8.EE.A.3Common CoreMathExpressions and EquationsGrade 8
8.EE.A.3: Estimating Very Large and Very Small Quantities With Powers of 10
In plain English: 8.EE.A.3 is the Common Core grade 8 math standard that asks students to estimate very large or very small quantities as a single digit times a power of 10, such as 3 × 10⁸ or 8 × 10⁻⁶. Students then compare two such estimates to say how many times as much one is than the other. It comes in Grade 8 Math, just before full scientific notation.
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 × 10⁸ and the population of the world as 7 × 10⁹, and determine that the world population is more than 20 times larger.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Work with radicals and integer exponents. Also written as 8.EE.3 · Official standard
Students learn to write huge and tiny quantities in a short form: a single digit times a power of 10, such as 3 × 10⁸ people or 8 × 10⁻⁶ m. A power of 10 is 10 multiplied by itself, like 10³ = 1,000, and a negative exponent (the small raised number) gives a number less than 1, like 10⁻³ = 0.001. Students round real quantities, from the seconds in a year to the width of a red blood cell, to this form.
The second half of the standard is comparison. Students divide one estimate by another to say how many times as much one quantity is than the other, working through the official example: the world population, 7 × 10⁹, is more than 20 times the US population, 3 × 10⁸. Full scientific notation with more digits, such as 3.4 × 10⁸, and operations with it belong to the next standard, 8.EE.A.4.
Learning Objectives
By the end of this lesson, students will be able to:
Estimate a very large quantity as a single digit times a positive power of 10
Estimate a very small quantity as a single digit times a negative power of 10
Find how many times as much one quantity is than another by dividing the digits and subtracting the exponents
Explain the official example: why the world population is more than 20 times the US population
Judge when a one-digit estimate is close enough and explain why it differs from the exact ratio
Prior Knowledge Required
Students should already be comfortable with:
Place value and patterns when multiplying by powers of 10 5.NBT.A.2
Writing and evaluating whole-number exponents 6.EE.A.1
Properties of integer exponents, including negative exponents and dividing powers 8.EE.A.1
Rounding whole numbers and decimals to a given place
Write two numbers on the board: 1,000,000 and 1,000,000,000. Ask students to guess first, then check with a calculator.
Warm-Up Prompt
"A million seconds from now, will it still be this month? A billion seconds from now, how old will you be?"
Collect guesses, then work it out together. A day has 60 × 60 × 24 = 86,400 seconds, so a million seconds is about 11.6 days. A billion seconds is about 31.7 years, so a 13-year-old would be about 45. The two numbers look alike on the board, yet one is a thousand times as large as the other. Tell students that today they will learn a short way to write and compare numbers like these.
Direct Instruction20 minutes
Define the key words one at a time and write each on an anchor chart:
Power of 10: 10 multiplied by itself some number of times. The small raised number is the exponent. 10³ = 10 × 10 × 10 = 1,000. Recall from 8.EE.A.1 that a negative exponent means "one over": 10⁻³ = 1/10³ = 0.001.
Single digit times a power of 10: a number written as d × 10ⁿ, where d is one of the whole numbers 1 to 9 and n is an integer (a whole number or its opposite). Examples: 3 × 10⁸ and 8 × 10⁻⁶.
Estimate: a rounded value that is close to the real one. Here we round a number to one nonzero digit and keep that digit in the same place.
How to estimate a large number: round to one nonzero digit, then count the places that digit is worth. 31,536,000 rounds to 30,000,000 = 3 × 10,000,000 = 3 × 10⁷.
How to estimate a small number: find the first nonzero digit after the decimal point and count its place. In 0.0000078, round to 0.000008. The 8 is in the sixth place after the point, so 0.000008 = 8 × 10⁻⁶.
How many times as much: divide the larger quantity by the smaller one. Divide the digits, and subtract the exponents (8.EE.A.1): (6 × 10⁸) ÷ (2 × 10³) = 3 × 10⁵.
Work through the four examples below. For each one, ask: is this a very large or a very small quantity, and should the exponent be positive or negative?
Estimating a very large quantity
How many seconds are in one year of 365 days? Estimate it as a single digit times a power of 10.
