GRADE 5 • UNIT 1

Place Value and Decimal Fractions

Table of Contents

Overview of Unit Topics and Lesson Objectives

Lessons

Topic 1: Multiplicative Patterns on the Place Value Chart 1-4

Lesson 1: Reason concretely and pictorially using place value understanding to relate adjacent base ten units from millions to thousandths.

Lesson 2: Reason abstractly using place value understanding to relate adjacent base ten units from millions to thousandths.

Lesson 3: Use exponents to name place value units and explain patterns in the placement of the decimal point.

Lesson 4: Use exponents to denote powers of 10 with application to metric conversions.

Topic 2: Decimal Fractions and Place Value Patterns 5-6

Lesson 5: Name decimal fractions in expanded, unit, and word forms by applying place value reasoning.

Lesson 6: Compare decimal fractions to the thousandths using like units and express comparisons with >, <, = .

Topic 3: Place Value and Rounding Decimal Fractions 7-8

Lesson 7–8: Round a given decimal to any place using place value understanding and the vertical number line.

Topic 4: Adding and Subtracting Decimals 9-10

Lesson 9: Add decimals using place value strategies and relate those strategies to a written method.

Lesson 10: Subtract decimals using place value strategies and relate those strategies to a written method.

Topic 5: Multiplying Decimals 11-12

Lesson 11: Multiply a decimal fraction by single-digit whole numbers, relate to a written method through application of the area model and place value understanding, and explain the reasoning used.

Lesson 12: Multiply a decimal fraction by single-digit whole numbers, including using estimation to confirm the placement of the decimal point.

Topic 6: Dividing Decimals 13-16

Lesson 13: Divide decimals by single-digit whole numbers involving easily identifiable multiples using place value understanding and relate to a written method.

Lesson 14: Divide decimals with a remainder using place value understanding and relate to a written method.

Lesson 15: Divide decimals using place value understanding including remainders in the smallest unit.

Lesson 16: Solve word problems using decimal operations.

RDW or Read, Draw, Write (a Number Sentence and a Statement)

You will often see in the directions for students to use the RDW strategy. This strategy is modeled and used daily in class with a word problem of the day.

1) Read. Carefully read the word problem once all the way through.

What do I see?

2) Draw and Label.Use the information given – you may need to read the problem again.

Can I draw something?

3) Write a number sentence (equation) and/or -Write a word sentence (statement).

What conclusions can I make from my drawing?

Unit 1: Vocabulary

Review Familiar Terms and Symbols1

> - greater than

< - less than

= - equal to

Base ten units - place value units

Centimeter - cm, a unit of measure equal to one hundredth of a meter

Digit - a numeral between 0 and 9

Expanded form – a number written with each digit expressing its place value (ex. 100 + 30 + 5 = 135)

Hundredths - as related to place value, the second place to the right of the decimal point

Number line - a line marked with numbers at evenly spaced intervals

Number sentence (ex. 4 + 3 = 7)

Place value - the numerical value that a digit has based on its position in a number

Product – the answer or total when two numbers (factors) are multiplied

Quotient – the answer when dividing one number (dividend) by another (divisor)

Standard form - a number written using only digits: 135

Sum – the answer or total when two numbers (addends) are added

Tenths - as related to place value, the first place to the right of the decimal point

Unit Form - place value counting,

(ex. 3.21 = 3 ones 2 tenths 1 hundredth or 32 tenths 1 hundredth or 321 hundredths)

Word form – a number written using only words (ex. one hundred thirty-five)

Bundling, making, renaming, changing, regrouping, trading

Unbundling, breaking, renaming, changing, regrouping, trading

Place Value Disks -

1 These are terms and symbols students have used or seen previously.

Unit form modeled with number disks: 7 hundreds 2 tens 6 ones = 72 tens 6 ones = 726 ones

New Vocabulary for 5th Grade Unit 1

Thousandths - related to place value, the third place to the right of the decimal point

Expanded form with decimals – a number written with each digit multiplied by its place value; decimals are

used to represent tenths (0.1), hundredths (0.01), and thousandths (0.001).

(ex. (1 x 100) + (3 x 10) + (5 x 1) + (2 x 0.1) + (6 x 0.01) + (7 x 0.001) = 135.267)

Expanded form with fractions– a number written with each digit multiplied by its place value; fractions are

used to represent tenths (1/10), hundredths (1/100), and thousandths (1/1000).

