6.EE.A.1Common CoreMathExpressions and EquationsGrade 6
6.EE.A.1: Writing and Evaluating Expressions with Exponents
In plain English: 6.EE.A.1 is the Common Core grade 6 math standard that asks students to write and evaluate numerical expressions with whole-number exponents. Students write repeated multiplication such as 5 × 5 × 5 as 5³, find the value of powers of whole numbers, fractions and decimals, and follow the order of operations when an exponent appears. It prepares for the exponent rules of grade 8.
Write and evaluate numerical expressions involving whole-number exponents.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions. Also written as 6.EE.1 · Official standard
Students learn a short way to write repeated multiplication. An exponent is a small raised number that tells how many times a number is used as a factor (a number being multiplied). The number being multiplied is the base. In 5³, the base is 5 and the exponent is 3, so 5³ = 5 × 5 × 5 = 125. The whole expression 5³ is called a power, and it is read "5 to the third power" or "5 cubed".
Students write numerical expressions (math phrases made of numbers and operation signs, with no letters) that use exponents, starting from repeated multiplication, from words and from situations such as square areas, phone trees and prime factorizations (a number written as a product of prime numbers). Then they evaluate the expressions, which means they find their values. They evaluate powers of whole numbers, fractions and decimals, and they use the order of operations (the agreed order for doing the operations in an expression) when an exponent appears together with other operations. All exponents are whole numbers, as the standard says. Negative exponents and the exponent rules come later, in grade 8.
Learning Objectives
By the end of this lesson, students will be able to:
Name the base and the exponent of a power and read the power aloud
Write repeated multiplication, word phrases and simple situations as numerical expressions with whole-number exponents
Evaluate powers of whole numbers, fractions and decimals
Evaluate numerical expressions that combine exponents with other operations, using the order of operations
Explain why a power such as 3² is not the same as 3 × 2
Prior Knowledge Required
Students should already be comfortable with:
Finding factor pairs and telling prime numbers from composite numbers 4.OA.B.4
Evaluating expressions with parentheses 5.OA.A.1
Writing simple expressions that record calculations 5.OA.A.2
Using whole-number exponents for powers of 10 5.NBT.A.2
Multiplying fractions 5.NF.B.4 and decimals 5.NBT.B.7
Hand each student a sheet of printer paper. Ask them to fold it in half, count the layers, then fold it in half again and count again. Then pose the question:
Warm-Up Prompt
"One fold makes 2 layers. Two folds make 4 layers. How many layers will 5 folds make? Write a multiplication that shows how you got your answer."
Collect answers. Each fold doubles the layers, so 5 folds give 2 × 2 × 2 × 2 × 2 = 32 layers. Some students will say 10, because they multiply 2 × 5. Ask them to count the layers after 3 folds: 2 × 3 = 6, but the paper shows 8. Tell students that mathematicians write 2 × 2 × 2 × 2 × 2 in a shorter way, 2⁵, and that today they will learn to write and find the value of expressions like it.
Direct Instruction20 minutes
Part 1: Base, exponent and power. Write 2⁵ on the board and label it. The base (2) is the factor that repeats. The exponent (5) tells how many times the base is used as a factor. The whole expression is a power. Read 2⁵ as "2 to the fifth power". Two powers have special names: a number to the second power is squared, because it gives the area of a square, and a number to the third power is cubed, because it gives the volume of a cube. Diagram 1 shows the squares. An exponent of 1 means the base is used once: 9¹ = 9.
Writing repeated multiplication with an exponent
Write 6 × 6 × 6 × 6 as a power, then evaluate it.
Equation: 6 is used as a factor 4 times, so 6 × 6 × 6 × 6 = 6⁴ = 36 × 36 = 1,296
Writing an expression from a situation
In a snow-day phone tree, the principal calls 3 teachers in round 1. In each later round, every person who was just called calls 3 new people. Write the number of calls in round 3 with an exponent.
Write 72 as a product of prime numbers, using exponents for repeated factors.
