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You're student(s) are going to love learning about fractions as we sing, eat pies, see birds in the skies, and even win an Olympic prize! Three new NUMBEROCK characters come to life in this informative and easy to understand introduction to fractions song. The numerator and denominator are shown in 3 real-world examples before introducing to students the idea of fractions on a number line, all the while reinforcing that fractions are equal parts of a whole.

Numberock's Fractions Song Lyrics:

There were five pieces in one whole pie;
the denominator of the whole pie was five.
The numerator of my part was three,
as three fifths of the pie was for me.
My mom made the apple pie out of Granny Smiths.
She asked me if I liked it. I said, "I plead the fifth..."

The numerator's the part,
the denominator's the whole,
and fractions are parts of a whole.

Three birds were on a telephone pole.
The denominator, three, was the group as a whole.
The numerator, two, out of the thirds,
described the two thirds that were blue birds.
The numerator, two, described how many were blue.
Then something fell on my head... it was bird poo!

I took four shots on the goal;
three went in, but one hit the pole.
The denominator was all four shots.
The numerator was the three goals I'd got.
The other team's record had taken a toll;
I'd got three fourths of my shots and won Olympic Gold.

A number line has every fraction between zero and one.
Right in the middle is one half; one fourth is half of that.
A fraction's parts, you realize, have to be the same size.
Hey, we're late for the game!
Hate to leave you, guys!

Learn More

This song targets TEKS and Common Core learning standards from both 2nd Grade (TEKS) and 3rd Grade (CCSS). Look into the relevant standards here, or dig deeper into the concept of fractions here.

If you are interested in getting ideas on how to plan a robust standards-aligned fractions lesson, we recommend checking out Instructure's recommendations for common core standards 1.MD.3, and 2.MD.7. These pages help break down standard language, lay out the grade-appropriate level of rigor for each concept, and offer a variety of suggestions for activities (lesson seeds) that help students achieve their learning targets.

To continue browsing Numberock's math video content library, click here. To gain access to Numberock's growing library of premium content, click here.

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NUMBEROCK Fractions Song | Intro to Fractions Video | 3rd Grade

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This whimsical video by NUMBEROCK introduces fractions as "parts of a whole". Caution: This song may drive redonkulous levels engagement.

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Join Benny in his school where he is investigating fractions on number lines. He gets some help from his friend Thania, and then shows his stuffed animal how to write 1/3 on the number line. While doing his homework later in the evening, Benny gets stuck on a problem about representing fractions greater than one, so he texts Thania (his friend, and apparently, tutor-on-demand), to get some help.

Fractions on a Number Line Lyrics:

Then I texted my friend, "What's to be done
when the numerator isn't one?"
She said, "Let's try to find five over nine
on the number line in - like - no time.
Simply partition nine parts from zero to one.
Starting from zero, count five, and at five-ninths we arrive."

Let's use the number line to describe this pie.
First, count the total pieces... there are five.
Five's the denominator (all the parts), so five goes underneath the fraction bar.
This number line goes from zero to two,
but the fraction's less than one, so here's what to do:
Make five equal spaces from zero to one.
Now the final step before we're done:
The numerator is the one piece that's mine.
After the first space I'll write one-fifth under the line.
Just remember this one tip in your minds:
Always count the spaces - never the lines.

Learn More

This song targets TEKS and Common Core learning standards primarily from 3rd Grade. Look into the relevant standards, or dig deeper into the concept of fractions or number lines.

For more information on planning a complete, standards-aligned fractions on a number line lesson, we advise checking out Instructure's recommendations for common core standard 3.NF.2. These pages help break down standard language, lay out the grade-appropriate level of rigor for the concept, and offer a variety of suggestions for activities (lesson seeds) that help students achieve their learning targets.

To continue browsing Numberock's math video content library, click on the green banner below. To gain access to Numberock's growing library of premium content, click here.

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Numberock's Fractions on a Number Line Song | 3rd Grade Math Video

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I guess nobody told NUMBEROCK that representing fractions on a number line shouldn't be this fun. Sorry. Not sorry. Enjoy the video!

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Equivalent fractions is the name... of fractions whole values are the same! With the added touch of a Latin video in the musical track, equivalent fractions will be easy to understand - it's just an added bonus that the music and animations are hard to beat in their level of awesomeness.

