Domain: Operations and Algebraic Thinking - 4.OA Vocabulary Examples, Activities, Responses Use the four operations with whole numbers to solve problems. 4.OA.1 Interpret a multiplication equation as a comparison. A multiplicative comparison is when one number is multiplied by a specific number to get a comparative value. In other words, in 4 x 5, the product (20) is 5 times greater than 4. Example: Interpret 35 = 5 x 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. Example: Anthony has eight crayons. Lydia has six times as many crayons as Anthony. How many crayons does Lydia have? 8 x 6 = 48 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Examples: Number of groups unknown: A new book costs $24 the same book used costs $8. How many times as much does the new book cost compared to the used book. 24 8 = n or 8 x n = 24. Group size unknown: A new book costs $24, which is three times more that the same book costs used. How much does the used book cost? 24 n = 3 or 3 x n = 24 Unknown product: A used book costs $8. The same book new costs three times as much. How much does a new book cost? 8 x 3 = n Multiplicative comparison: Matt read 3 chapters. Lynn read 5 times as many chapters. How many chapters did Lynn read? Additive comparison: Matt has 3 pencils. Lynn had 5 pencils. How many more pencils does Lynn have? 4.OA.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 Example: Sarah needed to collect 200 box tops for a school project. The first day, 3 of her neighbors gave her 6 box tops each. The second day, 7 neighbors gave her 5 box tops each. After two days of collecting, how many more box tops did Sarah need to collect? First, I multiplied 3 x 6 to get 18. 1
of answers using mental computation and estimation strategies including rounding. Then I multiplied 7 x 5 to get 35. Then I added 20 and 35 to get 55. Then I subtracted 2 to get 53. Next I subtracted 200-50 to get 150. Then I subtract 3 more to get 147. Example: Remainders should be placed in context and explained as left-over, rounded to the next number for approximate results, discarded leaving the whole number as the answer, or partitioned into fraction/decimal. Left over remainder: Lori has 34 apples. 6 apples fit into each bag. How many bags can Lori fill? How many apples will be left over? Rounded to next number: Lori has 34 apples. 6 apples fit into each bag. What is the fewest number of bags that Lori will need to hold all of the apples? 7 bags. Discarded remainder: Lori has 34 apples. 6 apples fit into each bag. How many bags can Lori fill completely? 5 bags. Partitioned remainder: If Ryan has 15 cookies and 2 friends to give them to, how many cookies will each friend get? 7 ½ cookies. Gain familiarity with factors and multiples. 4.OA.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. Factor: Any numbers multiplied to form a product. In the equation 4 x 3 = 12, 4 and 3 are factors and 12 is the product. The whole number factors of 12 are 1, 2, 3, 4, 6, and 12 because 1 x 12 = 12, 2 x 6 = 12, and 3 x 4 = 12. A prime number is a number that has exactly 2 factors, 1 and itself. 17 is a prime number because its only factors are 1 and 17. A composite number has more than 2 factors. 15 is a composite number because its factors are 1, 3, 5, and 15. 1 x 15 = 15 and 3 x 5 = 15. The number 1 has only one factor 1 x 1 = 1 therefore it is neither prime nor composite. Example: Is 51 prime or composite? Response: Composite (17 x 3 = 51, 1 x 51 = 51) 2
Generate and analyze patterns. 4.OA.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. Example: Pattern Rule Feature(s) 7, 10, 13, 16, 19, 22 Start with 7, add 3 The numbers alternate between odd and even 3, 8, 13, 18, 23, 28 Start with 3 add 5 The numbers alternate between 3 and 8 in the ones place, the numbers alternate between odd and even Example: Generate a pattern: Teacher: Start with 1 and multiply each successive number by 3. Write the first six numbers. Child writes 3, 9, 27, 81, 243, 729 and writes that all of the numbers are odd. Domain: Number and Operations in Base Ten - 4.NBT Generalize place value understanding for multidigit whole numbers. 4.NBT.1 Recognize that in a multi-digit number, a digit in one place represents ten times what it represents in the place to its right. 4.NBT.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. Example: How is the digit 3 in 739 similar to or different from the digit 3 in the number 793? Response: In 739 the digit 3 represents 30. In 793 the digit 3 represents 3. Example: Base ten numeral: 386: Number name: three hundred eighty-six Expanded form: 300 + 80 + 6 Example: Students should also recognize that 386 is 38 tens and 6 ones or 2 hundreds 18 tens and six ones, etc. Example: 196 < 619 because 196 only has one hundred and 619 has six hundreds. 3
