## The Concept of a Variable: Expressing a Pattern

### The Basic Problem

Consider this:
1 ---> 2
2 ---> 3
3 ---> 4
4 ---> 5

This is saying that when someone is given the number 1, they produce 2; when they are given 2, they produce 3; etc.

The student's task is to say what the pattern is. This is meant to be a very easy problem, which everyone can solve. Putting it into words might not be simple, but if you allow for a little inarticulation, everyone can do that too.

This is also designed to be a problem that the answer is a variable. Suppose the answer is something like "given some number, the person produces the next number." In this phrasing, "some number" is functioning exactly like a variable. All that remains is for you the teacher to attach this concept to some terminology. Explain that "some number" can be called n, and that the person, when given n, produces the number after n, also known as n + 1.

This is not an easy concept, unless the student is already familiar with it. It took mathematicians a long to invent it too. "The next number", by itself, doesn't count as a good enough answer to "What is the pattern?" You need to ask, "The next number after what?"

Students are commonly told that a variable is like a pronoun. We can applaud the effort to attach the concept of a variable to the student's existing mental model. However..... This problem actually elicits the appropriate mental model (if the student can be said to have the mental model before the problem is given). Also, when people use pronouns, they know what they are talking about. When you use n to represent some number in this problem, you do not know what number you are talking about.

The pattern above is the easiest possible pattern, and of course would be the best starting point for anyone who would have trouble noticing a pattern. But if there is no difficulty recognizing the pattern, a better problem for technical reasons is:
3 ---> 6
2 ---> 5
4 ---> 7
1 ---> 4

This avoids the problem that the person's answer can be predicted from the person's last answer. It also avoids the problem that the person's answer is "the next number", which is not easy to capture with a variable. Instead, the person's answer, given some number, is "that number plus three", which is easily expressed as n + 3.

### Other Problems in this Format

n ---> ?

3 ---> 6
2 ---> 4
4 ---> 8
1 ---> 2

n ---> ?

1 ---> 1
2 ---> 4
3 ---> 9
4 ---> 16

n ---> ?

1 ---> 5
2 ---> 8
3 ---> 11
4 ---> 14
5 ---> 17

n ---> ?

The last one was difficult for a bright sixth-grader.

### The "Ultimate" Problem

Suppose you ask your student to determine the sum of the numbers from 1 to 3. Once the student understands the problem, he/she will add up 1 + 2 + 3 and say six. Then you could ask for the sum of the numbers from 1 to 4, then you could ask for the sum of the numbers from 1 to 5. Now, each time the student could add up all of the numbers anew. But, perhaps because they are embedded in the actual problem and realize they are adding up the same numbers over and over again, at least some students will realize that they just need to keep a running total. Then, for example, if the sum of the numbers from 1 to 10 is 55 and you ask for the sum of the numbers from 1 to 11, they can calculate 55 + 11 = 66.

In other words, in terms of a pattern, (1,2) --> 3 (1,3) --> 6 (1,4) --> 10 (1,5) --> 15 (1,n-1) --> k (1,n) --> k + n

Suppose, when asked for three numbers that add to 20, your student has produced 1+2+17=20. And to subsequent problems the answers were 1+2+47=50, 1+2+97=100, and 1+2+197=200. There is a pattern here. (If your student has not fallen into a pattern, you can say that you produced those answers.)

This is not a difficult pattern. If the pattern wasn't produced by your student you can check that your student can predict the next answer, following the pattern.

The next problem is to put the pattern into words. This is not as easy as it looks, and you should be tolerant of poor communication. But verbalizing a pattern is an important skill.

Okay, the key problem goes like this. If the problem is to think of three numbers that add up to some number, what three numbers will be produced? Call this number, whatever it is, m. What three numbers will be produced to add up to m?

They probably won't get it. There are two conceptual problems. So you should back up to (or start with) a simpler pattern. Suppose the solutions are 0+0+20=20, 0+0+50=50, and 0+0+100=100. This problem is easier to express -- if the goal is three numbers that add up to m, the solution is 0+0+m=m.

You are probably going to have to show this to a young child. Apparently it is a tough concept. You are going to have to lecture too, but give the problem first.

