Why 5 x 3 = 5 + 5 + 5 Was Marked Wrong
Viral Math Problem Explained
A lot of you have been asking about WHY this viral math problem is marked incorrect. And that’s a great question! It seems a little absurd, doesn’t it?
Equals Versus Equivalency
Just because two things are equal doesn’t mean they’re equivalent.
Equal is defined as, “being the same in quantity, size, degree, or value.” Whereas equivalent is defined as, “equal in value, amount, function, or meaning.”
In the above problem 5 x 3 is equal to 5 + 5 + 5, but they’re not necessarily equivalent. Equivalence relates to meaning, so it depends on the meaning of multiplication, as the directions indicate.
I did something today I’ve never done before,
I looked up the definition of multiplication.
And as I suspected in the definition of multiplication, the first factor is is the number of copies and the second is the number being repeated.
So by definition, 5 x 3 is equivalent to 5 copies of 3, or 3 + 3 + 3 + 3 + 3 as the teacher marked. It is equal but not equivalent to 5 + 5 + 5 because 3 copies of 5 represents something different.
For example, 3 bundles of 5 bananas is different from 5 bundles of 3 bananas although they total to the same number of bananas. Their structures are different.
Here’s another example: 30 ÷ 2 is equal to 15. But does 30 ÷ 2 represent multiplication? Is it equivalent to repeated addition?
No, it represents division. Therefore, 5 + 5 + 5 is equal to 30 ÷ 2, but they are not equivalent.
I get the difference, but isn’t it a little harsh?
It depends. If the teacher has already taught the commutative property of multiplication (the law that says a x b = b x a), then this is a fine substitution to make.
If the teacher has not covered the commutative property, then it might be unwise to let a student continue with this line of thought.
Why?
It’s common for beginners to get confused as to when it is okay to switch the order of values in binary operations.
We know that the following are not equal.
But this is easily confused with the child who sees that sometimes it’s okay to switch the order and other times it’s not. By focusing on the meaning of these operations and relating them to their context, teachers are trying to prevent students from making these kinds of easy mistakes.
But It’s the Right Answer,
why does it matter if the meanings are slightly different?
It’s more important than ever for students to understand the difference between equal as a result and equivalence in meaning from a young age because it is a fundamental computer science concept.
In programming, there is a distinction between testing if two things are equal or equivalent (aka identical).
Equal means they have the same value, like 5 + 5 + 5 = 30 ÷ 2 = 15. Equivalent means not only are they equal, they are also of the same data type. In other words, they mean the same thing.
Depending on the language, numbers and expressions that look the same don’t always mean the same.
For example, in JavaScript if we test for equality with the == operator:
- “4” == 4 returns True
because both are referring to the number 4. But if we test for identity using the === operator:
- “4” === 4 returns False
because they mean different things. The first is a string whereas the second is a number, therefore they are not the same.
(Note: for a more in depth discussion on == vs === in JS, check out this stack overflow discussion.)
The Right Mindset for Matrix Multiplication
Notice that the second problem is marked incorrect as well. Why is it important to have 4 rows of 6, instead of 6 rows of 4?
Not only is this how multiplication is defined, it teaches students the correct order for diagramming matrices which is rows times columns.
Keeping rows and columns straight in matrix multiplication is vital. Matrices are labeled using a row by column notation, n x m.
To multiply matrices together, you multiply the rows of the first matrix by the columns of the second. The number of columns in the first matrix must equal the number of rows in the second, or else you cannot multiply them together.
For example, we can multiply a 2x3 matrix and a 3x4 matrix together. But if we swap the order and multiply a 3x4 matrix with a 2x3 matrix, there will not be sufficient rows and columns, and the operation cannot be performed.
It is crucial students learn the correct definition of multiplication and diagramming so they don’t get confused later on.
I know it’s frustrating but
Respect the Teachers
They are qualified experts on child education. They have the best intentions for the students in mind. This teacher made a decision based off a lot more information about the student and class setting than we can tell from a photo. We don’t have to agree with it, but we can respect it. If you are confused, ask them why they did something before you slam and discredit them on the internet.
Thanks for reading!
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