Equation: 365 × 24 × 60 × 60 = 31,536,000 seconds. Round to 30,000,000, which is 3 × 10⁷ seconds.
Estimating a very small quantity
A red blood cell is about 0.0000078 m across. Estimate its width as a single digit times a power of 10.
Equation: 0.0000078 rounds to 0.000008. The 8 is in the sixth decimal place, so the width is about 8 × 10⁻⁶ m.
The official example
Estimate the population of the United States as 3 × 10⁸ and the population of the world as 7 × 10⁹. How many times as large is the world population?
Equation: (7 × 10⁹) ÷ (3 × 10⁸) = (7 ÷ 3) × 10¹ ≈ 2.3 × 10 ≈ 23. The world population is more than 20 times the US population.
Comparing two small quantities
A human hair is about 7 × 10⁻⁵ m wide. A flu virus is about 1 × 10⁻⁷ m across. How many times as wide is the hair?
Equation: (7 × 10⁻⁵) ÷ (1 × 10⁻⁷) = 7 × 10⁻⁵⁻⁽⁻⁷⁾ = 7 × 10² = 700. The hair is about 700 times as wide.
Use Diagram 1 to show that each power of 10 is 10 times the one before it, so small lengths can be placed on one line. Use Diagram 2 for the official example: about 23 copies of the US population fit into the world population. Point out that the estimate 7 × 10⁹ comes from the standard; the real world population has grown past 8 billion, and the answer "more than 20 times" is still true.
Guided Practice15 minutes
Pairs work on four items, one at a time, and a pair explains each answer at the board.
Mount Everest is about 8,849 m tall. (8,849 rounds to 9,000, so about 9 × 10³ m.)
A dust mite is about 0.0003 m long. (The 3 is in the fourth decimal place: 3 × 10⁻⁴ m.)
The Sun is about 1,390,000 km across and Earth about 12,700 km. Estimate both, then compare. (1 × 10⁶ km and 1 × 10⁴ km, so the Sun is about 10² = 100 times as wide. The exact ratio is about 109.)
Estimate 0.00096 as a single digit times a power of 10. (9.6 ten-thousandths rounds up to 10 ten-thousandths, which is 0.001 = 1 × 10⁻³. Writing 10 × 10⁻⁴ is not allowed, because 10 is not a single digit.)
Listen for students who count zeros instead of places. 72,000,000 has six zeros, but the 7 is worth 10,000,000 = 10⁷.
Independent Practice10 minutes
Students work alone on five items, then compare with a partner:
A heart beating 70 times a minute for 80 years beats 70 × 60 × 24 × 365 × 80 = 2,943,360,000 times. Estimate this. (3 × 10⁹ beats.)
A small garden ant has a mass of about 0.000003 kg. Estimate it. (3 × 10⁻⁶ kg.)
An adult African elephant has a mass of about 6 × 10³ kg and a house mouse about 2 × 10⁻² kg. How many times as heavy is the elephant? ((6 ÷ 2) × 10³⁻⁽⁻²⁾ = 3 × 10⁵, or 300,000 times.)
Which is larger, 5 × 10⁻⁴ or 9 × 10⁻⁵? About how many times as large? (5 × 10⁻⁴ = 0.0005 is larger, and 0.0005 ÷ 0.00009 ≈ 5.6, so about 6 times.)
Pick one very large and one very small quantity from your life, estimate each as a single digit times a power of 10, and write one "how many times" sentence about them.
Closure5-10 minutes
Exit ticket: (1) Write 0.000042 as a single digit times a power of 10. (4 × 10⁻⁵.) (2) A city has about 2 × 10⁶ people and its country about 8 × 10⁸. How many times as many people live in the country? (4 × 10² = 400 times.) (3) Finish the sentence: "To find how many times as much one estimate is than another, I ..."
Differentiation Strategies
For Struggling Students
Give a place-value chart from 10⁷ down to 10⁻⁷ and have students write the digit in its column before writing the power of 10
Start with numbers that are already one digit followed by zeros, such as 40,000 or 0.0006, before numbers that need rounding
Keep a card on the desk: "Big number: positive exponent. Small number: negative exponent."
