When distinguishing multiplicative comparison from additive comparison, students should note that Additive comparisons focus on the difference between two quantities (e.g., Deb has 3 apples and Karen has 5 apples. How many more apples does Karen have?). A simple way to remember this is, “How many more?” multiplicative comparisons focus on comparing two quantities by showing that one quantity is a specified number of times larger or smaller than the other (e.g., Deb ran 3 miles. Karen ran 5 times as many miles as Deb. How many miles did Karen run?). A simple way to remember this is “How many times as much?” or “How many times as many?” Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: Use the four operations with whole numbers to solve problems. 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 of answers using mental computation and estimation strategies including rounding. Knowledge Targets “I Can” Statements Standard Interpretations
Distributive Property Inverse Operations Fact Family Topic: 2 Generate & Analyze Patterns Weeks: 2 Domain: Operations and Algebraic Thinking Cluster: 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. 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. Knowledge Targets “I Can” Statements Standard Interpretations Identify a number or shape pattern. I can create a number or shape pattern that follows a given rule I can notice and explain different features of a pattern once it is created by a rule Patterns involving numbers or symbols either repeat or grow. Students need multiple opportunities creating and extending number and shape patterns. Numerical patterns allow students to reinforce facts and develop fluency with operations. Patterns and rules are related. A pattern is a sequence that repeats the same process over and over. A rule dictates what that process will look like. Students investigate different patterns to find rules, identify features in the patterns, and justify the reason for those features. Examples: Reasoning Target Generate a number or shape pattern that follows a given rule. Analyze a pattern to determine features not apparent in the rule (always odd or even, alternates between odd and even, etc.) Performance Target
4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place. Knowledge Targets “I Can” Statements Standard Interpretations Round multi-digit whole numbers to any place using place value. I can round large whole numbers to any place When students are asked to round large numbers, they first need to identify which digit is in the appropriate place. Example: Round 76,398 to the nearest 1000. Step 1: Since I need to round to the nearest 1000, then the answer is either 76,000 or 77,000. Step 2: I know that the halfway point between these two numbers is 76,500. Step 3: I see that 76,398 is between 76,000 and 76,500. Step 4: Therefore, the rounded number would be 76,000. Reasoning Target Performance Target Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: 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 the standard algorithm. Knowledge Targets “I Can” Statements Standard Interpretations
have to take away any thousands. (Writes 2 in the thousands place of answer.) Note: Students should know that it is mathematically possible to subtract a larger number from a smaller number but that their work with whole numbers does not allow this as the difference would result in a negative number. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Topic 4: Addition and Subtraction of Whole Numbers Breaking Apart Counting On Compensation Commutative Property of Addition Associative Property of Addition Identity Property of Addition Inverse Operations
Knowledge Targets “I Can” Statements Standard Interpretations Divide whole numbers including division with remainders. I can use what I know about addition, subtraction, multiplication and division to solve multi-step word problems involving whole numbers I can represent word problems by using equations with a letter standing for the unknown number I can determine how reasonable my answers to word problems are by using estimation, mental math, and rounding I can interpret remainders when solving division word problems Students need many opportunities solving multistep story problems using all four operations. An interactive whiteboard, document camera, drawings, words, numbers, and/or objects may be used to help solve story problems. Example: Chris bought clothes for school. She bought 3 shirts for $12 each and a skirt for $15. How much money did Chris spend on her new school clothes? 3 x $12 + $15 = a In division problems, the remainder is the whole number left over when as large a multiple of the divisor as possible has been subtracted. Example: Kim is making candy bags. There will be 5 pieces of candy in each bag. She had 53 pieces of candy. She ate 14 pieces of candy. How many candy bags can Kim make now? (7 bags with 4 leftover) Kim has 28 cookies. She wants to share them equally between herself and 3 friends. How many cookies will Reasoning Targets Represent multi-step word problems using equations with a letter standing for the unknown quantity. Interpret multistep word problems (including problems in which remainders must be interpreted) and determine the appropriate operations (s) to solve. Assess the reasonableness of an answer in solving a multi-step word problem using mental math and estimation strategies (including rounding).
