Mathematics notes, unit 2

Addition, Subtraction, Multiplication and Division

Textbook Mathematics Year 3 (DLP), Part 1 and Part 2, pages Part 1, 27–80

What this chapter covers

You can add two or three numbers with a total up to 10 000, with and without regrouping (2.1)
You can subtract numbers up to 10 000, including subtracting two numbers one after another (2.2)
You can multiply a number by a one-digit number, and by 10, 100 and 1 000 (2.3)
You can divide a number by a one-digit number, and by 10, 100 and 1 000, with and without a remainder (2.4)
You can work out number sentences that mix addition and subtraction (2.5)
You can find the unknown in a number sentence (2.6)
You can create stories from number sentences and solve problems in daily life (2.7)

2.1Addition

Addition means putting numbers together. The answer is called the sum or the total. Words that tell you to add: add, plus, sum, total, altogether.

At the zoo there are 1 382 birds and 15 crocodiles. How many birds and crocodiles are there altogether? We can count on along a number line: jump 10, then jump 5.

Number line from 1 382 jumping +10 to 1 392 and +5 to 1 397 — 1 382 + 15 = 1 397 (textbook p. 29)

Adding in columns

For bigger numbers, write them in a place-value chart: thousands, hundreds, tens and ones. Always start adding from the ones.

Regroup (carry): when a column adds up to 10 or more, keep the ones digit and carry the 1 ten to the next column on the left.

Example 1: 3 215 + 1 997 (p. 34)

3 215 plus 1 997 in a place-value chart with carries, answer 5 212

Ones: 5 + 7 = 12. Write 2, carry 1 ten.
Tens: 1 + 9 + 1 = 11. Write 1, carry 1 hundred.
Hundreds: 2 + 9 + 1 = 12. Write 2, carry 1 thousand.
Thousands: 3 + 1 + 1 = 5.

3 215 + 1 997 = 5 212

Smart way: 1 997 is very near 2 000. Take 3 from 3 215 and give it to 1 997: 3 212 + 2 000 = 5 212.

Line up the digits by place value. If you write 15 under the thousands and hundreds of 1 382, you get 2 882, which is wrong. The 5 must sit under the ones.

Check with estimation (a quick, close guess). 104 + 1 382 is about 100 + 1 400 = 1 500. The real answer, 1 486, is close to 1 500, so it is reasonable (p. 29).

Finding a missing number. 5 602 tickets are sold. The target is 5 648. How many more? 5 602 + ☐ = 5 648. Take away: 5 648 − 5 602 = 46 (p. 31).

Using an abacus

On an abacus, the top bead is worth 5 and each bottom bead is worth 1 when it is pushed to the middle bar. The abacus below shows 2 406: 2 thousands, 4 hundreds, 0 tens and 6 ones (top bead + 1 bottom bead).

To add on an abacus, start from the thousands and move to the ones. When there are not enough bottom beads, use the little friends of 5 (4 and 1, 3 and 2) or the big friends of 10 (6 and 4, 7 and 3) (p. 33).

Adding three numbers

Add all three numbers in the same chart, column by column. You can also add two of the numbers first and then add the third.

Example 2: 169 + 3 547 + 49 (p. 37)

169 plus 3 547 plus 49 in a place-value chart, answer 3 765

Ones: 9 + 7 + 9 = 25. Write 5, carry 2 tens.
Tens: 6 + 4 + 4 + 2 = 16. Write 6, carry 1 hundred.
Hundreds: 1 + 5 + 1 = 7. Thousands: 3.

169 + 3 547 + 49 = 3 765

Did you know? Adding the two same digits first makes it easier: 9 + 9 = 18, then 18 + 7 = 25.

a) 4 563 + 2 718 = ? b) 1 205 + 386 + 2 470 = ?

Show answerHide answer

a) 7 281 (ones 3 + 8 = 11, carry 1) b) 4 061

2.2Subtraction

Subtraction means taking away. The answer is called the difference or the balance. Words that tell you to subtract: minus, take away, left, difference, how many more, how many less.

Start from the ones. If the top digit is too small, regroup: change 1 from the next column into 10.

Example 3: 1 283 − 790 (adult and teenage train passengers, p. 44)

1 283 minus 790 with regrouping, answer 493

Ones: 3 − 0 = 3.
Tens: 8 − 9 cannot. Change 1 hundred into 10 tens: 18 − 9 = 9. The hundreds digit becomes 1.
Hundreds: 1 − 7 cannot. Change 1 thousand into 10 hundreds: 11 − 7 = 4. The thousands digit becomes 0.

