Today we had two periods of math. During the first period we reviewed the strategies for solving problems involving input/output machines and recursive patterns.
For input/output machines, make sure that you find the difference in the output numbers. If it is increasing by the same amount each time, this is a clue that the input numbers are being multiplied by the amount that they are increasing by each time (this is because multiplying is a shortcut for repeated addition, i.e. if the output number increases by 4 each time, it means that the input number is being multiplied by 4).
| Input | Output | Difference | Test | Test –> Input |
| 1 | 5 | 1 x 3 = 3 | 3 +2 = 5 | |
| 2 | 8 | +3 | 2 x 3 = 6 | 6 + 2 = 8 |
| 3 | 11 | +3 | 3 x 3 = 9 | 9 + 2 = 11 |
| 4 | 14 | +3 | 4 x 3 = 12 | 12 +2 = 14 |
| 5 | 17 | +3 | 5 x 3 = 15 | 15 +2 = 17 |
| 6 | 20 | +3 | 6 x 3 = 18 | 18 + 2 = 20 |
| Means first operation is X3 | Multiplying by 3 along won’t make the output numbers | The second operation must be +2 |
So the input/output machine that created this pattern must have the following operations (x3) and (+2).
One you’ve found what the input numbers are being multiplied by, test your solution. Multiply the input number by the number the output numbers are increasing by. Do you get your output number? If not, consider what you might need to add or subtract in order to get to the output number.
For recursive patterns, the first step is again to find the difference between each of the numbers in the pattern. Sometimes the difference is increasing by a multiple (ie. First difference is 4, next difference is 12, next difference is 36, next difference is 108). If you can figure out what this multiple is, that will tell you the first part of the recursive pattern. In the example below, the change doubles each time, telling us that the recursive pattern involves doubling. We then need to follow up to figure out how we get each number in the pattern. In this case, after doubling, add 1 each time.
Recursive pattern: 3, 7, 15, 31, 63, 190
| Values | Change | Test | Test 2 | |
| 3 | ||||
| 7 | +4 | 3 x 2 =6 | 6 + 1 = 7 | |
| 15 | +8 | 4 x 2 = 8 | 7 x 2 = 14 | 14 + 1 = 15 |
| 31 | +16 | 8 x 2 = 16 | 15 x2 = 30 | 30 + 1 = 31 |
| 63 | +32 | 16 x 2 = 32 | 31 x 2 = 62 | 62 + 1 = 63 |
| 127 | +64 | 32 x 2 = 64 | 63 x 2 = 126 | 126 + 1 = 127 |
| This tells us that our recursive pattern involves doubling (or multiplying by 2 each time) | This tells us that after doubling, we must add 1 each time. |
During the second math period, we learned about divisibility rules.
- Today’s Lesson: Chapter 1, Lesson 3 - Patterns in Division (Pages 13-15)
- Learning Goals:
- Use patterns to explore divisibility rules
- Additional Resources:
- Classwork: Practice Questions 1-7
- Homework Book: Pages 6-7
