Singapore Math Approach Decimal Numbers

Please make a name tent

Some of the Tools used in Singapore Math Place Value Strips Place Value Disks Place Value Charts Number Bond Cards Part-Whole Cards Decimal Tiles Decimal Strips

The big thing to remember about decimals… It’s all about location, location, location.

Read the following number 34,156 456.7 456.259 Write the decimal numbers using both the common fraction and the decimal fraction form Elementary teachers – What would each number look like in expanded notation? Secondary teachers – What would each number look like in scientific notation?

Developing the place value concept All concepts begin with some concrete material such as base ten blocks.

Let’s Assume students have worked with base ten blocks. Concrete

Then we move onto drawing pictures of the base ten blocks (Pictorial / Representation) Concrete

Students then use words to describe the value. Concrete Word Twenty Three AND Two tens and Three

And Write the number in symbolic form (Abstract) Concrete The Number 23 Word Twenty Three AND Two tens and Three

Finally students should be able to go between any of the forms Concrete The Number 23 Word Twenty Three AND Two tens and Three

Developing the concept of a decimal What is 0.1? Point out that multiple shapes are used.

What is 0.5?

The first thing student have to do to begin a study on decimals is to accept that decimals are another way to write fractions.

Common Misconception to address The role of the decimal point is not to separate a number into two numbers. It is a warning that the single number is composed of a whole number part and a fractional part. The decimal point is a mark that lets a person know where the fractional part begins.

Scattered Grouped Exchange

How does that activity connect to the number line?

The number line helps to begin to prepare students to perform operations and compare decimals

Conceptual versus careless errors 63.5 We are trying to distinguish between the place, the name of the digit, and the value of the digit when in its particular place. What errors might you expect students to make when asked to name the digit in the tenths place or to give the value of the digit 6? The number in the tenths position is 5, not 0.5. The value of the digit 6 is 60, not 6 or 10. Would these errors be conceptual or careless? What might be appropriate pedagogical responses to such errors?

Last idea before we move on: Decimal equivalence to fractions The decimal form of 7/10 is easy, BUT give me an understanding how to get the decimal equivalence of ½ and 3/5.

Hundredths and Thousandths Bring in a meter stick to show this on a number line

Checking for an understanding of place value with Base Ten Blocks

100-Square

Number discs

Show these numbers using the number discs

Students need to understand that value also has a location. The number line is less concrete and begins the shift towards representation and abstract use. This also reinforces the importance of place value. To plot a decimal value we need to first locate the tenths, then the hundredths, etc.

This will help feed the addition algorithm Point out the use of the language one hundredths less than 52 hundredths

Here are six options for concrete materials for illustrating decimal fraction ideas. Paper strips Number discs Base-ten blocks Money 100-square grids Number line If you were to choose three for use in your classroom, which ones would you choose? Are there others that you use that are not in this list? On what basis did you make your choices?

Now What

Expanding on Conceptual Ideas from earlier studies aka: Making Connections

Hindu-Arabic numeration Additive property 75 = 70 + 5 Multiplicative property 75 = 7 x 10 + 5 x 1

What are some different ways to describe these numbers?

Place Value Charts

How many ten’s are there in 120? In the number 630 In the number 6.3 How many tens are there? How many ones are there? How many tenths are there? How many ones are there? How many tenths are there? How many thousandths are there?

Counting patterns These students have reverted to counting albeit in the tens place but counting by ones without thought to regrouping by tens. This carries over to decimals.

What should the student have written? What objective is being tested in each sequence? What error pattern do we observe? Did the student continue any of the sequences correctly? What does the student appear to know? What would you suggest as a follow-up? Do we need to return to the use of concrete materials?

Equivalence as a tool Before we can discuss ordering and comparing decimals and decimal fractions we need to establish the notion of decimal equivalences.

How do we get a decimal equivalence from a fraction?

The annexation of zero(s) If I annex a zero to the number 4 it becomes ten times bigger than the original. If I annex a zero to the number 0.4 it maintains its value even though it looks different 0.40. What’s the deal? What other concept has the student used in this situation? To provide a valid answer to the question, what should the student have said? How would you help the student re-evaluate the response?

