Double rules
This is a level 3 algebra activity from the Figure It Out series. It is focused on finding a number rule and using a ? symbol to express the rule. A PDF of the student activity is included.
About this resource
Figure It Out is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.
This resource provides the teachers' notes and answers for one activity from the Figure It Out series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.
Specific learning outcomes:
- Find a number rule.
- Use ? symbols to express the rule.
Double rules
Achievement objectives
NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.
Required materials
- Figure It Out, Level 3-4, Algebra, "Double rules", page 10
- two calculators
- a classmate
See Materials that come with this resource to download:
- Double rules activity (.pdf)
Activity
Most four-function calculators behave as described in the activity when a calculation is entered. However, Casio calculators need the operation button keyed in twice to activate the constant capability.
For example:
- 2 x = will set a doubling rule on most calculators, but on Casio calculators 2 x x = must be entered.
The Find the Function game described in this activity is an excellent way for the students to find relationships between numbers without having to work with spatial patterns and practical tasks. Encourage the students to play the game regularly with a classmate.
Research indicates that it is unwise to train students to find the function by inputting 1 first, then 2, then 3, and so on. Students appear to learn more from having to apply the processes of arithmetic to discover the relationships. Consider the example given in the activity. Suppose that Harmony has asked Rāwiri to enter three numbers and has these in/out pairs:
In |
6 |
2 |
9 |
|---|---|---|---|
Out |
15 |
7 |
21 |
Harmony realises that the function makes the input number larger by more than double. She might try experimenting with x 2 and x 3 as rules and then working out how much she would have to add or subtract to get the out number:
Using x 2
In |
6 |
2 |
9 |
|---|---|---|---|
X 2 |
12 |
4 |
18 |
Missing operation |
+ 3 |
+ 3 |
+ 3 |
Out |
15 |
7 |
21 |
Using x 3
In |
6 |
2 |
9 |
|---|---|---|---|
X 3 |
18 |
6 |
27 |
Missing operation |
- 3 |
+ 1 |
-6 |
Out |
15 |
7 |
21 |
The missing operation in the x 2 table is constant, so Harmony has discovered that Rāwiri is using the rule (? x 2) + 3.
If this activity does not work on your calculator, try pressing the x operations key twice: 2 x x =
1.
a. 7
b. 23
c. 9
d. 11
2.
Harmony realises that each number that comes out is twice the original number plus 3. So the first calculator has to have 2 x = in it and the second has to have 3 + =
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