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Lesson 7 — Activity 2: Making Predictions and Drawing Conclusions from Patterns
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Lesson 7 — Activity 2: Making Predictions and Drawing Conclusions from Patterns
Getting Ready
Now that you have practised solving patterns that involve numbers that come at the end of patterns, you will look at numbers missing within patterns.
Sometimes you will have only part of a pattern and be missing some of the middle terms of the pattern. You have to use the information you have been given to figure out the pattern and then use that information to fill in the blanks.
Watch the following video to see two examples of patterns that have missing numbers within the pattern. Please note that the person in the video uses the term “geometric progressions.” This is the same as geometric sequences.
Watch the following video to see two examples of patterns that have missing numbers within the pattern. Please note that the person in the video uses the term “geometric progressions.” This is the same as geometric sequences.
The video only deals with geometric sequences, but you can also have a
missing term in an arithmetic sequence. To find the missing term in
either an arithmetic sequence or a geometric sequence, you have to find
the pattern. Once you have done that, you can use the pattern to find
the missing value.
For example, if you had the pattern 3, 6, 9, __, 15, 18 and wanted to
know the missing number, you would have to follow the steps from L7 — A1 to find out what kind of a pattern you had.
Start by subtracting to see if you have a common difference.
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6 – 3 = 3
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9 – 6 = 3
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18 – 15 = 3
You have a common difference of 3, so you have an arithmetic sequence. To find the missing number, all you have to do is add 3 to 9. You can also find the missing number by subtracting 3 from 15.
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9 + 3 = 12
15 – 3 = 12
The missing number in the pattern is 12.
What about the pattern 2, 4, 8, 16, __, 64?
- 4 – 2 = 2
- 8 – 4 = 4
- 16 – 8 = 8
There is no common difference between the numbers. Nor is there an increase in the difference that is constant. Try dividing the numbers to see if the answers are the same.
- 4 ÷ 2 = 2
- 8 ÷ 4 = 2
- 16 ÷ 8 = 2
You get
the same answer when you divide consecutive numbers, so you have a geometric
sequence. In order to find the missing number, simply multiply 16 by 2. You can also find the missing number by dividing 2 into 64.
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16 x 2 = 32
64 ÷ 2 = 32
The missing number is 32.
Try This:
Find the missing number in this pattern:
3, 6, ____, 24, 48
The missing number is 12.
6 ÷ 3 = 2
48 ÷ 24 = 2
6 x 2 = 12 OR 24 ÷ 12 = 2
You can also make predictions using patterns. A prediction is a
statement about something you don't know for sure. You are making an
educated guess as to what will happen.
If you use the pattern from the
second example above (2, 4, 8, 16, __, 64) and want to figure out the next number in the
pattern, you can predict that it will be 2 times as much as 64.
What about if you were given the information that Person A travelled 100
kilometres in 1 hour and 200 kilometres in 2 hours and were asked to
predict how far they would go in 6 hours?
The first thing you should
notice is that they are travelling 100 kilometres per hour. In order to
predict how far they would go in 6 hours, you would simply multiply 100
by 6.
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100 × 6 = 600
You would predict that they will travel 600 kilometres in 6 hours.
What about conclusions? A conclusion is a decision on what is occurring. You had to make a conclusion in the above example before you could answer the question. You had to conclude that Person A was travelling consistently at 100 km/h before you could go on.
Conclusions can also be made using a whole set of numbers. For example, let's say that 1 person will eat 2 pieces of pizza; 2 people will eat 4 pieces of pizza, and 3 people will eat 6 pieces of pizza. From this information, you can conclude two things: that each person is eating 2 pieces of pizza and that each additional person will eat 2 pieces of pizza as well.
Taking that a step further, if you were asked how many pieces of pizza 10 people would eat, you could quickly multiply 10 people by 2 pieces of pizza.
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10 × 2 = 20
You can conclude that 10 people will eat 20 pieces of pizza.
