Bubble Splat

For 7 × 8 = 56, one 8 too few is 48 and one too many is 64. The others are 49 and 63 (the 7 times table either side), 15 (7 + 8) and 65 (56 with its digits swapped). Popping 48 or 64 usually means counting up in 8s and losing your place; popping 15 means the × was read as +.

Still bubbles and mini whiteboards turn one pupil’s game into retrieval for the whole room. Each wrong bubble is a real mistake (a neighbouring fact, the two numbers added, digits swapped), so a wrong answer on a whiteboard tells you which mistake it was.

Most pupils are slower on division at first, because they get there through the multiplication fact (what times 8 makes 56?). One wrong bubble is nearly always the number divided minus the divisor (56 − 8 = 48), which catches pupils who take away once instead of dividing.

Missing-number questions are division in disguise, and they lead on to equations such as 8x = 56. The wrong bubbles are the same as for division: the answer one or two either side, the number already in the question, and the product minus that number.

The multiplication tables check taken in Year 4 in England has 25 questions from the 2 to 12 times tables, with 6 seconds for each. Pupils type answers in the real check, so popping is practice for recall, not for the check’s format. Facts that time out go into Your tricky ones, and ▶ Practise these plays just those, twice each.

A practice round holds the facts that went wrong, each twice and never twice in a row, topped up to at least 6 questions with others from the same set-up, so pupils meet the hard facts again within a minute. The best record ignores practice rounds, so it stays a fair record of the full set-up.

37 + 63 = 100. The wrong bubbles are 73 (the tens made 10 and the ones made 10, so 37 + 73 = 110), 3 (only as far as the next ten, 40), 62 (the ones made 9: 37 + 62 = 99) and 36 (63 with its digits swapped). The tens of a bond to 100 make 9, not 10, because the ones make the tenth ten. A phone shows the first three: 73, 3 and 62.

For 30 + ☐ = 100 one wrong bubble is 7: the right digit, but 7 ones, not 7 tens. The others are 60 and 80 (one ten out either way) and 30 (the number in the question). Pupils who pop 7 know the bond to 10 and need to say it in tens: 3 tens and 7 tens make 10 tens.

Each subtraction has the same answer and the same wrong bubbles as its missing-number bond, so slower subtractions mean the pupil is counting back rather than using the bond. For 7 + ☐ = 20 the first wrong bubble is 3, a bond to 10 instead of 20; for 13 + ☐ = 20 it is 17, which makes 30.

0.37 + 0.63 = 1. The wrong bubbles are 0.73 (the tenths and the hundredths each made 10, so 0.37 + 0.73 = 1.1), 0.03 (only to the next tenth, 0.4), 0.62 (0.99) and 0.36 (digits swapped): the same mistakes as for 37 + ☐ = 100, one hundredth of the size. For a bond in tenths, such as 0.3 + ☐ = 1, one wrong bubble is 0.07, the right digit in the hundredths.

47 + 38 = 85. The wrong bubbles are 75 (the ten from 7 + 8 = 15 not carried), 84 and 86 (one out), 9 (47 − 38: took away instead), 715 (15 written in the ones column) and 95 (a ten too many). A phone shows the first three: 75, one of 84 and 86, and 9. Popping 75 or 715 points to the same gap: 15 ones is 1 ten and 5 ones.

52 − 27 = 25. 35 comes from taking the smaller digit from the larger in each column (7 − 2 = 5, 5 − 2 = 3), and also from exchanging a ten but not taking it away. It is the first wrong bubble whenever a subtraction needs an exchange. The others are 24 or 26 (one out) and 79 (52 + 27: added instead).

With Never, 43 + 5 and 48 − 5 stay inside the ten; with Always, every addition carries (47 + 8) and every subtraction exchanges (52 − 7). Crossing is the main difficulty dial for mental calculation. With Always, the first wrong bubble for every addition is the answer with the ten not carried (47 + 8 → 45), and for every subtraction it is the smaller digit taken from the larger (52 − 7 → 55).

370 + 50 = 420. 320 is the hundred not carried (7 tens + 5 tens = 12 tens, which is 1 hundred and 2 tens), and 375 is 5 ones added instead of 5 tens. 410 and 430 are one ten out. Saying “7 tens and 5 tens make 12 tens” out loud, as with ones, is often what fixes it.

305. The wrong bubbles are 3005 (300 and 5 written side by side, the way they sound), 35 (the zero that holds the tens place left out) and 350 (the zero in the wrong place), then 405, 304 and 505 as near misses. A phone shows the first three. Pupils who pop 3005 are writing what they hear; 3 hundreds, 0 tens and 5 ones in a place value chart, or on the Place Value Counters, is the usual way in.

