Smart Strategies: Properties and Relationships of Operations

Remember from Grade 4

In Grade 4 you learned the math rules called properties: you can add or multiply in any order, you can group numbers in a smart way, adding 0 or multiplying by 1 changes nothing, multiplying by 0 gives 0, and you can split a number to multiply. You also checked subtraction with addition and division with multiplication.

In Grade 5 you will use these rules with bigger numbers and with decimals, and use them to check your work.

Let's start with a puzzle

At the school bake sale, Noor buys a muffin for $1.25, a juice for $2.60 and a cookie for $0.75. How much does she spend?

You could add from left to right. But look: $1.25 + $0.75 = $2.00. That's a friendly pair! Then $2.00 + $2.60 = $4.60.

Noor changed the order and the grouping to make the math easy. That's allowed, because of the properties of addition.

Rule 1 and Rule 2: Any order, any grouping

When you add or multiply, you can change the order and choose which numbers to do first. The answer stays the same.

• 2.5 + 4.8 + 7.5 → (2.5 + 7.5) + 4.8 = 10 + 4.8 = 14.8
• 25 × 37 × 4 → (25 × 4) × 37 = 100 × 37 = 3700

Look for pairs that make a whole number, 10 or 100: 0.25 + 0.75 = 1, 0.6 + 0.4 = 1, 2 × 5 = 10, 4 × 25 = 100, 20 × 5 = 100.

Careful: these rules do not work for subtraction or division. 10 − 4 is not the same as 4 − 10, and 24 ÷ 4 is not the same as 4 ÷ 24.

Rule 3: Split it (the distributive property)

Split one factor into friendly parts. Multiply each part. Then put the parts back together.

12 × 15: split 15 into 10 + 5.

12 × 10 = 120 and 12 × 5 = 60. 120 + 60 = 180.

You can also split with subtraction. 9 × 99 is 9 groups of 100, take away 9 groups of 1:

9 × 100 = 900, then 900 − 9 = 891.

Rule 4: Make it friendly, then fix it (compensation)

Some numbers are almost friendly. Change them to a friendly number, do the math, then fix your answer.

• 4.99 + 2.35: 4.99 is 0.01 less than 5. Add 5 + 2.35 = 7.35. You added 0.01 too much, so take it away: 7.34.
• 5000 − 2387: take 1 from both numbers. 4999 − 2386 = 2613. (The difference stays the same when you move both numbers the same amount.)

Rule 5: Operations are partners (inverse operations)

• Adding and subtracting undo each other. If 6.25 − 2.8 = 3.45, then 3.45 + 2.8 = 6.25.
• Multiplying and dividing undo each other. If 144 ÷ 12 = 12, then 12 × 12 = 144.

So you can always check a subtraction with addition, and a division with multiplication. Mathematicians check their work. You should too!