Song Lyrics:

Once upon a time, there were nine canaries
On telephone wires, sittin' down in threes.
Each wire contained one third of the birds,
So three ninths is equivalent to one third.
Drawing 1/3 and 3/9 on a number line
Shows 1/3 equals three parts out of nine.
Therefore, equivalent is the word
That describes 3/9 and 1/3.

(CHORUS)
Equivalent fractions is the name
For fractions whose values are the same.

If there were four socks to wear,
Then we would say there'd be two in each pair;
And one of the two pairs, we can declare,
Equals two fourths of all the socks there.
Drawing one half and two fourths on a number line
Shows they can be described with an equal sign;
An equal value is what they're worth,
So we say one half is equivalent to two fourths.

(CHORUS)

If the numerator and denominator share a common factor,
Then you can divide the top and bottom by the same number.
On top and under, and say with satisfaction,
"We found an equivalent fraction."
And you can multiply the top and the bottom
By the same number on top and under,
Which is really a fraction whose value equals one,
And through this action we found an equivalent fraction!

Learn More

This song targets TEKS and Common Core learning standards from both 3rd Grade and 4th Grade. Look into the relevant standards here, or dig deeper into equivalent fractions here.

If you are interested in getting ideas on how to plan a robust standards-aligned telling time lesson, we recommend checking out Instructure's recommendations for common core standards 3.NF.A.3, 4.NF.A.1 and 4.NF.A.2. These pages help break down standard language, lay out the grade-appropriate level of rigor for each concept, and offer a variety of suggestions for activities (lesson seeds) that help students achieve their learning targets.

To continue browsing Numberock's math video content library, click here. To gain access to Numberock's growing library of premium content, click here.

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Equivalent Fractions Song | 3rd Grade - 4th Grade Video | NUMBEROCK

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There has never before been a more fun way to learn about equivalent fractions than with this catchy song with a Latin rhythm.

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Godzilla goes underwater in this video to chase submarines which are affixed with fractions that display simplified fractions and their unsimplified pair. Meanwhile, on land, numbers are being catapulted into the skyline where they fall down as parts of a fraction in its lowest term. Your students will love this cute little video about simplifying fractions!
Aligned With:
Common Core Standard ( CCSS )
3.NF.A.3
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
4.NF.A.1
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
5.NF.A.1
Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

TEKS Standard
Supporting Standard 5.3H
... represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations.

* UK Age Range:
Year 5 - Year 6
KS2 and KS3 Maths

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Simplifying Fractions Video, Game, Activities & Reducing to Simplest Form

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Godzilla goes underwater in this video to chase submarines which are affixed with fractions that display simplified fractions and their unsimplified pair. Meanwhile, on land, numbers are being catapulted into the skyline where they fall down as parts of a fraction in its lowest term.

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In a future time, when AI robots have taken over running pizzerias on a planet orbiting Alpha Centauri, the robot Zigon takes orders and then adds and subtracts fractions before the Pizzamatic-5000 serves up large pepperoni and cheese pizzas for its alien visitors! Adding Fractions has never been so stellar or astral.

Song Lyrics:

This is how the least common multiple's found:
write a list of each denominator's multiples down;
the smallest number you can find in both of them
is the least common multiple (a.k.a. the LCM.)

The least common multiple can also be
called the least common denominator - or LCD.
It stays the same in the difference or the sum
when you subtract or add two fractions.

Three-sixths of a pizza minus one-fourth:
find the LCM, which is twelve, of course.
(Twelve is the least common denominator.)
Make equivalent fractions (six-twelfths and three-twelfths).
Then, do the subtraction: six minus three is three...
three-twelfths of a pizza at the pizzeria.

Chorus:
Find the least common multiple; that's the first action.
Next, is finding equivalent fractions.
Then, do the addition or subtraction.
Simplify like an automatic reaction.

Two-fourths of a pizza plus one-fifth:
it smelled so good we had to give it a sniff.
The LCM was twenty (the new denominator).
We found equivalents quickly: ten-twentieths plus four-twentieths... fourteen-twentieths of a pizza at the pizzeria.

(Chorus)

Five-sixths of a pizza minus one-third:
I knew the LCM was six, and have you heard?
It's also called the least common denominator.
I found equivalent fractions (five-sixths and two-sixths).
Then, I did the subtraction: five minus two is three...
three-sixths of a pizza at the pizzeria.

