Maryland K–6 mathematics standards

Count to 100 by ones and by tens.

Count forward from any given number within 100.

Count backwards from any given number within 20.

Write numbers from 0 to 20.

Represent a number of objects with a written numeral within 20 (with 0 representing a count of no objects).

Use the relationship between numbers and quantities within 20 to count objects with one-to-one correspondence (arranged in a line, a rectangular array, a circle, or as many as 10 things in scatter configuration) and verbalize that the last number stated is the total when asked “How many?” (cardinality).

When counting objects within 20, recognize that each successive number name refers to a quantity that is one larger.

Recognize the number of objects in a set without counting (subitizing) with both unfamiliar patterns within 6 and familiar patterns within 10.

Understand 10 as a group, collection, or bundle of ten ones called a “ten.” a. Compose and decompose numbers from 11 to 19 into ten ones and some further ones by using objects or drawings, and equations (e.g., 13 = 10 + 3). b. Describe a given number as one ten and the correct number of ones (e.g., 13 is one ten and three ones).

Compare two quantities within 20 using greater than, equal to, or less than, with objects, location on number paths, and written numerals.

Decompose numbers within 10 in more than one way, by using objects or drawings, and record each decomposition with a drawing or equation (e.g., 10 = 4 + 6 and 10 = 4+5 + 1).

Represent addition and subtraction situations (presented with numerals or mathematical symbols) within 10 with objects, fingers, drawings, sounds (e.g., claps), acting out situations, using verbal explanations, and/or writing expressions and equations.

Solve addition and subtraction problems in context (add to, take from, put together/take apart) within 10 and represent by using objects, drawings, and/or equations.

Identify, extend, and create repeating patterns (AABAAB, ABCABC, or AABBCC) using concrete objects, drawings, sounds, or movements.

Describe measurable attributes of an object such as length, height, weight, and capacity using appropriate vocabulary (e.g. long, short, tall, heavy, light, wide, narrow, full, empty).

Directly compare two objects with a measurable attribute in common, using words such as “greater/less,” “more/fewer,” “longer/shorter,” “lighter/heavier,” or “taller/shorter.”

Order up to 3 objects by a measurable attribute (e.g. greatest to least in height).

Describe the relative positions of objects using terms such as above, below, beside, in front of, behind, and next to.

Identify and describe given shapes and shapes of objects in everyday situations including two-dimensional shapes (circle, triangle, rectangle, square, hexagon) and three- dimensional shapes (cone, cube, cylinder, and sphere) regardless of their orientation or overall size.

Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).

Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides, edges, and vertices/”corners”) and other attributes (e.g., having sides of equal length).

Draw two dimensional shapes and build or create models of three-dimensional shapes.

Compose simple shapes to form larger shapes (e.g. join two triangles with full sides touching to compose a rectangle).

Organize data by classifying objects into given categories and counting the number of objects in each category (Limit the total in any one category to maximum of 20).

Analyze data sets by ordering the categories by count.

Count forward and backward within 120. a. Count forward and backwards by ones starting with any number. b. Skip count forward and backwards by tens.

Read, write numerals, and represent a number of objects with a written numeral within 120.

Extend understanding of 10 as a group, collection, or bundle of ten ones, called a “ten,” to compose and decompose two-digit numbers. a. Compose and decompose two-digit numbers into tens and ones using objects or drawings, and equations (e.g., 10 + 4 = 14; 38 = 30 + 8). b. Describe a given number as the correct number of tens and ones (e.g., 14 is one ten and four ones; 38 is three tens and eight ones). c. Compose and decompose two-digit numbers in more than one way (e.g. 46 = 41 + 5 or 46 = 23 + 23).

Represent whole numbers as lengths from 0 on a number line (horizontal and vertical) with equally spaced points corresponding to whole numbers.

Compare two numbers within 100 by reasoning about values of tens and ones digits and the location of the numbers on a number line. Record the results of comparisons with the symbols , =, and .

Estimate the location of numbers on a number line by reasoning about their relationship to benchmark numbers (e.g. 10, 50, 100).

Apply the Commutative Property of Addition and Associative Property of Addition as a strategy to add.

Use the inverse relationship between addition and subtraction to subtract.

Recall or quickly derive addition and subtraction facts within 20. a. Use the count on and count back strategies to add and subtract (e.g., 4 + 1, 7 – 2, etc.). b. Use the make ten strategy to add and subtract with combinations of 10 (e.g., 7 + 3, 6 + 4, 10 – 3, 10 – 6, etc.). c. Use the ten more and ten less strategies to add and subtract (including those with a difference of 10). d. Use the doubles to add and subtract (e.g., 6 + 6 = 12, 14 – 7 = 7, etc.)

Add within 100 including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of 10. Understand that when adding two-digit numbers, tens are added to tens, ones are added to ones, and sometimes it is necessary to compose or decompose a ten. a. Use concrete models and drawings to add. b. Use counting strategies and strategies based on place value (e.g., partial sums, making tens, etc.) to add. c. Use properties of operations, and/or the inverse relationship between addition and subtraction to add. d. Represent and explain the calculation by connecting the strategy used to the meaning of addition.

