Decimal Foundations
Understanding Parts of a Whole
What is a Whole? 🎯
Before we can understand parts, we need to understand what "one whole" means.
🎯 One Complete Unit
A whole is one complete thing.
When we talk about "1" in math, we mean one complete unit — nothing missing, nothing extra.
Think about this: What are some examples of "one whole thing" in your everyday life?
🍕 Examples of Wholes
Here are some things we can call "one whole":
Turn to your partner: Can you think of 2 more examples of "one whole thing"?
📓 Journal Note
What is a Whole?
A whole is one complete unit.
Examples: 1 pizza, 1 dollar, 1 square
The number 1 represents one whole.
Everything we learn about fractions and decimals is about parts of a whole. That's why understanding "whole" comes first!
What is a Fraction? 🍰
Fractions show us how to describe parts of a whole.
🍰 Parts of a Whole
A fraction describes a part of a whole.
The WHOLE pizza
HALF of the pizza
When we take ONE thing and break it into EQUAL pieces, we create fractions.
If you cut a pizza into 8 equal slices and eat 3 of them, what fraction did you eat?
You ate 38 of the pizza!
🔍 The Parts of a Fraction
⬆️ Numerator (Top)
3 = How many parts we HAVE
"Numer" sounds like "number" — it's the number of parts!
⬇️ Denominator (Bottom)
4 = How many EQUAL parts make the whole
"Denom" sounds like "name" — it names what kind of parts!
34 means: "3 out of 4 equal parts"
📓 Journal Note
What is a Fraction?
A fraction shows a part of a whole.
Parts of a Fraction:
• Numerator (top) = how many parts we have
• Denominator (bottom) = how many equal parts in the whole
Example: 34 means "3 out of 4 equal parts"
✏️ Let's Practice!
Problem 1
13
Problem 2
34
Problem 3
25
Fractions = Division ➗
Here's a secret: every fraction is actually a division problem!
➗ The Fraction Bar Means "Divided By"
The line in a fraction is actually a division symbol!
If you have 1 cookie and you want to share it equally with 2 people...
You divide the cookie: 1 ÷ 2
Each person gets 12 (one half) of the cookie!
📓 Journal Note
Fractions ARE Division
The fraction bar means "divided by."
ab = a ÷ b
Examples:
• 12 = 1 ÷ 2
• 34 = 3 ÷ 4
• 710 = 7 ÷ 10
This is how calculators turn fractions into decimals! When you type 3 ÷ 4, you get 0.75, which equals 34.
Special Fractions: Tenths 🔟
When we divide a whole into 10 equal parts, we get tenths!
🔟 Dividing into 10 Equal Parts
When we divide ONE whole into 10 equal parts, each part is called a tenth.
This shows 110 — one tenth
110 means 1 out of 10 equal parts.
It also means 1 ÷ 10 (one divided by ten).
🔢 Counting Tenths
Let's count tenths from 0 to 1 whole:
Notice: 1010 equals 1 whole. Why does that make sense?
📓 Journal Note
Tenths
A tenth is what you get when you divide 1 whole into 10 equal parts.
110 = 1 ÷ 10 = one tenth
Important facts:
• 510 = 12 (half)
• 1010 = 1 (one whole)
Introducing Decimals 🎯
Decimals are just another way to write fractions!
🎯 What is a Decimal?
A decimal is a fraction written in a different form.
Instead of writing fractions like 110, we can write them as decimals:
Decimals make it easier to:
- Write money amounts ($1.50 instead of 112 dollars)
- Use calculators (they show 0.5 instead of 12)
- Compare numbers quickly
⚫ The Decimal Point
The decimal point separates the whole number from the fractional part.
(Whole) Point Tenths
(Part)
⬅️ Left of the Point
Whole numbers (1, 2, 3...)
Things you have completely
➡️ Right of the Point
Parts of a whole (fractions!)
Things that are less than 1
🔄 Tenths as Decimals
Here's how tenths look as fractions AND decimals:
110
0.1
210
0.2
310
0.3
410
0.4
510
0.5
= ½
610
0.6
710
0.7
810
0.8
910
0.9
1010
1.0
= 1
Look at the pattern. What do you notice about how the numerator relates to the decimal?
📓 Journal Note
What is a Decimal?
A decimal is a fraction written in a different form.
The decimal point separates:
• Left side = whole numbers
• Right side = parts of a whole (fractions)
Converting tenths to decimals:
110 = 0.1 510 = 0.5 910 = 0.9
✏️ Let's Practice!
Special Fractions: Hundredths 💯
What happens when we divide a whole into 100 equal parts?
💯 Dividing into 100 Equal Parts
When we divide ONE whole into 100 equal parts, each part is called a hundredth.
