Eighty-one times-table facts, forty-five to learn
Treat a x b and b x a as one fact and the pile halves. Which ones are last to stick, and what is gained by taking one table at a time.
A child answers 2 + 5 without pausing, then stops at 7 + 5.
Anyone who has sat beside a child practising addition has seen it. The numbers look only
slightly bigger, but between those two problems there is a step.
Here is everything that happens when 7 + 5 is worked out mentally.
It looks like a single addition. It is really one decomposition and two additions.
Where 2 + 5 takes one step, this takes three. The difficulty is not the
size of the numbers - it is the number of steps.
Japanese textbooks have children draw this split as a diagram - two branches under the 5, which is why it is nicknamed "cherry addition". What English-speaking classrooms call making ten, or bridging through ten, is the same procedure.
Steps 1 and 2 above depend entirely on knowing which pairs make ten: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. Five pairs - and the question is whether they arrive the instant the child looks at the problem.
If they do not, the child falls back on counting. That is usually what is happening when a child who counts on their fingers stalls just past ten: there are only ten fingers. Practising the bonds to ten on their own, before any carrying, clears the way for everything after. It is why English classrooms drill number bonds to ten first.
7 + 5 also comes out right by counting on: 8, 9, 10, 11, 12. The answer is
correct, so nothing looks wrong from the outside. But nobody counts their way through
57 + 35. A correct answer does not tell you whether the child is carrying
or counting.
Time is the tell. A child who makes ten answers with barely a pause; a child who counts takes as long as the counting takes. Same answer, but never at a steady speed. This is why timing an answer is worth as much as marking it - the clock separates two children who both got it right.
"Weak at addition" gives you nothing to act on. Record the sums that carry separately from those that do not, and a more useful sentence becomes available: "fast at two-digit sums without a carry, stops at single digits with one." That is a child stuck on an operation, not on the size of the numbers.
That is why this drill splits addition into 10 kinds - five steps without a carry and five with one, 3,934 problems in all. Every answer records which side of that line it fell on. The full breakdown across all four operations is in the 42 kinds.
29 + 1 carries. But none of the steps above are needed: the ones column
lands on zero, so the answer arrives without the child ever reaching for a bond to ten.
As carrying practice it is a miss.
So this drill does not ask them: no sum where one term is a single digit and the answer ends in zero. (The exclusion is inherited from the Android app this was ported from.) Generate problems by machine and sums like that always creep in. Being a valid calculation is not the same as being worth practising.
7 + 5) - where the three steps appear27 + 15) - the tens column joins inSkip a step and the answers stay correct for a while, because counting still works. The wall comes at two digits - by which point the real gap is three levels further down.
You can practise only the sums that carry. No sign-in needed.
For a walk-through with screenshots, see the full introduction.
More reading
Treat a x b and b x a as one fact and the pile halves. Which ones are last to stick, and what is gained by taking one table at a time.
A box plus five equals twelve wears the face of addition and asks for subtraction. It needs the equals sign read as a balance, not as a place to write the answer.
Wrong, slow, or not seen for a while - three faces of weakness weighed into one number, plus what has to be added so that practice stays bearable.
Where the four- and ten-second marks came from, and what to do about the case that timing children makes them afraid of maths.
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