-dimensional shapes. 4. Describe and compare edges, bases, faces and vertices of 3-dimensional shapes. 5. Describe the results of putting together or taking apart a two-dimensional shape and describe the attributes of the resulting shape or shapes. 6. Explore the concept of similar and congruent objects

16 NCTM STANDARD 3: Geometry Kindergarten Grade 2 3B: Specify locations and describe spatial relationships using coordinate geometry and other representational systems. Grade K Grade 1 Grade 2 1. Learn names of positions by moving their bodies in space (i.e. on, around, through, above, below, inside, outside). 2. Use the language of positions of objects in space. 1. Describe and name positions in space (i.e. near, far, between). 2. Move objects and be able to describe their position (i.e. above, below, behind, in front of). 3. Build and draw simple maps (i.e. playground, classroom, home). 4. Describe how to move from one location to another on a given map. 1. Make and/or interpret a simple map that includes direction and distance. 2. Find and name locations with simple relationships such as near to, inside, outside, beside, next to, left and right. 3. Transfer a simple geometric shape from one grid to another, as is

17 NCTM STANDARD 3: Geometry Kindergarten Grade 2 3C: Apply transformations and use symmetry to analyze mathematical situations. Grade K Grade 1 Grade 2 1. Practice moving shapes in different directions, as with tangrams. 2. Practice use of symmetry by tracing, cutting and folding seasonal shapes (i.e. apple, snowman). 1. Explore the effects of slides, flips and turns using manipulatives. 2. Recognize and build symmetrical patterns. 1. Demonstrate the turn (rotation), flip or slide of a given shape. 2. Identify shapes that are symmetric and draw in the lines of symmetry

18 NCTM STANDARD 3: Geometry Kindergarten Grade 2 3D: Use visualization, spatial reasoning, and geometric modeling to solve problems. Grade K Grade 1 Grade 2 1. Draw simple shapes, such as circle, triangle and rectangle. 2. Describe how to move from one location to another. 3. Make designs and demonstrate shapes changing positions using manipulatives. 4. Learn that shapes can appear in different sizes. 5. Identify shapes in the classroom and at home. 1. Describe how to move from one location to another. 2. Build a pattern, then describe it so another student can build it from description. 3. Recognize and extend a given pattern. 4. Use attribute blocks to compare shapes in different sizes: big, small. 5. Recognize various geometric shapes in environment. 1. Identify and/or describe a given shape. 2. Recognize geometric shapes and structures in the environment and specify their location

19 NCTM STANDARD 3: Geometry Grade 3 Grade 5 3A: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Grade 3 Grade 4 Grade 5 1. Define and classify 2- and 3-dimensional shapes using appropriate vocabulary. 2. Draw and build 2- and 3-dimensional shapes according to their properties. 3. Compare objects that are congruent and objects that are similar. 4. Explore the geometric properties of common shapes and related vocabulary (i.e. parallel, perpendicular). 5. Name, make and describe cylinders, cones, spheres and pyramids. 1. Identify, describe and draw models of rays, parallel and perpendicular lines, as well as right, straight, acute and obtuse angles. 2. Use geoboards to explore the properties of parallelograms, rhombuses and trapezoids. 3. Use technology or paper and pencil to draw figures and angles (i.e. Geometer s Sketch Pad). 4. Distinguish between congruence and similarity; recognize two figures that are congruent or similar. 5. Name, make and describe properties of cylinders, cones, cubes, spheres and pyramids. 6. Identify faces, edges and vertices of 3- dimensional figures. 1. Sort and/or classify by properties, a selection of plane figures. 2. Use appropriate vocabulary to describe properties of common 2- and 3- dimensional figures (i.e. edges, vertices, faces, etc.). 3. Justify why two or more complex figures are congruent or similar

20 NCTM STANDARD 3: Geometry Grade 3 Grade 5 3B: Specify locations and describe spatial relationships using coordinate geometry and other representational systems. Grade 3 Grade 4 Grade 5 1. Give and/or follow directions for moving from one location to another. 2. Locate points, create paths and measure vertical and horizontal distances within the first quadrant. 3. Explore and develop strategies for navigating on a grid (i.e. finding multiple paths from one location to another). 4. Use a coordinate pair to locate a point in the first quadrant. 1. Make and use coordinate systems to specify locations and describe points. 2. Given a point, identify the coordinate pair that describes its location in the first quadrant. 1. Recognize the characteristics that make up a Cartesian Plane (rectangular coordinate plane). 2. Plot points on a number line and on a Cartesian Plane. 3. Given a variety of points on a Cartesian Plane, identify the coordinate pair of each. 4. Use technology to explore graphs of linear and non-linear relationships to make predictions or form conclusions. 5. Find vertical and horizontal distances on coordinate systems