For Advanced Students
Ask students to find a pair of real quantities that are about 10⁶ times apart and justify both estimates with a source
Ask: can two numbers have the same single-digit estimate but differ by almost a factor of 2? Give an example
Preview (beyond this standard): write the world population with two digits, 8.1 × 10⁹, and ask how that changes the comparison, as students will do in 8.EE.A.4
Assessment Guidance
What to Look For
Listen for place value, not zero counting. A strong answer says where the digit is, for example "the 8 is in the sixth place after the decimal point, so 10⁻⁶." Watch for four errors: counting zeros instead of places, dropping the negative sign for small quantities, adding exponents when dividing, and dividing the smaller quantity by the larger. In "how many times" answers, check that students say which quantity is larger.
02
Classroom Activities
3 Activities
1
Powers-of-Ten Line: Card Sort
20 minPairs
Pairs get 8 cards, each with a real length written in standard form (the usual way, with all its digits). They estimate each length as a single digit times a power of 10 and tape the card on a paper strip marked from 10⁻⁶ m to 10⁷ m, one mark for each power of 10.
The 8 Cards (with answers)
Card 1: The Statue of Liberty, from the ground to the tip of the torch: about 93 m (answer: 9 × 10¹ m)
Card 2: A school bus: about 12 m long (answer: 1 × 10¹ m)
Card 3: The straight-line distance from New York City to Los Angeles: about 3,900,000 m (answer: 4 × 10⁶ m)
Card 4: The main span of the Golden Gate Bridge: about 1,280 m (answer: 1 × 10³ m)
Card 5: A classroom door: about 2.1 m tall (answer: 2 × 10⁰ m)
Card 6: The thickness of a US dime: about 0.00135 m (answer: 1 × 10⁻³ m)
Card 7: A pollen grain: about 0.00003 m across (answer: 3 × 10⁻⁵ m)
Card 8: An E. coli bacterium: about 0.000002 m long (answer: 2 × 10⁻⁶ m)
Procedure
Mark 14 evenly spaced points on the strip and label them 10⁻⁶ m, 10⁻⁵ m, and so on up to 10⁷ m
For each card, write the estimate on a sticky note before placing the card
Place each card at its power of 10, a little to the right for larger digits
Check with another pair and settle any card that is in a different place
Discussion Questions
Using the estimates, about how many school buses tall is the Statue of Liberty? The exact answer is 93 ÷ 12 = 7.75. Why is the estimate 9 bigger?
The New York to Los Angeles card and the door card are how many powers of 10 apart?
Which card is the only one between 1 m and 10 m? What power of 10 did you use for it?
Modification for Distance Learning
Put the strip on a shared slide. Pairs drag each card to its place and type the estimate in a text box beside it.
2
Population Face-Off
15 minGroups of 3
Groups compare the populations of real places. Each group estimates both populations as a single digit times a power of 10, then writes a "how many times as many" sentence and checks it against a calculator answer with the rounded 2024 figures below.
The Places (2024 figures, rounded)
Mexico: about 130,000,000 people (estimate: 1 × 10⁸)
Australia: about 27,000,000 people (estimate: 3 × 10⁷)
New York City: about 8,300,000 people (estimate: 8 × 10⁶)
Luxembourg: about 670,000 people (estimate: 7 × 10⁵)
A school of 820 students: about 820 people (estimate: 8 × 10²)
Procedure
Each group draws three pairs of places from a bag
One student estimates both populations, one divides the estimates, and one divides the full numbers on a calculator
Record both answers in a table, then rotate the roles
Sample Record
Australia and Luxembourg: (3 × 10⁷) ÷ (7 × 10⁵) ≈ 43; the full numbers give about 40. New York City and the school: (8 × 10⁶) ÷ (8 × 10²) = 10⁴, so the city has about 10,000 times as many people. Mexico and New York City: (1 × 10⁸) ÷ (8 × 10⁶) = 12.5; the full numbers give about 15.7.
Discussion Questions
In the sample record, which pair had the estimate furthest from the calculator answer? Which rounding caused it?