Given a two-digit number, identify 10 more or 10 less than the given number, without having to count, and explain the reasoning used.

Subtract a multiple of 10 from another multiple of 10 within 90. a. Use concrete models or drawings to subtract. b. Use counting strategies, properties of operations, and/or the inverse relationship between addition and subtraction to subtract. c. Represent and explain the calculation by connecting the strategy used to the meaning of subtraction.

Subtract a 1-digit number from a 2-digit number (without regrouping). a. Use concrete models or drawings to subtract. b. Use counting strategies, properties of operations, and/or the inverse relationship between addition and subtraction to subtract. c. Represent and explain the calculation by connecting the strategy used to the meaning of subtraction.

Partition circles and rectangles into equal shares (halves and fourths) recognizing that decomposing into more equal shares creates smaller shares. Determine how many equal shares are needed to make a whole (e.g., two halves).

Understand the meaning of the equal sign. Determine if equations involving addition and subtraction (on one or both sides of the equal sign) are true or false.

Add and subtract within 20 involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions (result unknown, change unknown, start unknown) by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem.

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers (e.g., 8 + ? = 13; 6 = – 2; 8 + 8 = ).

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Use non-standard units (with no gaps or overlaps) to measure the length of an object to the nearest whole unit.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size).

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Tell time in hours, half hours, and quarter hours using a digital clock or analog clock. Estimate time intervals (1 hour, 30 minutes, 15 minutes, 1 minute) for activities (e.g., sharpen pencil, soccer game, etc).

Identify and know the value of coins (penny, nickel, dime, and quarter) and bills ($1, $5, $10, and $20).

Ask and answer questions by collecting, organizing and summarizing data. a. Craft a question that can be answered by collecting categorical data. b. Collect and organize categorical data (into up to three categories) using surveys or observations. c. Represent data by creating tally charts and picture graphs. d. Summarize the data presented in tally charts using “most,” “least,” “greater than,” “less than,” and “equal to”.

Use understanding of 100 as a bundle of ten tens, called a “hundred,” and compose and decompose three-digit numbers. a. Compose and decompose three-digit numbers into hundreds, tens, and ones by using objects, drawings, and/or equations (e.g., 328 = 300 + 20 + 8 or 328 is thirty-two tens and 8 ones). b. Describe a given number as the correct number of hundreds, tens, and ones (e.g., 328 is three hundreds, two tens and eight ones). c. Compose and decompose three-digit numbers in more than one way (e.g., 439 = 400 + 39 or 439 = 435 + 4).

Count forward and backward within 1000 starting with any number. a. Skip-count forward and backwards by 2s, 5s, 10s, and 100s. b. Use skip-counting to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns.

Read and write numbers within 1000 using base-ten numerals, number names, and expanded form.

Compare two numbers within 1000 by reasoning about the values of the hundreds, tens, and ones digits and/or the location of the numbers on a number line. Record the results of comparisons with the symbols , =, and .

Estimate quantities by reasoning about their location on a number line, their relationship to benchmark numbers (e.g. 10, 50, 100), and to assess reasonableness of sums and differences.

Recall or quickly derive addition and subtraction facts within 20. a. Use counting, make ten, and ten more/less, doubling strategies to add and subtract. b. Use the use ten strategy (e.g., think of 9 + 1 + 6 to solve 9 + 7) and doubles plus one strategy (e.g., think of 4 + 4 + 1 to solve 4 + 5) to add and subtract. c. Use the inverse operation to add and subtract (e.g. think of 7 + ? = 15 to solve 15 – 7 = ?).

Represent whole number sums and differences within 100 using lengths on a number line.

Fluently add and subtract within 100. a. Use strategies based on counting (e.g., counting on, counting back) and place value (e.g., partial sums, making tens, etc.) to add and subtract. b. Use properties of operations, and the inverse relationship between addition and subtraction to add and subtract. c. Determine and explain when a strategy is most efficient.

Add and subtract within 1000, recognizing that when adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. a. Use concrete models or drawings to add and subtract. b. Use strategies based on place value (counting on, partial sums, making tens, etc.) to add and subtract. c. Use properties of operations, and/or the relationship between addition and subtraction to add and subtract. d. Represent and explain the calculation by connecting the strategy used to the meaning of addition and subtraction.

Mentally add and subtract 10 or 100 to a given number 100-900.

Partition circles and rectangles into equal shares (halves, thirds, and fourths) recognizing that equal shares do not need to have the same shape. Determine how many equal shares are needed to make a whole (e.g., three thirds).

