Topic Proficiency Scale Domain: Operations & Algebraic Thinking Topic: Multiplication Critical Area: Multi-Digit Multiplication

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1 Topic Proficiency Scale Domain: Operations & Algebraic Thinking Topic: Multiplication Critical Area: Multi-Digit Multiplication 4.0 I know all of the Simple and Complex Learning Goals and my understanding goes beyond the grade level target. COMPLEX 3.0 I know all of the Simple and Complex Learning Goals. Solve multi-step word problems using multiplication involving multiplicative comparisons and distinguish from an additive comparison. (4.OA.2) Multiply two, two-digit numbers using strategies based on place value and properties of operations. (Not standard algorithm) (4.NBT.5) Multiply a whole number of up to four digits by a one-digit whole number (using strategies based on place value and properties of operation; illustrate and explain (i.e,. equations, rectangular array, or area model). (4.NBT.5) Interpret and represent multiplication equations as comparisons.(4.oi.1) 2.5 I know all of the Simple Learning Goals plus some of the Complex Learning Goals. SIMPLE 2.0 I know all of the Simple Learning Goals. Find all factors pairs for a whole number within 100. ( 4.OA.4) Determine whether a given whole number within 100 is prime or composite. ( 4.OA.4) Determine whether a given number within 100 is a multiple of a one-digit number. (4.OA.4) Use the area formula for rectangles in real world and mathematical problems (A= L x W). (4.MD.3) Academic Vocabulary: multiple prime number composite number 1.5 Not applicable. 1.0 I know some of the Simple Learning Goals. 0.5 Not applicable. 0.0 No evidence of knowing the Learning Goals.

2 Topic Proficiency Scale Domain: Operations & Algebraic Thinking Topic: Multiplication Critical Area: Multi-Digit Multiplication 4.NBT.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. Use the four operations with whole numbers to solve problems. 4.OA.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. 4.OA.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. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 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. 4.OA.4 Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. 4.MD.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. 4.0 I know all of the Simple and Complex Learning Goals and my understanding goes beyond the grade level target. COMPLEX 3.0 I know all of the Simple and Complex Learning Goals. Solve multi-step word problems using multiplication involving multiplicative comparisons and distinguish from an additive comparison. (4.OA.2) Multiply two, two-digit numbers using strategies based on place value and properties of operations. (Not standard algorithm) (4.NBT.5) Multiply a whole number of up to four digits by a one-digit whole number (using strategies based on place value and properties of operation; illustrate and explain (i.e,. equations, rectangular array, or area model). (4.NBT.5) Interpret and represent multiplication equations as comparisons.(4.oa.1) 2.5 I know all of the Simple Learning Goals plus some of the Complex Learning Goals. SIMPLE 2.0 I know all of the Simple Learning Goals. Find all factors pairs for a whole number within 100. ( 4.OA.4) Determine whether a given whole number within 100 is prime or composite. ( 4.OA.4) Determine whether a given number within 100 is a multiple of a one-digit number. (4.OA.4) Use the area formula for rectangles in real world and mathematical problems (A= L x W). (4.MD.3) Academic Vocabulary: multiple prime number composite number 1.5 Not applicable. 1.0 I know some of the Simple Learning Goals. 0.5 Not applicable. 0.0 No evidence of knowing the Learning Goals.

3 Success Criteria Topic: Multiplication Guiding Questions: What do the numbers in the problem represent? What is the relationship of the quantities? Can you put this problem into context? What are some ways to represent the quantities? How would it help to create a diagram, graph, table...? Complex Goal: Solve multi-step word problems using multiplication involving multiplicative comparisons and distinguish from an additive comparison. (4.OA.2) Mathematical Practices: 2 and 4 Mastery: Students will understand that a symbol represents an unknown variable in an equation. Students will recognize division as the inverse of multiplication. Students will distinguish when a word problem is asking to solve using a multiplicative or additive comparison. Students can use concrete, pictorial, and abstract methods to demonstrate the relationship between multiplication and division. Students can distinguish the difference between multiplicative comparison (e.g. Maria has 9 stickers. Joe has 3 times as many stickers as Maria. How many stickers do they have altogether?) Solve a word problem by creating an equation using a variable or symbol to represent the unknown number. (e.g. a pink rod is 4 inches long. A green rod is 3 times as long as the pink rod. How long is the green rod? Let y represent the green rod. 3 x This standard calls for students to translate comparative situations into equations with an unknown and solve. Students need many opportunities to solve contextual problems. Refer to Table 2, Appendix, for more examples. Examples: Group Size Unknown: A book costs $18. That is 3 times more than a DVD. How much does a DVD cost? (18 p = 3 or 3 p = 18). Number of Groups Unknown: A red scarf costs $18. A blue scarf costs $6. How many times as much does the red scarf cost compared to the blue scarf? (18 6 = p or 6 p = 18). Unknown Product: A blue scarf costs $3. A red scarf costs 6 times as much. How much does the red scarf cost? (3 6 = p). Major Gain familiarity with factors and multiples. 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? Students need many opportunities to solve contextual problems. A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? In solving this problem, the student should identify $6 as the quantity that is being multiplied by 3. The student should write the problem using a symbol to represent the unknown. Teacher Note: Real-world situations can be teacher or student generated and can be one-step or multi-step.

