Multiplication Arrays Garden – Plant Rows, Find the Facts

What an Array Shows

An array is a set of things arranged in equal rows and equal columns: eggs in a tray, chairs in a hall, windows on a block of flats, stamps on a sheet. Arrays are one of the most useful pictures in primary mathematics because they make the meaning of multiplication visible. When a child sees three rows with four strawberries in each row, the question “how many altogether?” can be answered by counting one by one, by adding 4 + 4 + 4, or by knowing that 3 × 4 = 12. Moving from the first way to the last is exactly the journey children make between Primary 1 and Primary 3.

Arrays also connect ideas that children often meet separately. The same picture shows equal groups (each row is a group), repeated addition (adding the rows), multiplication (rows times columns), the commutative property (turning the array round), division (sharing the plants into equal rows), and finally area (each plant sits on one square unit). The garden on this page shows all of these at once, so that a child who changes one number can watch every other description change with it.

How to Use This Page

1. Plant an array. Drag a finger from the top left corner of the garden. The number of rows and columns follows your finger, up to 10 by 10. You can also tap a square, or use the + and − buttons beside Rows and Columns.
2. Read the facts. Beside the garden you see the array in words (3 rows of 4), the multiplication sentence, the sentence turned round, the rows and the columns added up, and the area in square units.
3. Turn it round. The rotate button swaps the rows and the columns. The total never changes, which is the commutative property in action.
4. Find the factor pairs. In the second box, choose any number of squares from 1 to 100 and see every rectangle that can be made from them.
5. Play a challenge round. Choose P2, P3 or P4 and answer 5, 10 or 20 questions. You earn up to three stars, and the best result for each level is remembered on this device.
6. Print a worksheet with arrays to describe, arrays to draw and factor pairs to find, followed by an answer page.

Turning the Array Round: Why 3 × 4 = 4 × 3

Many children learn that the order of the numbers in a multiplication does not matter, but few can say why. An array is the clearest explanation there is. Three rows of four and four rows of three are the same set of objects looked at from a different side, so there cannot be a different number of them. Press Turn it round a few times and ask your child to predict the new total before looking. Once they are sure it never changes, point out the practical payoff: they only need to learn about half of the times-table facts. If they know 7 × 3, they already know 3 × 7.

It is worth being careful with the words. In Singapore classrooms, “3 rows of 4” is usually written as 3 × 4, read as “three groups of four”. The page always writes the rows first, then the number in each row, so the first sentence matches the picture exactly and the turned-round sentence is shown separately.

From Repeated Addition to Multiplication

Before multiplication facts are fluent, children add equal groups again and again. The garden shows two repeated additions for every array: adding the rows (4 + 4 + 4) and adding the columns (3 + 3 + 3 + 3). Both reach the same total, and comparing them is a quick way to see why multiplication saves time. Ask which addition is shorter, and which is easier to do in your head. For the 5 and 10 times tables, encourage counting in fives and tens along the rows; for harder tables, children can split an array into two smaller ones, for example 7 rows of 6 as 5 rows of 6 plus 2 rows of 6, which is the idea behind the distributive property they will use later.

Arrays and Area

When the plants are replaced by squares, an array becomes a rectangle, and counting the squares gives its area in square units. This is the bridge from multiplication in P2 to the area of rectangles in P3 and P4. Children who have built arrays understand that the area of a rectangle is length × breadth because each row of squares is the length and there are as many rows as the breadth. The P4 challenge level asks for areas and for missing sides: if a rectangle has an area of 24 square units and one side is 6 units, the other side must be 4 units, because 6 × 4 = 24. That is division thinking hidden inside a rectangle.

Factor Pairs: All the Rectangles a Number Can Make

Give a child twelve square tiles and ask for every rectangle that uses all of them. They will find 1 by 12, 2 by 6 and 3 by 4, and nothing else. Those are the factor pairs of 12, and the numbers in them, 1, 2, 3, 4, 6 and 12, are its factors. The factor pairs box does this for any number up to 100 and draws every rectangle. Some numbers are worth exploring together:

• Prime numbers such as 7, 13 or 29 make only one rectangle, a single long row. That is what being prime means.
• Square numbers such as 16, 36 or 49 include a square, where both sides are the same. They are the only numbers with an odd number of factors.
• Numbers with many factors such as 24, 36, 48 and 60 make lots of rectangles, which is why they are useful for sharing and why there are 60 minutes in an hour.

A good habit is to search for factor pairs in order, starting at 1 and stopping once the first number would be bigger than the second. The page lists pairs in that order so children can see the pattern and know when they have found them all.

How the Levels Match the Singapore Syllabus

The levels follow the usual order in Singapore primary schools, though schools differ a little in pace, so move up or down freely. P2 questions use the 2, 5 and 10 times tables with arrays to count, missing numbers, repeated addition and arrays to plant. P3 uses the 3, 4, 6, 7, 8 and 9 times tables and adds questions about turning an array round; the 3 and 4 tables are often first met in P2, and they are included here again for revision alongside the harder tables. P4 moves on to area in square units, missing sides of rectangles, missing factors and counting factor pairs, which fits the work on factors and multiples in Primary 4.

Tips for Parents and Teachers

• Talk before you count. Ask “How many rows? How many in each row?” before asking for the total. Naming the two numbers is the real skill.
• Make arrays in real life. Muffin trays, egg boxes, Lego studs and tiles on the kitchen floor are all arrays. Ask for the multiplication sentence when you spot one.
• Mix up the questions. Ask for a total, then a missing number, then the turned-round fact, so children see the same idea from different directions.
• Use the stars as a goal, not a judgement. Three stars means every answer right first time. The Fix the ones I missed button gives a second chance at exactly the questions that went wrong.
• Draw on paper too. The printed worksheet asks children to draw arrays on dotted grids, which slows them down and makes them think about equal rows.

Printing the Worksheet

Choose a level and press Print worksheet. The first page has six arrays to describe with words and a multiplication sentence (P4 also asks for the area), three arrays to draw on dotted grids, and two or three numbers to find all the factor pairs of. The second page is the answer key, with every array drawn for checking. Press New questions for a fresh sheet whenever you need one; it makes a good warm-up or homework page.

Nothing to Install, Nothing Uploaded

The arrays garden runs entirely in your browser on any phone, tablet or computer, with no account and no download. Your best stars, last level and last array are kept only in your own browser; nothing your child does is sent to us.