Paramount Unified School District Educational Services

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1 Paramount Unified School District Educational Services Grade 4 Unit 4 Stage One Desired Results Unit 4: Multi-Digit Multiplication In this unit, students will: use place value understanding and visual representations to solve problems with multi-digit numbers multiply single-digit numbers by multiples of 10, 100, and 1,000 and two-digit multiples of 10 by two-digit multiples of 10 and reason between arrays and written numerical work to see the role of place value units in decompose numbers into base ten units to find products of single-digit by multi-digit numbers multiply two-digit by two-digit numbers using understanding of place value and of the area model to empower them to multiply by larger numbers ( connect the partial products appearing in the area model to the use estimation to check for reasonableness before computing apply understanding of addition, subtraction and to solving multi-step problems apply algebraic thinking by using a symbol to represent an unknown quantity Common Misconceptions: Students may: under generalizes the results of by powers of 10. To find products like 3 50 = 150 or = 1,500, she must work the product out using a long method of computation (see example). can state and give example s of properties of but does not apply them to simplify computations. For example, the student labors to find the product because he does not realize that he could instead perform the equivalent but much easier computation, misapplies the procedure for multiplying multi-digit numbers by ignoring place value. For example, student multiplies correctly by ones digit but ignores the fact that the 3 in the tens place means 30. Thus, = 1,800 (see example). knows how to multiply but does not know when to multiply (other than because he was told to do so, or because the computation was written as a problem). For example, the student cannot explain why he should multiply or connect to actions with materials. See 1

2 Unit 4 Overview: Multi-Digit Multiplication Transfer Goals 1) Demonstrate perseverance by making sense of a never-before-seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Standards NBT.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. NBT.3 Use place value understanding to round multi-digit whole numbers to any place. 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. OA.3 Solve multi-step word problems posed with whole numbers and having whole answers using addition, subtraction,, and division; including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Understandings Students will understand that Place value and properties play a critical role in number operations A variety of models and strategies can be used to multiply multi-digit numbers Estimation is important to use when problem solving because it helps to check the reasonableness of a solution Knowledge Students will know : Multiple, mental math, array, area model, product, partial products, estimate, round, about how many, approximately, reasonable, operation, variable Distributive Property Associative Property Basic facts for Inverse relationship Symbols can represent the unknown Meaning-Making Essential Questions Students will consider. How do arrays and area models develop an understanding of multi-digit? What is the role of place value and properties of operations in multi-digit? How do different strategies for solving multi-digit relate to each other (e.g., area model, partial products,? Why is estimation important to use when solving problems? Acquisition Skills Students will be skilled at and able to Use basic facts and patterns in zeros to mentally multiply a number by a multiple of ten Recognize that the digit in one place represents ten times what it represents in the place to its right Illustrate problems using place value, rectangular arrays and area models Estimate numbers and assess the reasonableness of answers Multiply a number up to 4-digits by a 1-digit number using partial products and Multiply a 2-digit number by a multiple of ten using the Associative Property Multiply a 2-digit number by a 2-digit number using partial products and Solve multi-step word problems involving addition, subtraction and Represent word problems using equations with a letter symbolizing the unknown quantity 2

3 Paramount Unified School District Educational Services Grade 4 Unit 4 Stage Two Evidence of Learning Transfer is a student s ability to independently apply understanding in a novel or unfamiliar situation. In mathematics, this requires that students use reasoning and strategy, not merely plug in numbers in a familiar-looking exercise, via a memorized algorithm. Transfer goals highlight the effective uses of understanding, knowledge, and skills we seek in the long run that is, what we want students to be able to do when they confront new challenges, both in and outside school, beyond the current lessons and unit. These goals were developed so all students can apply their learning to mathematical or real-world problems while simultaneously engaging in the Standards for Mathematical Practices. In the mathematics classroom, assessment opportunities should reflect student progress towards meeting the transfer goals. With this in mind, the revised PUSD transfer goals are: 1) Demonstrate perseverance by making sense of a never-before-seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and by using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Multiple measures will be used to evaluate student acquisition, meaning-making and transfer. Formative and summative assessments play an important role in determining the extent to which students achieve the desired results in stage one. Formative Assessment What constitutes evidence of understanding for this lesson? Through what other evidence during the lesson (e.g. response to questions, observations, journals, etc.) will students demonstrate achievement of the desired results? How will students reflect upon, self-assess, and set goals for their future learning? Discussions and student presentations Checking for understanding (using response boards) Ticket out the door, Cornell note summary, and error analysis Learn Zillion end-of-lesson assessments Check My Progress, teacher-created assessments/quizzes ST Math (curriculum progress, data reports, etc.) Unit 4: Multi-Digit Multiplication Aligning Assessment to Stage One Opportunities Summative Assessment What evidence must be collected and assessed, given the desired results defined in stage one? What is evidence of understanding (as opposed to recall)? Through what task(s) will students demonstrate the desired understandings? Unit assessments Teacher-created chapter tests or mid-unit assessments Challenge lessons Illustrative Mathematics tasks ( Performance tasks 3

