Memorising Multiplication Tables

To memorise the multiplication tables, cut the grid down to the fifteen facts that are actually hard, work most of those out from facts you already know, give the few that still will not stick a picture, and practise for five minutes a day with the answers covered. The cut comes from the swap rule (6 × 8 is the same fact as 8 × 6) and the easy tables of 1, 2, 5, 9 and 10, which between them leave only fifteen facts in a 10 × 10 table. Take 7 × 8. Double 7 three times, 14, 28, 56, and you have the answer by arithmetic; or picture a snowman slipping off a cliff edge until a fishing hook and an elephant's trunk haul him back, and you have it without arithmetic, because in the number-shape pictures this guide uses, a cliff edge is 7, a snowman is 8, a hook is 5 and an elephant's trunk is 6.

The rest of the guide takes those steps in order: the count that shows how little there really is to learn, the doubling routes that rebuild most of it, pictures for a child and Major System pegs for an older learner, then a daily routine, a week-by-week plan and the research behind each step. It is written for a parent helping a child, but every step works just as well for an adult who never quite learned the tables.

Shrink the job before memorising anything

A 10 × 10 grid has a hundred squares, and that number is what makes the tables look hopeless. Most of the squares, though, are duplicates or easy, and the first duplicate is built into multiplication itself: the order of the two numbers does not matter, so 6 × 8 and 8 × 6 are one fact written two ways. Schools teach this early. The American Common Core standards for third grade give the example "If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known," and England's national curriculum asks Year 2 pupils to "show that multiplication of 2 numbers can be done in any order". Teach the swap first, and teach it as a promise that every fact learned counts twice.

The easy tables come next, and each has a rule that does the work:

• The ones give back the number itself, so 1 × 7 is 7.
• The tens put a zero on the end, so 10 × 7 is 70.
• The twos are doubles, so 2 × 8 is 8 + 8, which is 16.
• The fives are half of the tens, so 5 × 8 is half of 80, which is 40, and every answer ends in 5 or 0.
• The nines are ten lots minus one lot, so 9 × 7 is 70 − 7, which is 63. Up to 9 × 10 the two digits of the answer add to 9, and the tens digit is one less than the number you multiplied by, so 9 × 7 starts with 6, and 6 + 3 = 9. The finger version does the same sum: hold up ten fingers, fold down the seventh, and read six fingers before the fold and three after it.
• The elevens, up to 11 × 9, repeat the digit, so 11 × 7 is 77; beyond that they are ten lots plus one lot, so 11 × 12 is 120 + 12, which is 132.

Now count what is left. For a 10 × 10 table, with factors from 1 to 10:

A 12 × 12 table, which is what England expects by Year 4, shrinks the same way. Its 144 squares merge into 78 facts (12 on the diagonal plus 66 pairs); setting aside every fact with a 1, 2, 5, 10 or 11 leaves 28 among 3, 4, 6, 7, 8, 9 and 12, and setting aside the seven nines (9 × 12 is 120 − 12, which is 108) leaves 21. Those 21 are the fifteen below plus six twelves: 12 × 3, 12 × 4, 12 × 6, 12 × 7, 12 × 8 and 12 × 12.

The fifteen facts that are actually hard in a 10 × 10 table are these:

Others have shrunk the table this way before. Wikipedia's article on the table notes that because multiplication is commutative, the grid for 1 to 9 "can be reduced to 45 entries", and the traditional Chinese nine-nine table has a short version of exactly forty-five sentences. In a 1998 study with three students who had learning disabilities, Wood, Frank and Wacker grouped the facts into "the zeros, ones, doubles, fives, and nines categories, and those remaining", taught a strategy for each group, and found "substantial and immediate effects".

If you are an adult relearning the tables, run the same count on yourself: say the fifteen aloud in a mixed order, and whichever do not come back instantly are your whole list.

Work out the rest from facts you know

Most of the fifteen can be rebuilt from facts a child already has, mainly by doubling. Common Core spells out the idea with 8 × 7: "Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56." A child does not need the brackets, but the moves are worth naming, because each one covers a family of facts rather than a single answer.

Fours are a double, doubled. For 4 × 7, double 7 to get 14 and double again to get 28. Every fact in the fifteen with a 4 in it falls to this.

Eights are the fours doubled. Since 4 × 6 is 24, 8 × 6 is 48; since 4 × 8 is 32, 8 × 8 is 64; and since 4 × 7 is 28, 8 × 7 is 56. That is where the three doublings of 7 in the opening came from.

Threes are a double and one more. For 3 × 7, double 7 to 14 and add another 7 to reach 21; for 3 × 8, 16 and 8 more is 24.

