345.6 = 3 × 100 + 4 × 10 + 5 × 1 + 6 × 0.1, so the 4 is in the tens and is worth 40. Number is on to start with; press N to hide it when you want pupils to read the digits first.

The 7 is worth 70, and 572.4 is 57.24 tens altogether. The ▲ presses give 582.4, 592.4 and 602.4: the third one carries into the hundreds. The keypad under the chart lets pupils type digits on a whiteboard without a keyboard.

340. Every digit moves one place left and a 0 fills the empty ones column; the decimal point never moves. Predict works for every button and every ▲ ▼, so the class always commits to an answer first.

999, then 1,000: the tool shows the thousands column and says so. Columns ▾ chooses the biggest and smallest columns, and a column with a digit in it can’t be hidden, so nothing is ever lost.

Yes: each mode keeps its own number, columns and undo, so switching never loses work. The number keys switch mode only when no column is selected; with a column selected they type a digit into it.

The grey 0 is a placeholder the written number needs; the pale zeros (the thousands) aren’t needed. Without the grey zero you write 47: the 4 drops from the hundreds to the tens.

Three thousand and sixty. No word stands for the empty hundreds: the zero is silent when we speak, which is why pupils who hear “three thousand and sixty” write 300060 or 360.

Ten thousand and ten, and ten thousand, one hundred. Only the place of the second 1 changes (tens or hundreds), and the difference is 90.

It is worth 0.4, then 4 after × 10: ten times as much. A digit’s value is multiplied by 10 each time it moves one place left, which is all × 10 does. (Clicking a cell again deselects it.)

4.32 hundreds and 432 ones. The tens digit is 3, but there are 43 whole tens in 432: mixing up “the tens digit” with “how many tens” causes errors in division (432 ÷ 10 = 43.2).

0.25 = 25/100 = 1/4, 0.5 = 5/10 = 1/2 and 0.125 = 125/1000 = 1/8; 0.2 (1/5) and 0.05 (1/20) also work. The denominator before simplifying is always 10, 100 or 1000: the column of the last digit.

4.5 × 10³, 10², 10¹, 10⁰, 10⁻¹ and 10⁻² (0.045); the next ÷ 10 would need ten-thousandths, which the chart doesn’t have. The 4.5 never changes: ÷ 10 just takes one off the power.

340, then 3,400: each digit moves one place left per press, so the 3 goes from tens to hundreds to thousands. Zeros fill the empty columns so the 3 and 4 stay in their new places.

405. The 0 moves from the tenths to the tens with the other digits and keeps holding the place; if it stayed behind you’d get 45.

36, then 3.6: the 6 crosses the decimal point on the second press. “Three point six”, not “three point sixty”, shows the 6 is now tenths.

7 → 70 → 700 → 7,000 → 7. ÷ 1000 moves every digit three places right, which undoes three × 10s. The History shows the whole chain.

7 presses reach 5,000,000 and the 8th is refused, because millions is the biggest column. From 5, three ÷ 10s reach 0.005 and the 4th is refused. The chart has 7 whole-number columns and 3 decimal columns.

123.456: the 4 moves from the ones (worth 4) to the tenths (worth 0.4), a tenth of the size. Every digit is worth a tenth as much; the 4 is just the one you can see cross the point.

4 × 1,000 + 5 × 100 + 2 × 1 becomes 4 × 10,000 + 5 × 1,000 + 2 × 10 = 45,020. Each digit stays the same and its column is worth ten times as much.

One press (÷ 1000 or × 0.001) reaches 0.005, but 5,000,000 needs two (× 1000 twice), because it is six places left and the biggest button moves three.

0.72, 7.2, then 72: “point” goes on the third press, when the 2 reaches the ones. Reading “zero point zero seven two” digit by digit shows each digit’s column.

Both give 45. Multiplying by 0.1 finds a tenth of the number, which is the same as dividing by 10; the digits move one place right either way.

56 × 0.1 = 5.6 and 56 ÷ 0.1 = 560. Dividing by 0.1 asks “how many tenths are in 56?”, and there are 560. Many pupils predict 5.6 for both because they believe division always makes smaller.

30: the digits move two places right. “Multiplying makes bigger” is only true when you multiply by a number greater than 1.

25 and 250: there are 25 tenths and 250 hundredths in 2.5. Dividing by any number between 0 and 1 gives an answer bigger than the number you started with.

× 0.1 = ÷ 10, × 0.01 = ÷ 100 and ÷ 0.01 = × 100 (and ÷ 0.1 = × 10). Each pair moves the digits the same number of places in the same direction.

400. The ones pass 9 so 1 carries into the tens, which pass 9 so 1 carries into the hundreds: three digits change. (v1 used 397 here, which doesn’t carry.)

999 + 1 = 1,000 changes four digits; 909 + 1 = 910 changes two. A carry keeps going only while the next digit is a 9.

5.99 + 0.01 = 6 changes three digits; 5.09 + 0.01 = 5.1 changes two. The carry stops at the first digit that isn’t a 9.

