B1.1 Represent and compare very large and very small numbers, including through the use of scientific notation, and describe various ways they are used in everyday life.
Activity 1: Representation of Small and Large Numbers
Write or project a large number on the board (written in 4 different ways).
Examples
- Three hundred five billion (number expressed in words)
- 305 000 000 000 (standard notation)
- \(3\; \times \;{10^{11}}\; + \;5\; \times \;{10^9}\)(expanded form using powers of 10)
- \(3.05\; \times \;{10^{11}}\) (scientific notation)
Ask students about these numbers.
- What do you notice?
- What is the largest number? The smallest?
- What are the similarities and differences between the representations?
- Do you encounter these representations in everyday life? If so, in what context?
Ensure that the student realizes that a number can be written using different representations and still represent the same quantity.
Then write and do the exercise with very small numbers.
- 0, 000 042
- 42 millionths
- \(4 \times \;{10^{{}^ - 5}}\; + \;2\; \times \;{10^{{}^ - 6}}\)
- \(4.2 \times {10^{{}^ - 5}}\)
Activity 2: Do Large Numbers Exist in Everyday Life? (Representing and Comparing Numbers)
Have students search the Internet for examples where very large numbers and very small numbers are found in everyday life. Students should list the different contexts and notice the notation(s) used. Afterwards, ask them to order the numbers found in descending order.
Examples
World economy (monetary unit), population, astronomy (distance, size of planets), biology (size of cells), science/chemistry (properties of atoms), computers (memory), provincial and national budget, medicine, etc.
Point out to students that these numbers are not used on a daily basis; they are mostly used in specific situations and in specialized areas.
Note: Students should note that scientific notation is the preferred representation for writing very large numbers and very small numbers.
Invite students to complete a table of values with very large numbers and a table of values with very small numbers based on their research.
Ask the student what area/context they took the numbers from to complete the table. The student can write, for example, five populations of countries (large numbers) and the mass of some insects/animals in kg (small numbers) to compare and represent in different ways.
*The first row of this table is an example of each of the representations sought.
| Numbers Expressed in Digits | Numbers Expressed in Words | Numbers Expressed in Expanded Form Using Powers of Ten | Numbers Expressed in Scientific Notation |
|---|---|---|---|
| \(950\;000\;000\) | Nine hundred fifty million | \(9\; \times \;{10^8}\; + \;5\; \times \;{10^7}\) | \(9.5\; \times \;{10^8}\) |
| 1) | |||
| 2) | |||
| 3) | |||
| 4) | |||
| 5) |
| Numbers Expressed in Digits | Numbers Expressed in Words | Numbers Expressed in Expanded Form Using Powers of Ten | Numbers Expressed in Scientific Notation |
|---|---|---|---|
| 0,025 | Twenty-five thousandths | \(2\; \times \;{10^{{}^ - 2}} + \;5\; \times \;{10^{{}^ - 3}}\) | \(2.5\; \times \;{10^{{}^ - 2}}\) |
| 1) | |||
| 2) | |||
| 3) | |||
| 4) | |||
| 5) |
Once the table of values is completed, ask students to compare the numbers. Ask them about the different representations.
- Which representation makes it easier to compare numbers?
- Which one is the biggest?
- Which one is the smallest?
- How do you know that? Explain.
Activity 3: Looking for Patterns
Choose the largest number from the first table of values.
- Why did you choose this number? How do you know it's the biggest?
Example
12 350 000
The teacher asks the student questions about this number.
- What does the number 2 represent in this number? The number 5?
- Can you write a number that is 10 times greater than this number? 100 times? 1 000 times? 1 000 000 times larger? How do you do it? What do you notice as the number grows?
- What happens if I divide this number by 10 (10 times smaller)? By 100?
- What do you notice?
End with an analysis of the numbers found, trying to highlight certain patterns.
Do the same exercise with the smallest number in the2nd table of values.