How Would You Visualize a Fraction Divided by a Fraction

Gamification

Team FunTrove

October 16, 2022

How Would You Visualize a Fraction Divided by a Fraction

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I detest “Keep, Flip, Change.” When we educate college students tips as a substitute of quantity sense the result’s typically that college students fail to grasp what they’re doing. In Mathematical Mindsets by Jo Boaler she states that math is “Creative and Visual.” Instead of educating tips, contemplate having college students visualize and clarify the fraction. So how would you visualize a fraction divided by a fraction?

What Does Divide Mean?

Instead of going straight to the principles of dividing fractions… which many individuals don’t perceive… lets take a while to consider division.

How Many Ways Can You Describe It?

It can assist rather a lot to have college students share out alternative ways to precise what the division means. What does 20 ÷ 4 imply?

Divide 20 by 4. Create 20 circles and break into 4 groups. Circle each group of 5.

Divide is to interrupt into teams
How many are in every group
20 ÷ 4 is to interrupt into 4 teams. How many are in ONE group?
What is ONE of the 4 teams?
Create 4 teams. Evenly divide the 20 items into all of the teams
How many will every group have when all 20 items are evenly break up
What are different methods you may say this?

By not all the time presenting it or saying it in the identical manner college students assist to grasp the idea of division.

20 times one fourth is the same as 20 divided by 4

Why is 20 occasions one fourth the identical as 20 ÷ 4 ?

When you divide by 4 you specific what number of are in ONE of the teams.

What Would Change if You Divided by One Fourth?

What if as a substitute of 20 occasions one fourth, you had 20 divided by one fourth?

Compare and Contrast

How is dividing by 1/4 completely different than dividing by 4?

Each piece is split right into a fourth. 1/4 of every piece is a small piece.

Think of taking a bar of chocolate (that has segments, like a Hershey™ bar) and breaking it up into every chunk. You went from 1 piece (bar) to 12 items.

You have 20 items and also you divide each bit into quarters (1/4th) … then every huge piece turns into 4 little items … for a complete of 80 items.

What in the event you had 20 hours of yard obligation for the semester. No one desires to do yard obligation so it’s agreed to divide it up into 15 minute (quarter of an hour) spots. How many individuals are wanted to cowl the 20 hours of obligation? One particular person is just doing a fraction of an hour. So if there are 20 whole hours of yard obligation within the semester it should take extra than 20 folks to cowl this. Each hour has 4 quarters… so that’s 4 folks wanted every hour. Four folks every hour for 20 hours is … 80 folks. Or 80 obligation slots to be lined.

Fraction Divided by a Whole Number

So after we had 20 divided by one fourth we ended up with 80 small items. But what if we began with a fraction and wished to divide it up. I selected divide by 2 as a result of most of us intuitively know which means 1/2. YOU KNOW THAT ÷2 equals 1/2

one fourth divided by 2. Visualize as 4 pieces and you want ONE of the four pieces. Now cut that in half.

Cut every of these 1/4 items in half. You need ONE out of the TWO items which might be created by reducing the piece up.

So first you’re taking 1/4, which suggests you chop the entire thing into 4 items. Then you’re taking 1 out of the 4 items (1/4) and divide that into two items. You need ONE out of the TWO smaller items. Breaking it up into smaller items means you have got extra items. So the entire thing would have a complete of 8 items however you solely have 1 out of the 8 smaller items.

You began with one piece.
Broke into 4 items
and Broke that into 8 items.
And you have got one out of the 8 items
This is one eighth

You need half of the one fourth piece.

Fraction Divided by a Fraction

Let’s examine that to dividing by one half.

This isn’t the identical math drawback. I’m NOT dividing each bit into 2 items. I’m dividing each bit into half a bit.

Remember how 20 items divided into 1/4 measurement items ended up with 80 smaller items.

