How to use a multiplication chart

Reading the chart, step by step

1. Find the first factor in the shaded column on the left. That is your row.
2. Find the second factor in the shaded row across the top. That is your column.
3. Follow the row and the column until they meet. The number in that square is the product.

A multiplication chart also shows equal groups. Common Core standard 3.OA.A.1 describes the idea:

"Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each."

Column 7 counts those groups for you: 7, 14, 21, 28, 35. The fifth number down, in row 5, is 35, so 5 groups of 7 make 35.

Pattern 1: the two halves match (the commutative property)

7 × 8 and 8 × 7 sit opposite each other across the diagonal and hold the same number, so the numbers above the diagonal repeat the numbers below it. Common Core 3.OA.B.5 gives the example: "If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.)" The English national curriculum asks Year 2 pupils to "show that multiplication of 2 numbers can be done in any order (commutative) and division of 1 number by another cannot".

Pattern 2: square numbers run down the diagonal

The diagonal holds the square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. They are the only facts with no matching partner. Another pattern: starting at a square number, move one square up and one square to the right; the result is one less. 8 × 8 = 64, and 7 × 9 = 63. It works for any number n because (n − 1) × (n + 1) = n × n − 1. The English curriculum expects Year 5 pupils to "recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)".

Pattern 3: the 9s

• Down the column for 9, the tens digit goes up by one and the ones digit goes down by one: 09, 18, 27, 36, 45, 54, 63, 72, 81, 90.
• The digits of each product from 9 × 1 through 9 × 10 add up to 9 (63: 6 + 3 = 9). The rule stops at 9 × 11 = 99, whose digits add to 18.
• Multiplying by 9 is the same as multiplying by 10 and then subtracting the number once: 9 × 7 = 70 − 7 = 63. This one works for any number.
• Try the finger trick for 9 × 7: hold up all ten fingers and fold down the seventh finger from the left. The six fingers to its left are tens and the three to its right are ones, which makes 63. It only covers 9 × 1 to 9 × 10, because you only have ten fingers.

Pattern 4: the 5s and the 10s

Every multiple of 10 ends in 0. Every multiple of 5 ends in 5 or 0, and multiplying a number by 5 gives half the result of multiplying it by 10: 5 × 8 is half of 80, which is 40.

Pattern 5: odd and even products

A product is odd only when both factors are odd. Shade every even product and a grid pattern appears: every square in an even row or an even column is even. This is the pattern Common Core 3.OA.D.9 uses as its example:

"Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends."

Pattern 6: doubling links the 2s, 4s and 8s

Row 4 is row 2 doubled, and row 8 is row 4 doubled: 2 × 7 = 14, 4 × 7 = 28, 8 × 7 = 56. The English curriculum notes for Year 3 say: "Through doubling, they connect the 2, 4 and 8 multiplication tables." The same trick links the 3s and 6s: 6 × 7 is double 3 × 7 (21), so it is 42.

Pattern 7: break a hard fact into easy ones

If you know 5 × 7 = 35, then 6 × 7 is one more group of 7: 35 + 7 = 42. This is the distributive property, and Common Core 3.OA.B.5 demonstrates the same strategy: "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. (Distributive property.)"

On the chart it means stepping down one row and adding the column number. It is also how the 12s work: 12 × 6 = 10 × 6 + 2 × 6 = 60 + 12 = 72.

Pattern 8: the 11s

From 11 × 1 to 11 × 9, the answer repeats the digit: 11 × 4 = 44. After that, the repeat rule stops (11 × 10 = 110, 11 × 12 = 132), so use the break-apart rule: 11 × 12 = 10 × 12 + 12 = 132.

Pattern 9: factor pairs and multiples

Each place a number appears on the chart shows one of its factor pairs. The number 24 appears 6 times on a 12 × 12 chart: 2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3, 12 × 2. So 24's factor pairs up to 12 are 2 and 12, 3 and 8, 4 and 6 (1 and 24 are the other pair, off the edge of this chart). Common Core 4.OA.B.4 asks fourth graders to "Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number."

Dividing with the chart

1. Find the number you are dividing by in the left column (for 56 ÷ 7, that is 7).
2. Move along that row until you reach the number being divided, 56.
3. Go up to the top of that column. The number there, 8, is the answer.

Common Core 3.OA.B.6 describes division the same way: "Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8." If the number is not in the row (38 ÷ 7), stop at the largest number in the row that is smaller than 38. That is 7 × 5 = 35, so the answer is 5 with 3 left over. The division chart groups all 144 division facts up to 144 ÷ 12 by the number you divide by.

How much of the chart are children expected to know?

• United States (Common Core): 3.OA.C.7 says, "Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers." The largest one-digit number is 9, so that is a 9 × 9 chart: 45 facts, counting matching pairs once. The 10s, 11s and 12s go beyond what the standard requires.
• England: Year 2 covers the 2, 5 and 10 tables, Year 3 the 3, 4 and 8 tables, and Year 4 asks pupils to "recall multiplication and division facts for multiplication tables up to 12 × 12". Year 4 pupils then take the multiplication tables check. Its framework says, "Each form consists of 25 questions worth one mark each." It also says, "A time limit of 6 seconds per item has been set for the MTC."

Practice these facts with a blank or partly blank chart or a times tables worksheet; both print with an answer key.