Grade by Grade
B. Number · Operations: Numbers to 100 000, Decimals, Fractions, Ratios and Rates

Adding and Subtracting Whole Numbers to 100 000

B2.4 B2.1 A1About 45 minutes
Learning goals
  • I can estimate before I add or subtract big numbers.
  • I can add and subtract numbers up to 100 000 with the column method (an algorithm).
  • I can regroup across several places, including across zeros.
  • I can choose a strategy and check my answer with the inverse operation.

Words to know

algorithm
A set of steps that always works, like the column method for adding.
regroup
Trade 10 of one place for 1 of the next bigger place (when adding), or 1 for 10 of the next smaller place (when subtracting).
ten thousands
The place to the left of the thousands. In 84 295, the 8 means 8 ten thousands, or 80 000.
sum / difference
The answer to an addition / the answer to a subtraction.

Remember from Grade 4

In Grade 4 you added and subtracted numbers up to 10 000 with the column method, regrouped across zeros, counted up to subtract, and checked with the inverse operation. In Grade 5 the numbers get one place bigger: up to 100 000. The steps are exactly the same. You just keep going one more column!

Let's start with a story

The town library lent 38 476 books last year and 45 819 books this year. How many books is that altogether?

Estimate first: 38 476 is about 38 000 and 45 819 is about 46 000. 38 000 + 46 000 = 84 000. So the answer should be close to 84 000.

Now let's add exactly.

OTHThTTh1116748391854+59248
38 476 + 45 819 = 84 295. The small 1s show where we regrouped. TTh = ten thousands, Th = thousands, H = hundreds, T = tens, O = ones.
Worked example

Example 1: Add with regrouping

Solve 38 476 + 45 819.

  1. Line up the digits by place value. Start with the ones: 6 + 9 = 15. Write 5, regroup 1 ten.
  2. Tens: 1 + 7 + 1 = 9.
  3. Hundreds: 4 + 8 = 12. Write 2, regroup 1 thousand.
  4. Thousands: 1 + 8 + 5 = 14. Write 4, regroup 1 ten thousand.
  5. Ten thousands: 1 + 3 + 4 = 8.
  6. Compare with the estimate: 84 295 is close to 84 000. ✓

38 476 + 45 819 = 84 295 books

Subtracting with the column method

Start at the ones. If the top digit is smaller than the bottom digit, regroup: take 1 from the next place to the left and trade it for 10.

If the next place is 0, keep going left until you find a digit you can take from.

Worked example

Example 2: Regroup across a zero

Solve 72 305 − 46 158.

  1. Ones: 5 − 8? Not enough. The tens digit is 0, so go to the hundreds.
  2. Take 1 hundred from 3 hundreds (2 are left). Trade it for 10 tens. Now take 1 ten (9 tens are left) and trade it for 10 ones: 15 ones.
  3. Ones: 15 − 8 = 7. Tens: 9 − 5 = 4. Hundreds: 2 − 1 = 1.
  4. Thousands: 2 − 6? Regroup 1 ten thousand (6 are left): 12 − 6 = 6.
  5. Ten thousands: 6 − 4 = 2.
  6. Check with addition: 26 147 + 46 158 = 72 305. ✓

72 305 − 46 158 = 26 147

A shortcut for numbers like 60 000

Subtracting from a number with lots of zeros means lots of regrouping. Instead, take 1 from both numbers. The difference stays the same, and now you don't need to regroup at all!

60 000 − 23 874 is the same as 59 999 − 23 873.

Worked example

Example 3: Take 1 from both

Solve 60 000 − 23 874.

  1. Take 1 from both numbers: 59 999 − 23 873.
  2. Subtract each place, no regrouping needed: 9 − 3 = 6, 9 − 7 = 2, 9 − 8 = 1, 9 − 3 = 6, 5 − 2 = 3.
  3. That gives 36 126.
  4. Check: 36 126 + 23 874 = 60 000. ✓

60 000 − 23 874 = 36 126

2000030000400005000060000+126+6000+30 000
Another way: count up from 23 874. +126 to 24 000, +6000 to 30 000, +30 000 to 60 000. 126 + 6000 + 30 000 = 36 126.
Worked example

Example 4: A comparing story

A maple syrup farm sold 12 480 L of syrup one year and 15 925 L the next year. How much more did it sell the second year?

