Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10 and the population of the world as 7 times 10, and determine that the world population is more than 20 times larger.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Numbers like 7,800,000,000 or 0.00004 are hard to picture and harder to compare. Rounding them to a single digit times a power of ten tames them: about 8 × 10⁹ and about 4 × 10⁻⁵. The single digit gives the rough size, and the exponent tells you how many places the decimal point has moved. In eighth grade this is an estimation tool first and a notation second.
Once two quantities are in that form, comparing them becomes a matter of dividing the digits and subtracting the exponents. A city of about 2 × 10⁶ people and a town of about 5 × 10³ people differ by a factor of (2 ÷ 5) × 10³, which is 0.4 × 1000 = 400. Students learn to express answers as how many times as much one quantity is than another, which is the kind of reasoning scientists use to talk about distances in space or the size of cells.
For 0.0006 students often count three zeros and write 6 × 10⁻³. The decimal point must move four places to reach 6, so it is 6 × 10⁻⁴.
Asked how many times as large 6 × 10⁸ is compared with 2 × 10⁵, some find the difference. The question is about a ratio, so divide: 3 × 10³ = 3000 times.
4 × 10⁻² is 0.04, a small positive amount. The negative sign describes a tiny size, not a value below zero.
Students may say 9 × 10³ is bigger than 2 × 10⁵ because 9 is bigger than 2. The power of 10 matters far more: 9000 is much smaller than 200,000.
A large reservoir holds about 9 × 10⁹ liters of water. A swimming pool holds about 3 × 10⁶ liters. About how many times as much water does the reservoir hold?
Answer: The reservoir holds about 3000 times as much water as the pool.
Collect real estimates the class finds interesting, such as the number of people in the state, the thickness of a sheet of paper or the distance to the Moon, and have students round each to one digit times a power of 10. Ordering these cards on a long strip of paper builds intuition about orders of magnitude far better than worksheets of conversions.
This standard leads straight into full scientific notation in 8.EE.A.4. Assessment items usually give two estimates and ask how many times larger one is, so practice the divide-the-digits, subtract-the-exponents routine until it is automatic, and always ask whether the answer is sensible.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: A) 4 × 10⁷
41,900,000 is about 40,000,000. Moving the decimal point seven places gives 4 × 10⁷.
Answer: D) 0.005
10⁻³ = 1/1000, so 5 × 10⁻³ = 5/1000 = 0.005.
Answer: 300
6 ÷ 2 = 3 and 10⁷ ÷ 10⁵ = 10². So the country has 3 × 100 = 300 times as many people.
Answer: -4
0.0008 = 8 × 10⁻⁴, because the decimal point moves four places to the right to reach 8. The exponent is -4.
Answer: A) 3 × 10⁵
Compare powers first: 10⁵ beats 10⁴ and 10³. Of the two 10⁵ numbers, 3 × 10⁵ = 300,000 is larger than 1 × 10⁵.
Answer: 2000
4 ÷ 2 = 2 and 10⁻² ÷ 10⁻⁵ = 10³. So the pebble is about 2 × 1000 = 2000 times heavier.
A full lesson with slides, activities and an exit ticket on estimating with powers of 10, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.EE.A.3 with an answer key, ready in about a minute.
Make a worksheet →Turn estimating with powers of 10 into a quiz students answer online that marks itself, with a class summary for you.
Build a test →8.EE.A.3 uses a single digit times a power of 10 for estimates and comparisons. 8.EE.A.4 then works with full scientific notation, such as 4.37 × 10⁶, and with calculations in that form.
With very large or very small numbers, a difference is hard to interpret. A ratio such as 'about 300 times larger' gives a clear sense of scale.