Teaching Arrays for Multiplication in 3rd Grade Small Groups
An array organizes equal groups into rows and columns, so 4 rows of 3 objects each show the same total as 4 groups of 3 arranged loosely. Students need to be able to identify the number of rows, the number of columns, and the number in each row or column, and connect that structure to a multiplication equation. Arrays make it possible to see, rather than just calculate, why rearranging the same array (say, turning 4 rows of 3 into 3 rows of 4) gives the same total even though which dimension is called "rows" and which is called "columns" changes. This is a common 3rd grade instructional focus that builds directly on equal groups, and it sets up the area model used for multi-digit multiplication in later grades. Specific grade-level expectations and number ranges vary by state and curriculum.
Where This Skill Fits
Equal groups → Arrays, where equal groups are organized into rows and columns → Multiplication fact fluency and the area model for multi-digit multiplication, introduced in 4th grade.
Prerequisite Skills
- Essential: Understanding of equal groups (same amount repeated a set number of times)
- Essential: Skip counting fluency for the relevant numbers
- Helpful but not required: Experience organizing objects into rows when counting
Common Student Thinking / Misconceptions
- Student may think: "Rows and columns are the same thing, so it doesn't matter which one I count."
What this may reveal: This can suggest the student hasn't yet distinguished the two directions of the array, which may indicate confusion about which number the problem is asking for.
Possible teacher response: Trace a full row with a finger while saying "row," then trace a full column while saying "column," and have the student repeat this before counting. - Student may think: "Turning the array 4 rows of 3 into 3 rows of 4 changes the total."
What this may reveal: One possibility is that the student is treating the rotated array as a new, unrelated arrangement rather than the same objects reorganized.
Possible teacher response: Physically rotate a built array 90 degrees and count the total both before and after to confirm it stays the same, while naming that rows and columns swapped. - Student may think: The number of rows and the number in each row are the same number as long as the array "looks square."
What this may reveal: This can indicate the student is estimating visually rather than counting each dimension directly.
Possible teacher response: Ask the student to count the rows and the columns separately and compare, even when the array looks close to square. - Student may think: Counting all the individual squares one at a time is the only reliable way to find the total.
What this may reveal: The student may not yet trust skip counting or multiplication as a faster, equally accurate way to find the same total.
Possible teacher response: Have the student count one row, then skip count by that row's amount for the remaining rows, and compare the result to counting one by one. - Student may think: An array with uneven rows (some longer than others) still counts as a valid array for a multiplication equation.
What this may reveal: This may reveal that the student has not yet connected the "equal groups" requirement to the array model specifically.
Possible teacher response: Ask the student to check that every row has the same number of objects before writing the multiplication equation.
Visual Models
Arrays — organizing counters, tiles, or grid squares into rows and columns can help students see the multiplication structure at a glance and supports skip counting by row or column. A limitation is that very large arrays can become difficult to build and count with physical materials, which is part of why students eventually move to more abstract representations.
Area Models — this is the next representation students will use, applying the same rows-and-columns structure to multi-digit multiplication by breaking a large array into smaller labeled rectangles.
Small-Group Teaching Sequence
A roughly 15–30 minute sequence: Activate Prior Knowledge (2–4 min) by reviewing an equal-groups problem; I Do (4–6 min) building and labeling one array with rows, columns, and the matching equation; We Do (5–8 min) building a second array together and rotating it to explore the commutative property; You Do (4–8 min) building and recording arrays independently; Quick Check (2–3 min) with one new array problem.
I Do Example
"I'm going to build an array with 4 rows and 3 columns." (Teacher arranges counters into 4 rows, 3 counters in each row.) "Let's count the rows: 1, 2, 3, 4 rows. Now let's count how many are in each row: 1, 2, 3. I can skip count by 3s, once for each row: 3, 6, 9, 12. This array shows 4 rows of 3, which I can write as 4 × 3 = 12." (Teacher then rotates the array 90 degrees.) "Now I have 3 rows of 4. Watch — the rows and columns swapped, but let's count again: 4, 8, 12. The total is still 12. 3 × 4 = 12 as well, because it's the same objects, just turned."
We Do Example
"Let's build an array with 5 rows and 2 columns together. How many rows will we make? How many will go in each row?" (Students build 5 rows of 2.) "What do you notice about each row? How does the model show that they're equal? What multiplication equation matches this array?" (5 × 2 = 10.) "Now let's rotate it. What changed? What stayed the same? How do you know the total didn't change?"
"Let's try one more: an array with 3 rows and 6 columns. What's the total? What equation matches?"
You Do Example
- Build an array with 4 rows and 5 columns. Write the matching multiplication equation. (4 × 5 = 20)
- Build an array with 6 rows and 2 columns. Write the matching multiplication equation.
- Draw an array with 3 rows and 7 columns. Find the total.
- Build an array for 8 × 3, then rotate it and write the new equation it shows.
Quick Check
Ask the student to build an array with 4 rows and 6 columns, name the total, and then rotate it to explain what changed and what stayed the same.
If the student demonstrates understanding → move toward multiplication fact fluency work and an early look at the area model for multi-digit multiplication.
If the student is not yet secure → return to equal groups with concrete objects and rebuild the connection between equal groups and rows/columns before reintroducing arrays.
If Students Are Ready
Move toward multiplication fact fluency and an introduction to the area model, which uses the same rows-and-columns structure to multiply larger numbers by breaking an array into smaller, labeled rectangular parts.
If Students Need More Support
Return to loose equal groups with concrete counters before reorganizing them into rows, use smaller numbers (arrays no larger than 4 by 4), and prompt the student to trace and count one row or column at a time rather than scanning the whole array at once.
Related Skills
- Before: Equal Groups
- Current: Arrays (rows and columns)
- Next: Multi-Digit Multiplication (area model)
Related Misconceptions
Related Visual Models
Relevant SMS Resources
- 3rd Grade Multiplication Arrays & Fact Families | Small Group Routine
- 3rd Grade Multiplication Foundations Small Group Routine Arrays & Fact Families
- 3rd Grade Multiplication & Division Small Group Routines Bundle
Browse more 3rd grade multiplication and division routines, task cards, and quick checks in the full catalog.
Teacher FAQ
What's the difference between teaching equal groups and teaching arrays?
Equal groups can be arranged loosely, while arrays organize those same equal groups into rows and columns, which makes the structure easier to see and count and previews the area model.
How should I introduce the commutative property with arrays?
Build one array, count the total, physically rotate it 90 degrees, and count again. Name explicitly that the total stayed the same while which dimension is called "rows" and which is called "columns" switched.
Do students need to know the word "commutative"?
Not necessarily at this stage. What matters more is that students can show and explain, using the array, why the total doesn't change when the array is rotated.
My student can build the array but struggles to write the equation. What should I do?
Have the student point to and count the rows first, then the number in one row, before writing each number in the equation, so the equation stays connected to what they built.
How do arrays connect to the area model taught later?
The area model breaks a larger array into smaller labeled rectangular parts, so students who are comfortable identifying rows, columns, and totals in a simple array have an easier time interpreting the area model when it's introduced.