B2.1 Properties and the × / ÷ connection
Multiplying in any order gives the same answer: 10 × 6 = 6 × 10. Division undoes multiplication: if 10 × 6 = 60, then 60 ÷ 6 = 10. Use that to check.
Example
Write the fact family for 10, 6 and 60.
- 10 × 6 = 60
- 6 × 10 = 60
- 60 ÷ 6 = 10
- 60 ÷ 10 = 6
B2.2 Facts of 2, 5 and 10
×2 is doubling. ×5 always ends in 0 or 5. ×10 puts a 0 on the end. Learn these until they are automatic, and their division partners come free.
Example
Find 10 × 6 and 60 ÷ 10 quickly.
- Skip-count by 10: 10, 20, 30 … 6 jumps lands on 60.
- Division asks: how many 10s in 60? 6.
- Same fact, two directions.
B2.3 Mental math to 1000
To add a number that is one short of a hundred, add the whole hundred, then take 1 back. Estimate first so you know the ballpark.
Example
Add 365 + 299 in your head.
- Estimate: about 365 + 300 = 665.
- 299 is 1 less than 300. Add 300: 365 + 300 = 665.
- Take 1 back: 665 − 1 = 664.
B2.4 How the written algorithm works
The written method for adding is the blocks method in columns: add the ones, then the tens, then the hundreds. When a column reaches 10, regroup — trade ten ones for a ten, ten tens for a hundred.
Example
Add 365 + 275 in columns, and say what each step means with blocks.
- Ones column first: add the ones; if 10 or more, carry a ten.
- Tens column: add the tens plus any carried ten; regroup if needed.
- Hundreds column: add the hundreds. Total: 640.
- Every carry is a trade of ten small blocks for one bigger block.
B2.5 Solving problems to 1000
Example
A school collected 365 cans in September and 275 cans in October. How many altogether? How many more in September?
- Altogether: 365 + 275 = 640.
- How many more: 365 − 275 = 90.
- Check the subtraction by adding back: 90 + 275 = 365. ✓
- Read it twice; what is asked?
- Joining → add. Comparing or taking away → subtract.
- Estimate.
- Solve with columns or a mental strategy.
- Check with the opposite operation.
B2.6 Arrays: multiplying and dividing to 10 × 10
Example
How many cookies are in 7 rows of 7? How many rows does 49 ÷ 7 make?
- Multiply: 7 × 7 = 49.
- Turn it sideways: 7 rows of 7 is also 49.
- Divide: 49 ÷ 7 = 7 rows. The array shows both.
B2.7 Groups of a half, a third, a fourth
Example
What is one third of 12 cookies? What is 3 groups of one half?
- One third of 12: split 12 into 3 equal groups. Each is 4.
- 3 groups of one half: half + half + half = 1 and a half.
- A fraction of a set is a division; groups of a fraction is repeated adding.
B2.8 The numerator counts unit fractions
Example
Write three eighths as repeated addition of unit fractions.
- .
- The bottom number names the piece size; the top counts the pieces.
- Adding the bottoms is wrong: is not — the pieces stay eighths.
B2.9 Ratios of 1 to 2, 1 to 5 and 1 to 10
Example
1 dime is 10 cents. How many cents are 6 dimes? If 1 box holds 5 pencils, how many in 6 boxes?
- 1 to 10: 6 dimes → 6 × 10 = 60 cents.
- 1 to 5: 6 boxes → 6 × 5 = 30 pencils.
- 1 to 2: 6 pairs → 6 × 2 = 12. Scale up: multiply.
Summary
- Any order for ×; division undoes it.
- Facts of 2, 5, 10 by heart.
- Mental math: use the nearby hundred, then fix.
- Columns = blocks; carry = trade.
- Arrays show × and ÷ together; ratios scale up.