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Algebra I · Lesson 1.3

When Order Matters

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You can rewrite an expression in a different order, or group its numbers differently. Sometimes the value stays exactly the same, so the rewrite is free and just makes the arithmetic easier. Sometimes it changes, and you land on a different number. This lesson is about telling the two apart.

Problem
A running total to knock out in your head. Find 38+65+62+35. Straight down the line this is a grind, but the four numbers were chosen so that a little rearranging pairs them into round hundreds. What is the sum?
Show a hint
  • Two of these four addends already sit close to a hundred. Can you find a pairing that lands on round numbers?
Show the full solution
Reorder to 38+62=100 and 65+35=100, so the sum is 100+100=200. Reordering addends never changes a sum, so it is always worth scanning a list for pairs that make round numbers.
Problem
Now a product built the same way. Find the value of 8⋅17⋅125 by choosing which two factors to multiply first.
Show a hint
  • You may multiply the factors of a product in any order. Which two of these three make a power of ten?
Show the full solution
Pair 8⋅125=1000 first, then 1000⋅17=17000. Starting with 8⋅17=136 lands in the same place, but the arithmetic is much uglier.
Order flips, the sum holds3 + 88 + 311Grouping moves, the total holds(3 + 8) + 53 + (8 + 5)16
Two laws in one picture. On top the order flips, 3+8 and 8+3, and the sum stays 11. Below the order holds but the grouping moves, (3+8)+5 and 3+(8+5), and the total stays 16. Commutative is about order, associative is about grouping, and both hold for addition and for multiplication.
Problem
A classmate evaluates 40−12−6. Instead of working left to right, they group the last two numbers, compute 12−6=6 first, and report 40−6=34. That is not what the expression says. Worked correctly, what is 40−12−6?
Show a hint
  • Subtraction is done left to right. Grouping a different pair first is a change, not a shortcut. Does it give the same number?
Show the full solution
Left to right, 40−12=28 and 28−6=22. The classmate's regrouping computes 40−(12−6)=34, a different value, because subtraction is not associative.
Problem
Rewrite every subtraction in 58−24+42−8 as adding a negative, then reorder to pair friendly numbers. What is the value?
Show a hint
  • Turn the chain into 58+(−24)+42+(−8). Now it is one long sum, so reorder freely.
  • Pair the two that reach a round number, then add the two negatives.
Show the full solution
Rewritten, 58+(−24)+42+(−8) is a single sum with each minus stuck to its number, so reorder to 58+42=100 and (−24)+(−8)=−32. Then 100+(−32)=68.
Problem
A student wants to reorder 30−18+5 and writes 30−5+18, swapping the 18 and the 5 but leaving the signs in place. That gives 43, which is wrong, because the minus belongs to the 18. Evaluated correctly, what does 30−18+5 equal?
Show a hint
  • Read it as 30+(−18)+5. When you move a number, its sign has to move with it.
Show the full solution
Write it as 30+(−18)+5. Left to right, 30−18=12 and 12+5=17. The minus has to travel with the 18. Writing 30−5+18 flips both signs, which is how it lands on 43.
Problem
Rewrite the division in 34⋅19÷34 as multiplying by a reciprocal, then reorder the factors so a pair cancels. What is the value?
Show a hint
  • Read it as 34⋅19⋅134. Which two factors undo each other?
Show the full solution
As 34⋅19⋅134, the factors reorder to (34⋅134)⋅19=1⋅19. The 34 and its reciprocal cancel, leaving 19.
Problem
Two clever pieces, one minus sign between them. Find the value of 25⋅63⋅4−(88+47+12+53). Reorder inside the product and inside the sum, but do not move a piece across the minus sign.
Show a hint
  • Inside the product, 25⋅4 is a round number. Inside the parentheses, look for two pairs that each make 100.
  • The freedom to reorder lives within each level. You may not pull a factor out of the product and add it to the sum.
Show the full solution
In the product, 25⋅4=100, so 25⋅63⋅4=100⋅63=6300. In the sum, 88+12=100 and 47+53=100, so the parentheses hold 200. Then 6300−200=6100. The shuffling stays inside each piece, since the factors and the addends sit at different levels.

So order and grouping never change a sum or a product, while both can change a difference or a quotient. That changes once you rewrite the subtraction as adding a negative or the division as multiplying by a reciprocal, since the expression becomes a single chain of additions or multiplications that you can reorder freely. The next lesson is about turning word descriptions into expressions.

Practice these ideas

Practice
Add 46+29+54+21 by rearranging the four numbers into round pairs. What is the sum?
Show the solution
Reorder to 46+54=100 and 29+21=50, so 100+50=150.
Practice
Find the value of 2⋅39⋅50 by choosing which two factors to multiply first.
Show the solution
Pair 2⋅50=100, then 100⋅39=3900.
Practice
Rewrite each subtraction as adding a negative in 73−26+27−14, then pair friendly numbers. What is the value?
Show the solution
As one sum, 73+27=100 and (−26)+(−14)=−40, so 100+(−40)=60.
Practice
Rewrite the division as multiplying by a reciprocal, then find the value of 61⋅7÷61.
Show the solution
As 61⋅161⋅7=1⋅7, the 61 cancels its reciprocal, leaving 7.
Practice
A classmate reads 70−35−8 by grouping the last two numbers, computing 35−8=27 first, and reporting 70−27=43. Evaluated correctly, left to right, what does 70−35−8 equal?
Show the solution
Left to right, 70−35=35 and 35−8=27. The classmate computed 70−(35−8)=43, a different value, because subtraction is not associative.
Practice
Among the four operations +, −, ⋅, and ÷, exactly two always let you swap the two numbers with no change in value. One of them is addition. Name the other one.
Show the solution
The other one is multiplication, since a⋅b=b⋅a. Subtraction and division fail the swap, because 7−2 is not 2−7 and 20÷4 is not 4÷20.
Practice
Add 71+48+29+22 by rearranging into round pairs. What is the sum?
Show the solution
Reorder to 71+29=100 and 48+22=70, so 100+70=170.
Practice
Division is not associative, so the grouping in 96÷8÷2 matters. Reading it correctly, left to right, what is its value? (Grouping as 96÷(8÷2) would give the different answer 24.)
Show the solution
Left to right, 96÷8=12 and 12÷2=6. Grouping 8÷2=4 first gives 96÷4=24, which is why division needs a fixed reading order.
Practice
Rewrite the divisions as multiplying by reciprocals, then reorder to cancel. Find the value of 15⋅46÷15÷2.
Show the solution
As 15⋅115⋅46⋅12, the 15 cancels, leaving 46⋅12=23.
Practice
Insert exactly one pair of parentheses into 24−8−6−2 to make the result as large as possible. Enter that largest value.
Show the solution
Group as 24−(8−6−2). The inside is 8−6−2=0, so the result is 24−0=24. Every other placement does worse, since 24−(8−6)−2=20, 24−8−(6−2)=12, and the ungrouped chain is only 8.