4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place. Example: One book contained 367 pages. Another book contained 294 pages. A third book contained 34 pages. How many pages were in all three books? Response 1: I noticed that 367 is about 400. Then I saw that 294 is about 300 which together makes 700. Then rounded 34 to 30 to get about 730. The correct answer will be about 730. Response 2: I noticed that 367 and 34 make about 400. Then I rounded 294 to 300. I added 400 and 300 to find that the correct answer will be about 700. Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.4 Fluently add and subtract multi-digit whole numbers using standard algorithm. Example: This is the first grade level that students are expected to be proficient at using the standard algorithm for addition and subtraction. 2467 +5296 Response: I added 7 ones and 6 ones and got 13 ones. I wrote down 3 ones and noted the ten from thirteen ones in the tens column. Then I added nine tens and one ten to get ten tens, then I added six tens to get sixteen tens. I wrote down the six tens and then noted the remaining ten tens as one hundred in the hundreds column. Then I added four hundreds, two hundreds, and one hundred to get 7 hundreds. Finally, I added two thousands and five thousands to get 7 thousand for a total of seven thousand, seven hundred, sixty-three. 4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two twodigit 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. Example: There are 25 dozen eggs in the grocery store display case. What is the total number of eggs in the store? Response 1: I broke 25 into 10, 10 and 5. I then multiplied 12 x 10 to get 120. I multiplied 12 x 10 again to get another 120. 120 + 120 = 240. 12 x 5 = 60. So, 240 + 60 = 300 eggs. Response 2: I broke 12 into 10 and 2. 25 x 10 = 250. 25 x 2 = 50. 200 + 50 = 300 eggs. 4
Response 3: I broke 25 into 20 and 5 and then I broke 12 into 12 and 2. 20 5 10x5=50 10 x 20 = 200 2 x 20 = 40 2x5=10 200 + 50 + 40 + 10 = 300 eggs. 4.NBT.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 buy using equations, rectangular arrays, and/or area models. Example: There are 220 children on the playground for Play Day. The children must be divided into four equal groups to rotate from activity to activity. How many children will be in each group? Response 1 Using place value: I broke 220 into 200 and 20. 220 4 = (200 4) + (20 4). 200 4 = 50. 20 4 = 5. 50 + 5 = 55 children. Response 2 Using multiplication: 4 x 50 = 200. 4 x 5 = 20, so 50 + 5 = 55. So 220 4 = 55 children. Response 3 Area model: 4 4 220 200 20 I broke 220 into 200 and 20. 220 4 = (200 4) + (20 4). 200 4 = 50. 20 4 = 5. 50 + 5 = 55 children. 5
Domain: Number and Operations Fractions - 4.NF Extend understanding of fraction equivalence and ordering. 4.NF.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. Example: 1/2 x 2/2 = 2/4, 1/2 x 6/6 = 6/12 (Recognize that 2/2 = 1 and 6/6 = 1 and that any number multiplied by one results in the same number.) 1/2 1/2 = 2/4 1/2 = 6/12 4.NF.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. Example using visual fraction model: Two same size pizzas were served at Tom s birthday party. 1/2 of the pepperoni pizza was left over. 4/12 of the cheese pizza was left over. Which pizza has more left? cheese pepperoni Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 1/2 > 4/12 So, there was more pepperoni pizza left. 6
4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.3.a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 4.NF.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. Example: Understand that 2/5 = 1/5 + 1/5. Example: 2/5 + 3/5 = (1/5 + 1/5) +(1/5 +1/5 +1/5) = 5/5 = 1 Example: 3/8 = 1/8 + 1/8 + 1/8 3/8 = 1/8 + 1/8 + 1/8 Example: 3/8 = 1/8 + 2/8 3/8 = 1/8 + 2/8 4.NF.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. Example: Josh and Charlie need 9 3/4 feet of wood to complete a project. Josh checked his garage and found 3 1/4 feet of wood. Charlie found 6 3/4 feet of wood in his basement. Do the boys have enough wood? Response 1: 3 1/4 + 6 3/4 I know that 3 + 6 = 9 and that 1/4 + 3/4 = 4/4 or 1. So, the boys have 10 feet of wood which is enough because it s more than 9 3/4 feet. Response 2: 1 2 3 4 The above figure represents 4/4 or one whole. 3 1/4 = 13/4 1 2 5 6 9 10 13 3 4 7 8 11 12 1 2 3 4 5 6 3/4 1 2 5 6 9 11 13 3 4 7 8 10 12 15 6 3/4 = 27/4 14 16 17 19 21 23 25 26 18 20 22 24 27 13/4 + 27/4 = 40/4 which is equal to 10 feet of wood. 10 is more than 9 3/4, so the boys have enough wood. 7