4 The following pages address how a given skill may be assessed. Assessment guidelines, examples and possible question types have been provided to assist teachers in developing formative and summative assessments that reflect the rigor of the standards. These exact examples cannot be used for instruction or assessment, but can be modified by teachers. Skill Standard Assessment Guidelines Example Possible Question Type(s) Use basic facts and patterns in zeros to mentally multiply a number by a multiple of ten Recognize that the digit in one place represents ten times what it represents in the place to its right NBT.1 The student is prompted to explain the difference between the values of the same digit in different place values. In each item, the numbers presented differ by a factor of 10 (e.g., 8 and 80, or 1725 and 17,250). Items should be equally distributed across these number bands: small numbers (up to 1,000), medium numbers (from 1,000 up to 100,000), and large numbers (from 100,000 up to 1,000,000). Select the statement that explains how the values of the numbers 420 and 4200 are different. A is 1000 times as large as 420. B is 100 times as large as 420. C is 10 times as large as 420. D is 1 time as large as 420. Multiple Choice, Single Correct Response Multiply a number up to 4-digits by a 1- digit number using partial products and NBT.5 The student is prompted to multiply two whole numbers Item difficulty can be adjusted by using: One factor is a multiple of 10, 100, or 1000 Enter the product x 4 Equation/Numeric 4

5 Skill Standard Assessment Guidelines Example Possible Question Type(s) Multiply a 2-digit number by a 2-digit number using partial products and NBT.5 The student is presented with a expression in which properties of operations have been used as strategies for, with one unknown number The student is presented with a expression in the stem and expressions reflecting use of the or decomposition of factors in the answer choices. The student is presented with a problem and four vertically recorded partial solutions. 36 x 94 = (30 + 6) x ( + 4) 36 x 94 = Which expression is equal to 36 x 94? A. (30 x 90) + (6 x 4) B. (30 x 6) + (90 x 4) C. (30 x 6) x 94 + (30 + 6) x 4 D. (30 x 90) + (30 x 6) + (90 x 6) + (90 x 4) Which strategy for multiplying 36 and 94 should result in the correct product? Equation/Numeric Multiple Choice, Single Correct Response The student is presented with a expression in the stem and expressions reflecting use of the or decomposition of factors in the answer choices. Which expression is equal to 36 94? A. (30 90) + (6 4) B. (30 + 6) (90 + 4) C. (30 + 6) 94 + (30 + 6) 4 D. (30 90) + (30 6) + (90 6) + (90 4) The student is presented with a expression in which properties of operations have been used as strategies for, with one unknown number. Enter the unknown number that makes the equation true = (30 + 6) ( + 4) 5

6 Paramount Unified School District Educational Services Grade 4 Unit 4 Stage Three Learning Experiences & Instruction Unit 4: Multi-Digit Multiplication Prior to planning for instruction, it is important for teachers to understand the progression of learning and how the current unit of instruction connects to previous and future courses. Teachers should consider: What prior learning do the standards and skills build upon? How does this unit connect to essential understandings of later content? How can assessing prior knowledge help in planning effective instruction? What is the role of activating prior knowledge in inquiry? Looking Back In Grade 3, students: Used place value understanding to round whole numbers to nearest 10 or 100. Multiplied one-digit whole numbers by multiples of 10 in the range (e.g., 9 80, 5 60) using strategies based on place value and properties of operations. Used 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. Determined the unknown whole number in a 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 =? Looking Ahead In Grade 5, students will: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Use place value understanding to round decimals to any place. Fluently multiply multi-digit whole numbers using the standard algorithm. Add, subtract, multiply, and divide decimals to hundredths, 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 and explain the reasoning used. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Applied properties as strategies to multiply and divide. Solved two-step word problems using the four operations. Represented problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 6

