This problem solving activity has an algebra focus.
Paul is talking to Pesi on the phone.
He is trying to describe a pattern to Pesi but he can’t find the words.
If Paul’s pattern is: 3, 7, 11, 15, …how can he describe to Pesi how to get any member of the number pattern?
How can Paul tell Pesi how to get the 50th number in as simple a way as possible?
- Find a rule to describe any member of a number sequence and express it in words.
- Devise and use problem solving strategies to explore situations mathematically (systematic, guess and check).
In this problem students must describe a sequence in words, and state a rule for any term in the sequence, using an expression for the nth term of a sequence. Note that the focus is on students using their own language to describe a rule. They are not expected to create a rule using variables (e.g. letters). This could be an extra focus for students ready for further extension. Students exploring the rule in further detail could be encouraged to use a table, or create a graph, to show the continuation of the pattern.
Other related Level 3 Algebra problems include: Toothpick Squares and Race To 100.
The Problem
Paul is talking to Pesi on the phone. He is trying to describe a pattern to Pesi but he can’t find the words.
If Paul’s pattern is:
3, 7, 11, 15, …how can he describe to Pesi how to get any member of the number pattern?
How can Paul tell Pesi how to get the 50th number in as simple a way as possible?
Teaching Sequence
- Introduce the lesson with a game of "who belongs". In this game the students try to guess terms (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 students 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. - Pose the problem to the class.
- As the students work, ask them questions that require them to describe the patterns in their own words.
How could you describe this pattern?
Could you describe it another way?
What happens to each member of the pattern?
Is there a rule that could help us find the 50th member of the pattern? - If the students are having problems with finding the patterns encourage them to explore the size of the "jumps" between the numbers. Students ready for extension could create the rule using a table, and use it to find other members of the pattern.
- As the students work to find the 50th number remind them to find the easiest way. Although the students could continue the sequence 50 times, this is time consuming and does not draw attention to the need to find a rule for any term in the pattern.
- Share descriptions of the number patterns.
Extension
Pesi has a pattern. It is 3, 6, 12, 24, …
How can Pesi describe to Paul how to get any member of the number pattern? How can Pesi tell Paul how to get the 50th number in as simple a way as possible?
Solution
Paul’s sequence is 3, 7, 11, 15, …
The first term is 3. From there, Paul keeps adding 4. Paul can say "Pesi, you start at 3 and keep adding 4. That way you’ll get all members of my pattern."
The 50th term can be found by ‘adding 4’ until Pesi gets to the 50th term.
The more efficient way is to tell Pesi, "You just take 3 and add 49 4s." To which Pesi replies "Great, so the 50th term is 3 + (49 x 4) = 199."
Using variables, this rule could be expressed as 3 + 4(T - 1) where T represents the term number in the pattern.
Solution to the Extension
Pesi's sequence is 3, 6, 12, 24, …
He starts with 3 and doubles each time to get the next number in the sequence. He says, "Paul, take 3 and keep doubling."
Since doubling is done one less time than the number of the term in the sequence, he tells Paul, "Take 3 and double it, then double it again and keep doing this for 49 doublings. So the 50th term is 3 x 2 x 2 x … x 2, where there are 49 2s." A calculator may be needed to work that one out. The number 3 x 249 is very big! It’s roughly 3 with 15 zeros after it!