Chapter Notes: Calculation, Missing Number Problems

Missing Number Problems

• Example: If □ - 7 = 13 (cookies eaten and remaining), then □ = 13 + 7 = 20 (total cookies).
• Example: If 4 × □ = 700 (four glasses fill a jug), then □ = 700 ÷ 4 = 175 ml (each glass).
• Example: If 50 - □ = 5 (change from an apple), then □ = 50 - 5 = 45 cents (apple cost).
• Example: If 3a + o = 200 (three apples and an orange cost $2), and assuming equal costs, then 4a = 200, so a = 50 cents, o = 50 cents.

Terms like double, half, triple, sum, and difference describe relationships:
Example: p + k = 15 (plums and kiwis sum to 15).

Multiple solutions may exist for some problems:
Example: flower + leaf = 30 and flower - leaf = 18:

• Add equations: 2 × flower = 48, so flower = 24, leaf = 6.
• Other integer solutions depend on constraints (e.g., positive numbers).

Order of Operations

Sample calculations:

• 20 - 3 × 3 = 20 - 9 = 11.
• 5 × 3 - 6 = 15 - 6 = 9.
• 9 + 10 - 5 = 19 - 5 = 14.
• 36 - 18 ÷ 6 = 36 - 3 = 33.
• 6 - 5 × 3 = 6 - 15 = -9.

Multiplication and Division

Multiplication finds the product of numbers, often using the column method for larger numbers.
Example: 365 × 24(days in 24 years):

• Compact method: 365 × 20 = 7,300, 365 × 4 = 1,460, total 7,300 + 1,460 = 8,760.
• Expanded method: 300 × 20 = 6,000, 60 × 20 = 1,200, 5 × 20 = 100, 300 × 4 = 1,200, 60 × 4 = 240, 5 × 4 = 20, total 6,000 + 1,200 + 100 + 1,200 + 240 + 20 = 8,760.

Division finds how many times a divisor fits into a number, possibly with a remainder.

• Example: 497 ÷ 7 = 71 (weeks in 497 days).
• Example: 350 ÷ 7 = 50.
• Example: 371 ÷ 7 = 53.
• Example: 695 ÷ 7 = 99 r 2.
• Example: 58 ÷ 7 = 8 r 2.

Estimates guide calculations:

• Example: 33 months ≈ 33 × 30 = 990 days.

Inverse operations check results:

• Example: For 72 ÷ n = k (cutting 72 cm string), possible divisors n = 2, 3, 4, 6, 8, 9 yield k = 36, 24, 18, 12, 9, 8 cm.

Adding and Subtracting Decimal Numbers

• Decompose: 1.35 = 1 + 0.3 + 0.05, 1.24 = 1 + 0.2 + 0.04.
• Add: 1 + 1 = 2, 0.3 + 0.2 = 0.5, 0.05 + 0.04 = 0.09.
• Total: 2 + 0.5 + 0.09 = 2.59.

Regrouping method:

• 1.35 = 1 + 0.35, 1.24 = 1 + 0.24.
• 0.35 + 0.24 = 0.59, so 1 + 1 + 0.59 = 2.59.

Example: 3.86 - 1.53:

• Decompose: 3.86 = 3 + 0.8 + 0.06, 1.53 = 1 + 0.5 + 0.03.
• Subtract: 3 - 1 = 2, 0.8 - 0.5 = 0.3, 0.06 - 0.03 = 0.03.
• Total: 2 + 0.3 + 0.03 = 2.33.

Example calculations:

• 4.35 + 2.21 = 6.56.
• 4.21 + 2.35 = 6.56.
• 4.35 + 2.77 = 7.12.
• 6.48 - 3.25 = 3.23.
• 6.48 - 3.29 = 3.19.
• 6.25 - 3.48 = 2.77.

Multiplying Decimal Numbers

Multiplying decimals uses place value and known multiplication facts.
Example: 0.6 × 4:

• 0.6 = 6 tenths, so 6 tenths × 4 = 24 tenths = 2.4.
• Since 0.6 is 10 times smaller than 6, 0.6 × 4 = (6 × 4) ÷ 10 = 24 ÷ 10 = 2.4.

Patterns in multiplication:

• 3 × 5 = 15, 0.3 × 5 = 1.5 (10 times smaller).
• 7 × 4 = 28, 0.7 × 4 = 2.8.
• 6 × 8 = 48, 0.6 × 8 = 4.8.
• 9 × 7 = 63, 0.9 × 7 = 6.3.
• 12 × 2 = 24, 1.2 × 2 = 2.4.

Estimates sort products:

• Example: 0.5 × 5 ≈ 2.5 (product less than 3).
• Example: 6 × 2.7 ≈ 12 (product more than 4).
• Example: 3 × 3 = 9 (whole number).