What Is the Associative Property? Make-a-Ten Math Explained
Your child's worksheet says to “make a ten” in 2 + 6 + 4. That's the associative property at work. Here's what it means, which grades teach it, how it differs from the commutative property, and a free worksheet that checks every answer.
Have you ever asked your child to make a ten when solving a problem on a homework sheet, such as 2 + 6 + 4? One way to solve it is to add 6 and 4 first, then add 2.
It may sound like a trick, but it's an application of the associative property. When you add or multiply three numbers, you can choose which pair of numbers to calculate first and still get the same answer.
For example, 2 + (6 + 4) = 2 + 10 = 12. Children who learn to recognize number pairs that make ten can develop stronger mental math skills and make fewer mistakes when adding.
In this guide, you'll learn what the associative property is, which grade levels introduce it, how it differs from the commutative property, when it doesn't apply, and how to help your child practice it at home with simple examples and a free worksheet.
Try It Yourself: Group It Two Ways. Work out each problem with both groupings, then press Check My Answer. The rest of this guide explains why both answers always match.
- What Is the Associative Property?
- Associative Property of Addition (Grade 1)
- Associative Property of Multiplication (Grade 3)
- Associative vs Commutative Property: The One-Question Test
- Why Subtraction and Division Aren't Associative
- How the Associative Property Makes Mental Math Faster
- 3 Associative Property Mistakes Kids Make
- Associative Property Worksheet: Try It Yourself
- Math Games to Play After the Worksheet
- How Cuemath Tutors Teach the Associative Property
- FAQs
What Is the Associative Property?
The associative property is a math rule that says you can change how you group three or more numbers when you add or multiply, and the answer stays the same.
Parentheses show which two numbers to work out first. In (2 + 6) + 4, you add 2 + 6 first to get 8, then add 4 to get 12. In 2 + (6 + 4), you add 6 + 4 first to get 10, then add 2. That's also 12. In symbols, it's written as (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
The word comes from “associate,” which means to group or pair up. The numbers pair up differently, but the answer doesn't change.
The rule works for addition and multiplication, but not for subtraction or division:
| Operation | Grouped one way | Grouped the other way | Same answer? |
|---|---|---|---|
| Addition | (2 + 6) + 4 = 12 | 2 + (6 + 4) = 12 | Yes |
| Multiplication | (3 × 5) × 2 = 30 | 3 × (5 × 2) = 30 | Yes |
| Subtraction | (8 − 3) − 2 = 3 | 8 − (3 − 2) = 7 | No |
| Division | (24 ÷ 4) ÷ 2 = 3 | 24 ÷ (4 ÷ 2) = 12 | No |
A quick way to check any operation: work it out with both groupings. If the answers match, it's associative.
Try It Yourself: Same Answer Both Ways?. Work out each problem with both groupings, then press Check My Answer.
Associative Property of Addition (Grade 1)
The associative property of addition says that (a + b) + c = a + (b + c). When you add three numbers, you can add any two of them first, and the total stays the same.
Common Core standard 1.OA.B.3 uses this example: “To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12.”
Here's how first graders use it to make a ten:
Same three steps every time: find the 10, group it, add the rest.
The same idea works when one number has to be split first. For 8 + 6, your child splits the 6 into 2 + 4, so the problem becomes 8 + 2 + 4. Then they group the 8 and the 2 to make 10: (8 + 2) + 4 = 10 + 4 = 14. Common Core lists this exact “making ten” strategy in standard 1. OA. C.6.
Splitting the 6 and grouping 8 + 2 turns a hard fact into 10 + 4.
By Grade 2, kids use the same move with bigger numbers. For 38 + 17 + 3, they group 17 + 3 to make 20: 38 + (17 + 3) = 38 + 20 = 58.
Number bonds show how a number splits into two parts, which is the first step of making a ten. Cuemath's free Grade 1 math curriculum has more addition practice by topic.
