Division with Objects
What is Division with Objects?
When kids first meet division, it usually comes in two forms:
Splitting into equal groups
Sharing into equal parts
Imagine you have 35 apples. If you want to divide them into groups of 7, you can place 7 apples into a basket, then another 7, and so on. After filling 5 baskets, you’ve used all 35 apples. That shows us:
35 ÷ 7 = 5
Now, let’s look at it the other way. What if you want to share 35 apples equally among 7 baskets? First, give 1 apple to each basket (7 total), then 2 apples to each (14 total), then 3 apples (21), 4 apples (28), and finally 5 apples in each basket — all 35 are gone. Again, we see:
35 ÷ 7 = 5
Division into equal parts can also be shown with a rectangle diagram. Picture a grid with 7 rows. We start filling it step by step: first one square in each row (7 in total), then two in each (14), then three (21), four (28), and finally five in each row — adding up to 35. Each row now has 5 squares, which shows that 35 ÷ 7 = 5.
But what if the numbers don’t work out evenly? Say you have 39 apples and want to share them into 7 baskets. Each basket will get 5 apples, and 4 apples will be left over. We write that as:
39 ÷ 7 = 5 remainder 4
If you cut those 4 apples into 7 equal slices, you can even write the answer as a fraction:
39 ÷ 7 = 5 4/7
Why Does It Matter for Kids?
Division is more than just “splitting things up.” It’s:
Sharing fairly
Making groups of a given size
The reverse of multiplication (to solve 24 ÷ 6, kids find the number that, multiplied by 6, makes 24).
Research shows that the best way to build these ideas is by starting with manipulatives — blocks, counters, apples — and keeping the numbers small at first, usually within 20. This helps kids form an intuitive sense of division long before they start working only with numbers. Once that sense is in place, it’s much easier to refine and formalize later.
But manipulatives are just the first step. Pictures, diagrams, and even hand gestures linked to “sharing” and “grouping” help kids bridge the gap from concrete objects to abstract math.
For students in grades 3–4 who struggle with math, going back to hands-on division with objects can be surprisingly powerful. In fact, research suggests that the older the student, the more effective this “return to basics” can be in boosting both understanding and accuracy.
How Do We Teach?
We start simple — by splitting things in half and making pairs. One pie can’t be split into two whole pies, so it has to be cut into two equal halves. But what about two pies? When can a number be divided in half evenly, and when can’t it?
Kids soon notice a pattern: the numbers that can be divided in half are exactly the same ones made up of twos. These are the even numbers. On the number line, even and odd numbers alternate back and forth.
In one snowman game, kids practice seeing a number as a product. For example, they arrange 24 snowballs into a neat rectangle.
Next come numbers that can be divided by 3. Here, kids see that when you divide by 3, there may be 1 or 2 items left over. In money-themed games, they try problems like: How many candies costing 7 each can you buy with 100? That’s an early step toward understanding division with remainders.
Kids also work with picture-equations where they have to figure out how to share candies equally into boxes. In harder challenges, they first split the candies into one type of box, then put the leftovers into another — an important move toward multi-step division.
Finally, when objects are cut into equal parts, fractions come onto the scene. Kids solve simple comparison problems, like: Who gets more — the group that split a pizza three ways or the one that split it five ways?
First Steps
Halves and Pairs
The simplest kind of division is splitting things evenly between two people. Kids can hand out items one by one — one for you, one for me, another for you, another for me — or they can look at the whole set and try lining it up into two equal rows.
Halves and multiples
Halves
Children also practice spotting whether a pie or a chocolate bar in a picture has really been divided into equal halves. Each person’s share might be made up of several smaller pieces, but the goal is for both to get the same total amount.
Another way to think about dividing by two is making pairs. Kids who help sort laundry quickly learn that even if there’s a pile of socks, they don’t always match up neatly. Sometimes one sock gets left out. Eventually, they discover that this depends on the number itself. If there are 5 shoes, you can’t pair them all up evenly. That’s how the idea of even and odd numbers starts to take shape.
Counting with pictures
Halves
Halves and multiples
Halves
Counting with pictures
Halves
Counting with pictures
Odd and even numbers
Deep Understanding
Divisible or Not?
At this stage, kids begin to tell just by looking at a number whether it can be split into two equal parts — in other words, whether it’s even or odd.
They notice a simple pattern: if you share objects evenly between two people and then add just one more, it can’t be divided fairly anymore. On the number line, every odd number sits right between two evens.
Counting with pictures
Odd and even numbers
Kids experiment: How many ways can you split 4 objects? Into 2 parts or 4 parts. What about 6? You can’t make 4 equal groups, but you can make 2, 3, or 6.
In the snowman game, kids build rectangles with the right “area” of snowballs. If the snowman asks for 24 snowballs, they might arrange them as 4 rows of 6 or 6 rows of 4.
While sharing objects equally, kids also notice something important: when dividing by 2, there can only ever be 1 leftover. But when dividing by 3, there might be 1 or 2 left over. This becomes an early step toward understanding division with remainders.
Halves and multiples
Halves
Division by 2 and 3
Equal sets
Multiplication up to 6
Multiplication within 5-30 (objects only)
Division by 2 and 3
Division by 3 with a remainder within 15 (objects and numbers)
Confident Mastery
Remainders, Equations, and Fractions
Money problems are a natural way to introduce division with remainders. For example: How many candies that cost 20 each can you get with 110?
Counting with a twist
Division with remainders
If three boxes hold the same number of candies and we know the total, kids can figure out how many are in each box. All they need to do is find three equal numbers that add up to the total — in other words, divide the total by 3.
As kids progress, they encounter more complex picture-equations. Sometimes the answer takes two steps: first find how many candies are in the green box, and then use that to figure out how many are in the red one.
At a visual level, kids are also introduced to fractions. They practice comparing equal shares: Who gets more — the group that splits one piece of cheese three ways, or the group that splits two pieces four ways? This is where fraction symbols begin to appear.
Addition & subtraction
Еquations
Addition & subtraction
Systems of equations
Halves and multiples
Halves
Fractions and pie-charts
Visual fractions
Big Ideas
The idea of divisibility — and division with remainders — has turned out to be a powerful one in number theory. The concept of a prime number (a number that can only be divided evenly by 1 and itself) is central in modern coding and cryptography. Prime numbers make it possible to create codes that can only be unlocked by someone who knows the secret prime.
And when a number doesn’t divide evenly? We can still take the leftovers, cut them into smaller parts, and share them equally. That leads to the concept of rational numbers — fractions that represent one whole number divided by another. Rational numbers are the backbone of solving proportions.
The ancient Greeks believed that everything in the world was built on harmony and proportion. That’s why the discovery that the diagonal of a square cannot be expressed as a simple ratio of integers — that it’s incommensurable with its side — came as such a shock. It was their first glimpse into the world of irrational numbers.
