Teaching unit · Grade 6 (ages 11 to 12)

Exponents and algebraic expressions

Writing and evaluating exponents, writing algebraic expressions from words, and generating equivalent expressions

About four lessons of 45 to 60 minutes

Start here · hook

How do you write '2 used as a factor 4 times' without writing 2 x 2 x 2 x 2 every time?

Some numbers, like the total squares on a chessboard doubling pattern or the ways to arrange a deck of cards, involve the same factor multiplied by itself again and again. Writing 2 x 2 x 2 x 2 every time is slow and easy to miscount, so mathematicians write it as 2 to the power of 4, a compact exponent notation you have already met with powers of ten.

Letters do a similar compacting job in algebra: instead of writing out 'a number, plus 5' in words every time, you write n + 5, where n stands in for any number. This unit builds both compact-writing skills, then a third: rewriting the same expression in a different but equal form, which is the engine behind almost every algebra shortcut later on.

2 x 2 x 2 x 2 x 2written as 2 to the power of 5, which equals 32
5 more than twice a number nwritten as the algebraic expression 2n + 5
Evaluate 3x + 7 when x = 43 x 4 + 7 = 19
4(x + 3) and 4x + 12two different-looking but equal (equivalent) expressions

Learning objective

What students will be able to do

Students will write and evaluate numerical expressions involving whole-number exponents, write, read, and evaluate algebraic expressions in which a letter stands for an unknown number, and apply the distributive property and combining like terms to generate equivalent algebraic expressions.

Success criteria

I can write a repeated multiplication using exponent notation and evaluate it.
I can translate a word phrase into an algebraic expression using a variable.
I can evaluate an algebraic expression by substituting a given value for the variable.
I can use the distributive property to expand an expression such as 4(x + 3).
I can factor an expression such as 6x + 9 by pulling out a common factor.

Curriculum anchor

Standards this unit teaches

6.EE.A.1Common Core (US)
Write and evaluate exponents
Write and evaluate numerical expressions that involve whole number exponents.
6.EE.A.2Common Core (US)
Write algebraic expressions
Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.3Common Core (US)
Generate equivalent expressions
Apply properties of operations to write equivalent algebraic expressions, such as expanding or factoring.
AC9M7N02Australian Curriculum v9 (ACARA)
Primes and exponent notation (Year 7)
Write natural numbers as products of powers of their prime factors using exponent (index) notation. Australia's formal exponent-notation descriptor sits at Year 7, so this unit's exponent work runs about a year ahead of the ACARA placement.
AC9M7A02Australian Curriculum v9 (ACARA)
Build algebraic expressions (Year 7)
Write algebraic expressions from words using constants, variables, operations and brackets. Australia's matching descriptor for writing algebraic expressions also sits at Year 7, so this unit runs a year ahead there too.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Exponent: the small raised number showing how many times the base is multiplied by itself
Base: the number being multiplied by itself in an exponent expression
Variable: a letter that stands in for an unknown or changing number
Algebraic expression: a combination of numbers, variables, and operations, with no equals sign, such as 3x + 7
Evaluate: to find the value of an expression by substituting a number for each variable
Equivalent expressions: two expressions that always give the same value for every value of the variable, even though they look different
Distributive property: the rule a(b + c) = ab + ac, used to expand or factor an expression

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Exponents: a compact way to write repeated multiplication

Concrete

5 to the power of 3 means 5 x 5 x 5, three copies of 5 multiplied together: 5 x 5 = 25, then 25 x 5 = 125. The base, 5, is the number being repeated, and the exponent, 3, counts how many times. Unlike 5 x 3 (which is just 15), an exponent means repeated multiplying, not repeated adding.

Exponents fit into the order of operations right after grouping symbols and before multiplication and division, so 2 + 3 to the power of 2 means 2 + 9 = 11, evaluating the exponent before the addition.

Worked example

Evaluate 4 to the power of 3.

