Arrays: rows and columns of objects
Corn plants or toy blocks set out in equal rows and equal columns can be counted a row at a time. Why do the rows and the columns always give the same total?
Where you are 16 in Multiplication and division
Why this exists
Thabo’s family grows corn. In spring they plant the seeds in straight rows, the same number of seeds in every row, and each seed comes up as one corn plant. Thabo wants to know how many plants are in the small patch by the gate. He could walk from plant to plant and count one by one, but it is easy to skip a plant or count one twice.
The rows help. Thabo can count one row and find 5 plants. Every row has the same number, so he can add 5 again for each row instead of counting every plant. Three rows give 5 + 5 + 5 = 15. One long count has become a short sum in which the same number is added again and again.
Objects set out like this, in equal rows that also line up into equal columns, form an array. Arrays are all around a child: eggs in a carton, chairs set out for a show, stickers on a sheet, corn in a field, toy blocks pushed into neat lines on a mat. Each one turns “how many in all?” into “how many in a row, and how many rows?”
The array matters beyond counting. A sum of the same number again and again is the first step toward multiplying, and the array is the picture that later work on multiplying is built on. Two rows of an array are also the plainest picture of an even number.
Read everything here aloud with the child, and let them build it with blocks, counters or linking cubes on a mat or a table.
Definition
An array is a set of objects arranged in rows and columns so that every row has the same number of objects and every column has the same number of objects.
- A row runs across, from left to right.
- A column runs up and down.
We describe an array by its rows: “3 rows of 5” means 3 rows with 5 objects in each row. That array holds
objects, one 5 for each row.
A sum in which every number being added is the same, like 5 + 5 + 5, is a sum of equal addends. (An addend is one of the numbers being added.)
The arrays in this lesson have at most 5 rows and at most 5 objects in each row. That limit keeps the sums short enough to add; it is not part of what an array is.
Intuition
The lesson runs over four days. Part 1 counts an array row by row. Part 2 counts the same array column by column and shows why the two counts agree. Part 3 builds arrays for a given number of objects. Part 4 counts arrays with some objects missing or hidden.
Part 1: counting row by row
Here is the corn patch by the gate.
Count the first row: 5 plants. The second row has 5 more, so two rows make 5 + 5 = 10. The third row has 5 more again, so the whole patch is
The sum has one 5 for each row: three rows, three fives. Counting by fives says the same thing with one number for each row, 5, 10, 15 (see skip counting). A child who counts every plant one by one also reaches 15, which is a good check while the idea is new.
What makes this work is that every row is the same. Suppose a child lines up blocks in rows of 3, 4 and 5. The blocks can still be counted by rows, 3 + 4 + 5 = 12, but the addends are not equal and the rows are not an array. “So many rows of so many” only describes rows that match.
The rows must also line up. On a mat, push each block so it sits straight under the block above it. Then the blocks form columns too, and every column has the same number of blocks. Part 2 uses those columns.
A sum of equal addends can be added in any convenient order. Two equal rows together make a double, so an array with four rows can be added two rows at a time: two rows, then the same again (Example 2).
Part 2: counting column by column
The same corn patch can be read down the columns.
There are 5 columns, and each column has 3 plants. Adding one 3 for each column gives
Counting by threes says the same: 3, 6, 9, 12, 15.
Two different sums, 5 + 5 + 5 and 3 + 3 + 3 + 3 + 3, gave the same total. That is not luck. Both sums count the same 15 plants. Every plant sits in exactly one row and exactly one column. Counting by rows takes in every plant once; counting by columns takes in every plant once too, only grouped a different way. So the totals must agree, for this array and for every array.
Look at how the two sums are built. The row sum has three addends, each 5: as many addends as there are rows, each the size of a row. The column sum has five addends, each 3: as many addends as there are columns, each the size of a column. The numbers swap places. A child can watch it happen by turning the mat a quarter turn: the rows become columns and the columns become rows, and no block is added or taken away.
An array can be read as an equation, and an equation can be read as an array. The sum 4 + 4 is 2 rows of 4. The sum 2 + 2 + 2 + 2 is 4 rows of 2, which is the same array read down its columns. When an array has as many rows as columns, such as 3 rows of 3, the row sum and the column sum are the very same sum, 3 + 3 + 3.