(Chorus)

Learn More

This song targets TEKS and Common Core learning standards from both 4th Grade and 5th Grade. Look into the relevant standards here, or dig deeper into Subtracting Fractions here.

If you are interested in getting ideas on how to plan a robust standards-aligned telling time lesson, we recommend checking out Instructure's recommendations for common core standards 4.NF.4, and 5.NF.1 . These pages help break down standard language, lay out the grade-appropriate level of rigor for each concept, and offer a variety of suggestions for activities (lesson seeds) that help students achieve their learning targets

To continue browsing Numberock's math video content library, click here. To gain access to Numberock's growing library of premium content, click here.

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Adding & Subtracting Fractions Song by NUMBEROCK

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Numberock's robotic chef works through adding and subtracting fractions as he serves up large pepperoni and cheese pizzas for the pizza parlor's patrons.

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Did you know that scientists now agree that there was a dinosaur called the Brontosaurus? Does that have anything to do with multiplying fractions? You'll have to watch this video to find out. This song's got a catchy chorus and some pretty clever lyrics that will capture the attention of every student in your class.

Copy and paste the following text onto your teacher, school, or homeschool website to give your students a chuckle.

Multiplying Fractions Song Lyrics:

Three of us found treasure on a scuba trip,
So we pulled our treasure onto the ship.
Since we were so blessed by this rarity,
We decided to give half of it to charity.
We took the rest and split it up three ways;
One third of the rest was still a huge payday!
One third for me, one half for charity,
So what's my share? Let's multiply and see...

When we need to multiply two fractions,
This is our reaction:
Multiply the numerators,
Then multiply the denominators.

My friend and I were digging for dinosaur bones.
We decided to search in a rectangular zone;
We split up equally between the two of us.
The first day I found a bone from a brontosaurus!
I found a few more by the end of day two,
When three eighths of my half had been searched through.
Of my half, I had explored three eighths;
how much of the entire space did I excavate?

Learn More

This song targets TEKS and Common Core learning standards from 4th Grade to 6th Grade. Look into the relevant standards here, or dig deeper into multiplying fractions here.

If you are interested in getting ideas on how to plan a robust standards-aligned Multiplying Fractions lesson, we recommend checking out Instructure's recommendations for common core standards 4.NF.4, and 5.NF.4 . These pages help break down standard language, lay out the grade-appropriate level of rigor for each concept, and offer a variety of suggestions for activities (lesson seeds) that help students achieve their learning targets.

To continue browsing Numberock's math video content library, click here. To gain access to Numberock's growing library of premium content, click here.

Steal Our Joke.

Copy and paste the following text onto your teacher, school, or homeschool website to give your students a chuckle.

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Multiplying Fractions Worksheets, Word Problems, and Games for 5th, 6th

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Did you know that scientists now agree that there was a dinosaur called the Brontosaurus? Does that have anything to do with multiplying fractions? You'll have to watch this video to find out. This song's got a catchy chorus and some pretty clever lyrics that will capture the attention of every student in your class.

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Multiplying Fractions by Whole Numbers shows up in your daily life all the time, but perhaps you just don't know it yet. Multiplying Fractions and Whole Numbers together is often termed finding the fractional part of a whole number. We teach you why in this song. Enjoy the Multiplying Fractions by Whole Numbers video!
Aligned With:
Common Core State Standards ( CCSS )
4.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
A. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
B. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
C. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
Note: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

5.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

TEKS Supporting Standards
Supporting Standard 5.3I
... represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models.

* UK Age Range:
| Year 5 - Year 6 - Year 7 |
KS2 and KS3 Maths

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Multiplying Fractions by Whole Numbers with Word Problems, Worksheets

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Multiplying Fractions by Whole Numbers shows up in your daily life all the time, but perhaps you just don't know it yet. Multiplying Fractions and Whole Numbers together is often termed finding the fractional part of a whole number. We teach you why in this song. Enjoy the Multiplying Fractions by Whole Numbers video!

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Follow Davey around as he asks his best friend Steve how to convert fractions to decimals. The bell's about to ring, and Steve gives him the perfect advice in the nick of time! Later on, Davey helps correct the misconceptions of his personified stuffed animal by showing him that fractions are division.