Use addition and subtraction within 100 to solve one-step problems in context involving an unknown in any situation or position by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Use addition and/or subtraction within 100 to solve two- step problems in context involving any situation or position of an unknown by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Determine whether a group of objects within 20 has an odd or even number of members (e.g., by pairing objects or counting them by 2s). Write an equation to express an even number as a sum of two equal addends.

Measure the length of an object to the nearest whole unit by using appropriate tools (e.g., rulers, yardsticks, meter sticks, and measuring tapes).

Measure the length of an object twice using different length units. Compare the two measurements relative to the size of the unit chosen.

Estimate lengths using units of inches, feet, centimeters, and meters.

Measure to determine how much longer one object is than another, expressing the difference in length in terms of a standard-length unit.

Identify and draw shapes (triangles, quadrilaterals, pentagons, hexagons, and cubes) having specific attributes, such as a given number of angles or a given number of equal faces.

Explain that a line of symmetry for a two-dimensional figure is a line across the figure such that the figure can be folded along the line into matching parts. Identify line- symmetric figures and draw lines of symmetry.

Tell and represent time to the nearest minute using multiple representations (e.g., digital clocks, analog clocks, number lines).

Estimate and calculate elapsed time to the hour and half hour (e.g. If the school day begins at 8am and ends at 3 pm, the school day lasts 7 hours).

Count mixed sets of coins and bills to solve problems in context (within $20.00) involving bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Ask and answer questions by collecting, organizing and summarizing data. a. Craft a question that can be answered by collecting categorical and numerical data. b. Collect and organize data using surveys, making observations, or measuring. c. Represent categorical data using picture graphs and bar graphs. Represent numerical data using line plots. d. Summarize the data presented in data visualizations using “most,” “least,” “greater than,” “less than,” and “equal to” and determine what questions can be answered with a given data set.

Apply and extend place value understanding to whole numbers within 10,000. a. Read and write whole numbers using base-ten numerals, number names, and expanded form. b. Compose and decompose whole numbers. c. Estimate quantities by reasoning about their location on a number line, their relationship to benchmark numbers, and rounding to nearest 10 or 100. d. Compare two whole numbers by reasoning about the values of the digits and the location of the numbers on a number line. Record the results of comparisons with the symbols , =, and .

Represent and interpret multiplication of two factors (0- 10) as equal groups or as a composed unit that can be repeated/iterated. Explain the relationship between the factors and products.

Represent and interpret whole-number quotients (0-10) as dividing an amount into known number of groups (partitive) or as dividing an amount into groups of known size (quotative). Explain the relationship between the quotient, divisor, and dividend.

Identify and apply the Commutative Property of Multiplication, Associative Property of Multiplication, and Distributive Property of Multiplication as strategies to multiply.

Explain and use the inverse relationship between multiplication and division to determine the unknown whole number in a multiplication or division equation involving three whole numbers (e.g., find 32 ÷ 8 by thinking of unknown-factor problem and finding the number that equals 32 when multiplied by 8).

Estimate sums and differences within 1,000 using benchmark numbers and front-end estimation. Determine when an estimate is appropriate and when an exact answer is needed.

Fluently add and subtract within 1,000. a. Use computational strategies efficiently (e.g., decomposition, partials sums, compensation, make ten/hundred/thousand) to add and subtract. b. Use properties of operations, and/or the inverse relationship between addition and subtraction. c. Determine and explain when a strategy is most efficient.

Multiply one-digit whole numbers by multiples of 10 in the range of 10-90 (e.g., 9 × 80;60 × 5) using strategies based on place value and properties of operations.

Recall or quickly derive multiplication and division facts within 100 (e.g., factors less than or equal to 10 and quotients less than or equal to 10). a. Skip count (2s, 5s, 10s) and apply properties of operations (0s, 1s) to derive foundational facts. b. Derive unknown facts from known facts using double facts (e.g., to solve 4 × 3, double 3 to get 6 and double 6 to get 12) to multiply and divide. c. Use properties of operations (e.g., to solve 8 × 7, think of (5 × 7) + (3 × 7)) to multiply. d. Use the inverse relationship between multiplication and division (e.g., think of to solve ) to multiply and divide.

Recognize and explain that a fraction represents a part of a whole. a. Partition shapes into parts with equal areas and express the area of each part using a unit fraction. b. Explain that a fraction represents one part of a whole that is divided into d equal parts, and that a fraction represents c parts of size .

Explain a fraction of a set as the quantity formed by c items as part of the given set of d objects.

Represent a fraction as a number on the number line. a. Represent a fraction on a number line by defining the interval from 0 to 1 as the whole and partitioning it into d equal lengths. Recognize that each length measures and that the endpoints of the length starting at 0 represents the number on the number line. b. Represent a fraction on a number line by marking off c lengths of from 0. Recognize that the resulting interval has size and that the endpoint of the length locates the number on the number line.

