6.EE.A.3Common CoreMathExpressions and EquationsGrade 6
6.EE.A.3: Using Properties of Operations to Write Equivalent Expressions
In plain English: 6.EE.A.3 is the Common Core grade 6 math standard that asks students to apply the properties of operations to write equivalent expressions. Students use the distributive property to expand 3(2 + x) into 6 + 3x and to factor 24x + 18y into 6(4x + 3y), and they combine like terms, such as y + y + y into 3y. It prepares for work with linear expressions in grade 7.
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
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.3 · Official standard
Students learn to rewrite an expression in a new form that always has the same value. An expression is a group of numbers, letters and operation signs, such as 5n + 7, with no equal sign. A variable is a letter that stands for a number. Two expressions are equivalent when they name the same number for every value of the variable. Students generate equivalent expressions with the properties of operations: the commutative, associative, identity and distributive properties they already use with whole numbers.
The lesson works through the three official examples. Students expand 3(2 + x) into 6 + 3x, factor 24x + 18y into 6(4x + 3y) with the greatest common factor, and rewrite y + y + y as 3y. Area models on grid paper and tiles make each step visible. All numbers stay positive, and the coefficients are whole numbers, simple fractions or decimals, as expected in grade 6. Deciding whether two given expressions are equivalent is the focus of 6.EE.A.4, the next standard.
Learning Objectives
By the end of this lesson, students will be able to:
Name the commutative, associative, identity and distributive properties and use each one to rewrite an expression
Use the distributive property to expand an expression such as 3(2 + x) into a sum
Use the greatest common factor and the distributive property to factor a sum such as 24x + 18y into a product
Combine like terms, such as y + y + y or 5a + 3a, to write an equivalent expression with fewer terms
Explain each step of a rewrite by naming the property it uses
Prior Knowledge Required
Students should already be comfortable with:
Using properties of operations as strategies to multiply and divide 3.OA.B.5
Using parentheses in numerical expressions and evaluating them 5.OA.A.1
Writing and reading expressions with variables, and naming terms and coefficients 6.EE.A.2
Finding the greatest common factor of two whole numbers 6.NS.B.4
Post the question and give students 3 minutes to answer it in two different ways without a calculator.
Warm-Up Prompt
"Four friends go to a movie. Each friend buys a ticket for $9 and a popcorn for $6. Write two different ways to find the total cost. Do both ways give the same total?"
Collect both methods. Some students add first: 4 × (9 + 6) = 4 × 15 = $60. Others multiply first: 4 × 9 + 4 × 6 = 36 + 24 = $60. Both give $60 because of the distributive property: multiplying a sum gives the same result as multiplying each part and then adding. Now replace the prices with letters. If a ticket costs t dollars and a popcorn costs p dollars, the two methods become 4(t + p) and 4t + 4p. These are equivalent expressions: they give the same total for any prices.
Direct Instruction20-25 minutes
Part 1: Words for the parts of an expression. In 5n + 7, the parts joined by + are called terms: 5n and 7. The number multiplied by the variable, 5, is the coefficient. A term with no variable, 7, is a constant. Like terms have the same variable part: 4a and 9a are like terms, but 4a and 4b are not. In algebra, 3x means 3 × x, and 3(2 + x) means 3 × (2 + x).
Part 2: The properties. Each property below is true for every number, so it can be used with variables too.
Properties of operations used in this lesson
Property
With numbers
With variables
Commutative property of addition
7 + 5 = 5 + 7
a + b = b + a
Commutative property of multiplication
4 × 9 = 9 × 4
ab = ba
Associative property of addition
(2 + 6) + 4 = 2 + (6 + 4)
(a + b) + c = a + (b + c)
Associative property of multiplication
(3 × 5) × 2 = 3 × (5 × 2)
(ab)c = a(bc)
Distributive property
5 × (10 + 2) = 5 × 10 + 5 × 2
a(b + c) = ab + ac
Identity property of multiplication
1 × 8 = 8
1a = a
Part 3: Worked examples. Work each example on the board and name the property at every step. The first three are the official examples of the standard. For the first two, draw the area models in Diagram 1: a rectangle whose height is the number outside the parentheses and whose width is split into the parts inside them. The area of the whole rectangle equals the sum of the areas of its parts.