Part 3: building arrays for a number
Give a child 10 blocks and ask for an array. They might try 2 rows. Sharing the blocks out between two rows, one block at a time, gives 5 in each row, and 5 + 5 = 10, so 2 rows of 5 works. They might try 3 rows. Rows of 3 use 3 + 3 + 3 = 9 blocks and leave one over; rows of 4 would need 4 + 4 + 4 = 12 blocks, two more than there are. So 3 rows does not work. Each try is a sum of equal addends, compared with 10. Example 5 finishes the search.
Some numbers make several arrays, some make only one within the limit of 5 rows and 5 in a row, and some, like 7, make none with more than one row and more than one column.
Arrays with exactly two rows are special. In a two-row array, each column holds one block on top and one block just under it: a pair. So a two-row array is a set of pairs with nothing left over, and its total is a double, such as 4 + 4. The numbers of objects that can be set out in two equal rows are exactly the even numbers (Even and odd numbers). An odd number of blocks always leaves one block on top with no partner under it.
Part 4: when some objects are missing or hidden
An array is decided by two numbers: how many rows, and how many in each row. So a child does not have to see every object to know the total. Two situations come up.
Gaps. Some seeds never come up. The field was planted as an array, so the empty spots still show where the plants should be. Count the plants by adding each row as it is, or count the full array and take away the empty spots. Both give the same total (Example 7).
Covered objects. A towel lies over part of an array. If the rows that show tell how many are in each row, and the total is known, keep adding the row amount, or count on by it, until the sum reaches the total. The number of addends is the number of rows (Example 8). If instead the number of rows is known, the row amount is added once for each row, whether the rows are covered or not.
Results
Counting by rows. An array holds the number in one row, added once for each row. For 3 rows of 5 that is 5 + 5 + 5.
Counting by columns. The same array holds the number in one column, added once for each column. For 3 rows of 5 that is 3 + 3 + 3 + 3 + 3.
Rows and columns agree. For any array, the row sum and the column sum have the same total.
Two rows make an even number. A two-row array always holds an even number, the double of the number in a row. Every even number of objects can be set out in two equal rows.
Why it is true
Take any array. Every object in it sits in exactly one row: it cannot be in two rows at once, and it cannot be in none. Each row holds the same number of objects, so adding the row amount once for each row adds up every object exactly once. That is the first result.
Every object also sits in exactly one column, and each column holds the same number, so adding the column amount once for each column also adds up every object exactly once. That is the second result. Both sums count the same objects, each one once, so they have the same total. That is the third result.
For the last result, look at a two-row array. Each column has exactly two objects, one on top and one below, and no object is in two columns. So the objects come in pairs with none left over, and the number is even. The total is the number in the top row added to the same number in the bottom row, which is a double. The other way round: an even number of objects pairs up with none left over. Put one object of each pair in a top row and its partner just below it, and the pairs become the columns of a two-row array. \blacksquare
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.
Thabo’s corn patch
The corn patch by the gate has 3 rows with 5 plants in each row. Write a sum of equal addends for the rows, and find how many plants there are.
SolutionEach row has 5 plants, and there are 3 rows, so the sum has one 5 for each row:
To add it, 5 + 5 = 10 and 10 + 5 = 15. Counting by fives, one number for each row, also lands on 15: 5, 10, 15. The patch has 15 plants.
The number of addends, 3, is the number of rows. The size of each addend, 5, is the number of plants in a row. Reading an array always means finding those two numbers.
Adding the rows two at a time
Naledi sets out toy blocks on a mat in 4 rows, with 4 blocks in each row. How many blocks are on the mat?
SolutionOne addend for each row gives
The first two rows together are 4 + 4 = 8, a double. The last two rows are another 8. So the whole array is
Naledi has 16 blocks. Adding two rows at a time is allowed because addends can be grouped in any way without changing the sum. Counting by fours checks it: 4, 8, 12, 16, one number for each row.
The array has 4 columns with 4 blocks in each, so the column sum is 4 + 4 + 4 + 4 as well. When the rows and columns match in number, the two readings give the same sum.
Down the columns
Look again at Thabo’s patch, 3 rows of 5. Write a sum of equal addends for its columns, and check that it gives the same total as the rows. (The second picture in Part 2 shows the columns.)
