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43 votes

Student asked me if it is necessary to simplify fractions at the end of answering a question. I'm not sure how to respond

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 ...
Vaekor's user avatar
  • 630
29 votes

Why do some students struggle so much with fractions?

There are many reasons why fractions are so hard for students to learn. Mostly, they're taught gibberish and assessed according to such gibberish. Example 1 You are a 12-year-old student who has ...
WeCanLearnAnything's user avatar
26 votes

Don't these word problems seem designed to be confusing?

I think there's a countervailing issue that the book you're describing is trying to deal with. I teach college students, so I don't know what the particular approach it's taking is age appropriate ...
Henry Towsner's user avatar
25 votes

What is the rationale for distinguishing between proper and improper fractions?

added Oct 6 The reason mixed numbers are found in US education is that mixed numbers are found outside of school in the US, so the children need to learn to understand them. Mixed numbers are found ...
Gerald Edgar's user avatar
  • 7,607
23 votes
Accepted

Is it meaningful to add a number to itself a fractional number of times?

For the product $a\times b$, I intentionally don't use the phrase "add $a$ to itself $b$ times", but rather I prefer something like "start with zero and add $b$ (copies) of the number $...
Nick C's user avatar
  • 9,639
22 votes

Student asked me if it is necessary to simplify fractions at the end of answering a question. I'm not sure how to respond

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 ...
A. Goodier's user avatar
  • 1,725
19 votes
Accepted

Student asked me if it is necessary to simplify fractions at the end of answering a question. I'm not sure how to respond

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 ...
Xander Henderson's user avatar
  • 8,225
17 votes
Accepted

How to teach students when they can and can't cancel factors in a fraction?

Instead of presenting "cancelling" as an arbitrary rule (which is often how students have seen it — or at least how they learned it — before), explain it in a way that shows what's actually going on. ...
Daniel Hast's user avatar
  • 4,893
16 votes
Accepted

Is there a way to extend the analogy that fractions means "x out of y" to show that fractions are also dividing?

I think the heuristic \begin{align*} \frac{8}{4} \quad \longleftrightarrow \quad 8 \textrm{ out of every group of } 4 \end{align*} still makes sense if you emphasize that you want $8$ slices of pizza ...
Justin Skycak's user avatar
14 votes

How to teach sum of fractions to students?

First, I would have them really understand equivalent fractions. There are a lot of ways to write the number represented by the fraction $\frac23$. We can call it $\frac23,\frac46,\frac{20}{30},\frac{-...
G Tony Jacobs's user avatar
13 votes

Don't these word problems seem designed to be confusing?

I feel like I always post the same thing in these threads, but this again sounds like an issue of blocking vs interleaving. In this case, the textbook may have started interleaving different problems ...
WeCanLearnAnything's user avatar
13 votes

Student asked me if it is necessary to simplify fractions at the end of answering a question. I'm not sure how to respond

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 ...
Chris Cunningham's user avatar
12 votes

How to explain the difference between the fraction a / b and the ratio a : b?

There are several possible fractions one could associate with a given ratio. Say that a recipe for lemonade calls for $2$ cups of lemon juice and $5$ cups of water. This could be expressed with the ...
Steven Gubkin's user avatar
11 votes

How to explain fractions to 7 year old kid

Comparing fractions only works when the whole is the same size. Here's a few examples to get help the 7 year old understand what happens when things aren't the same size: 1/2 will be greater than 2/4 ...
Amy B's user avatar
  • 8,017
11 votes

How is $\frac{a}{b}$ interpreted?

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 $\...
Steven Gubkin's user avatar
11 votes

Composite fraction?

I like your term. The wikipedia article on fractions also mentions they are called complex fractions or compound fractions. Personally, I dislike the term complex fraction as it is obviously going to ...
James S. Cook's user avatar
11 votes

What is the rationale for distinguishing between proper and improper fractions?