1 283 − 790 = 493

Do not just take the smaller digit from the bigger one in each column. For 1 283 − 790 that gives 1 513, which is wrong. When the top digit is smaller, you must regroup.

Example 4: 8 000 − 1 203 (p. 45)

8 000 minus 1 203 with regrouping across zeros, answer 6 797

There are no ones, tens or hundreds in 8 000, so regroup from the thousands: 8 thousands become 7 thousands, 9 hundreds, 9 tens and 10 ones. Then subtract each column.

8 000 − 1 203 = 6 797

Smart way: take 1 from both numbers. 7 999 − 1 202 = 6 797. No regrouping needed!

Check subtraction by adding: 6 797 + 1 203 = 8 000. Correct!

Subtracting one after another

There are 2 458 young plants at a nursery. One customer buys 103, another buys 12. How many are left? (p. 48)

Method 1: one at a time	Method 2: add what was taken first
2 458 − 103 = 2 355 2 355 − 12 = 2 343	103 + 12 = 115 2 458 − 115 = 2 343

Both ways give 2 343 plants left.

a) Dev's father has 6 000 catfish and sells 846. How many are left? b) 4 267 − 29 − 301 = ?

Show answerHide answer

a) 6 000 − 846 = 5 154 catfish b) 4 267 − 29 = 4 238; 4 238 − 301 = 3 937

2.3Multiplication

Multiplication is quick adding of equal groups. The sign is ×. Words that tell you to multiply: times, multiply, each … how many in all, equal groups.

Three boxes with 12 oranges each — 3 boxes of 12 oranges: 12 + 12 + 12 = 3 × 12 = 36 (p. 60)

Did you know? 3 × 12 = 12 × 3. You can swap the numbers and the answer stays the same.

Example 5: 6 school buses with 44 pupils each (p. 64)

44 times 6 in a place-value chart, answer 264

Ones: 6 × 4 = 24 ones. Write 4, carry 2 tens.
Tens: 6 × 4 = 24 tens, plus the 2 carried = 26 tens. Write 6 tens and 2 hundreds.

6 × 44 = 264 pupils.

Remember to add the number you carried. 15 × 3: 3 × 5 = 15 (write 5, carry 1), then 3 × 1 = 3, plus 1 = 4. The answer is 45, not 35.

Example 6: 576 × 8 by place value (p. 65)

Break 576 into 500 + 70 + 6 and multiply each part by 8:

500 × 8 = 4 000 70 × 8 = 560 6 × 8 = 48

4 000 + 560 + 48 = 4 608

Multiplying by 10, 100 and 1 000

Multiply by	What happens	Examples
10	Add one 0 at the end	9 × 10 = 90 596 × 10 = 5 960
100	Add two 0s	9 × 100 = 900 78 × 100 = 7 800
1 000	Add three 0s	9 × 1 000 = 9 000 10 × 1 000 = 10 000

a) 4 × 2 193 = ? b) 35 × 100 = ?

Show answerHide answer

a) 8 772 (4 × 3 = 12, carry 1; 4 × 9 = 36 + 1 = 37, carry 3; 4 × 1 = 4 + 3 = 7; 4 × 2 = 8) b) 3 500

2.4Division

Division means sharing equally or putting into equal groups. The sign is ÷. Words that tell you to divide: share equally, each gets, how many groups, packed in fours.

24 small stones are put equally into 2 jars. 24 ÷ 2 = 12 stones in each jar (p. 66).

Long division

For bigger numbers we use long division. Start from the digit on the left (the biggest place value).

Example 7: 628 doughnuts packed in fours (p. 67)

Long division 628 divided by 4 equals 157

Hundreds: 6 ÷ 4 = 1, and 6 − 4 = 2 left. Bring down the 2 to make 22.
Tens: 22 ÷ 4 = 5, and 22 − 20 = 2 left. Bring down the 8 to make 28.
Ones: 28 ÷ 4 = 7, and 28 − 28 = 0.

628 ÷ 4 = 157 packets. Check: 157 × 4 = 628.

Division with a remainder

The remainder is the amount left over after sharing equally.

29 pieces of chalk shared into 3 boxes of 9 with 2 left over — 29 ÷ 3 = 9 remainder 2 (p. 70)

Check: 3 × 9 = 27, and 27 + 2 = 29.

The remainder must be smaller than the number you divide by. 29 ÷ 3 = 8 remainder 5 is wrong, because 5 is bigger than 3. Put one more in each box.