Equation: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
Part 2: Writing expressions. Model three ways an expression with an exponent can start: from repeated multiplication (the first example), from a situation (the phone tree in the second example) and from a prime factorization, which writes a number as a product of prime numbers (the fifth example). Point out that parentheses matter when you write from words. "The sum of 3 and 5, squared" is (3 + 5)², but "3 plus 5 squared" is 3 + 5².
Part 3: Evaluating expressions. The order of operations is the agreed order for doing the operations in an expression: first work inside parentheses, then evaluate exponents, then multiply and divide from left to right, then add and subtract from left to right. Show why the order matters: (3 × 5)² = 15² = 225, but 3 × 5² = 3 × 25 = 75. The exponent belongs only to the number right before it, unless parentheses group more. For powers of 10, the exponent counts the 10s being multiplied, and it also counts the zeros: 10² = 100 and 10³ = 1,000. For fractions, multiply the numerators and multiply the denominators. For decimals, count the decimal places in all the factors.
Guided Practice15 minutes
Pairs copy and complete the table, then check with the class. Ask one pair to explain each row aloud, using the words base and exponent.
Complete each row: the power, the repeated multiplication and the value.
Power
Repeated multiplication
Value
7²
?
?
?
2 × 2 × 2 × 2
?
10³
?
?
?
0.1 × 0.1 × 0.1
?
Answers: 7² = 7 × 7 = 49; 2 × 2 × 2 × 2 = 2⁴ = 16; 10³ = 10 × 10 × 10 = 1,000; 0.1 × 0.1 × 0.1 = 0.1³ = 0.001. Then work two evaluations together: 20 - 2³ × 2 = 20 - 8 × 2 = 20 - 16 = 4, and (5 - 2)³ + 1 = 3³ + 1 = 27 + 1 = 28. Finish with a writing problem: a square patio is 11 feet on each side. Write its area in square feet with an exponent and evaluate it (11² = 121 square feet). Listen for students who multiply the base by the exponent, and ask them to write the repeated multiplication first.
Independent Practice10 minutes
Students work alone on six problems, then compare with a partner. (1) Write 9 × 9 × 9 × 9 as a power and evaluate it. (9⁴ = 6,561.) (2) Evaluate 3² + 4². (9 + 16 = 25.) (3) Evaluate 50 - 6² ÷ 4. (50 - 36 ÷ 4 = 50 - 9 = 41.) (4) Evaluate 0.2³. (0.008.) (5) Write 200 as a product of primes with exponents. (2³ × 5².) (6) What is the value of 1⁸? Explain. (1, because 1 × 1 × ... × 1 is always 1.)
Closure5 minutes
Exit ticket: (1) Write 8 × 8 × 8 × 8 × 8 with an exponent. (8⁵.) (2) Evaluate 4 + 3 × 2³. (4 + 3 × 8 = 4 + 24 = 28.) (3) A classmate says 10² = 20. Explain the mistake. (10² means 10 × 10 = 100, not 10 × 2.)
Differentiation Strategies
For Struggling Students
Have students always write the repeated multiplication under a power before they evaluate it, for example 4³ = 4 × 4 × 4
Give an order-of-operations card with four rows (parentheses, exponents, multiply and divide, add and subtract) and have students rewrite the expression after each step
For squares, let students draw the square on grid paper and count the unit squares
For Advanced Students
Ask which is greater, 2⁸ or 8², and have them explain without a calculator (2⁸ = 256 is greater than 8² = 64)
Ask students to find the last digit of 3¹⁰ without finding the whole number, by looking for a pattern in the last digits of 3¹, 3², 3³, 3⁴ and so on (the last digits repeat 3, 9, 7, 1, so the last digit is 9)
Have students write 1,000,000 as a power of 10 and as a power of 100 (10⁶ and 100³)
Assessment Guidance
What to Look For
Check that students can name the base and the exponent and that they write repeated multiplication, not multiplication by the exponent, when they evaluate a power. When they write expressions from words or situations, look for parentheses where a sum or product is raised to a power. When they evaluate, look for exponents done before multiplication and division, and for correct decimal places in powers of decimals. Ask students to explain one answer aloud with the words base, exponent and power.