Part 4: Factoring. To factor a sum means to write it as a product. The second example uses the greatest common factor (GCF), the largest whole number that divides both coefficients. Other common factors also give equivalent expressions: 24x + 18y = 2(12x + 9y) = 3(8x + 6y). With the GCF, the numbers left inside the parentheses (4 and 3) have no common factor except 1, which is how students can tell the factoring is complete.
Part 5: Combining like terms. The third example shows why y + y + y is 3y: each y is 1y by the identity property, and the distributive property adds the coefficients. Diagram 2 shows the same idea with a tape diagram (a bar split into equal parts) and with tiles for the fourth example. Point out that 3y is not y³: y + y + y is repeated addition, while y³ means y × y × y.
Guided Practice15 minutes
Pairs solve four problems, one at a time. For each one, they write every step and the property it uses, and they sketch an area model for problems 1 and 2. After each problem, one pair presents.
Problem 1: Expand 5(x + 4). (5x + 20, by the distributive property.)
Problem 2: Factor 12 + 30n with the GCF. (The GCF of 12 and 30 is 6, so 6(2 + 5n).)
Problem 3: Write b + b + b + b + 6 with two terms. (4b + 6: identity property, then distributive property.)
Problem 4: Rewrite 7 × (2w) as a single term. (By the associative property, (7 × 2)w = 14w.)
Listen for students who write 5(x + 4) as 5x + 4. Ask them to draw the area model: the second part of the rectangle is 5 by 4, so its area is 20, not 4.
Independent Practice10-15 minutes
Students work alone on five problems and name one property for each. (1) Expand 8(2k + 1). (16k + 8.) (2) Factor 36p + 27 with the GCF. (9(4p + 3).) (3) Write 3r + 4 + 2r + 1 with two terms, then factor the result. (5r + 5, which is 5(r + 1).) (4) A class orders 6 lunch boxes. Each box has a sandwich that costs s dollars and a $2 drink. Write the total cost as a product and as a sum. (6(s + 2) and 6s + 12.) (5) Write 4(c + 3) + 2c with two terms. (6c + 12.)
Closure5 minutes
Exit ticket: (1) Expand 4(3 + 2z). (12 + 8z.) (2) Factor 10a + 15 with the GCF. (5(2a + 3).) (3) Which property shows that 6h + 3h = (6 + 3)h? (The distributive property, used to combine like terms.) Collect the tickets and sort them into three piles: correct, distributed to only one term, and other errors.
Differentiation Strategies
For Struggling Students
Give an area model template with the height box and two width boxes already drawn, so students fill in numbers and multiply each part
Let students draw arrows from the number outside the parentheses to each term inside before they multiply
Use paper tiles (tall strips for the variable, small squares for 1) to combine like terms before writing the result
For Advanced Students
Ask for three different factored forms of 36x + 48 and which one uses the GCF (12(3x + 4))
Ask students to expand 1/2(10a + 4b + 6) and to explain why the distributive property works with three terms (5a + 2b + 3)
Have students write a four-step rewrite of their own that uses four different properties, and trade with a partner who names each property
Assessment Guidance
What to Look For
Check that students multiply the number outside the parentheses by every term inside, not only the first. When factoring, look for the GCF and for a check by expanding back to the starting sum. When combining like terms, students should add only coefficients of like terms and keep the variable, writing 7m and not 7m² or 7. Ask students to name the property for each step: a student who can explain why each step is allowed can rewrite expressions they have never seen.
02
Classroom Activities
3 Activities
1
Area Models on Grid Paper
20 minPairs
Pairs draw area models to expand and to factor. Each rectangle shows one expression as a product (height times total width) and as a sum (the areas of the parts).
Expression Cards (6 cards)
Card 1, expand: 2(x + 5). (2x + 10)
Card 2, expand: 4(3 + y). (12 + 4y)
Card 3, expand: 3(2a + 1). (6a + 3)
Card 4, factor: 8m + 12. (4(2m + 3))
Card 5, factor: 15 + 10k. (5(3 + 2k))
Card 6, factor: 18p + 6q. (6(3p + q))
Procedure
For an expand card, draw a rectangle on grid paper. Write the number outside the parentheses as the height, and split the width into one part for each term inside
Write the area of each part inside it, then write the sum of the areas
For a factor card, write the two areas first, then find the largest height that divides both of them: the GCF. Write each width as area ÷ height
Write the finished equation under each rectangle, for example 2(x + 5) = 2x + 10
Discussion Questions
On Card 1, why is the area of the part with width x equal to 2x and not x + 2?
