Trevor Mattea
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Math

Somehow it's O.K. for people to chuckle about not being good at math. Yet if I said, "I never learned to read," they'd say I was an illiterate dolt. -- Neil deGrasse Tyson

Overview

Prior to working for High Tech High, I taught at Stevenson PACT Elementary School, where my teaching partner and I taught to the Common Core State Standards for Mathematical Practice. We primarily used the EngageNY curriculum, although I also drew on resources from Investigations, YouCubed, Dan Meyer, Graham Fletcher, Duane Habecker, and Mike Lawler. In my current position, our grade-level team continues to use those standards and resources to inform our work. With my students, I have developed success criteria for all classroom practices, including math assignments, and it is available here.


2015-2016 Slides | 2015-2016 Application Problems and Success Criteria
2014-2015 Second Grade Slides | 2014-2015 Third Grade Slides
2013-2014 Slides

Common Core State Standards for Mathematics

Second Grade

Operations and Algebraic Thinking
  • Represent and solve problems involving addition and subtraction.
    • CCSS.MATH.CONTENT.2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
      • Khan Academy Resources
  • ​Add and subtract within 20.
    • CCSS.MATH.CONTENT.2.OA.B.2  Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.
      • Khan Academy Resources
  • Work with equal groups of objects to gain foundations for multiplication.
    • CCSS.MATH.CONTENT.2.OA.C.3  Determine whether a group of objects (up to 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.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.OA.C.4  Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
      • Khan Academy Resources
Number and Operations in Base Ten
  • Understand place value.
    • CCSS.MATH.CONTENT.2.NBT.A.1  Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases"
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.NBT.A.1.A  100 can be thought of as a bundle of ten tens — called a "hundred."
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.NBT.A.1.B The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.2.NBT.A.2  Count within 1000; skip-count by 5s, 10s, and 100s.
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.2.NBT.A.3  Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.NBT.A.4  Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
      • Khan Academy Resources
  • ​Use place value understanding and properties of operations to add and subtract.
    • CCSS.MATH.CONTENT.2.NBT.B.5 ​Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.2.NBT.B.6  Add up to four two-digit numbers using strategies based on place value and properties of operations.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.NBT.B.7  Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in 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.
      • ​Khan Academy Resources
    • CCSS.MATH.CONTENT.2.NBT.B.8 ​Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.NBT.B.9  Explain why addition and subtraction strategies work, using place value and the properties of operations.
      • Khan Academy Resources
Measurement and Data
  • Measure and estimate lengths in standard units.
    • CCSS.MATH.CONTENT.2.MD.A.1 ​Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.2.MD.A.2  Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.MD.A.3 ​Estimate lengths using units of inches, feet, centimeters, and meters.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.MD.A.4  Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
      • Khan Academy Resources
  • Relate addition and subtraction to length.
    • CCSS.MATH.CONTENT.2.MD.B.5 ​Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.MD.B.6  Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.
      • Khan Academy Resources
  • Work with time and money.
    • CCSS.MATH.CONTENT.2.MD.C.7  Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.MD.C.8  Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?
      • Khan Academy Resources
  • Represent and interpret data.
    • CCSS.MATH.CONTENT.2.MD.D.9 ​Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.MD.D.10 ​Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems1using information presented in a bar graph.
      • Khan Academy Resources​
Geometry
  • Reason with shapes and their attributes.
    • CCSS.MATH.CONTENT.2.G.A.1​Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
      • Khan Academy Resources
    • ​​​CCSS.MATH.CONTENT.2.G.A.2 ​Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.2.G.A.3 ​Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
      • Khan Academy Resources

