Nys Common Core Mathematics Curriculum Answers Grade 4 Module 6

GRADE

New York State Common Core

Mathematics Curriculum

GRADE 4

• MODULE

Module 6:

Decimal Fractions

Date:

1/28/14

commoncore.org

Table of Contents

GRADE 4 • MODULE 6

Decimal Fractions

Module Overview

......................................................................................................... i Topic A: Exploration of Tenths .............................................................................. 6.A.1 Topic B: Tenths and Hundredths ............................................................................ 6.B.1 Topic C: Decimal Comparison ................................................................................. 6.C.1 Topic D: Addition with Tenths and Hundredths ..................................................... 6.D.1 Topic E: Money Amounts as Decimal Numbers...................................................... 6.E.1

Module Assessments

............................................................................................. 6.S.1

Lesson

Module 6:

Decimal Fractions

Date:

1/28/14

commoncore.org

New York State Common Core

Module Overview

NYS COMMON CORE MATHEMATICS CURRICULUM

Grade 4

•

Module 6

Decimal Fractions

OVERVIEW

This 20-day module gives students their first opportunity to explore decimal numbers via their relationship to decimal fractions, expressing a given quantity in both fraction and decimal forms. Utilizing the understanding of fractions developed throughout Module 5, students apply the same reasoning to decimal numbers, building a solid foundation for Grade 5 work with decimal operations. Previously referred to as whole numbers, all numbers written in the base ten number system with place value units that are powers of 10 are henceforth referred to as decimal numbers, a set which now includes tenths and hundredths, e.g. 1, 15, 248, 0.3, 3.02, and 24.345. In Topic A, students use their understanding of fractions to explore tenths. At the opening of the topic, they use metric measurement to see tenths in relationship to different whole units: centimeters, meters, kilograms, and liters. Students explore, creating and identifying tenths of various wholes, as they draw lines of specified length, identify the weight of objects, and read the level of liquid measurements. Students connect these concrete experiences pictorially as tenths are represented on the number line and with tape diagrams as pictured to the right. Students express tenths as decimal fractions and are introduced to decimal notation. They write statements of equivalence in unit, fraction, and decimal forms, e.g., 3 tenths =

 

= 0.3 (

4.NF.6

Next, students return to the use of metric measurement to investigate decimal fractions greater than 1. Using a centimeter ruler, they draw lines that measure, for example,



 



 

centimeters. Using the area model, students see that numbers containing a whole number and fractional part, i.e., mixed numbers, can also be expressed using decimal notation provided that the fractional part can be converted to a decimal number (

4.NF.6

). Students use place value disks to represent the value of each digit in a decimal number. Just as they wrote whole numbers in expanded form using multiplication, students write the value of a decimal number in expanded form using fractions and decimals, e.g., 2 ones 4 tenths =



 

= (2



1) + (4



 



and 2.4 = (2



1) + (4



0.1). Additionally, students plot decimal numbers on the number line. Students decompose tenths into 10 equal parts to create hundredths in Topic B. Through the decomposition of a meter, students identify 1 centimeter as 1 hundredth of a meter. As they count up by hundredths, they realize the equivalence of 10 hundredths and 1 tenth and go on to represent them as both decimal fractions

and as decimal numbers (

4.NF.5

). Students use area models, tape diagrams, and number disks on a place value chart to see and model the equivalence of numbers involving units of tenths and hundredths. They express the value of the number in both decimal and fraction expanded forms.

Lesson

New York State Common Core

Module Overview

NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6:

Decimal Fractions

Date:

1/28/14

iii

commoncore.org

Close work with the place value chart helps students see that place value units are not symmetric about the decimal point

—

a common misconception that often leads students to

mistakenly believe there is a "oneths" place.

They explore the placement of decimal numbers to hundredths and recognize that the place value chart is symmetric about the ones column. This understanding helps students recognize that, even as we move to the units on the right side of the decimal on the place value chart, a column continues to represent a unit 10 times as large as that of the column to its right. This understanding builds on the place value work done in Module 1 and enables students to understand that 3.2, for example, might be modeled as 3 ones 2 tenths, 32 tenths, or 320 hundredths. Topic B concludes with students using their knowledge of fraction equivalence to work with decimal numbers expressed in unit form, fraction form, and decimal form (

4.NF.6

). The focus of Topic C is comparison of decimal numbers (

4.NF.7

). To begin, students work with concrete representations of measurements. They see measurement of length on meter sticks, of mass using a scale, and of volume using graduated cylinders. In each case, students record the measurements on a place value chart and then compare them. They use their understanding of metric measurement and decimals to answer questions such as,

"Which is greater? Less?

Which is longer? Shorter? Which is heavier? Lighter

Comparing the decimals in the context of measurement supports students' justification of

their comparisons and grounds their reasoning, while at the same time setting them up for work with decimal comparison at a more concrete level. Next, students use area models and number lines to compare decimal numbers and use the <, >, and = symbols to record their comparisons. All of their work with comparisons at the pictorial level helps to eradicate the common misconception that is often made when students assume a greater number of hundredths must be greater than a lesser number of tenths. For example, when comparing 7 tenths and 27 hundredths, students recognize that 7 tenths is greater than 27 hundredths because, in any comparison, one must consider the

size of the units

. Students go on to arrange mixed groups of decimal fractions in unit, fraction, and decimal forms in order from greatest to least or least to greatest. They use their understanding of different ways of expressing equivalent values in order to arrange a set of decimal fractions as pictured below.