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I hate “Preserve, Flip, Swap”. After we educate college students methods as a substitute of quantity sense, the result’s usually that college students don’t perceive what they’re doing. In Jo Boaler’s Mathematical Mindsets, he says that arithmetic is “inventive and visible.” As a substitute of educating methods, take into account having college students visualize and clarify the fraction. So how would you visualize a fraction divided by a fraction?
What does divide imply?
As a substitute of going straight into the principles for dividing fractions… that lots of people do not perceive… let’s take a while to consider division.
What number of methods are you able to describe it?
It may be very useful for college students to share alternative ways of expressing what division means. What does 20 ÷ 4 imply?
 To divide is to divide into teams
 what number of are in every group
 20 ÷ 4 is to divide into 4 teams. What number of are in ONE group?
 Which is ONE of the 4 teams?
 Create 4 teams. Evenly divide the 20 items into all teams.
 What number of will every group have when the 20 items are divided equally?
 What different methods are you able to say this?
By not at all times presenting or saying it the identical method, college students assist to know the idea of division.
Why is 20 instances 1 / 4 the identical as 20 ÷ 4?
What would change when you divided by a fourth?
What if as a substitute of 20 instances 1 / 4, you had 20 divided by 1 / 4?
evaluate and distinction
How is dividing by 1/4 completely different from dividing by 4?
Each bit is split right into a fourth. 1/4 of every piece is a small piece.
Consider taking a sweet bar (which has segments, like a Hershey™ bar) and breaking it up into each bit. You went from 1 piece (bar) to 12 items.
You have got 20 items and also you divide each bit into quarters (1/4)…then every massive piece turns into 4 small items…for a complete of 80 items.
What when you had 20 hours of yard work throughout the semester? No one needs to work within the backyard, so it’s agreed to divide it into 15minute slots (1 / 4 of an hour). How many individuals are wanted to cowl 20 hours of service? An individual is barely doing a fraction of an hour. So if there’s a whole of 20 hours of yard work within the semester, it can take plus of 20 individuals to cowl this. Every hour has 4 rooms… so 4 individuals are wanted every hour. 4 individuals each hour for 20 hours is… 80 individuals. Or 80 time slots to cowl.
Fraction divided by an entire quantity
So after we had 20 divided by 1 / 4, we ended up with 80 small items. However what if we began with a fraction and wished to divide it? I selected to divide by 2 as a result of most of us intuitively know meaning 1/2. YOU KNOW THAT ÷2 is the same as 1/2
Lower every of these 1/4 items in half. You need ONE of TWO items which might be created by slicing the piece.
So first you are taking 1/4, which implies you chop every little thing into 4 elements. Then you definately take 1 of the 4 items (1/4) and divide it into two items. You need ONE of the TWO smaller items. Breaking it into smaller items means you’ve got extra items. So every little thing would have a complete of 8 items, however you solely have 1 of the 8 smallest items.
You began with one piece.
broke into 4 items
and broke it into 8 items.
And you’ve got one of many 8 items
that is an eighth
You need half of the fourth piece.
Fraction divided by a fraction
Let’s evaluate that to dividing in half.
This isn’t the identical math downside. I’m NOT dividing each bit into 2 items. I’m dividing each bit right into a half piece.
Bear in mind how 20 items divided into 1/4 dimension items ended up with 80 smaller items.
20 divided by 1/4 dimension is 80. (Discover how I maintain rephrasing it! It is actually vital to maintain rethinking alternative ways of claiming what it means. Making sense of it’s math apply #1.) What number of quarter cups of flour are in 20 cups of flour?
Of the 20 items, every was minimize into 4 smaller items.
Of all of the 1/4 items, every was minimize into 2 smaller items.
Clearly 2 of those newly created smaller items collectively would make 1/4 piece. There are 4 of the newly created 1/8 items.
Visually put all of the triangles collectively and you may find yourself with 4 of the 8 items… or half of every little thing.
three of 4
What number of 1/4 are there in 20?
What number of three fourths are there in 20?