Sixes are five lots and one more. Five sevens are 35, and one more seven makes 42, so 6 × 7 is 42; in the same way 6 × 8 is 40 + 8, or 48, and 6 × 6 is 30 + 6, or 36. There is a pattern to point out here, too: six times an even number ends in that number's digit, as in 6 × 2 = 12, 6 × 4 = 24, 6 × 6 = 36 and 6 × 8 = 48.

Squares make good anchors. Once 7 × 7 = 49 is known, both of its neighbours follow: 6 × 7 is one seven less, 42, and 7 × 8 is one seven more, 56.

Twelves are ten lots and two lots. For 12 × 7, ten sevens are 70 and two sevens are 14, so the answer is 84; for 12 × 12 it is 120 + 24, or 144.

Here is one route for each of the fifteen. Any route that a child trusts is a good one, so treat these as defaults rather than rules.

These routes are a bridge, not the destination. The Common Core standard ends with the words "By the end of Grade 3, know from memory all products of two one-digit numbers," and a fact that takes ten seconds of doubling is not yet known from memory. The bridge is still worth building. In a 2006 study by John Woodward, 58 fourth-graders were taught multiplication facts either with timed practice drills alone or with strategies plus timed drills; both approaches helped the children reach automaticity, but the strategy group "generally performed better" on tests that applied the facts to what the study calls extended facts and to approximation. Keep using a route until the answer arrives before the working does, and whichever facts are still slow after a week or two are the ones that need a picture.

Number-shape pictures for the stubborn few

A number-shape list gives each digit an object it looks like, so that a number can become a picture. Mind Tools describes it plainly: "each number from 1-10 is assigned an image, based on its shape." The pictures vary from list to list. This guide uses the same ten as the site's peg system guide, and each of them appears in at least one published list:

Art of Memory's list offers all of them except the cliff edge, which comes from Mind Tools. Other lists choose differently: Wikipedia's list of visual mnemonics has an egg for 0 and a pair of pouted lips for 3, and Mind Tools gives part of a love heart for 3. None of them is the correct list, because shapes are a personal choice. What matters is that each digit keeps one picture for good, and Mind Tools' advice is that you "can pick anything that comes to mind, as long as it's not too similar to one of the other images you've chosen." Let a child swap in any picture they like better, and then keep it.

Building a scene for a fact takes three habits. The two numbers being multiplied start the scene and the digits of the answer finish it, in reading order, tens first. The pictures act on each other rather than standing side by side, because, as the link method guide explains, interaction is what makes a join stick. And the child says the whole fact aloud while picturing it, then covers it and answers it both ways round, 6 × 8 and 8 × 6, since the swap rule makes them one fact.

Each scene below is a mnemonic, a made-up story to hang the fact on, not a reason the fact is true. The arithmetic that proves it comes after each one.

6 × 8 = 48. An elephant (6) scoops up a snowman (8) with its trunk and sets him down in a sailboat (4), where a second snowman (8) is already waiting at the tiller. The snowman in the question also ends the answer, which matches the sixes pattern above. In arithmetic, five eights are 40 and one more eight makes 48.

7 × 8 = 56. A snowman (8) teeters on a cliff edge (7) and slips, a fishing hook (5) snags his scarf, and an elephant (6) on the clifftop hauls him back up with its trunk. There is also a counting run hidden in this fact: 5, 6, 7, 8 gives 56 = 7 × 8. The arithmetic is 4 × 7, which is 28, doubled to 56.

6 × 7 = 42. An elephant (6) stands at the cliff edge (7) and blows through its trunk, pushing a sailboat (4) out to sea with a swan (2) as its only passenger. The arithmetic is five sevens, 35, and one more seven, 42.

7 × 7 = 49. Two cliff edges (7 and 7) face each other across a gorge, and a sailboat (4) rises slowly out of the gap, lifted by a balloon on a string (9) tied to its mast. In arithmetic, five sevens are 35 and two more sevens are 14, which make 49.

8 × 8 = 64. Two snowmen (8 and 8) lead an elephant (6) down to the harbour and squeeze it into a sailboat (4), which sinks to the waterline. The arithmetic is 4 × 8, which is 32, doubled to 64.

12 × 12 = 144. Twelve is a candle and a swan, so picture two swans, each with a lit candle on its back, towing one giant candle (1) across a lake towards two sailboats (4 and 4) moored side by side. A word helps here as well: twelve twelves is a dozen dozen, which has its own name, a gross, and a gross is 144. In arithmetic, ten twelves are 120 and two twelves are 24, which make 144.

One fact needs care. Both 3 × 8 and 4 × 6 make 24, so a scene for either must begin with its own two numbers; a picture of the answer alone cannot tell them apart.