For 14 the first wrong bubble is “forty”, and for 40 it is “fourteen”: the two sound alike, especially said quickly, and the same goes for 13 to 19 and 30 to 90. For 7 the first two are “seventeen” and “seventy”. Stressing the ending (fourTEEN, FORty) and checking with tens and ones helps pupils who mix them up.

4.17 is “four point one seven”: 4 ones, 1 tenth and 7 hundredths. “Four point seventeen” reads the digits after the point as one number, and it is the first wrong bubble whenever there are two digits after the point and the first is not zero. The others for 4.17 are “four point seven one”, “forty-one point seven” and “four hundred and seventeen”. Pupils who say “point seventeen” often also think 4.17 is bigger than 4.2.

2,050. Writing each part as it is heard gives 200,050 (2,000 then 50). The other wrong bubbles are 250 (the zeros left out), 2,015 (fifty and fifteen sound alike), 2,500 and 2,005 (a zero in the wrong place) and 20,050 (one zero too many). A phone shows 200,050, 250 and 2,015. The big bands lean on zeros: every pattern of zero and non-zero digits comes up, because the zeros are where the mistakes are.

405. The wrong bubbles are 305 (the tens went from 9 to 0 but the hundred wasn’t carried), 495 (100 more instead of 10), 396 (1 more), 385 (10 less), 415 (20 more) and 395 (no change). A phone shows the first three: 305, 495 and 396. Saying “9 tens and 1 ten make 10 tens, which is 1 hundred” out loud, as with the ones, is often what fixes 305.

998. 1,098 comes from taking 1 ten from 0 tens by turning the 0 into a 9 without exchanging a hundred, and it is the first wrong bubble for every “less” question that crosses a boundary. The others are 908 (100 less instead) and 1,007 (1 less). Zero in the column you are changing is the hard case: there is nothing to take, so the next column has to give one up.

With Never, the digit in the column you change is never 9 going up or 0 going down, so only that one digit changes (345 + 10 = 355). With Always it is, so the next column changes too (395 + 10 = 405, 1,008 − 10 = 998). Crossing is the main difficulty dial: a pupil who is quick with Never and slow with Always is counting on in their head rather than thinking in tens and hundreds.

4.05. 3.105 writes 9 tenths and 1 tenth as “10 tenths” after the point, as if tenths were a separate whole number; it is a wrong bubble whenever 0.1 is added to a 9 in the tenths (or 0.01 to a 9 in the hundredths: 2.49 → 2.410). The others for 3.95 are 3.05 (the 1 not carried into the ones), 4.95 (1 more instead of 0.1) and 3.96 (0.01 more). Ten tenths make one whole, just as ten ones make a ten.

76. The wrong bubbles are 616 (each digit doubled on its own: 6 and 16 side by side), 66 (16 written as 6, the ten carried left out), 40 (2 added instead of doubling), 19 (halved instead) and 68 (only the 3 doubled). A phone shows 616, 66 and 40. Partitioning makes the carry visible: double 30 is 60, double 8 is 16, and 60 + 16 = 76.

28. 23 halves each digit on its own: half of 5 is “2” and half of 6 is 3, so the ten left over from the 5 tens is lost. It is the first wrong bubble whenever a digit before the last is odd, apart from numbers such as 10 and 110, where it would give 0. The next two are 54 (2 taken away instead of halving) and 112 (doubled instead). Splitting 56 into 50 and 6, or into 40 and 16, keeps the odd ten.

For half of 57 the wrong bubbles start with 28 (the half left off), 23 (each digit halved on its own) and 29 (rounded up), then 55 (2 taken away) and 114 (doubled). Whenever the half goes one place further than the number (apart from half of 1, 10, 0.1 and 0.01, where it would leave 0), the first wrong bubble leaves that half off: half of 350 → 170 (not 175), half of 7.5 → 3.7 (not 3.75). Pupils who pop 28 often know “57 is odd” but stop at the whole number.

Double 3.7 is 7.4. 6.14 doubles the 3 and the 7 separately and writes 6 and 14 side by side; 6.4 leaves out the 1 carried from 14 tenths; 5.7 adds 2. For double 0.45 = 0.9, the same mistakes give 0.810 and 0.80. Every double where a digit after the first is 5 or more has the each-digit-doubled bubble first (for 5 or 50 it gives the right answer, so it is not shown). Fourteen tenths is one whole and four tenths, just as fourteen ones is a ten and four ones.

For 3.4 × 10 = 34 the first wrong bubbles are 3.40 (a zero added on the end), 30.4 (the zero added to the whole number only), 340 (one place too many) and 0.34 (divided instead). A phone shows the first three. Every decimal multiplied by 10, 100 or 1,000 has the zero-on-the-end bubble first: 3.40 is still 3.4, because “add a zero” only works for whole numbers. On a place value chart the digits move one place to the left for each 10, and the point stays where it is.