7 Unit 3: Multi-Digit Multiplication Timeframe: Nov. 14-Jan. 19 Course Textbook: McGraw Hill, My Math Transfer Goals ST Math Objectives: Multi-digit Multiplication 1) Demonstrate perseverance by making sense of a never-before-seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Understandings: Students will understand that Essential Question(s): Students will consider. Place value and properties play a critical role in number How do arrays and area models develop an understanding of multi-digit? operations A variety of models and strategies can be used to multiply multidigit What is the role of place value and properties of operations in multi-digit? numbers Estimation is important to use when problem solving because it How do different strategies for solving multi-digit relate to each other (e.g., area model, partial products,? helps to check the reasonableness of a solution Why is estimation important to use when solving problems? Time Skills Learning Goal Lesson/Activity/Resource Knowledge Focus Questions for Lessons Teacher Notes Use basic facts and patterns in zeros to mentally multiply a number by a multiple of ten Recognize that the digit in one place represents ten times what it represents in the place to its right Use basic facts to mentally multiply a number Observe patterns in problems to mentally multiply by a multiple of ten Use place value to multiply by multiples of 10s, 100s and 1000s 5 friends go to a carnival. They each buy 3 tickets. How many total tickets do they have? What if they had 30 tickets each? And 300 tickets each? What do you observe is happening to the total of tickets? Chapter 4 Lesson 1 Multiples of 10, 100, and 1,000 : Multiple Mental math Why does the product of a multiple of 10 always have a zero in the ones place? What is the relationship between a digit in one place and the place to its right? In grade 3, students multiplied one-digit whole numbers by multiples of 10 in the range LearnZillion video: Multiply by powers of 10 (TP6RXQ4) Investigation Look at these problems. See if you can find any patterns that will help you solve for d). a) 4 x 8= 32 b) 4 x 80 = 320 c) 4 x 800 = 3,200 d) 4 x 8000 = (patterns may include adding the zero when multiplying by a number that is ten times larger as well as the number of digits in the product.) 7

8 Time Skills Learning Goal Lesson/Activity/ Resource Knowledge Focus Questions for Lessons Teacher Notes Independent practice with transfer goals: Illustrative Mathematics Thousands and Millions of 4 th Graders Estimate numbers and assess the reasonableness of answers Estimate products of a 2-digit number by a 1- digit number Lena loves playing video games at the arcade. She played 4 games and scored 23 points each time. About how many points did she score in total? Estimate Round About how many Approximately Reasonable Why is it important to use estimation when solving problems? LearnZillion video: Use estimation to check reasonableness of products (LZ70) 4 days Illustrate problems using place value (unifix cubes), rectangular arrays and area models Multiply a number up to 4-digits by a 1- digit number using partial products and distributive property (2 digit by 1 digit) Use cubes to create arrays that represent a problem Use the area model to represent a problem Use the area model to multiply larger numbers Use the distributive property to multiply Use partial products to multiply See Lesson 2 for additional practice Inquiry Lesson: Arrays and Area Model Inquiry Lesson: Distributive Property and Partial Products Array Area model Product Partial products Distributive Property What is the role of place value when using arrays and area model to multiply larger numbers? When multiplying multi-digit numbers, how do arrays, area model, partial products and relate to each other? LearnZillion videos: Use an array to multiply a twodigit number by a one digit number (GENCU7W) Multiply multidigit whole numbers by single digit whole numbers using an area model (4VFHCJK) Use place value understanding to multiply three and four digit numbers (9HNTY63) 8

9 Time Skills Learning Goal Lesson/Activity/ Resource Illustrate Estimate products problems using place value (unifix cubes), rectangular arrays and area models 4 days Represent word problems using equations with a letter symbolizing the unknown quantity Select a strategy to solve a word problem (e.g., area model, partial products or ) Represent the product of a word problem using a letter Knowledge Area model Partial products Distributive Property Focus Questions for Lessons What is the best strategy for solving multi-digit? Why? Cumulative Review and Error Analysis of Unit 3 Extended Constructed Responses Introduce students to the 4-point Extended-Constructed Response rubric. Use this opportunity to get students familiar with rubric. Possible activities include evaluating their own work, peer feedback, whole-class discussion about displayed exemplars, reflecting on next steps, etc. Estimate numbers and assess the reasonableness of answers Illustrate problems using place value (unifix cubes), rectangular arrays and area models Multiply a number up to 4-digits by a 1-digit number using partial products and Estimate the product of a 3-digit number by a 1-digit number Use area model to multiply a 3-digit number by a 1- digit number Use and partial products to multiply a 3-digit number by a 1-digit number Use and partial products to multiply a 3-digit number (containing a 0) by a 1-digit number DEPENDING ON YOUR STUDENTS, YOU CAN REPEAT THIS SAME PROGRESSION TO MULTIPLY 4-DIGIT BY 1-DIGIT NUMBERS OR STUDENTS CAN SELECT WHICH STRATEGY THEY WOULD LIKE TO USE TO SOLVE 4-DIGIT BY 1-DIGIT PROBLEMS. Exploration Adrian has four boxes of action figures. There are 12 in each box. Alex has 21 action figures in each of his 3 boxes. Who has more action figures? How do you know? Lena goes back to the arcade. She played 4 games but this time she scored 275 points each time! She thinks she has about 800 total points. Do you agree or disagree with Lena? Why? LearnZillion lesson: Multiplying larger numbers with arrays and base-ten blocks (XC2886R) Diller Elementary is collecting money to donate to a charity. $1023 is collected each month. How much money is collected in 4 months? Area model Partial products Distributive Property What is the role of place value when using arrays, area model, partial products and the to multiply larger numbers? How can I apply my understanding of place value when multiplying factors with zeros? See Lessons 9 and 11 for additional practice Teacher Notes Students do not need to solve multidigit problems using the standard algorithm. They can use any of the strategies they have been learning about to solve. The standard states that students should multiply a number up to 4- digits. The textbook does not include these kind of problems so teachers will need to incorporate them during instruction and practice (see SBAC example). LearnZillion video: "Use place value understanding to multiply three and four digit numbers" (3W5DT6U) 9

