Scott Baldridge

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:

1. 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.

1. 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

The sentence “3 girls + 2 boys = 3 kids + 2 kids” is not a number sentence.

First, let’s recall the what a number sentence is.  For that, we have to briefly describe numerical expression. (See “What is a variable? Part II: Expressions.”)  A numerical expression is an algebraic expression involving only numbers (no variables) that evaluates to a single number.  More generally, it is an expression that denotes a single quantity (see “What is a Quantity“).

A number sentence is a statement of equality between two numerical expressions.

In mathematics, a number sentence is a complete thought and must be either true or false. That means that both numerical expressions must be valid expressions (not 3/0, for example) and that the statement can be clearly and unequivocally assigned a truth value.  Undetermined sentences, even if they contain numbers, are not number sentences.

Back to: 3 girls + 2 boys = 3 kids + 2 kids.

This sentence is undetermined and cannot be given a truth value of either true or false.  To see why, ask, “Given 3 kids and 2 kids, would I know exactly which kids are girls and which are boys?”  It may be that, of the 3 kids and 2 kids, 4 of them are girls and 1 is a boy.  I don’t know whether “3 kids + 2 kids” means “4 girls + 1 boy” or “3 girls + 2 boys,” so I can’t tell whether the sentence is true or false.  The use of the = symbol in a number sentence should always be used for definitive statements about numbers or quantities—that all equations involving only numbers or quantities have well-defined truth or false values.  By starting with this premise in 1st grade, and maintaining the same level of precision in grades 1-8, students develop one coherent picture of the use of the = symbol from the very beginning.

Maybe you read the previous paragraph and registered the thought, “Well, a word problem gives context to the sentence.”  This is absolutely true.  But the beauty of a mathematical number sentence involving the = symbol is that it is always true or always false regardless of any surrounding context.  We are trying to build that sense of beauty and absolute resoluteness in the minds of our students as they progress through the elementary grades.  The binary true-false nature of number sentences is crucial for students to make sense out of the meaning of a solution set in algebra—in fact, much of algebra basically rests on the meaning and use of the = symbol and true-false number sentences.

Still not convinced?  Here is a pedagogical reason to reconsider for first graders: too much is going on in that (non-number) sentence.  If we ignore the main point above for the moment, and say that “we are just renaming units from boys and girls to kids,” then, mathematically, the sentence above is like the following (non-1st grade) true number sentence,

3 x (7/1) + 2 x 700% = 3 sevens + 2 sevens,

since 7/1 and 700% are two ways to name seven.  In this number sentence you can see that there is a lot to digest since you have to recall several definitions at the same time.  While the boys-and-girls sentence is much simpler, we should be careful to view the original sentence through the eyes of a first grader and how they might perceive it (which may be more like the more complicated sentence for us).  However, that doesn’t mean you can’t have a discussion about number sentences with first graders.  Excellent first grade teachers have told me that first graders can handle a discussion about the difference between a sentences with numbers and a number sentence, and I believe them.

CHANNEL: Engineering School Mathematics

Does Textbook School Mathematics really exists?  Absolutely!  Can you spot why the equations in the “Follow the path” question in the picture lead students to conclude that the equal sign means “compute” rather than its true meaning? Why is this wrong? I look forward to your answers below!

Hint: What does the equal symbol mean?

CHANNEL: Engineering School Mathematics

This post is a continuation of “What is a variable? Part I: Symbols.”

We can now precisely define the terms algebraic expression (both rational and polynomials), and numerical expressions in terms of constants (numbers), variables, operators, and functions.

For example, the expression ((3×x)×y)+5 is recursively built by noting:

1. 3, x, y, and 5 are all expressions.
2. (3×x) is an expression by placing 3 and x in the blanks of a multiplication symbol.
3. ((3×x)×y) is an expression by placing (3×x) and y into the blanks of another multiplication symbol, using parenthesis () to keep track of the order of we used to create the expression.
4. (((3×x)×y)+5) is an expression by placing ((3×x)×y) and 5 into the blanks of an addition symbol.

