Grade 3 · 3.OA.D.9

Patterns in the multiplication table

The multiplication table holds a hundred products. Why do its rows, columns and diagonal repeat in ways that let you predict a number before you work it out?

You need

Facts for two, five and tenGrade 3

Where you are 4 in Multiplication facts

Number9

Arithmetic5

Multiplication and division16

Multiplication facts4

Facts for two, five and ten

Patterns in the multiplication table

Facts for three, four and six Facts for seven, eight and nine

Why this exists

By now every product up to 10 \times 10 has been found, one family at a time, in Facts for two, five and ten, Facts for three, four and six and Facts for seven, eight and nine. Written as a hundred separate facts they are a lot to carry. Laid out in one grid they stop being separate. The same numbers turn up in two places, one row is twice another, a column counts up in its tens and down in its ones, and the odd numbers sit in a neat checkerboard.

Each of those patterns has a reason, and the reason is always the meaning of multiplication: equal groups, or an array of rows and columns, as in Arrays. That is the point of this lesson. A pattern that is only noticed is a guess about the next number. A pattern that is explained is a fact about every number it covers, and it can be used to predict a product, to check one, or to rebuild one that has slipped from memory.

The lesson takes three days, one part each. Part 1 reads the table along its rows and across its diagonal. Part 2 compares whole rows and looks down the nines column. Part 3 asks which products are even and what the rows add up to.

Definition

The multiplication table is a grid with the factors 1 to 10 along the top and down the left side. The cell in row a and column b holds the product a \times b. So row 3, column 7 holds 3 \times 7 = 21.

A row of the table is the ten products a \times 1, a \times 2, \dots, a \times 10: the skip count by a. A column is the same thing read downward.

The diagonal is the line of cells running from the top left corner to the bottom right, where the row number and the column number are the same. Its cells hold 1 \times 1, 2 \times 2, \dots, 10 \times 10. A product of a number with itself is called a square number, because a rows of a tiles make a square.

The mirror cell of row a, column b is row b, column a: the cell you reach by folding the table along the diagonal.

A number is even when it splits into twos with nothing left over, and odd when one is left over, as in Even and odd numbers.

A pattern in the table is a statement about a whole set of cells (a row, a column, the diagonal, every cell) that holds for every one of them.

Intuition

Figure 1The table to 10 \times 10. The shaded diagonal holds the square products, and the two halves on either side of it are mirror images.

Part 1: rows step evenly, and the table folds onto itself

Read any row from left to right. Each cell is one more group than the cell before it: 3 \times 5 is 5 groups of 3, and 3 \times 6 is those same 5 groups and one more. So every step along row 3 adds 3, every step along row 8 adds 8, and so on. A row is a skip count, and its step is the row number. The same is true of a column read downward.

Now fold the table along its shaded diagonal. Row 3, column 7 lands on row 7, column 3, and both hold 21. That happens in every cell. An array of 3 rows of 7, turned on its side, is an array of 7 rows of 3, with the same tiles. This is the commutative property, and in the table it shows up as a mirror: the part above the diagonal is the part below it, flipped. Only the cells on the diagonal have no partner, because folding leaves them where they are.

The diagonal has its own pattern. Its numbers are

\begin{aligned} &1, \ 4, \ 9, \ 16, \ 25, \\ &36, \ 49, \ 64, \ 81, \ 100 \end{aligned}

and the gaps between them are 3, 5, 7, 9, 11, 13, 15, 17, 19: the odd numbers, in order. Tiles show why. A 5 by 5 square becomes a 6 by 6 square by adding a column of 5 down the side, a row of 5 along the bottom, and one tile in the corner.

Figure 2Growing a 5 by 5 square to a 6 by 6 square adds a column of 5, a row of 5 and one corner, so the next square is 11 more.

So the step from 5 \times 5 to 6 \times 6 is 5 + 5 + 1 = 11. Every step is built the same way, a double and one more, and a double plus one is always odd.