10 Time Skills Learning Goal Lesson/Activity/ Resource Independent practice with transfer goals/questions to Ask: How many different ways can you solve 289 x 8? 4 days Knowledge How does the order of the digits in the factors impact the product? (e.g., 452 x 7 compared to 425 x 7) How are the values of the numbers 420 and 4200 different? Multiply a 2-digit number by a multiple of ten using the Associative Property Estimate numbers and assess the reasonableness of answers Illustrate problems using place value, rectangular arrays and area models Multiply a 2-digit number by a 2-digit number using partial products and Decompose 2-digit numbers that end in zero into a number times 10 Multiply a 2-digit number by a multiple of ten using the Associative Property Estimate the product of a 2-digit number by a 2-digit number Use area model to multiply a 2-digit number by a 2-digit number (decompose both factors) Use and partial products to multiply a 2-digit number by a 2-digit number (decompose both factors)* A hummingbird flies 12 miles per hour. If it flies a total of 40 hours, how far did it fly? Chapter 5 Lesson 1 Multiply by Tens A concert ticket costs $48. About how much will tickets cost for a group of 22 people? Lesson 2 Estimate Products LearnZillion lesson: Using place value strategies to multiply two double digit numbers (Quick Code: 4XF9HY4) LearnZillion lesson: Multiplying larger numbers using the area model (Quick Code: 62G8AZ6) Associative Property Estimate Round About how many Approximately Reasonable See Lessons 3-4 for additional practice Focus Questions for Lessons How can we use decomposition and the Associative Property to multiply larger numbers? Joey is finding 67 x 40. Explain why he can think of 67 x 40 as 67 x 4 x 10? How can you use area model, and partial products to multiply two 2- digit numbers? What is the role of place value when using area model, and partial products to multiply two 2- digit numbers? Teacher Notes LearnZillion video: Solve 2- by 2- digit problems using partial products (LZ22) For LearnZillion lessons: only use examples with 2- digit by 2-digit and 4-digit by 1-digit examples (not 4 digit by 2-digit) *Note: Students should decompose both factors when solving using the area model although the textbook only decomposes one factor Independent practice with transfer goals/learnzillion Application Task: Zoo Donation: Using place value strategies to solve multi-digit problems (Quick Code: S75ZE5R) 10

11 Time Skills Learning Goal Lesson/Activity/ Resource Knowledge Focus Questions for Lessons Teacher Notes Jan days Review after Winter Break 3 days Solve multi-step word problems involving addition, subtraction and Represent word problems using equations with a letter symbolizing the unknown quantity Solve a one-step word problem Solve a multi-step word problem (teacher adds a second step to the original onestep problem) Solve a multi-step word problem (see Challenge Lesson) Use a letter to represent the unknown sum Use a letter to represent an unknown within the context of the story At the pet store, each small dog weighs 35 pounds. If there are 4 small dogs, how many pounds do they weigh together? If there are 6 large dogs and each large dog weighs 60 pounds, how much do they weigh together? What if I wanted to know the total weight of small dogs and large dogs how could I do this? See Lesson 5 for additional practice Challenge Lesson: School Fundraising Operation Variable How can I use equations to model real-world problems? How can you solve problems with more than one operation? How can the unknown be represented in an equation? Students should have experiences in which the unknown appears in different places in a story or in an equation. Jan days Review and Administer Unit 4 Assessment Review: How many different ways can you solve 94 x 64? Create two sentences that could create a product between 500 and 600? Is the product of 29 x 34 over or under 900? Explain how you know. Think of an example in life when you might multiply two numbers? An example is, when might you multiply two two-digit numbers? Or a three-digit number by a one-digit number? Common Core Practices Instruction in the Standards for Mathematical Practices Use of Talk Moves Writing in math (e.g. math notes, prompts, journals) Use of manipulatives Use of technology Use of real-world scenarios Project-based learning Number Talks 11