(ex. (1 x 100) + (3 x 10) + (5 x 1) + (2 x 1/10) + (6 x 1/100) + (7 x 1/1000) = 135.267)

Exponent - the number of times a number is to be used as a factor in a multiplication expression

(e.g. 102 = 10 x 10 = 100 the two is the exponent)

Millimeter - a metric unit of length equal to one thousandth of a meter

Equation - a statement that the values of two mathematical expressions are equal, has an equal sign =, also

called a number sentence (ex. 12 = 4 x 2 + 4)

Lesson by Lesson Suggestions Lessons 1 - 4: Using Multiplication & Division to Change the Value of Digits Using Place Value

Parent Tip: When multiplying or dividing by 10, 100 or 1,000 it is important to explain and understand that each digit either increases or decreases in value. This is why we start by using place value charts and eventually place value disks to help students visualize this concept. Students use place value disks and a place value chart to build the place value/base ten system from millions to thousandths. The language we are looking to teach and for students to use includes “10 times as large” and “1/10 as large” or 1/10th the value of”. This will build a solid understanding of the place value system including decimals.

For example:

The 3 in the answer is 10 times greater than the 3 in the original number in the top row.

The 4 in the answer is 1/10 as larger or 1/10 the value of the 4 in the original number in the top row. *It is important to not fall back on the “just add a zero” rule and here is why. 8.7 x 10 = 87, if we were to just “add a zero” students may answer 8.70 which has no change in value. If we teach students to first understand the change in the value of the numbers they eventually discover the “add a zero trick” but are able to correctly only apply this to whole numbers.

Lesson 4: Use place value to convert measurement units.

1,000 millimeters/milliliters = 1 meter/liter

100 centimeters = 1 meter

1,000 meters/liters/grams = 1 kilometer/kiloliter/kilogram

Lesson 5: See Vocabulary for explanation on the different ways to write numbers.

Standard Form: 2,174,506.983 Expanded Form: 2,000,000 + 100,000 + 70,000+ 4,000 + 500 + 6 + 0.9 + 0.08 + 0.003

w/ fractions: (2 x 1,000,000) + (1 x 100,000) + (7 x 10,000) + (4 x 1,000) + (5 x 100) + (6 x 1) + (9 x 1/10) + (8 x 1/100) + (3 x 1/1000)

w/ decimals: (2 x 1,000,000) + (1 x 100,000) + (7 x 10,000) + (4 x 1,000) + (5 x 100) + (6 x 1)

+ (9 x 0.1) + (8 x 0.01) + (3 x 0.001)

Word Form: two million, one hundred seventy-four thousand, five hundred six and nine

hundred eighty-three thousandths

Thousands

Kilo-

Hundreds

Hecto-

Tens

Deka-

(Units) Ones

Meter Liter Gram

Tenths

Deci-

Hundredths

Centi-

Thousandths

Milli -

2 1 7 4 5 0 6 9 8 3

Lessons 7-8: Rounding using vertical number lines. *see videos in reference section*

Round 2.345 to the nearest whole number.

(use the ones place)

- 2.345 is between 2 and 3 wholes

- 2.5 is the midpoint between 2 and 3

- 2.3 is below the midpoint of 2.5

- round down: the answer is 2

Round 2.345 to the nearest tenth.

- 2.345 is between 23 tenths (2.3) and 24 tenths (2.4)

- 2.35 is the midpoint between 2.3 and 2.4

- 2.34 is below the midpoint of 2.35

- round down: the answer is 2.3

Round 2.345 to the nearest hundredth.

- 2.345 is between 234 hundredths (2.34) and 235 hundredths (2.35)

- 2.345 is the midpoint between 2.34 and 2.35

- 2.345 is at the midpoint of 2.345

- round up: the answer is 2.35

Step 1: Identify the value of the place that you are rounding to:

(ie: 2.345 rounded to the nearest tenth – there are 23 tenths in 2.345)

Step 2: Label the bottom of the number line using the number from step 1.

Step 3: Add one more of the same unit and label the top of the number line.

Step 4: Find the midpoint the number in between the top and bottom

(usually matches the bottom number with a 5 in the next place value position.

Step 5: Put an x where the original number would go on the number line.

Answer:

- if the number is below the midpoint - round down (circle the bottom value),

- if the number is at the midpoint or above - round up (circle the top value).

3

2.5

2.3

2

24 tenths = 2.4

2.35 = 23.5 tenths

2.34 23 tenths = 2.3

X

235 hundredths = 2.35

2.345 = 234.5 tenths

234 hundredths = 2.34

X

X

Lessons 9-10: Adding and Subtracting Decimals Based on Place Value Students will be introduced to adding and subtracting decimals with a focus on place value using unit form.

