# Valid Equalities?

## Task

Decide if the equations are true or false. Explain your answer.

- $2+5 = 6$
- $3 + 4 = 2 + 5$
- $8 = 4 + 4$
- $3 + 4 + 2 = 4 + 5$
- $5 + 3 = 8 + 1$
- $1 + 2 = 12$
- $12 = 10 + 2$
- $3 + 2 = 2 + 3$
- $32 = 23$

## IM Commentary

The purpose of this task is to help broaden and deepen students understanding of the equals sign and equality. For some students, an equals sign means "compute" because they only see equations of the form $$4 + 3 = 7.$$ In this task, students must attend to the meaning of the equal sign by determining whether or not the left-hand expression and the right hand expression are equal. This task helps students attend to precision (as in Standard for Mathematical Practice 6).

Many of these problems naturally lead to discussions of topics such as place value and properties of addition.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This particular task helps illustrate Mathematical Practice Standard 6, Attend to precision. Students must use mathematical symbols correctly and describe the meaning of the symbols they use. In this case, they understand that the equal sign denotes that the quantities on either side have the same value and use this understanding flexibly to identify and express equivalences. When crafting their explanations, they learn to communicate their reasoning by using precise mathematical vocabulary describing each quantity accurately. Even though the main focus may be attending to precision, this task also supports MP.3, Construct viable arguments and critique the reasoning of others as well as MP.7, Look for and make use of structure.

## Solution

While the question asks for simple "true" or "false" answers, complete solutions include some valid explanation. There are many possible explanations, so we give a variety of kinds of explanations in these solutions.

- False. $2+5$ equals $7$ and not $6$.
- True. Both sides equal $7$.
- True. Since $4 + 4 = 8$, $8 = 4 + 4$.
- True. We can combine the three and the two on the left to get $5$, and then after reordering both sides are $4 + 5$.
- False. $3+5$ is 8 but $8 +1$ is 9.
- False. $1 + 2= 3$, which is less than $12$.
- True. If you count up two from $10$ you get $12$. (Alternately, $12$ means one ten and two ones.)
- True. You can always change the order of numbers being added.
- False. $32$ is 3 tens and 2 ones. $23$ is 2 tens and 3 ones.

## Valid Equalities?

Decide if the equations are true or false. Explain your answer.

- $2+5 = 6$
- $3 + 4 = 2 + 5$
- $8 = 4 + 4$
- $3 + 4 + 2 = 4 + 5$
- $5 + 3 = 8 + 1$
- $1 + 2 = 12$
- $12 = 10 + 2$
- $3 + 2 = 2 + 3$
- $32 = 23$