Equals signs

$$== \: \: \: \: \: = \: \: \: \: \: ==$$

The classic equals sign question

What number would you put in the box to make this a true number sentence?

$$8+4 = \square + 5$$

Elementary student responses

Source: Thinking Mathematically

Percentage of various responses per grade level

 Response 7 12 17 12 and 17 Grade 1st and 2nd 5 58 13 8 3rd and 4th 9 49 25 10 5th and 6th 2 76 21 2

The correct answer is certainly $7$,

but $12$ is not exactly an unreasonable response

Introducing $x$

What's the difference between $$8+4 = \square + 5$$ and $$8+4 = x + 5?$$

Not much that I can see.

A systematic approach

to solving $x+5=12$

Subtract $5$ from both sides.

If $x+5=12$, then $(x+5)-5=12-5$ so $x=7$.

Relational thinking

In relational thinking, we consider the relationship between quantities. Often, we can solve algebraic expressions without doing significant computation.

Example

What value of $x$ makes the following statement true?

$$123-115 = 54-x$$

Sometimes, we simply think about whether an expression is true or false.

Examples

• $3-8 = 42-47$
• $2x-7x = x-6x$
• $2x-7 = x-6$
• $2x-7 = 2x-6$

Computer languages

$=$ vs $==$

Set a variable

Javascript uses = to set a variable to a value.

In :
x = 2;
x
Out:
2
In :
x + 3;
Out:
5

Test against a value

Javascript uses == to check if a variable has a certain value. The result is either true or false.

In :
x == 2
Out:
true
In :
x + 3 == 4;
Out:
false

You don't have to take my word for any of this!

A mathematical example with two different equals signs

Let $x=3$. Then, $x+7=10$.