Study Guides
Guides to common stumbling blocks in elementary school math — how to teach carrying, times tables, fractions and more, with free practice worksheets.
How Accurate Is Photo Grading? What 714 Handwritten Answers Showed
diglabo grades a finished worksheet from a photo. But what happens when the machine misreads? Here are the results of measuring 714 handwritten answers — and the things I still haven't fixed.
Teaching Multiples and Divisors — Why 1 and 12 Go Missing From "The Divisors of 12"
Ask your child to list the divisors of 12, and they stop at 2, 3, 4, 6 — 1 and 12 never show up. The cause isn't calculation skill. It's that your child hasn't settled on how far to search for divisors. Here's how to teach them to see it as a set.
Why Speed Is a Step Harder Than Percentage — Turning Two Different Kinds of Quantity Into One
Which is faster, 300 metres per minute or 12 kilometres per hour? When your child's hand stops, it isn't about calculation skill. It's because combining two different kinds of quantity — length and time — into one is a new kind of thinking, the first of its kind in elementary math. Here's how that idea gets built up.
Why Kids Who Know Their Times Tables Still Trip Up — The Steps of Column Multiplication
Just because a child can say their times tables doesn't mean column multiplication comes easily. Plenty of children get the times-table part right and still stumble on the steps of writing it in column form. A mistake in column multiplication tends to get blamed on the times tables. Usually, though, it's a different step that's the problem.
Column Addition and Subtraction — Why 23 + 4 Goes Wrong
Ask your child to write 23 + 4 in column form, and they write the 4 under the tens digit — the answer comes out 63. This is a new stumble that shows up in grade 2, right after a child has learned addition with carrying in grade 1. Here's how to explain what lining up place value means, and how to split practice into stages by the number of carries.
What's Really Behind "Multiply and It Gets Smaller" — Fraction Multiplication and Division
Multiply, and the answer gets smaller. Divide, and it gets bigger. The sense that worked reliably with whole numbers keeps flipping on its head. Here's how that sense of meaning gets rebuilt, and how to avoid getting stuck at the mixed-number stage.
Common Denominators and Reducing Fractions — When "Same Size, Different Shape" Doesn't Land
Trouble with fractions isn't one thing. There's the grade 3 struggle to grasp what a fraction even is, and there's the grade 5 struggle with the mechanics of finding a common denominator. Rewriting the denominator and leaving the numerator behind belongs to the second kind. Here's how to show why the size doesn't change.
"That's Not a Triangle If It's Facing the Wrong Way" — How to Tell Shapes Apart by Their Properties
A pointed corner facing sideways is enough for a child to say, "that's not a triangle." It's a sign they've learned shape names by overall look rather than by their properties. Here's how to teach children to tell shapes apart by the number and length of their sides, in order from grade 2 through grade 4.
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 seam between the counting and the formula is where things come apart later, on combined shapes and on unit conversions. Here's how to keep the two connected.
Long Division — Fixing "Place, Multiply, Subtract, Bring Down"
Have you ever looked at a long division problem and wondered where to even start explaining "place, multiply, subtract, bring down"? Most children who stumble on long division aren't struggling with their times tables. There are four steps — place, multiply, subtract, bring down — and every time, they're just skipping one of them somewhere.
These Worksheets Follow the Japanese Curriculum — What That Means If You're Outside Japan
diglabo's worksheets are built to Japan's national curriculum. Here's what differs from other countries, what's the same everywhere, and why they're still usable wherever you are.
How to Teach Addition with Carrying — Clearing the Hurdle with "Groups of 10"
8 + 5, and your child runs out of fingers and freezes. The cause is that composing and decomposing 10 isn't automatic yet. Here's how it's taught in Japan, and the order to practise it at home.
How to Teach Subtraction with Borrowing — Why It's Harder Than Addition
13 − 5 is a significantly harder calculation than 8 + 5. Here's why, how to build the "subtract from 10" reflex at home, and how to use the practice worksheets.
Learning the Times Tables — Beyond Rote Chanting, and the Rows That Trip Children Up
Chanting alone tends to stall at the 6, 7 and 8 rows. Here's the order that works, what the hardest facts have in common, and how to run chanting and understanding in parallel at home.
Teaching a Child to Tell Time — Getting Past "The Long Hand Is on 3, So Why Is It 15?"
A clock face carries two scales at once: the 3 means both three o'clock and fifteen minutes. Here's what makes it hard, and the order to build it up in everyday life.
Learning Units of Length and Capacity — Build a Feel for Quantity Before Conversion
They know 1 L = 10 dL but freeze on the problem. Unit trouble isn't a memory gap — it's the missing feel for what 1 dL or 1 cm actually is. Here's how to build it at home.
Where Division with Remainders Goes Wrong — Fixing "The Remainder Is Too Big"
Mistakes like "17 ÷ 5 = 2 remainder 7" follow a clear pattern. Here are the three causes, the link to the times tables, and how to handle remainders in word problems.
Why Fractions Confuse Children at the Start — "Cut in Two" Isn't the Same as "One Half"
Fraction trouble starts in grade 3, not grade 5. The missing word "equally", the shifting "half of what", the reversed sense of size — and how to build the base that carries through to common denominators.
Place Value in Decimals — Preventing "0.7 + 0.5 = 0.12"
"0.7 + 0.5 = 0.12". "2.5 × 10 = 2.50". Decimal errors come from place value. Here's how to teach 0.1 as "one tenth of 1", and what to secure at each grade.
Teaching Percentage to a Child Who Doesn't Get It — "See It as 1" Before Any Formula
Percentage is the hardest topic in elementary math. Why memorising the formula doesn't work, how to build the sense of "seeing the base amount as 1", and how to extend it to percentages and rates.
What Children Who Can't Do Word Problems Have in Common — Behind "But the Arithmetic Is Fine"
Full marks on calculation sheets, and then the hand stops on word problems. The cause isn't vague "reading ability" but specific, identifiable stumbles. Here's how to tell them apart and what to do at home.