Answer: No, if you mean by the = symbol, “every inch corresponds to 13 feet.”
It cannot be stressed enough how important it is that the “=” symbol be used correctly. It is one of the most important (if not the most important) symbols used in all of mathematics. The meaning and use of this symbol in number sentences should not vary, even in the slightest, over the 14 years students come in contact with it. When we use this symbol to mean other things in number sentences, like setting up a scale factor by stating “1 inch = 13 feet,” we dilute the real meaning of the symbol that we are so disparately trying to get students to understand. In fact, part of getting students to understand the meaning of the “=” symbol is done by using it correctly throughout PK-12—it takes a long time for the precise meaning of this symbol to sink in and become “rigid” in students’ minds.
Here’s a little background about how the “=” signed is used when writing number sentences (and how it is used in mathematics in general). First, the “=” symbol translates into “is.” It literally is the verb in all number sentences. For example, “3+4 = 7,” translates into “Three plus four is seven.” In English, the verb “is” can sometimes be used to set up sentences that are either absolutely true or absolutely false.
- The Earth is a planet. (True.)
- The domesticated cat is a member of the canis lupus species. (False.)
However, most English sentences that use the word “is” cannot be given truth values:
- The sun, seen from the surface of the Earth, is yellow. (True or False depending on the time of day.)
- The car is fast. (This statement is subjective.)
In mathematics, things are different. The “=” is always used in number sentences or number sentences with quantities to make statements that have well-defined truth values. The following examples show correct ways to interpret the “=” sign:
- 3+4=7. (True.)
- 2+2=5. (False.)
- 100 cm = 1 m. (True.)
- 1 inch = 13 feet. (False.)
As you can see from the examples above, the use of the “=” sign is actually much broader than “a symbol used to state facts.” The equal symbol in a number sentence is used to set up well-defined assertions about two numbers or quantities. If the assertion made by the number sentence is true, then it is a fact. For example, in grades 1-5, asking in a problem for students to, “Write a true number sentence for…,” is better than, “Write a number sentence…,” because it emphasizes the truth value of the number sentence. Of course, we are mostly interested in writing down facts (true assertions) and so we don’t usually make a big deal about the truth value of a number sentence every time we write one, but the truth value is still there lurking in the background and teachers should always be aware of it.
Let’s return to setting up a scale factor. In mathematics, the statement, “1 inch = 13 feet,” has only ONE interpretation: It is a false assertion about two quantities. Any other interpretation of that number sentence is a misuse of the “=” symbol. Again, misusing this symbol puts all of mathematics on wobbly ground for students. Here are two more examples that misuse the symbol—one that is obvious and the other isn’t so obvious:
- Wrong use of the = Symbol: 3+4+5+6 = 7+5 = 12+6 = 18.In this case, the “=” is being used wrongly to mean “compute,” that is, “3+4 computes to 7, then 7+5 computes to 12, then 12+6 computes to 18.” (Old calculators use to reinforce this notion because the compute button was labeled with an “=,” now many of them label the compute button with “Enter.”) The correct way to interpret the number sentence above is as the following false assertion: 18 = 12 = 18 = 18.
- Subtle wrong use of the = Symbol: 2 boys + 3 girls = 2 kids + 3 kids (See my post on this sentence). This sentence seems acceptable, after all, we are just converting boy and girl units into a “common” unit kids (yes, think “common denominator”). However, this conversion is not 1-to-1. For example, the expression “3 girls” converts into “3 kids” just fine, but “3 kids” may mean, “1 boy and 2 girls,” or, “2 boys and 1 girl,” or, “3 girls.” Hence the equivalent sentence, “2 kids + 3 kids = 2 girls + 3 boys,” may be true or it may be false. Since the truth value cannot be determined, this sentence isn’t a proper use of the “=” symbol.
Note that this example can be fixed easily by just using numbers, “2 + 3 = 5,” or by using kids, as in, “2 kids + 3 kids = 5 kids.” In general, stick to using units that have 1-to-1 conversions and this problem will never even come up. Here are some examples of some 1-to-1 conversions of quantities (and valid uses of =):
- 200 cm + 3 m = 2 m + 3 m,
- 1 kg + 292 g = 1000 g + 292 g = 1 kg + 0.292 kg,
- 3 tens + 2 ones = 30 ones + 2 ones,
- 3 fourths + 3 eighths = 6 eighths + 3 eighths.
In the next week or two, I will explain how the “=” sign is used in equations and why it is so important for understanding equations that number sentences have this special property of always having a well-defined truth value.
CHANNEL: Engineering School Mathematics
You state in the linked blog post “=” means “is”. What do you think about “=” means “the number on the left is the same point on the number line as the number on the right”? This formulation would run into problems, I think, when including units, as in your example 2.
The point of the article is that a number sentence involving the equal sign should always have a truth value (that is either true or false). I show examples of an equal sign that is being used incorrectly in a statement for two reasons: (1) invalid interpretation for the sentence, or (2) the statement does not have a well-defined truth value.
In the first case, “1 inch = 13 feet,” the only valid interpretation that can be given to the sentence is, “It is a false number sentence about two quantities.” It is a number sentence involving quantities; it is just a false one. Of course, what the user of such a sentence was really trying to do was state a proportional relationship, which can be correctly stated as, “The proportional relationship of inches to feet is represented by the ratio 1:13.”
The second case is subtler: the statement, “2 kids + 3 kids = 2 boys + 3 girls,” cannot be given a truth value, i.e., there is no interpretation where we can definitively say this statement is absolutely true or absolutely false.
That brings us to the heart of your question: how do we determine the truth value of a number sentence like ¼ = 0.25? One semantic answer is as you say: Locate the two numbers as points on a number line, and if they both represent the same point, then the number sentence is true. This interpretation is maybe a bit sophisticated for students in grades K-2, but starting in grade 3 (cf. CCSS 3.NF.A.3a) this does become an important way to recognize when two numbers are equal. Note: the fact that ¼ = 0.25 is true is nontrivial for kids at this age!
The properties of a number sentence and the interpretation used to determine the truth value of a number sentence are two different things. We really can’t mix the two ideas and this is probably what led to the confusion.
Finally, to other readers reading this response: The information in this response is just a small facet of the knowledge that curriculum writers need to know to construct a curriculum. The information isn’t presented here in a way that students in elementary school will see when using the Eureka Math curriculum (students are certainly not asked to talked about these issues at this level of sophistication). However, it is information that (adult) writers and teachers need to be cognizant of to make sure we present a coherent picture of mathematics for our students in all grades.
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“the fact that ¼ = 0.25 is true is nontrivial for kids at this age!”
Ain’t that the truth! Thanks for your thoughtful reply.
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I agree with your post. How would you suggest “1in=13ft” be written symbolically? My inclination would be to write define 1in:=13ft.
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Good question! I was hoping someone would ask it in the comments—I guess that means you win a prize? 🙂 I would avoid using the “='” symbol in this case. This is because we are setting up a proportional relationship, not an equation.
I’d say it using ratio language such as, “A plan of house is drawn on a scale of 1 inch to 13 feet,” or “A blueprint has a scale that is 1 inch to 13 feet.” You could also use rate language, “The blueprint scale is given by 13 feet per inch.”
BTW: The notation “:=” is usually (but not always) reserved for definitions or for naming an object by another name. So “1in:=13ft” is usually read as “1 inch is another name for 13 feet,” which it is not. The meaning of the symbol “:=” is a bit ambiguous in mathematics, so mathematicians usually specify the meaning of it first before using it in an article.
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