When is a quick estimate good enough, and when would you want the full numbers?
3
Stack and Divide: Measuring Thin Things
20 minGroups of 4
One sheet of paper or one penny is too thin to measure well with a ruler, but a stack is not. Groups measure a stack, divide by the number of items, and write the thickness of one item as a single digit times a power of 10 in meters.
Materials per Group
A stack of 200 sheets of printer paper
20 pennies
A centimeter ruler
Sample Data (invented)
An invented group measured the paper stack as 2.0 cm and the stack of 20 pennies as 3.0 cm. One sheet: 2.0 cm ÷ 200 = 0.01 cm = 0.0001 m = 1 × 10⁻⁴ m. One penny: 3.0 cm ÷ 20 = 0.15 cm = 0.0015 m. The digits 1.5 are exactly halfway, and rounding half up gives 2 × 10⁻³ m.
Procedure
Squeeze each stack flat and measure it to the nearest millimeter
Divide to find one item, change centimeters to meters, and write the estimate
Use the estimates to find how many times as thick a penny is as a sheet of paper, then repeat with the unrounded values
Discussion Questions
The estimates say a penny is about 20 times as thick as a sheet. The unrounded values say 15. Which rounding made the difference?
Why does measuring a stack give a better answer than measuring one sheet?
Challenge Variation
Groups measure a stack of 10 sticky notes or a pile of 50 grains of rice laid end to end, and add the results to the powers-of-ten line from Activity 1.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Small Lengths on a Powers-of-Ten Line
Seven small lengths from this lesson placed on a line where each mark is 10 times the one to its left. The positions are drawn to scale for this kind of line, so the red blood cell (8 × 10⁻⁶ m) sits most of the way from 10⁻⁶ to 10⁻⁵. A length one mark to the right is 10 times as long.
Diagram 2: The Official Example, World Population and US Population
The US population estimate, 3 × 10⁸, is drawn as one block. The world population estimate, 7 × 10⁹, is drawn to the same scale: it holds 23 whole blocks and one third of another, so the world population is more than 20 times the US population.
04
Homework Assignment
~30 min
8.EE.A.3 Homework: Estimating and Comparing With Powers of 10
Directions: Show your work. For every estimate, write the rounded number first and then the single digit times a power of 10. For every comparison, say which quantity is larger and how many times as large it is.
Part 1: Estimating Quantities (Problems 1-3)
Write each quantity as a single digit times a power of 10. (a) The average distance from Earth to the Moon, about 384,400 km. (b) The number of seconds in one day, 86,400.
Write each quantity as a single digit times a power of 10. (a) A strand of spider silk is about 0.000004 m thick. (b) A grain of rice has a mass of about 0.000022 kg.
Maya estimated three numbers: 0.00058 ≈ 6 × 10⁴, 83,000,000 ≈ 8 × 10⁶ and 0.0031 ≈ 3 × 10⁻³. Which estimates are right? For each wrong one, name her mistake and write the correct estimate.
Part 2: How Many Times as Much? (Problems 4-6)
The distance around Earth at the equator is about 40,075 km. Use your estimate of the Earth-to-Moon distance from Problem 1 to find about how many trips around the equator equal one trip to the Moon.
A honeybee has a mass of about 0.0001 kg and a fruit fly about 0.0000002 kg. Write both masses as a single digit times a power of 10, then find how many times as heavy the honeybee is.
In 2024, Japan had about 124,000,000 people and Iceland about 390,000. (a) Estimate both populations as a single digit times a power of 10. (b) Use your estimates to find how many times as many people live in Japan. (c) The full numbers give a ratio of about 318. Explain why your estimate is lower.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Estimates
Every estimate uses one digit from 1 to 9 and the correct power of 10
One or two estimates off by one power of 10
Most estimates wrong or not in the form
Signs of Exponents
Positive for large and negative for small quantities every time
One sign error
Several sign errors
Comparisons
Divides the larger by the smaller and subtracts exponents correctly
Method right with one arithmetic slip
Adds exponents or divides the wrong way
Explanations
Names Maya's mistakes and explains the Japan-Iceland difference clearly
Explanations vague or incomplete
No explanations
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which number is written as a single digit times an integer power of 10?