21 NCTM STANDARD 3: Geometry Grade 3 Grade 5 3C: Apply transformations and use symmetry to analyze mathematical situations. Grade 3 Grade 4 Grade 5 1. Explore and discuss figures with multiple lines of symmetry. 2. Create designs that show rotational symmetry. 3. Determine the transformations needed to convert an original figure to its new orientation using technology (i.e. The Factory Deluxe). 4. Describe a motion or series of motions that will show that two shapes are congruent. 1. Draw a shape that has been slid (translated) a given distance or flipped (reflected) about a given line. 2. Recognize that figures may be congruent even though their orientations are different. 3. Use computer software to explore rotations and reflections of 2- and 3-dimensional objects. 4. Draw in all lines of symmetry in a given figure. 1. Draw a shape that has been rotated about a point or flipped (reflected) about a given line. 2. Describe a series of transformations that will give two congruent shapes the same orientation. 3. Identify and describe line and rotational symmetry. 4. Use technology to explore translations, reflections and rotations of 2- and 3- dimensional shapes

22 NCTM STANDARD 3: Geometry Grade 3 Grade 5 3D: Use visualization, spatial reasoning, and geometric modeling to solve problems. Grade 3 Grade 4 Grade 5 1. Construct simple three-dimensional shapes from two-dimensional nets (plans). 2. Recognize that by altering a shape, its properties are also altered. 1. Create complex shapes from a set of three or more simple shapes (i.e. pattern blocks, etc.). 2. Separate a given shape into smaller shapes (i.e. rectangle into smaller rectangles or two or more triangles). 3. Select a three-dimensional object given illustrations from two or more different perspectives. 1. Create a given shape from a set of tangrams or other manipulatives. 2. Separate a given two-dimensional complex shape into simpler shapes. 3. Construct common three-dimensional shapes using manipulatives and/or twodimensional nets. 4. Construct a three-dimensional object from a set of two-dimensional plans. 5. Draw a two-dimensional representation of a three-dimensional object

23 NCTM STANDARD 4: Measurement Kindergarten Grade 2 4A: Understand measurable attributes of objects and the units, systems, and processes of measurement. Grade K Grade 1 Grade 2 1. Explore topics in measurement such as weight, area, time, length, volume and money through block play, housekeeping, puzzles and the use of calendar. 2. Use appropriate comparison words (i.e. heavy, light, long and short). 3. Order objects by attribute of weight, volume and length. 4. Explore measurement with non-standard and standard devices. 5. Explore measurement using metric and customary units. 6. Explore use of tools and units used for different kinds of measurement (i.e. ruler for measuring length). 7. Recognize penny, nickel and dime. 1. Recognize and describe the attributes of length, volume, weight, area, time and money using manipulatives. 2. Compare and order objects by their attributes (i.e. shortest to tallest). 3. Measure length using non-standard units. 4. Measure length, time and temperature using standard units. 5. Recognize the value of a penny, nickel, dime and quarter. 6. Find the value of combinations of pennies, nickels and dimes. 7. Explore the many ways to represent a given amount of money. 1. Recognize and describe the attributes of length, volume, weight, area, perimeter, time and money using manipulatives. 2. Order objects according to an attribute. 3. Compare objects in terms of more than one attribute. 4. Demonstrate how to measure length and volume using non-standard and standard units. 5. Recognize the value of a penny, nickel, dime, quarter, half-dollar and dollar. 6. Find the value of combinations of pennies, nickels, dimes, quarters, half-dollars and dollars. 7. Demonstrate the many ways to represent a given amount of money

