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Grade 4 Mathematics Module 3, Topic E: Division of Tens and Ones with Successive Remainders

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Title:

Grade 4 Mathematics Module 3, Topic E: Division of Tens and Ones with Successive Remainders

URL:

https://www.engageny.org/resource/grade-4-mathematics-module-3-topic-e-overview

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 3, Topic E, students synthesize their Grade 3 knowledge of division types (group size unknown and number of groups unknown) with their new, deeper understanding of place value. Students focus on interpreting the remainder within division problems both in word problems and long division (4.OA.3). A remainder of 1, as exemplified below, represents a leftover flower in the first situation and a remainder of 1 ten in the second situation. While we have no reason to subdivide a remaining flower, there are good reasons to subdivide a remaining ten. Students apply this simple idea to divide two-digit numbers unit by unit: dividing the tens units first, finding the remainder (the number of tens unable to be divided), and decomposing remaining tens into ones to then be divided.  Lesson 14 begins Topic E by having students solve division word problems involving remainders. In Lesson 15, students deepen their understanding of division by solving problems with remainders using both arrays and the area model. Students practice dividing two-digit dividends with a remainder in the ones place using place value disks in Lesson 16 and continue that modeling in Lesson 17 where the remainder in the tens place is decomposed into ones. The long division algorithm is introduced in Lesson 16 by directly relating the steps of the algorithm to the steps involved when dividing using place value disks. Introducing the algorithm in this manner helps students to understand how place value plays a role in the steps of the algorithm. The same process of relating the standard algorithm to the concrete representation of division continues in Lesson 17.  Lesson 18 moves students to the abstract level by requiring them to solve division problems numerically without drawing. In Lesson 19, students explain the successive remainders of the algorithm by using place value understanding and place value disks. Finally, in Lessons 20 and 21, students use the area model to solve division problems and then compare the standard algorithm to the area model (4.NBT.6). Lesson 20 focuses on division problems without remainders, while Lesson 21 involves remainders.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 4
3. Determine and justify solutions for multi-step word problems, including problems where remainders must be interpreted.

a. Write equations to show solutions for multi-step word problems with a letter standing for the unknown quantity.

b. Determine reasonableness of answers for multi-step word problems, using mental computation and estimation strategies including rounding.

Unpacked Content
Evidence Of Student Attainment:
Students:
When given multi step word problems,
  • Solve a variety of multistep word problems involving all four operations on whole numbers including problems where remainders must be interpreted.
  • Explain and justify solutions using connections between the problem and related equations involving a single (letter) unknown.
  • Evaluate the reasonableness of solutions using estimation strategies.
Note: Multi step problems must have at least 3 steps.
Teacher Vocabulary:
  • Operation
  • Multi Step problem
  • Remainder
  • Unknown quantity
  • Equation
  • Rounding
  • Mental strategy
  • Partition
  • Estimation
  • Reasonableness
Knowledge:
Students know:
  • Context situations represented by the four operations.
  • How to calculate sums, differences, products, and quotients.
  • Estimation strategies to justify solutions as reasonable.
Skills:
Students are able to:
  • Solve multi-step word situations using the four operations.
  • Represent quantities and operations physically, pictorially, or symbolically.
  • Write equations to represent the word problem and use symbols to represent unknown quantities.
  • Use context and reasoning to interpret remainders.
  • Use estimation strategies to assess reasonableness of answers by comparing actual answers to estimates.
Understanding:
Students understand that:
  • Using problem solving strategies will help them determine which operation to use to solve a problem.
  • Remainders must be interpreted based on the context, and remainders are sometimes ignored, rounded up, or partitioned.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.4.3.1: Use multiplication 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.
M.4.3.2: Solve single-step word problems.
M.4.3.3: Recognize key terms to solve word problems.
Examples: in all, how much, how many, in each.
M.4.3.4: Solve division problems without remainders.
M.4.3.5: Recall basic addition, subtraction, and multiplication facts.

Prior Knowledge Skills:
  • Demonstrate computational understanding of multiplication and division by solving authentic problems with multiple representations using drawings, words, and/or numbers.
  • Identify key vocabulary words to solve multiplication and division word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
  • Recall basic multiplication facts.
  • Add and subtract within 20.
  • Represent repeated addition, subtraction, and equal groups using manipulatives.
  • Distinguish between rows and columns.
  • Use repeated addition to solve problems with multiple addends.
  • Count forward in multiples from a given number.
    Examples: 3, 6, 9, 12; 4, 8, 12, 16.
  • Recall doubles addition facts.
  • Model written method for composing equations.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.4.1 Solve one-step word problems involving real-life situations using the four operations within 100 without regrouping and select the appropriate method of computation when problem solving.


Mathematics
MA2019 (2019)
Grade: 4
12. Use strategies based on place value, properties of operations, and/or the relationship between multiplication and division to find whole-number quotients and remainders with one-digit divisors and up to four-digit dividends.

a. Illustrate and/or explain quotients using equations, rectangular arrays, and/or area models.
Unpacked Content
Evidence Of Student Attainment:
Students:
When given division problems with one-digit divisors and up to four-digit dividends,
  • Find quotients with remainders using strategies based on place value, properties of operations, and the relationship between multiplication and division.
  • Illustrate quotients using a rectangular array and/or area model, and explain the connection of the visual model to the equation.
Teacher Vocabulary:
  • Quotient
  • Dividend
  • Divisor
  • Divide
  • Multiply
  • Multiple
  • Equation
  • Remainder
  • Area model
  • Greatest multiple
  • Decompose
  • Compose
  • Array
  • Properties of operations
  • Unknown factor
  • Partial quotient
Knowledge:
Students know:
  • How to decompose and compose numbers in a variety of ways using place value and the properties of operations to demonstrate a variety of strategies for division.
  • Division can be described as an unknown factor problem.
  • A variety of contextual situations can be represented with a division equation.
Skills:
Students are able to:
  • Use strategies based on place value to find whole number quotients and remainders.
  • Use the properties of operations to find whole number quotients and remainders.
  • Use arrays and area models to find whole number quotients and remainders.
  • Illustrate division situations with rectangular arrays and area models.
  • Write an equation to represent a division situation.
Understanding:
Students understand that:
Division expressions represent
  • The number of objects in each group when the total number is partitioned evenly into a given number of groups.
  • The number of groups when the total number is partitioned into groups that each contain a given number.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.4.12.1: Define fraction, numerator and denominator.
M.4.12.2: Recognize fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts.
M.4.12.3: Identify the parts of a fraction a/b as the quantity formed by a parts and size 1/b.
M.4.12.4: Recognize fractions as numerals that may represent division problems.
M.4.12.5: Label numerator, denominator, and fraction bar.
M.4.12.6: Identify parts of a whole with two, three, or four equal parts.
M.4.12.7: Recognize that equal shares of identical wholes need not have the same shape.
M.4.12.8: Distinguish between equal and non-equal parts.

Prior Knowledge Skills:
  • Recognize fractions as lengths from zero to one.
  • Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
  • Identify a number line.
  • Recognize whole numbers as lengths from zero to one.
  • Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
  • Identify a number line.
  • Label the fractions on a pre-made number line diagram.
  • Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
  • Recognize a number line diagram with equally spaced points.
  • Compare length using non
  • standard units to determine which is longer.
Tags: divide, dividends, divisor, equations, operations, place value, quotients, remainders, word problems
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There are eight lessons in this topic.

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Author: Hannah Bradley