Introducing Decimals

There’s a place for every number!

Grades 3 – 5

Module 31

Welcome & Overview

•Welcome & Introductions • Participants will engage in an

ice breaker activity (e.g., Jeopardy Game)• Review objectives, and ground

rules.

NGSSS Benchmarks

• MA.4.A.2.1 Use decimals through the thousandths place to name numbers between whole numbers.• MA.4.A.2.2 Describe decimals as an

extension of the base-ten number system.• MA.4.A.2.3 Relate equivalent fractions and

decimals with and without models, including locations on a number line.

Show what you know!

• Participants will be given a 10 item review exercise on fractions, decimals comparison, estimation and place value. • This exercise will be used to assess

participants’ understanding of the teaching and learning of decimals.

Our Numeration System

Question:Although we say, “Our” numeration system, is it really ‘ours’?–Have you ever thought where we got our

number system from?Let’s find out about this Numeration System!

Participants will research and present information about the origin of ‘Our Numeration System’ and its characteristics. (Internet Access)

Base 10/Decimal System

DecimalSystem

Additive Property

Place value

Base of 10

Use of Zero

Using Technology

NLVM: Use Blocks to represent a number in different basesClick on the following link and follow the instructions.

http://nlvm.usu.edu/en/nav/frames_asid_152_g_2_t_1.html?from=topic_t_1.html

1. Begin working with base 10. 2. Now try working with 2 other bases. Discuss results!3. Don’t go to the decimals; we’ll return to this later.

Extra: FunBrain - Place value Puzzler http://www.funbrain.com/cgi-bin/tens.cgi?A1=s&A2=0

Place Value Models

• Base 10 BlocksRepresenting numbers with base ten blocks

http://www.ulm.edu/~esmith/250/31/repbase10.html

• Place Value Mat

• Number Strips 1 + 2 tenshundred + 5 ones

1 + 5+ 2

Activity 2: Fractions & Place Value

Brainstorming ideas that may be used to encourage students to develop communication skills and express understanding of fractions and place value.

Questions: How can you relate fractions, mixed numbers and

tenth/hundredths? How can you model tenths and hundredths?How can you find the value of a digit using its place-

value position?

Activity 3: Introducing decimals

• Where does the word “decimal” come from?• How many different kinds of decimal units you can

form out of a whole. • Counting numbers smaller than one - why you need

a decimal point, and where it goes. • Find out six ways you can use decimals in everyday

life, as well as two reasons that decimals can make math easier.

Decimals: The decimal point

• The zero and the counting numbers (1,2,3,...) make up the set of whole numbers. But not every number is a whole number.

• Our decimal system lets us write numbers of all types and sizes, using a clever symbol called the decimal point.

• Click on the link below and move your mouse cursor over the decimal number to see its parts.

• http://www.math.com/school/subject1/lessons/S1U1L2GL.html

Visualizing and developing understanding of decimals

• Download and view the video clip “What is a decimal?

• http://www.onlinemathlearning.com/decimal.html

Question:

How best can we help students develop understanding of the concept of decimals?

Activity 4Developing vocabulary power

Purpose: Practice hearing and writing decimals• Introduce the words for the session • Divide participants into 4 groups for instruction.• Each group works on a strategy for developing the

mathematical language: » (i) make a word chart; »(ii) Vocabulary flash cards; »(iii) Tree Diagram; »(iv) Semantic mapping/ a Number Line.

Strat

egie

s

!

!! Comparing Decimals !!

Which is more >, or less <, or equal

=?How would they

prove this?

Dar

e to

com

pare

!Tens Ones Tenths Hundredths

Activity 5: Comparing decimals

Too big or too small: Estimating Fractions Between 0 and 1

• Provide a model for participants to make that can be used to help them evaluate students' understanding of fractions between 0 and 1.

• Begin by making estimates using familiar benchmarks (e.g., ½, ¼, ¾; 0.5, 0.25, 0.75).

• Provide opportunity for participants to suggest alternative strategies.

Str

ate

gi

es!

Strategy: Act it OutAlligator Jam Game

This activity will require three students each time the game is played.• Make the connection that the greater than or less than symbols resemble an

alligator’s mouth.