Try It Yourself: Make a Ten First. Group the two numbers that make 10, add them, then add the rest.
Associative Property of Multiplication (Grade 3)
The associative property of multiplication says that (a × b) × c = a × (b × c). When you multiply three numbers, you can multiply any two neighbors first, and the product stays the same.
Common Core standard 3.OA.B.5 uses this example: 3 × 5 × 2 can be found as 3 × 5 = 15, then 15 × 2 = 30, or as 5 × 2 = 10, then 3 × 10 = 30.
Kids can see why it works with a box of cubes. A box with 2 layers, where each layer has 3 rows of 4 cubes, holds 2 × 3 × 4 cubes. Count it by layers: each layer has 3 × 4 = 12 cubes, and 2 layers make 24. Count it by rows: there are 2 × 3 = 6 rows of 4 cubes, and 6 × 4 = 24.
Same box, same 24 cubes. Cuemath's associative property of multiplication page has more worked examples, and our 3rd grade math problems put multiplication to work in word problems. For a full lesson list, see the free Grade 3 math curriculum.
State math tests cover this standard too. Grade 3 tests like Smarter Balanced and NJSLA include properties of multiplication, so it's worth getting right before testing season in spring.
Try It Yourself: Find the 10 First. Multiply the two numbers that make 10, then multiply by the third.
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Associative vs Commutative Property: The One-Question Test
To tell the two properties apart, ask one question: did the numbers change places, or did only the parentheses move?
- If the numbers changed places, it's the commutative property: 2 + 7 = 7 + 2.
- If the numbers stayed in the same order and only the parentheses moved, it's the associative property: (2 + 7) + 3 = 2 + (7 + 3).
| Problem | Did the numbers change places? | Property |
|---|---|---|
| 4 + 9 = 9 + 4 | Yes | Commutative |
| (4 + 9) + 1 = 4 + (9 + 1) | No, only the parentheses moved | Associative |
| 3 + (7 + 1) = 3 + (1 + 7) | Yes, inside the parentheses | Commutative |
| (5 × 2) × 6 = 5 × (2 × 6) | No, only the parentheses moved | Associative |
| 6 × 8 = 8 × 6 | Yes | Commutative |
The third row trips up a lot of kids. It has parentheses, so it looks associative, but the 7 and the 1 swapped places. That makes it commutative.
Kids often use both properties together. For 7 + 8 + 3, your child moves the 3 next to the 7 (commutative), then groups them to make 10 (associative): (7 + 3) + 8 = 10 + 8 = 18. Our guide to the commutative property covers the first half of that move.
Try It Yourself: Move It, Then Group It. First move the numbers that make 10 next to each other (commutative), then add them first (associative).
Also Read
Why Subtraction and Division Aren't Associative
Subtraction and division don't follow the associative property. If you move the parentheses, the answer changes.
Try it with 8 − 3 − 2. Grouped as (8 − 3) − 2, you take away 3, then take away 2 more, and 3 is left. Grouped as 8 − (3 − 2), you work out 3 − 2 = 1 first, so you only take away 1, and 7 is left.
Division works the same way. (24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3, but 24 ÷ (4 ÷ 2) = 24 ÷ 2 = 12.
A good rule to teach your child: when a problem has only subtraction or only division and no parentheses, work from left to right. Don't regroup.
Try It Yourself: Work Left to Right. Only subtraction or only division? Don't regroup. Work from left to right.
How the Associative Property Makes Mental Math Faster
The associative property lets your child pick the easiest pair to work out first. That's how many adults add and multiply in their heads without knowing the rule's name.
Adding (Grades 1–2). For 45 + 38 + 2, group 38 + 2 to make 40: 45 + (38 + 2) = 45 + 40 = 85.
Multiplying (Grade 3 and up). For 7 × 25 × 4, group 25 × 4 to make 100: 7 × (25 × 4) = 7 × 100 = 700. Working left to right instead means 7 × 25 = 175, then 175 × 4, which is much harder to do in your head.
