Number sense, operations, and order of operations

Place value and operations with whole numbers

You need to move comfortably and quickly with whole numbers because everything else in math stacks on top of this.

Place value: how big is a digit really?

Each place in a whole number tells you how many of a certain size “bundle” you have.

• Ones: $1 each
• Tens: $10 each
• Hundreds: $100 each
• Thousands: $1000 each

Example:
4 582

• 2 is in the ones place → 2 × 1 = 2
• 8 is in the tens place → 8 × 10 = 80
• 5 is in the hundreds place → 5 × 100 = 500
• 4 is in the thousands place → 4 × 1000 = 4000

So 4 582 = 4000 + 500 + 80 + 2.

Place value is what lets you do operations (add, subtract, multiply, divide) column by column, instead of feeling random.

Adding and subtracting whole numbers (column method)

Key idea: line up place values and work from right to left.

Example 1 – addition
347 + 285

Write it stacked:

text 347 + 285 -----

Step by step:

1. Ones: 7 + 5 = 12 → write 2, carry 1 to tens.
2. Tens: 4 + 8 = 12, plus carried 1 = 13 → write 3, carry 1.
3. Hundreds: 3 + 2 = 5, plus carried 1 = 6.

Result:

text 347 + 285 ----- 632

Example 2 – subtraction with borrowing
503 − 268

text 503 - 268 -----

Working right to left:

1. Ones: 3 − 8 → can’t, so borrow 1 ten from the tens place.
   But tens place is 0, so first borrow from hundreds:
   • Hundreds: 5 becomes 4
   • Tens: gets 10, then we borrow 1 ten from there:
     • Tens: 10 becomes 9
     • Ones: 3 becomes 13
2. Ones: now 13 − 8 = 5
3. Tens: 9 − 6 = 3
4. Hundreds: 4 − 2 = 2

Answer: 503 − 268 = 235.

Multiplying whole numbers (long multiplication)

Key idea: multiply by each digit and use place value (tens, hundreds) to position answers.

Example: 47 × 36

text 47 × 36 ------

1. Multiply by 6 (ones digit of 36):

   • 6 × 7 = 42 → write 2, carry 4
   • 6 × 4 = 24, plus 4 = 28

   First row: 282

2. Multiply by 3 (tens digit of 36, really 30):

   • 3 × 7 = 21 → write 1, carry 2
   • 3 × 4 = 12, plus 2 = 14

   Because 3 is in tens place (30), this row is shifted one place to the left:

   Second row: 1410

3. Add the rows:

text 282 + 1410 ------ 1692

So 47 × 36 = 1 692.

Dividing whole numbers (long division)

Key idea: see how many times the divisor “fits into” each part of the number from left to right.

Example: 784 ÷ 7

text 112 7⟌784

1. 7 into 7 → 1 time, write 1 above the 7.
2. Multiply 1 × 7 = 7, subtract: 7 − 7 = 0. Bring down 8.
3. 7 into 8 → 1 time, write 1 above the 8.
4. Multiply 1 × 7 = 7, subtract: 8 − 7 = 1. Bring down 4.
5. 7 into 14 → 2 times, write 2 above the 4.
6. Multiply 2 × 7 = 14, subtract: 14 − 14 = 0.

Answer is 112.