The Astuzo blog
Number lines, arrays and area models: Ontario math methods
Your child’s math homework uses number lines, arrays or boxes for 23 × 14? What each Ontario method is, which grade uses it, and how to help.
It’s homework time, and your child’s worksheet shows a line with loops jumping along it. Or a grid of dots. Or a rectangle cut into four boxes, for what you’d have solved as 23 × 14 in a column. It isn’t how you learned math, and your child is waiting for help.
These methods aren’t random. They come from Ontario’s 2020 math curriculum, and each one has a job. This guide shows the ones you’re most likely to meet, what each looks like, which grade uses it, and how to help when you’ve never seen it before.
Why Ontario teaches more than one way
Two ideas run through the curriculum:
- Kids explain their thinking. From Grade 1 to Grade 8, students use mental math strategies “and explain the strategies used”. A child who can say why a method works can use it on a problem they haven’t seen.
- Strategies connect to the written method. The column method you learned is still in the curriculum. In Grade 3, students learn algorithms for adding and subtracting by “making connections” to the other strategies. In Grade 5, they multiply with the area model and an algorithm, and connect the two.
So the drawings don’t replace the method you know. They come first, or alongside, so the steps of the written method make sense instead of being memorized.
Making 10 (Grades 1 and 2)
What it looks like: 8 + 5 becomes 8 + 2 + 3. Get to 10 first, then add what’s left: 10 + 3 = 13.
Why: 10 is an easy number to add to. Kids who can split numbers to make 10 add quickly in their head, which is what “mental math strategies” means in the early grades.
How to help: play “how many more make 10?” with fingers, dice or cards. If your child says “I just know 8 + 5 is 13”, that’s fine too.
Splitting numbers (Grades 1 to 5)
What it looks like: 47 + 25 becomes 40 + 20 = 60, and 7 + 5 = 12, so 60 + 12 = 72. The curriculum calls this decomposing (breaking apart) and composing (putting back together).
Why: it’s place value in action: tens with tens, ones with ones. It’s the same idea as the column method, written out in the open.
How to help: ask “What are the tens? What are the ones?” and let your child do the adding.
Number lines
What it looks like: to add 47 + 25, your child starts at 47 and jumps: +3 to 50, +20 to 70, +2 to 72. The line doesn’t need every number marked. It’s a sketch of the jumps.
For subtraction, the same line finds the gap. For 72 − 47, start at 47 and jump to 72: 3, then 20, then 2. The jumps add up to 25.
Why: jumping to a “friendly” number like 50 or 70 is easier than carrying in your head, and the drawing shows each step, so your child (and the teacher) can see where the thinking went.
Where it shows up: number lines are a common tool for adding and subtracting in the early grades. In Grade 6 the curriculum names them for placing negative numbers, and they stay handy when kids add and subtract integers in Grade 7.
How to help: ask “Where would you jump first to get to a friendly number?” rather than counting the jumps for them.
Arrays (Grades 3 and 4)
What it looks like: 4 × 6 drawn as 4 rows of 6 dots. Grade 3 uses arrays for multiplication facts up to 10 × 10 and division up to 100 ÷ 10. Grade 4 uses them for bigger numbers, like a two- or three-digit number times a one-digit number.
Why: an array shows that multiplication is equal groups, and that 4 × 6 is the same as 6 × 4 (turn the picture sideways). It also lets kids split a hard fact into easy ones: 7 × 6 is 5 rows of 6 plus 2 rows of 6, so 30 + 12 = 42.
How to help: if your child is stuck on a fact, ask “Can you split it into facts you know?” For division, an array works backwards: 24 dots in 4 equal rows, how many in each row?
The area model (Grade 5)
What it looks like: 23 × 14 becomes a rectangle split into four boxes. Split 23 into 20 + 3, and 14 into 10 + 4:
| × | 20 | 3 |
|---|---|---|
| 10 | 200 | 30 |
| 4 | 80 | 12 |
Add the four boxes: 200 + 30 + 80 + 12 = 322.
Why: each box is a multiplication your child can do in their head. The curriculum asks Grade 5 students to multiply two-digit numbers “using the area model and using algorithms”, and to connect the two methods. Here’s the connection: the column method does the same thing. 23 × 4 = 92 (the bottom row of boxes, 80 + 12), and 23 × 10 = 230 (the top row, 200 + 30). 92 + 230 = 322.
Grade 5 also uses the area model for division, such as a three-digit number divided by a two-digit number.
How to help: ask “How could you split each number into tens and ones?” and “How many boxes will that make?” Our Grade 5 guide has more on what Grade 5 multiplication looks like.
The written method (the one you know)
What it looks like: stacking the numbers in columns, carrying and borrowing (now usually called regrouping), and long multiplication and division.
Where it shows up: algorithms for adding and subtracting are in Grade 3, and for multiplying and dividing larger numbers in Grades 5 and 6. Teachers usually bring them in after the drawings, so your child knows why “carrying the 1” works: it’s 10 ones becoming one ten.
How to help: if the homework asks for a specific method, use that one, even if the column method would be faster. The question is usually checking the method, not just the answer.
When it’s a method you’ve never seen
- Ask your child to teach you. “Show me how your teacher does it.” Explaining a method is one of the best ways to understand it, and you’ll often spot the step they’re missing.
- Look for a class example. Notebooks, handouts and the textbook often show a worked example of the method.
- Don’t switch them to your way. Two methods at once is confusing, and the teacher may be building towards something. You can check the answer your way, quietly, to see whether it’s right.
- Ask small questions, not the next step. “What did you do first? What comes next?” keeps your child doing the thinking. Our post on helping with math homework without giving the answer has more questions to try.
- Write a short note to the teacher if you’re both stuck. It tells them the class may need another look at it.
Which methods your child will meet, by grade
| Grade | Methods named in the curriculum (2020) |
|---|---|
| 1–2 | Mental math strategies (like making 10), splitting and combining numbers |
| 3 | Arrays and equal groups for multiplication and division; algorithms for adding and subtracting, connected to other strategies |
| 4 | Arrays for multiplying and dividing two- or three-digit numbers by one-digit numbers |
| 5 | The area model and algorithms for two-digit × two-digit multiplication and for division |
| 6 | Number lines for negative numbers; mental math for percents |
| 7–8 | Mental math with percents (Grade 7) and with powers of ten (Grade 8) |
For the full picture of a grade, with typical homework and a hint for each, open the guide: Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7 or Grade 8.
Where Astuzo fits in
We built Astuzo for exactly this moment: your child is stuck on a method you don’t recognise. They snap a photo of the homework, and Spark, our friendly robot tutor, gives hints using the strategies Ontario classrooms use, like number lines, arrays and area models, and never just gives the answer. You get a short email each evening they use it, so you know what they worked on.
Astuzo is in pilot now, with math for Grades 1 to 8, made for Ontario. Join the waitlist to hear when it’s ready, or start with our plain-language guides to the Ontario math curriculum.
Curriculum details are from The Ontario Curriculum: Mathematics, Grades 1–8 (2020).