Subtraction Algorithms (Part 2)

10 min

Narrative

The purpose of this True or False? is to elicit insights students have about how the commutative property applies to addition and to multiplication, but does not apply to subtraction. The reasoning of students here helps to deepen their understanding of the properties of operations and how they apply to subtracting within 1,000. It also will be helpful later when students recognize the need to decompose a hundred or a ten to get more tens or ones.

This is the first time students experience the True or False? routine in grade 3. Students are familiar with this routine from a previous grade, however, they may benefit from a brief review of the steps involved.

Launch

  • Display one equation.
  • “Give me a signal when you know whether the equation is true and can explain how you know.”
  • 1 minute: quiet think time
Teacher Instructions
  • Share and record answers and strategies.
  • Repeat with each equation.

Student Task

Decide if each statement is true or false. Be prepared to explain your reasoning.

  • 4×5=5×44 \times 5 = 5 \times 4
  • 125+200=200+125125 + 200 = 200 + 125
  • 300−100=100−300300 - 100 = 100 - 300
Solution Steps (3)
  1. 1
    Evaluate 4×5=5×4
    →4×5=20, 5×4=20 → TRUE (multiplication commutes)
  2. 2
    Evaluate 125+200=200+125
    →125+200=325, 200+125=325 → TRUE (addition commutes)
  3. 3
    Evaluate 300-100=100-300
    →300-100=200, but 100-300 impossible (not enough) → FALSE (subtraction does not commute)

Sample Response

  • True: Both products are 20. It doesn’t matter if we change the order of the factors, as the product is the same.
  • True: Both sides of the equal sign add up to 325. Rearranging the numbers that are added doesn't change the sum.
  • False: We don’t have enough on the right side to take away 300.
Activity Synthesis (Teacher Notes)
  • “What is different about the last equation?” (If we switch the order in subtraction, then both sides of the equal side aren't the same. If we switch the order when we subtract, we don't get the same number.)
  • Consider asking:
    • “Who can restate _____'s reasoning in a different way?”
    • “Does anyone want to add on to _____'s reasoning?”
    • “Can we make any generalizations based on the statements?”
Standards
Building On
  • 2.NBT.7·Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
  • 2.NBT.B.7·Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
Building Toward
  • 3.NBT.2·Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
  • 3.NBT.A.2·Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

15 min

20 min