44

The short answer to your question is: everyone is right. I agree with people here that in many contexts, $0.75$ or $\frac{3}{4}$ would be a more desirable answer than $\frac{45}{60}$. I also agree with several here that when the context is "out of 50 people", an unreduced fraction like $\frac{40}{50}$ makes perfect sense. Certainly, when I write ...


25

Coming from the perspective of someone who reteaches this material at the college level, neither the graph perspective nor the list of properties perspective really translate into a deep understanding later. For the former, all you get is an understanding that the graph goes up, but not a lot, and is too vague to really be useful. For the latter, they don't ...


21

In real-world applications, the typical case is that the domain is neither implicit in an expression we write down, nor explicitly stated along with the expression. Rather, one uses knowledge of the real world to decide what numbers make sense as inputs to the function. So for example, if $x$ represents an investment, and $f(x)=x-3$ represents the profit on ...


20

The answer to your question depends on the pedagogical goal of the exercise, and what learning outcomes you have identified. It basically comes down to the following question: Is manipulating fractions one of the skills which you are emphasizing in this class? If one of the goals of your class is to get students to more handily work with fractions, then ...


20

I am a GCSE Maths examiner. For a question like this, any correct equivalent decimal, percentage or fraction, whether simplified or not, would receive full marks. It is only specifically if it says in the question that the answer should be simplified that a simplified fraction is required to obtain the final accuracy mark. The reason for this is that the ...


15

How do you teach students about the operator $\sqrt[3]{}$? It's a similar operator in many ways; when dealing with cube roots I try to show them these things: $\sqrt[3]{x}$ asks "which base to the third power gives us $x$?" $f(x) = \sqrt[3]{x}$ is a one-to-one function, and $y=f(x)$ has a graph of a certain shape. $\sqrt[3]{}$ is the inverse ...


13

We give high-schoolers many different explanations of the word "function." Here are a few that are either implied or outright stated at various points in a student's education: A function is an expression in terms of $x$. This is pretty unusual all by itself, but it may appear in conjunction with (2) or (3) (i.e. students may reason that $f(x)$ or ...


12

I tell my students this story when this issue comes up: Imagine you are answering the phone at the local pizza place. Someone on the other end says "Yes, I'd like to place an order. I'd like twelve thirds pizzas." What would you think? They often come up with the following explanations: It is a prank call. Maybe I misheard them? Maybe they don't ...


9

In my high school days (1970s), the set of all numbers for which an expression is defined (and, implicitly, real) was called the “natural domain” of the function defined by the expression. The domain of a function with an unspecified domain was to be understood to be its natural domain, a convenient assumption. I’m not sure this is an accurate answer to why ...


8

Í am a physics teacher, not a mathematics teacher, but I would reward the mark. It was a multipart question and you did not show us the other parts. But when I formulate tests I try to have all parts independently of each other, so a mistake in a previous part does not impact the latter parts. Assuming that the parts are independent, the student does have to ...


8

i don’t think they are. In fractions there are (at least) 3 analogies: set (discrete objects), area (or volume), and length. Your 1st is set, 2nd maybe area (more like length) and 3rd is length. You could re-write #2 so that your filling a glass or jug or jar with something (or using 1/5 of an amount) But, i think they are all conceptually the same. a ...


6

This depends on when in the "fractions curriculum" this happens. If the children know that "a fraction of" really means to multiply by this fraction, then all problems are equal enough that one single student will be able to solve either all or none of them. If the problems are used to teach "fraction of is multiplication by", ...


6

I would start with explaining exponentials as repeated multiplication. For example, we look at the sequence $2^1=2$, $2^2=2\times 2$, $2^3=2\times 2\times 2$... and call it one 'two', two 'twos', three 'twos'. Then show the rule for multiplication goes like $(2\times 2)\times (2\times 2\times 2)$ and say that we find the total number of twos by adding up the ...


5

Let me start with an analogous question on phil-SE: Teacher: What is 2+2? Student: 2+2 is 2+2 What is wrong with this answer? Now it may seem to be an issue with using tautologies for communication. However as I point out law-of-identity/tautology is a red-herring to address this issue. To see that, let's change the exchange slightly: Q : What is 2+3 A :...


5

Instant intuition giver: Logarithm computes the number of digits. One could start by letting the students think how such a function should behave. One possible idea: That's just for whole numbers. Is there a way to interpolate? (Turns out yes, there is -> give the formal definition for decadic logarithm.) Another one: Different number systems. Binary ...


