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 (  and ). 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).

• “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.)

• “What new units did you have to compose in this problem?”
• “Where would it make the most sense to you to record the newly composed 10? Where would it make the most sense to you to record the newly composed 100?”