4 to the power of 3 means 4 x 4 x 4.
4 x 4 = 16.
16 x 4 = 64.

Answer: 4 to the power of 3 = 64.

Check for understanding, ask

What does the exponent in 5 to the power of 3 count?
Is 2 to the power of 4 the same as 2 x 4? Why or why not?

2. Writing and evaluating algebraic expressions

Pictorial

An algebraic expression uses a letter, a variable, to stand for a number that is unknown or that can change. Translate word phrases the same careful way you write numerical expressions: 'twice a number n' is 2n, and 'the product of 6 and a number' is 6n. Watch for 'more than', which reverses word order: '5 more than twice a number n' is 2n + 5, not 5n + 2.

To evaluate an expression, substitute the given number for the variable everywhere it appears, then follow the order of operations. Evaluate 2x² + 1 when x = 3: first 3 to the power of 2 is 9, then 2 x 9 = 18, then 18 + 1 = 19.

Worked example

Write an algebraic expression for 'the product of 6 and a number, plus 3', then evaluate it when the number is 5.

'The product of 6 and a number' is 6n. 'Plus 3' attaches + 3: 6n + 3.
Substitute n = 5: 6 x 5 + 3.
6 x 5 = 30. Then 30 + 3 = 33.

Answer: The expression is 6n + 3, and when n = 5 it evaluates to 33.

Check for understanding, ask

Write an algebraic expression for 'half of a number, decreased by 7'.
Why does '5 more than twice a number n' become 2n + 5 and not 5n + 2?

3. Generating equivalent expressions

Abstract

Two expressions are equivalent when they give the same value for every possible value of the variable, even though they look different. The distributive property, a(b + c) = ab + ac, is the main tool for rewriting one form into the other: 4(x + 3) expands to 4x + 12, and they are equivalent because substituting any value of x into either gives the same result.

Factoring runs the distributive property in reverse: to factor 6x + 9, find the greatest common factor of 6 and 9, which is 3, and pull it out: 6x + 9 = 3(2x + 3). Combining like terms is another way to generate an equivalent, simpler expression: 3x + 5x = 8x, and 7y + 2 - 3y = 4y + 2.

Worked example

Expand 5(y + 2), and factor 8x + 12.

Expand using the distributive property: 5(y + 2) = 5 x y + 5 x 2 = 5y + 10.
Factor 8x + 12: the greatest common factor of 8 and 12 is 4.
8x + 12 = 4(2x + 3).

Answer: 5(y + 2) = 5y + 10 (expanded). 8x + 12 = 4(2x + 3) (factored).

Check for understanding, ask

How do you check that two expressions are truly equivalent, using a value for the variable?
What is the first step in factoring 6x + 9?

Watch for

Common misconceptions and how to address them

MisconceptionAn exponent means multiply the base by the exponent, so 5 to the power of 3 is 5 x 3 = 15.

Why it happens: The exponent notation looks similar to multiplication and students have not yet internalised that it represents repeated multiplication, not a single multiplication by the exponent.

How to address it: Write the exponent out in full every time until it is trusted: 5 to the power of 3 is 5 x 5 x 5 = 125, not 5 x 3. Contrast 5 to the power of 3 (125) directly against 5 x 3 (15) so the difference is unmistakable.

MisconceptionAny algebraic expression with 'more than' keeps the same word order when translated, so '5 more than twice a number n' becomes 5n + 2.

Why it happens: Students translate word by word in reading order rather than untangling which quantity is actually described as 'more than' the other.

How to address it: 'A more than B' always means B + A -- the second-named quantity is the base, and the first-named quantity is added to it. '5 more than twice a number n' means (twice n) + 5, so 2n + 5. Practise several 'more than' and 'less than' phrases side by side to build the pattern.

Misconception4(x + 3) and 4x + 3 are the same expression, since expanding just means 'multiply the first number in'.

Why it happens: Students apply the distributive property to only the first term inside the parentheses and forget it must be applied to every term.