Look for pairs that make 10 or 100.
Pairs worth knowing: when you multiply, 2 and 5 make 10, 4 and 25 make 100, and 2 and 50 make 100. When you add any two ones digits that add to 10 (1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5), they make a round number.
For daily practice ideas that don't feel like drills, see our guide to math facts practice.
Spot the Easy Pair in 5 Minutes a Day in the Cuemath App
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3 Associative Property Mistakes Kids Make
Most associative property mistakes come from using the rule where it doesn't belong. Here are the three you'll see most on homework.
Mistake 1: Regrouping a subtraction. For 15 − 6 − 4, your child sees 6 and 4 and works out 15 − (6 − 4) = 15 − 2 = 13. The right answer, working left to right, is (15 − 6) − 4 = 5.
Mistake 2: Regrouping across two different operations. The associative property only works when every sign in the problem is the same. (2 + 3) × 4 = 20, but 2 + (3 × 4) = 14. Moving the parentheses across a + and a × changes the answer.
Mistake 3: Calling it the wrong property. On a “name the property” question, your child sees parentheses and writes “associative,” even though the numbers swapped places inside them, as in 3 + (7 + 1) = 3 + (1 + 7). Ask the one-question test: did the numbers change places? If yes, it's commutative.
Associative Property Worksheet: Try It Yourself
Pick a tab and have your child answer 10 questions: addition for Grades 1–2, multiplication for Grade 3, or naming the property for Grades 3–6. When they finish, press Check My Answers to see their score and which questions to go over together.
For printable sheets, Cuemath also has associative property of addition worksheets, and the free Smarter Balanced practice curriculum has more Grade 3 practice.
Pick a tab, answer all 10, then press Check My Answers. The score shows which questions to go over together.
Math Games to Play After the Worksheet
Finished the worksheet? These 6 Cuemath games start with addition and multiplication facts, then move on to a Grade 3 fraction topic your child meets next. Pick one and play it right here on the page.
Each game opens right here and takes a few seconds to load. On a phone, turn it sideways for the best view.
Want to practice without a screen? These take about five minutes each:
- Make-ten hunt. Roll three dice. Your child looks for two numbers that add to 10 and adds those first. Roll 4, 3, 6: (4 + 6) + 3 = 13.
- Easy pair first. Say a three-number problem like 9 + 7 + 1. Your child says which pair to group first and why: “9 and 1, because they make 10.”
- Box of cubes. Build a box from blocks or sugar cubes: 2 layers of 3 rows of 4. Count it by layers, then by rows. Both ways should give 24.
- Shopping basket. Write three prices that pair up, like $7, $5, and $3. Your child adds the pair that makes 10 first: ($7 + $3) + $5 = $15.
How Cuemath Tutors Teach the Associative Property
Cuemath tutors teach the associative property by having your child solve the same problem two ways and explain why the answer didn't change. A worksheet only shows whether the answer is right. A tutor can see whether your child spots the easy pair or still counts everything out one by one.
| How Cuemath helps | What it means for your child |
|---|---|
| How it works | In a 1:1 live class, the tutor writes a problem like 2 + 6 + 4 on the Cueboard (Cuemath's shared digital whiteboard). Your child groups it one way, then the other, and explains why both give 12. Cuemath calls this “cue, don't tell.” The tutor asks questions instead of giving the answer. |
| Who it helps | Grade 1–3 kids who still count every number one by one, and Grade 3–6 kids who mix up the associative and commutative properties or regroup subtraction by mistake. |
| What it won't fix by itself | If your child only needs the property names for a quiz, the one-question test and the worksheet above are faster than a class. |
| Verdict | For a child who counts everything out instead of looking for the easy pair, one-on-one practice builds the habit of grouping first. Cuemath classes start at $15/class on the 12-month plan. Meet Cuemath's Grade 1, Grade 2 and Grade 3 math tutors, or see how Grade 3 online math classes work. |
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- Math Facts Practice
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FAQs
What is the associative property?