Part 2: one row from another, and the nines column

Put row 2 and row 4 side by side:

column	row 2	row 4
1	2	4
2	4	8
3	6	12
4	8	16
5	10	20
6	12	24
7	14	28
8	16	32
9	18	36
10	20	40

Each number in row 4 is double the number above it. An array of 4 rows cuts across the middle into two arrays of 2 rows, so 4 of anything is 2 of it twice.

Figure 3Four rows of 6 are two rows of 6 twice, so 4 \times 6 is double 2 \times 6.

The same cut shows that row 8 is double row 4, row 6 is double row 3, and row 10 is double row 5. It works for any two rows that add to a third: an array of 7 rows cuts into 3 rows and 4 rows, so each number in row 7 is the number in row 3 plus the number in row 4. That is the The distributive property at work across a whole row at once. Doubling is the case where the two pieces are the same size.

Next, read the nines column downward:

fact	product	fact	product
9 \times 1	9	9 \times 6	54
9 \times 2	18	9 \times 7	63
9 \times 3	27	9 \times 8	72
9 \times 4	36	9 \times 9	81
9 \times 5	45	9 \times 10	90

The tens digit counts up 0, 1, 2, \dots, 9 while the ones digit counts down 9, 8, 7, \dots, 0, and the two digits always add to 9. The reason is the step. Each cell is one more group of 9 than the cell above, and adding 9 is the same as adding a ten and taking away a one, because 9 = 10 - 1. Adding the ten puts the tens digit up by one. Taking away the one puts the ones digit down by one. One digit goes up as the other goes down, so their sum never changes, and it starts at 0 + 9 = 9 (think of 9 as 09). How to find a nines fact quickly is the work of Facts for seven, eight and nine; here the point is why the column looks the way it does.

Part 3: even and odd products, and row totals

Look at the even rows of the table, 2, 4, 6, 8, 10: every number in them is even. Look at the even columns: every number in them is even too. The only odd numbers in the table are where an odd row crosses an odd column.

The reason is pairing. If one factor is even, the array splits into twos. Say the rows are even in length: each row splits into pairs, so the whole array does. Say instead the number of rows is even: pair the rows off, the first with the second and so on, and each tile has a partner in the other row of its pair. Either way, nothing is left over.

If both factors are odd, every row of the array is some pairs and one tile over.

Figure 4Each row of 5 is two pairs and one tile over. Three tiles over make one pair and one more, so 3 \times 5 is odd.

There are as many tiles over as there are rows, and that is an odd number, so the tiles over make some pairs and one more. One tile is left without a partner, and the product is odd.

Rows can also be added up. Row 1 is 1 + 2 + 3 + \dots + 10 = 55. A column read downward is a skip count by its column number, so going from row 1 to row 2, column b grows by b. So the whole row grows by 1 + 2 + \dots + 10, which is 55. The same is true from every row to the next, so the row totals are 55, 110, 165, 220, \dots, growing by 55 each time.

Results

Rows step evenly. Along row a, each product is a more than the one before it:

a \times (b + 1) = a \times b + a

The mirror. For every row a and column b,

a \times b = b \times a

so the table is the same on both sides of its diagonal. A number off the diagonal appears in mirror pairs.

Square steps. The diagonal holds the square numbers, and each is the one before it plus a double plus one:

(n + 1) \times (n + 1) = n \times n + n + n + 1

Rows that add. If row a and row c add to row a + c, their products add too:

(a + c) \times b = a \times b + c \times b

In particular row 4 is double row 2, row 8 is double row 4, row 6 is double row 3 and row 10 is double row 5.

The nines column. For n from 1 to 10, the product 9 \times n has tens digit n - 1 and ones digit 10 - n, so its two digits add to 9. Each step down the column adds a ten and takes away a one.

Even and odd. A product is even when at least one factor is even, and odd only when both factors are odd.