4

Although I am not a teacher, I would recommend going with the historical perspective. Logarithms were invented by John Napier (see https://en.wikipedia.org/wiki/Logarithm) basically to make multiplication easier. Instead of having to multiply several large numbers, you looked up their logarithms in the table, then only had to add them and consult the table ...


4

All of the math explanations here are very good. And they all use math to explain math. Sometimes it is helpful to understand the rationale; concept or the "Why was this invented / Why does someone need this;" or even the 'real world' explanation behand the math. I had a story I used to tell. It went something like this: Suppose you need to count ...


4

tl;dr: the answer should be given in the reduced form. Say there are min. 40 students who are taking the exam. Would you want to find the reduced version of the fraction $394.784176044 / 493.480220054$ 40 times? (assuming every time the answer will be given differently but of same complexity) Another issue is that the rational number has infinitely many ...


4

(Adding this because I want to - the first version of this answer is near the end after a horizontal line.) I think that the acceptable answers should succeed in communicating with whoever is reading the answer. And the context (who is reading, what the answer is supposed to tell to the reader,...) should be taken into account. Here $40/50$ is ok, ...


4

I then stopped for pause and admitted, maybe they're right, but I'm not sure if they would get the mark if they do not fully simplify the fraction or give the answer as a decimal. Life is not about getting points. A person who doesn't automatically simplify 40/50 to 4/5 (or 0.8) is being silly and annoying, or showing a lack of competence. This person may ...


4

At A-level (16+ in the UK) students are never asked questions such as "what is the domain of $f(x)=x^2$ in formal examination questions. In my teaching I mention that such a question is meaningless, the domain is part of the definition of the function and the function isn't fully defined until a domain is given. Question papers are careful always to ...


3

One point where C is different from A and B is that a distance can scaled (or split) by any fraction. So 20km can be split into 6 equal parts, even if the result cannot (yet) be represented by the numbers the students know. But 20 problems or 20 pages cannot be split into 6 equal parts. At least nothing simple comes to mind. In other words, C is a problem ...


3

Yes, I think your general understanding of the purpose is correct. In my view, it has four goals. For the weaker student, it gives them a chance to find the footing to start asking questions. For the average student, it gives them a chance to start making connections to prior knowledge. For the gifted student, it gives them a chance to hypothesize and ...


3

At the secondary level, students tend to think of "real numbers" as being literal: any number that isn't in the real number system isn't a "real number", and any time you're given a variable, the default is that it can be any real number, and the default is that a function applies to all real numbers. The idea of a function that takes ...


2

The test should indicate in what format the answer is expected to be given. This should be clear both to you and to the students. Otherwise grading becomes subjective. If you accept 40/50, would you accept another student giving 120/150, and another 468/585? The test covers basic math skills. Simplifying fractions seems to fit the overall material covered ...


2

Slide rules are a good tactile method of exploring logs. You can use them to calculate the $log_ba$ and $b^a$, and $a\cdot b$. They also visually illustrate the properties of logs, such as $\log(xy)=\log(x)+\log(y), \log(x^y)=y\log(x), \log_yx=\frac{\log_bx}{\log_by}$.


2

Your question is somewhat broad, and answerable, but this answer should not be considered as providing advice. Your question may have gone unanswered for two reasons. First, people may not know of the available resources for obtaining a copy of your book of interest for review. And second, nobody wants you to have a bad experience based on what happened ...


2

I think it would be more useful to teach logarithms as the inverse of exponents1, and compare most of the properties of logarithms with the properties of exponents, because most of the properties of logarithms are related to those of exponents, or those of other logarithmic properties. Here’s a list of common (as in often talked about, and not that ...


1

Log-linear and log-log graph paper can help visualization. (If paper is passé, you can find templates on the web.) Plot the same function on the different scales. Examples help. Familiar examples. E.g., the signal strength bars on mobile phones are decibels and are logarithmic. E.g., the Richter scale. It may not be so familiar unless you live in California, ...


1

The general purpose of simplification or writing an expression in some "standard form" is to be able to have a unique canonical answer, and thereby easily compare student work to an official answer for grading, to spot-check against an answer at the back of a book, to compare between two cooperating students, etc. So generally the protocol in a ...


Only top voted, non community-wiki answers of a minimum length are eligible