On Card 6, 3(6p + 2q) is also equivalent to 18p + 6q. Why is 6 the better height? (6 and 2 inside the parentheses still share a factor of 2)
How can you check a factored answer without drawing? (Expand it and see if you get the starting sum)
Modification for Distance Learning
Share the cards on a slide with a grid background. Pairs drag rectangles and text boxes to build each model, then post a screenshot with the finished equations.
2
Property Relay Card Sort
20 minGroups of 3-4
Each group gets 13 step strips from three rewrites (Chains A, B and C) mixed together, and 4 property cards. The group puts each chain in order and places a property card between steps to explain each move.
Step Strips (13 strips)
Chain A (5 strips): 4(n + 2) + 3n, then 4n + 8 + 3n, then 4n + 3n + 8, then (4 + 3)n + 8, then 7n + 8
Chain B (3 strips): 2 × (6 × g), then (2 × 6) × g, then 12g
Chain C (5 strips): w + 5 + w + w, then w + w + w + 5, then 1w + 1w + 1w + 5, then (1 + 1 + 1)w + 5, then 3w + 5
Property Cards (4 cards)
Commutative property of addition
Associative property of multiplication
Distributive property
Identity property of multiplication
Groups reuse the property cards by writing the name on a sticky note when a property appears more than once.
Procedure
Sort the 13 strips into three chains by the letters they use (n, g and w)
Put each chain in order from the starting expression to the shortest one
Between every two strips, place the property that allows the move
Check each chain by substituting a small number, such as 2, into the first and last strips
Discussion Questions
Chain A uses the distributive property twice. What does it do the first time, and what does it do the second time? (First it expands 4(n + 2); then it adds the coefficients of 4n and 3n)
Which chain uses the identity property, and why is that step needed? (Chain C: writing w as 1w shows the coefficients to add)
With n = 2, the first and last strips of Chain A both equal 22. Does that one check prove the chain is right? Why do the properties matter?
Challenge Variation
Groups write their own chain of at least four strips that starts with 5(2 + d) + d and ends with 10 + 6d, then trade with another group, which must add the property cards.
3
One Sum, Many Products
20 minPairs
Pairs factor the same sum in every possible way, then decide which way uses the greatest common factor. This builds the habit of checking that factoring is complete.
Sums to Factor (4 cards)
24x + 36
20 + 30t
16a + 40b
45m + 18
Procedure
List every whole number greater than 1 that divides both coefficients. For 24x + 36, the list is 2, 3, 4, 6 and 12
Write the factored form for each number on your list, for example 4(6x + 9)
Expand each factored form to check it
Circle the form that uses the GCF: 12(2x + 3), 10(2 + 3t), 8(2a + 5b) and 9(5m + 2)
Discussion Questions
Which card has the most factored forms? (24x + 36, with 5 forms; 20 + 30t and 16a + 40b each have 3, and 45m + 18 has 2)
Look at the numbers inside the parentheses of each circled form. What do you notice? (Inside each circled form, the two numbers have no common factor other than 1)
Is 4(6x + 9) wrong? (No, it is equivalent to 24x + 36, but it can still be factored: 6 and 9 share a factor of 3)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Area Models for Expanding and Factoring
Left: an area model for 3(2 + x). The rectangle is 3 tall, and its width is split into 2 and x. The parts have areas 6 and 3x, so 3(2 + x) = 6 + 3x. Right: the same idea in reverse for 24x + 18y. The height 6, the GCF of 24 and 18, divides both areas, which leaves the widths 4x and 3y, so 24x + 18y = 6(4x + 3y).
Diagram 2: A Tape Diagram and Tiles for Like Terms
Top: three bars of length y laid end to end are as long as one bar of 3 equal parts, so y + y + y = 3y for any length y. Bottom: tiles for 5a + 2 + 3a + 7. Moving the tall a tiles together and the unit squares together does not change the total, which gives 8a + 9.
04
Homework Assignment
~30 min
6.EE.A.3 Homework: Writing Equivalent Expressions
Directions: Show every step. For each step, name the property you used (commutative, associative, identity or distributive). Check each answer by substituting a small whole number into the starting expression and your answer.
Part 1: Expanding with the Distributive Property (Problems 1-2)
Use the distributive property to rewrite each expression without parentheses: (a) 7(y + 3) (b) 5(4 + 2d) (c) 1/2(12x + 2). Draw an area model for part (a).