Row totals. Row 1 adds to 55, and each row adds to 55 more than the row above it.

Why it is true

Rows step evenly. a \times b counts b groups of a. Moving one column to the right makes it b + 1 groups, which is one more group of a.

The mirror. An array of a rows of b tiles, turned a quarter turn, is an array of b rows of a tiles. Nothing is added or removed by turning, so a \times b = b \times a. In the table, row a, column b and row b, column a are mirror cells, so they hold the same number. A cell on the diagonal is its own mirror.

Square steps. Start with n rows of n tiles. Add one tile to the end of each row: that is n tiles, and the array is now n rows of n + 1. Add one more row of n + 1 tiles along the bottom: that is n tiles under the old square and one in the corner. The result is n + 1 rows of n + 1, and the tiles added were n + n + 1.

Rows that add. An array of a + c rows of b cuts across into an array of a rows and an array of c rows, both of length b. The cut moves no tile, so the two pieces together hold all of (a + c) \times b. When a = c the two pieces match, and the product is a double.

The nines column. Going from 9 \times n to 9 \times (n + 1) adds one more group of 9. Adding 9 gives the same total as adding 10 and then taking away 1. In the column, 9 \times 1 is 0 tens and 9 ones. Each step adds one ten and removes one one, so after n - 1 steps there are n - 1 tens and 9 - (n - 1) = 10 - n ones. The ones never run out before 9 \times 10, whose ones digit is 0, so no step needs to break a ten. The digit sum is (n - 1) + (10 - n) = 9 every time.

Even and odd. If the row length is even, each row splits into pairs. If the number of rows is even, the rows pair off, and each tile in a pair of rows is matched with the tile below it. Either way the array splits into twos, so the product is even. If both are odd, each row is pairs and one tile over, and the tiles over, one per row, are an odd number of tiles: pairs and one more. Exactly one tile is left, so the product is odd.

Row totals. Row a + 1 holds (a + 1) \times b in column b, which is a \times b + b: by the mirror, it is b \times (a + 1), one more group of b than b \times a. Adding the whole row, the total is row a’s total plus 1 + 2 + \dots + 10, and that sum is 55.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3, 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.

Example 1

Every place a number sits

Find every cell of the table that holds 24, and explain why the cells come in mirror pairs.

Solution

Go row by row and ask whether 24 is in that row’s skip count.

Row 1 ends at 10, and row 2 ends at 20, so neither reaches 24.
Row 3: 3 \times 8 = 24. Yes.
Row 4: 4 \times 6 = 24. Yes.
Row 5: its numbers end in 0 or 5, so no.
Row 6: 6 \times 4 = 24. Yes.
Row 7 goes 21, 28 and steps over 24.
Row 8: 8 \times 3 = 24. Yes.
Row 9 goes 18, 27 and steps over 24.
Row 10: its numbers end in 0, so no.

So 24 sits in four cells: 3 \times 8, 4 \times 6, 6 \times 4 and 8 \times 3. They pair off as mirror cells, 3 \times 8 with 8 \times 3 and 4 \times 6 with 6 \times 4, because turning an array on its side keeps its tiles. None of the four is on the diagonal: 4 \times 4 = 16 and 5 \times 5 = 25, so no number times itself is 24.

Example 2

From one square to the next

Phindile knows 8 \times 8 = 64. Find 9 \times 9 from it without the table, and say how big the step along the diagonal is.

Solution

Picture 8 rows of 8 tiles. To make 9 rows of 9, add one tile to the end of each of the 8 rows, then a new row of 9 along the bottom: 8 tiles under the old square and 1 in the corner. That adds

8 + 8 + 1 = 17

tiles, so

9 \times 9 = 64 + 17 = 81

The step from 8 \times 8 to 9 \times 9 is 17, a double and one more. Check: the nines column gives 9 \times 9 = 81 too, with digits 8 + 1 = 9.

