# Valid Equalities? (Part 2)

Alignments to Content Standards: 3.OA.B.5

1. 4 x 5 = 20
2. 34 = 7 x 5
3. 3 x 6 = 9 x 2
4. 5 x 8 = 10 x 4
5. 6 x 9 = 5 x 10
6. 2 x (3 x 4) = 8 x 3
7. 8 x 6 = 7 x 6 + 6
8. 4 x (10 + 2) = 40 + 2

## IM Commentary

On the surface, both tasks can be completed with sound procedural fluency in addition and multiplication. However, these tasks present the opportunity to delve much more deeply into equivalence and strategic use of mathematical properties. These tasks add clarity to the often misunderstood or neglected concept of equivalence. Students often understand the equal sign as the precursor to writing the answer. Class discussion should be carefully guided to ensure that students come to the understanding that the equal sign indicates equivalence between two expressions. Though these tasks can be completed by evaluating each expression on either side of the equal sign, they present deliberate next levels of reasoning that invite students to look for different approaches.

Anyone facilitating a conversation about this task should constantly ask, "Is there another way to know whether this equation is true?" Consider 5 x 8 = 10 x 4. Students will likely know these facts relatively quickly and come to the conclusion that both sides are equal to 40, thus this equation is true. When pressed to see other options, students may reason that the 8 can be broken down into 4 x 2. The equation becomes 5 x (2 x 4) = 10 x 4. Through the associative property, this becomes (5 x 2) x 4 = 10 x 4. We can see that these expressions are equivalent because we know that 5 x 2 has the same value as 10. The same opportunity presents itself in part f. Part g presents an opportunity for students to think critically about the meaning of multiplication.

Third graders interpret multiplication as equal sized groups. Students might reason that 8 x 6 means 8 groups of 6. Thus 7 x 6 + 6 would mean 7 groups of 6 with another group of 6. Students might recognize that extra 6 as the "8th group of 6," thereby making the two expressions equivalent.