I do not know of any relevant research. Here are my own not-research-informed ideas. Most people refer to fractions as parts of a whole. If someone says "I lost a fraction of a pound on my diet&...
Steven Gubkin's user avatar
11 votes

Is there an agreed upon difference between how we represent $\frac{a}{b}$ and $a \cdot \frac{1}{b}$?

The common core state standards definition of the fraction $\frac{N}{D}$ of a unit is to subdivide the unit into $D$ equal sized pieces. Each of these pieces is defined to be $\frac{1}{D}$ of the ...
Steven Gubkin's user avatar
11 votes

Is it meaningful to add a number to itself a fractional number of times?

Frame challenge: I think your verbiage "adding (whole number) to itself (whole number) times" is misleading and incorrect and exhibits an off-by-one error. Think about the example $(4×1)$. ...
shoover's user avatar
  • 816
9 votes

How do you explain the whole integer and fractions subject to a kid in 6th grade?

I think this problem is much easier to handle if you refer throughout to the number of students who study each subject, rather than the fraction, only expressing the final answer as a fraction at the ...
mweiss's user avatar
  • 17.4k
9 votes
Accepted

How do you explain the whole integer and fractions subject to a kid in 6th grade?

You could try keeping 30 as the denominator throughout, that is, observing that $\frac{1}{3} = \frac{10}{30}$ and $\frac{1}{5} = \frac{6}{30}$, so the portion of the class that doesn't study chemistry ...
Daniel Hast's user avatar
  • 4,893
9 votes

Don't these word problems seem designed to be confusing?

I would say that this book is trying to get students to think about what the calculations mean, rather than simply execute an algorithm, and I strongly believe that is something that should be done ...
mweiss's user avatar
  • 17.4k
9 votes

What is the standard for "simplifying your answer"?

Rigid criteria for simplification seem to me largely a bad idea if they are not motivated by contextual considerations. The idea that $\sqrt{2}/2$ should be preferred to $1/\sqrt{2}$ struck me as ...
Dan Fox's user avatar
  • 5,869
9 votes

Concrete way to teach addition and subtraction of fractions

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 ...
Amy B's user avatar
  • 8,017
9 votes
Accepted

Analyzing an answer to the following problem: Give meaning to $\frac{4}{5} + \frac{2}{3}$

The student who designed this problem wasn't thinking about the different wholes. IN your students problem, there are 3 different wholes. Anna's flowers - The whole is 5 flowers and $\frac{4}{5}$ are ...
Amy B's user avatar
  • 8,017
8 votes

Negative Denominator in Fractions; Importance and Applications

I would say you're doing your student a disservice if you were to seriously disallow a negative denominator. A fraction is simply a ratio of two integers (where the denominator is not allowed to be ...
A.Ellett's user avatar
  • 390
8 votes
Accepted

How to explain multiplying and dividing by fractions with real-world examples

We have two cookies. We divide them into pieces of 1/2 cookie each and end up with four pieces. Thus 2 divided by 1/2 equals 4. We have two cookies. We take 1/2 of the collection which is one cookie. ...
Rory Daulton's user avatar
  • 2,582
8 votes
Accepted

Why do some students struggle so much with fractions?

As for research on fractions education... there's a TON. As for research on fractions education that rigorously measures cause and effect through randomized controlled experiments and long-term ...
WeCanLearnAnything's user avatar
8 votes

Analyzing an answer to the following problem: Give meaning to $\frac{4}{5} + \frac{2}{3}$

The word you are looking for is mediant. The mediant of two fractions $\frac{a}{c}$ and $\frac{b}{d}$ is $\frac{a+b}{c+d}$. According to Wikipedia, It is sometimes called the freshman sum, as it is a ...
JRN's user avatar
  • 10.8k
8 votes
Accepted

What is the expected fluency with fractions at UK key stage 3?

I tutor maths as a full-time job, and many of the students I have tutored over the past few years are not fluent in their times tables up to $10$ or arithmetic with fractions, or both. And it's not ...
Adam Rubinson's user avatar

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