4.2 Applying Place Value Strategies to Multiply and Divide ...



4th Grade Mathematics

Unit #2 : Applying Pl ace Value Strategies to M ultiply a n d Di vide Multi--Digit Numbers

Pacing: 46 Days

Unit Overview

In this unit, students will illustrate multi-digit multiplication and division using a variety of strategies and models. The goal of this unit is for students to develop a deep conceptual understanding of these operations and strategies in order to equip them with the schema necessary to tackle real world problems that require them to apply their skills in new contexts. Students who develop flexibility in breaking numbers apart 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.

In fourth grade, students build on their third grade work with division within 100 by extending into four digit by one digit long division. Students need opportunities to develop their understandings by using problems in and out of context. New to the 4th grade, students will see multi-digit division requires working with remainders. Students must see

remainders in context to interpret and address why and what is left over or to partition into fractions or decimals. Students will use previous understandings of rounding to the nearest whole number to interpret the remainder. Use of the standard algorithm for multi-digit division is not an expectation in 4th grade. Students need to build on their

understanding of the inverse relationships between multiplication and division to deepen their conceptual understanding and practice good habits of checking their work. Area of

rectangles provide one context for developing such understanding.

Prerequisite Skills

Vocabulary

Mathematical Practices

1) Division fact fluency through 12s 2) Multiplication fluency through 12s 3) Understand and demonstrate basic multiplication

problems as repeated addition 4) Decompose multi-digit numbers based on place value 5) Represent multi-digit numbers using base ten blocks 6) Visually represent a basic multiplication problem ( __

groups of ___) 7) Represent an unknown quantity using a variable in

order to solve a basic equation 8) Differentiate between perimeter and area 9) Know common benchmarks for units of length, weight

and capacity

Commutative Property Distributive Property Identity Property Partial Product Area Model Equation Product Factor Quotient Dividend Divisor Remainder Interpret Context

Quotative/Partitive Models Equivalent Convert Dimensions Perimeter Compute Estimate Rectangular Array Strategy Area Formula Square Unit Length/Width Distance

MP.1: Make sense of problems and persevere in solving them MP.2: Reason abstractly and quantitatively MP.3: Construct viable arguments and critique the reasoning of others MP.4: Model with mathematics MP.5: Use appropriate tools strategically MP.6: Attend to precision MP.7: Look for and make use of structure MP.8: Look for and express regularity in repeated reasoning

Additional Standards (10%)

Supporting

Standards

(20%)

Major

Standards (70%)

Common Core State Standards

4.MD.3 Area and Perimeter

4.NBT.5: Multiply Whole Numbers 4.NBT.6: Divide Whole Numbers 4.OA. 2: Multiplication & Division 4.OA.3: Multi-step Word Problems

According to the PARCC Model Content Framework, Standard 4.NBT.5 and 4.NBT.6 should serve as opportunities for in-depth

focus:

"When students work toward meeting this standard, they combine prior understanding of multiplication with deepening understanding of the base-ten system of units to express the product of two multi-digit numbers as another multi-digit number (4.NBT.5)"...and "...to find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors (4.NBT.6)."

Progression of Skills

3rd Grade

4th Grade

5th Grade

3.OA.4: products

oInf twerhpAorleectcording4n.uNtmoBbTethr.5oe:fMPupuAlttRiopflCyouCarwdMhigooiltesdbeyl

aCon5m.tNuelBtnipTtl.yF5:mrFuallmtui-edeniwtgliyot wrkho,le

numbers, e.g., iSnttearnprdetardo3n.eN-dFigi.t2wshholoeunuldmbseer,ravnedas annoumppbeorrstuusinngittyhefor in-

5 ? 7 as the totdalepth focumsu:ltiply two two-digit

standard algorithm.

number of objects in 5 numbers, using strategies based

groups of 7 objects

each.

on place value and the properties of operations. Illustrate and

explain the calculation by using

equations, rectangular arrays,

and/or area models.

3.OA.3: Interpret whole-number quotients of whole numbers, e.g., interpret 56 ? 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

3.OA.8: Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers. 3.MD.5: Recognize area as an attribute of plane figures and understand concepts of area measurement.

3.MD.7: Relate area to the operations of multiplication and addition.

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.

4.OA.3: Solve multistep word problems posed with whole numbers and having wholenumber answers using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers.

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.

5.NBT.6: Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.

N/A

5.MD.5: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

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Big Ideas

Students Will...