24 NCTM STANDARD 4: Measurement Kindergarten Grade 2 4B: Apply appropriate techniques, tools and formulas to determine measurements. Grade K Grade 1 Grade 2 1. Measure with multiple copies of a unit (i.e. how many blocks to measure a rug). 2. Measure something larger than the unit with repetition of a unit. 3. Use various measuring tools (i.e. cups, cubes, rulers, balance scales). 4. Recognize the appropriate unit for the attribute being measured. 1. Select an appropriate unit and tool for the attribute being measured. 2. Measure and record length and weight with multiple copies of non-standard units. 3. Measure and record the length of something larger than the unit with repetition of a single non-standard or standard unit. 4. Recognize the starting and end point of a measurement tool. 5. Read a clock in terms of hours and half hours. 6. Read a calendar in terms of the day of the week, the month. 7. Correctly apply the terms yesterday, today and tomorrow. 8. Understand and apply the terms degree, pound, day, week, inches, feet, centimeter. 1. Select and apply an appropriate tool for the attribute being measured and record the measurement using the appropriate unit. 2. Measure and record length, weight and area with multiple copies of non-standard units. 3. Measure and record length of something larger than the unit with repetition of a single non-standard or standard unit. 4. Read a clock in terms of hours, half hours, quarter hours and five-minute increments. 5. Read a calendar in terms of the day of the week, the month, the number days in a week, the number of months and days in a year. 6. Measure to the nearest inch. 7. Use common vocabulary as it relates to measurement

25 NCTM STANDARD 4: Measurement Grade 3 Grade 5 4A: Understand measurable attributes of objects and the units, systems, and processes of measurement. Grade 3 Grade 4 Grade 5 1. Explore measuring area of rectangles on a coordinate grid. 2. Explore measuring the perimeter of rectangles. 3. Distinguish between the area and perimeter of a rectangle. 4. Explain the attributes of length, area, weight and volume. 5. Justify the need for measuring with standard units and apply standard units in the customary and metric systems. 6. Recognize a right angle. 7. Determine the appropriate unit of measurement for a given object. 8. Convert between the notations (i.e. March 12, 2003 to 3/12/03). 1. Select appropriate units of measure to measure various objects (i.e. centimeter for the length of a pencil). 2. Estimate and measure area of simple polygons on a coordinate grid. 3. Estimate and measure the perimeter of simple polygons. 4. Distinguish between the area and perimeter of simple polygons. 5. Recognize right, obtuse and acute angles. 6. Determine the appropriate unit of measurement for a given object. 7. Recognize the reasonableness of an estimate Select an appropriate metric or customary unit to measure length and the area of a common figure. 2. Describe the attributes and the units of measure for length, area, weight, volume and angles. 3. Estimate volume, weight and temperature using metric and customary units. 4. Convert units within the same system, metric and customary. 5. Do simple conversions between metric and customary systems. 6. Convert among units of time (i.e. seconds, minutes, hours, days, weeks, years). 7. Estimate and measure the approximate area of an irregular polygon or complex figure. 8. Estimate and measure the perimeter of an irregular polygon or complex figure. 9. Explore standard and non-standard units of measure for length, volume, weight and temperature. 10. Understand measurements are approximations and how differences in units affect precision. 11. Recognize and apply the use of reasonable estimates.

26 NCTM STANDARD 4: Measurement Grade 3 Grade 5 4B: Apply appropriate techniques, tools, and formulas to determine measurements. Grade 3 Grade 4 Grade 5 1. Tell time to the nearest minute. 2. Measure length of objects to nearest half inch, quarter inch and centimeter. 3. Use a ruler to measure the sides of a rectangle to explore the concept of perimeter and area. 4. Select and apply appropriate standard units and tools to measure length, weight, time and temperature. 5. Use non-standard units to estimate measurements (i.e. the length of a bus seat is similar to a meter). 6. Given specified amount of money, determine what could be bought and/or the amount of change received from a purchase. 7. Given a specified time, determine what time it will be one hour and/or one half hour earlier or later. 1. Measure various objects using appropriate units (i.e. cm for pencil length, m for building length). 2. Given a specified amount of money, determine what could be bought and/or the amount of change received from a purchase. 3. Add or subtract units of length that may or may not require renaming units (i.e. m to cm). 4. Convert inches to feet or centimeters to meters. 5. Estimate and measure capacity in liters, quarts, and gallons. 6. Choose the appropriate tool to find perimeter of a given figure. 7. Explore the addition of minutes to a specified time. 8. Estimate perimeters and areas of standard figures in metric and customary units. 9. Given a standard unit, estimate and measure the area of rectangular regions. 10. Recognize and apply the appropriate formula to determine area of rectangles and volume of rectangular prisms. 11. Make appropriate estimates of size, quantity, temperature and passage of time. 12. Estimate length and area prior to calculating the actual measurement. 1. Make accurate measurements using appropriate tools for length, area, volume, weight, time, temperature and angles. 2. Use formulas for finding the area of rectangles, related triangles and parallelograms. 3. Find the perimeter of polygons and complex shapes. 4. Use formulas to find the volume of rectangular solids. 5. Develop strategies to determine the surface area of rectangular solids. 6. Given a specified amount of money, determine what could be bought and/or the amount of change received from a purchase

