# Module 1

Version 1.3.1

### Place Value, Rounding, and Algorithms for Addition and Subtraction

In this 25-day Grade 4 module, students extend their work with whole numbers.  They begin with large numbers using familiar units (hundreds and thousands) and develop their understanding of millions by building knowledge of the pattern of times ten in the base ten system on the place value chart (4.NBT.1).  They recognize that each sequence of three digits is read as hundreds, tens, and ones followed by the naming of the corresponding base thousand unit (thousand, million, billion).[1]

In this 25-day Grade 4 module, students extend their work with whole numbers.  They begin with large numbers using familiar units (hundreds and thousands) and develop their understanding of millions by building knowledge of the pattern of times ten in the base ten system on the place value chart (4.NBT.1).  They recognize that each sequence of three digits is read as hundreds, tens, and ones followed by the naming of the corresponding base thousand unit (thousand, million, billion).[1]

The place value chart is fundamental to Topic A.  Building upon their previous knowledge of bundling, students learn that 10 hundreds can be composed into 1 thousand, and therefore, 30 hundreds can be composed into 3 thousands because a digit’s value is 10 times what it would be one place to its right (4.NBT.1).  Students learn to recognize that in a number such as 7,777, each 7 has a value that is 10 times the value of its neighbor to the immediate right.  One thousand can be decomposed into 10 hundreds; therefore 7 thousands can be decomposed into 70 hundreds.

Similarly, multiplying by 10 shifts digits one place to the left, and dividing by 10 shifts digits one place to the right.

3,000 = 10 × 300                 3,000 ÷ 10 = 300

In Topic B, students use place value as a basis for comparing whole numbers.  Although this is not a new concept, it becomes more complex as the numbers become larger.  For example, it becomes clear that 34,156 is 3 thousands greater than 31,156.

Comparison leads directly into rounding, where their skill with isolating units is applied and extended.  Rounding to the nearest ten and hundred was mastered with three-digit numbers in Grade 3.  Now, Grade 4 students moving into Topic C learn to round to any place value (4.NBT.3), initially using the vertical number line though ultimately moving away from the visual model altogether.  Topic C also includes word problems where students apply rounding to real life situations.

In Grade 4, students become fluent with the standard algorithms for addition and subtraction.  In Topics D and E, students focus on single like-unit calculations (ones with ones, thousands with thousands, etc.), at times requiring the composition of greater units when adding (10 hundreds are composed into 1 thousand) and decomposition into smaller units when subtracting (1 thousand is decomposed into 10 hundreds) (4.NBT.4).  Throughout these topics, students apply their algorithmic knowledge to solve word problems.  Students also use a variable to represent the unknown quantity.

The module culminates with multi-step word problems in Topic F (4.OA.3).  Tape diagrams are used throughout the topic to model additive compare problems like the one exemplified below.  These diagrams facilitate deeper comprehension and serve as a way to support the reasonableness of an answer.

A goat produces 5,212 gallons of milk a year.

A cow produces 17,279 gallons of milk a year.

How much more milk does a goat need to produce to make the same amount of milk as a cow?

17,279 – 5,212 = _____

A goat needs to produce _____ more gallons of milk a year.

The Mid-Module Assessment follows Topic C.  The End-of-Module Assessment follows Topic F.

____________________

[1]Grade 4 expectations in the NBT standards domain are limited to whole numbers less than or equal to 1,000,000.

Module 1

If pacing is a challenge, consider omitting Lesson 17 since multi-step problems are taught in Lesson 18.  Instead, embed problems from Lesson 17 into Module 2 or 3 as extensions.  Since multi-step problems are taught in Lesson 18, Lesson 19 could also be omitted.

Module 2

Although composed of just five lessons, Module 2 has great importance in the Grade 4 sequence of modules.  Module 2, along with Module 1, is paramount in setting the foundation for developing fluency with the manipulation of place value units, a skill upon which Module 3 greatly depends.  Teachers who have taught Module 2 prior to Module 3 have reportedly moved through Module 3 more efficiently than colleagues who have omitted it.  Module 2 also sets the foundation for work with fractions and mixed numbers in Module 5.  Therefore, it is not recommended to omit any lessons from Module 2.

To help with the pacing of Module 3’s Topic A, consider replacing the Convert Units fluencies in Module 2, Lessons 1­3, with area and perimeter fluencies.  Also, consider incorporating Problem 1 from Module 3, Lesson 1, into the fluency component of Module 2, Lessons 4 and 5.

Module 3

Within this module, if pacing is a challenge, consider the following omissions.  In Lesson 1, omit Problems 1 and 4 of the Concept Development.  Problem 1 could have been embedded into Module 2.  Problem 4 can be used for a center activity.  In Lesson 8, omit the drawing of models in Problems 2 and 4 of the Concept Development and in Problem 2 of the Problem Set.  Instead, have students think about and visualize what they would draw.  Omit Lesson 10 because the objective for Lesson 10 is the same as that for Lesson 9.  Omit Lesson 19, and instead, embed discussions of interpreting remainders into other division lessons.  Omit Lesson 21 because students solve division problems using the area model in Lesson 20.  Using the area model to solve division problems with remainders is not specified in the Progressions documents.  Omit Lesson 31, and instead, embed analysis of division situations throughout later lessons.  Omit Lesson 33, and embed into Lesson 30 the discussion of the connection between division using the area model and division using the algorithm.

