1. Scientific notation

2. Very large numbers
Use your calculator to find the answer to the calculation 40,000,000 × 50,000,000.
Your calculator may display the answer like this:
What does the 15 mean?
Teacher notes
Different models of calculator may show the answer in different ways. Many will leave out the ×10 and will have EXP before the power or nothing at all.
Discuss how many zeros there will be in the answer. 4 × 5 is 20. There are 7 zeros in 40,000,000 and 7 zeros in 50,000,000. That means that the answer will have 14 zeros plus the zero from the 20, making 15 zeros altogether.
Mathematical Practices
5) Use appropriate tools strategically.
This slide encourages students to use their calculators for calculations involving very large numbers, and to correctly interpret the result displayed on the screen.
The 15 means that the 2 is multiplied 15 times by 10.
2 × 1015
= 2,000,000,000,000,000

3. Very small numbers
Use your calculator to find the answer to the calculation
÷ 30,000,000.
Your calculator may display the answer like this:
What does the –12 mean?
Teacher notes
Point out that if we include the 0 before the decimal point the answer has 12 zeros altogether.
Mathematical Practices
5) Use appropriate tools strategically.
This slide encourages students to use their calculators for calculations involving very small numbers, and to correctly interpret the result displayed on the screen.
The –12 means that the 15 is divided 12 times by 10.
1.5 × 10-12 =

4. Scientific notation
2 × 1015 and 1.5 × are examples of numbers written in scientific notation.
Numbers written in scientific notation have two parts:
A number between 1 and 10
A power of 10 ×
This way of writing a number is also called the standard exponent form.
Teacher notes
Point out that the numbers between 1 and 10 do not include the number 10.
Any number can be written using scientific notation, however it is usually used to write very large or very small numbers.

5. 5.97 × 1024 kg Writing large numbers
We can write very large numbers using powers of ten.
The mass of the planet earth is about
5,970,000,000,000,000,000,000,000 kg.
We can write this in scientific notation as a number between 1 and 10 multiplied by a power of 10.
5.97 × 1024 kg
A number between 1 and 10
A power of ten

6. Writing large numbers
How do we write these numbers in scientific notation?
80,000,000 =
8 × 107
230,000,000 =
2.3 × 108
724,000 =
7.24 × 105
Teacher notes
Discuss how each number should be written in scientific notation.
Notice that for large numbers the power of ten will always be one less than the number of digits in the whole part of the number.
6,003,000,000 =
6.003 × 109 =
× 102

7. Writing large numbers
These numbers are written in scientific notation. How can they be written as ordinary numbers?
5 × 1010 =
50,000,000,000
7.1 × 106 =
7,100,000
4.208 × 1011 =
420,800,000,000
Teacher notes
Discuss how each number written in scientific notation should be written in full.
2.168 × 107 =
21,680,000
× 103 =
6764.5

8. 1.3 × 10–4 m. Writing small numbers
We can write very small numbers using negative powers of ten.
For example, the width of this shelled amoeba is m.
We write this in scientific notation as:
1.3 × 10–4 m.
Photo credit: The image of a shelled amoeba has been reproduced with the kind permission of Wim van Egmond © Microscopy UK
A number between 1 and 10
A negative power of 10

9. Writing small numbers
How can we write these numbers in scientific notation? =
6 × 10–4 =
7.2 × 10–7 =
5.02 × 10–5
Teacher notes
Notice that the power of ten is always minus the number of zeros before the first significant figure including the one before the decimal point. =
3.29 × 10–8 =
1.008 × 10–3

10. Writing small numbers
These numbers are written in scientific notation. How can they be written as ordinary numbers?
8 × 10–4 =
0.0008
2.6 × 10–6 =
9.108 × 10–8 =
Teacher notes
Again, notice that the power of ten tells us the number of zeros before the first significant figure including the one before the decimal point.
7.329 × 10–5 =
× 10–2 =

11. Ordering numbers in scientific notation
Write these numbers in order from smallest to largest:
5.3 × 10–4, × 10–5, × 10–3, × 10–4.
To order numbers written in scientific notation, first compare the powers of 10.
Remember, 10–5 is smaller than 10–4, so 6.8 × 10-5 is the smallest number in the list.
When two or more numbers have the same power of ten we can compare the number parts. 5.3 × 10–4 is larger than 1.5 × 10–4 so the correct order is:
6.8 × 10–5, 1.5 × 10–4, 5.3 × 10–4, 4.7 × 10–3