Explain equivalence of fractions and compare fractions by reasoning about their size, recognizing that comparisons are valid only when the two fractions refer to the same whole. a. Represent two fractions as equivalent if they are the same size, or they represent the same point on the number line. b. Identify and generate equivalent fractions, (e.g., , .) and explain why the fractions are equivalent using concrete materials, drawings, number lines, or equations. c. Express whole numbers as fractions, and identify fractions that are equivalent to whole numbers, (e.g., express 3 in the form ; recognize that ; locate and 1 at the same point on a number line). d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Record the results of comparisons with the symbols , =, or , and justify the conclusions (e.g., by using concrete materials, drawings, number lines, and/or equations).

Use multiplication and division within 100 to solve one- step problems in context involving equal groups, arrays, and area by using concrete materials, drawings, and/or equations.

Use the four operations to solve two-step problems in context (within number limits for each operation as stated in 3.NOS.B and 3.NOS.D). a. Represent these problems using equations with a letter standing for the unknown quantity. b. Assess the reasonableness of answers in terms of context, using mental computation and estimation strategies including, rounding, front- end estimation, and benchmark numbers.

Identify arithmetic patterns and explain them using properties of operations (e.g., observe the pattern that 4 times a number is always even and explain why 4 times a number can be decomposed into two equal addends).

Measure the length of an object to the nearest half and quarter unit by using appropriate tools (e.g., rulers, yardsticks, meter sticks, and measuring tapes).

Solve problems in context that involve addition and subtraction of time intervals in minutes (e.g., represent and solve using a number line).

Measure and estimate liquid volumes and masses of objects using standard units of liters(l), grams (g), and kilograms (kg).

Add, subtract, multiply, or divide to solve one-step problems in measurement contexts involving masses or volumes that are given in the same units.

Explain that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category of shapes (e.g., quadrilaterals).

Identify area as an attribute of a two-dimensional figure and measure area of rectangular space by counting square units (non-standard equal-sized units, square centimeters, square meters, square inches, square feet).

a. Determine the area of a rectangle with whole- number side lengths by tiling and connecting the visual representation to multiplication of the side lengths to solve problems in context. b. Use tiling and area models to show that the area of a rectangle with whole number side lengths b and c + d is the sum of b × c and b × d, representing the Distributive Property of Multiplication. c. Determine the area of composite rectilinear figures by decomposing them into non- overlapping rectangles and adding the areas.

Identify perimeter as an attribute of two-dimensional figures and measure the distance around the outside of a figure by counting units or adding lengths.

Solve problems in contexts involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and finding rectangles with the same perimeter and different areas or with the same area and different perimeters.

Ask and answer questions by collecting, organizing and summarizing data, recognizing the importance of context when analyzing data. a. Create a scaled data visualization (e.g., picture graph, bar graph for categorial data; line plot with fraction units of halves and fourths for numerical data) to display collected or given data. b. Summarize data presented in scaled data visualizations (e.g., picture graph, bar graph for categorial data; line plot for numerical data) by describing the story the data is telling. c. Identify patterns in the data and evaluate whether these patterns might apply to different contexts or situations.

Apply and extend place value understanding to multi- digit whole numbers within 1,000,000. a. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. b. Estimate quantities by reasoning about their location on a number line, their relationship to benchmark numbers, and rounding to any place. c. Compare two multi-digit numbers by reasoning about the values of the digits and the location of the numbers on a number line. Record the results of comparisons with the symbols , =, and .

Explain that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right (e.g., recognize that 700 ÷ 70 = 10 by applying concepts of place value and division).

Fluently add and subtract multi-digit whole numbers within 1,000,000. a. Apply estimation strategies to estimate sums and differences. b. Use computational strategies (e.g., decomposition, partials sums) efficiently. c. Use a standard algorithm d. Determine and explain when a strategy or algorithm is most efficient.

Extend place value understanding and basic facts to multiply a multi-digit whole number by a multiple of ten.

Multiply a whole number of up to four digits by a one- digit whole number, and multiply two two-digit numbers. a. Apply estimation strategies to estimate products. b. Use strategies based on place value (e.g., partial products) to multiply. c. Use properties of operations (e.g., use doubling and halving strategy to think of 12 × 32 as 6 × 64) to multiply. d. Represent and explain the computation by connecting rectangular arrays, area models, and/or equations to the meaning of multiplication.

Divide whole numbers with up to four-digit dividends and one-digit divisors with and without remainders. a. Apply estimation strategies to estimate quotients. b. Use strategies based on place value (e.g., partial quotients) to divide. c. Use properties of operations and/or the inverse relationship between multiplication and division to divide. d. Represent and explain the computation by connecting rectangular arrays, area models, and/or equations to the meaning of division.

Recall or quickly derive multiplication and division facts within 100 (e.g., factors less than or equal to 10 and quotients less than are equal to 10). a. Skip count (2s, 5s, 10s) and apply properties of operations (0s, 1s) to derive foundational facts. b. Derive unknown facts from known facts using double facts (e.g., to solve 4 × 3, double 3 to get 6 and double 6 to get 12) to multiply and divide. c. Use properties of operations (e.g., to solve 8 × 7, think of (5 × 7 + (3 × 7)) to multiply. d. Use the inverse relationship between multiplication and division (e.g., think of to solve ) to multiply and divide.