The Major System for older learners

A teenager or adult, or anyone who already uses the Major System, can turn each answer into one word instead of two pictures. In Wikipedia's words, "The system works by converting numbers into consonants, then into words by adding vowels." On this site's chart 5 is an L sound and 6 an sh sound, which makes 56 a leash, and 48, an r and an f, is a roof. The single digits have words too: 6 is shoe, 7 is key and 8 is ivy.

56, Major System?leashA dog catching a scent and lunging, the leash going from slack to rigid in a half-second. L is the length of it unspoiling, Sh is the hiss of nylon through the clasp.

7 × 8 = 56. A key (7) lies buried under ivy (8) on the garden path, so you hook the dog's leash (56) through the key's bow and let the dog haul it free.

6 × 8 = 48. A shoe (6) tangled in ivy (8) is flung up onto the roof (48), where it hangs from the gutter by a tendril.

12 × 12 = 144. Twelve is a tin on the chart, and 144 is drawer on the site's table of three-digit pegs, so two tins (12 and 12) are jammed into a drawer (144) that will not close.

Every hard product already has a peg word on the chart. Here they are for the fifteen facts and the six twelves:

The Major System is the better tool for someone who will keep using it for phone numbers, dates and PINs, because the hundred words learned for the tables then serve everything else. For a seven-year-old with fifteen facts to learn it is far more than the job needs, and the ten shape pictures are enough.

Practice that sticks

A picture gets a fact in, and practice is what keeps it. Four habits do most of the work.

A few minutes a day, with the answer covered. Practice means producing the answer before seeing it, not reading the table aloud. Karpicke and colleagues tested this with 88 children, average age ten, who studied word lists and then either restudied them or practised recalling them; in all three experiments the children "showed robust retrieval practice effects." Spreading the practice out matters as well. In a 2015 study by Schutte and colleagues, 48 third-graders did four one-minute timed practices on addition facts each day for 19 days; children who split the four minutes into two or four sittings across the day "showed significantly higher fluency growth rates" than those who did all four in a row, although the total time was the same. Those were addition facts, not times tables, but they are a reason to split ten minutes of practice into five before school and five after dinner.

Come back at widening gaps. Ask each new fact once within the hour, as part of learning it; the first spaced review is the next day, and to keep the fact for good, use gaps of about 1, 6 and 16 days, then monthly. The spaced repetition guide explains why the first review is tomorrow rather than tonight and why the gaps grow. The most direct evidence for times tables that we found is a 1985 classroom study by Cornelius Rea and Vito Modigliani, in which 44 grade-three children learned five facts, 8 × 5 to 8 × 9, in a single session. Its gaps were seconds long, widening from none to four five-second distractions between tests, yet children tested this way scored almost twice as well on a test a minute after the session as children tested back to back, though both groups had scored perfectly while learning. The authors add that an expanding series "maintains a feeling of success throughout," because the first test comes soon enough to be passed, and each success helps carry the fact across the longer gap that follows.

Mix the facts. Reciting a table in order lets a child count on from the last answer instead of recalling the next one, so ask the facts in a mixed order once they are past the first day. The large trial on mixing that we found used older pupils and harder problems: in a 2020 trial by Doug Rohrer and colleagues, 54 seventh-grade maths classes did either mixed or blocked practice for four months, and a month later the mixed group outscored the blocked group "61% versus 38%" on an unannounced test. Mixing in that study forced pupils to choose a strategy for each problem from the problem itself, which is exactly what a mixed set of facts asks of a child still using doubling routes.

Ask both ways, and backwards. The Common Core standard ties multiplication to division: "knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8." Once 7 × 8 is solid, ask 8 × 7, then 56 ÷ 8, then "what times 7 makes 56?"

The simplest tool is a set of paper cards, with the question on the front and the answer, the route and the picture on the back. Ask a card, and put it in a pile for a few days later if the answer comes at once, or keep it in today's pile if it was slow or wrong. Any spaced-repetition app works the same way. Whatever you use, start each session with the cards that are due and add new facts last.

Fifteen facts in four weeks

This is our plan, not a study's. It assumes five to ten minutes a day on school days, split into two short sittings where you can, the easy tables first, and about one new hard fact a day, which leaves most of each session for review.

Each day runs the same way: yesterday's new facts first, then anything due from earlier in the week, then one new fact with its route and, if it needs one, its picture. If a week's facts are not reliable by the weekend, repeat the week rather than adding more, since four weeks that stick beat two weeks that have to be redone. By the end of the plan each fact has been recalled on several different days in a mixed order, and the reviews have thinned to a few a week.