20 divided into 1/4 sizes is 80. (discover how I hold rewording it! So essential to maintain rethinking alternative ways to say what it means. Sense making is mathematical follow #1) How many quarter cups of flour are in 20 cups of flour?

Of all of the 20 items, every was lower into 4 smaller items.
Of all of the 1/4 items, every was lower into 2 smaller items

Obviously 2 of those newly created smaller items pushed again collectively would make the 1/4 piece. There are 4 of the newly created 1/8 items.
Visually, push all of the triangles collectively and you find yourself with 4 out of the 8 items… or half the entire thing.

Three Out of Four

How many 1/4’s are there in 20?

How many three fourths are in 20?
You have 20 cups of flour and you utilize a 3/4 cup measurer. How many 3/4’ths cups are there?

Now keep in mind you have got ALL 20 cups of flour. You are simply making smaller baggies of flour that solely have 3/4 of a cup of flour in them. How many baggies will you small baggage of flour will you have got? 20 + 6 + two out of three

If you wished to take the 26 baggage and put them into third measurement baggage so you have got a standard denominator (improper fraction) then every of these 26 baggage in thirds can be a complete of 78 third sized baggage.

78 third sized baggage + 2 third sized bag = 80 half sized baggage.

26 baggies and a couple of/3 of a baggie.

Now With Fractions

How about 1/4 divided by 3/4?

This is NOT three fourths of the 1/4. This is One Fourth divided up into 3/4 sized items. You ought to find yourself with a bigger variety of items.

I find yourself not with 3 sweet bars.. .however quite THREE one fourth sized chunks of a sweet bar.

The reply is THREE however the measurement modified. Let’s give it some thought as 3 enjoyable sized sweet bars!

How About 3/5 Divide 1/4

I’ve 3/5 of a cup of flour. I need to divide this into 1/4 (not of a cup) sized baggies. How many 1/4ths are in 3/5ths?

I’ve 2 and a couple of/5 teams

Another Way

Even after you’ve figured it out… what’s one other technique to specific it? The extra methods you must specific an issue the extra versatile you might be with numbers in numerous conditions.

Three fifths is three … 1/5ths. Or three teams of 1/5. Being versatile to interrupt up fractions makes many math issues rather a lot simpler!

Thinking of three/5 as THREE 1/5’s permits me to regroup the unique query. Can you break numbers aside? Regroup? Use the Associative and Commutative properties to rethink how numbers can work together?

Using the Commutative Property I swapped the 1/5 and three.

When I’m breaking down numbers I typically will swap the numbers fully so I can see how different numbers work together after which come again to the unique set of numbers and apply the sample I found. This is mathematical follow #6 and mathematical follow #7. Unsure what I can do with this regrouping I’m going to take a look at some extra acquainted numbers:

12 Divided by 3 Times 4

Let’s check out the jerk math drawback I’d all the time give my highschool college students. WHY would I give them 12 divided by 3 occasions 4? Because I knew they might get it fallacious. MY SOLE goal for placing this on an evaluation was to … take factors off? Prove to them they’re unhealthy at math? Complain later that children can’t do easy order of operations?

What it proved was… college students should not have quantity sense. NOT that they’re unhealthy at math.

I do NOT must go left to proper. The Commutative Property says {that a}•b•c = c•a•b. SO if I’ve multiplication I can swap up the order. However, division is the multiplication of a fraction. Start studying the divide image as fraction. This won’t solely aid you (and your college students) be higher at fractions, it opens up a complete new chance for the right way to simplify expressions.

12 divide 3 occasions 4 is 12 fraction 3 occasions 4 or 12 occasions 1/3 occasions 4

Flatly, it’s not 3 occasions 4 in any respect. The divide clearly places the three within the denominator. Let’s have that dialog. WHAT is being divided. Instead of a rule that claims “Left to Right”… BUT WHY?

The fact is, most individuals do not know WHY. The response I get after I ask that’s overwhelmingly “because that is what my teacher told me.”