27 NCTM STANDARD 5: Data Analysis and Probability Kindergarten Grade 2 5A: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. Grade K Grade 1 Grade 2 1. Collect data in response to simple questions (i.e. What kind of pets do you have?). 2. Discuss ways to organize the data. 3. Display data using objects, pictures or drawings. 4. Explore coloring in a given bar graph. 1. Pose questions relevant to themselves, the environment or the school. 2. Collect data for questions that have multiple responses (i.e. What month is your birthday?). 3. Organize data in a tally sheet or with manipulatives. 4. Display data in terms of tally sheets, pictographs and simple bar graphs. 1. Pose questions relevant to themselves, the environment or the school. 2. Collect data from a single or multiple sources using tally sheets. 3. Display data as a bar graph or pictograph with paper and pencil and technology. 4. Explore the idea that scale can influence the interpretation of the data

28 NCTM STANDARD 5: Data Analysis and Probability Kindergarten Grade 2 5B: Select and use appropriate statistical methods to analyze data. Grade K Grade 1 Grade 2 1. Describe a given set of data in terms of most or least. 1. Recognize the numerical value represented by each bar on a bar graph. 2. Recognize the numerical value represented by each image on a pictograph. 1. Explore the relationship between the size of the sample and the accuracy of the information

29 NCTM STANDARD 5: Data Analysis and Probability Kindergarten Grade 2 5C: Develop and evaluate inferences and predictions that are based on data. Grade K Grade 1 Grade 2 1. Discuss how data answers the question posed. 1. Share ideas about what the data represents. 2. Determine which event is most likely or least likely to happen given appropriate information. 1. Write story problems using information from their graphs. 2. Predict how sets of data will likely/not likely change with the introduction of new data

30 NCTM STANDARD 5: Data Analysis and Probability Kindergarten Grade 2 5D: Understand and apply basic concepts of probability. Grade K Grade 1 Grade 2 1. Discuss probability in terms of seasonal weather (i.e. Is it likely to snow today?) 1. Discuss probability in terms of seasonal weather. 2. Predict the likelihood of an event based on recorded data. 1. Explore the probability of an event happening through experimentation (i.e. flipping a coin, tossing a die and spinning a spinner). 2. Write about the outcomes of their experiments

32 NCTM STANDARD 5: Data Analysis and Probability Grade 3 Grade 5 5B: Select and use appropriate statistical methods to analyze data. Grade 3 Grade 4 Grade 5 1. Recognize a set of data as a whole. 2. Compare related data sets. 3. Describe the distribution of data. 4. Explain the meaning of the terms median and mode. 5. Given a set of ordered numbers, identify the median and the mode. 1. Explain the meaning of the terms mean (average), median and mode. 2. Identify mean, median and mode for a given set of data. 3. Compare various representations of data and select the most suitable one. 1. Explain the meaning of the terms mean (average), median, mode and range. 2. Identify mean, median, mode and range for a given set of data with and without technology. 3. Compare data displayed various ways and evaluate how well it represents the data

33 NCTM STANDARD 5: Data Analysis and Probability Grade 3 Grade 5 5C: Develop and evaluate inferences and predictions that are based on data. Grade 3 Grade 4 Grade 5 1. Write story problems using information collected from a graph. 2. Interpret picture and bar graphs made by classmates or found in books or newspapers. 3. Provide a written or oral interpretation of the information represented by a graph. 4. Use line graphs to make predictions in terms of increasing/decreasing trends. 1. Write brief descriptions of graphs. 2. Interpret and analyze bar, picture, circle and line graphs. 3. Write story problems using information from a graph. 4. Use line graphs to make predictions in terms of increasing/decreasing trends. 1. Explore the concepts of sampling and biased data. 2. Propose and justify conclusions and predictions based on data represented as bar, line, circle and pictographs. 3. Use the mean, median, mode and range to make conclusions for a given set of data. 4. Explain how mean, median and mode can be used to define the middle of a set of data. 5. Generate additional questions based on their conclusions and predictions