4 continued... Complex Goal: Mastery: Multiply two, two-digit numbers using strategies based on place value and properties of operations. (Not standard algorithm) (4.NBT.5) Multiply a whole number of up to four digits by a one-digit whole number (using strategies based on place value and properties of operation; illustrate and explain (i.e,. equations, rectangular array, or area model). (4.NBT.5) Students who develop flexibility in breaking numbers apart (decomposing numbers) have a better understanding of the importance of place value and the distributive property in multi-digit multiplication. Students use base ten blocks, area models, partitioning, compensation strategies, etc. when multiplying whole numbers and use words and diagrams to explain their thinking. They use the terms factor and product when communicating their reasoning. Multiple strategies enable students to develop fluency with multiplication and transfer that understanding to division. Use of the standard algorithm for multiplication and understanding why it works, is an expectation in the 5th grade. This standard calls for students to multiply numbers using a variety of strategies. Example: There are 25 dozen cookies in the bakery. What is the total number of cookies at the baker?

5 continued... Use of place value and the distributive property are applied in the scaffold examples below. To illustrate 154 x 6 students use base 10 blocks or use drawings to show 154 six times. Seeing 154 six times will lead them to understand the distributive property, 154 X 6 = ( ) x 6 = (100 x 6) + (50 X 6) + (4 X 6) = = 924. The area model shows the partial products. Students explain this strategy and the one below with base 10 blocks, drawings, or numbers. Matrix Model: This model should be introduced after students have facility with the strategies shown above. Example: What would an array area model of 74 X 38 look like? Complex Goal: Mastery: Interpret and represent multiplication equations as comparisons.(4.oa.1)

6 A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity (e.g., a is n times as much as b ). Students should be able to identify and verbalize which quantity is being multiplied and which number tells how many times. Students should be given many opportunities to write and identify equations and statements for multiplicative comparisons. It is essential that students are provided many opportunities to solve contextual problems. Multiplicative Comparison Example: 5 x 8 = 40. Sally is five years old. Her mom is eight times older. How old is Sally s Mom? 5 x 5 = 25 Sally has five times as many pencils as Mary. If Sally has 5 pencils, how many does Mary have? Students need experiences that allow them to connect mathematical statements and number sentences or equations. This allows for an effective transition to formal algebraic concepts. They represent an unknown number in a word problem with a symbol. Word problems which require multiplication or division are solved by using drawings and equations. Students need to solve word problems involving multiplicative comparison (product unknown, partition unknown) using multiplication or division. They should use drawings or equations with a symbol for the unknown number to represent the problem. Students need to be able to distinguish whether a word problem involves multiplicative comparison or additive comparison (solved when adding and subtracting in Grades 1 and 2). Present multistep word problems with whole numbers and whole-number answers using the four operations. Students should know which operations are needed to solve the problem. Drawing pictures or using models will help students understand what the problem is asking. They should check the reasonableness of their answer using mental computation and estimation strategies. Incomplete District Resources: My Math: Chapter 2 lessons 1,2,3,4,5,6,7,8,9 Chapter 5 lesson 5 District Resources: 4.OA.2 Mc Graw- Hill My Math Chapter 3 Lesson 3, 4, and 8 District Resources: 4.NBT.5 Mc Graw-Hill My Math Chapter 3 Lesson 1, 5, 6 - Chapter 4 Lesson 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11- Chapter 5 Lesson Exemplar Sample:

7 Incomplete (4.OA.1) Exemplar: There are almost 40 thousand fourth graders in Mississippi and almost 400 thousand fourth graders in Texas. There are almost 4 million fourth graders in the United States. Part A: We write 4 million as 4,000,000. How many times more fourth graders are there in Texas than in Mississippi? How many times more fourth graders are there in the United States than in Texas? Use the approximate populations listed above to solve. Part B: There are about 4 thousand fourth graders in Washington, D.C. How many times more fourth graders are there in the United States than in Washington, D.C.? Student Response: 40,000 is 10 times 4, ,000 is 10 times 40,000. 4,000,000 is 10 times 400, ,000 = 10 40,000, and there are about 10 times as many fourth graders in Texas as there are in Mississippi. And, 4,000,000 = ,000, so there are about 10 times as many fourth graders in the US as there are in Texas. Then I went from 4,000 to 4,000,000, I have to multiply by 10 three times. So =10 100=1000 So there are about 1,000 times as many fourth graders in the US as there are in Washington DC. (4.OA.2) Exemplar: A red umbrella costs $8.00. A green umbrella costs 3 times as much as the red umbrella. How much does the green umbrella cost? Student Response: The answer is $24. If the red umbrella is $8 and the green one is 3 times as much, then you just need to multiply 8 times 3 to get the answer. (4.NBT.5) Exemplar: Which expression best shows how to solve 52 x 18? A. (50 x 2 ) x (10 x 8) B C D Student Response: The answer is B. If I break 52 into 50 and 2 and break 18 into 10 and 8, I can multiply much easier. 50 x 10 = 500 and 50 x 8 = 400. Then multiply 2 x 8, which is 16 and 2 x 10, which is 20. Then add all of the products, , which gives me 936.