And finally, Davey texts his friend Alana before their school dance (where NUMBEROCK Live is performing) who helps him clear up one more question he has about the conversion process. In the end, Alana, Steve, and Davey all make it to the school dance where they rock out on stage to the performance they've all been waiting for!

Fractions to Decimals Song Lyrics:

I asked my friend, "Which way is optimal
to convert a fraction to a decimal?
I gotta know now; I can't wait 'til later."
She said "Divide the numerator by the denominator."
So I asked, "Huh, well how can that be done?
As far as I know, two doesn't go into one."
She said, "Listen, here's what you need to know:
just write a decimal point, followed by a zero.
And you know two goes into ten five times;
then the decimal point rises above the line!"

The numerator becomes the dividend.
Then write a decimal point and a zero in the tenths.
Divide and write the decimal point in the quotient.
So, one half and five tenths are equivalent.

Later on I was hanging with my stuffed animal,
showing him how to convert a fraction to a decimal.
I divided the numerator by the denominator
as he sat there real quiet like a spectator.
I bet he was thinking, "Five can't go into two!?"
But luckily for him I knew just what to do.
I said, "Listen, here's how it's got to go:
just write a decimal point, followed by a zero.
Then five goes into twenty (four times),
and the decimal point jumps up on the line!"

The numerator becomes the dividend.
Then write a decimal point and a zero in the tenths.
Divide and write the decimal point in the quotient.
So, two fifths and four tenths are equivalent.

I had to get my homework done before the school dance.
I had to convert a fraction - couldn't leave it to chance.
I divided the numerator by the denominator,
but to my surprise, there was a remainder.
I texted my friend, "I don't want to be late.
Can you tell me how to divide one by eight?"
She said, "Write two more zeroes to the right of the dividend.
The value will be unchanged; it's equivalent."
I divided the two numbers and got point one two five.
We arrived just in time to see NUMBEROCK Live!

Learn More

This song targets TEKS and Common Core learning standards from both 4th Grade and 5th Grade. Look into the relevant standards here, or dig deeper Fractions here.

If you are interested in getting ideas on how to plan a robust standards-aligned telling time lesson, we recommend checking out Instructure's recommendations for common core standards 4.NF.6, and 5.NF.3 . These pages help break down standard language, lay out the grade-appropriate level of rigor for each concept, and offer a variety of suggestions for activities (lesson seeds) that help students achieve their learning targets.

To continue browsing Numberock's math video content library, click here. To gain access to Numberock's growing library of premium content, click here.

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NUMBEROCK Fractions to Decimals Song | Video with Worksheets

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Follow Davey around as he asks his best friend Steve how to convert a fraction to a decimal in this fun song.

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Fractions are Fun: How Fractions are used in Everyday Life

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Math Antics - Adding and Subtracting Fractions

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Math Antics - Converting Base-10 Fractions

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Growth Mindset (1 of 5) | Social Emotional Learning for kids

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The Mindset of a Champion | Carson Byblow | TEDxYouth@AASSofia

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LearnStorm Growth Mindset: The Truth About Your Brain

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An Important Message About Growth Mindset (with captions)

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Developing a Growth Mindset! | Dr. Nagler's Laboratory

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What difference does it make if you think you can do it and if you think you can't?

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What difference does it make if you think you can do it and if you think you can't?

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What is a Fraction and When Do We Use Them?

Fractions are a way of expressing those numbers that aren't whole, natural, or integral numbers. However, every number can be expressed in fractional form. For example, if we take number '5', apparently, it is just five, no denominator. But it can be written as '5/1'. However, we usually don't do that. We don't write natural, whole, or integral numbers in fractional form. Fractions are used to represent only those numbers that have a fractional or decimal part. A fraction consists of two parts that are a numerator that is written above the division and a denominator that is written below the division. There are two types of fractions: proper and improper. The denominator divides the numerator, and the numerator is the number that is being divided. Proper fractions have numerators smaller than the denominators. Improper fractions are their opposite. Fractions are used to show the outcome of a division.

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Fractions are a way of expressing those numbers that aren't whole, natural, or integral numbers. However, every number can be expressed in fractional form. For example, if we take number '5', apparently, it is just five, no denominator. But it can be written as '5/1'. However, we usually don't do that.