Example 3

The eights row from the fours row

Unathi writes row 4 and row 8 of the table. Show that every number in row 8 is double the number in row 4, and use it to find 8 \times 7.

Solution

column	row 4	row 8
1	4	8
2	8	16
3	12	24
4	16	32
5	20	40
6	24	48
7	28	56
8	32	64
9	36	72
10	40	80

Why it must be so: 8 \times b is an array of 8 rows of b. Cut it across the middle into two arrays of 4 rows of b. Each half is 4 \times b, and the two halves hold every tile once. So 8 \times b = 4 \times b + 4 \times b for every column b.

For 8 \times 7, take 4 \times 7 = 28 and double it:

8 \times 7 = 28 + 28 = 56

The step along row 8 is 8, double the step of 4 along row 4, which is the same fact seen another way.

Example 4

Two rows make a third

Xolani says each number in row 7 is the number in row 3 plus the number in row 4. Explain why, and use it to find 7 \times 6.

Solution

An array of 7 rows of 6 cuts across into 3 rows of 6 on top and 4 rows of 6 below, because 3 + 4 = 7. The cut moves no tile, so

7 \times 6 = 3 \times 6 + 4 \times 6

The same cut works in every column, which is why the whole of row 7 is row 3 plus row 4. This is the The distributive property applied to a full row.

Now 3 \times 6 = 18 and 4 \times 6 = 24, so

7 \times 6 = 18 + 24 = 42

Check with the mirror: column 7 of row 6 also holds 42, since 6 \times 7 = 7 \times 6.

Example 5

Reading the nines column

Which of 56, 54 and 45 is 9 \times 6? Use the pattern of the nines column, not the fact.

Solution

In the nines column the tens digit is one less than the number of nines, and the two digits add to 9. For 9 \times 6 the tens digit is 6 - 1 = 5, and the ones digit is 9 - 5 = 4.

56 has digits adding to 5 + 6 = 11, so it is not in the nines column at all.
45 has digits adding to 9, but its tens digit is 4, so it is 9 \times 5.
54 has tens digit 5 and ones digit 4.

So 9 \times 6 = 54. Check by the step: 9 \times 5 = 45, and adding 9 means adding a ten and taking away a one, 45 \to 55 \to 54.

Example 6

Odd or even, without multiplying

Decide whether each product is odd or even before working it out: 5 \times 7, 4 \times 9, 3 \times 9, 8 \times 5. Then count the odd numbers in the whole table.

Solution

A product is odd only when both factors are odd.

5 \times 7: both odd, so odd. Indeed 5 \times 7 = 35.
4 \times 9: 4 is even, so even. Indeed 4 \times 9 = 36.
3 \times 9: both odd, so odd. Indeed 3 \times 9 = 27.
8 \times 5: 8 is even, so even. Indeed 8 \times 5 = 40.

For the whole table, an odd product needs an odd row and an odd column. The odd rows are 1, 3, 5, 7, 9, five of them, and each crosses the five odd columns. That is 5 rows of 5 cells, so

5 \times 5 = 25

of the hundred cells hold odd numbers, and the other 75 hold even ones.

Example 7

Row totals

Xolani adds row 1 and gets 55. Unathi adds row 2 and gets 110. Explain why each row adds to 55 more than the row above, and find the total of row 4. Check it by adding.

Solution

Line up row 1 and row 2. Column b read downward is a skip count by b, so from row 1 to row 2 the number grows by 1 in column 1, by 2 in column 2, and so on up to 10 in column 10. The row total grows by all of those together:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

The same is true from any row to the next, since every step down a column adds the column number. So the totals are 55, 110, 165, 220: row 4 adds to 220.

Check by adding row 4 in pairs from the two ends, 4 + 40, 8 + 36, 12 + 32, 16 + 28, 20 + 24. Each pair is 44, and five 44s make 220.

