Study Guides
Understanding Area — Connecting Counting Squares to the Formula
On a grid worksheet, finding area starts with counting squares one at a time. Then, one day, a child who used to count starts writing down just the formula: "length 4 × width 6." Once you stop seeing their finger trace the squares, it's easy to feel like the unit has simply moved on. But ask what the formula means, and sometimes there's no answer.
The commentary on the curriculum guidelines states: "It is important to make it possible to think about a way of finding area by calculation, rather than by laying out unit squares." Putting the experience of counting squares first isn't a detour — it's the foundation the formula rests on. Once I knew that, I ordered the worksheets as: area of rectangles and squares, then combined shapes, then converting between units.
The root cause: the experience of counting squares and the formula aren't connected
Cause 1: the formula skips past "why this equation"
The area of a square or rectangle is found with length × width. Where that equation comes from is the arrangement of the grid squares itself. Japan's Ministry of Education, in its commentary on the national curriculum guidelines for elementary mathematics (2017), states: "In squares and rectangles, unit squares line up regularly along the sides, so using multiplication lets you efficiently find how many there are." (Grade 4, Measurement) For instance, when a row of 5 squares across repeats 7 times down, you don't have to count one by one — 5 × 7 gives the total straight away. It's because the number of squares in a row repeats down by the number of rows that multiplication finds it. Skip this and just memorise the formula, and a child stalls the moment an L-shape or a staircase shape — a combined shape — shows up. A combined shape is a problem where the child has to find, on their own, where to cut it to get shapes they already know. Without a sense of where to cut, they can't even draw the first line.
Cause 2: mixing up area with perimeter
A common mistake is looking at the length and width and adding where you should multiply. Area is "the space inside"; perimeter is "the distance around the outside" — they're asking for different things.
Cause 3: getting the place value wrong when converting units
Converting cm² to m² is a common place to slip up: thinking of 1 m² as 100 cm² and multiplying by 100. The correct multiplier is 10,000. If the answer comes out two digits off, this mix-up is the cause.
How to teach it at home
Step 1: Have your child "split" combined shapes to count them
For an L-shape or a staircase shape, split it into two rectangles, find each one with length × width, and add the results. There's more than one way to split a shape. Trying "does a different split give the same answer?" is what makes the formula's meaning sink in.
Step 2: Show the relationship between cm² and m² with a grid
Look at a 1 m by 1 m square as a grid of 1 cm by 1 cm squares: 100 squares across, 100 squares down. 100 × 100 is 10,000. That's why 1 m² is 10,000 cm². Thinking of it not as "times 100" but as "both the length and the width become 100 times bigger" makes sense of the particular way area's place value jumps.
Step 3: Don't add too many units at once
Area has a lot of units — cm², m², a, ha, km² — and the commentary itself says to "flexibly choose the size of the unit while paying attention to the relationships between units" (Grade 4, Measurement). When school adds two more units, that's the moment to pause. Confirming the relationship with the previous unit before moving on sticks better than cramming several in at once. Learning to choose the right unit is itself the goal here — I only found that out once I looked into it. diglabo's worksheets also split converting between cm²/m²/a and converting up to ha/km² into separate steps.
What to avoid
If your child forgets the formula, going straight back to re-memorising it isn't the best move. Going back to counting squares instead usually brings the formula back with it.
How to practise
Practice starts with the area of rectangles and squares, moves into finding a side length or a perimeter, then into combined shapes — L-shapes, T-shapes, staircase shapes. From there it covers converting between cm², m², and a, then larger units up to ha and km², and problems that ask you to find the area and then convert the unit. The Area unit runs as one continuous sequence in this order, so if your child gets stuck partway, go back one step to the shape before it and check from there.
Frequently asked questions
Q. Once they've memorised the formula, is counting squares still worth practising?
A. It's reassuring to keep it as a place to return to when an unfamiliar shape shows up. If your child is stuck on how to split a combined shape, going back to the idea of counting squares can be exactly what unlocks it.
Q. Wouldn't it be faster to just drill unit conversion over and over?
A. That tends to turn into guessing how many zeros to add. It holds up better later if you first confirm your child can explain, in their own words, how many cm² make up 1 m² by picturing the grid.
Q. My child split a combined shape a different way from the answer key. Should I mark it wrong?
A. If the final number is right and each piece they split it into is a proper rectangle or square, there's no need to correct a different way of splitting it. Noticing that the same area can be found more than one way builds understanding more than following one fixed procedure does.
Area is a unit where counting and calculating stay connected the whole way through. Don't rush straight to the formula — leave room to go back to the grid, and the whole unit starts to hang together.
This article is written by a parent who builds diglabo. I am not a teacher or an education specialist. The general guidance here is based on how the topic is taught in Japanese primary schools, checked against Japan's national curriculum guidelines.
diglabo's worksheets follow the Japanese curriculum. What that means if you are outside Japan