SolutionThere are 5 columns, and each column has 3 plants, one from each row. One 3 for each column gives
To add it, 3 + 3 = 6, 6 + 3 = 9, 9 + 3 = 12 and 12 + 3 = 15. The rows gave 5 + 5 + 5 = 15 in Example 1, and the columns give 15 too.
They must agree. The rows and the columns are two ways of sorting the same 15 plants, and each plant is counted once either way. Notice that the rows gave three addends of 5, and the columns give five addends of 3: the two numbers trade jobs.
From a sum to an array
Ayaan writes 4 + 4 + 4 + 4 + 4. Build an array of blocks to match it. Write the sum its columns give, and find the total both ways.
SolutionEach addend is one row, so the array has 5 rows with 4 blocks in each row. Adding the rows, count by fours: 4, 8, 12, 16, 20, so
The array has 4 columns. Each column has one block from each of the 5 rows, so each column has 5 blocks. Adding the columns, count by fives: 5, 10, 15, 20, so
Both sums describe the same 20 blocks. A child may find the second sum easier, since counting by fives is quicker than counting by fours. Choosing the easier reading of an array is a real use of the fact that the two readings agree.
Ten blocks on the mat
Naledi has 10 blocks. Find every array she can make with 2 to 5 rows and 2 to 5 blocks in each row.
Try each number of rows in turn, and for each one ask what row size uses exactly 10 blocks.
- 2 rows: 5 + 5 = 10. So 2 rows of 5 works.
- 3 rows: 3 + 3 + 3 = 9 is one short, and 4 + 4 + 4 = 12 is two too many. No row size in between, so 3 rows does not work.
- 4 rows: 2 + 2 + 2 + 2 = 8 is too few, and 3 + 3 + 3 + 3 = 12 is too many. No.
- 5 rows: 2 + 2 + 2 + 2 + 2 = 10. So 5 rows of 2 works.
There are two arrays: 2 rows of 5 and 5 rows of 2. They are the same blocks turned a quarter turn, which is why one sum is 5 + 5 and the other is 2 + 2 + 2 + 2 + 2. One row of 10 is a line of blocks rather than an array of rows and columns, and it is longer than the row limit here.
Which numbers make two rows?
Ayaan has 7 blocks, Thabo has 8 and Naledi has 9. Which of them can set out their blocks in an array with exactly 2 rows?
SolutionTwo equal rows means the total is a double: some number added to itself.
- 7: the doubles near it are 3 + 3 = 6 and 4 + 4 = 8. Nothing in between, so 7 is not a double. Ayaan cannot make 2 equal rows.
- 8: 4 + 4 = 8. Thabo can make 2 rows of 4.
- 9: the doubles near it are 4 + 4 = 8 and 5 + 5 = 10. So 9 is not a double, and Naledi cannot.
With real blocks the child sees why. Put blocks on the mat one column at a time, a block on top and then one under it. With 8 blocks every column fills: 4 columns, 4 pairs. With 7 or 9, the last block sits on top with no partner under it. That is the test for even and odd: 8 is even, and 7 and 9 are odd.
Gaps in the corn
Thabo’s family planted another patch as 4 rows of 5. Three seeds never came up. How many corn plants are growing?
The empty spots still sit in their rows, so add the rows as they are. Reading the picture, the rows hold 5, 4, 5 and 3 plants:
To add it, 5 + 5 = 10 and 4 + 3 = 7, then 10 + 7 = 17.
A second way: if every seed had come up, the patch would be a full array, 5 + 5 + 5 + 5 = 20 plants. Three spots are empty, so there are 3 fewer:
Both ways give 17 plants. The second way needs only the size of the planned array and the number of gaps, which is why the array shape helps even when it is not complete.
Under the towel
Naledi has 25 blocks set out in an array with 5 blocks in each row. She lays a towel over part of it, and 2 rows still show. How many rows are under the towel?
Every row has 5 blocks, so the rows that show hold 5 + 5 = 10 blocks. Keep adding rows of 5, counting by fives from 10, until the count reaches 25:
That is 3 more fives after the 10, so 3 rows are under the towel, and the whole array is 2 + 3 = 5 rows. Check by adding all the rows:
The question gave the size of each addend and the total, and asked how many addends there are. Counting on by the row amount answers it without lifting the towel.