02
Classroom Activities
3 Activities
1
Paper Folding and Grid Squares
15 minPairs
Pairs fold paper to build a table of powers of 2, then draw squares on grid paper to see why the second power is called "squared". They write every result as a power and evaluate it.
Procedure
Fold a sheet of printer paper in half again and again. After each fold, count the layers and record them in a table with the columns: number of folds, repeated multiplication, power, layers
Stop when the paper is too thick to fold. Predict the layers for the next 2 folds by writing the power and evaluating it
On grid paper, draw squares with sides of 1, 2, 3, 4, 5 and 6 units. Under each, write the area as a power and its value, for example 4² = 16
Discussion Questions
After how many folds would the number of layers first be more than 100? (7 folds: 2⁷ = 128, while 2⁶ is still less than 100)
Compare 2³ and 3². Which is greater? (3² = 9 is greater than 2³ = 8, so a bigger exponent does not always give a bigger value)
Why is a square with side 6 made of 6 × 6 unit squares and not 6 × 2?
Modification for Distance Learning
Students fold paper at home and share their table in a shared document. For the squares, they use a digital grid or draw on graph paper and hold it up to the camera.
2
Exponent Match-Up Card Sort
15 minGroups of 3-4
Each group gets 15 cards: 4 sets of three matching cards (a power, its repeated multiplication and its value) and 3 trap cards that show a common mistake. Groups build the 4 sets and explain why each trap card does not belong.
Card Set (15 cards)
Set 1: 2⁵, 2 × 2 × 2 × 2 × 2, 32
Set 2: 5², 5 × 5, 25
Set 3: 4⁴, 4 × 4 × 4 × 4, 256
Set 4: 1⁷, 1 × 1 × 1 × 1 × 1 × 1 × 1, 1
Trap cards: 10, 16 and 7
Procedure
Shuffle the cards and deal them face up
Take turns placing a card into a set and explaining the match in words, for example "4 is used as a factor 4 times"
For each trap card, write which power it could be mistaken for and the error that gives it
Discussion Questions
Which trap card comes from 2 × 5? (10, a mistake for 2⁵) Which comes from 4 × 4? (16, a mistake for 4⁴) Which comes from 1 × 7? (7, a mistake for 1⁷)
Sets 1 and 2 use the same two digits. Which power is greater? (2⁵ = 32 is greater than 5² = 25)
Why is 1 to any whole-number power, 1 or more, always 1?
Challenge Variation
Each group makes one new set of three cards and one trap card for another group to sort. The new set must use a base or an exponent that is not already in the deck.
3
Order of Operations Relay
15 minPairs
Partners evaluate expression cards one step at a time. Partner A does one step and writes the new expression, partner B checks it and does the next step, and so on until the value is found. The cards come in pairs that look alike but have different values.
Before starting, each pair circles the part of the expression that must be done first
Write each step on a new line, so a mistake is easy to find
When a pair finishes a card, they compare with another pair and fix any step that differs
Discussion Questions
Cards A and B use the same numbers. Why are their values different? (In Card B the parentheses make 1 + 5 the base; in Card A only 5 is squared)
In Card C, which operation comes first, the multiplication or the exponent? (The exponent)
Which card has the greatest value? (Card D, 196)
Challenge Variation
Target number: use the numbers 2, 3 and 4 once each, with one of them as an exponent, to write an expression whose value is 19. (Two answers are 3 + 4² and 2⁴ + 3.)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Squares Show What "Squared" Means
Squares with sides of 1 to 5 units, drawn to scale on a grid. A square with side n is made of n × n unit squares, which is n². For example, the square with side 3 has 3² = 3 × 3 = 9 unit squares. The exponent 2 means the base is used as a factor 2 times, not multiplied by 2.
Diagram 2: A Phone Tree Grows by Powers of 3
A snow-day phone tree. The principal calls 3 people in round 1, and every person called then calls 3 new people. Round 1 has 3¹ = 3 calls, round 2 has 3² = 9 calls and round 3 has 3³ = 27 calls. Each new round uses 3 as a factor one more time.
