To introduce the notation for recursive patterns, begin by asking
students to find patterns and predict the next number in the following
- 5, 8, 11, 14, …
- 4, 6, 8, 10, 12, …
- 10, 20, 30, 40, …
Give students some time to work through these examples and
ask them to write down how they got their answers. Have students share
their answers with the class. Students may have different answers, so
allow students who have different methods to explain how they got their
answers. Correct any misconceptions.
Discuss students' responses with the following:
- The next number in the first pattern is 17 (add 3 to the previous number).
17 = 14 + 3
Ask them how to get the next number: 20 = 17 + 3
Students should be able to explain that to get the next number, you add 3 to the previous number.
In mathematical language this is An = An – 1 + 3 and is called a recursive rule. Recursion is the root of recursive.
|Recursion is defining the next (subsequent) term in a pattern (sequence) by using the term(s) that came before it.|
The terms subsequent and sequence are formal
language that can be substituted into the definition if a formal
definition is needed. If students are not used to these formal words,
it may be helpful to define math terms using their vocabulary.
- The next number in the second pattern is 14 (add 2 to the previous number): 14 = 12 + 2
Ask them how to get the next number: 16 = 14 + 2
Tell them that in order to get the next number, you keep adding 2 to the previous term. In mathematical language this is An = An – 1 + 2
- In the third pattern, the next number is 50 (add 10 to the term before).
Ask students to write this pattern using the new recursion notation.
Check that students know what the subscripts refer to.
At your discretion, you may choose to talk about the first term as the
“0th” term or as the “1st” term. Note that students easily grasp that
the first term is the “1st” term or n = 1, but may have trouble with n = 0 as the 1st term.
There is a rule that can be applied to all 3 of the patterns. Wait for students to think about this. If necessary, suggest the next number in any pattern is double the term before, minus the term before that.
Ask students to try it out for each one. Check that they see how this works.
- From pattern 1: 2 × 14 – 11 = 17
- From pattern 2: 2 × 12 – 10 = 14
- From pattern 3: 2 × 40 – 30 = 50
Challenge the class find the recursive rule of An = 2An – 1 – An – 2 and explain that n – 2 refers to the term that is 2 before.
To check for understanding, ask them to write a rule for 3, 7, 11, 15, ….
Ask students to find the next term and to write 2 recursive rules. An = An – 1 + 4 and An = 2An – 1 – An – 2 will both work. If they need more examples, use either or both of the following:
- 6, 11, 16, 21, …
- 22, 25, 28, 31, …
Before moving on, make sure to explain recursion again, and make sure students have a working definition written down.
Finding Recursive Patterns Using Trains
Explain to students that they will use two lengths of cars to form trains:
Show students the trains below and explain that even though they use the same cars, they are two different trains.
| || || |
|a 4-train made from 2 cars of length 1 and 1 car of length 2 || ||a 4-train train made from 1 car of length 1, 1 car of length 2, and another car of length 1 |
Explain that the train below is a train of length 5 made from 1 car of length 1 and 2 cars of length 2.
Give students an opportunity to ask questions, and allow them to
hold the trains if it helps them to explain their thinking. Emphasizing
the difference between train length and car length at
the beginning helps students to talk about the different types of
trains. It gives them a common language. To assess their understanding,
hold up a train and ask students to describe it to you. Repeat this
until all students understand. Then, have students work in groups
of 2–4 to build the trains. This will decrease the number of train
combinations you will have to check.
Distribute the following: train sets; Counting the Trains
activity sheet; grid paper; and colored pencils, markers, or crayons.
Before the lesson, prepare set of Cuisenaire train cards (or paper
strips if you are using them as substitutes). Make sure that each set
has 50 cars of length 1, 20 cars of length 2, 15 cars of length 3,
10 cars of length 4, and 5 cars of length 5.
Students who can’t see certain colors could put a number in each
car. Explain that studnets should use the grid paper and the colors to
record their trains. Remind students that they must build and record
their trains. It might help to ask, "If I were to take your cars away,
would you still be able to tell me what ALL the combinations you built
were by reading what is on your paper?" Suggest that they use black to
represent the white trains to avoid problems with recording white
Make sure that students have built and recorded all of the
different trains they need to complete Question 1 on the activity
sheet. When students say they have them all, be sure to ask how they
know they have them all. If they are missing trains, you can tell them
that they are missing some. You might need to point out another train
that has the same cars or point to another train of a different length
that looks similar. When all else fails, give them the pieces they need
and let them build the train(s) they are missing.
If some students finish building all the trains sooner than others,
encourage them to move on to the subsequent questions. Students who
complete the activity sheet before others are done can always do the Extension activities listed below.
Lead a whole group discussion, allowing students to present how they
figured out the number of trains of length 5 and including the
Students should be able to see that the trains of length 5 are made
from adding the trains of length 3 and length 4, and see that A5 = A4 + A3. Ask them to explain how they made trains of length 6 and to show you the recursive rule (A6 = A5 + A4).
At this point, ask students how to find the trains of length n. They should be able to tell you that you take the train before (n – 1) and the train before that (n – 2), which leads to the rule An = An – 1 + An – 2.
Ask students to present or explain their table and scatter plot. Have
students discuss any difficulties they ran into in making the table.
You may want to point out that train length should be on the x‑axis
since it is the independent variable (save this discussion if this is
not part of your curriculum). When discussing the scatter plot, ask
students if they think the graph is linear. Have them explain their
reasoning. Depending on the students’ exposure to exponential
functions, they may be able to explain why the function is exponential.
The common ratio for this function is approximately (1+√5)/2
(which is about 1.618) and is directly related to Fibonacci numbers. A
discussion of finding a regression line and a common ratio may be
appropriate, depending on students’ prior experience with regression
and exponential functions. You may want to ask students to calculate an
exponential regression for the data in Questions 4 and 5 on their
Directions for calculating a regression line on the TI-83/TI-84 calculator are available on the Using the TI-83 or TI-84 for Regression sheet. The calculator gives y = 0.685 · 1.632x as a regression line with an r‑squared
value of 0.998, which indicates that it is a very good fit. If students
use more data points, the regression comes closer and closer to
To help get at why this is, use the Golden Ratio lesson in the Extensions section. The Fibonacci Rabbits
activity sheet from that lesson can also be used as an assessment.
However, More Trains, the next lesson in this unit, involves another
pattern that looks exponential and will lead students to a better
understanding of lines of best fit.