What Is an Area Model?

An area model is a rectangle used to multiply: the two numbers are its side lengths, the bigger number is split into tens and ones, and the answer is the areas of the parts added together. To work out 4 × 23, draw a rectangle 4 by 23 and split the 23 into 20 and 3. The two parts are 4 × 20 = 80 and 4 × 3 = 12, and 80 + 12 = 92. It is the same multiplication as the carrying method most parents learned, with the place values written down instead of hidden in the carrying.

Children meet the area model in third grade, and fourth grade uses the same boxes for bigger numbers. The most useful thing to do at home is to ask your child what each box means: a child who says “4 times 20”, not “4 times 2”, understands it. Six free practice sheets with an answer key are at the end of this guide.

What is an area model?

An area model is a rectangle drawn to multiply two numbers, with one number along its short side and the other along its long side, split by place value. For 4 × 23, the 4 goes on the short side and the 23 is split into 20 and 3 along the long side, which makes two boxes. Each box holds one multiplication, 4 × 20 = 80 and 4 × 3 = 12, and the boxes added together, 80 + 12 = 92, are the answer.

The name comes from the area of a rectangle, which is its length times its width. A rectangle 4 units by 23 units has an area of 92 square units, and cutting it into a 4-by-20 piece and a 4-by-3 piece cannot change the total. Common Core’s third-grade standard 3.MD.C.7c asks children to “use area models to represent the distributive property in mathematical reasoning.” The distributive property is the rule that 4 × 23 is the same as 4 × 20 plus 4 × 3.

The boxes are not drawn to scale. The tens box only needs to look bigger than the ones box, and nobody counts squares inside it. Some worksheets call the same drawing the box method or partial products multiplication, and the Texas standards for third grade name “area models” outright in TEKS 3.4(K).

An area model is the step after an array, which draws every object in rows and columns. Drawing every dot stops working past a few dozen, so the area model keeps the rectangle and writes the side lengths instead. The Common Core writers’ progressions document for base ten describes it as “imagining things inside the square units to make an array (but not drawing them).”

Area model multiplication, step by step

To multiply with an area model, draw a rectangle, split the bigger number into tens and ones, multiply in each box, and add the boxes. The five steps work two problems side by side: 7 × 13, whose answer is inside 100, the range a Common Core third grader works in, and 6 × 34, whose answer is past it.

1. 1

   Draw the rectangle

   Draw a rectangle and write the one-digit number along its short side: 7 for 7 × 13, and 6 for 6 × 34.

2. 2

   Split the bigger number into tens and ones

   Write the tens and the ones of the two-digit number along the long side: 13 is 10 and 3, and 34 is 30 and 4. Draw a line down the rectangle so each part has its own box, and make the tens box the wider one.

3. 3

   Multiply in each box

   Write one multiplication in each box: 7 × 10 = 70 and 7 × 3 = 21. Say a tens box in tens: 6 × 30 is 6 × 3 tens, which is 18 tens, or 180. The ones box is 6 × 4 = 24.

4. 4

   Add the boxes

   Add the numbers in the boxes to get the answer: 70 + 21 = 91, so 7 × 13 = 91, and 180 + 24 = 204, so 6 × 34 = 204.

5. 5

   Check it another way

   Check that the answer is sensible. 7 × 13 sits between 7 × 10 = 70 and 7 × 15 = 105, so 91 fits. 6 × 34 is a little more than 6 × 30 = 180, so 204 fits.

Step 3 is the move third-grade standard 3.NBT.A.3 asks for: multiplying a one-digit number by a multiple of 10 from 10 to 90 “using strategies based on place value.” The progressions document works 3 × 50 as “3 groups of 5 tens, which is 15 tens, which is 150.” A multiple of ten on its own needs only one box: 4 × 30 is 4 × 3 tens, which is 12 tens, or 120.

The tens and ones in step 2 are the same parts children build with base ten blocks: 34 is 3 rods of ten and 4 single cubes.

Area model vs. the standard algorithm: is the old way wrong?