Answer: C
5 × 10⁻³ has one digit from 1 to 9 and a power of 10 with an integer exponent. Choice A uses the two-digit number 12. Choice B uses 0.5, which is not a whole digit from 1 to 9. Choice D uses a power of 2, not a power of 10.
Question 2 of 20 · Multiple Choice
In 2024, Brazil had about 212,000,000 people. Which is the best estimate as a single digit times a power of 10?
Answer: A
212,000,000 rounds to 200,000,000, and the 2 is worth 100,000,000 = 10⁸. Choice B counts only the six zeros in 212,000,000. Choice D counts all nine digits instead of the places after the first digit. Choice C uses 21, which is not a single digit.
Question 3 of 20 · Multiple Choice
A grain of fine sand is about 0.00018 m across. Which is the best estimate of its width?
Answer: B
0.00018 rounds to 0.0002, and the 2 is in the fourth decimal place, so the estimate is 2 × 10⁻⁴ m. Choice A counts only the three zeros after the decimal point. Choice D counts all five decimal places, including the 8. Choice C drops the negative sign, which would make the grain 20,000 m wide.
Question 4 of 20 · Multiple Choice
Which is the best estimate of 0.0000097 as a single digit times a power of 10?
Answer: D
9.7 millionths rounds up to 10 millionths, which is 0.00001 = 1 × 10⁻⁵. Choice A cuts off the 7 instead of rounding. Choice B rounds up to 1 but forgets to move to the next power of 10. Choice C moves two places, to 0.0001.
Question 5 of 20 · Multiple Choice
A city has about 3 × 10⁵ people and its state has about 9 × 10⁶ people. How many times as many people live in the state?
Answer: B
(9 × 10⁶) ÷ (3 × 10⁵) = (9 ÷ 3) × 10⁶⁻⁵ = 3 × 10¹ = 30. Choice A adds the exponents instead of subtracting them. Choice C divides the city by the state, the wrong way around. Choice D divides the digits and ignores the powers of 10.
Question 6 of 20 · Multiple Choice
A dust particle is about 6 × 10⁻⁶ m across and a raindrop about 2 × 10⁻³ m across. About how many times as wide is the raindrop?
Answer: C
(2 × 10⁻³) ÷ (6 × 10⁻⁶) = (2 ÷ 6) × 10³ ≈ 0.33 × 10³ ≈ 333, which is about 3 × 10². Choice A divides 6 by 2 instead of 2 by 6. Choice B divides 6 by 2 and adds the exponents -6 and -3. Choice D divides the dust particle by the raindrop, the wrong way around.
Question 7 of 20 · Multiple Choice
In 2024, the United States had about 340,000,000 people and Texas about 31,000,000. Estimate both as a single digit times a power of 10. About how many times as many people live in the whole country?
Answer: A
The estimates are 3 × 10⁸ and 3 × 10⁷, and (3 × 10⁸) ÷ (3 × 10⁷) = 1 × 10¹ = 10. Choice B writes Texas as 3 × 10⁶ by counting its six zeros. Choice C compares only the digits 3 and 3. Choice D divides Texas by the whole country.
Question 8 of 20 · Multiple Choice
Which number is the largest?
Answer: D
1 × 10⁻² = 0.01 is the largest; the others are 0.004, 0.0009 and 0.00007. Choice B looks largest because 9 is the biggest digit, but its power of 10 is smaller. Choice C is chosen by students who think the exponent -5 is bigger than -2 because 5 is bigger than 2.
Question 9 of 20 · Multiple Choice
How many times as large is 6 × 10⁷ as 2 × 10⁴?
Answer: B
(6 ÷ 2) × 10⁷⁻⁴ = 3 × 10³ = 3,000. Choice A adds the exponents. Choice C divides the smaller number by the larger one: (2 × 10⁴) ÷ (6 × 10⁷) ≈ 0.0003. Choice D subtracts the digits instead of dividing them.
