Decimals are fractions. 0.60 means 60/100ths which is also 6 tenths 6/tenths Compare Compare Compare

You do not drop or ignore zeroes... You explain the equivalence. 6/10 + 4 hundredths is the same as 60 hundredths + 4 hundredths 0.6 + 0.04 is the same as 0.60+0.04

Imagery for Decimals

Decimals are fractions.

If decimals are fractions, you can fold one piece of paper to see them.

Decimals are always related to dividing by ten. So start with a piece of paper and fold it into ten parts or pieces to see tenths.

Decimals are fractions. Express decimals with fraction vocabulary. Compare them to one unit from the hundred grid.

This is a one.

It came from here.

Divide 100 into 10 equal parts.

Fold into 10 equal parts. 100 ÷ 10 = 10.

Divide 10 into 10 equal parts by folding

You have one. 1

Now divide 1 into 10 equal parts by folding….. 1 ÷ 10 = ?

Can you see it?

Let’s magnify it so you can see.

Fold into 10 equal parts 1 ÷ 10 =

1 ÷ 10 = one tenth 𝟏 𝟏𝟎 You have one tenth of the paper showing

Open it up and see there are ten tenths in one. 1 10

In decimal fractions you write 1.0 In fractions you write 𝟏𝟎 𝟏𝟎 In decimal fractions you write 1.0 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

Each of the pieces is one tenth of the whole one. You can write that as 𝟏 𝟏𝟎 or 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

1 ÷ 10 = 𝟏 𝟏𝟎 1 ÷ 10 = 0.1 𝟏 𝟏𝟎 = 0.1 Show your partner how to explain each of these using the model.

𝟑 𝟏𝟎 0.7 𝟒 𝟏𝟎 = 0.4 Show your partner how to explain each of these using the model.

How much of the 1 (1.0) is brown?

𝟑 𝟏𝟎 = 0.3 0.1 0.1 0.1

How much is white?

7 10 or 0.7 Explain….

7 10 + 3 10 = 10 10 0.7 + 0.3 = 1.0 Can you explain how these equations are true?

10 10 - 7 10 = 3 10 1.0 - 0.7= 0.3 Can you explain how these equations are true?

Explain this? What’s brown? What’s white?

5 10 = 0.5 3 10 + 2 10 = 5 10 0.3 + 0.2 = 0.5

5 10 + 5 10 = 10 10 10 10 = 1 0.5 + 0.5 = 1.0

How about this?

When I am working in decimals I remember that 1 is the same as ten tenths by writing it with a decimal point 10 10 = 1.0 1.0 = 1

What if we divide tenths into ten equal pieces?

0.1 ÷ 10 = ? 1 10 ÷ 10 = ? Read these two equations out loud.

This could be one tenth divided into 10 equal parts.

If you divide all the tenths into 10 parts, how many parts have you created?

Each of those parts is one hundredth of what?

𝟏 𝟏𝟎𝟎 = 0.01 Where is 𝟏 𝟏𝟎𝟎 or 0.01?

𝟏 𝟏𝟎𝟎 = 0.01 Where is 𝟓 𝟏𝟎𝟎 or 0.05?

𝟏 𝟏𝟎𝟎 = 0.01 Where is 𝟓𝟎 𝟏𝟎𝟎 or 0.50?

This is another way to see the one divided into hundredths. Now where is 𝟏 𝟏𝟎𝟎 or 0.01?

This is another way to see the one divided into hundredths. Now where is 𝟓 𝟏𝟎𝟎 or 0.05?

This is another way to see the one divided into hundredths. Now where is 𝟓𝟎 𝟏𝟎𝟎 or 0.50?

Compare hundredths to tenths. 0.1 is not 0.01 1 10 is not 1 100

How many hundredths are in one tenth? Prove to your partner.

How many hundredths are in 3 tenths? Prove to your partner.

One tenth has ten hundredths in it so we can write these equalities 0.1 = 0.10 𝟏 𝟏𝟎 = 𝟏𝟎 𝟏𝟎𝟎

Three tenths have 30 hundredths in them. 0.3 = 0.30 𝟑 𝟏𝟎 = 𝟑𝟎 𝟏𝟎𝟎

How many tenths? How many hundredths?

𝟔 𝟏𝟎 = 𝟔𝟎 𝟏𝟎𝟎 0.6 = 0.60

𝟔 𝟏𝟎 = 𝟔𝟎 𝟏𝟎𝟎 0.6 = 0.60

How many tenths? How many hundredths?

𝟓 𝟏𝟎 + 𝟑 𝟏𝟎𝟎 5 tenths, 3 hundredths

𝟓𝟑 𝟏𝟎𝟎 5 tenths, 3 hundredths

Can you see the tenths in your mind? The hundredths in your mind?

Can you imagine in thousandths?

1/100 is inside 1/10 is inside 1

Where would you put the decimal point?

Where would you put 62 hundredths? Explain.

What decimal fraction describes the brown? Write it in the folder.

Does it have any hundredths in it? How do we show that?

Does it have any thousandths in it? How do we show that?

What decimal fraction describes the brown? Write it in the folder.

Does it have any hundredths in it? How do we show that?

Does it have any thousandths in it? How do we show that?

What decimal fraction describes the brown? Write it in the folder.

Does it have any hundredths in it? How do we show that?

Does it have any thousandths in it? How do we show that?

What decimal fraction describes the brown? Write it in the folder.

Does it have any hundredths in it? How do we show that?

Does it have any thousandths in it? How do we show that?

What decimal fraction describes the brown? Write it in the folder.

What decimal fraction is this Write it in the folder.

What decimal fraction is this Write it in the folder.

As a number 0.1 or 1 10 has a position on the numberline. You have to fit ten in the space between 0 and 1 0 1 10

I enlarged the numberline to see .1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9 0 1 2 3 I enlarged the numberline to see 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

So where are the hundredths? .1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9 0 1 2 3 So where are the hundredths?

Imagine showing all the hundredths in between zero and one? 0.01 hundredths 0.1 tenths 0 1 2 3 Imagine showing all the hundredths in between zero and one?

Can you understand this number line? Head this way to 1 .1 .2 .3 .01 .02 .03 .04 .05 .06 .07 .08 .09 Can you understand this number line?

0 1 2 3 Where is 0.7? Where is 0.07? Where is 2.7? Where is 2.07?

Where is 0.7? Where is 0.07? Where is 2.7? Where is 2.07? 0 1 2 3 0.07 0.7 2.07 2.7 0 1 2 3 Where is 0.7? Where is 0.07? Where is 2.7? Where is 2.07?

Students need to practice designing numberlines that they can put decimals on. Stretchable items like bungee or elastic might help.