Boundaries

The table stops at 10. Every pattern here is about the cells from 1 \times 1 to 10 \times 10. The reasons behind most of them keep working past 10, but the nines pattern in particular changes shape: 9 \times 11 = 99, whose digits add to 18, because there the step from 90 has no one to take away and must break a ten.
A pattern is a check, not a proof of an answer. A digit sum of 9 shows a number could be in the nines column; it does not say which fact it is. The tens digit settles that. An even answer to 4 \times 7 passes the even test whether it is 28 or 26.
Only the diagonal folds onto itself. Every other cell has a different mirror cell with the same number. A number that appears an odd number of times in the table must sit on the diagonal.
Rows add only when the row numbers add. Row 3 plus row 4 is row 7 because 3 + 4 = 7. Row 3 times row 4 is not row 12, and the table has no row 12.
Zero is left out. The table here starts at 1. A row of 0 would be all zeros: no groups is nothing.

Where this goes next

These patterns are the first look at structure in arithmetic: a statement that holds for a whole family of numbers, with a reason. The row-adding pattern is the The distributive property, which soon multiplies numbers far past the table. The mirror is the commutative property, which halves the table to learn. Reading the table backward, a product and one factor give the other, which is division.

The digit sum of the nines column grows into a test for any number, however large, in Tests for three and nine. And looking for a rule behind a row of numbers, then explaining it, is the work of Number and shape patterns from a rule.

Exercises

Exercises 1 to 7 go with Part 1, 8 to 15 with Part 2, and 16 to 21 with Part 3. The reasoning group draws on all three.

Part 1: rows, the mirror and the diagonal

What is the step along row 8? Write row 8 from 8 \times 1 to 8 \times 10.
Find every cell of the table that holds 30.
Find every cell of the table that holds 16. How many are on the diagonal?
Use 6 \times 6 = 36 to find 7 \times 7 without the table.
Phindile has a square of tiles 3 by 3. How many tiles must she add to make it 4 by 4, and how many tiles does the new square have?
Which cell is the mirror of row 4, column 9? What number do both cells hold?
Which numbers on the diagonal are odd? Say what they have in common.

Part 2: rows from rows, and the nines

Use row 5 to find 10 \times 7 and 10 \times 9.
Use row 4 to find 8 \times 5 and 8 \times 8.
Use row 4 and row 5 to find 9 \times 6.
Write 9 \times 3 and 9 \times 8 from the nines pattern, and check that each digit sum is 9.
Which of 63, 72 and 64 is 9 \times 7? Use the pattern.
Unathi knows 9 \times 4 = 36. What is 9 \times 5, and how did each digit change?
Use row 2 and row 4 to find 6 \times 3.
A number in the nines column has tens digit 1. What is the number, and which fact is it?

Part 3: even, odd and totals

Without multiplying, say whether each is odd or even: 3 \times 7, 6 \times 5, 7 \times 9, 2 \times 9.
How many numbers in row 7 are odd? Name them.
How many numbers in row 6 are odd?
Find the total of row 9, and check it by adding the row.
Row 5 adds to 275. Without adding, find the total of row 6.
The first five numbers of row 1 add to 15, and the first five of row 2 add to 30. How much does this sum grow from one row to the next? Find it for row 6 and check by adding.

Reasoning

Explain with an array why the table is the same on both sides of its diagonal.
Explain why an odd number times an odd number is always odd.
Explain why every number in row 8 is double the number in row 4.
Explain why a number that is not on the diagonal appears an even number of times in the table.