Boundaries
- The rows must be equal. Rows of 3, 4 and 5 can be added, 3 + 4 + 5 = 12, but they are not an array, and no single “rows of” description fits them. The row sum is a sum of equal addends only when every row holds the same number.
- The rows must line up into columns. Blocks in rows of the same length can be pushed out of line, each row sliding along a little. The rows still count correctly, but the columns no longer hold the same number each, so counting by columns fails. Reading both ways needs straight columns.
- A gap breaks the array. A patch with empty spots is not an array of plants. The planned array is still useful, as in Example 7, but its sum counts the spots, not the plants, until the gaps are taken away.
- Every object counts as one. An array counts objects, not how big they are. A big block and a small block in an array are one object each.
- Two equal rows need an even number. Seven or nine blocks cannot fill two equal rows, because only even numbers are doubles. Some numbers, such as 7, make no array at all with more than one row and more than one column.
- The limit of five is a choice. Arrays can have any number of rows and any number in a row. The limit of 5 rows and 5 in a row keeps the sums short; it is not a rule about arrays.
Where this goes next
Each row of an array is one of the equal groups of What multiplication means, set out in a straight line. Later the same pictures get a new sign: in Arrays the array becomes the main picture for multiplying, so that 5 + 5 + 5 is written more shortly. The fact that the row and column readings agree becomes the commutative property of multiplication. Example 8, which asked how many equal rows make a total, is the first look at division.
Arrays also turn into shapes. When the objects are squares pushed edge to edge, the array fills a rectangle, which is Splitting a rectangle into rows and columns of squares, and counting those squares by rows is how area is measured later on.
Exercises
Part 1: counting row by row
For Exercises 1 to 3, use the arrays in the picture. Write a sum of equal addends for the rows and find the total.
- Array A.
- Array B.
- Array C.
- Thabo plants 3 rows of corn with 4 plants in each row. Write a sum for the rows and find how many plants he plants.
- Ayaan lines up blocks in 3 rows: 3 blocks, then 4, then 5. Is this an array? How many blocks are there? Can he move one block to make an array with 3 rows?
- Naledi sets out 4 rows of 3 blocks. Find the total by adding the rows two at a time.
Part 2: counting column by column
- An array has 2 rows of 4. Write a sum for its rows and a sum for its columns, and find the total.
- An array has 3 rows of 2. Write a sum for its rows and a sum for its columns, and find the total.
- Draw an array for 3 + 3 + 3. What sum do its columns give?
- An array has 4 rows of 2. Thabo writes 2 + 2 + 2 + 2 for it, and Naledi writes 4 + 4. Do both sums describe the array? What is the total?
- An array has 3 columns with 4 objects in each column. How many rows are there, and how many objects are in each row? How many objects are there in all?
- Ayaan has an array of 2 rows of 3. He adds one more row of 3 at the bottom. Write the row sum and the column sum before and after.
Part 3: building arrays for a number
For Exercises 15 to 18, an array has 2 to 5 rows and 2 to 5 objects in each row.
- Can 6 blocks make an array with exactly 2 rows? If so, how many are in each row?
- Can 11 blocks make an array with exactly 2 rows?
- Find every array that 9 blocks can make.
- Find every array that 8 blocks can make.
- Find every array that 6 blocks can make.
- Ayaan has 7 blocks. Can he make any array at all?
Part 4: missing and hidden objects
- A corn patch was planted as 3 rows of 4. In one row, one seed never came up. How many plants are growing?
- Naledi’s array has 3 blocks in each row and 12 blocks in all. A towel covers part of it, and 2 rows still show. How many rows are under the towel?
- A patch was planted as 5 rows of 4, and 2 seeds never came up. How many plants are growing?
- Ayaan’s array has 4 rows. A towel covers every row but the top one, which has 2 blocks. How many blocks are there in all?
Reasoning
- Explain why adding the rows of an array and adding its columns always give the same total.
- Can a two-row array ever hold an odd number of objects? Explain.
- Without adding, how can an array show that 3 + 3 + 3 + 3 and 4 + 4 + 4 have the same total? Then find the total.
Answers
- Array A has 2 rows of 3: 3 + 3 = 6.