You have got 20 cups of flour and you utilize a 3/4 cup measurer. What number of 3/4′ cups are there?
Now bear in mind that you’ve ALL 20 cups of flour. You might be simply making smaller flour sachets that solely have 3/4 cup of flour. What number of small luggage of flour will you’ve got? 20 + 6 + two of three
For those who wished to take all 26 luggage and put them into thirdsize luggage to get a standard denominator (improper fraction), then every of these 26 luggage in thirds could be a complete of 78 thirdsize luggage.
78 luggage of the third dimension + 2 luggage of the third dimension = 80 luggage of the medium dimension.
26 luggage and a couple of/3 of a bag.
now with fractions
How about 1/4 divided by 3/4?
That is NOT three quarters of 1/4. This can be a quarter divided into 3/4 dimension items. You must find yourself with a bigger variety of items.
I find yourself not with 3 chocolate bars… however with THREE items of 1 / 4 of a chocolate bar.
The reply is THREE however the dimension modified. Let’s consider it as 3 enjoyable sized chocolate bars!
How about 3/5 divides 1/4?
I’ve 3/5 cup of flour. I need to divide this into 1/4 (not cup) sachets. What number of 1/4 are there in 3/5?
In any other case
Even after you have figured it out… what’s one other approach to put it? The extra methods it’s important to categorical an issue, the extra versatile you’re with numbers in several conditions.
Three fifths is three… 1/5. Or three teams of 1/5. Being versatile about dividing fractions makes many math issues a lot simpler!
Pondering of three/5 as THREE 1/5 permits me to regroup the unique query. Are you able to separate the numbers? Regroup? Use the associative and commutative properties to rethink how numbers can work together?
Utilizing the commutative property I modified the 1/5 and the three.
After I’m breaking down numbers, I usually change the numbers totally so I can see how different numbers work together after which return to the unique set of numbers and apply the sample I found. That is math apply #7 and math apply #8. I am unsure what I can do with this regrouping. I am going to have a look at some extra acquainted numbers:
12 divided by 3 instances 4
Let’s check out the idiotic math downside I’d at all times give my highschool college students. WHY would you give them 12 divided by 3 instances 4? As a result of I knew they’d be fallacious. MY ONLY goal for placing this on a quiz was…take away factors? Present them they’re unhealthy at math? Complain later that the youngsters cannot do a easy order of operations?
What he confirmed was…college students haven’t got quantity sense. It’s NOT that they’re unhealthy at math.
I do NOT must go from left to proper. The commutative property says {that a}•b•c = c•a•b. So if I’ve multiplication, I can change the order. Nevertheless, division is multiplication of a fraction. Begin studying the division image as fraction. Not solely will this assist you (and your college students) get higher at fractions, it opens up an entire new risk for the best way to simplify expressions.
12 divides 3 by 4 it’s 12 fraction 3 instances 4 both 12 instances 1/3 instances 4
Flat out, it isn’t 3 instances 4 in any respect. The division clearly places the three within the denominator. Let’s have that dialog. WHAT is dividing. As a substitute of a rule that claims “From left to proper”… BUT WHY?
The reality is, most individuals don’t know WHY. The reply I get after I ask that’s overwhelmingly “as a result of that is what my trainer informed me.”
Get within the behavior of issues in several methods. Examine and distinction. Why is that this resolution completely different from one other (related) downside?
I do not find out about you, however I get some concepts by switching loosely between the division image and a fraction. Being snug with equal expressions is having a greater quantity sense.
Again to three/5 Divided by 1/4
Dividing by 5 means: “What number of GROUPS of dimension 5 are you able to create?” So first we break all three into smaller items. Divide by 1/4 says to divide every into fourths. This creates 12 items. Now we need to make teams of dimension 5. I can create 2 full teams with 2 additional 5’s. So 12 items divided by 5 (12/5) or 2 and a couple of/5.
Google Jam Board
To see the Google Jamboard I made to discover visualizing fractions:

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How would you visualize a fraction divided by a fraction?

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How Would You Visualize a Fraction Divided by a Fraction