34 NCTM STANDARD 5: Data Analysis and Probability Grade 3 Grade 5 5D: Understand and apply basic concepts of probability. Grade 3 Grade 4 Grade 5 1. Explore probability as a measurement of the likelihood of an event. 2. Explore probability through experiments with a few outcomes. 3. Given appropriate information, determine which is most likely or least likely to happen. 1. Determine which is most likely or least likely to happen, whether one event is more likely than another and explain the reasoning. 2. Explore the empirical probability of a simple event and record it as a ratio. 3. Explore notion of probability with computer software. 1. Recognize an event as either certain or impossible. 2. Use zero to describe an event that is impossible and one if it is certain. 3. Describe an event by its degree (i.e. certain, equally likely, impossible). 4. Recognize that the probability of an event is a number between zero and one. 5. Identify all possible outcomes for a simple sample space (i.e. tossing a die). 6. Conduct experiments to quantify the empirical (experimental) probability of simple events

35 NCTM STANDARD 1: Numbers and Operations Grade 6 1A: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. 1. Read, write and use physical models to represent positive rational numbers. 2. Convert a given fraction to equivalent fractions. 3. Compare two fractions or decimals using models. 4. Convert fractions to decimals and decimals to fractions. 5. Express percents as their fractional and decimal equivalents. 6. Express a composite number in terms of its prime factors. 7. Identify multiples and factors of whole numbers. 8. Use and apply the divisibility rules of 2, 3, 4, 5, 6, 8, 9 and 10. 1B: Understand meanings of operations and how they relate to one another. 1. Use models to demonstrate an understanding of the four operations with fractions. 2. Use models to demonstrate an understanding of the four operations with decimals. 3. Use models to express percents as parts of Demonstrate an understanding of the associative, commutative and distributive properties in a problem-solving situation. 5. Demonstrate an understanding of multiplication as repeated addition and division as repeated subtraction. 6. Understand the conversion of fractions to equivalent fractions with a common denominator. 7. Understand the conversion of fractions with base 10 denominators to decimals. 8. Understand the multiplicative and additive properties of 0 and Make predictions about products when multiplying different combinations of numbers greater than or less than one. 10. Understand that the power of a number is repeated multiplication (i.e. 2 3 is 2x2x2). 11. Explore the order of operations

36 NCTM STANDARD 1: Numbers and Operations, continued Grade 6 1C: Compute fluently and make reasonable estimates. 1. Demonstrate fluency in all four operations using decimals. 2. Demonstrate fluency in all four operations using fractions. 3. Find any whole number percent of any whole number or decimal. 4. Add and subtract integers using number lines. 5. Apply the associative, commutative and distributive properties to simplify computations. 6. Use estimation and mental computation strategies to determine the reasonableness of answers. 7. Select the appropriate computational method for a given situation (i.e. mental math, pencil and paper, estimation or calculator)

37 NCTM STANDARD 2: Algebra Grade 6 2A: Understand patterns, relations, and functions. 1. Represent, analyze, generalize and extend numeric and geometric patterns using tables, graphs, equations and words. 2. Represent simple linear equations using concrete materials, tables or graphs. 2B: Represent and analyze mathematical situations and structures using algebraic symbols. 1. Demonstrate an understanding of different uses of variables. 2. Given a graph of two linear equations, identify the trends represented. 3. Determine the operations needed to solve a word problem. 4. Solve one-step equations involving addition, subtraction, multiplication and division. 5. Write equations to model and solve word problems. 2C: Use mathematical models to represent and understand quantitative relationships. 1. Generate a table and a graph to model the relationship between two variables. 2. Describe what the variables represent in the context of a problem

38 NCTM STANDARD 2: Algebra, continued Grade 6 2D: Analyze change in various contexts. 1. Use graphs to analyze the nature of change in quantities in linear relationships. 2. Identify the graph of a constant rate of change and the graph of a variable rate of change. 3. Explore linear relationships to describe how a change in one quantity results in a change in another