30.5. 35 crosses off a zero to divide by 10, but the zero in 305 is not on the end, so crossing it off changes the tens and the ones instead. It is the first wrong bubble for every 3-digit number with a zero in the middle divided by 10 (105, 305, 907 and so on). The next two for 305 ÷ 10 are 3.05 (one place too many) and 3,050 (multiplied instead). Moving the digits, not crossing off zeros, works for every number.

For 3.4 × ☐ = 340 the wrong bubbles are 10 and 1,000 (one place out), then 0.01 (which would make it smaller), 10,000, 0.1 and 0.001: a wide board shows every power of ten from 0.001 to 10,000, so pupils have to decide whether the number got bigger and by how many places. For ☐ × 100 = 340 the answer is 3.4, and the wrong bubbles are 34 and 0.34 (one place out) and 34,000 (multiplied instead of undoing the × 100).

34 × 0.1 = 3.4: multiplying by 0.1 is the same as dividing by 10. The first wrong bubble for every × 0.1 and × 0.01 question is the one that made it bigger (34 × 0.1 → 340), and for every ÷ 0.1 and ÷ 0.01 question the one that made it smaller (3.4 ÷ 0.1 → 0.34). The next two are one place out: for 34 × 0.1 they are 0.34 and 34. Pupils who pop 340 believe multiplying always makes bigger, which is only true when multiplying by a number greater than 1.

−3 − 5 = −8. The wrong bubbles are 2 (−3 + 5: added instead of subtracting), 8 (the right size with the wrong sign) and −2 (3 − 5: the signs ignored), plus −9, −7 and −10 as near misses. A phone shows only the first three: 2, 8 and −2.

3 − (−5) = 8 and 3 + (−5) = −2. Each one’s answer is the other’s first wrong bubble: −2 (3 − 5) for the first and 8 (3 + 5) for the second, so the game checks whether pupils read both signs. Subtracting a negative usually causes more wrong pops; zero pairs on the Double-Sided Counters show why taking away −5 adds 5.

For −3 + 5 = 2, the bubble 1 catches pupils who count the numbers −3, −2, −1, 0, 1 instead of five jumps. On a wide screen every + or − question that crosses zero, with a jump bigger than 1, has this bubble. Negatives only in the first number keeps to Year 6 work: counting on and back across zero, and the difference between temperatures.

The first wrong bubble for every × and ÷ question is the answer with its sign changed: −24 for (−4) × (−6) = 24, and −4 for (−24) ÷ (−6) = 4. The next two are neighbouring facts with the right sign (two of 18, 20, 28 and 30 for (−4) × (−6); 3 and 5 for (−24) ÷ (−6)), so popping the sign-changed bubble means the times-table fact was right and the sign rule wasn’t. A phone shows just these three.

The sign errors are the same as within ±10 (−45 − 38 = −83, with 7, 83 and −7 as the first three wrong bubbles), but the calculation underneath is now a 2-digit addition or subtraction. Slow answers to questions such as −45 + 38 usually come from working out 45 − 38, not from the sign.

49. 14 is 7 × 2: the little 2 read as “times 2”, and it is the first wrong bubble for every square except 2² (where 2 × 2 is the right answer) and the multiples of 10 (10² is led by 10, one zero only). The others for 7² are 36 and 64 (6² and 8², the squares either side), 42 and 56 (7 × 6 and 7 × 8) and 9 (7 + 2). A phone shows 14, 36 and 64. Drawing 7² as a 7 by 7 square of dots shows why it is 49.

8. 32 halves 64, as if the square root sign meant “divide by 2”; it is the first wrong bubble for every square root of an even square number (except √4, where half of 4 is the right answer, and √100, led by 100 and 1). The next are the roots either side (7 and 9, since 7² = 49 and 9² = 81), then 64 itself. For an odd square such as √49 the first wrong bubbles are 6 and 8. Asking “what number times itself makes 64?” usually fixes 32.

30² = 900: 30 × 30 is 3 × 3 × 10 × 10, so two zeros. The wrong bubbles are 90 (only one zero), 9,000 (too many) and 60 (× 2). For √900 = 30 they are 300 (both zeros kept), 3 (the zeros dropped) and 450 (halved). Cubes have three zeros: 20³ = 8,000, with 800, 80 and 60 as the first wrong bubbles, and ∛8,000 = 20 with 200, 2 and 2,000.

−3² = −9, because only the 3 is squared and the minus sign stays in front; (−3)² = (−3) × (−3) = 9. Each one’s answer is the other’s first wrong bubble, so the game checks whether pupils read the brackets. A negative cubed stays negative: (−2)³ = −8, and the first wrong bubble is 8. ∛−8 = −2, with 2 first. Calculators agree: typing −3² usually gives −9.