In practice, of course, teachers and students often skip the building-up process and just write down the expression ((((3×x)×y)+5)), using parentheses to indicate the order in which the expression was created.  Unfortunately, the seemingly squirrelly topic of “order of operations” actually hides this absolutely clear building-up process from students, often causing confusion for students.  Textbook School Mathematics exacerbates this problem with its emphases of “order of operations” as a set of rules to be memorize.   On the other hand, parentheses can quickly become quite annoying for expressions with lots of symbols, like

((((((x+3)×3)×(2+3))×4)×5)×5),

so any reduction in the use of parentheses afforded by “order of operations” and other tricks (like using 3x instead of 3×x) is welcome relief.

In school, there is a curriculum sequence building up to the full definition of algebraic expression.  Initially (grade 7 and below in the CCSS), students are basically working with algebraic expressions that are built out of the four operators, possibly with some simple exponentials (square, cube, etc.) included.  In grade 8, students learn the rules of exponentials and increase their knowledge of algebraic expressions to include any integer exponents.  Also in grade 8, students begin to study simple fractional exponents (using the square root and cubed root symbols).  In Algebra I and more extensively in Algebra II, students are introduced to rational expressions (algebraic expressions generated only by the four operators, or equivalently, the four operators together with exponentiation by an integer power).  Algebra I students study an important subset of the rational expressions—polynomials (expressions generated by numbers, variables, addition and multiplication).    Two years later, in Algebra II, students work with the full definition of algebraic expression when they study rational expressions as well as expressions involving nth roots.

As the term expression grows in meaning, so does the term, numerical expression.  In fact, it usually precedes algebraic expression.  Numerical expression is introduced and/or the idea is used from 1st grade onward:

For example:

•  “3” when evaluated is the number three.
• “3×(4+17)” when evaluated is the number sixty-three.
• “3+” is not a numerical expression, nor is “3÷0”.
• In grade 11, “4 sin(30)” is a numerical expression.

Equivalent Expressions

A numerical expression is a way to represent or “express” a number.  Algebraic expressions containing variables are incomplete thoughts in the sense that they are waiting for numbers to assigned to the variables (after which they become numerical expressions).  For example, x+3 is not a number until we substitute a number, say 5, in for x.  The result then denotes a number, in this case 8.

It is easy to tell when two numerical expressions are equivalent—just evaluate both of them to see if they are the same number.  Two expressions are equivalent if, whenever the same number is assigned to the corresponding variable in each expression for all variables, then the resulting numerical expressions are equivalent.

We investigate equality of two expressions in Part III.

CHANNEL: Engineering School Mathematics

The Common Core State Standards (CCSS) use the term “proportional relationship” throughout the standards.  Fortunately, the standards clearly describe what is meant by that term.  To put the definition into proper context, we need some background terms.  Based upon the definitions in the Appendix of the Ratio and Proportional Relationships Progression:

• A ratio is an ordered pair of numbers, A:B, which are not both zero.
• The value of a ratio A:B is the quotient A/B as long as B is not zero.
• Two ratios A:B and C:D are equivalent if there is a number, c, such that C=cA and D=cB.
• (Easily inferred from the Progressions document)  A ratio relationship is a collection of ratios that are mutually equivalent to each other.

A ratio A:B determines a ratio relationship C.  For example, if C is the ratio relationship determined by 2:3, then

C = {X:Y such that X:Y is equivalent to 2:3}.

In particular, C is the set {2:3, 4:6, 6:9, …} while 2:3 is a ratio in that set.  Adults often blur the distinction between a ratio and a ratio relationship; we often use the ratio 2:3 to refer to the ratio relationship.  Thus it is important for teachers to make the distinction between ratio and ratio relationship until students can derive which meaning is being used in the context on their own.  Note: Students learn what a ratio relationship is first, and use that knowledge to understand its sister term proportional relationship.

