Pair Products

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Age

14 to 16

Challenge level

Secondary

ALGEBRA

Creating & Manipulating Linear & Quadratic Expressions

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Pair Products printable worksheet

Choose four consecutive whole numbers.

Multiply the first and last numbers together.

Multiply the middle pair together.

Choose several different sets of four consecutive whole numbers and do the same.

What do you notice?

Can you explain what you have noticed? Will it always happen?

Click below to see how Charlie and Alison explained what they noticed.

Charlie said:

I noticed that the product of the outer pair was always $2$ less than the product of the inner pair.

I can explain this by labelling the four consecutive numbers $n, n+1, n+2, n+3$.

Outer pair: $n(n+3) = n^2 + 3n$

Inner pair: $(n+1)(n+2) = n^2 + 3n + 2$

Alison said:

I drew a diagram, in which the product of each pair is represented by the area of a rectangle:

The outer pair is represented by the red rectangle.

The inner pair is represented by the blue rectangle.

The purple area is common to both.

The area of the red strip will always be two units less than the area of the blue strip.

Therefore, the product of the outer pair is always two less than the product of the inner pair.

Instead of doing lots of calculations, can you use these representations to compare the product of the first and last numbers with the product of the second and penultimate numbers, when you have:

$5$ consecutive whole numbers
$6, 7, 8, \ldots x$ consecutive whole numbers
$4$ consecutive even numbers
$4$ consecutive odd numbers
$5, 6, 7, 8, \ldots x$ consecutive even or odd numbers
$4$ consecutive multiples of $3, 4, 5 \ldots $

Decimals that differ by $1$, such as $1.2, 2.2, 3.2, 4.2$
Four numbers going up in $3$s, such as $2, 5, 8, 11$
Four numbers going up in $\frac{1}{2}$s, such as $4, 4\frac{1}{2}, 5, 5\frac{1}{2}$

Make up a few similar questions of your own. Impress your friends by giving them a calculator and 'predicting' what will happen!

Click here for a poster of this problem.

Barbora from Parklane International School in the Czech Republic wrote about the patterns in the numbers:

I have noticed that the product of the middle pair is always 2 larger than the product of the outer pair. I have also noticed that the sum of the 1st and 4th number is equal to the sum of the 2nd and 3rd number. Furthermore, the 3rd number is always 2 larger than the 1st number and the 4th number is always 2 larger than the 2nd number. I think that these things will always happen because the numbers are arranged in the same order each time, so there isn't any variation in the pattern other than the integers used. The pattern is always: $n, n + 1, n + 2, n + 3$

Lyle, Caden and Michael from Cockburn Laurence Calvert Academy used Barbora's idea to explain what happens to the products. This is Lyle's work:

4 consecutive numbers	Multiply first and last numbers	Multiply middle pair	Notes
1, 2, 3, 4	1 $\times$ 4 = 4	2 $\times$ 3 = 6	difference of 2
2, 3, 4, 5	2 $\times$ 5 = 10	3 $\times$ 4 = 12	difference of 2
10, 11, 12, 13	10 $\times$ 13 = 130	11 $\times$ 12 = 132	difference of 2
1000, 1001, 1002, 1003	1000 $\times$ 1003 = 1003000	1001 $\times$ 1002 = 1003002	difference of 2

Solution

$\begin{align}&n\hspace{3mm}\times \hspace{3mm}n+3&\hspace{15mm}&n+1\hspace{3mm}\times \hspace{3mm}n+2\\

&n^2+3n& &n^2+3n+2\end{align}$

Difference of 2

Ben from Comberton Village College in the UK, Andrew from Kellett in Hong Kong and Ella from Parklane International School also used the same algebra as Lyle. Andrew repeated the process for sets of 6 and 10 consecutive numbers (click on the image to open a larger version):

Matouš and Matyáš from Parklane International School in the Czech Republic wondered what the difference would be for other numbers of consecutive numbers, like 154. Here is their work:

For any number of consecutive numbers:

$x$ = total number of consecutive numbers

$y$ = the difference between the two products (1st $\times$ last and 2nd $\times$ penultimate)

Eg. If we use all the consecutive numbers between and including 1 and 154.