0.25. 0.4 puts the denominator after the point, 0.14 puts the numerator and the denominator there, and 1.4 reads the fraction bar as a decimal point; these are the three a phone shows. The others are 25 (the percentage), 4 (4 ÷ 1, the division the wrong way round) and 0.75 (the rest of the whole). For ⅗ the first is 0.35. Asking “what is 1 ÷ 4?” or shading a quarter of a 10 by 10 grid shows why it is 25 hundredths.

30%. 3% takes the digit after the point as the percentage, the first wrong bubble for every decimal with one decimal place; for two places (0.37) the same mistake happens to give the right answer, which is why it survives. Next come 0.3% (the decimal with % on the end) and 300% (× 1,000). The reverse shows up too: 3% as a decimal has 0.3 first, and ½ as a percentage has 2%, 12% and 0.5%. Writing 0.3 as 0.30 first, 30 hundredths, helps.

Only 0.3̇, with the dot on the 3: the 3s go on for ever. 0.3 stopped too soon and 0.33 is rounded, so neither is exact; they are the first two wrong bubbles. As a percentage ⅓ is 33⅓%, with 30% and 33% first. ⅙ is 0.16̇ (only the 6 recurs), and one of its wrong bubbles is 0.1̇6̇, where both digits recur: that is 16/99. Dividing 1 by 3 and 1 by 6 by short division shows where each dot goes.

¼, because 25% is 25 hundredths. 1/25 uses the 25 as the denominator (the first wrong bubble for every whole-number percentage up to 100% except 1% and 10%), and 2/5 makes the two digits into a fraction. 5% → 1/5 is the same mistake as 1/25, and 40% → ¼ uses the tens digit as the denominator. Fractions are always in their simplest form, wrong ones included, so the simplest one is never a giveaway.

Each board has 2 to 5 multiples of 4, and the rule says how many are left to find. The traps are one either side of a multiple (27 and 29 next to 28), multiples of 2 that aren’t multiples of 4 (14, 30), numbers with a 4 in them (43, 54) and multiples of 3 or 5, the tables either side. A multiple of 4 is a multiple of 2 twice over: halve it and the answer is still even. The board moves on only when every multiple has been popped.

1, 2, 3, 4, 6, 9, 12, 18 and 36. 8 and 16 look like factors because they share a factor with 36 (36 ÷ 8 = 4.5, 36 ÷ 16 = 2.25), and 72 is a multiple, not a factor; 35 and 37 sit next to 36. 1 and 36 are factors too and are often left unpopped. Listing factor pairs (1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6) finds them all and shows 8 isn’t one.

1 isn’t prime: a prime has exactly two factors, and 1 has one. 2 is prime, the only even one, and is often left on the board. 91 = 7 × 13, 51 = 3 × 17 and 57 = 3 × 19 are odd and don’t end in 5, so they look prime. Most of the wrong numbers on a prime board are like this (odd, not ending in 5: 9, 21, 27, 33, 49), with 1 among them. If one is popped, the end card gives its factors.

14 is 7 × 2, the little 2 in 7² read as “times 2”. 8 is 2³, a cube, not a square; 24 and 26 sit next to 25. 1 is a square (1 × 1) and is often missed. For a quick check, a square number has an odd number of factors (16: 1, 2, 4, 8, 16), because its middle factor pairs with itself. 64 and 1 are both squares and cubes.

8 is always a wrong bubble for −3 − 5 = −8. Pupils who have just answered (−3) × (−5) = 15 often carry “two negatives make a positive” over to −3 − 5. Taking turns means each question needs its own rule, which is exactly what a block of only × questions hides.

Mixed rounds are usually slower and feel harder, because each question needs you to recognise what kind it is before you answer. Studies of interleaved practice (for example Rohrer and Taylor, 2007) found it helped pupils choose the right method later, even though practice felt harder. Use a mix after each game has been practised on its own.

Both are the same bond, one hundredth of the size: 37 and 63 make 100, so 0.37 and 0.63 make 1. Their wrong bubbles match as well (73 and 0.73, 3 and 0.03). Pupils who are fast on 100 and slow on 1 have not yet linked hundredths to the bond they already know.

Each side is dealt its own questions from the same set, so neither player can copy the other. The end card shows both players’ times and wrong pops, so a fast player who popped wildly can be compared with a slower, careful one.

With Clear the board the answers to the next few questions share one board and leave when they are popped, so the last question on each board is a choice between two bubbles: it is quick and satisfying, but partly elimination rather than recall. New bubbles every question gives each question its own wrong answers, so the tenth question is as hard as the first. A sensible routine is Clear the board to warm up, then New bubbles to check.

The best for a sprint is the most right answers, then the fewest wrong pops. Pupils who tap everything get a high wrong count straight away, which makes the end card a talking point about accuracy as well as speed.