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Let's Learn Fractions - Understanding Math for Kids

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Exploring Fraction Constructs and Proportional Reasoning

Fractions are a beast of a concept that causes struggles for many adults and students alike. While we all come to school with some intuition to help us with thinking fractionally and proportionally, the complexity quickly begins to increase as we move from concrete, to visual, to symbolic and from identifying, to comparing, to manipulating. Fractions are formally introduced in the Ontario Math Curriculum when students begin dividing whole objects into pieces and identify these pieces using fractional names (e.g.: halves; fourths or quarters) and continues to promote the development fractional fluency concretely through each grade. Interestingly enough, it is often said that students struggle much more with numbers represented in fraction notation than those represented in decimal notation, yet the word "fraction" appears in the curriculum document 99 times beginning in grade 1, while the word "decimal" appears only 69 times beginning three years later in the 4th grade. While it might be true that fractions tend to intimidate, I wonder if our dependence on the calculator has tricked us into believing we are more fluent with quantities represented in decimal form than is reality.

Paying Attention to Fractions

As I did with the Progression of Proportional Reasoning, I'd like to reference the Paying Attention To Mathematics series released by the Ontario Ministry of Education Literacy and Numeracy Secretariat (LNS) called Paying Attention to Fractions. These guides are a great start to help you wrap your head around big ideas in mathematics and thus, this post will attempt to expand on the ideas shared in this particular document.

Why Fractions?

As if the struggles our students experience when working with fractions aren't enough justification, I like this quote shared in the guide:

"No area of elementary school mathematics is as mathematically rich, cognitively complicated, and difficult to teach as fractions, ratios, and proportionality. These ideas all express mathematical relationships: fractions and ratios are 'relational' numbers. They are the first place in which students encounter numerals like ' 3/4 ' that represent relationships between two discrete or continuous quantities, rather than a single discrete ('three apples') or continuous quantity ('4 inches of rope')."

(Litwiller & Bright, 2002, p. 3)

What Is A Fraction?

A fraction is a number.

While many think of fractions and fractional notation as a means to represent quantities that are not whole, all quantities can be represented as a fraction.

While these descriptions are simple on the surface, they do not appropriately communicate the complexity throughout the progression of fractions and the many big ideas connected to this progression.

Connecting Proportional Reasoning and Fractions

If you recall from my last post, proportional reasoning takes place when one compares two quantities in relative terms rather than absolute terms. Thus, it can be said that a fraction provides a means for representing that relationship between two numbers. However, it is important not to confuse that while the numerator and denominator can be thought of as two quantities that are related, a fraction is a single quantity that can be represented on a number line.

So where does the progression of fractions begin?

As it turns out, pretty early on!

Young learners begin their journey through the progression of fractions in tandem with the progression of proportional reasoning as they make comparisons between two (or more) quantities such as:

this bag is heavier than that one;

I'm "this much" shorter than you are; and,

there are 5 more red candies than green.

While these early comparisons are likely absolute (i.e.: additive) in nature, we are one step closer to comparing these quantities relatively (i.e.: multiplicatively). This multiplicative comparison between two or more quantities is called a ratio.

In these early stages of the progression of fractions, young learners often begin thinking fractionally as they explore geometric quantities like volume and area; comparing a part relative to its whole.

Counting Cartons: Part-Whole Relationships

Since the progression of fractions begins for most by thinking of a part relative to a whole as a means to measure as well as the lack of conceptual clarity related to fractions and proportional reasoning, it should be no surprise that many think of fractions narrowly as a part-whole relationship alone.

Let's start our exploration of fraction constructs beginning with the most common part-whole fraction.

Have a look at the image below:

I might ask students:

How many eggs are there?
How many different ways can you represent that quantity?

While the question is clearly asking students to determine the quantity of eggs, asking students to share how many different ways they can represent the quantity encourages students to potentially think beyond counting relative to 1 egg. While we can expect that many students will be able to communicate that there are 18 eggs, others may get creative and unitize (i.e.: 3 groups of 6; 9 groups of 2; etc.) or even reference the number of eggs relative to egg cartons. By asking students to represent the quantity in different ways, we are promoting proportional reasoning by changing the whole that the quantity of eggs is being related to.

After students share out with a partner, I might ask students more specifically:

How many cartons of eggs are there?

This prompt maintains the same "part" (18 eggs), but changes the whole (1 carton rather than 1 egg).