  1. "How much more" means compare, so subtract: 15 925 − 12 480.
  2. Estimate: 16 000 − 12 000 = about 4000.
  3. Ones: 5 − 0 = 5. Tens: 2 − 8? Regroup a hundred: 12 − 8 = 4. Hundreds: 8 − 4 = 4. Thousands: 5 − 2 = 3. Ten thousands: 1 − 1 = 0.
  4. That gives 3445. It's close to the estimate of 4000. ✓

3445 L more

Big-number tips

  • Estimate first so you can spot a big mistake.
  • Write the numbers neatly in columns. Grid paper helps.
  • Choose your strategy: column method, counting up, or taking 1 from both.
  • Check subtraction with addition. If your check doesn't match, don't worry. Finding and fixing a mistake is what good mathematicians do!
Just for fun

If you counted out loud from 1 to 100 000, saying one number every second and never stopping, it would take you more than 27 hours. (100 000 seconds is about 27 hours and 47 minutes!)

Hands-on activity

Big number race

You need: a deck of cards with 10s and face cards removed (aces = 1), plus paper and a pencil.

  1. Each player draws 5 cards and arranges them to make a 5-digit number, with their smallest card in the ten thousands place.
  2. Add the two players' numbers together. Estimate first, then use the column method. (If your estimate is over 100 000, skip the adding and go to step 3 for this round.)
  3. Next round, subtract the smaller number from the bigger one. Check with addition.
  4. Bonus round: try to make two numbers whose sum is as close to 100 000 as possible.

Math talk: "Where did you have to regroup? How did your estimate help?"

Your turn

1

24 365 + 18 452 = ?

2

56 708 + 27 596 = ?

3

83 492 − 51 270 = ?

4

70 315 − 28 647 = ?

5

50 000 − 12 345 = ?

6

Estimate 39 870 + 20 150.

7

A town planted 12 480 tulip bulbs in the fall and 9 765 in the spring. How many bulbs in all?

More practice: warm-up

1

30 000 + 40 000 = ?

2

45 000 − 5000 = ?

3

12 500 + 2500 = ?

4

9999 + 1 = ?

5

65 000 − 15 000 = ?

More practice: on your own

1

46 829 + 37 594 = ?

2

91 006 − 47 538 = ?

3

12 345 + 23 456 + 34 567 = ?

4

A stadium had 28 750 fans on Saturday and 31 480 on Sunday. How many fans in total?

5

Check with addition: is 64 200 − 18 975 = 45 225 correct?

Challenge

1

Find the missing number: □ + 36 845 = 90 000.

2

A town wants to plant 50 000 trees. It planted 18 640 the first year and 21 375 the second year. How many trees are still to be planted?

3

Find two numbers whose sum is 100 000 and whose difference is 20 000.

Check your understanding

Quiz

1.

What is 45 678 + 21 432?

2.

What is 80 000 − 34 500?

3.

Which number sentence checks 72 400 − 28 100 = 44 300?

4.

What is the best estimate for 49 820 + 30 105?

5.

What is 33 333 + 66 667?

6.

A school collected 52 640 cans one year and 27 390 cans the next. How many more cans were collected the first year?

For parents and teachers

Big idea: The Grade 5 curriculum (B2.4) asks students to solve problems involving addition and subtraction of whole numbers that add up to no more than 100 000, using appropriate tools, strategies and algorithms. Your child already knows the column method from Grade 4; now the numbers have one more place.

Common difficulties:

  • Misaligned digits in long numbers. Use grid paper and say the place names.
  • Regrouping across zeros (like 72 305 − 46 158). Model it slowly, or use "take 1 from both" for numbers like 60 000.
  • Subtracting the smaller digit from the bigger one in every column (8 − 5 instead of 5 − 8). Ask "Is the top digit big enough?"
  • Skipping the estimate. Estimating first catches most big errors.

How to help: Use real big numbers: car odometer readings, populations of nearby towns, steps on a fitness tracker. Ask your child to check every subtraction with addition. Praise checking and fixing mistakes; that perseverance is part of the A1 expectations.