12. Space ship’s speed
How long would it take a space ship traveling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?
distance
distance
Rearrange speed =
to give time =
time
speed
8.32 × 107
Teacher notes
Remind students that 107 ÷ 103 = 104 because the indices are subtracted when dividing. To extend the activity, ask students to convert the answer in hours into years.
Mathematical Practices
4) Model with mathematics.
Students should apply their knowledge of performing calculations with numbers in scientific notation to solve this problem based on speed.
6) Attend to precision.
Students should be careful about stating any formulas they use and correctly stating units where necessary. Here they should remember to include units in their final answer, and realize that since the speed in the question is given in km/h, their answer (a time) should be in hours.
Photo credit: © Stephen Girimont, Shutterstock.com 2012
Time to reach Mars =
2.6 × 103
= (8.32 ÷ 2.6) × (107 ÷ 103)
= 3.2 × 104 hours

13. How many years?
It would take the space ship 3.2 × 104 hours to reach Mars. How long is this in years? Use your calculator
Enter 3.2 × 104 ÷ 24 into your calculator to give the equivalent number of days.
Divide by 365 to give the equivalent number of years.
3.2 × 104 = 32, ,000 ÷ 24 = ÷ 365 ≈ 3.65
Teacher notes
Make sure that students are able to enter numbers given in standard form into their calculators.
A 16 year old will enter: 16 × 365 [to find days] × 24 [to find hours] × 60 [to find minutes] = 8.4 × 106 minutes
Mathematical Practices
4) Model with mathematics.
Students should apply their knowledge of performing calculations with numbers in scientific notation to solve this problem based on converting hours into years. They should be able to contextualize their final answer; remember to interpret the answer 3.65 as “over 3 and a half years”.
6) Attend to precision.
This slide demonstrates how students should communicate precisely when solving problems. They should be careful about specifying units of measure in their work, and choose a degree of accuracy appropriate for the context of the question. For the second question about their age, they should be reasonable about the amount of accuracy they use. Using “16 years old” as the starting point is sensible. Using “16 years and 5 months old” is reasonable. Using anything more specific than this is slightly unnecessary.
Photo credit: © Provasilich, Shutterstock.com 2012
3.2 × 104 hours is over 3½ years!
Now figure out approximately how old you are in minutes! Write it using scientific notation.

14. Atoms and molecules The table shows some masses of very small items.
Which weighs more…
Item
Mass
Hydrogen atom
1.67×10-27 kg
Water molecule
2.99×10-26 kg
Silver atom
1.79×10-25 kg
Lead atom
3.45×10-25 kg
Small grain of sand
3.5×10-10 kg
Large grain of sand
1.1×10-5 kg
1 euro coin
0.008 kg
3.56 × 1020 Hydrogen atoms or 4.3 × 1019 water molecules?
5 million small grains of sand or 200 large grains?
How many atoms of silver are needed to have the same weight as a 1 Euro coin?
Teacher notes
Students will need to be encouraged to show their calculations.
Answers:
Hydrogen: 3.56 × 1020 × 1.67 × 10–27 = 5.95 × 10-7 kg Water: 4.3 × 1019 × 2.99 × 10–26 = × 10-6 kg so the water weighs more
Small sand: 5 × 106 × 3.5 × 10–10 = 1.75 × 10–3 kg Large sand: 200 × 1.1 × 10–5 = 2.2 × 10–3 kg so the large sand weighs more
4.47 × 1022 atoms of silver
Moles: hydrogen 0.001kg (or 1g), water 0.018kg, silver 0.108kg, lead 0.208kg.
Mathematical Practices
Make sense of problems and persevere in solving them.
These questions test how well students can work with numbers in standard form and understand what type of calculation is needed to answer each question.
4) Model with mathematics.
This slide tests how well students can work with numbers in standard form in a real-world context. They should be able to identify the kind of calculation is needed and then extract their answer in the context of the question.
A mole of a substance is 6.02 × 1023 atoms or molecules of that substance. What does a mole of hydrogen, water, silver and lead weigh?