4.NF.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. Example: A smoothie recipe calls for 3/4 cup of orange juice, 1/4 cup of raspberry juice, and 2/4 cup of cranberry juice. How much juice was needed to make the smoothie? Response: 3/4 + 1/4 + 2/4 = 6/4, 6/4 = 1 2/4 1 2 3 1 2 1 2 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF4.a Understand a fraction a/b as a multiple of 1/b. Example: Use a visual fraction model to represent 5/8 as the product 5 (1/8), recording the conclusion by the equation 5/8 = 5 (1/8). 5/8 = 1/8 + 1/8 + 1/8 + 1/8 + 1/8, 5/8 = 5 x (1/8) 0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 9/8 10/8 1 4.NF.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. (In general, n (a/b) = (n a)/b.) Example: Use a visual fraction model to express 3 (2/5) as 6 (1/5), recognizing this product as 6/5. Response: 2/5 2/5 2/5 1 2 3 4 5 6 1/5 2/5 3/5 4/5 5/5 1/5 2/5 3/5 4/5 5/5 1/5 1/5 1/5 1/5 1/5 1/5 3 x 2/5 = 6 x 1/5 = 6/5 4.NF.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. 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 8
your answer lie? Response: 3/8 3/8 3/8 1 2 3 1 2 3 1 2 3/8 3/8 3 1 2 3 1 2 3 Understand decimal notation for fractions and compare decimal fractions. 4.NF.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. 4.NF.6 Use decimal notation for fractions with denominators 10 or 100. 4.NF.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. 15/8 lb. of meat will be needed. That s between 1 and 2 pounds. Example: Express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. Example: Rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. Example: 0.62 > 0.49 (See visual model below) 0.62 0.49 9
Domain: Measurement and Data - 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 4.MD.1 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. 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. Feet Inches 1 12 2 24 3 36 4 48 5 60 n n x 3 4.MD.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. 4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. 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. Area = length x width (l x w) 8 ft.? 48 sq. ft. 8 x? = 48 10
Represent and interpret data. 4.MD.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. Example: From a line plot, find and interpret the difference in length between the longest and shortest specimens in an insect collection. x x x x x x x x x x x x 0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 How many insects are1/8 inch long? 5/8 inch? What is the total length of all the insects? What is the length of the longest insect? Geometric measurement: understand concepts of angle and measure angles. 4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. 4.MD.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. Example: Although the area between the two rays is different for each circle, the angle measure remains the same. A circle contains 360. 4.MD.5.b An angle that turns through n one-degree angles is said to have an angle measure of n degrees. Example: A spot light can rotate one hundred eighty degrees, how many one-degree turns is that? 11
4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. Example: Student should measure and sketch angles using a protractor. The angle below measures 120 degrees. 4.MD.7 Recognize 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. Example 1: Sam turned the dial on the radio 35 then another 55. How many degrees did Sam turn the dial in all? 35 55 Example 2: Rita turned the sprinkler 35. How many more degrees does Rita need to turn the sprinkler to reach 90? 35 + n = 90. 90 35 = n. Domain: Geometry 4.G Draw and identify lines and angles, and classify shapes by properties of their angles. 4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. Right angle: Ray: Acute angle: Line: 12
Obtuse angle: Parallel Lines Line segment: Point Perpendicular Lines: 4.G.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. Example: Draw and name a four-sided figure that has exactly one set of parallel sides. Responses: Trapezoid 4.G.3 Recognize a line of symmetry for a twodimensional 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. Example: For each figure, draw all the lines of symmetry. 13