Explain and apply concepts of factors, multiples, and prime and composite numbers for whole numbers in the range of 1-100. a. Identify factor pairs for a whole number in the range of 1–100. b. Determine whether a given whole number in the range of 1–100 is a multiple of a one-digit number. c. Distinguish between factors and multiples and explain how they relate to a given number. d. Determine whether a given whole number in the range of 1–100 is prime, composite, or neither.

a. Identify and generate equivalent fractions using representations and the Identity Property of Multiplication. b. Explain why fractions are equivalent ( ).

Compare two fractions with different numerators and different denominators, understanding that comparisons are valid only when the two fractions refer to the same whole. a. Use benchmark numbers (e.g., , 1), common numerators, and common denominators to compare. b. Record comparisons with symbols , =, . c. Justify comparisons using representations and reasoning.

Apply understanding of a fraction with d >1 as a sum of unit fractions ( ) to decompose a fraction (including fractions greater than 1) into a sum of fractions with the same denominator in more than one way, recording each decomposition as an equation. Justify decompositions using visual fraction models and equations (e.g., or or or ).

Apply and extend previous understanding of addition and subtraction of whole numbers to add and subtract fractions with like denominators. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Estimate sums and differences by reasoning about benchmark numbers and assess reasonableness of answers (e.g., will be greater than 1 because both fractions are greater than ). c. Apply and extend whole number addition and subtraction strategies (e.g. counting on, making a whole, partial sums, compensation, properties of operations, the inverse relationship between addition and subtraction) to add and subtract fractions. d. Solve problems in context involving addition and subtraction of fractions using visual fraction models and equations to represent the problem.

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number ( ). a. Estimate products by reasoning about benchmark numbers and assess reasonableness of answers (e.g., will be less than 4 because I have 4 groups of something less than 1). b. Use understanding of multiplication as equal groups to multiply a unit fraction by whole number (e.g., interpret as b groups of ). c. Use understanding of multiplication as equal groups to multiply a fraction by whole number (e.g., interpret as b groups of when is greater than or less than 1). d. Solve problems in context involving multiplication of a fraction by a whole number, using visual fraction models and equations to represent the problem.

Express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 to add two fractions with respective denominators of 10 and 100.

Use decimal notation for fractions with denominators 10 and 100 (e.g., rewrite 0.62 as or describe a length as 0.62 meters or locate 0.62 on a number line).

Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols , =, , and justify the conclusions, (e.g. using a visual model).

Interpret a multiplication situation as a comparison (e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5). Represent verbal statements of multiplicative comparisons as multiplication equations.

Multiply or divide to solve multiplicative comparison problems in context by using drawings and equations with a symbol for the unknown number to represent the problem

Distinguish multiplicative comparison from additive comparison.

Solve multistep problems in context involving whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. a. Represent these problems using equations with a letter standing for the unknown quantity. b. Assess the reasonableness of answers in terms of context, including interpreting remainders.

Apply the relationship between measurement units within a given measurement system (customary: in, ft, yd, oz, lb, sec, min, hr; metric: cm, m, km, g, kg, mL, L) to convert measurements from a larger unit to smaller unit.

Use the four operations to solve problems in context involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving common fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.

Apply the area and perimeter formulas for rectangles in context, including rectangles with missing dimensions.

Relate the area of a triangle to the area of a rectangle using decomposition and visual models. Derive and apply the formula to solve problems in contexts involving triangles positioned within or formed from the decomposition of rectangles

Identify angles as geometric figures formed by two rays that share a common endpoint and describe angle size as the amount of rotation between the two rays measured in degrees with reference to a circle.

Measure angles in whole-number degrees using a protractor. Estimate and sketch angles of specified measures.

Explain that when an angle is decomposed into non- overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in context (e.g., by using an equation with a symbol for the unknown angle measure).

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category and identify right triangles.

Ask and answer questions by collecting, organizing and summarizing data, recognizing the importance of context when analyzing data. a. Create scaled data visualization (e.g. bar graphs for categorical data; line plots with fraction units of halves, fourths, and eighths for numerical data) to display data to communicate an idea. b. Summarize data presented in scaled data visualizations (e.g. bar graphs, line plots) and draw conclusions about the data. c. Compare and contrast different data visualizations of the same data by varying attributes (e.g., reordering bars, changing the scale) and explain how changing the attributes affects the interpretation.

Explain that in a multi-digit number, a digit in one 5.NBT.A.1 place represents 10 times as much as it represents in the place to its right and of what it represents in the place to its left.

Explain patterns in the number of zeros of the product 5.NBT.A.2 when multiplying a number by powers of 10 and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10.

