Another Addition Algorithm

10 min

Narrative

The purpose of this Warm-up is to elicit observations about patterns in the sums of two- and three-digit addends in an addition table. The table is partially completed to highlight some properties of operations. For example, the sums in the table can illustrate the commutative property (99+98=19799+98=197 and 98+99=19798+99=197). The numbers also prompt students to notice patterns in the sums of odd and even numbers. For example, the sum of an odd number and an even number is always odd.

While students may notice and wonder many things about the addition table, focus the discussion on the patterns in the table and possible explanations for them. When students make sense of patterns in sums and explain them in terms of the features of the addends and how they are added, they notice and use regularity in repeated reasoning (MP7).

Launch

  • Groups of 2
  • Display the table.
  • “What do you notice? What do you wonder?”
  • 1 minute: quiet think time
Teacher Instructions
  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Share and record responses.

Student Task

What do you notice? What do you wonder?

+ 98 99 100 101 102
98 197 199
99 197 199 201
100 ? ?
101 199 201 203
102 201 203
Solution Steps (4)
  1. 1
    Understand table structure
    →Addition table with headers 98, 99, 100, 101, 102 - adds row header + column header
  2. 2
    Notice patterns in filled cells
    →Sums go up by 2 each column/row, all sums are odd (even+odd), diagonals show commutative property
  3. 3
    Find first ? (row 100, col 99)
    →100 + 99 = 199
  4. 4
    Find second ? (row 100, col 101)
    →100 + 101 = 201

Sample Response

Students may notice:

  • The same numbers are across the top and along the left side.
  • Every other cell is filled.
  • There are two question marks in the table.
  • The sums go up by 2 from left to right, from top to bottom, and diagonally from upper left to lower right.
  • The sums that make a diagonal line from lower left to upper right are the same number.
  • All the sums are odd numbers.

Students may wonder:

  • What numbers go in the cells with a question mark?
  • Why do the numbers in the table go up by 2 from left to right and from top to bottom?
  • Why are the sums odd numbers?
  • Are there other patterns in the table?
Activity Synthesis (Teacher Notes)
  • “How do you think the table works?” (The table shows addition. Each number in the row at the top is added to each number in the first column on the left.)
  • Ask questions to invite students to explain some of the patterns they noticed in the table. For each question, give students a minute of quiet think time. Record their responses, with equations, if possible.
  • If students do not notice the following patterns, consider asking:
    • “What numbers would go in the cells with a question mark?” (199 and 201) “How do you know?”
    • “Why do the values in the table go up by 2 from left to right and top to bottom?” (Because of the skipping. The number being added in a row or a column goes up by 2, so the sum also goes up by 2.)
    • “What patterns do you notice with even and odd numbers in the table? Why do you think this happens?” (I notice all the sums in the table are odd. It’s because 1 addend is even and the other is odd. I notice that the empty squares should be even because they are the sums of adding two even numbers or two odd numbers.)
    • “What patterns do you notice in numbers that make a diagonal line going down to the right or going up to the right? Why do you think this happens?” (Going up to the right, the numbers are the same because 1 addend is decreasing by 1 and the other is increasing by 1. Going down to the right, it’s adding 2 each time because both addends are increasing by 1.)
Standards
Addressing
  • 3.OA.9·Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <em>For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.</em>
  • 3.OA.D.9·Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <span>For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.</span>

15 min

20 min