Consider the numbers: ½ 0.5 2/4 0.50 Are these numbers equivalent? What other equivalent numbers could we add to the collection? Mary and Juan added the numbers 3/6 and 10/20 to the set. Are they correct? Here is a number but the denominator has been smudged. 50/. What is the denominator? Ali added 0.5000 to the set of numbers. Is he correct? How would that number be expressed as a fraction?

**** Building an Algorithm **** Two basic fractional equivalency to begin to build the algorithm. ½ = 5/10 and ¼ = 25/100 How do we know what 1/8 is in decimal form from previous work? What about connecting the first two to develop a pattern?

Ordering and comparing decimals Which is bigger 0.3 or 0.299? How did you approach this comparison? What reasoning would students use to correctly answer the question, but the reasoning be incorrect? How would reasoning that 300 is bigger than 299 be incorrect reasoning? How would you help a student struggling with comparing decimal values?

Compare and order the numbers 5.237 and 5.291

This method help over come the misguiding notion that bigger numbers are longer.

Staircases

Staircases

Activity: Number Card Line Which is largest / smallest? How do you know? Which card is 0.01 less than 0.88? Is one card exactly 0.01 more than another? How could you use the idea of number cards in your class?

Clothesline

Rounding off decimals to the nearest whole number Application of what it means to be nearer. Conversation about place value and its influence on rounding based on the number line argument.

Using the number line to teach Rounding to other decimal places.

Round the number 6. 48 to the nearest whole number Round the number 6.48 to the nearest whole number. How would you help susie?

Key Benchmark - Closeness

Key Benchmark – Between-ness Name a fraction between 2/7 and 4/7. Name a fraction between 1/3 and ½. Name two different (non-equivalent) fractions between 1/3 and ½. How many fractions are between 1/3 and ½? Name a decimal fraction between 0.3 and 0.5. Name a decimal fraction between 0.3 and 0.4. Name two decimal fractions between 0.3 and 0.4. How many decimal fractions are between 0.3 and 0.4?

What time is it?

Addition of decimals Students who have a good understanding of addition by place value have no problem learning to add decimals.

Add 5.365 to 42.6

Misconceptions The importance of rounding and estimating

Add 4.1 and 5.68 4.1 + 5.68 6.09 0.609 The importance of the annexation of zero

Subtraction

Design a task Plan a lesson for the example “subtract 0.4 from 5”. What materials will you use? How will you address the missing decimal point? When would you expect students to annex the zero? What is the sequencing in your steps to provide an exemplary sequencing of the steps?

Multiplication Multiplication always begins as repeated addition.

Bar model and area tools What is the product of 0.5 and 0.25?

Using the 100-grid square The square can be used to show the product of 0.2 x 0.5. I can also use it to show 2 x .32.

The sequencing of multiplying decimals Decimal Number Discs Number Line Area Model What tool could be use for each?

Area Models to Arrays Concrete model: Base-ten blocks Pictorial: Area Array

Multi-digit multiplication Step 1: Multiplying by 10, 100, 1000

Multi-digit multiplication Step 2: Multiplying by Multiples of 10, 100, 1000

Multi-digit multiplication Step 3: Multiplying by any two-digit number

Where’s the Point?

Word Problems

What types of problems are represented by division? Equal Sharing – How many in a group? Measurement – How many groups? Missing factor

Division of decimals as sharing

Divide 0.35 by 7

3 methods for dividing 2.4 by 40

Thinking about misconceptions

Division as repeated Subtraction

A visual model of this strategy. 0.5 ÷ 6 The question to be asked is, “In 6, how many 0.5s are there?” or “How many times can we subtract 0.5 from 6?”

Types of real world problems that this could help students think about.

What about dividing by a decimal? Divide 3.2 by 6.5 Divide 6.12 by 0.4

Does multiplication always make a number bigger Does multiplication always make a number bigger? Does division always make a number smaller?

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