This past week Long-View mathematicians were presented the following problem:
9 – 3 ÷1/3 + 1
Do you know the correct answer?
This expression went viral in Japan a few years ago and only 60% of Japanese 20-year-olds could solve it correctly.
So why is it that most of the Long-View students solved this problem correctly (during the 6th week of school, no less)? Long-View kids know to look at a complex expression and evaluate the “terms.” Because of this understanding, the kids did not simplify the above expression left to right, or attempt to recall a silly mnemonic like “Please excuse my dear Aunt Sally.” Instead, they read the expression, knew how to analyze the structure of the expression through the lens of “terms,” and correctly simplified it.
As you may recall, a “term” is a mathematical expression that may form a separate part of an equation or of another, more complex, expression. “Terms” are typically not introduced until kids are in Algebra I. At Long-View, however, an understanding of the idea of terms is a part of the earliest lessons Long-View mathematicians have, and the idea of “terms” is integral to much of the mathematics work we do daily.
Regardless of a child’s age level or experience level, the first lesson Long-View kids experience is to learn that as mathematicians, we must always evaluate what is being counted (or numerated) and also evaluate how many of the quantity we have. For example, when our students see the numeral 5, they first think, “What am I counting?” (ones) and “How many do I have?” (five). We notate this as 5(1) and read it as “5 ones” or “five of one.” This is the Multiplicative Identity Property at work.
Once students understand the idea that they are always counting something, we move to evaluating an expression such as 5 + 3, which we read as “five ones and three ones” and notate as 5(1) and 3(1). In evaluating this expression, a Long-View child would immediately notice that we have two “terms,” as they know addition and subtraction symbols separate terms. Thus, a child might say, “I know we are counting ones in the first term and we have 5 of them and in the second term we are counting ones, and we have 3 of them.” Additionally, he or she would know that because we are counting the same thing in both terms, we have “like terms” and therefore can combine these terms.
This is very powerful understanding for a young child and much more effective as an early arithmetic lesson than just left to be introduced for the first time in an Algebra lesson. Our children can read an expression, and know that simplifying an expression isn’t merely a means of just enacting a series of rote procedures, but rather evaluating the terms of an expression, seeking like terms, and then combining. Their very understanding of arithmetic is directly tied to the way they will be asked to think in a formal Algebra course, as our proprietary mathematics curriculum (The Number Lab) unifies the concepts of arithmetic and Algebra, providing a bridge that supports what we call “number reasoning.”
After the early lessons described above, Long-View mathematicians would then be presented with an expression such as:
5(1/2) + 3(1/2)
Again, our kids would immediately notice that we have two terms and they are “like terms,” as we are counting the same thing in each term (we are counting halves). Thus, a young mathematician could combine these two terms and simplify this expression to 8(1/2), or utilize the Substitution Principle to simplify to 4(1). As a next step, we then might present the child with an expression that does not have like terms, such as:
3 + 4(1/2)
In this expression, we have unlike terms (because we are counting ones in the first term and halves in the second term) and thus we must first create like terms before we can combine them.
This series of lessons assists students in constructing the understanding that the act of combining number values is only possible when like terms are present.
Just to review, “terms” are parts of an expression and separated by “+” and “–“ signs. A term is either a single number or variable, or numbers and variables multiplied together, that are separated by “+” and “–“ signs. Additionally, it’s important to remember to use the word “expression” correctly. If there is not an “=” sign, then we will use the word “expression” as opposed to “equation.”
Now that we have a solid understanding of terms, let’s go back and take a look at the expression presented in that viral Japanese video:
9 – 3 ÷1/3 + 1
Long-View students would first identify the terms in the expression. They would underline each of the three terms and circle the separators. They would then tell you that the first term is counting ones and there are nine of them. In the 2nd term, they would tell you that three is to be divided into groups of one third. Finally, they would determine that the 3rd term is also counting ones and that there is only one.
9 – 3 ÷1/3 + 1
=9 – 9 + 1
=0 + 1
And there you have it…we are ready for a video to go viral of a Long-View child simplifying this math expression!