Question 10 of 20 · Multiple Choice
A large football stadium holds about 7 × 10⁴ people. About how many such stadiums would it take to seat 3 × 10⁸ people?
Answer: C
(3 × 10⁸) ÷ (7 × 10⁴) = (3 ÷ 7) × 10⁴ ≈ 0.43 × 10⁴ ≈ 4,300, which is about 4 × 10³. Choice A finds 3 ÷ 7 ≈ 0.4 but writes it as 4 without moving to the next lower power of 10. Choice B divides 7 by 3 and adds the exponents. Choice D divides the stadium by the people, the wrong way around.
Question 11 of 20 · Multiple Choice
A mosquito has a mass of about 0.000003 kg. About how many mosquitoes have the same mass as a 60 kg adult?
Answer: A
The mosquito is 3 × 10⁻⁶ kg and the adult is 6 × 10¹ kg. (6 × 10¹) ÷ (3 × 10⁻⁶) = 2 × 10¹⁻⁽⁻⁶⁾ = 2 × 10⁷, or 20 million mosquitoes. Choice B adds the exponents 1 and -6. Choice C divides the mosquito by the adult. Choice D writes the mosquito as 3 × 10⁻⁵ by counting its five zeros after the decimal point.
Question 12 of 20 · Multiple Choice
Two quantities are estimated as 4 × 10⁹ and 4 × 10⁵. Which statement is true?
Answer: D
(4 × 10⁹) ÷ (4 × 10⁵) = 1 × 10⁴ = 10,000. Choice A treats the difference of the exponents, 4, as the answer instead of 10⁴. Choice B multiplies the digit 4 by that difference. Choice C looks only at the digits.
Question 13 of 20 · Multiple Choice
A camera flash lasts about 0.001 s. A blink of an eye lasts about 0.3 s. How many times as long as the flash is the blink?
Answer: C
The flash is 1 × 10⁻³ s and the blink is 3 × 10⁻¹ s. (3 × 10⁻¹) ÷ (1 × 10⁻³) = 3 × 10² = 300. Choice A writes 0.3 as 3 × 10⁻² by counting one zero as a place. Choice B divides the flash by the blink. Choice D adds the exponents -1 and -3 to get 3 × 10⁻⁴.
Question 14 of 20 · Multiple Choice
Which number is equal to 7 × 10⁻⁴?
Answer: B
10⁻⁴ = 0.0001, and 7 × 0.0001 = 0.0007. Choice A writes four zeros after the decimal point instead of putting the 7 in the fourth place. Choice C ignores the negative sign. Choice D puts the 7 in the third place.
Question 15 of 20 · Short Answer
The Sahara Desert covers about 9,200,000 km². Write this area as a single digit times a power of 10.
9,200,000 rounds to 9,000,000, and the 9 is worth 1,000,000 = 10⁶. The area is about 9 × 10⁶ km².
Question 16 of 20 · Short Answer
Red light has a wavelength of about 0.0000007 m. Write this length as a single digit times a power of 10.
The 7 is in the seventh place after the decimal point, so the wavelength is about 7 × 10⁻⁷ m.
Question 17 of 20 · Short Answer
An adult blue whale can have a mass of about 1 × 10⁵ kg. A house cat has a mass of about 4 kg. About how many times as heavy is the whale?
Write 4 kg as 4 × 10⁰ kg. (1 × 10⁵) ÷ (4 × 10⁰) = 0.25 × 10⁵ = 25,000 times, or 2.5 × 10⁴.
Question 18 of 20 · Short Answer
A ladybug is about 6 × 10⁻³ m long. A tardigrade, a tiny animal also called a water bear, is about 3 × 10⁻⁴ m long. How many times as long is the ladybug?
The Nile River is about 6,650 km long. A charity fun run is 5 km. Estimate the length of the Nile as a single digit times a power of 10, then find about how many fun runs laid end to end would equal it.
6,650 km is about 7 × 10³ km. (7 × 10³) ÷ 5 = 1.4 × 10³, so about 1,400 fun runs. (The full numbers give 6,650 ÷ 5 = 1,330.)
