Eleventh Heaven

If students can identify the pattern in multiplying a two-digit number by 11, they should be able to use the rule to find the product of other two-digit numbers multiplied by 11. The pattern is explained in Toby’s example. The students should first consider some examples of their own, using their calculators, and then discuss findings. Finally, they should make a conjecture based on the results.
The students can use a spreadsheet to see all the two-digit numbers multiplied by 11:
In cell A1, enter 10.
In cell A2, enter =A1+1.
Fill down to cell A99.
In cell B1, enter 11.
Fill down to cell B99.
In cell C1, enter =A1xB1.
Fill down to cell C99.
The students will then be able to see the pattern. Question 3b encourages the students to extend the pattern for two-digit numbers that add to 10 or more.
You could also discuss with them the strategy of multiplying a factor by 10 and then adding that factor. For example, 12 x 11 = 12 x 10 + 12. (As an extension, the students could apply the multiply-by-11 system to factors with more than two or three digits. For example, 11 x 5 218 is 10 x 5 218 + 5 218 = 52 180 + 5 218.)
After this, encourage the students to discuss and share ways of mentally calculating other two-digit and “teen times” tables. They will be able to multiply using closely related facts. Encourage them to split factors into suitable part-whole combinations and to share their results.
For example:

• 13 x 12 = (13 x 10) + (13 x 2) = 130 + 26

• 19 x 7= (20 x 7) – 7 = 140 – 7

• Work out the 12 times table by adding the two times table facts to the 10 times table facts.

• For the even teen times tables, halve the teen number and multiply the product by 2.

For example:
14 x 4 = (7 x 4 ) x 2
= 28 x 2
= 56
Practising with a classmate, with one student checking on the calculator and creating questions, should help improve confidence over time.
As a variation, see if the students can predict the results of:
1 x 1 =
11 x 11 =
111 x 111 =
1 111 x 1 111 =
11 111 x 11 111 =

(This is also discussed in “Investigating 11” in the Connected 2 2000 teachers’ notes, page 19.)

Answers to Activity

1. a. 165, 275, 385, and 495
b. The ones and the hundreds digits are the same as the two digits in the original factor (for example, 25 x 11 = 275). These two digits add up to the tens digit (2 + 5 = 7).
2. a. 3_6: 3 + 6 = 9. The answer is 396.
b. 4_3: 4 + 3 = 7. The answer is 473.
c. 5_2: 5 + 2 = 7. The answer is 572.
3. a. The pattern works in this way:
Think about any two-digit number, for example, 35. When the 5 is multiplied by
11, the result is 55, which is 5 tens and 5 ones. When the 30 is multiplied by 11, the
result is 330, which is 3 hundreds and 3 tens.
So, the ones digit of the answer must be 5 because this is the only way ones are
created. The hundreds digit must be 3 because this is the only way hundreds are
created. The tens digit in the answer will be the result of adding together 5 tens and 3
tens, that is, 8 tens.
3. b. If the two digits of the number add up to 10 or more, an extra step is necessary.
For example, for 57, 5 + 7 = 12. Using Toby’s method:
57 x 11 = 5_7.
But 5 + 7 = 12, and you can’t put 12 in the tens column, so you have to add an extra 1 to the hundreds digit:

Another way of showing this is

4. Answers will vary.
For example, the 19 times tables are “one less than” the 20 times tables.
7 x 19 = (7 x 20) – 7
= 140 – 7
= 133
12 x 19 = (12 x 20) – 12
= 240 – 12
= 228
You could also use the 10 times table.
For example:
12 x 19 = (10 x 19) + (2 x 19)
= 190 + 38
= 228
Ways of remembering the 15 times table are given on page 2 of the student booklet.
The 13 times table is a combination of the 10 times and 3 times tables. For example:
9 x 13 = (9 x 10) + (9 x 3)
= 90 + 27
= 117
16 x 13 = (16 x 10) + (16 x 3)
= 160 + 48
= 208
This method could be used for other tables. For example:
2 x 14 = (2 x 10) + (2 x 4)
= 20 + 8
= 28
For “teen” times tables that are even numbers, you could use the “halve and double” method.
For example, the 14 times table is double the 7 times table, the 16 times table is double the 8 times table, and so on.