Read, write, and compare decimals to thousandths. 5.NBT.A.3 5.NBT.A.4 a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form (e.g. 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × ( ) + 9 × ( ) + 2 × ( )). b. Estimate decimal quantities by reasoning about their location on a number line, their relationship to benchmark numbers (e.g. 0, 0.25, 0.5, 0.75, 1) and rounding to any place. c. Compare two decimals to thousandths by reasoning about the values of the digits and the location on the number line. Record the results of comparisons with the symbols , =, .

Fluently multiply multi-digit whole numbers. a. Apply estimation strategies to estimate products. b. Use computational strategies (e.g., partial products, doubling and halving) efficiently to multiply. c. Use a standard algorithm to multiply. d. Determine and explain when a strategy or algorithm is most efficient.

Divide multi-digit whole numbers with up to four-digit dividends and two-digit divisors. a. Apply estimation strategies to estimate quotients. b. Use partial quotients efficiently to divide. c. Use properties of operations and/or the inverse relationship between multiplication and division to divide. d. Represent and explain the computation by connecting rectangular arrays, area models, and/or equations to the meaning of division.

Apply and extend previous understanding of addition and subtraction of whole numbers to add and subtract decimals to hundredths. a. Apply estimation strategies to estimate sums and differences of decimals. b. Use concrete models, drawings, strategies based on place value (e.g., partial sums), properties of operations and/or the inverse relationship between addition and subtraction to add and subtract decimals. c. Use an algorithm to add and subtract decimals. d. Determine and explain when a strategy or algorithm is most efficient.

Apply and extend previous understanding of multiplication and division of whole numbers to multiply and divide decimals to hundredths in context. a. Apply estimation strategies to estimate products and quotients. b. Use concrete models, drawings, and arrays to multiply and divide. c. Use strategies based on place value (e.g., partial products, partial quotients) to multiply and divide. d. Use properties of operations and/or the inverse relationship between multiplication and division to multiply and divide. e. Represent and explain the computation by connecting concrete models, drawings, and/or equations to the meaning of multiplication and division.

Add and subtract fractions with unlike denominators. a. Estimate sums and differences by reasoning about benchmark numbers and assess reasonableness of answers (e.g., will be more than 1 because both fractions are greater than ). b. Apply and extend whole number addition and subtraction strategies (e.g. counting on, making a whole, partial sums, compensation, properties of operations, the inverse relationship between addition and subtraction) to add and subtract fractions. c. Use equivalent fractions to produce an equivalent sum or difference of fractions with like denominators (e.g. . In general, ).

Solve problems in context involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators using visual fraction models and/or equations to represent the problem.

Interpret a fraction as division of the numerator by the denominator ( ) in context using visual fraction models and equations to represent the problem (e.g., interpret as the result of dividing 3 by 4 so when 3 wholes are shared equally among 4 people each person has a share of size ).

Apply and extend previous understandings of multiplication to multiply a whole number by a fraction ( ). a. Estimate products by reasoning about benchmark numbers and assess reasonableness of answers (e.g., will be less than 2 because I am creating a fractional part of 2). b. Multiply a whole number by a fraction (e.g., interpret as of b when is greater than or less than 1).

Apply and extend previous understandings of multiplication to multiply a fraction by a fraction. a. Estimate products by reasoning about benchmark numbers and assess reasonableness of answers (e.g., will be less than because I will have one half of ). b. Apply and extend whole number multiplication strategies (e.g. area models). c. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths and show that the area is the same as would be found by multiplying the side lengths.

Apply and extend previous understandings of multiplication to multiply a fraction by a fraction, with one or both factors greater than 1. a. Estimate products by reasoning about benchmark numbers and assess reasonableness of answers. b. Apply and extend whole number multiplication strategies (e.g. area model, partial products).

Interpret multiplication as scaling (sizing) in context. a. Compare the size of a product to the size of one factor based on the size of the other factor, without performing the indicated multiplication. b. Explain why multiplying a given number by a fraction greater than a 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relate the principle of fraction equivalence to the Identity Property of Multiplication.

Solve problems in contexts involving multiplication of fractions and mixed numbers, (e.g., by using visual fraction models or equations to represent the problem).

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions in context. a. Apply estimation strategies to estimate quotients and assess reasonableness of answers. b. Use visual fraction models and equations to interpret division of a unit fraction by a non-zero whole number and compute quotients. c. Use visual fraction models and equations to interpret division of a whole number by a unit fraction and compute quotients. d. Represent and explain the computation by connecting visual fraction models and/or equations to the meaning of division with unit fractions.

Interpret and evaluate numerical expressions with grouping symbols (e.g. parentheses, brackets, or braces).

Write expressions that record calculations with numbers and interpret numerical expressions without evaluating them.

Generate and analyze number patterns. a. Generate a number pattern that follows a given rule and identify relationships between the numbers within the pattern. b. Generate two numerical patterns using two given rules and identify relationships between corresponding terms. c. Use tables, ordered pairs, and graphs to represent the relationship between quantities.

