Interactive number patterns
How good are you at finding the formula for a number pattern ?
Problem
This problem involves making a connection between the pattern in a sequence of numbers and the algebra that expresses the general term.
You can right-click and save to get your own copy of each resource: Interactive Number Patterns 1 and Interactive Number Patterns 2
Open the spreadsheet called Interactive Number Patterns 1 and experiment with the control buttons. You should discover that the red and blue numbers in the boxes, and also the algebra, can be made to show or not show. You should also discover that the numbers and the algebra can be changed.
Adjust the view size if necessary using Zoom on the View drop-down menu.
Challenge One has three stages :
- Hide the blue numbers in the boxes, change the algebra and the red numbers and check you can successfully predict the hidden blue values.
- Next hide the red numbers instead and repeat the challenge. Explain to a friend how you manage this. You could even work in pairs to set challenges for each other.
- Finally hide the algebra. Change it without showing it, then predict the formula. You have succeeded with this task when you can explain your method to someone else.
- Predict the hidden blue
- Predict the hidden red, explaining your method
- Predict the hidden algebra, again explaining your method and especially why it works
And the "solutions" we would love to receive from you are your best ways of justifying this important method.
Even when we cannot immediately explain why a method is valid it is good to keep it in mind as something we very much want to know.
Getting Started
Interactive Number Patterns 1
- Start with a formula that has only multiples of n, nothing added or subtracted, for example : $2n$, $3n$, $4n$ or $5n$
- Look at the sequence of blue numbers and notice the pattern - explain why it happens.
- Next include in the formula something added and watch how that changes the pattern.
- When you can explain what happens, switch to a formula where something is subtracted.
Interactive Number Patterns 2
The hidden blue comes from substitution, but predicting the hidden red needs you to solve an equation. Perhaps you can solve it using algebra or maybe you could guess and test, improving your guess until you hit the right value. Could there be more than one value that works as a solution to your equation?
This spreadsheet file has a second sheet (see the tab called 'Patterns with differences' at the bottom of the work area). There are a lot of numbers to watch, so take your time.
- What are these numbers ? Perhaps begin by explaining the numbers in line Difference 1 and in line Difference 2. Use the buttons to change the formula so that you can check you were right.
- Do you see a connection between Difference 2 and the formula? Change the formula to check that you were right about the connection.
- Now see how the numbers in the bottom boxes work and the purple numbers between them. Take it row by row, changing the formula to be sure you understand where each value comes from.
- Finally, use those numbers in the bottom half of the screen to help you explain why the connection you discovered in Difference 2 actually works.
Student Solutions
We would love to hear from more people about the best ways of justifying this important method.
Teachers' Resources
This problem, and the Excel interactivities, have been created to deepen students' understanding of the 'difference method'.
The Hints section contains additional direction for students engaging with this task.
Students often possess the method but without any sense of why it's valid. Hopefully this activity will not only help them to explore this method's validity but will also encourage in them a questioning attitude and a desire to establish the validity of other common methods as they meet them.
It may be valuable to notice that the quadratic expression and the red values can be set on either sheet and will then automatically update on the other.
There are a large number of values displayed on the sheet called 'Patterns with differences' (see the tab at the bottom of the work area on Interactive Number Patterns 2). Students need to be given the time to understand all parts of the display.
The following may be one useful way to use this resource with a class.
Below the formula box are three rows. These display the values of the quadratic, linear and constant terms respectively. Change the terms in the formula box to help students grasp what each row shows, and verify that the three values added together match the blue values. In the two rows of differences above the formula box and verify that the data showing is the correct difference for the blue values, and use the slider to change the red values, allowing students to verify the new values displayed within the sheet.
Draw attention to the first difference which changes, and the second difference which does not. Why is that?
The constant term contributes nothing to the first difference, the linear term contributes to the first difference but not the second, and the coefficient, or multiple, of the quadratic term is the only coefficient which influences the second difference.