- Array B has 3 rows of 3: 3 + 3 + 3 = 9. Count by threes: 3, 6, 9.
- Array C has 4 rows of 2: 2 + 2 + 2 + 2 = 8. Count by twos: 2, 4, 6, 8.
- One 4 for each of the 3 rows: 4 + 4 + 4 = 12. Thabo plants 12 plants.
- No, because the rows are not equal. There are 3 + 4 + 5 = 12 blocks. Moving one block from the row of 5 to the row of 3 makes 3 rows of 4, and 4 + 4 + 4 = 12.
- One 3 for each row: 3 + 3 + 3 + 3. Two rows make the double 3 + 3 = 6, and the other two rows make another 6, so 6 + 6 = 12 blocks.
- Rows: 4 + 4 = 8. The array has 4 columns of 2, so columns: 2 + 2 + 2 + 2 = 8. The total is 8.
- Rows: 2 + 2 + 2 = 6. The array has 2 columns of 3, so columns: 3 + 3 = 6. The total is 6.
- 3 rows of 3. It has 3 columns of 3, so the columns give the same sum, 3 + 3 + 3 = 9.
- Yes, both. 2 + 2 + 2 + 2 adds the 4 rows of 2, and 4 + 4 adds the 2 columns of 4. The total is 8 either way.
- Each column has one object from each row, so there are 4 rows. There are 3 columns, so each row has 3 objects. The total is 3 + 3 + 3 + 3 = 12, and the columns agree: 4 + 4 + 4 = 12.
- Before: rows 3 + 3 = 6, columns 2 + 2 + 2 = 6. After: 3 rows of 3, so rows 3 + 3 + 3 = 9, and each column grows by one block to 3, so columns 3 + 3 + 3 = 9. The new row added one more addend to the row sum and made every addend of the column sum one bigger.
- Yes. 3 + 3 = 6, so 2 rows of 3.
- No. The doubles nearest 11 are 5 + 5 = 10 and 6 + 6 = 12, so 11 is not a double, which means it is odd. One block is always left without a partner.
- Only 3 rows of 3, since 3 + 3 + 3 = 9. Two rows fail because 9 is odd. Four rows give 2 + 2 + 2 + 2 = 8 or 3 + 3 + 3 + 3 = 12, and five rows give 2 + 2 + 2 + 2 + 2 = 10, so neither makes 9.
- 2 rows of 4 (4 + 4 = 8) and 4 rows of 2 (2 + 2 + 2 + 2 = 8). Three rows give 2 + 2 + 2 = 6 or 3 + 3 + 3 = 9, and five rows give 2 + 2 + 2 + 2 + 2 = 10, so neither makes 8.
- 2 rows of 3 (3 + 3 = 6) and 3 rows of 2 (2 + 2 + 2 = 6). Four rows of 2 make 8 and five rows of 2 make 10, both too many.
- No. Two rows give 3 + 3 = 6 or 4 + 4 = 8. Three rows give 2 + 2 + 2 = 6 or 3 + 3 + 3 = 9. Four rows of 2 make 8 and five rows of 2 make 10. None of these is 7.
- Adding the rows as they are: 4 + 4 + 3 = 11. Or the full array is 4 + 4 + 4 = 12, and 12 - 1 = 11. There are 11 plants.
- The rows that show hold 3 + 3 = 6 blocks. Counting on by threes, 9, 12, takes 2 more rows. So 2 rows are under the towel, and the array has 4 rows: 3 + 3 + 3 + 3 = 12.
- The full array is 4 + 4 + 4 + 4 + 4 = 20 spots. Two are empty, so 20 - 2 = 18 plants are growing.
- Every row has 2 blocks, hidden or not, and there are 4 rows: 2 + 2 + 2 + 2 = 8 blocks.
- Every object sits in exactly one row and in exactly one column. Adding the rows counts each object once, and adding the columns counts each object once. Both sums count the same objects, so they have the same total.
- No. In a two-row array each column is a pair, one object on top and one below, so every object has a partner and none is left over. That makes the number even. It is also a double: the top row added to the bottom row, which is the same number twice.
- Set out 4 rows of 3. Its rows give 3 + 3 + 3 + 3, and it has 3 columns of 4, so its columns give 4 + 4 + 4. Both sums count the same blocks, so they are equal. The total is 12.