Math · Pre-K – Grade 5

Multiplication

Equal groups, arrays, and repeated addition used to find a total efficiently.

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Multiplication is counting equal groups. Four groups of six is 24 whether the groups are cupcakes, rows of floor tiles, or steps of six along a number line. The facts are worth memorizing, but only once a child can build the model, because word problems and division both ask for the model. Expect the facts themselves to take most of a school year.

Equal groups, arrays, and why the array matters most

Start with equal groups: four plates with six raisins on each. Say it the way it is written, four groups of six, and let him count the slow way once. Then arrange the same 24 raisins into a rectangle of four rows of six, and you have an array. The array is worth more than the plates because it can be turned. Rotate it and four sixes becomes six fours without a single raisin moving.

That one observation cuts the list of facts to learn roughly in half. Many children already own the twos, fives, and tens from skip-counting, and turning the array removes every duplicate fact from the grid. What genuinely remains to be memorized is a far shorter list than the wall chart suggests, and it is worth telling a child that plainly before he starts.

The order to teach the tables

Do not go in numerical order. Take the groups that are easy or half-known first, so each stage ends in a win, and leave the small stubborn set for last with real time on it. Introduce one group, practice it out of order until it is quick, then mix it with everything already learned before adding the next one.

  1. Twos, fives, and tens, which most children can already skip-count.
  2. Ones and zeros, understood with a model rather than taught as a trick.
  3. Threes, then fours by doubling the twos.
  4. The squares: 6 x 6, 7 x 7, 8 x 8, 9 x 9. They are memorable and they anchor their neighbors.
  5. Nines, using the pattern in the digits once the array has explained why it works.
  6. What remains: 6 x 7, 6 x 8, 7 x 8. There are very few, and they need daily contact.

Arrays you already own

A muffin tin gives twelve equal groups. An egg carton gives two rows of six. Interlocking bricks are arrays already, so a four-by-six plate is the fact sitting in his hand. The tile floor, a window's panes, an ice-cube tray, and the squares of a chocolate bar can all be counted two ways. Skip-counting aloud while climbing stairs or bouncing a ball puts the sequences into memory during time you were not using anyway.

Out of order is the real test

The test of understanding is not speed but order. A child who owns the fives answers 5 x 7 straight away, out of sequence, without starting at five and climbing. He can draw the array for a fact he has forgotten and rebuild the answer from it, and he can look at a word problem and say it is equal groups before he calculates anything.

  • He recites the fives fluently but has to run the whole sequence to reach 5 x 7.
  • He treats 4 x 6 and 6 x 4 as two unrelated facts to learn separately.
  • He cannot draw or build an array for a fact he has just answered correctly.
  • He adds when the word problem calls for multiplying.
  • Facts he knew in the spring are gone after the summer, which usually means they were learned as sounds rather than built on arrays.

When the facts will not stick

Come back to arrays with one factor at two, five, or ten, and build the stuck fact with real objects instead of telling him the answer. If the trouble is general rather than a few facts, check addition fluency first, because a child still counting on for 8 + 7 has no attention left over for products. An older child who never memorized the tables follows that same list in that same sequence, ten minutes a day, with material that does not look babyish.

How long the tables really take

The concept, equal groups and arrays, usually takes a few weeks and often lands in late second grade. The facts are the long part: for most children, daily practice across most of third grade, with fluency arriving in third or fourth. Multi-digit multiplication follows in fourth and fifth and leans on place value as much as on facts. Building Number Sense is sequenced this way and is still growing at the upper grades.

First, your child should already…

This skill builds directly on:

What your child is learning

Multiplication begins as equal groups and arrays before it becomes fluent facts. Children should understand the model before memorizing isolated products.

Signs your child is ready

  • Skip-counts accurately
  • Can build equal groups
  • Recognizes repeated addition

The common stumbling point

Memorizing facts without understanding equal groups, which causes confusion with word problems and division.

Practice this skill

Worksheets and decodable practice for Multiplication are in the library.

Every set opens its first three worksheets to everyone; the rest are included with All Access.

Browse the whole practice library

How to know it’s mastered

The child represents multiplication with equal groups or arrays and solves grade-level facts and problems accurately.

Common questions

What order should I teach the times tables?
Twos, fives, and tens first, because skip-counting has already half taught them. Then ones and zeros, then threes and fours by doubling, then the squares, then nines, and finally the handful that remain: six sevens, six eights, and seven eights. Grouping facts by how they are learned rather than by number keeps every stage short and leaves only a few genuinely hard facts at the end.
How long does it take to learn all the times tables?
For most children, most of a school year of short daily practice, ten minutes or so, mixed rather than one table at a time. Some finish in a term and some need two years, and both arrive at the same place by fifth grade. What does not work is one long weekly session; contact on most days matters more here than the total number of minutes.
Are timed tests a good idea?
They measure recall, and they are fine for a child who already has it. For a child who does not, a timer usually produces anxiety and guessing rather than learning, and it disguises which particular facts have gone. Practice untimed until the answers come, then use a timer briefly if he finds it motivating. If it makes him tense, drop it; nothing about the math is lost.
My fourth grader still does not know his facts. Do we go back to the beginning?
Go back to arrays for the facts he is missing, not to the start of the year. Give him a mixed page untimed, mark only the ones he had to work out, and drill that written list for ten minutes a day while the rest of his math continues. Stopping everything until the facts are perfect costs more than it buys, and facts stick better while they are being used for something.

What comes next

Once this is solid, move on to:

Division →

Once this is solid, move on to:

Fractions and Decimals →