Third Grade

Operations and Algebraic Thinking
  • Represent and solve problems involving multiplication and division.
    • CCSS.MATH.CONTENT.3.OA.A.1  Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.OA.A.2  Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
      • Khan Academy Resources
  • Additional Resources on Khan Academy
    • CCSS.MATH.CONTENT.3.OA.A.3  Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.3.OA.A.4  Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?
      • Khan Academy Resources
  • ​Understand properties of multiplication and the relationship between multiplication and division.
    • CCSS.MATH.CONTENT.3.OA.B.5 ​Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.3.OA.B.6  Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
      • Khan Academy Resources
  • ​Multiply and divide within 100.
    • CCSS.MATH.CONTENT.3.OA.C.7  Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
  • Solve problems involving the four operations, and identify and explain patterns in arithmetic.
    • CCSS.MATH.CONTENT.3.OA.D.8 ​Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.OA.D.9 ​Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
      • Khan Academy Resources​
Number and Operations in Base Ten
  • Use place value understanding and properties of operations to perform multi-digit arithmetic.
    • CCSS.MATH.CONTENT.3.NBT.A.1  Use place value understanding to round whole numbers to the nearest 10 or 100.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.NBT.A.2 ​Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
    • CCSS.MATH.CONTENT.3.NBT.A.3 ​Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
      • Khan Academy Resources​
Number and Operations - Fractions
  • Develop understanding of fractions as numbers.
    • CCSS.MATH.CONTENT.3.NF.A.1  Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.3.NF.A.2​Understand a fraction as a number on the number line; represent fractions on a number line diagram.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.NF.A.2.A  Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.NF.A.2.B ​Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.NF.A.3 ​Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.NF.A.3.A  Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.NF.A.3.B  Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.NF.A.3.C ​Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.NF.A.3.D ​Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
      • Khan Academy Resources
Measurement and Data
  • Solve problems involving measurement and estimation.
    • CCSS.MATH.CONTENT.3.MD.A.1 ​Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.3.MD.A.2 ​Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).1 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
  • Represent and interpret data.
    • CCSS.MATH.CONTENT.3.MD.B.3 ​Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.
      • Khan Academy Resources
    • ​CCSS.MATH.CONTENT.3.MD.B.4  ​Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
      • Khan Academy Resources
  • Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
    • CCSS.MATH.CONTENT.3.MD.C.5 ​Recognize area as an attribute of plane figures and understand concepts of area measurement.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.MD.C.5 A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.MD.C.5.B A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.MD.C.6  Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.MD.C.7​Relate area to the operations of multiplication and addition.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.MD.C.7.A​​​Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.MD.C.7.B Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.MD.C.7.C Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.MD.C.7.D Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
      • Khan Academy Resources
  • Geometric measurement: recognize perimeter.
    • CCSS.MATH.CONTENT.3.MD.D.8​​Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
      • Khan Academy Resources
Geometry
  • Reason with shapes and their attributes.
    • CCSS.MATH.CONTENT.3.G.A.1​Understand 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 (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
      • Khan Academy Resources
    • CCSS.MATH.CONTENT.3.G.A.2​Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.
      • Khan Academy Resources​

Fourth Grade

Operations and Algebraic Thinking
  • Use the four operations with whole numbers to solve problems.
    • CCSS.MATH.CONTENT.4.OA.A.1  Interpret a multiplication equation 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.
    • CCSS.MATH.CONTENT.4.OA.A.2  Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1
    • CCSS.MATH.CONTENT.4.OA.A.3  Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
  • Gain familiarity with factors and multiples.
    • CCSS.MATH.CONTENT.4.OA.B.4  Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.
  • Generate and analyze patterns.
    • CCSS.MATH.CONTENT.4.OA.C.5  Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.​
Number and Operations in Base Ten
  • Generalize place value understanding for multi-digit whole numbers.
    • CCSS.MATH.CONTENT.4.NBT.A.1  Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
    • CCSS.MATH.CONTENT.4.NBT.A.2  Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
    • CCSS.MATH.CONTENT.4.NBT.A.3  Use place value understanding to round multi-digit whole numbers to any place.
  • Use place value understanding and properties of operations to perform multi-digit arithmetic.
    • CCSS.MATH.CONTENT.4.NBT.B.4  Fluently add and subtract multi-digit whole numbers using the standard algorithm.
    • CCSS.MATH.CONTENT.4.NBT.B.5  Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
    • CCSS.MATH.CONTENT.4.NBT.B.6 ​Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Number and Operations in Fractions
  • Extend understanding of fraction equivalence and ordering.
    • CCSS.MATH.CONTENT.4.NF.A.1 ​Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
    • CCSS.MATH.CONTENT.4.NF.A.2  Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
  • Build fractions from unit fractions.
    • CCSS.MATH.CONTENT.4.NF.B.3​Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
    • CCSS.MATH.CONTENT.4.NF.B.3.A  Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
    • CCSS.MATH.CONTENT.4.NF.B.3.B  Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
    • CCSS.MATH.CONTENT.4.NF.B.3.C​Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
    • CCSS.MATH.CONTENT.4.NF.B.3.D ​Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
    • CCSS.MATH.CONTENT.4.NF.B.4  Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
    • CCSS.MATH.CONTENT.4.NF.B.4.A  Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
    • CCSS.MATH.CONTENT.4.NF.B.4.B  Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
    • CCSS.MATH.CONTENT.4.NF.B.4.C ​Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
  • Understand decimal notation for fractions, and compare decimal fractions.
    • CCSS.MATH.CONTENT.4.NF.C.5 ​Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
    • CCSS.MATH.CONTENT.4.NF.C.6 ​Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
    • CCSS.MATH.CONTENT.4.NF.C.7  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 >, =, or <, and justify the conclusions, e.g., by using a visual model.
Measurement and Data
  • Solve problems involving measurement and conversion of measurements.
    • CCSS.MATH.CONTENT.4.MD.A.1 ​Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
    • CCSS.MATH.CONTENT.4.MD.A.2 ​Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
    • CCSS.MATH.CONTENT.4.MD.A.3 ​Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
  • Represent and interpret data.
    • CCSS.MATH.CONTENT.4.MD.B.4​Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
  • Geometric measurement: understand concepts of angle and measure angles.
    • CCSS.MATH.CONTENT.4.MD.C.5 ​Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
    • CCSS.MATH.CONTENT.4.MD.C.5.A An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.
    • CCSS.MATH.CONTENT.4.MD.C.5.B An angle that turns through n one-degree angles is said to have an angle measure ofn degrees.
    • CCSS.MATH.CONTENT.4.MD.C.6​Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
    • CCSS.MATH.CONTENT.4.MD.C.7Recognize angle measure as additive. 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 real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.​
Geometry
  • Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
    • CCSS.MATH.CONTENT.4.G.A.1​ ​Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
    • CCSS.MATH.CONTENT.4.G.A.2 ​​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.
    • CCSS.MATH.CONTENT.4.G.A.3​Recognize a line of symmetry for a two-dimensional figure as 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.