Answers

The step is 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80. Each cell is one more group of 8.
Four cells: 3 \times 10, 5 \times 6, 6 \times 5 and 10 \times 3. Rows 1 and 2 end below 30, and rows 4, 7, 8 and 9 step over it.
Three cells: 2 \times 8, 4 \times 4 and 8 \times 2. One of them, 4 \times 4, is on the diagonal; the other two are mirror cells.
49. Going from 6 by 6 to 7 by 7 adds 6 + 6 + 1 = 13, and 36 + 13 = 49.
She adds 3 + 3 + 1 = 7 tiles, a column of 3, a row of 3 and a corner. The new square has 9 + 7 = 16 tiles, which is 4 \times 4.
Row 9, column 4. Both hold 36, since 4 \times 9 = 9 \times 4 = 36.
1, 9, 25, 49 and 81. They are 1 \times 1, 3 \times 3, 5 \times 5, 7 \times 7 and 9 \times 9: an odd number times itself, and odd times odd is odd.
Row 10 is double row 5. 10 \times 7 = 35 + 35 = 70 and 10 \times 9 = 45 + 45 = 90, doubling 5 \times 7 = 35 and 5 \times 9 = 45.
8 \times 5 = 20 + 20 = 40 and 8 \times 8 = 32 + 32 = 64, doubling 4 \times 5 = 20 and 4 \times 8 = 32.
54. Row 9 is row 4 plus row 5, because 4 + 5 = 9, so 9 \times 6 = 4 \times 6 + 5 \times 6, which is 24 + 30 = 54.
9 \times 3 = 27, tens digit 3 - 1 = 2, and 2 + 7 = 9. 9 \times 8 = 72, tens digit 8 - 1 = 7, and 7 + 2 = 9.
63. The tens digit of 9 \times 7 is 7 - 1 = 6 and the ones digit is 9 - 6 = 3. 72 is 9 \times 8, and 64 has digit sum 10, so it is not in the nines column.
9 \times 5 = 45. The tens digit went up from 3 to 4 and the ones digit went down from 6 to 5, since adding 9 is adding a ten and taking away a one.
18. Row 6 is row 2 plus row 4, because 2 + 4 = 6, so 6 \times 3 = 2 \times 3 + 4 \times 3, which is 6 + 12 = 18.
18, which is 9 \times 2: the tens digit 1 is one less than 2, and the ones digit is 9 - 1 = 8.
3 \times 7 is odd (both odd, 21). 6 \times 5 is even (6 is even, 30). 7 \times 9 is odd (both odd, 63). 2 \times 9 is even (2 is even, 18).
Five: 7, 21, 35, 49 and 63, the odd columns 1, 3, 5, 7, 9, since 7 is odd.
None. Every product with the even factor 6 is even.
495. The totals grow by 55 a row: 55, 110, 165, 220, 275, 330, 385, 440, 495. Check by adding row 9 in pairs from the ends: 9 + 90, 18 + 81, 27 + 72, 36 + 63 and 45 + 54 are each 99, and five pairs of 99 make 495.
330. Each row adds to 55 more than the row above, so 275 + 55 = 330.
It grows by 15 each row, because the five numbers grow by 1 + 2 + 3 + 4 + 5 = 15. For row 6 that is six 15s, which make 90. Check: 6 + 12 + 18 + 24 + 30 = 90.
The cell in row a, column b holds an array of a rows of b. Its mirror cell, row b, column a, holds b rows of a. Turning the first array a quarter turn gives the second with the same tiles, so the two cells hold the same number. Folding along the diagonal puts every cell on its mirror cell, so the two halves match.
Make an array with an odd number of rows, each of odd length. Each row is some pairs and one tile over. The tiles over, one per row, are an odd number of tiles, so they make some pairs and one more. All the other tiles are in pairs, so exactly one tile is left without a partner, and the total is odd.
8 \times b is an array of 8 rows of b. Cut it across the middle into two arrays of 4 rows of b; each half is 4 \times b, and together they hold every tile once. So 8 \times b = 4 \times b + 4 \times b in every column.
If the number sits in row a, column b with a different from b, it also sits in the mirror cell, row b, column a, which is a different cell. So the cells holding it pair off into mirror pairs, and any number of pairs is an even number of cells. Only a cell on the diagonal is its own mirror and has no partner.