How to address it: The number outside the parentheses multiplies EVERY term inside, not just the first: 4(x + 3) = 4 x x + 4 x 3 = 4x + 12. Check by substituting a number for x in both the original and expanded form to confirm they match.

Do it together

Guided practice (with answers)

1. Evaluate 4 to the power of 3.
Answer: 64, since 4 x 4 x 4 = 64.
2. Evaluate 2x² + 1 when x = 3.
Answer: 19, since 2 x 9 + 1 = 18 + 1 = 19.
3. Write an algebraic expression for 'the product of 6 and a number, plus 3'.
Answer: 6n + 3.
4. Expand 5(y + 2).
Answer: 5y + 10.
5. Factor 8x + 12.
Answer: 4(2x + 3), since the greatest common factor of 8 and 12 is 4.
6. Combine like terms: 6a + 3a - 2.
Answer: 9a - 2.

On their own

Independent practice worksheets

Set the matching ChalkBee algebra and order-of-operations worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with evaluating exponents, then move to writing and manipulating algebraic expressions.

Grade 6 order of operations worksheet (Set B)exponents inside expressions, extending Grade 5's grouping symbols Algebra: Equations & Expressions (Grade 6, Set A)solving equations, evaluating expressions and function tables Grade 6 square numbers worksheet (Set A)exponent and squaring practice All algebra worksheetsthe full set across Grades 4 to 6

Reach every student

Differentiation

Support

Write out every exponent expression in full multiplication form (5 to the power of 3 = 5 x 5 x 5) before evaluating, until the notation is trusted.
Use a consistent translation table for word phrases (product means multiply, sum means add, more than means add after) as a reference during writing-expressions practice.
Model the distributive property with an area-model rectangle (width a, length split into b and c) so 'multiply every term' has a visual reason, not just a rule.
Keep early evaluate-the-expression tasks to a single variable and a single substitution value before mixing variables and exponents together.

Extension

Evaluate expressions with two variables, substituting given values for both.
Factor expressions where the greatest common factor includes both a number and a repeated variable, such as 6x² + 9x.
Prove two expressions are equivalent by evaluating both at several different values of the variable and confirming they always match.
Preview solving equations (the next unit) by finding what value of x makes an expanded and a factored form of the same expression equal.

Check it stuck

Assessment: exit ticket

A four-question exit ticket for the last five minutes, sampling exponents, evaluating, expanding, and word-to-expression translation.

1. Evaluate 6 to the power of 2.
Answer: 36, since 6 x 6 = 36.
2. Evaluate 3n - 4 when n = 5.
Answer: 11, since 3 x 5 - 4 = 15 - 4 = 11.
3. Write an algebraic expression for 'half of a number, decreased by 7'.
Answer: n / 2 - 7, or equivalently n/2 - 7.
4. Expand 3(2x + 5).
Answer: 6x + 15.

For the teacher

Teacher notes and timings

Rough timing across four lessons: Lesson 1 exponents (section 1), Lesson 2 writing and evaluating algebraic expressions (section 2), Lesson 3 equivalent expressions (section 3), Lesson 4 mixed practice and the exit ticket.
Language to keep saying: an exponent counts repeated multiplication not a single multiply, translate 'more than' by adding after the base quantity, the outside number multiplies every term inside the parentheses. These target the three main misconceptions directly.
This unit deliberately keeps exponents to whole-number bases and small exponents (2 or 3), and expressions to a single variable, so the notation itself is the focus rather than complex arithmetic.
Curriculum note: this US Grade 6 standard bundles three skills the Australian Curriculum places together a year later, at Year 7: exponent (index) notation (AC9M7N02) and building algebraic expressions from words (AC9M7A02), so this unit runs about a year ahead of the ACARA placement for both halves.
Present mode and print both work: use Present to build the area-model rectangle for the distributive property live with the class, then print for independent expression practice.

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