Doing it on this site

This site has no multiplication drill of its own, but its custom drill can run a deck of facts that you make yourself. Save the facts as a .csv file with the header row front,back,note and one fact per row, such as 7 x 8,56,snowman on a cliff edge with a hook and a trunk. The question goes on the front with a letter x, the answer on the back and the picture in the note, which the importer joins to the back. A spreadsheet saved as .csv takes care of any commas in the notes. Upload the file at the import page, which also takes .tsv, Anki's .txt export and .apkg files. There you confirm which column is the front and which the back, and the import creates a deck named after the file in My Dict. Keep the x on the front: if every front in a file is a bare number of one to three digits, the importer reads the file as a list of Major System pegs rather than as plain cards, so a reversed deck with 56 on the front will not import the way you expect.

The custom drill then runs it. Pick the deck, five, ten or twenty cards, and a memorise window of fifteen seconds to three minutes. During the window the cards appear one at a time with their backs showing, and the recall that follows covers only the cards you saw, so let the timer run, or step through every card with "next", before you press "ready to recall". Each question then appears alone: say the answer, press "show note" and grade yourself miss, hard or easy. A missed card comes back at the end of the session. Due cards come first, most overdue first, then new or edited ones and then the rest, in the order of the rows in your file. The drill never shuffles, so write the rows in a mixed order rather than table by table. Review dates are kept in the browser you drill in. Because the memorise window shows every answer before the test, the drill suits facts that are still being learned; for a quick check of facts already known, paper cards are faster.

For parents, a game rather than a test

The method works best when it feels like play, and most of what makes it playful costs nothing. Let your child choose or draw the ten pictures, and draw each scene on the back of its card. Act a scene out at the table. Ask facts in odd moments, a few at a time and mixed, and let your child quiz you and catch your mistakes. Keep the list of fifteen on the fridge and cross off each fact as it becomes instant, because a list that shrinks is easier to face than a grid of a hundred squares.

Be careful with the clock. Jo Boaler, a professor of mathematics education at Stanford, wrote in 2014 that "Evidence strongly suggests that timed tests cause the early onset of math anxiety for students across the achievement range." That is her reading of the research rather than a single finding, and timed practice does not always go badly in the studies: in Woodward's study above, the timed drills helped. What a study of 154 first- and second-graders by Gerardo Ramirez and colleagues did show is that maths anxiety can already be measured at that age, and that more anxious children scored lower in maths if their working memory was higher, though not if it was lower. In England the Year 4 multiplication tables check is timed, giving children 6 seconds a question for 25 questions, but the government's guidance for parents says "There is no pass mark for the check" and "you do not need to do anything additional to prepare your child for the check."

A sensible middle path is to aim for accuracy first and let speed come from recall rather than from a stopwatch. Keep sessions short enough to end on a success; Rea and Modigliani built their widening gaps on the same idea, that a child should pass nearly every test along the way. If a session ends in tears, it was too long or too hard, so cut it back to the facts your child nearly knows.

What the research says

The research on mnemonics for multiplication facts is small, mostly done with children who were struggling, and encouraging.

Gary Greene in 1999 taught 23 elementary and middle-school students with learning disabilities "14 difficult-to-memorize multiplication facts" with a mix of mnemonic and traditional teaching, and found that "mnemonic training enhanced learning and these benefits were retained over time." Wood, Frank and Wacker's 1998 study grouped the facts and taught a mnemonic strategy for each group to three students with learning disabilities, whose accuracy after instruction "was often 100%". Most recently, Matthias Grünke and colleagues in 2025 taught a pegword method to three sixth-graders with persistent difficulties in mathematics and saw "a noticeable, gradual improvement" in their recall of multiplication facts involving two-digit numbers, and the students were keen to keep using it.

The evidence on practice is broader. Strategies plus timed drills generally did better than drills alone on applying the facts (Woodward, 2006); tests at expanding gaps nearly doubled short-term retention of five multiplication facts (Rea and Modigliani, 1985); spreading addition practice across the day sped up fluency growth (Schutte and colleagues, 2015); retrieval helped ten-year-olds remember word lists (Karpicke and colleagues, 2016); and mixed practice raised seventh-graders' maths scores a month later (Rohrer and colleagues, 2020).

Put together, a strategy for each family of facts, a mnemonic for the hardest few and short, spaced, mixed practice each has some support, though the mnemonic studies are small and the one large trial, on mixing, used older pupils. We found no study that tested number-shape pictures for times tables specifically, or any of this with typical children practising at home, so treat the method as well-founded rather than proven, and judge it by whether the facts come back the next morning.

Tonight

Write the fifteen facts on cards and ask your child each one once, in a mixed order. Put the instant ones aside, because those are already done. From the rest, pick the two slowest, find a route for each in the table above, and build one picture together for whichever still will not stick. Tomorrow before school, ask those two again with the answers covered. If they come back, you have the method: add one fact a day, and put the reviews first.