04
Homework Assignment
~30 min
6.EE.A.1 Homework: Writing and Evaluating Powers
Directions: Show your work for every problem. When you evaluate a power, write the repeated multiplication first. When an expression has more than one operation, write one step per line.
Part 1: Writing Expressions with Exponents (Problems 1-2)
Write each product as a power, then evaluate it. (a) 17 × 17 (b) 6 × 6 × 6 × 6 × 6 (c) 1.5 × 1.5 (d) 3/4 × 3/4 × 3/4
(a) A square garden is 21 feet on each side. Write its area in square feet as a power and evaluate it. (b) A cube-shaped tissue box is 11 centimeters long on each edge. Write its volume in cubic centimeters as a power and evaluate it. (c) Write 108 as a product of prime numbers, using exponents for repeated factors.
Evaluate each expression. Show one step per line. (a) 6 + 2⁴ ÷ 8 (b) (7 - 4)³ × 2 (c) 5 × 2³ - 3² (d) 90 - 2 × 3³
Part 3: Exponents in Context and Error Analysis (Problems 5-6)
Maya saves money by doubling. She puts 1 cent in a jar on day 1, 2 cents on day 2, 4 cents on day 3, and so on. Write the number of cents she puts in on day 10 as a power of 2 and evaluate it. How much is that in dollars? Her brother says the answer is 2¹⁰. Is he right? Explain.
(a) Jordan says 2 × 5² = 100. Find his mistake and give the correct value. (b) Jordan also says 5⁴ = 20. Find his mistake and give the correct value. (c) Write 1,024 as a power of 2 and as a power of 4.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Writing Powers
Base and exponent correct for every product and situation, primes written with exponents
One base or exponent wrong
Most powers written incorrectly or missing
Evaluating Powers
All values correct, including fractions and decimals
One or two values wrong
Base multiplied by the exponent, or most values wrong
Order of Operations
Every step shown and done in the correct order
Correct order with one arithmetic slip
Operations done left to right without regard to exponents
Reasoning
Day 10 and the error analysis explained clearly
Correct answers with weak or missing explanations
Explanations missing or incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order, without a calculator. 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 expression means 7 × 7 × 7 × 7?
Answer: C
7 is used as a factor 4 times, so the base is 7 and the exponent is 4: 7⁴. Choice A swaps the base and the exponent: 4⁷ means 4 × 4 × 4 × 4 × 4 × 4 × 4. Choice B multiplies the base by the number of factors, which gives 28 instead of 2,401. Choice D adds the base and the number of factors.
Question 2 of 20 · Multiple Choice
What is the value of 3⁵?
Answer: B
3⁵ = 3 × 3 × 3 × 3 × 3 = 9 × 9 × 3 = 243. Choice A multiplies the base by the exponent: 3 × 5 = 15. Choice C swaps the base and the exponent: 5³ = 125. Choice D adds the base and the exponent: 3 + 5 = 8.
Question 3 of 20 · Multiple Choice
Which power has a value of 400?
Answer: D
20² = 20 × 20 = 400. Choice A treats squaring as doubling: 200 × 2 = 400, but 200² = 200 × 200 = 40,000. Choice B uses 20 as a factor one time too many: 20³ = 8,000. Choice C swaps the base and the exponent of 20²: 2²⁰ = 1,048,576, which is more than one million.
Question 4 of 20 · Multiple Choice
Which shows 48 as a product of prime numbers, using exponents?
Answer: A
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3, and 2 and 3 are both prime. Choice B equals 48, but 4 is not a prime number, so it is not a prime factorization. Choice C equals 8 × 9 = 72. Choice D has one factor of 2 too many: 32 × 3 = 96.
Question 5 of 20 · Multiple Choice
What is the value of 0.4²?
Answer: D
0.4² = 0.4 × 0.4 = 0.16. There are 2 decimal places in the factors, so there are 2 in the product. Choice A multiplies by the exponent: 0.4 × 2 = 0.8. Choice B moves the decimal point one place too few: 16 with 1 decimal place. Choice C uses 3 decimal places instead of 2.