Find whole number quotients and remainders with up to four-digit dividends and one-digit divisors. I can find whole number quotients and remainders with up to four-digit dividends and one-digit divisors, and explain using equations, arrays, or models In fourth grade, students build on their third grade work with division within 100. Students need opportunities to develop their understandings by using problems in and out of context. Examples: A 4th grade teacher bought 4 new pencil boxes. She has 260 pencils. She wants to put the pencils in the boxes so that each box has the same number of pencils. How many pencils will there be in each box? Using Base 10 Blocks: Students build 260 with base 10 blocks and distribute them into 4 equal groups. Some students may need to trade the 2 hundreds for tens but others may easily recognize that 200 divided by 4 is 50. Using Place Value: 260 ÷ 4 = (200 ÷ 4) + (60 ÷ 4) Using Multiplication: 4 x 50 = 200, 4 x 10 = 40, 4 x 5 = 20; 50 + 10 + 5 = 65; so 260 ÷ 4 = 65 Students may use digital tools to express ideas. Continued on next page Using an Open Array or Area Model After developing an understanding of using arrays to divide, students begin to use a more abstract model for division. This model connects to a recording process that will be formalized in the 5th grade. Example: 150 ÷ 6 Students make a rectangle and write 6 on one of its sides. They express their understanding that they need to think of the rectangle as representing a total of 150. Students think, 6 times what number is a number close to 150? They recognize that 6 x 10 is 60 so they record 10 as a factor and partition the rectangle into 2 rectangles and label the area aligned to Reasoning Targets Use the strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using written equations, rectangular arrays, and/or area models.
Add and subtract fractions with like denominators. I can understand that improper fractions have a greater numerator than denominator I can understand addition and subtraction of fractions as joining and separating parts of the same whole I can decompose a fraction into a sum of fractions with the same denominator I can add and subtract mixed numbers with like denominators I can solve word problems involving addition and subtraction of fractions with like denominators using models and equations Refer to 4.NF.a Reasoning Targets Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, by using visual fraction models and equations to represent the problem. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.4a Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 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 x ¼, recording the conclusion by equation 5/4 = 5 x (1/4) Knowledge Targets “I Can” Statements Standard Interpretations
Knowledge Targets “I Can” Statements Standard Interpretations Add, subtract, multiply, and divide fractions and decimals. Express measurements given in a larger unit in terms of a smaller unit. I can use the four operations (addition, subtraction, multiplication, division) to solve word problems involving measurement of distance, time, liquid volume, mass and money, with calculations including simple fractions and decimals I can represent measurement amounts using diagrams like number lines with measurement scales Examples: Division/fractions: Susan has 2 feet of ribbon. She wants to give her ribbon to her 3 best friends so each friend gets the same amount. How much ribbon will each friend get? Students may record their solutions using fractions or inches. (The answer would be 2/3 of a foot or 8 inches. Students are able to express the answer in inches because they understand that 1/3 of a foot is 4 inches and 2/3 of a foot is 2 groups of 1/3.) Addition: Mason ran for an hour and 15 minutes on Monday, 25 minutes on Tuesday, and 40 minutes on Wednesday. What was the total number of minutes Mason ran? Subtraction: A pound of apples costs $1.20. Rachel bought a pound and a half of apples. If she gave the clerk a $5.00 bill, how much change will she get back? Multiplication: Mario and his 2 brothers are selling lemonade. Mario brought one and a half liters, Javier brought 2 liters, and Ernesto brought 450 milliliters. How many total milliliters of lemonade did the boys have? Number line diagrams that feature a measurement scale can represent Reasoning Targets Solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money. Solve word problems involving measurement that include simple fractions or decimals. Solve word 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.
measurement quantities. Examples include: ruler, diagram marking off distance along a road with cities at various points, a timetable showing hours throughout the day, or a volume measure on the side of a container. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Topic 11: Fraction Equivalence and Ordering Prime Number
Number line diagrams that feature a measurement scale can represent measurement quantities. Examples include: ruler, diagram marking off distance along a road with cities at various points, a timetable showing hours throughout the day, or a volume measure on the side of a container. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
Cluster: 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, ¼, 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. Knowledge Targets “I Can” Statements Standard Interpretations Add and subtract fractions. I can make a line plot to show measurements involving fractions I can solve problems involving addition and subtraction of fractions by using information presented in line plots Example: Ten students in Room 31 measured their pencils at the end of the day. They recorded their results on the line plot below. X X X X X X X X X X _________________________________________ 3 ½” 4” 4 ¼” 5 1/8” 5 1/2” Possible questions: What is the difference in length from the longest to the shortest pencil? If you were to line up all the pencils, what would the total length be? If the 5 1/8” pencils are placed end to end, what would be their total length? Reasoning Targets Analyze and interpret a line plot to solve problems involving addition and subtraction of fractions. Product Target Create a line plot to display a data set of measurements given in fractions of a unit Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
sketch angles that measure approximately 45º and 30º. They use appropriate terminology (acute, right, and obtuse) to describe angles and rays (perpendicular). Determine which scale on the protractor to use, based on the direction the angle is open. Determine the kind of angle based on the specified measure to decide reasonableness of the sketch. Performance Skills Targets Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. Make sense of problems and preserver in solving them. Reason abstractly and quantitatively Construct viable arguments and critiques the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Cluster: Geometric measurement: Understand concepts of angle and measure angles. 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. Knowledge Targets “I Can” Statements Standard Interpretations