Of course, the solution to the first problem is 1+2+(m-3)=m. It is hard enough to think of m as being a number. The next conceptual leap is realizing that m-3 is a number.

Once they have seen what an answer is like, you can give as many problems as needed until your student is comfortable with this. For example, the pattern could be 2+3+(m-5)=m or 0+1+(m-1)=m. Of course, you just give examples and your student has to say the pattern.

If your child is comfortable with division, patterns can use division. Actually, the most common pattern for young children, to the problem of two numbers that add up to something, is 5+5=10, 10+10=20, 50+50=100. Given the pattern 10+5+5=20, 50+25+25=100, and 100+50+50=200, my daughter produced the answer of m/2+m/2/2+m/2/2=m. Close enough.

### Using the Pattern

Once your student is comfortable with this, you should reverse the problem. In other words, given that a person is following the pattern 1,2,(m-3)=m, what three numbers will a person produce when given the number 50? Again, you might want to start with an easier pattern.

This segues into difficult answers. For example, given the pattern 1+2+(m-3)=m, what three numbers will be produced to add up to 3? If your child knows negative numbers, what three numbers will be produced to add up to 2? Patterns involving division can be manipulated to lead to fractions.

When asked for two numbers that add to 10, Sarah said 5 + 5. When asked for two numbers that add to 100, Sarah said 50 + 50. When asked for two numbers that add to 82, Sarah said 41 + 41.

What is Sarah's pattern? Call the number that she is supposed to add to n. The two numbers she produces are n/2 and n/2. Of course, you can call that number whatever you want, but it is convenient to use a single letter, and that is what mathematicians usually do. If you called the number q, then Sarah is producing q/2 and q/2.

If Sarah was asked for two numbers that add up to 9, she might have changed her pattern. But if she had followed her pattern, she would have produced 9/2 and 9/2 (which is also known as 4 1/2). To follow her pattern for two numbers that add up to -10, she would have to produce -5 and -5.

When asked for three numbers that add to 12, Celina produces 4 + 4 + 4. When asked for three numbers that add up to 30, Celina produced 10 + 10 + 10.

What is Celina's pattern?

If Celina was asked for three numbers that add to 36, what would she produce?

If Celina was asked for three numbers that add to 1, what would she produce?

If Celina was asked for three numbers that add to 5, what would she produce?

If Celina was asked for three numbers that add to -6, what would she produce?

When asked for three numbers that add to 20, Canaan produced 2 + 3 + 15. When asked for three numbers that add up to 30, Canaan produced 2 + 3 + 25. When asked for three numbers that add up to 100, Canaan produced 2 + 3 + 95.

What is Canaan's pattern?

If Canaan was asked for three numbers that add to 36, what would he produce?

If Canaan was asked for three numbers that add to 1000, what would he produce?

If Canaan was asked for three numbers that add to 1,000,000, what would he produce?

If Canaan was asked for three numbers that add to 5, what would he produce?

If Canaan was asked for three numbers that add to 1, what would he produce?

If Canaan was asked for three numbers that add to -6, what would he produce?

Call the goal k. When asked for three numbers that add up to k, Lindsay always produced three numbers following the pattern 0, 1, k-1.

When asked for three numbers that add up to 100, what three numbers did Lindsay produce?

When asked for three numbers that add up to 1,000,000,000, what three numbers did Lindsay produce?

When asked for three numbers that add up to 4 1/3, what three numbers did Lindsay produce?

When asked for three numbers that add up to -7, what three numbers did Lindsay produce?

When asked for three numbers that add up to 1/2, what three numbers did Lindsay produce?

Call the goal n. When asked for three numbers that add up to n, Gregory always produced three numbers following the pattern n, -n, and n.

What three numbers did Gregory produce to add up to 10?

What three numbers did Gregory produce to add up to 47?

What three numbers did Gregory produce to add up to 1,000,000?

What three numbers did Gregory produce to add up to 4.379?

What three numbers did Gregory produce to add up to -10?

### Content

The content area is discovering patterns, expressing patterns, and learning the concept of a variable.