999. There are no ones to take, so the tool exchanges 1 thousand for 10 hundreds, 1 hundred for 10 tens and 1 ten for 10 ones first. The expanded form goes from one term to three.

300: the tens go from 9 to 10, and 10 tens make 1 hundred. A ▲ does exactly what that column’s + button does.

15: four − 100s, five − 10s and six − 1s. With subtract buttons only you can’t beat the digit sum, because each press takes 1 from one column.

1.5. After 9 tenths, the 10th carries into the ones, so 15 tenths is 1 one and 5 tenths. 0.15 would be 15 hundredths.

0.7 > 0.65: comparing from the left, the first difference is in the tenths and 7 > 6. Pupils who think “longer is bigger” see 65 > 7; Fill with zeros writes 0.7 as 0.70, so it’s 70 hundredths against 65.

3.7 > 3.07. The 0 in 3.07 keeps the 7 in the hundredths. A zero at the end of a decimal changes nothing (3.70 = 3.7); a zero in the middle does.

1,009 > 999: the first difference is the thousands (1 against 0). With zeros filled, 999 becomes 0999, which shows why the number with more whole-number digits is bigger.

0.054, 0.405, 0.45, 0.5. With zeros filled they are 54, 405, 450 and 500 thousandths, which are easy to order.

Make the thousandths 1: 4.561 is 0.001 bigger. Pupils often change the last digit they can see (4.57, which is 0.01 bigger), forgetting there are smaller columns.

0.3 = 3/10 = 30/100, which is bigger than 29/100. Rewriting both in hundredths is the same idea as Fill with zeros.

350. The decider is the ones digit, 5, which rounds up; the number line shows 345.67 is past halfway (345).

346 (the decider is the 6 tenths) and 345.7 (the 7 hundredths). The decider is always the digit just to the right of the column you round to.

4.0. Rounding up 9 tenths makes 10 tenths, which carries into the ones. Keep the 0: it shows the answer is correct to 1 dp. Watch for 3.10 and 3.100.

2.4 directly, but 2.45 and then 2.5 in steps. Rounding twice is wrong because 2.45 has already been rounded up; always round the original number.

0.60. The first significant figure is the 5 in the tenths, so 2 sf means rounding to hundredths. Write the 0: 0.6 is only 1 significant figure.

7,000, 7,000 and 7,050 (the 3 sf decider is the 5 ones, so round up). In a whole number the zeros after the significant figures hold places, so they stay.

345 is the smallest, because halfway rounds up. With 1 dp the largest is 354.9, since 355 rounds to 360. This leads to error intervals: 345 ≤ x < 355.

10,000: all five digits change. Rounding up carries through every 9, just like + 1 does, and the answer has one more digit than the number.

25. 2.50 is the same number as 2.5; “add a zero” only works for whole numbers. The digits move one place left and the point stays still.

“Zero point one two” = 12/100 = 3/25, which is less than 0.5 = 50/100. Saying “point twelve” treats the decimal part as a whole number.

4,006 is four thousand and six. The pupil wrote 400 then put the 6 after it; in 406 the 6 takes the place of the last zero.

3.45. 0.05 is 5 hundredths, not 5 tenths. The pupil lined the digits up from the right instead of by place value.

4.4: the decider is 8, so the 3 rounds up. Chopping off (truncating) always rounds down, which is right only when the decider is 0 to 4.

0.05. They moved the digit the wrong way, as if they had multiplied. Dividing moves digits right, towards the smaller columns.

One press is × 100 (or ÷ 0.01). There are 12 two-press routes, all pairs of × and ÷ buttons that move the digits two places left in total, such as × 1000 then ÷ 10, or ÷ 0.1 then ÷ 0.1. ↺ Start again resets the count.

Two presses, such as ÷ 1000 then ÷ 10, or ÷ 100 twice. The digits have to move four places right, and the biggest button moves three.

No: 8 presses is the best, which the tool confirms by checking every route. × and ÷ can’t help, because they don’t change the digits, and you still have to make a 3 and a 5.

The best is 4: three + 1s, then × 10 (or × 10, then three + 10s). Using all four kinds of button takes at least 7 presses.

2: + 1 makes 1,000, then × 1000. Spotting that 999 is one away from a power of ten is the whole puzzle.

The best is hidden until the target is reached or 👁 Best is pressed. Targets are made from a few random moves, and the tool checks every route to find the true best, which is sometimes shorter than the moves used to make it.

1,999 → 2,000 changes 4 digits and 29,999 → 30,000 changes 5. The most is 7: 999,999 + 1 = 1,000,000. 9,999,999 + 1 would be too big for the chart.

“Point” goes after 3 presses (1). After 9 presses you reach 1,000,000, and the 10th is refused.

Yes, every time: ÷ 10 undoes × 10, and so does × 0.1, because both move the digits one place right. The only exception is when the chart refuses the × 10.

× and ÷ by powers of 10 never change the digits, so never the digit sum. + 1 adds 1 to it unless there is a carry; each carry turns a 9 into a 0, so the sum drops by 8, 17, 26 and so on.

The link stores the mode, the number, columns, displays and Predict, but not the undo history. Good questions ask for a prediction first and say which displays to keep off.