? How does place value help me

Know/Understand

Be Skilled At...

find the product in a

? Commutative property helps students understand that ? Connecting diagrams of areas or arrays to numerical

multiplication problem?

regardless of order the product will be the same.

work.

? Identity property helps students understand that any ? Using a variety of strategies, such as base ten blocks,

? How do the properties of multiplication help illustrate my understanding of multiplication and division strategies?

factor times one is itself. ? The distributive property is the sum of two numbers

times a third number is equal to the sum of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3. ( ? Distributive property helps decompose numbers into

area models, partitioning, and rectangular arrays when multiplying two-digit numbers by two-digit numbers and one-digit number by a whole number of up to four digits. ? Developing flexibility in breaking numbers apart and make connections to place value and the distributive

base ten units and single digit multiplications and

property

? What do the quotient,

products of multiples of 10, 100 etc.

? Illustrating multiplication using graph paper, arrays,

dividend, divisor and remainder represent in a division problem? Why is it

? The properties of division. ? Visual models can be used to show multiplication:

arrays, area models, combining equal sized sets,

and/or area models. ? Explaining the strategy they used to solve

multiplication and division problems

important to understand the

groups, and repeated addition.

? Using place value and base-ten units to decompose the

context of a division problem in order to interpret what each of these components represent?

? ?

That multiplication and division are inverse operations. That division problems can be solved using equations, rectangular arrays, and/or area models.

?

dividend into like base-ten units and find the quotient unit by unit, starting with the largest unit and continuing on to smaller units. Dividing four-digit dividends and one-digit divisors using a variety of strategies in real world context.

? Why do the formulas for finding perimeter and area

? How to group (quotative model) or fair share (partitive model) as it relates to single-digit division.

? Solving for unknown rectangular side lengths using the area and perimeter formulas.

work? How can I reason about ? What the remainder means in a division problem

? Using the area and perimeter formulas to solve for real

these measurements to make

? Remainders should be put into context for

world problems involving perimeter and area of

sense of the formulas?

interpretation. Ways to address remainders o Remain as a left over

rectangles.

o Partitioned into fractions or decimals

o Discarded leaving only the whole number

o Increase the whole number answer up one

o Round to the nearest whole number for an

approximate result

? the formula for perimeter = 2(length + width).

? the area for a rectangle formula as area = length x

width.

? the difference between area and perimeter and their

formulas.

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Unit Sequence

Student Friendly Objective SWBAT...

Key Points/ Teaching Tips

Exit Ticket

1 Relate the dimensions of a ? Focus on area and having students use? See ET_4.2_L5

rectangle to the formulas

strategies for multiplication to solve

for finding its perimeter

real world problems.

and area.

? This should be a hands on lesson using

graph/grid paper

? Struggling students may compose rectangles with tiles to "build" the

area

2 Model and measure area of ? Continue to connect area of a figure See ET_4.2_L6

a figure.

around a rectangle with multiplication

strategies.

?

3 Make sense of and

? Key Point: Relate tiling to an array

persevere in solving real

and record your area as the product of

world problems involving

the two dimensions: length (rows) x

area and perimeter

width (column)

? Task-Based Problem to Start Lesson: Ms. McCrary wants to make a rabbit pen in a section of her lawn. Her plan for the rabbit pen includes the following:

? It will be in the shape of a rectangle.

? It will take 24 feet of fence material to make.

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Instructional Resources

Engage NY Lesson 3.1 (Appendix C)

My Math Chapter 13 Lesson 1

Engage NY Lesson 3.2 (Appendix C)

My Math Chapter 13 Lesson 3

Engage NY Lesson 3.3 (Appendix C)

"Chris' Garden Dilemma" (Appendix C)

? Each side will be longer than 1 foot.

? The length and width will measure in whole feet.

a. On your grid, draw 3 different rectangles that can each represent Ms. McCrary's rabbit pen. Be sure to use all 24 feet of fence material for each pen.

Now write down the length and width of each rabbit pen you drew. Then, find the area of each pen. Be sure to label each measurement with either feet or square feet.

Rabbit pen 1: ____ x ___

Rabbit pen 2: ____ x ____

Rabbit pen 3: _____ x _____

*Note: struggling students may use

tiles to complete the task

?

4 Look for and make use of ? Allow students to use hundreds charts See ET_4.2_L1

structure by using number

and base ten blocks at first in order to

patterns to multiply

deduce a pattern

efficiently by multiples of

10.

5 Multiply two-digit

?

multiples of ten by two-

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My Math Chapter 4 Lesson 1

Engage NY Lessons 3.4 & 3.5 (Appendix C)

Engage NY Lesson 3.6

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