Notice that the set C is also equal to the set:

C = {x:y such that y = (3/2)x},

where k = 3/2 is called the constant of proportionality.  When a ratio relationship is thought of in this way it is called a proportional relationship.

A set of ordered pairs is a proportional relationship if there is a number  so that every ordered pair  in the set satisfies the equation .

Finally, let’s compare the use of the terms “proportion” and “proportional” in the CCSS and the progression documents.  The term proportional by itself sets up a proportional relationship.  Here are some examples of how that is done:

• It specifies a ratio A:B, which can then be used to specify a proportional relationship.
• It specifies a the constant of proportionality, which then determines a proportional relationship.
• It doesn’t specify a ratio or a constant of proportionality, but does state that a proportional relationship exists.  For example, “The number of total number of candies C is directly proportional to the number of boxes b.“

A proportion, on the other hand, is an equation stating that the values of two ratios are well-defined and equal.  Like all equations, a proportion involving only numbers (no variables) can be either true or false.  So, thinking of ratios, 2:3 and 3:5, the proportion “2/3=3/5” is a perfectly acceptable false proportion!  In solving problems, however, we are interested in finding solutions to equations, that is, finding the value of x for which 2/x = 3/5 is true.

CHANNEL: Engineering School Mathematics

A better initial question is: What is a symbol, and what are the types of symbols used in mathematics?  A symbol is an object, a mark, a sound, etc. that represents an idea, a process, or a physical entity.  Simple examples include stop signs, clock faces, company logos, the music used for the olympics games, badges, red roses for love, etc.

Symbols in mathematics are usually marks on paper, or special characters or pictures on computers.   One of the most important set of symbols in mathematics are numerals, which are symbols for numbers.  For example, the numeral “3” stands for and names the idea of “threeness” (for mathematicians: “threeness” is the equivalence class of all sets that have the same order as the set {*,**,***}).  You could also use three tick marks, | | |, to denote that number, but denoting the number 3,431 using tick marks would be very unproductive!

There are three important types of symbols used in mathematics: constant symbols, placeholder symbols, and operator symbols.

• Constant symbols.  These symbols denote numbers, as in

 0,   1,   2,   3,  4,  5,  -12,  3.14,    π,  .

The key for understanding constant symbols is that they specify a unique number or   object.  For example, if we say, “Let  stand for the number 3,” then  is being used as a constant symbol—not as a variable or placeholder symbol.

• Placeholder or variable symbols.  Letters such as , , and  are often used as placeholders for where we might expect to see or place a number.  (Note that ONLY 1 number is expected—a variable, contrary to popular belief, does not “vary.”)

A variable symbol is “waiting” to be supplied with a unique number.  Context is everything in distinguishing between variables and constants.  By analogy, the word he in, “He is the President of the United States,” is like a variable.  With no context or clarifying sentences  surrounding this sentence, we cannot determine who “he” is, and so we cannot determine if the sentence is true or false (determining when a sentence is true or false becomes very important in later posts on variables).  However, a little context and the word he becomes a constant symbol, “Look son, there goes Barack Obama.  He is the President of the United States.”  The word he is no longer a variable.

• Operator symbols.   These symbols include the addition symbol “+” and the multiplication symbol “×”.  There are usually one or two blank spaces associated to operator symbols in mathematics.  For example, the addition symbol in a mathematical sentence is really  “___ + ___” with the two blank spaces to be filled in with constants or variables.   Here are some other familiar operator symbols:

__ – __      |__|      -(__).

Maybe surprisingly, the division symbol is not considered an operator symbol in formal  mathematical logic because not ALL numbers can be placed in the blank spaces associated to the division symbol.  For example, placing a 0 in the blank of 3÷___ is not allowed.  However, the division symbol is defined in terms of the multiplication symbol, i.e., when ,  a÷b=c if and only if a=b×c.  In school mathematics, it is customary to include the division symbol in the list of operators with the following caveat: the set of numbers that can be substituted must now be specified (like, 3 ÷ x for all x ≠ 0).