The old way is not wrong: the area model and the standard algorithm, the carrying method most parents learned, do the same multiplications and get the same answer. The area model writes the place values down, and the standard algorithm keeps them in the child’s head. Here is 6 × 34 both ways:

Step	Standard algorithm (the old way)	Area model
Ones	6 × 4 = 24: write 4, carry 2	6 × 4 = 24 in the ones box
Tens	6 × 3 = 18, plus the carried 2 makes 20: write 20	6 × 30 = 180 in the tens box
Answer	204	180 + 24 = 204

The 18 in the standard algorithm is 18 tens, which is the 180 in the area model’s tens box. The carried 2 is the 20 inside 24. Both methods multiply 6 by 30 and 6 by 4 and add the results. The standard algorithm writes those partial products in a shorter form.

Common Core teaches the two in order. Third graders use area models for the distributive property (3.MD.C.7c). Fourth graders multiply a number of up to four digits by a one-digit number, and one two-digit number by another. Their standard 4.NBT.B.5 asks them to “illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.” Fluency with the standard algorithm for multiplication is a fifth-grade standard, 5.NBT.B.5. Texas starts the algorithm sooner: its third-grade standard 3.4(G) asks children to multiply a two-digit number by a one-digit number with “strategies and algorithms, including the standard algorithm.” It lists partial products among the strategies (TEKS grade 3).

The progressions document gives the reason for the order. By connecting drawings to written work, students “can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities.” The area model comes first so that the shortcut, when it arrives, is already explained. The grade at which schools teach each of the old methods is in new math explained for parents.

Show the standard algorithm to a child who can already explain the boxes, as a shorter way to write the same thing. Check with the teacher first, because a test may ask for the model. Lattice multiplication, a grid with diagonals that some parents also learned, is another written method, and the standards do not name it.

Why do schools teach the area model?

Schools teach the area model because it shows place value, it draws the distributive property, and it works for bigger numbers without changing.

It shows place value. In an area model a child writes 6 × 30 = 180, not 18, so a dropped zero is on the page where a teacher can see it. The standard algorithm hides that step: the progressions document points out that a 0 in a written method is there “because the whole row of digits is produced by multiplying by 30 (not 3).”

It draws the distributive property. Common Core’s own example in third-grade standard 3.OA.B.5 is 8 × 7 = 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56, which is an area model with two boxes. Children use it to work out a fact they do not know from two they do, such as 7 × 8 from 5 × 8 and 2 × 8.

It works for bigger numbers without changing. The same rectangle carries two-digit by two-digit multiplication and division in fourth grade, and fractions in later grades.

The research evidence is about drawings and objects in math help for children who find math hard, not about the area model itself. The What Works Clearinghouse practice guide Assisting Students Struggling with Mathematics (2021) gives a strong level of evidence, from 28 studies, to using “a well-chosen set of concrete and semi-concrete representations” connected to the written math.

Which grade uses the area model?

Children meet the area model in third grade, and fourth grade uses the same model for bigger numbers and for division. Arrays come the year before: second grade’s 2.OA.C.4 asks children to add up objects “arranged in rectangular arrays with up to 5 rows and up to 5 columns.”

Grade	What children do	Common Core	Texas (TEKS)
2nd	Arrays: rows and columns of objects, added up (the step before)	2.OA.C.4	None
3rd	Area as rows of unit squares; the area model for the distributive property; one digit × multiples of ten; products within 100	3.MD.C.7, 3.OA.B.5, 3.NBT.A.3	3.4(G) two digits × one digit, including the standard algorithm; 3.4(K) area models; 3.6(C) area
4th	Up to four digits × one digit, and two digits × two digits (four boxes); division with the area model	4.NBT.B.5, 4.NBT.B.6	4.4(D) two digits × two digits
5th	The standard algorithm, fluently	5.NBT.B.5	Earlier, in grades 3 and 4

In Texas, third graders already multiply a two-digit number by a one-digit number, with partial products or the standard algorithm (3.4(G)). In Common Core states that is fourth grade, and third grade stays within 100 and multiples of ten.