• The alligator is always eating the bigger number!!!! Therefore the

• open part of the symbol will face the bigger number.

• Have two students holding number cards and one student being

• the < or > symbol holding the alligator symbol. The one holding the

alligator has the job of eating the bigger number by opening the

mouth towards the bigger number. After the alligator has made the

decision the audience will decide if the action is right.

• Provide enough opportunities and number cards for each student to have a turn.

Is Bigger Better?Directions: For three players.

Mix up the cards. Place them face down on your desk. Each player draws one card. The player with the smallest decimal tells how he/she knows his/her decimal is the smallest. The smallest decimal wins the round and keeps all three cards. Draw cards until there are no cards left. Each player counts his or her total cards. The player with the most cards wins.

Activity 6Equivalent fractions and decimals

• Participants will use the number line to model equivalent fractions and decimals.

• They will suggest and use other models that allow students to name decimals and place value (NLVM). This online activity provides a concrete way to introduce equivalent forms of fractions and decimals.

• Participants will construct models to represent a fraction or a decimal.

• Focus on ordering decimals from least to greatest.

What those decimals mean!

Purpose: to examine the problems and judge by observation where the error occurred in the decimal placement of the answers.

Given a set of decimals, participants will identify common errors with decimals.

In groups of 3, participants will identify the

error pattern(s), design an activity to remediate the identified problem(s), and present their findings and activity.

Student work Sample

Error Patterns

Strengths

Review & Evaluation

• Review content: Communication/Letter writingParticipants write a letter to a friend

describing an aspect of the lesson that was significant to them.

• Participants will complete an Evaluation survey

Thank you for participating in this session

Remember to submit all suggestions.

Module 32

The extended …Decimal family!

Module Objectives

Participants will:

• Demonstrate understanding of the connection between fractions, and decimals by using prior knowledge of dollars and cents.

• Identify effective strategies for teaching equivalent fractions, writing and decimals/fractions/percent comparisons.

• Examine common errors/difficulties encountered by students when adding and subtracting decimals.

• Identify teacher-specific instructional tools and methods for teaching decimal/fraction/percents comparisons (e.g., models, place value or properties)

• Explore web-based educational resources designed to reinforce learning of writing and relating decimals to common fractions and percents.

Review Benchmarks

• MA.4.A.2.1 Use decimals through the thousandths place to name numbers between whole numbers.

• MA.4.A.2.3 Relate equivalent fractions and decimals with and without models, including locations on a number line.

• MA.4.A.2.4 Compare and order decimals, and estimate fraction and decimal amounts in real-world problems.

Developing Conceptual Understanding of Decimal Numbers

Discuss: How can we help students develop a robust understanding of concepts and symbols?

Provide opportunities for students to:Communicate its meaning, its use, the results of its

application, and its implications for a given contextMake conjectures and justify conclusionsRepresent decimals in a variety of waysConnect decimals to other ideas in and outside mathematicsKnow when and how to apply it to solve problems in

mathematics and other contexts

Bills! Bills! Bills!

Fractions and Decimals in real-life situations

One strategy that may be used to emphasize the importance of decimals is the use of decimals in monetary amounts.

Making ConnectionsThis connection must be made before introducing hundredths: Model& symbol; symbol & word; word model

Model

Symbol (0.3 or 3/10) Word (Three tenths)

Representing Decimal Numbers

• Use physical objects like base-10 blocks to represent decimals

• Draw diagrams to represent decimals• Connect representations to word form and

symbolic form

Connecting Fractions, Decimals and Percents

We are all one!

1/10 75%

1.25 ¾ 0.25 1%

25/100 0.1 75/100

0.75 25% 125/100 12.5%

Percents

To build conceptual understanding:• Connect percents to decimals and fractions,

especially with representations.• Connect to meanings that students have.• Use referents that make sense to students,

like money and test scores.

What should students knowabout decimals and percents?

• Understand what the numbers mean• Understand equivalence and relate decimals

to fractions to percents• Compare decimals and decide the relative

size of decimals• Compute with decimals and percents• Solve problems involving decimals and

percents

Percent Designs

Participants create artful designs that are all one color or another.

• Participants predict the percent of the picture covered by each color, and then they discuss their predictions.

• Participants use grids to figure out the percentages.