Operator symbols are also called function symbols—they are, in fact, shorthand notation for functions (recall that a function is a rule that assigns a unique output of the range set for each input or inputs from a domain set).

Next Up:  Part II: Expressions.

CHANNEL: Engineering School Mathematics

Well….it makes sense that “3 liters” is a quantity, but what is it really?  Suppose you had a bowl of applesauce.  That’s a quantity too.  There may be 3 liters of applesauce in the bowl.  But when we refer to a quantity are we referring to the “3 liters” or the actual applesauce in the bowl?

Answer: (You guessed it) Depends upon whether we are thinking colloquially or mathematically.

In common speech, I can certainly have a quantity of applesauce, maybe to share at a Thanksgiving dinner perhaps.  But quantity in mathematical modeling has a precise, non-collquial, meaning.  It is the precision of the definition that allows us to generalize quantity to far more interesting objects.  (And unfortunately, Textbook School Mathematics blurs the uses of the word together, hoping the colloquial definition is “good enough,” which it isn’t.)

Some of these interesting objects appear in school mathematics.  While this post is not about rates per se, let’s point out an interesting dilemma about rates that shows that quantity isn’t a definition that can be easily glossed over.  Clearly, a rate should be a quantity.  If you are walking 3 mph and you increase your speed by 3 mph to a run, you are now running at 6 mph.  But what is the unit, “miles per hour?”  It’s easy enough to compute with, for example, if you walked 6 miles in 2 hours at a constant speed, then you’ve averaged (6 miles)/(2 hours) = 3 mph during that trip.  But unlike the physical distance of 1 mile, there is no physical way to divide the length given by 1 mile by the time it takes for 1 hour to pass, so there is no physical unit “miles/hour.”  If mph is not a physical unit, then what is it and why do we have no problems calculating with it?

The answer is that a quantity, when using it in a mathematics, is an element of a set that models a type of measurement.  Model is a keyword for it implies that the things we are measuring do not have to physically exist.  Also, the model can handle very large or very small quantities that are not easy to observe physically, like an object that is 12,022,403 miles long.

Now we are ready to get precise.  We base our definition on the work of Hassler Whitney, who described the idea of quantity in two articles in the American Mathematical Monthly some 50 years ago (Article 1 and Article 2).  We encourage you to read these two articles to see how he puts the definition of quantity on a solid mathematical foundation.  Here is a brief, non-technical summary:

• A quantity structure is a set that models a type of measurement and satisfies certain properties (see Hassler Whitney for descriptions of the properties).  The quantity structure that is the set of all lengths is an example.  Other examples: the set of all volumes, the set of all masses (weight), the set of all times, etc.    Note the term “model” in the first sentence—we are not talking about physical measurements, just a model of such measurements. For mathematicians: The simplest quantity structure is mathematically the set of real numbers thought of as an ordered, 1-dimensional real vector space with R multiplication. (See Hassler Whitney for details.)

• A quantity is an element in a quantity structure.  For example, in the set of all lengths, the quantity represented by “3 meters” is an element of that set.  Note that the quantity is independent of how it is measured, so that same quantity can also be represented by 300 cm.

• A unit in a quantity structure is just a choice of a (usually positive) quantity that all other quantities are compared to.  Examples include cm, m, in, ft, liters, sec but these are just standard choices, any choice will do.   For mathematicians: a choice of a unit is really a choice of a basis element in the vector space.

• With a choice of a unit, all other quantities can be represented by a number times that unit.  Therefore, “3 meters” is a representation of a length quantity.   The description, “3 meters,” is short for “3 x (1 meter).”  The numerical part of this expression is 3 and the unit part of the expression is 1 meter.

Note: the “type of measurement” in the definition of quantity structure above is not as important as the fact that the set satisfies certain properties.  It is the properties that define the quantity structure.  Therefore, since the set of all velocities satisfies the properties outlined by Hassler Whitney, we can rightfully work with velocity as a quantity structure and think of 5 mph as a quantity.

We end with this question:  Can numbers be thought of as quantities?

CHANNEL: Engineering School Mathematics