Question 6 of 20 · Multiple Choice
What is the value of (1/2)³?
Answer: C
(1/2)³ = 1/2 × 1/2 × 1/2 = 1/8: multiply the numerators (1 × 1 × 1 = 1) and the denominators (2 × 2 × 2 = 8). Choice A multiplies the denominator by the exponent: 2 × 3 = 6. Choice B multiplies the fraction by the exponent: 1/2 × 3 = 3/2. Choice D uses 1/2 as a factor only 2 times.
Question 7 of 20 · Multiple Choice
Evaluate 2 + 5 × 3².
Answer: B
Exponent first: 3² = 9. Then multiply: 5 × 9 = 45. Then add: 2 + 45 = 47. Choice A adds first: (2 + 5) × 9 = 63. Choice C multiplies before squaring: 2 + 15² = 2 + 225 = 227. Choice D treats 3² as 3 × 2: 2 + 5 × 6 = 32.
Question 8 of 20 · Multiple Choice
Evaluate 36 ÷ 3² × 2.
Answer: A
Exponent first: 3² = 9. Then divide and multiply from left to right: 36 ÷ 9 = 4, and 4 × 2 = 8. Choice B multiplies before dividing: 36 ÷ (9 × 2) = 2. Choice C ignores the exponent: 36 ÷ 3 × 2 = 24. Choice D treats 3² as 3 × 2: 36 ÷ 6 × 2 = 12.
Question 9 of 20 · Multiple Choice
Evaluate (4 + 2)³.
Answer: D
Parentheses first: 4 + 2 = 6. Then 6³ = 6 × 6 × 6 = 216. Choice A cubes only the 2: 4 + 2³ = 12. Choice B squares the sum instead of cubing it: 6² = 36. Choice C multiplies the sum by the exponent: 6 × 3 = 18.
Question 10 of 20 · Multiple Choice
Which expression shows "the sum of 6 and 4, squared"?
Answer: B
The sum 6 + 4 is squared as one group, so it needs parentheses: (6 + 4)² = 10² = 100. Choice A squares only the 4, which is "6 plus 4 squared". Choice C squares each number before adding. Choice D multiplies the sum by 2 instead of squaring it.
Question 11 of 20 · Multiple Choice
A square poster is 18 inches on each side. Which expression gives its area in square inches?
Answer: A
The area of a square is side × side = 18 × 18 = 18², which is 324 square inches. Choice B would be the volume of a cube with 18-inch edges. Choice C is the perimeter, the distance around the poster. Choice D multiplies the side by the exponent 2 instead of using it as a factor twice.
Question 12 of 20 · Multiple Choice
Which number is equal to 10⁵?
Answer: C
10⁵ = 10 × 10 × 10 × 10 × 10 = 100,000, a 1 followed by 5 zeros. Choice A multiplies 10 by 5. Choice B has only 4 zeros, which uses 10 as a factor 4 times. Choice D has 6 zeros, which uses 10 as a factor 6 times.
Question 13 of 20 · Multiple Choice
Which is greater, 2⁶ or 6²?
Answer: A
2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64 and 6² = 6 × 6 = 36, so 2⁶ is greater. Choice B finds 6² correctly but treats 2⁶ as 2 × 6 = 12. Choice C multiplies the base by the exponent in both powers. Choice D judges by the base only, but the exponent matters just as much.
Question 14 of 20 · Multiple Choice
In a video game, level 1 has 3 coins, and each level has 3 times as many coins as the level before. Which gives the number of coins on level 4?
Answer: C
Level 1 has 3 coins, level 2 has 3 × 3, level 3 has 3 × 3 × 3, and level 4 has 3 × 3 × 3 × 3 = 3⁴ = 81 coins. Choice A multiplies 3 by the level number (12 coins) instead of using 3 as a factor 4 times. Choice B swaps the base and the exponent. Choice D, 3³ = 27, is the number of coins on level 3.
Question 15 of 20 · Short Answer
Write 7 × 7 × 7 as a power. Then evaluate it.