Comparing and Ordering Decimals

Skill with comparing and ordering is tied to understanding ideas of equivalence, and to being able to imagine representations for decimals.

Comparison Activities/Strategies

• Activity 2. Fraction Game: less than or equal This applet allows students to individually practice working with relationships among fractions and ways of combining fractions. http://illuminations.nctm.org/ActivityDetail.aspx?ID=18

Activity 3 Fractional Clothes line

• http://illuminations.nctm.org/LessonDetail.aspx?ID=L784

• This classroom number line will be used to show that all proper fractions are grouped between 0 and 1, and that improper fractions or mixed numbers are all grouped above 1.

Convert FromDecimal To a Fraction

http://webmath.com/dec2fract.html• Participants clink on the above link to access

the Decimal converter. The page shows how to convert a decimal into its equivalent fraction.

Decimals and Percent Videos

• http://www.youtube.com/watch?v=RvtdJnYFNhc&feature=player_embedded

• Fractions to decimalshttp://www.youtube.com/watch?v=Gn2pdkvdbGQ&feature=channel• Factions to Decimals to Percents

http://www.youtube.com/watch?v=VlqhVldVces&feature=related or http://www.youtube.com/watch?v=VlqhVldVces

Activity 4 Selecting a Problem Solving Strategy

Estimating

• Participants will be given a problem that requires them to use an appropriate strategy to estimate the total cost of several items. They will then justify their answers.

Problem Solving Strategy

Draw a diagram• Participants use a problem solving strategy to

locate decimals, mixed numbers, and fractions on a number line, and convert decimals to fractions.

Problem Solving Strategy

Summarizing or restating

Participants will be given a real-life problem that requires them to connect math and reading.

Participants solve the problem by summarizing or restating the information given.

Participants provide alternative strategies.

Problem Solving Strategy(RQSCAL)

Using Prompts (RQSCAL)

1. Read the question. Read

2. What does it mean? Question

3. What will I do to solve it? Strategies

4. Do the maths. Calculate

5. Write the answer. Answer

6. Look back at the original Problem Look

Activity 6Exploring interesting patterns

Using a calculator

Group 1:Participants will be provided with a calculator and the family of ninths, and ask to find the decimal equivalents (1/9, 2/9, 3/9, 4/9 …).Group 2:Participants will use paper and pencil to calculate decimal equivalents for ninths.

Groups discuss the rationale for using each method.

Similarly have participants look at the meaning of 1/3 to help them reinforce the idea that 1/3 is not equal to 0.33, 0.333, or 0.333 (i.e., 1/3 + 1/3 + 1/3 = 3/3 or the whole)

Games Fractions, Decimals and Percents

Purpose:To reinforce and practice calculating equivalent fractions and learning the underlying concepts.

PoemRounding

• Find your number

• Look right next door

• 4 or less, just ignore

• 5 or more add one more!

Example: Round 123.649 to the nearest 100th

123.649 (Find your number)

Look right next door

5 or more add 1 more ( 4 + 1= 5)

Response: 123.649 to the nearest 100th is 123.67

Conceptual Knowledge

It’s Not!Learning decimal place value names as verbal labels does not qualify as conceptual knowledge.

It is!If place value names are related to common fractions, diagrams, and objects of appropriate size, the knowledge is conceptual.

Module 33

Computation:Adding

and Subtracting Decimals!

Module Objectives

• Discuss appropriate benchmarks related to decimals.• Examine common error patterns.• Explore instructional strategies to help students understand decimals

and percents.• Identify effective strategies for teaching adding and subtracting

decimals.• Identify teacher-specific instructional tools and methods for teaching

addition and subtraction of decimals, and addition and subtraction of fractions with like and unlike denominators (e.g., models, place value or properties)

• Explore web-based educational resources designed to reinforce learning of estimating and calculating decimals in tenths, hundredths, and thousandths and relate them to common fractions.

Module 33 Benchmarks

• MA.5.A.2.1: Represent addition and subtraction of decimals and fractions with like and unlike denominators using models, place value, or properties.