Look ahead to the Pacing Suggestions for Module 4.  Consider partnering with the art teacher to teach Module 4’s Topic A simultaneously with Module 3.

Module 4

The placement of Module 4 in A Story of Units was determined based on the New York State Education Department Pre-Post Math Standards document, which placed 4.NF.5–7 outside the testing window and 4.MD.5 inside the testing window.  This is not in alignment with PARCC’s Content Emphases Clusters (http://www.parcconline.org/mcf/mathematics/content-emphases-cluster-0), which reverses those priorities, labeling 4.NF.5–7 as Major Clusters and 4.MD.5 as an Additional Cluster, the status of lowest priority.

Those from outside New York State may want to teach Module 4 after Module 6 and truncate the lessons using the Preparing a Lesson protocol (see the Module Overview, just before the Assessment Overview).  This would change the order of the modules to the following:  Modules 1, 2, 3, 5, 6, 4, and 7.

Those from New York State might apply the following suggestions and truncate Module 4’s lessons using the Preparing a Lesson protocol.  Topic A could be taught simultaneously with Module 3 during an art class.  Topics B and C could be taught directly following Module 3, prior to Module 5, since they offer excellent scaffolding for the fraction work of Module 5.  Topic D could be taught simultaneously with Module 5, 6, or 7 during an art class when students are served well with hands-on, rigorous experiences.

Keep in mind that Topics B and C of this module are foundational to Grade 7’s missing angle problems.

Module 5

For Module 5, consider the following modifications and omissions.  Study the objectives and the sequence of problems within Lessons 1, 2, and 3, and then consolidate the three lessons.  Omit Lesson 4.  Instead, in Lesson 5, embed the contrast of the decomposition of a fraction using the tape diagram versus using the area model.  Note that the area model’s cross hatches are used to transition to multiplying to generate equivalent fractions, add related fractions in Lessons 20 and 21, add decimals in Module 6, add/subtract all fractions in Grade 5’s Module 3, and multiply a fraction by a fraction in Grade 5’s Module 4.  Omit Lesson 29, and embed estimation within many problems throughout the module and curriculum.  Omit Lesson 40, and embed line plot problems in social studies or science.  Be aware, however, that there is a line plot question on the End-of-Module Assessment.

Module 6

In Module 6, students explore decimal numbers for the first time by means of the decimal numbers’ relationship to decimal fractions.  Module 6 builds directly from Module 5 and is foundational to students’ Grade 5 work with decimal operations.  Therefore, it is not recommended to omit any lessons from Module 6.

Module 7

Module 7 affords students the opportunity to use all that they have learned throughout Grade 4 as they first relate multiplication to the conversion of measurement units and then explore multiple strategies for solving measurement problems involving unit conversion.  Module 7 ends with practice of the major skills and concepts of the grade as well as the preparation of a take-home summer folder.  Therefore, it is not recommended to omit any lessons from Module 7.

Focus Standards for Mathematical Practice

MP.1          Make sense of problems and persevere in solving them.  Students use the place value chart to draw diagrams of the relationship between a digit’s value and what it would be one place to its right, for instance, by representing 3 thousands as 30 hundreds.  Students also use the place value chart to compare large numbers.

MP.2          Reason abstractly and quantitatively.  Students make sense of quantities and their relationships as they use both special strategies and the standard addition algorithm to add and subtract multi-digit numbers.  Students decontextualize when they represent problems symbolically and contextualize when they consider the value of the units used and understand the meaning of the quantities as they compute.

MP.3          Construct viable arguments and critique the reasoning of others.  Students construct arguments as they use the place value chart and model single- and multi-step problems.  Students also use the standard algorithm as a general strategy to add and subtract multi-digit numbers when a special strategy is not suitable.

MP.5          Use appropriate tools strategically.  Students decide on the appropriateness of using special strategies or the standard algorithm when adding and subtracting multi-digit numbers.

MP.6          Attend to precision.  Students use the place value chart to represent digits and their values as they compose and decompose base ten units.

#### Distribution of Minutes

• Fluency Practice
• Application Problem
• Concept Development
• Student Debrief
1
 13 5 35 7
2
 12 6 33 9
3
 15 6 32 7
4
 13 6 26 15
5
 14 6 30 10
6
 12 4 33 11
7
 15 6 27 12
8
 12 6 32 10
9
 12 8 30 10
10
 12 6 30 12
11
 12 7 30 11
12
 12 5 34 9
13
 12 5 35 8
14
 10 6 35 9
15
 11 6 32 11
16
 12 5 30 13
17
 10 8 35 7
18
 10 5 33 12
19
 12 5 30 13