Task
Below is a table showing how to add numbers from $1$ to $3$:
Cut out the table and fold it over the dotted line. Notice that the blue squares match up and so do the orange squares. Notice that the squares that match up have the same numbers in them. We say that the squares that match up when you fold along the line are "mirror images" of each other.
The table below shows how to add numbers from $1$ to $9$. Two squares are shaded blue and two are green:

Are the blue squares mirror images of each other? Explain why the numbers in the blue squares are equal.

Are the green squares mirror images of each other? Explain why the numbers in the green squares are equal.

Shade the rest of the mirror image squares with the same color. Why are the mirror image numbers always equal?
IM Commentary
The goal of this task is to help students understand the commutative property of addition by examining the addition facts for single digit numbers. This is important as it gives students a chance, at a young
age, to do more than memorize these arithmetic facts which they will use
throughout their education. The approach taken here is a geometric one, associating the commutative property of addition with the symmetry of the
addition table. Students don't "officially" study symmetry until 4th grade (see 4.G.3), so this task defines the particular symmetry for them.
Two additional interesting aspects of the table worth noting are:

The first nine even numbers appear on the shaded diagonal. This comes
from the fact, for example, that $4+4 = 2 \times 4$ and so gives students
a chance to think about the meaning of multiplication. This is discussed
for a smaller addition table in the task "Addition Patterns'' but the teacher
might also prompt the students to address this here.

The symmetry of the table about the diagonal essentially cuts in half the number of facts which
the students need to memorize if they think flexibly about addition of single
digit numbers.
This task is mainly intended for instructional purposes. There will be
ample opportunities to test whether or not the students have learned their
single digit addition tables and the goal of this task is to facilitate the process
by revealing patterns and structure in the table.
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