Problem
Paul is on the phone to his friend Pesi. He had this nice pattern that he was trying to describe to Pesi but he couldn’t find the words.

If Paul’s pattern was:
3, 7, 11, 15, …
how could he describe to Pesi how to get any member of the number pattern? How could Paul tell Pesi how to get the 50th number in as simple a way as possible?

Then Pesi thought of a pattern. Pesi’s pattern was
3, 6, 12, 24, …
how could Pesi describe to Paul how to get any member of the number pattern? How could Pesi tell Paul how to get the 50th number in as simple a way as possible?

What is this problem about?
The aim here is to get control over terms in a sequence. In this context that means to be able to state a rule for any term in the sequence. What we are aiming to do here is to get an expression for the nth term of a sequence. In Algebra Level 3 there are some other problems like this that can now be tackled from a more sophisticated point of view (see Toothpick Squares and Race To 100).

Achievement Objectives
Algebra (Level 4)
- find a rule to describe any member of a number sequence and express it in words.

Mathematical Processes
- devise and use problem solving strategies to explore situations mathematically (systematic, guess and check)

Resources
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- use their own words to describe a number sequence
- use a rule to find a member of a number sequence

1. Introduce the lesson with a game of "who belongs". In this game the children try to guess members of a number sequence. As they guess the teacher sorts the numbers into 2 lists on the board – "does belong" and "doesn't belong".
   As the lists develop the children attempt to guess the rule – however they keep this rule secret until the end of the game when rules are shared.
   For example: The rule is multiples of 4
   Do belong 12,  24,  8,  4
   Do not belong 1, 0, 5, 7, 6, 13, 15
2. Pose the problem to the class.
3. As the children work ask them questions that require them to describe the patterns in their own words.
4. If the children are having problems with finding the patterns encourage them to explore the size of the "jumps" between the numbers.
5. As the children work to find the 50^th number remind them to find the easiest way. Although the children could continue the sequence 50 times this is time consuming and does not require them to find a rule for any term in the pattern.
6. Share descriptions of the number patterns – these could be written for display on the wall.

Solution to the problem
In Paul’s sequence, the first term is 3. To get from the first to the second term, he had to add 4. The same thing has to be done in going from the second term to the third term. And if he adds another 4 he gets the fourth term. So to tell Pesi how to generate the pattern, he only has to say "Pesi, you just start at 3 and keep adding 4. That way you’ll get all members of my pattern."

Now that doesn’t help us with the 50th term. Of course, Pesi could now do the ‘adding 4’ until he got to the 50th term but can he do better than that?

Well, to get the first term Paul took 3 and added no 4s. To get the second term, Paul took 3 and added one 4. To get the third term he took 3 and added two 4s. To get the fourth term he took 3 and added three 4s. The number of 4s that Paul adds is always one less than the number of the term. So for the sixth term Paul will take 3 and add five 4s.

So Paul can now tell Pesi how to get the 50th term quickly. "Pesi, you just take 3 and add 49 4s." To which Pesi replies "Great, so the 50th term is 3 + 49 x 4 = 199. Thanks, I needed that."

Then now it’s Pesi’s turn. Remember his sequence is 3, 6, 12, 24, … So here he is starting with 3 and doubling each time to get the next number in the sequence. So he says just that. "Paul, just take 3 and keep doubling." Well maybe that’s not too accurate but it will do for now.