A soccer club buys 9 team kits. Each kit has a jersey that costs j dollars and a pair of socks that costs $4. Write the total cost as a product and as a sum. Then use both expressions to find the total cost when a jersey costs $18.
Part 2: Factoring with the GCF (Problems 3-4)
Use the greatest common factor to rewrite each sum as a product: (a) 14a + 21 (b) 32 + 48w (c) 30p + 25q. Check each answer by expanding it.
Ravi rewrote 40x + 16 as 4(10x + 4). Is his expression equivalent to 40x + 16? Did he use the greatest common factor? Write the factored form that uses the GCF and explain how you know it is complete.
Part 3: Combining Like Terms (Problems 5-6)
Rewrite each expression with as few terms as possible, and name the properties you used: (a) m + m + m + m + m (b) 6c + 9 + c + 2 (c) 3(2h + 5) + 4h.
The perimeter of a rectangular garden with length L meters and width 6 meters is L + 6 + L + 6. Write two shorter equivalent expressions for the perimeter: one as a sum of two terms and one as a product. Then find the perimeter when L = 10 meters.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Expanding
Every term inside the parentheses is multiplied, and the area model matches
One term missed or one arithmetic error
Expressions not rewritten or not equivalent
Factoring
Uses the GCF, checks by expanding, explains why the factoring is complete
Equivalent product with a smaller common factor, or no check
Product not equivalent to the sum
Combining Like Terms
Adds coefficients of like terms only and keeps the variable
One error, such as adding a constant to a coefficient
Unlike terms combined throughout
Naming Properties
Every step names the correct property
Most steps named correctly
Properties missing or wrong
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 expression is equivalent to 6(x + 5)?
Answer: C
The distributive property multiplies 6 by each term inside the parentheses: 6 × x + 6 × 5 = 6x + 30. Choice A multiplies only the x by 6 and leaves the 5 alone. Choice B multiplies only the 5. Choice D adds 6 to 5 instead of multiplying.
Question 2 of 20 · Multiple Choice
Which expression is equivalent to 4(2b + 7)?
Answer: A
Multiply 4 by both terms: 4 × 2b = 8b and 4 × 7 = 28, so the result is 8b + 28. Choice B multiplies only the first term. Choice C adds 4 to each number (2 + 4 and 7 + 4) instead of multiplying. Choice D multiplies only the second term.
Question 3 of 20 · Multiple Choice
Which expression is equivalent to 0.5(6y + 10)?
Answer: B
0.5 × 6y = 3y and 0.5 × 10 = 5, so the result is 3y + 5. Choice A multiplies only the first term by 0.5. Choice C adds 0.5 to each number instead of multiplying. Choice D divides each term by 0.5, which doubles it, instead of multiplying by 0.5.
Question 4 of 20 · Multiple Choice
Which expression uses the greatest common factor to rewrite 18a + 42 as a product?
Answer: C
The factors of 18 are 1, 2, 3, 6, 9 and 18, and 6 is the largest of them that also divides 42. So 18a + 42 = 6 × 3a + 6 × 7 = 6(3a + 7). Choices A and B are equivalent to 18a + 42, but they use smaller common factors (2 and 3), and the numbers left inside still share a factor. Choice D divides only the first term by 6: it expands to 18a + 252.
Question 5 of 20 · Multiple Choice
Which expression is equivalent to 35 + 20k?
Answer: A
The GCF of 35 and 20 is 5, and 35 = 5 × 7 and 20k = 5 × 4k, so 35 + 20k = 5(7 + 4k). Choice B divides only the 35 by 5. Choice C subtracts 5 from each number instead of dividing. Choice D puts the variable on the wrong term: it expands to 35k + 20.
Question 6 of 20 · Multiple Choice
Which expression is equivalent to p + p + p + p?
Answer: B
Each p is 1p, so p + p + p + p = (1 + 1 + 1 + 1)p = 4p. Choice A adds 4 to p instead of taking 4 groups of p. Choice C means p × p × p × p, which is repeated multiplication, not repeated addition. Choice D counts the 3 plus signs instead of the 4 terms.
Question 7 of 20 · Multiple Choice
Which expression is equivalent to 7g + 3 + 2g + 5?
