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Learn how to break a rectangle into same-size squares, then count them all up!
Have you ever looked at a waffle? A waffle has little squares all in a row. People have been splitting shapes into smaller pieces for a very, very long time! Let's take a peek at how this idea grew.
The big question people have always asked is: "How many squares fit inside this shape?" That is exactly what we will learn today!
Before we start drawing and counting, let's learn four important words and ideas. These will help everything make sense.
Here is a rectangle that has been split into 3 rows and 4 columns of same-size squares. Look at how the squares fill up the whole rectangle with no gaps and no overlaps.
Can you see it? The rectangle has 3 rows going across. It has 4 columns going up and down. If you count every square, you get 12! Each row has 4 squares. There are 3 rows: 4 + 4 + 4 = 12.
There are two great ways to find out how many squares are inside a rectangle. Let's learn both!
You can point to each square and count: 1, 2, 3, 4, 5 … all the way to the end. This always works! But it can take a while if there are lots of squares.
This is the faster way. Count how many squares are in one row. Then count how many rows there are. Add the rows together!
For our rectangle with 3 rows and 4 columns:
You can also do it by columns. Count the squares in one column, then add that number for every column:
Rectangles come in many sizes! Some are tall. Some are wide. Some are small. Let's look at a few and see how the number of squares changes.
| Rectangle | Rows | Columns | Total Squares |
|---|---|---|---|
| A (short and wide) | 2 | 3 | 6 |
| B (longer) | 2 | 5 | 10 |
| C (a perfect square!) | 4 | 4 | 16 |
| D (imagine one!) | 1 | 5 | 5 |
| E (tall and skinny) | 5 | 2 | 10 |
Do you see something cool? Rectangle B has 2 rows and 5 columns. Rectangle E has 5 rows and 2 columns. They both have 10 squares! The order does not change the total.
Here is a problem we will solve step by step. Follow along!
We learned that you can count squares one by one, or you can use rows (or columns) and add. Let's compare these two ways.
| Way of Counting | Good Things | Hard Things |
|---|---|---|
| One by one | Easy to understand. You can point and count. | Slow if there are many squares. Easy to lose count! |
| Using rows & adding | Much faster! Works great for big rectangles. | You need to know how to add the same number many times. |
For a small rectangle like 2 rows and 2 columns, counting one by one is fine. But for a big rectangle like 5 rows and 5 columns, that's 25 squares. Adding is much faster: 5 + 5 + 5 + 5 + 5 = 25!
The cool thing about splitting rectangles into squares is that it leads to bigger ideas! Here is a sneak peek at what you'll learn later.
| What You Know Now | What You'll Learn Soon |
|---|---|
| Count squares by adding: 4 + 4 + 4 | Multiplication! 3 × 4 = 12 (same thing, but faster!) |
| Rows and columns of squares | Arrays — a math word for rows and columns of things |
| Counting all the squares inside | Area — how much space is inside a flat shape |
When you split a rectangle into squares and count them, you are actually finding the area of the rectangle! Area tells us how much space a flat shape covers. So you're already doing big-kid math. How amazing is that?
Try these problems. Take your time! Click "Show Answer" when you're ready to check your work.
Today we learned how to take a rectangle and fill it with same-size squares that sit in neat rows and columns. A row goes left to right, and a column goes top to bottom. Every square must be the same size, and they fill the whole rectangle with no gaps and no overlaps.
To find the total number of squares, you can count one by one, or use the faster way: count the squares in one row, then add that number for every row. You can also count by columns the same way. This idea is the beginning of learning about area and multiplication — big math ideas you'll explore soon!