39 NCTM STANDARD 3: Geometry Grade 6 3A: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Students will engage in problem solving, communicating, reasoning, connecting, and representing as they 1. Make models to explore the properties of two-dimensional geometric shapes. 2. Use the properties of two-dimensional geometric shapes to generate definitions of the shapes. 3. Draw and construct two-dimensional geometric shapes based on their definitions. 4. Use definitions to classify triangles according to measures of their sides and angles. 5. Use definitions to classify quadrilaterals (i.e. parallelogram, rhombus, trapezoid, square, rectangle). 6. Recognize the names of polygons of five or more sides. 7. Identify the characteristics of parallel and perpendicular lines. 8. Know the properties of similar objects with respect to the length of their corresponding sides, corresponding angles, perimeters, areas and volumes. 9. Know that two congruent shapes have equal corresponding sides and equal corresponding angles. 10. Create and critique inductive and deductive arguments concerning geometric ideas and relationships such as congruence and similarity. 11. Apply the property that the sum of the angles of a triangle equals 180 degrees. 12. Explore the properties of supplementary and complementary angles. 3B: Specify locations and describe spatial relationships using coordinate geometry and other representational systems. Students will engage in problem solving, communicating, reasoning, connecting, and representing as they 1. Use coordinate geometry to represent and examine the properties of geometric shapes (i.e. regular polygons and quadrilaterals). 2. Draw figures on a coordinate grid from a given set of coordinate pairs. 3. Use technology to explore graphs of linear and non-linear relationships to make predictions or form conclusions

40 NCTM STANDARD 3: Geometry, continued Grade 6 3C: Apply transformations and use symmetry to analyze mathematical situations. Students will engage in problem solving, communicating, reasoning, connecting, and representing as they 1. Describe sizes, positions and orientation of shapes under informal transformations such as flips, turns, slides and scaling from given pictures. 2. Create a tessellation using slides. 3. Identify transformations by name (i.e. translations, reflections, rotations and dilations). 4. Demonstrate the congruence and similarity of shapes using transformations. 5. Identify and draw lines of symmetry for given shapes. 6. Recognize objects or designs that exhibit rotational symmetry. 3D: Use visualization, spatial reasoning, and geometric modeling to solve problems. Students will engage in problem solving, communicating, reasoning, connecting, and representing as they 1. Draw geometric shapes with specified properties such as side length or angle measure. 2. Use geoboards to model geometric figures and identify their properties. 3. Use a protractor to construct and measure angles. 4. Construct the perpendicular bisector of a line segment using a compass and straight edge. 5. Make two-dimensional sketches of cubes. 6. Interpret the perspective from which a drawing was made of a three-dimensional object. 7. Recognize the equal faces of rectangular prisms. 8. Recognize nets that result in rectangular prisms. 9. Compute the volume of a cube. 10. Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science and everyday life

41 NCTM STANDARD 4: Measurement Grade 6 4A: Understand measurable attributes of objects and the units, systems, and processes of measurement. Students will engage in problem solving, communicating, reasoning, connecting, and representing as they. 1. Estimate length, mass and volume using customary and metric units. 2. Understand relationships among units of length and convert from one unit to another within the same system. 3. Understand, select and use the appropriate customary or metric unit for length, area or perimeter to measure an object. 4. Label angles using degrees. 4B: Apply appropriate techniques, tools, and formulas to determine measurements. Students will engage in problem solving, communicating, reasoning, connecting, and representing as they 1. Develop common benchmarks to estimate length and volume in customary units. 2. Use benchmarks to check reasonableness of measurements. 3. Measure objects using a ruler to the nearest eighth of an inch or millimeter. 4. Measure and draw angles using a protractor. 5. Measure to determine the area and perimeter of basic geometric figures. 6. Measure to determine the volume of a cube. 7. Measure the area of a geometric figure by dividing it into squares. 8. Apply formulas for the area and perimeter of squares and rectangles. 9. Apply formulas for the area of triangles and parallelograms. 10. Apply formulas for the circumference and area of a circle. 11. Determine the surface area of rectangular prisms. 12. Apply formulas for the volume of rectangular prisms. 13. Use a scale factor to create a scale drawing. 14. Explore simple problems involving rates of time and distance

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