Answer: D
Use the commutative and associative properties to group like terms: (7g + 2g) + (3 + 5) = 9g + 8. Choice A adds every number, 7 + 3 + 2 + 5 = 17, and attaches g, but 3 and 5 are not like terms with 7g and 2g. Choice B multiplies the variables (g × g) when it should add like terms. Choice C multiplies the coefficients and the constants (7 × 2 and 3 × 5) instead of adding them.
Question 8 of 20 · Multiple Choice
Which property does 3 × (4 × n) = (3 × 4) × n show?
Answer: B
The three factors stay in the same order, and only the grouping (the parentheses) changes, so this is the associative property of multiplication. Both sides equal 12n. Choice A would change the order, as in 3 × n = n × 3. Choice C needs a sum inside the parentheses. Choice D is about multiplying by 1.
Question 9 of 20 · Multiple Choice
Which property explains why (x + 4) + 6 is equivalent to x + (4 + 6)?
Answer: C
The three addends x, 4 and 6 stay in the same order, and only the parentheses move. Changing the grouping of addends is the associative property of addition, and it lets you write the sum as x + 10. Choice A would change the order of the addends, as in x + 4 = 4 + x. Choice B needs a number multiplied by a sum, and nothing is multiplied here.
Question 10 of 20 · Multiple Choice
Maya rewrote 2(5w + 3) + w in three steps. Step 1: 10w + 6 + w. Step 2: 10w + w + 6. Step 3: 11w + 6. Which property did she use in Step 2?
Answer: D
In Step 2, the terms 6 and w swap places so that the like terms 10w and w sit together. Changing the order of addends is the commutative property of addition. Choice A is the property Maya used in Step 1 (and again in Step 3, when she adds the coefficients 10 + 1). Choice B would regroup factors, and there are no factors to regroup in Step 2.
Question 11 of 20 · Multiple Choice
Which expression is equivalent to 3(4r + 2) + 5r?
Answer: A
First expand: 3(4r + 2) = 12r + 6. Then combine like terms: 12r + 5r + 6 = 17r + 6. Choice B multiplies only 4r by 3 and leaves the 2. Choice C treats 5r as the number 5 and adds it to 6. Choice D adds 12 + 6 + 5 and attaches r to the total, combining unlike terms.
Question 12 of 20 · Multiple Choice
A backpack costs b dollars and a lunch bag costs $12. A teacher buys 5 backpacks and 5 lunch bags. Which expression does NOT give the total cost?
Answer: C
Choice C counts the lunch bags as costing $12 in all, not $12 each, so it is not equivalent to the total 5(b + 12). Choice A adds the price of one backpack and one lunch bag, then multiplies by 5. Choice B is choice A after the distributive property. Choice D is choice B with the terms reordered and written with the commutative property.
Question 13 of 20 · Multiple Choice
An area model has a height of 7. Its width is split into two parts, x and 3. Which pair of expressions both give the total area?
Answer: D
Height times total width is 7(x + 3). The two parts have areas 7 × x = 7x and 7 × 3 = 21, so the total is also 7x + 21. Choice A forgets to multiply the 3 by the height. Choice B adds the lengths instead of multiplying, which gives a sum of lengths, not an area. Choice C multiplies all three numbers together as if the rectangle had no parts.
Question 14 of 20 · Multiple Choice
Leo says that 5(n + 2) + 3 is equivalent to 5n + 13. Who is right, and why?
Answer: A
Expand first: 5(n + 2) = 5n + 10. Then add the constants: 5n + 10 + 3 = 5n + 13, so Leo is right. Choice B adds the 3 inside the parentheses, as if the 5 multiplied it too. Choice C multiplies only the n by 5 and leaves the 2 alone. Choice D adds the 3 to the coefficient 5, but 3 is a constant, not a like term with 5n.
Question 15 of 20 · Short Answer
Use the distributive property to rewrite 9(2t + 5) without parentheses.
9 × 2t = 18t and 9 × 5 = 45, so the result is 18t + 45. Check with t = 1: 9 × 7 = 63 and 18 + 45 = 63.
Question 16 of 20 · Short Answer
Use the greatest common factor to rewrite 28x + 35y as a product.
The GCF of 28 and 35 is 7. Since 28x = 7 × 4x and 35y = 7 × 5y, the product is 7(4x + 5y). Inside the parentheses, 4 and 5 have no common factor other than 1, so the factoring is complete.
Question 17 of 20 · Short Answer
Rewrite n + 6 + n + n + 1 with as few terms as possible. Name the properties you used.