Fourth grade takes the same model further. For 23 × 14, split 23 into 20 and 3 along one side and 14 into 10 and 4 along the other. The rectangle has four boxes: 20 × 10 = 200, 20 × 4 = 80, 3 × 10 = 30, and 3 × 4 = 12, which add to 322. Division on an area model is fourth grade too: standard 4.NBT.B.6 asks children to explain division “by using equations, rectangular arrays, and/or area models.” Texas reaches two digits by two digits in fourth grade as well, in 4.4(D) (TEKS grade 4).

Common area model mistakes (and how to fix each one)

Each of these six area model mistakes is something you can see on the page, and each one has a fix you can make on the spot.

• Splitting by digits instead of by place value. The child writes 23 as 2 and 3, so 4 × 2 = 8 lands in the tens box. Have your child say the tens part aloud, “twenty”, and write 20, never 2.
• Dropping the zero in a tens box. The child writes 6 × 30 as 18. Say it in tens, “6 times 3 tens is 18 tens”, then write 180.
• Stopping before the add. The boxes are filled in and the answer is left as two numbers. Write the plus sign on the page before filling the boxes.
• Adding inside a box. The child writes 4 + 20 instead of 4 × 20. Write the × in each box first.
• Drawing to scale. The child measures the tens box or counts squares in it. Remind your child that the boxes only show the parts, and the numbers do the work.
• Slipping on the final addition, especially past 100, as in 180 + 24. Line the parts up by place value, or add the tens and then the ones.

Which side the one-digit number goes on is not a mistake. 4 × 23 and 23 × 4 are the same rectangle turned on its side, which is the commutative property Common Core lists in 3.OA.B.5.

Free area model worksheets

Our free area model worksheets are one PDF of six practice pages and an answer key, and the support steps down from page to page. The first page multiplies by multiples of ten in one box, such as 5 × 60. The next three split a one-digit by two-digit problem into tens and ones: first with the rectangle split and labeled, then split for your child to label, then a plain rectangle your child splits and labels. The last two pages have bigger numbers, with answers from 100 to 999.

The first four pages suit a Common Core third grader: the first multiplies by multiples of ten, and on the next three every answer is within 100. The last two suit a Texas third grader or a fourth grader starting out. Each page has four problems.

The problems are not numbered, and the key matches each answer to its rectangle by position. The PDF ends with a page of instructions for grown-ups on why to use it, what it is good for, and how to use it. Every page prints in black and white, with no account and no email.

Free PDF · no sign-up

Free printable area model worksheets

One free PDF in black and white, with no account and no email, in teaching order: multiplying by tens, three pages that step the support down, then two pages of bigger numbers. Start with the page your child can already do. It has an answer key, and the PDF ends with a page of instructions for grown-ups on why, what for, and how to use it.

1. Area Model Multiplication (PDF)24 problems · answer key included · 8 pagesDownload

How can parents help with area model homework?

Help with area model homework by letting your child explain the boxes first, and by keeping to the method on the page. These five moves fit in 10 minutes at the table.

• Ask what each box means. “What does this box say?” A child who answers “4 times 20” understands the model, and a child who answers “4 times 2” needs the place-value fix.
• Keep the method the homework asks for. Carrying skips the stage the teacher is marking.
• Draw it once on graph paper. A 4-by-13 rectangle of real squares makes the “area” literal: count the 40 squares in the tens part and the 12 in the ones part. After that, the boxes stand for the squares and need no grid.
• Find an everyday rectangle. A patio of tiles 6 rows by 14 is 6 × 10 = 60 and 6 × 4 = 24, so 84 tiles. Common Core’s 3.MD.C.7b asks children to find areas “in the context of solving real world and mathematical problems.”
• Practice the times-ten facts aloud, such as 4 × 30 and 7 × 20, because every tens box depends on them.

If the single-digit facts are the hard part, an area model will not fix that, and fact practice comes first.