B2.1 Properties and multi-step problems
Operations come in inverse pairs: adding undoes subtracting, multiplying undoes dividing. Multiplication is commutative (6 × 8 = 8 × 6) and distributive (6 × 8 = 6 × 5 + 6 × 3). Use these to solve in steps and to check.
Example
A team buys 6 packs of 8 balls and gives away 12. How many are left? Check your answer.
- Step 1: 6 × 8 = 48.
- Step 2: 48 − 12 = 36.
- Check backwards: 36 + 12 = 48, and 48 ÷ 8 = 6. ✓
B2.2 Facts to 10 × 10
Example
Find 6 × 8 using a fact you know, and write the related division facts.
- Split: 6 × 8 = 6 × 5 + 6 × 3 = 30 + 18 = 48.
- Division partners: 48 ÷ 8 = 6 and 48 ÷ 6 = 8.
- Practise until the fact is instant; the array is the proof.
B2.3 Mental math: ×10, ×100, ×1000, ÷10, and tenths
Multiplying by 10 shifts every digit one place left — a 0 appears at the end. By 100, two zeros; by 1000, three. Dividing by 10 does the reverse. For tenths, add the tenths first: 0.7 + 0.5 = 1.2, because 7 tenths and 5 tenths make 12 tenths.
Example
Compute in your head: 6 × 10, 6 × 100, 6 × 1000, 60 ÷ 10, and 0.7 + 0.5.
- 6 × 10 = 60; 6 × 100 = 600; 6 × 1000 = 6000.
- 60 ÷ 10 = 6.
- 0.7 + 0.5: 7 tenths + 5 tenths = 12 tenths = 1.2.
B2.4 Adding and subtracting to 10 000 and tenths
Example
A stadium sold 2468 tickets on Friday and 1826 on Saturday. How many in all? How many more on Friday?
- Estimate: about 2468 + 1826, near 4294.
- Add in columns, ones to thousands, regrouping: 2468 + 1826 = 4294.
- Difference: 2468 − 1826 = 642. Check: 642 + 1826 = 2468. ✓
B2.5 Multiplying by a one-digit number
Example
Multiply 48 × 3 with an area model.
- Split 48 by place value: 48 = 20 + 28.
- 3 × 20 = 60; 3 × 28 = 84.
- Add the parts: 60 + 84 = 144. The same idea works for 3-digit numbers and for ×10, ×100, ×1000.
B2.6 Dividing, with the remainder as a fraction
Example
Share 21 marbles among 4 friends. Then divide 87 ÷ 7.
- 21 ÷ 4 = 5 remainder 1. Each gets 5, and the last marble stays whole — or is split: 5 1/4 each.
- 87 ÷ 7: 7 × 12 = 84, remainder 3. So 87 ÷ 7 = 12 3/7.
- The remainder over the divisor is the fraction part.
B2.7 A whole number times a unit fraction
Example
What is **2 × **?
- Repeated addition: added 2 times.
- 2 pieces of size one 6th: .
- Multiplying a unit fraction by a whole number just counts pieces — the piece size does not change.
B2.8 Whole-number rates
Example
Notebooks cost $8 each. What do 6 cost?
- A rate is a per-one amount: $8 per notebook.
- Multiply by the count: 6 × 8 = 48.
- The table shows the pattern: each extra notebook adds $8.
Summary
- Inverse pairs check each other.
- Facts to 10 × 10; split a hard fact into easy ones.
- ×10 shifts digits; tenths add like tenths.
- Area model for multiplying; remainder over divisor for dividing.
- Unit fraction × whole counts pieces; rates multiply per one.