Patterns are Predictable

I made the point in previous posts (The Study of Patterns and Next-Near-Far-Any ) that algebraic reasoning is based on our ability to notice patterns and generalise from them. As our students become more proficient at spotting patterns we must get them to realise that patterns are predictable: if you have spotted the pattern rule correctly, then you will be able to predict any term with 100% accuracy. 
This differs from the predictions we make in, say, handling data, where our predictions (even though they might be well thought out) aren’t guaranteed to hold. 
The Next-Near-Far-Any continuum is also useful in helping our students make predictions. First get them to predict what might come next. Then predict a ‘near’ term such as the tenth (this might be done continuing the pattern one at a time). From junior grades onwards, students might then be asked to predict a ‘far’ term such as the hundredth. As it is now impractical to continue the pattern one at a time, students will now begin to generalise patterns between the position number and the term number. This is more easily done when we use visual patterns. From intermediate grades onwards, students might be asked to predict any term: this is true generalising. In other words, for position number n, what is the term number in terms of n?
A great source of pattern prompts can be found at Fawn Nguyen's Visual Patterns site. These type of questions are 'Here are the first three terms: figure out what happens next.' They can be relatively straightforward with colour being used as a clue to how the pattern grows:

to more complex such as this one which asks 'How many visible spots in this view are there?' (The first term has 33 visible dots):

I love these types of questions but I like to balance them questions where students are given the chance to make their own patterns: I am always blown away by how creative they can be. Last week I was in a Grade 5/6 class and we asked the students to make their own patterns. They then had to look at someone else's pattern and predict the tenth and hundredth terms. Here is a selection of what we saw:

Notice how some of these combine growing patterns with repeating patterns. For example, to predict the hundredth term of the first pattern above, you would need to notice that the vertical line increases by one each time and that the colours rotate in a cycle of four. Since every fourth term is a blue vertical line with a single green square, we can predict that the hundredth term will be 100 vertical blue squares attached to a single green square. It was great seeing the students make and justify their predictions.
A variation on this is where you give a single term and ask the students to fill in the pattern from this single piece of information. This one (from Marian Small) is my favourite and I've used it with students from grades 4 to 11.

This is the fifth term. Make the first four terms. 
Students always come up with a variety of patterns: some increasing; some decreasing; some linear; some non-linear. I like to plot these patterns on a single graph and ask students to explain why these plots cross at the same point.



And if you need further evidence of the creativity of students, a colleague gave the dice problem above to his computing class and before long, one of his students had written the following program (in Turing, I believe) to solve the problem. It is, I hope you agree, a thing of beauty.

function dice (term : int) : int
    var dotnum : int := 0
    dotnum += 10 %Tops of yellow and red dice *
    dotnum += term * 15 %Sides of yellow and red dice *
    dotnum += 5 * term ** 2 %Tops of white dice *
    dotnum += 3 * term ** 2 %Sides of white dice *
    dotnum += 6 * term ** 2 %Sides of white dice
    dotnum -= 6 * term %Covered sides of white dice
    result dotnum
end dice

Bar Models 2

Following on from my last post, here are a couple of further examples of bar models being used.
The first is in a Grade 6 class and we wanted to specifically give them a multiplicative comparison problem. These problems tend to be more difficult for students compared to equal groups problems; I think this may be because students have a hard time visualising what is happening. Bar models can certainly help here.
Our question was:

Mr. Jacobs eats some candies. Mrs. Delaney eats four times as many candies as him. Together, they eat a total of 260 candies. How many candies did each of them each?

A typical response is shown below.

By using bars to represent how many candies Mr. Jacobs has and how many candies Mrs. Delaney has, the calculations needed to be performed are clearer to see: the combined five bars must represent the total 260 candies. Thus one bar must be 260÷5=52.

In another example (this time from a Grade 2/3 class) the student uses a bar model to visualise this part-part-whole problem:

A school has 83 water bottles. If 29 are filled, how many are not filled?

A typical response is shown below:

What I like about this, is that the student drew a bar model first and used this to help decide what number sentence could be written. Once this was done, the student decided to use an empty number line to calculate the difference. So whilst this might be thought of as a subtraction problem (when the unknown isolated, you get 83-29) it can be solved by addition. The beautiful thing about bar models is that they allow students to visualise both ways.
I'm not sure if students will become experts at using bar models to solve problems unless we as teachers model them effectively and consistently. In many ways a whole school approach is necessary. In primary, teachers should model how to use bar models to solve additive thinking problems. In junior grades, teachers need to model how to use bar models to solve multiplicative thinking problems. In intermediate grades, teachers need to model how to use bar models to solve proportional reasoning problems.

The pay off will be massive.