Question 20 of 20 · Short Answer
Priya says that 9 × 10³ is greater than 2 × 10⁴, because 9 is greater than 2. Is she right? Find about how many times as large the larger number is.
No. 9 × 10³ = 9,000 and 2 × 10⁴ = 20,000, so 2 × 10⁴ is larger. Compare the powers of 10 first, then the digits. (2 × 10⁴) ÷ (9 × 10³) = 20,000 ÷ 9,000 ≈ 2.2, so it is about 2 times as large.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.EE.A.3 mean?
8.EE.A.3 means students can write a very large or very small quantity as one digit times a power of 10, such as 3 × 10⁸ people or 8 × 10⁻⁶ m, and use those estimates to say how many times as much one quantity is than another. The point is quick, sensible comparison, not exact values.
Is 8.EE.A.3 the same as scientific notation?
Not quite: 8.EE.A.3 is the first step toward it. Here every number is rounded to a single digit, like 7 × 10⁹. In the next standard, 8.EE.A.4, students use full scientific notation, a number from 1 up to (but not including) 10 times a power of 10, such as 7.9 × 10⁹, and they add, subtract, multiply and divide with it.
How do you round a number to a single digit times a power of 10?
Round to one nonzero digit, then count the place that digit is in. 4,860,000 rounds to 5,000,000, and the 5 is in the millions place, so it is 5 × 10⁶. For 0.00031, the 3 is in the fourth place after the decimal point, so it is 3 × 10⁻⁴. If the digit rounds up to 10, move to the next power: 0.096 becomes 1 × 10⁻¹.
How do you find how many times as much one quantity is than another?
Divide the larger quantity by the smaller one. With single-digit estimates, divide the digits and subtract the exponents. For example, (6 × 10⁹) ÷ (2 × 10³) = 3 × 10⁶, so the first is 3,000,000 times the second. If the digit answer is less than 1, rewrite it: (3 × 10⁵) ÷ (6 × 10¹) = 0.5 × 10⁴ = 5 × 10³.
Why does the official example say the world population is "more than 20 times" the US population?
Because 7 × 10⁹ divided by 3 × 10⁸ is 7/3 × 10, which is about 23. The standard rounds this down to a safe statement, "more than 20 times." The estimates come from the standard. Today the world has more than 8 billion people, and the comparison is still more than 20 times.
What do negative powers of 10 mean for small quantities?
A negative exponent means a number less than 1: 10⁻¹ = 0.1, 10⁻² = 0.01, and 10⁻⁶ = 0.000001. So 5 × 10⁻⁶ m is 5 millionths of a meter. Students learn negative exponents in 8.EE.A.1, and 8.EE.A.3 uses them to describe cells, dust and other tiny things.
What mistakes do students make with 8.EE.A.3?
A common mistake is counting zeros instead of places: 72,000,000 has six zeros but is 7 × 10⁷, because the 7 is worth 10,000,000. Other errors are adding exponents when dividing, dividing the smaller number by the larger, dropping the negative sign on small numbers, and writing 12 × 10⁵, which is not a single digit.
Why do estimates give a different answer than the exact numbers?
Rounding to one digit can move each number by up to about half a step. If one number is rounded up and the other down, the ratio changes more. That is fine for a first comparison. When a closer answer matters, students keep more digits, which is the work of 8.EE.A.4.
What should students know before starting 8.EE.A.3?
Students should know place value and powers of 10 from grade 5 (5.NBT.A.2), whole-number exponents from grade 6 (6.EE.A.1), and the rules for integer exponents from 8.EE.A.1, especially dividing powers by subtracting exponents and what a negative exponent means.
How can parents help with 8.EE.A.3 at home?
Look up big and small numbers together: the distance to a vacation spot, the population of your town, the thickness of a coin. Ask your child to round each one to a single digit times a power of 10 and then ask "how many times bigger?" Checking the answer with a calculator turns it into a game.
07
Related Standards
5 standards
These standards connect to 8.EE.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.NBT.A.2Prerequisite
Explain patterns when multiplying or dividing by powers of 10; use exponents for them
Lesson coming soon
6.EE.A.1Prerequisite
Write and evaluate numerical expressions with whole-number exponents