Common Core State Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
  • Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
  • Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
  • Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
CCSS.MATH.PRACTICE.MP4 Model with mathematics.
  • Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
  • Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
CCSS.MATH.PRACTICE.MP6 Attend to precision.
  • Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
  • Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
  • Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x2 + x + 1), and (x - 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Math Triangle Exemplars

Prior to our units on addition, subtraction, multiplication, and division, I ask students to keep a set of math triangles in the classroom and at home in order to study fact families, such as 2 + 3 = 5, 3 + 2 = 5, 5 - 2 = 3, and 5 - 3 = 2. Patterns for addition and subtraction triangles are available here. Patterns for multiplication and division are available here.
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Hundreds Chart Exemplars

During our unit on multiplication, I ask students to color a hundreds chart with multiples of 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, 10s, 11s, and 12s. Blank hundreds charts are available here.

Addition Word Problem Exemplars

At the conclusion of our addition unit, I ask students to write their own story problems along with their preferred addition strategies. The success criteria for this assignment are available here.

Subtraction Word Problem Exemplars

At the conclusion of our subtraction unit, I ask students to write their own story problems along with their preferred subtraction strategies. The success criteria for this assignment are available here.

Addition with Regrouping Word Problem Exemplars

At the conclusion of our addition with regrouping unit, I ask students to write their own story problems along with their preferred addition strategies. The success criteria for this assignment are available here.

Subtraction with Regrouping Word Problem Exemplars

At the conclusion of our subtraction with regrouping unit, I ask students to write their own story problems along with their preferred subtraction strategies. The success criteria for this assignment are available here.

Addition and Subtraction with Unknowns Word Problem Exemplars

At the conclusion of our addition and subtraction units, I ask students to write their own story problems with unknowns along with their preferred addition and subtraction strategies. The success criteria for this assignment are available here and here.

Multiplication Word Problem Exemplars

At the conclusion of our multiplication unit, I ask students to write their own story problems along with their preferred multiplication strategies. The success criteria for this assignment are available here.

Division Word Problem Exemplars

At the conclusion of our division unit, I ask students to write their own story problems along with their preferred division strategies. The success criteria for this assignment are available here.

Multiplication and Division with Unknowns Word Problems Exemplars

At the conclusion of our multiplication and division units, I ask students to write their own story problems with unknowns along with their preferred multiplication and division strategies.

Time Measurement Word Problem Exemplars

At the conclusion of our time measurement unit, I ask students to write their own story problems with their preferred time measurement strategies.
I believe that education is a process of living and not a preparation for future living. -- John Dewey
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