Apply the relationship between measurement units within a given measurement system to convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m) and use these conversions in solving multi-step problems.

Explain that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category (e.g., all rectangles have four right angles and squares are rectangles, so all squares have four right angles).

Classify two-dimensional figures based on properties. a. Classify triangles based on properties (e.g., angle measure, side lengths). b. Classify quadrilaterals in a hierarchy based on properties (e.g., side lengths, angle measures, presence of parallel or perpendicular sides).

Identify volume as an attribute of three-dimensional figures and measure volume by counting cubic units (e.g., non-standard equal-sized units, cubic centimeters, cubic inches, and cubic feet) that fill a figure without gaps or overlaps.

Relate volume to the operations of multiplication and addition and solve problems in context. a. Find the volume of a right rectangular prism with whole-number side lengths by filling it with unit cubes in layers and show that the volume is the same as would be found by multiplying the edge lengths, or by multiplying the height by the area of the base. Represent products of three whole numbers as volumes, (e.g., to represent the associative property of multiplication). b. Apply the formulas V = (l)(w)(h) and V = (B)(h) for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths to solve problems in the context. c. Determine the volumes of composite figures by decomposing them into non-overlapping right rectangular prisms and adding their volumes together to solve problems in context

Define the coordinate system as a pair of perpendicular lines called axes that intersect at the origin. Use understanding that the coordinate values represent the distance from the origin on the x-axis and y-axis to graph and name coordinate points in the first quadrant using ordered pairs.

Represent problems by graphing points in the first quadrant of the coordinate plane and interpreting coordinate values of points in the context of the situation.

Ask and answer questions by collecting, organizing and summarizing data, recognizing the importance of context when analyzing data. a. Create scaled data visualization (e.g. bar graphs for categorical data; line plots with fraction units of halves, fourths, and eighths for numerical data) to display data and communicate relationships or support a claim. b. Summarize data presented in scaled visualizations (e.g. bar graphs, circle graphs, line plots, line graphs) by identifying the mode, the range, and any gaps in data and draw conclusions. c. Evaluate whether a data visualization accurately represents the data and allows for accurate interpretation given the context.

Divide fractions by fractions in context. a. Extend estimation strategies to estimate and assess the reasonableness of quotients. b. Use and connect concrete and visual fraction models (e.g., linear, regional/area, and set models) and equations to divide. c. Represent and explain the calculation by connecting fraction models, and/or equations to the meaning of division.

Fluently divide multi-digit numbers. a. Extend estimation strategies to estimate and assess the reasonableness of quotients. b. Use strategies (e.g., partial quotients, inverse relationship between multiplication and division) to divide. c. Use a standard algorithm to divide. d. Determine and explain when a strategy or algorithm is most efficient.

Fluently multiply and divide multi-digit decimals to the thousandths in context. a. Extend estimation strategies to estimate and assess the reasonableness of products and quotients. b. Generalize whole number strategies to multiply and divide decimals. c. Use a standard algorithm to multiply and divide decimals. d. Determine and explain when a strategy or algorithm is most efficient.

Identify the common factors and multiples of two whole numbers. a. Identify the greatest common factor of two whole numbers within 100. b. Identify the least common multiple of two whole numbers within 12. c. Express the sum of two whole numbers within 100 using the distributive property to factor out the common factor of the addends (e.g., express 36 + 8 as 4(9 + 2)).

Use understanding of rational numbers as positive and negative numbers to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge). Use rational numbers to represent quantities in context and explain the meaning of 0 in each situation.

Represent a rational number as a point on vertical or horizontal number lines. Represent points on a number line with negative numbers. a. Estimate quantities by reasoning about their location on a number line, their relationship to benchmark numbers, and by rounding. b. Locate and interpret the positions of rational numbers on a number line. c. Recognize and locate opposite signs of rational numbers as indicating locations on opposite sides of 0 on the number line.

Intersect perpendicular number lines at the point (0,0) as the x-axis and y-axis to introduce the four quadrants of the Coordinate Plane. Represent points on the plane with positive and negative coordinates. a. Use understanding of the relationship between signs and value of coordinates in ordered pairs to reason about the location of an ordered pair without plotting. b. Locate and interpret the positions of rational numbers on the Coordinate Plane, including how changes in the signs of ordered pairs reflect their location into different quadrants.

Order and identify the absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line (e.g., interpret −3 > −7 as a statement that –3 is located to the right of –7 on a horizontal number line). b. Write, interpret, and explain statements of order for rational numbers in context (e.g., write −3℃ > −7℃ to express the fact that −3℃ is warmer than −7℃). c. Use understanding of the absolute value of a rational number as its distance from 0 on the number line to interpret absolute value as magnitude for a positive or negative quantity in context (e.g., for an account balance of –30 dollars, write to describe the size of the debt in dollars).