• MA.5.A.2.2: Add and subtract fractions and decimals fluently, and verify the reasonableness of results, including in problem situations

• MA.5.A.2.3: Make reasonable estimates of fraction and decimal sums and differences, and use techniques for rounding

Review pre-requisite skills

Brainstorming• Participants will discuss why developing

competency in adding and subtracting whole numbers must be taught before teaching decimals.

• Instructor reviews addition of whole numbers, with and without regrouping …

• Participants will define the words regroup and ungroup by using each word in the context and by matching visuals to the definition. (E.g., regrouping means the same as ‘trading in’)

Addition of Fraction Error Pattern

Activity:Distribute activity sheet. Participants may work in small groups/with a partner.Participants identify student error patterns, and uses the pattern to compute two problems involving addition of fractions with unlike denominators.Participants discuss the procedure/error and describe 2 instructional activities that may be used to help the student add unlike fractions correctly.

Developing Math Language

Activity 2. Using a Tree Map to review the vocabulary Vocabulary for this section: fractions, decimals, comparing, finding equivalents, estimate, adding, subtracting, budget, shopping, sums, differences

Activity 3: Word Search/Word Puzzle The instructor will encourage participants to integrate Language Arts and Math by developing a word search or word puzzle activity to reinforce student grasp of the vocabulary.

Error & Remediation StrategyAddition and Subtraction

Activity 4: Difficulty with presentation/format of decimal

• Present an example of a problem/difficulty student encounter when adding or subtracting decimals Yesterday, John ran 5.12 meters, then drove 49.635 meters. What distance did he cover altogether? (e.g., 5. 12 + 49. 635).

Solution: 100.835

Participants identify the problem and suggest remediation strategies

Activity 5Problem Solving Strategy

Make a Table

Question: How can making a table help you organize and keep a track of your bank account balance?

Participants will respond to this question by providing examples on the whiteboard to support their claims.

Role Playing

• Examine ways in which students may actively participate in developing a better understanding of the concepts.

1. Given a menu from a local restaurant and a specified amount of spending money, they will select items from a menu and consider whether they want o spend all of the money, some, and the amount of change they should receive.OR2. Role play buying and making change. Participants will be divided into two groups (customers and staff). The customer will select the items and place their orders. The staff takes the order and presents the customer with a bill. The customer then checks the bill before paying.

• Discusses the potential benefits of using these activities in the classroom.

Activity 6Manipulative materials

(Base 10 blocks, 10s and 100s grids)

• Instructor explains the importance of having students work with manipulative materials to develop their understanding of addition and subtraction of decimals

• Participants will explore the use of Base 10 blocks. Encourage the use of both models: Online and Hands-on manipulative Base 10 blocks.

• http://nlvm.usu.edu/en/nav/frames_asid_264_g_2_t_1.html?from=grade_g_2.html

NLVM: Add and subtract decimal values using base blocks.

Activity 7Problem of the day: Track Event

Khandi is training to be a long distant runner. She runs once every day. On the first day, Khandi ran a distance of approximately 5 .94 miles. For the next two days she ran an average of 6. 1 miles. Today she ran for the fourth time. Her distance was 6.354 miles. What is the total distance Khandi ran over the four days?

Participants explain what being a long distance runner might entail.

Participants describe how they will assist students in unlocking the problem in order to solve it.

• Discuss with participants how they can use the commutative and associative properties of addition, and the importance of aligning place values when adding or subtracting decimals.

• Participants select a method to add or subtract decimal problems, and discuss the reasonableness of their selection.

Activity 8Adding and subtracting decimals to thousandths

• The instructor will model presenting problems in both formats: vertical and horizontal. Vertical format should be emphasized to ensure proper placement of the decimal point.

• Participants will be presented with a student’s work sample on addition and subtraction of decimals. They will identify the error pattern and describe how they will help students overcome such problems. They will include the use of manipulative materials/technology in the remediation plan.

Activity 9 Selecting a Method

• Distribute the activity sheet ‘Selecting a method’.

• Participants select a method for each problem and explain the reasonableness of their selection. (E.g., Mental math, paper and pencil, and calculators)

• Estimate to check reasonableness of• answer when using a calculator.

Wrapping it Up!

• Review content Individuals present highlights of the session

• Participants complete 10 Item Diagnostic Test and Evaluation survey.

Thank you for being such a great Team!!