7 is used as a factor 3 times, so the power is 7³. Its value is 7 × 7 = 49, then 49 × 7 = 343.
Ana says 5 + 3² = 64. What mistake did she make, and what is the correct value?
Ana added first and then squared: (5 + 3)² = 8² = 64. With no parentheses, the exponent comes before the addition, so 5 + 3² = 5 + 9 = 14.
Question 20 of 20 · Short Answer
A cube-shaped gift box is 9 inches long on each edge. Write an expression with an exponent for its volume in cubic inches, then evaluate it.
The volume of a cube is edge × edge × edge, so the expression is 9³. 9 × 9 = 81 and 81 × 9 = 729 cubic inches.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.EE.A.1 mean?
6.EE.A.1 means students can write and evaluate numerical expressions that use whole-number exponents. Writing means turning repeated multiplication, words or a situation into an expression such as 2³ × 5. Evaluating means finding its value, for example 2³ × 5 = 8 × 5 = 40, and following the order of operations when there are other operations too.
Is 6.EE.A.1 a grade 6 or grade 8 standard?
6.EE.A.1 is a grade 6 standard. Students first meet exponents in grade 5, but only for powers of 10 (5.NBT.A.2). In grade 8, students learn the properties of integer exponents, including negative exponents (8.EE.A.1). Grade 6 is the year students write and evaluate powers of any base.
What is the difference between the base and the exponent?
The base is the number that is multiplied, and the exponent tells how many times it is used as a factor. In 6², the base is 6 and the exponent is 2, so 6² = 6 × 6. The exponent is written smaller and raised, to the upper right of the base.
Why is 3² not equal to 6?
Because the exponent is not a factor: 3² means 3 × 3 = 9, not 3 × 2 = 6. Multiplying the base by the exponent is a common mistake. Writing out the repeated multiplication before evaluating, or drawing a 3-by-3 square, helps students avoid it.
Where do exponents go in the order of operations?
Exponents come right after parentheses and before multiplication and division. For example, 4 × 2³ = 4 × 8 = 32. If a sum or product is inside parentheses, work it out first and then apply the exponent: (1 + 2)² = 3² = 9.
Can the base be a fraction or a decimal in grade 6?
Yes. The standard limits the exponents to whole numbers, but the base can be a fraction or a decimal. For example, (3/4)² = 9/16 and 0.1² = 0.01. Students use what they know about multiplying fractions and decimals from grade 5.
Does 6.EE.A.1 include zero or negative exponents?
Negative exponents are not part of 6.EE.A.1; they come in grade 8 (8.EE.A.1). Zero is a whole number, but the reason a nonzero number to the zero power equals 1 is usually explained in grade 8 with the exponent rules. In grade 6, problems use exponents of 1 or more.
How are exponents used in prime factorization?
Exponents give a short way to write repeated prime factors. For example, 40 = 2 × 2 × 2 × 5 = 2³ × 5. This connects to finding greatest common factors and least common multiples (6.NS.B.4), where students compare the prime factors of two numbers.
What are real-life examples of exponents?
Area and volume are the most familiar: a square with 5-meter sides has an area of 5² square meters, and a cube with 2-foot edges has a volume of 2³ cubic feet. Doubling and tripling patterns, such as a message that each person shares with 2 friends, also use exponents. Place value uses powers of 10: 3,000 = 3 × 10³.
How can parents help with exponents at home?
Ask your child to explain what a power means before finding its value, for example "4³ means three 4s multiplied". Fold a piece of paper together and count the layers after each fold. Look for square tiles on a floor and count how many make a square, then write that number as a power.
07
Related Standards
6 standards
These standards connect to 6.EE.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.OA.A.1Prerequisite
Use parentheses, brackets or braces in numerical expressions and evaluate them
Lesson coming soon
5.NBT.A.2Prerequisite
Explain patterns when multiplying by powers of 10, using whole-number exponents
Lesson coming soon
Alongside
6.EE.A.2Parallel
Write, read and evaluate expressions in which letters stand for numbers