1100 = one hundredth (1 tiny square out of 100)
How many hundredths would make one tenth? (Hint: Look at one row of the grid!)
10 hundredths = 1 tenth (10100 = 110)
🔢 Hundredths as Decimals
Hundredths go in the second place after the decimal point:
1100
0.01
5100
0.05
25100
0.25
99100
0.99
0.1 (one tenth) = 0.10 (ten hundredths) — They're the same!
📓 Journal Note
Hundredths
A hundredth is what you get when you divide 1 whole into 100 equal parts.
1100 = 0.01 = one hundredth
Place value: 0.TH
• T = Tenths (first place after decimal)
• H = Hundredths (second place after decimal)
Important: 10 hundredths = 1 tenth (0.10 = 0.1)
Place Value Review 🏠
Let's remember how place value works for whole numbers first.
🏠 Whole Number Place Values
In the number 352, each digit has a different value based on its place:
300 + 50 + 2 = 352
In the number 847, what is the value of the 4? What place is it in?
🔟 The Pattern of 10
Each place is 10 times bigger than the place to its right!
Hundreds
100
Tens
10
Ones
1
Going the other direction: each place is ÷10 (one-tenth) of the place to its left.
100 ÷ 10 = 10 10 ÷ 10 = 1
What do you think comes AFTER the ones place if we keep dividing by 10?
Decimal Place Value 📍
Now let's extend the pattern to the RIGHT of the decimal point!
➡️ Extending the Pattern
If we keep dividing by 10, we go past the ones place...
Tens
10
Ones
1
Tenths
0.1
Hundredths
0.01
1 ÷ 10 = 0.1 (one tenth)
0.1 ÷ 10 = 0.01 (one hundredth)
The pattern continues forever to the right!
📊 The Complete Place Value Chart
Here's how the number 24.37 looks in a place value chart:
20 + 4 + 0.3 + 0.07 = 24.37
In 24.37, which digit is worth more: the 3 or the 7? Why?
📓 Journal Note
Decimal Place Value
Place Value Chart:
Tens | Ones | . | Tenths | Hundredths
The pattern:
• Each place to the LEFT is ×10 bigger
• Each place to the RIGHT is ÷10 smaller
Example: In 3.45
• 3 is in the ones place (value = 3)
• 4 is in the tenths place (value = 0.4)
• 5 is in the hundredths place (value = 0.05)
✏️ Let's Practice!
Money Connection 💰
You already use decimals every day — with money!
💵 Dollars, Dimes, and Pennies
Money is the perfect example of decimals!
• 10 pennies = 1 dime
• 10 dimes = 1 dollar
• 10 hundredths = 1 tenth
• 10 tenths = 1 whole
📓 Journal Note
Money and Decimals
Money uses decimal place value:
• $1.00 = 1 whole dollar
• $0.10 = 1 dime = 110 of a dollar (tenth)
• $0.01 = 1 penny = 1100 of a dollar (hundredth)
Example: $3.47 means
3 dollars + 4 dimes + 7 pennies
3 ones + 4 tenths + 7 hundredths
✏️ Let's Practice!
Reading Decimals 🗣️
Learn how to say decimal numbers correctly!
🗣️ How to Read Decimals Correctly
Read the number after the decimal point, then say the place value of the LAST digit.
0.7
Say: "seven tenths"
0.07
Say: "seven hundredths"
0.35
Say: "thirty-five hundredths"
2.4
Say: "two and four tenths"
⚠️ Watch Out! 1.5 vs 1.05
These look similar but are VERY different!
one and five TENTHS
= 1 + 510
= $1.50 💵
one and five HUNDREDTHS
= 1 + 5100
= $1.05 💵
1.5 has 5 dimes (50 cents) after the dollar
1.05 has 5 pennies (5 cents) after the dollar
1.5 is MUCH bigger than 1.05!
📓 Journal Note
How to Read Decimals
1. Read the whole number part
2. Say "and" for the decimal point
3. Read the decimal part as a number
4. Say the place value of the LAST digit
Examples:
• 0.3 = "three tenths"
• 0.25 = "twenty-five hundredths"
• 4.7 = "four and seven tenths"
• 12.09 = "twelve and nine hundredths"
Remember: 1.5 ≠ 1.05 (very different values!)
🧱 Connecting to Base Ten Pieces
Remember the base ten pieces from class? Here's how they connect to decimals:
If you have: 2 mats, 3 strips, and 5 small squares
That's: 2 ones + 3 tenths + 5 hundredths = 2.35
Amazing Work!
You now understand the foundations of decimals! Remember: decimals are just fractions written in a different way. They help us describe parts of a whole using place value.