For example: 7 tens + 8 tens = 15 tens = 150

7 tenths + 8 tenths = 15 tenths = 1.5.

The use of unit form provides the connection that allows students to use what they know about general methods for addition and subtraction with whole numbers to reason about decimal addition and subtraction.

Place value charts and disks are used to provide a bridge to relate place value understandings to a written method or standard algorithm.

It is important to point out that students should be focused on adding or subtracting digits of the same unit (place value), which will cause the decimal points to line up. However the focus is on the digits lining up and not “lining up the decimal point”.

Example 1: 2 tenths 5 thousandths + 6 hundredths

- draw the correct number of dots for each unit in its correct place/column.

- add up each column to record the answer.

- this example does not require regrouping

Example 2: 1.8 + 13 tenths

- notice there are 11 dots in the tenths

- when more than 10 dots are in a column they must be bundled/regrouped to create one more of the next greater

unit

Example 3: 4.083 – 1.29

- the same procedure is followed for subtraction, but student will borrow/unbundle/regroup to create additional units if needed to subtract.

Lessons 11-12: Multiplying Decimals Using Place Value Disks These lessons will connect Grade 4 work with multiplication and the standard algorithm by focusing on an intermediate step—reasoning about multiplying a decimal by a one-digit whole number. The area model, introduced first in Grade 3, is reintroduced.

The goal for 5th Grade is to acquire fluency with multi-digit multiplication. Mastery will not be met after these 2 lessons and is not expected – these lessons just help to bridge past-learning and work towards this goal.

Place value understanding of whole number multiplication, place value disks and use of an area model of the distributive property is used to help students build direct parallels between whole number products and the products of one-digit multipliers and decimals.

Students will also use an estimation-based strategy to confirm the reasonableness of the product once the decimal has been placed through place value reasoning.

Example: 43 hundredths x 2 (2 copies/rows of 43 hundredths)

The area model can be considered a graphic organizer. It organizes the partial products.

Example with regrouping: 423 thousandths x 4

(4 copies/rows of 423 thousandths)

Lessons 13-16: Dividing Decimals Using Place Value Disks

The last set of lessons in Module 1 has a similar exploration of division of decimal numbers by one-digit whole number divisors. Students solidify their skills with and understanding of the algorithm before moving on to long division involving two-digit divisors in Module 2.

Example 1: 0.8 ÷ 4 = 0.2 - Students will draw 8 disks in the tenths column,

then break the disks into 4 equal groups. 2 disks can be placed into each group. Using unit form: 8 tenths ÷ 4 = 2 tenths

Example 2: 6.72 ÷ 3

- Students will start by drawing the dividend (6.72) on the chart & make 3 rows below.

- Starting with the greatest place of value students will try to make 3 equal groups using each of the place value units. Students will regroup any leftover disks that cannot be shared equally. (see sample work below)

o 6 ones can be divided into 3 groups – placing 2 ones in each row on the bottom o 7 tenths can be divided by 3 but will have 1 tenth extra that will be regrouped into

hundredths o there are now 12 hundredths divided by 3 = 4 hundredths in each group

Recommended Resources Lesson 4: Converting metric measurement units. http://youtu.be/xxI5O-LihIY http://youtu.be/GElbzvu02Cs Lesson 5: Standard & Expanded Form http://youtu.be/nt2aov1t3Js Expanded Form using Fractions https://learnzillion.com/lessons/429-write-decimals-in-expanded-notation#quickcode-modal https://www.youtube.com/watch?v=TjkfpeqqI4M&feature=youtu.be

Lesson 7 & 8 - Rounding Using a Number line: http://commoncore.org/maps/math/video-gallery/vertical-number-line-rounding http://www.showme.com/sh/?h=nJiqWIa *(This example shows 5th grade expectations with decimals)

Lessons 9-15: Adding, Subtracting, Multiplying, & Dividing Decimals Based on Place Value

& Using Place Value Charts & Disks

http://vimeo.com/71596082

This video shows how to add, subtract, multiply & divide using place value disks. Here are the times in the video that each operation is shown: Add: 0:00 Subtract: 2:12 Multiply: 5:20 Divide: 7:00 Lesson 16: Word Problems & Tape Diagrams

http://vimeo.com/71593727

**This site has a video providing guidance for every homework page.** http://www.oakdale.k12.ca.us/ENY_Hmwk_Intro_Math

(Click on 5th Grade – Select the Module to match the unit – Select the lesson)