The associative property is the rule that changing how you group three numbers doesn't change the answer in addition or multiplication. For example, (2 + 6) + 4 = 2 + (6 + 4) = 12, and (3 × 5) × 2 = 3 × (5 × 2) = 30. In symbols, (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
What is the associative property of addition?
The associative property of addition says that (a + b) + c = a + (b + c): you can add any two of three numbers first, and the sum stays the same. Common Core's Grade 1 example is 2 + 6 + 4 = 2 + 10 = 12 (1.OA. B.3).
What is the associative property of multiplication?
The associative property of multiplication says that (a × b) × c = a × (b × c): you can multiply any two neighboring numbers first, and the product stays the same. Kids learn it in Grade 3 (Common Core 3.OA.B.5), with examples like 3 × 5 × 2 = 3 × 10 = 30.
What is the difference between the commutative and associative property?
The commutative property is about order, and the associative property is about grouping. Commutative: 2 + 7 = 7 + 2, where the numbers swap places. Associative: (2 + 7) + 3 = 2 + (7 + 3), where only the parentheses move. Both work for addition and multiplication only.
How do you know if it's commutative or associative?
Ask one question: did the numbers change places? If yes, it's commutative, even when there are parentheses, as in 3 + (7 + 1) = 3 + (1 + 7). If the numbers stayed in the same order and only the parentheses moved, it's associative.
What grade learns the associative property?
Kids use the associative property of addition in Grade 1 and the associative property of multiplication in Grade 3, under Common Core standards 1.OA.B.3 and 3.OA.B.5. Common Core says young students “need not use formal terms,” so first graders often learn it as “make a ten.” It comes back in Grades 6 and 7, when students use properties to rewrite expressions (6.EE.A.3, 7.EE.A.1).
Does the associative property work for subtraction or division?
No, the associative property doesn't work for subtraction or division. (8 − 3) − 2 = 3, but 8 − (3 − 2) = 7, and (24 ÷ 4) ÷ 2 = 3, but 24 ÷ (4 ÷ 2) = 12. When a problem has only subtraction or only division, work from left to right.
What is an example of the associative property?
An example of the associative property is (4 + 6) + 9 = 4 + (6 + 9) = 19. A multiplication example is (2 × 5) × 7 = 2 × (5 × 7) = 70. In both, the numbers stay in the same order and only the grouping changes.
What are the four properties of math?
The four properties most often taught together are the commutative, associative, distributive and identity properties. Commutative is about order, associative is about grouping, distributive splits a number to multiply (3 × 14 = 3 × 10 + 3 × 4), and identity says adding 0 or multiplying by 1 leaves a number unchanged.
How can I help my child practice the associative property at home?
Help your child practice the associative property with short games that ask them to find the easy pair first: roll three dice and make a ten, add three prices on a shopping list, or count a box of cubes by layers and by rows. Five minutes a day is enough. If your child still counts everything out, Cuemath's free evaluation shows which concept to work on.
Sources
- Grade levels and standards (1.OA.B.3, 1.OA.C.6, 2.NBT.B.5, 2.NBT.B.9, 3.OA.B.5, 6.EE.A.3, 7.EE.A.1, “need not use formal terms”)
- Common Core Grade 1 OA: Grade 1, Operations & Algebraic Thinking
- Common Core Grade 2 NBT: Grade 2, Number & Operations in Base Ten
- Common Core Grade 3 OA: Grade 3, Operations & Algebraic Thinking
- Common Core Grade 6 EE: Grade 6, Expressions & Equations
- Common Core Grade 7 EE: Grade 7, Expressions & Equations
- Reviews
- Cuemath on Trustpilot: 4.9/5 from 12,000+ reviews
All pages checked October 27, 2026.
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