Introducing Tenths: Relating $1/10$ Fractions to Decimal Notation $(0.1$)

Hello mathematicians! 👋 Today, we're going on an exciting journey to explore tenths – a special way to talk about parts of a whole. We'll connect what you already know about fractions to a new way of writing numbers called decimal notation. Get ready to visualize, understand, and master the concept of $1/10$ and its decimal friend, $0.1$! 🚀

Fraction Foundation: Reviewing and Modeling $1/10$

Let's start by remembering what a fraction represents: a part of a whole that has been divided into equal parts. Think of sharing a pizza or a chocolate bar equally! 🍕

Imagine we have a whole unit, like a long, rectangular strip. If we divide this whole strip into 10 equal columns or parts, what fraction would represent just one of those parts? 🤔

You got it! It's $\mathbf{1/10}$.

Let's quickly review the parts of a fraction:

• The numerator [the top number] tells us how many parts we have or are talking about.
• The denominator [the bottom number] tells us how many total equal parts make up the whole.

So, in $1/10$, the $1$ means we have one part, and the $10$ means the whole was divided into ten equal parts.

Look at the image below [imagine a tenths grid here, or instruct students to draw one]. If you color in 1 part of a model that's divided into 10 equal sections, you are visually representing $1/10$.

Let's practice writing and modeling a few more tenths fractions to make sure our foundation is super strong:

• If you color in 3 parts of the 10-part model, you are showing $3/10$.
• If you color in 7 parts of the 10-part model, you are showing $7/10$.

The denominator stays $10$ because the whole is still divided into 10 equal parts. Only the numerator changes to show how many of those parts we have! ✅

Transitioning to Place Value: Introducing the Tenths Place

Now, let's connect our visual fraction understanding to our number system. We're used to seeing place value charts with Ones, Tens, Hundreds, and so on for whole numbers.

Imagine a standard place value chart:

Hundreds | Tens | Ones

But what if we want to talk about parts of a whole, like our fractions $1/10$? Our current chart only shows whole numbers!

To represent these parts, we need to extend our place value system. We do this by introducing a very important symbol: the decimal point. ⚫

The decimal point acts as a separator. It sits directly to the right of the Ones place and clearly divides the whole numbers from the parts [fractions].

To the immediate right of the decimal point, we introduce a new place value column: the Tenths place.

Our extended place value chart now looks like this:

Hundreds | Tens | Ones . Tenths

Think back to our physical model: if the whole rectangle is $1$, and we divided it into $10$ equal parts, each of those parts now "fits" perfectly into the Tenths column. Just as we use the Ones column to count how many whole units we have, we use the Tenths column to count how many tenths we have!

Formalizing Decimal Notation: Relating $1/10$ to $0.1$

We know that $1/10$ represents one out of ten equal parts. Now, let's learn how to write this using decimal notation.

Let's place $1/10$ onto our extended place value chart.

• Do we have any whole units? No, $1/10$ is less than a whole. So, we put a $0$ in the Ones place.
• Do we have any tenths? Yes, we have one tenth. So, we put a $1$ in the Tenths place, right after the decimal point.

This gives us:

Ones . Tenths
$0 \quad . \quad 1$

The formal decimal notation for $1/10$ is $\mathbf{0.1}$.

It's super important to remember that $1/10$ and $0.1$ are just two different ways of writing the exact same quantity! They both mean "one tenth."

Let's practice reading these:

• The fraction $1/10$ is read as "one tenth."
• The decimal $0.1$ is also read as "one tenth." [Sometimes people say "zero point one," but reading it by its place value name helps us connect it to fractions!].

The leading zero in $0.1$ is very important. It tells us that there are zero whole units before we get to the parts of the whole. It helps prevent confusion and makes the decimal point easier to see.

Let's try converting a few more tenths fractions to decimal notation:

• For $4/10$: We have zero whole units and four tenths. So, it's $0.4$.
• For $9/10$: We have zero whole units and nine tenths. So, it's $0.9$.

Notice a pattern? The numerator of the fraction becomes the digit in the tenths place! ✨

Guided Practice and Application

Time to put your new knowledge to work! Remember, practice makes perfect. 💪

You'll encounter problems where you need to convert between fraction and decimal forms for tenths.

• Example: "Write the decimal for $6/10$." 👉 You should write $0.6$.
• Example: "Write the fraction for $0.3$." 👉 You should write $3/10$.

We can also use visual tools to help us. Imagine interactive activities where you have to match:

• A tenths model [like a grid with 5 out of 10 squares colored]
• A fraction card [$5/10$]
• A decimal notation card [$0.5$]

All three represent the same amount!

Another great tool is base ten blocks. If a large square "flat" block represents one whole unit ($1$), then a long "rod" block, which is $1/10$ of the flat, represents $1/10$ as a fraction. If you have, say, $4$ rods, you have $4/10$ of a whole, which is $0.4$.

When you're doing these exercises, remember these key points:

• Always place the decimal point correctly to separate the whole numbers from the parts.
• Read decimals using their place value names [e.g., "$0.7$" is "zero and seven tenths," not "zero point seven"] to reinforce the connection to fractions.

At the end of this lesson, you should be able to explain in your own words how the fraction $1/10$ is exactly the same quantity as the decimal $0.1$. That's how you know you've truly understood it! 🌟

Detailed Examples

Time for a Quick Check! 🧠

Lesson Summary 🏆

Congratulations! You've successfully been introduced to tenths, both as fractions and as decimals! 🎉

Here's a quick recap of what we've learned:

• A fraction like $1/10$ represents one part out of ten equal parts of a whole. The numerator [$1$] is the number of parts we have, and the denominator [$10$] is the total number of equal parts.
• Our standard place value system can be extended to include parts of a whole using the decimal point. This point separates whole numbers from fractional parts.
• The first place value to the right of the decimal point is the Tenths place.
• The fraction $1/10$ is equivalent to the decimal $0.1$. Both are read as "one tenth."
• The leading zero in decimals like $0.1$ signifies that there are zero whole units.
• To convert a fraction like $n/10$ to a decimal, you write $0.$ followed by the numerator $n$ [e.g., $3/10 = 0.3$].
• To convert a decimal like $0.n$ to a fraction, you write $n/10$ [e.g., $0.7 = 7/10$].

You now have a solid understanding of tenths and how to represent them in both fraction and decimal forms. Keep practicing, and soon you'll be a decimal and fraction pro! Great job! 🌟