Dividing by 10, 100 and 1 000

Number sentence	Answer
690 ÷ 10	69
8 600 ÷ 100	86
7 000 ÷ 1 000	7
682 ÷ 10	68 remainder 2

Dividing by 10, 100 or 1 000 takes off one, two or three 0s. If the last digits are not 0, they become the remainder: 9 768 ÷ 100 = 97 remainder 68.

a) 376 ÷ 6 = ? b) 840 storybooks are shared equally by 5 orphanages. How many does each get?

Show answerHide answer

a) 62 remainder 4 (check: 62 × 6 = 372, 372 + 4 = 376) b) 840 ÷ 5 = 168 storybooks

2.5Mixed addition and subtraction

Some number sentences have both + and −. Work from left to right, one step at a time.

Example 8: 8 728 − 524 + 39 (p. 52)

Left first: 8 728 − 524 = 8 204.
Then: 8 204 + 39 = 8 243.

8 728 − 524 + 39 = 8 243

Example: there are 15 red hats and 20 blue hats. 3 hats are worn. 15 + 20 − 3 = 35 − 3 = 32 hats are not worn (p. 51).

6 240 + 517 − 389 = ?

Show answerHide answer

6 240 + 517 = 6 757; 6 757 − 389 = 6 368

2.6Finding the unknown

An unknown is a number we do not know yet. We show it with a box ☐, a shape such as ▲ or ★, or a letter.

Story	Number sentence	Unknown
7 fish in the tank. Father adds some. Now there are 10.	7 + ☐ = 10	3
12 cupcakes. Some are eaten. 9 are left.	12 − ☐ = 9	3
3 comics with the same pages. 150 pages in all.	3 × ☐ = 150	50
20 papers shared equally. Each pupil gets 5.	20 ÷ ☐ = 5	4

A bar model helps you see the unknown. At a supermarket car park, 40 spaces are occupied and 596 are empty. What is the total? (p. 58)

Bar model: 40 occupied and 596 empty, total unknown

☐ = 40 + 596 = 636 parking spaces.

Use the opposite operation to find the unknown. Add ↔ subtract, multiply ↔ divide. For ☐ ÷ 4 = 30, do 30 × 4 = 120.

Shanti sells some quail eggs. Then she sells 90 more. She sells 105 eggs altogether. Write the number sentence and find the unknown.

Show answerHide answer

☐ + 90 = 105; ☐ = 105 − 90 = 15 eggs

2.7Create stories and solve problems

Create a story

A number sentence can become a story. Example: 3 × 24 = 72. Puan Zurina gives bags of souvenirs to 3 group leaders. Each leader gets 24 bags. She gives out 72 bags in all. (p. 73)

A good story has a place or people, the numbers from the sentence, the right action word (shares, buys, gives away) and a question or the answer at the end.

Solve problems in 4 steps

Understand: underline what is given and what you must find.
Plan: draw a picture or a bar model and choose +, −, × or ÷.
Solve: write the number sentence and calculate.
Check: use the opposite operation or estimate.

Key words	Operation
total, altogether, more are added	+
left, difference, how many more, given away	−
each … how many in all, times as many	×
shared equally, each gets, packed into	÷

Example 9: water for Sports Day (p. 56)

A school has 1 340 bottles of mineral water. The PTA gives another 960 bottles. 1 550 bottles are given to guests. How many bottles are left?

Two bar models: 1 340 plus 960 makes 2 300; 2 300 minus 1 550 leaves the unknown

1 340 + 960 = 2 300
2 300 − 1 550 = 750

750 bottles are left. Check: 750 + 1 550 = 2 300.

Example 10: soursop juice (p. 76)

A factory makes 9 507 bottles of soursop juice. 8 bottles go in each box. How many boxes? How many bottles are left over?

Long division 9 507 divided by 8 equals 1 188 remainder 3

9 507 ÷ 8 = 1 188 remainder 3. There are 1 188 boxes and 3 bottles left over. Check: 1 188 × 8 + 3 = 9 507.

Rita puts 18 flowers in a vase. How many flowers are in 6 vases?

Show answerHide answer

6 × 18 = 108 flowers

Summary

Operation	Remember	Example
Addition +	Start from the ones; carry when a column is 10 or more	3 215 + 1 997 = 5 212
Subtraction −	Start from the ones; regroup when the top digit is smaller; check by adding	1 283 − 790 = 493
Multiplication ×	Equal groups; add the carried number; × 10, 100, 1 000 adds 0s	6 × 44 = 264
Division ÷	Start from the left; remainder is smaller than the divisor; check by multiplying	628 ÷ 4 = 157
Mixed + and −	Work from left to right	8 728 − 524 + 39 = 8 243
Unknown ☐	Use the opposite operation	☐ + 90 = 105, ☐ = 15