$x$ = 154

1 $\times$ 154 = 154 (1st $\times$ last)

2 $\times$ 153 = 306 (2nd $\times$ penultimate)

$y$ = 306 $−$ 154 = 152

Have they made any assumptions?

Freya from Comberton Village College applied the same method to numbers that are not consecutive, but go up in threes:

The sequence I chose goes up in threes. For my example, I used 2,5,8,11.

Outer pair: 2$\times$11=22

Inner pair: 5$\times$8=40

40$-$22=18

To explain this, I used algebra.

$n(n+9)=n^2+9n$

$(n+3)(n+6)=n^2+9n+18$

$(n^2+9n+18)-(n^2+9n)=18$

Imaan from Walton High School in England investigated several different numbers of numbers and distances between the numbers. Here is Imaan's work (click on the image to open a larger version):

Sophia and Bianca from Kellet and Jackson from Cockburn Laurence Calvert Academy tried out different differences between the numbers in a systematic way. This is Jackson's work, which includes a graph (click on the image to open a larger version):

Alain, Teodor and Fabian from Parklane International School and Andrew Chambers's students at British International School Phuket in Thailand both wrote up lots of ideas, rules and proofs - including things which could be used to explain the shape of Jackson's graph. The British International School Phuket students really enjoyed this task and spent 2 full lessons working on it. They worked out the difference between the products of the first and last and the the middle two numbers, rather than the first and last and the second and penultimate numbers:

Click here to see the British International School Phuket students' work, and click here to see Alain, Teodor and Fabian's work. Both also include explorations of when the numbers are all even, or all odd.

Venya from the British International School Phuket considered a very general case, where there are $n$ numbers that go up in steps of $q$:

Zara from St James Senior Girls School carried out a slightly different investigation into whether the products were even or odd (click on the image to open a larger version).

Why do this problem?

This problem provides a purpose for practising the routine algebraic procedure of expanding brackets.

For students who are unfamiliar with algebra, this problem is an excellent context for observing, conjecturing and thinking about proof. It can be a good introduction to the power of algebra, and to a related geometrical interpretation.

Possible approach

This printable worksheet may be useful: Pair Products.

"Choose four consecutive numbers, multiply the outer pair and the inner pair. What were your two answers?"

Write a selection of students' responses on the board.

"What do you notice?" The inner pair product is always two more than the product of the outer pair.

"Will this always happen? Can you explain why?"

Give students some time to discuss with their partner why the answers always differ by two. Circulate and listen out for interesting insights.

Bring the class together and share any explanations they have found. Perhaps share Charlie's and Alison's representations from the problem if they haven't emerged.

"We've worked out what happens when you find the product of the inner and outer pair of a set of four consecutive numbers. What questions do you think a mathematician might ask next?"

Write students' suggestions up on the board. If suggestions are not forthcoming, introduce some of the ideas listed in the problem.

"You should be able to work out what will happen in the situations you've suggested using one of the powerful representations we've looked at, without having to try out lots of numerical examples first. Of course, if you want, you can use a numerical example to verify what you've done."

Students could be offered a choice of which situations to work on. Alternatively, you may want everybody to work on a series of related problems (5, 6, 7 ... n consecutive numbers, for example) that will lead to a generalisation.

One nice plenary activity is to challenge students to work out quickly what the difference in pair products will be for a randomly chosen sequence of numbers.

One blogger wrote a piece on his views of the problem, which you can read here.

Key question

Is there a way to represent the pair products that will explain the patterns you noticed?

Possible support

This problem could also be approached purely numerically, as an exercise in developing fluency with multiplication tables while looking for pattern and structure.

In g etting started, there is a dynamic image of Alison's representation that could be used to help students to see why the difference will always be 2.

Possible extension

This problem only operated on the end numbers and the 'end but one' numbers. Invite students to generalise further by looking at other pairs within the sequence.

For example, if you have an odd number of consecutive numbers, what's the difference between the product of the end numbers and the square of the middle number?

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