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A US specific answer: The Common Core State Standards define $\frac{1}{b}$ by saying it is one of $b$ equal parts making up a whole $1$. $\frac{a}{b}$ is then defined as $a$ of these. Connecting $\frac{a}{b}$ to $a \div b$ requires some reasoning. For instance $\frac{5}{3}$ of a candy bar means you take your one candy bar, divide it into 3 equal sized ...


7

Lots of different fractions all represent the same rational number. So if someone hands you two fractions, it can be difficult to tell if they represent the same number or not. For instance, you might need to do some work to determine if the fractions $\frac{18399}{30665}$ and $\frac{20991}{34985}$ represent the same number. However, all rational numbers ...


7

I made a handout with 10 true false questions, titled Algebra Temptations. I've put it on google drive here. I have students work in groups to decide which are true. I think it makes a good activity, though there is the risk mentioned in another answer of reinforcing the wrong idea.


5

A key idea in maths education, that at least becomes more visible at university level, is that people tend to start thinking of new ideas as processes, but to do more advanced maths they need to move on to thinking of the same things as objects. That is, we are taught to think of $2+3$ as an instruction to take $2$ and add $3$ to it. Most people will write $...


4

A common example is that students think that $\sqrt{ab}=\sqrt a \sqrt b$ for all $a,b$; try tricking them by saying that $1=\sqrt{1}=\sqrt{-1\cdot -1} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$. Similarly students will not often realize that $f(x) = \sqrt{x^2}$ is NOT $x$, but rather $|x|$. Extraneous solutions (i.e. solutions that pop out when solving but are ...


4

By definition, a rational number is a number that can be expressed as the quotient of two integers. This quotient is called fraction and is written as $\frac{a}{b}$. Hence, division and fraction are the same, at least in the context of dividing integers and turning them into rationals. This concept is expanded in middle school for dividing irrationals and ...


4

The ratio $a:b$ and the fraction $a/b$ are generally not synonymous and should not be treated as such, at least without making clear their interpretations. In many contexts, $a:b$ corresponds to the fraction $a/(a+b)$ or $b/(a+b)$. For example, if one cuts a pizza into $6$ slices, one of which has anchovies, the fraction of slices with anchovies is $1/6$, ...


2

For kids, I always use the pie (or pizza) example, cutting it into pieces. This way it is very visual and easy to understand that 1/2 is the same as 2/4. With the fraction program you can do any "cutting" of a circle interactively. Just set 1/2 and then 2/4. (Disclaimer: I am the developer of the website, it is in German.) Following an example how you can ...


2

My mathematical teaching experience is limited to helping my kids with their homework, but have you considered getting her to draw the problems? Pizzas, chocolate bars, stickmen, whatever - depending on the context.


2

This is the only way I've ever seen it taught or read it described in textbooks - except in my classes. I just teach students to multiply the numerator and denominator of the first fraction by the denominator of the second and vice versa. No prime factorizations, least common multiples, etc. are necessary. This is based on the idea that you don't need the ...


1

Use a piece of paper as your whole. To teach $3/2 + 4/3$ do the following. Give each child/group of children 6 pieces of paper. One piece of paper should be left as a whole - the students can write 1 whole on the paper. Have the students fold 2 of the pieces in half lengthwise, label each half as $1/2$, and cut out the halves. Have the students fold 2 ...


1

I wouldn't necessarily call them "trap" questions, but I wouldn't be surprised if a vocabulary quiz on middle/high school math could be difficult or low scoring. Ask students to define or to give examples for "degenerate triangle", "extraneous solution", "rational number", "principle square root", "inverse function", "quotient", "numerator", etc. Even ...


1

Relation between division and fraction have to be understood to understand the relationship between fraction and ratio: 1. Division: finding how many times of divisor in the dividend, or the value of a part 1/5 means 5 is 0.20 times in 1; value of one part is 0.20. 2. Fraction: It is how many times the denominator in numerator or how many parts of ...


1

I have no teaching experience so I don't know the answer. Here's a guess on a method that will work. Take the problem "What is $\frac{3}{4} \times \frac{5}{6}?$" Then explain how once you know the answer to $\frac{3}{4} \times \frac{1}{6}$, you can then use that to calculate $\frac{3}{4} \times \frac{5}{6}$. Then you could ask "How do you calculate $\frac{3}{...


1

$\times n$ is an action. It moves each point on the number line to a different point. $\div n$ is the opposite action, meaning: "$\times n$ then $\div n$" doesn't move any points at all. It's the same as doing nothing. Let's start with an action and rewrite it in several steps: $\div \frac{a}{b}$ the opposite of $\times \frac{a}{b}$ the opposite of "$\...


1

As I said in my comments to the OP, I think this is just a bit of a tricky topic. A little like quantum mechanics, where you sort of have to get used to it, versus understand it immediately like kinematics. So hoping for some secret key to unleash their groking may not work (for this item, other topics there may be a nice mental key). A couple practical ...


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