Commutative property of addition: n + n + n + 6 + 1. Identity property: 1n + 1n + 1n + 6 + 1. Distributive property: (1 + 1 + 1)n + 7. The result is 3n + 7.
Question 18 of 20 · Short Answer
Write three expressions equivalent to 20q + 30: one product that uses the GCF, one product that uses a smaller common factor, and one sum with the terms in a different order.
With the GCF 10: 10(2q + 3). With a smaller common factor: 2(10q + 15) or 5(4q + 6). With the commutative property: 30 + 20q. Expanding each product gives 20q + 30 again.
Question 19 of 20 · Short Answer
Tickets to a school play cost a dollars for an adult and $5 for a student. A family buys 3 adult tickets and 3 student tickets. Write two equivalent expressions for the total cost: one with parentheses and one without.
Each adult-and-student pair costs a + 5 dollars, and there are 3 pairs: 3(a + 5). By the distributive property, this equals 3a + 15. For example, if an adult ticket costs $8, both give $39.
Question 20 of 20 · Short Answer
Show, step by step, that 4(3 + k) + 2k is equivalent to 12 + 6k. Name the property you use at each step.
6.EE.A.3 means students use the properties of operations to rewrite an expression in a new form with the same value. The official examples are expanding 3(2 + x) to 6 + 3x, factoring 24x + 18y to 6(4x + 3y), and writing y + y + y as 3y. Students should be able to name the property behind each step.
Is 6.EE.A.3 taught in grade 6 or grade 7?
6.EE.A.3 is a grade 6 standard. Grade 6 uses positive whole numbers, fractions and decimals. In grade 7, 7.EE.A.1 extends the same properties to expressions with negative numbers and rational coefficients, such as subtracting and factoring linear expressions.
Which properties of operations do grade 6 students use?
They use the commutative properties (order does not matter for addition or multiplication), the associative properties (grouping does not matter), the identity property of multiplication (1 times a number is that number), and the distributive property, a(b + c) = ab + ac. Students met all of these with whole numbers in grades 3-5.
What is the difference between 6.EE.A.3 and 6.EE.A.4?
6.EE.A.3 is about making equivalent expressions: students start with one expression and use properties to write another. 6.EE.A.4 is about recognizing them: students decide whether two given expressions name the same number for every value of the variable. Most lessons teach them close together.
What are like terms?
Like terms have exactly the same variable part. 6d and 11d are like terms, so 6d + 11d = 17d. The terms 6d and 6 are not like terms, and neither are 6d and 6e, so those sums cannot be written as one term.
How do you factor an expression with the distributive property?
Find the greatest common factor of the coefficients, then divide each term by it. For 16x + 20, the GCF of 16 and 20 is 4, so 16x + 20 = 4(4x + 5). Check by expanding: 4 × 4x + 4 × 5 = 16x + 20. This is the same idea as 6.NS.B.4, where students write 36 + 8 as 4(9 + 2).
What are common mistakes with the distributive property?
A common mistake is multiplying only the first term, for example writing 3(m + 4) as 3m + 4 instead of 3m + 12. Another is adding coefficients of unlike terms, such as writing 2x + 3 as 5x. An area model helps with both mistakes, because every part of the rectangle needs its own area.
Why is y + y + y equal to 3y and not y³?
Adding y three times is repeated addition, which is multiplication by 3: 3y. The expression y³ means y × y × y, which is repeated multiplication. For y = 4, y + y + y = 12 but y³ = 64, so they are different.
How can students check that their rewritten expression is equivalent?
They can substitute a few values for the variable into both expressions and compare the results. If any value gives different results, the expressions are not equivalent. Matching values are a good sign but not a proof, so students should also be able to name the property behind each step.
How does 6.EE.A.3 prepare students for algebra in high school?
Expanding and factoring are used constantly in later courses. In grade 7, students expand and factor expressions with negative numbers (7.EE.A.1). In Algebra I, students use the structure of an expression to rewrite it (HSA.SSE.A.2) and multiply polynomials with the same distributive property.
07
Related Standards
6 standards
These standards connect to 6.EE.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
3.OA.B.5Prerequisite
Apply properties of operations as strategies to multiply and divide
Lesson coming soon
5.OA.A.1Prerequisite
Use parentheses, brackets or braces in numerical expressions and evaluate them
Lesson coming soon
Alongside
6.NS.B.4Parallel
Find the GCF and LCM, and use the distributive property to factor a sum