Solve problems in context by graphing points (with the same first coordinate or the same second coordinate) in all four quadrants of the Coordinate Plane. a. Explain that points with the same x-coordinate or y-coordinate are located on same vertical or horizontal line respectively. b. Use absolute value to calculate length of vertical and horizontal lines on Coordinate Plane.

Use ratio language in context (e.g., “__ to __,” “for every,” “per”) to describe a ratio relationship between two quantities, including part to part and part to whole (e.g., "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." )

Represent and use unit rates, written as fractions ( ) with whole number numerators and denominators, in context. a. Use unit rate language in the context of a ratio relationship (e.g., “The recipe has a ratio of 3 cups of flour to 4 cups of sugar so there is cup of flour for each cup of sugar.”) b. Solve unit rate problems, for example those involving unit pricing and constant speed (e.g., “If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?”).

Use ratio and rate reasoning to solve problems in context by reasoning about tables of equivalent ratios, tape diagrams, double number lines, or the Coordinate Plane. a. Generate, compare and find missing values of equivalent ratios and unit rates using multiple representations. b. Plot the pairs of values on the Coordinate Plane (in quadrant 1) to represent unit rates and make connections between representations. c. Use ratio reasoning to convert measurement units (e.g., money, time, length within the same system); manipulate and transform units appropriately when multiplying or dividing quantities.

Find a percent of a quantity in context as a rate per 100 (e.g., 30% of a quantity means times the quantity), including finding the whole given a part and the percent, by using tables, tape diagrams, and double number lines.

Write numerical expressions involving whole number exponents and positive rational number bases. Use technology to evaluate numerical expressions involving whole number exponents and positive rational number bases.

Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers (e.g., express the calculation "Subtract y from 5" as 5 − y). b. Identify parts of an expression using mathematical language (e.g., sum, term, product, factor, quotient, coefficient) and view one or more parts of an expression as a single unit (e.g., describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single unit and a sum of two terms). c. Evaluate expressions given specific values of their variables in context. Include expressions that involve arithmetic operations and whole number exponents.

Apply the properties of operations (e.g., distributive, associative, commutative, etc.) to generate equivalent algebraic expressions and to identify when two expressions are equivalent (e.g., apply the distributive property to the expression 3(2 + 𝑥) to produce the equivalent expression 6 + 3𝑥; apply the distributive property to the expression 24𝑥 +18𝑥 to produce the equivalent expression 6(4𝑥 + 3𝑥); apply properties of operations to 𝑥 + 𝑥 + 𝑥 to produce the equivalent expression 3y.)

Solve problems in context by writing and solving equations and interpreting the solutions in context. a. Explain that a variable can represent an unknown number and use substitution to determine whether a given number makes an equation true. b. Write and solve one-step equations of the form for cases in which p, q and x are all nonnegative rational numbers. c. Represent solutions of such equations on a number line.

Write an inequality of the form 𝑥 > 𝑎 (using ) represent a constraint or condition in context. Recognize that inequalities of this form have infinitely many solutions; represent solutions of such inequalities on number lines.

Solve problems in context by writing and solving inequalities and interpreting solution sets in context. a. Explain that a variable can represent an unknown number in a set and use substitution to determine whether a given number in a set makes an inequality true. b. Write and solve one-step inequalities of the form (using ) where p and q are nonnegative rational numbers. c. Represent solutions of such inequalities on a number line.

Use variables to represent two quantities in context that change in relationship to one another. a. Write an equation (one-step) to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. b. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation (e.g., in a problem involving motion at constant speed, list and graph ordered pairs of distances and times and write the equation d = 65𝑠 to represent the relationship between distance and time).

Find the area of triangles, quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and quadrilaterals to solve problems.

Find the volume of a right rectangular prism with fractional edge lengths. a. By packing prisms with unit cubes and using estimation to show that the volume can be found by multiplying the edge lengths of the prism. b. Apply the formulas and (B is the area of the base) to find the volumes of right rectangular prisms with fractional edge lengths in context.

Draw polygons in the Coordinate Plane given coordinates for the vertices; use coordinates to find the length of a side joining endpoints with the same first coordinate or the same second coordinate. Apply these techniques in context.

Represent three-dimensional figures (triangular prism, rectangular prism, pyramid) using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures to solve problems.

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers (e.g., "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages).

Use the language of probability (e.g., likely, unlikely, certain, impossible) to describe the likelihood of possible outcomes and to anticipate variability in data collected from statistical questions. Explain how the likelihood of outcomes helps predict which responses may be more or less common in the data and justify predictions using probability terms.

Explain how a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

Recognize that a measure of center (median and/or mean) for a numerical data set summarizes all the values with a single number, while a measure of variation describes how its values vary with a single number.

Interpret numerical data in plots on a number line, including line plots, histograms, and box plots.

Summarize numerical data sets in relation to their context. a. Report the number of observations. b. Describe the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Give quantitative measures of center (median and/or mean) and variability (interquartile range), as well as describing outliers with reference to the context in which the data were gathered. d. Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.