# Grade 4 Multi-Digit Multiplication and Division with One Divisor Unit of Instruction

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1 Grade 4 Multi-Digit Multiplication and Division with One Divisor Unit of Instruction This is a progressive unit of instruction beginning with students investigating the concrete area model of fraction equivalence and ordering. Students progress through the concrete, representational and abstract model of understanding within the lessons. The unit progresses to decimal understanding relating to money and fractional representation. Once your students have a solid understanding of fraction and decimal equivalence, they are ready to add, subtract and finally multiply fractions. (whole times a fraction) Those lessons also begin with concrete, move to representational and finally the abstract. Number and Operations in Base Ten (MAFS.4.NBT.2.4 is not addressed in this unit of study because it is Fluently add and subtract multi-digit whole numbers using the standard algorithm this unit only addresses multiplication and division) Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalization methods to compute products of multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalization procedures to find quotients involving multi-digit dividends. They select and accurately apply A bibliography of children's literature with a focus on multiplication is provided, which can be integrated so that students can connect through literature. 1. Ten Times Better, Richard Michelson 2. A Remainder of One, Elinor J. Pinczes 3. One Hundred Hungry Ants, Elinor J. Pinczes 4. The King s Chess Board, David Birch 5. In the Next Three Seconds Predictions for the Millennium, Comp. Rowland Morgan 6. Counting on Frank, Rod Clement

2 appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context. Cluster 1: Generalize place value understanding for multi-digit Whole numbers Cluster 2: Use place value understanding and properties of operations to perform multi-digit arithmetic Parent Resources Multiplying: 2-digit number times 2-digit number (using distributive property) 586 Multiplying: 2-digit number times a 2-digit number (area model) 587

3 10X Bigger! In this lesson students will move from constructing concrete models of what happens to a number when you multiply the number by 10 into drawing their own pictorial representations of the same process with numbers up to 1,000. They will develop an understanding of the rule 'a digit in one place represents ten times what it represents in the place to its right' and apply this rule to a variety of situations. MAFS.4.NBT.1.1: 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 = 10 by applying concepts of place value and division. What s My Value? This lesson will allow students to deepen their knowledge in place value. The teacher will use modeling, guided math groups, and independent practice to help students understand that the digit in one place represents ten times what it represents in the place to its right. Students will use base 10 blocks to build numbers and make general observations about the values of numbers. Students will also draw and label models, to make understanding concrete. In groups of 4-5, students will work together and discuss (using discussion questions) to deepen their knowledge on the topic. MAFS.4.NBT.1.1: 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 = 10 by applying concepts of place value and division.

4 Amazing Arrays 2 X 1 This is a hands-on lesson for extending and practicing drawing arrays using area models that show a 2-digit number times a 1-digit number. Students are also required to use the distributive property of multiplication and the equations they represent. Amazing Arrays 3 X 1 or 1 X 3 This lesson is the third lesson in a unit beginning with Amazing Arrays and Amazing Arrays 2X1. In this lesson students solve a multiplication problem by drawing arrays and segment the areas in several ways to solve the problem. Students will also apply the distributive property, explore rotations of area models to demonstrate the commutative property of multiplication, and match a word problem with its array.

5 Boxing Math Using the Area Model for Multiplication A common mistake students make when learning to multiply is treating multiplication like addition, and multiplying ones by ones and tens by tens. In this lesson, your students will avoid that mistake as they learn to use the area model to do double digit multiplication. After group practice, students are taught a game to reinforce their learning. Multiplying Around the Block The students will build upon their understanding of the place-value system and multiplying using base-ten models to build their understanding of multiplying with two-digit by two-digit numbers using area models. They will work with partners during the learning process to help them develop the usage of mathematical language when explaining their thinking and calculations to others.

6 2- Digit Array Multiplication This lesson explores a conceptual approach to multiplying two 2-digit numbers. Students will create, explore, describe and record arrays built with place value pieces. The lesson supplies the understanding that will make multiplying multidigit numbers easy to do. Students should use graph paper to draw their arrays and show the factor pairs. Concrete Representational step. Area Model of Multiplication Using Base 10 Manipulatives (Double digit multiplication) "Students will use base 10 manipulatives to build a rectangular array to represent double digit multiplication. Students will make the connection between the standard algorithm and the rectangular array" Students should use graph paper to draw their arrays and show the factor pairs. Concrete Representational step.

7 Modeling Multiplication for Mastery In this lesson, students will work to multiply multi-digit numbers using various strategies. Students will use arrays, array frames with base ten blocks, and area models to explore and justify their solutions. Array for Charity Students will use array frames to find out how many pennies each of the classes in their school collected for a charity drive. Students will demonstrate and explain the array frame as well as determining how many pennies will go to each of the seven charities for which they have collected pennies.

8 Model Multiplication This concept based, hands-on lesson is intended to help you assess how well your students can use a variety of strategies and representations of 2 two-digit multiplication. Aaron and Anya s Quilt Challenge: Problem Solving and Interpreting Remainders In this situational story, Aaron and Anya find a large piece of brightly colored fabric. They decide to cut it into squares to make a quilt. Students will find the area of the fabric by multiplying two digits by

9 two digits. They will explore factors as they figure out the largest quilt square that can be cut for 25 students. There will be fabric left over; students will have to determine and justify remainders based on several different scenarios. Finally, students will create their own quilt square using grid paper. Students should use graph paper to draw their arrays and show the factor pairs. Concrete Representational step. MAFS.4.NBT.2.6: 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 MAFS.4.OA.1.2: 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.

10 MAFS.4.OA.1.3: 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. Share and Share Alike MAFS.4.MD.1.3: Apply the area and perimeter formulas for rectangles in real world and mathematical problems. This inquiry-based 5E lesson provides a framework for students to investigate division of a multi-digit number by a 1-digit divisor using hands-on materials. Students will have the opportunity to work with manipulatives such as base-10 blocks to physically divide a multi-digit number into equal groups. Students will discover the idea of remainders through their exploration. The lesson is designed to be chunked and can easily be spread over 2, 45-minute sessions. MAFS.4.NBT.2.6: 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

11 I See! Division with the Distributive Property In this lesson, students will use visual models to represent division using the distributive property as a strategy. Students will have an understanding of how to decompose numbers in the context of division problems using an area model. MAFS.4.NBT.2.6: 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

12 Supply List Suggested class supplies for each student Math Notebook/Journal Math Folder Scissors Construction paper Colored pencils Pencils Index cards Place value charts thru hundredths Hundreds charts Number lines (can use string and index cards) Number strips Protractors Rulers Number and Operations in Base Ten (class set) place value blocks painter s tape (class set) student white boards (class set) white board markers Sock for erasing (1 each student) (teacher set) magnetic place value blocks graph paper

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