After students have some time to think independently, I would have them share out with their table groups. Some interesting responses are sure to come up such as:

You may notice that by asking students to state how many cartons of eggs there are, there is room for interpretation. Some students might simply state that there are two egg cartons without considering the number of eggs present, some might give a quantity relative to two egg cartons, while others might state the number of eggs relative to a single carton.

After students have been given an opportunity to defend their thinking, I might ask students:

How many cartons are full of eggs?

This fraction construct would be an example of a part-whole relationship. Students might consider how many whole and partial cartons there are in the form of a mixed fraction (1 and 1/2), improper fraction (3/2) or in decimal form (1.5). Some common representations for these different fractional forms might be using a set model, area model or number line. The following animated gif shows each of these representations, sometimes more than one simultaneously:

You might have noticed the double-number line included in the animated gif. This idea can be very useful when making the jump from thinking fractionally to proportional reasoning.

Doritos Roulette: Part-Part Relationships
Note: This section was updated on February 27th, 2021 to better communicate my current and more clarified understanding of fractions and proportional relationships that I continue to develop.
In the Doritos Roulette 3 Act Math Task, students are asked questions that stem from the relationship of "hot" chips to "not hot" chips based on the image on the front of the bag:

I might ask students to consider the image above and ask them this question:

What ratio of hot chips to not hot chips are there in a bag of Doritos Roulette?

and,

Can I represent this ratio as a fraction?

This string of questioning is helpful for emerging some of the muddiness that exists when it comes to understanding fractions and proportional relationships.

Despite the explicit request for a part-part relationship between "hot" chips (1) and "not hot" chips (6), we often see and/or hear many share the relationship between "hot" chips (1) and the whole (7).

When asking a group of students or educators to discuss with their neighbours about whether a part-part ratio relationship can be represented a fraction or not. Great (and sometimes heated) discussions can arise from this question, which makes this exciting to facilitate.

Based on the explanation given in the Paying Attention to Fractions document referenced earlier, I used to believe that a ratio is a fraction. However, after much learning and finally coming across the work of John Van De Walle et al and their book Elementary and Middle School Mathematics: Teaching Developmentally, I now realize that while a ratio can be represented using a fraction bar, it is not a fraction.

It is worth noting that while a part-part ratio can be represented using the fraction bar, note that this construct is representing two separate quantities where the fraction bar is used as an operator to divide.

In other words, when we see fraction notation used to represent this part-part ratio as 1/6, what this is representing is:

1 hot chip ÷ 6 not hot chips

Partitively dividing these two quantities in ratio will reveal a rate:

1 hot chip per not hot chip

The rate above represents a single quantity with a compound unit of hot chips per not hot chips.

Fun stuff, right?

I know!

All of this thinking is what fuelled me to create a full self-paced online proportional relationships course to help unpack the roadmap of proportional reasoning from spatial comparison all the way to proportional relationships.

After exploring part-whole fraction construct and representing a part-part ratio using the fraction bar for the purpose of dividing, you may begin to notice how important clarity around what a given fraction represents prior to attempting any comparison or manipulation of that fraction.

While the use of a set model and area model remain fairly similar regardless of whether you are dealing with a part-whole or part-part relationship, the use of a number line changes significantly.

Recall how the number line used in the part-whole counting cartons example made identifying the part and the whole fairly intuitive:

When we attempt to use a number line with a part-part ratio relationship, you'll notice that we cannot locate a single value on the number line because a part-part relationship is two separate quantities representing two different places along the number line:

Note that in the representation above, two parts are clearly shown and the sum of those two parts represents the whole.

Although the number line representation might not be as intuitive in the part-part case, you can visualize both parts and their sum represents the whole:

Extending from a single number line to a double-number line can help the learner see the part-part ratio relationship and easily scale both quantities in tandem. At any point along the double number line (or later, the ratio table) you can use partitive division to divide the parts to reveal a rate:

1/6 hot chips per not hot chip or 6 not hot chips per hot chip

In this case, there are one-sixth as many "hot" chips as there are "not hot" chips. Alternatively, there are 6 times as many "not hot" chips as there are "hot" chips.

Baking Brownies: Fraction as Quotient

As mentioned above, a part-part ratio relationship can be represented with a fraction bar when the intention is for two separate quantities to be divided. In other words, when we are using fraction as a quotient. The Paying Attention to Fractions document uses an example of kids fair sharing some brownies, so we'll stick to that and add some visuals to help.

Suppose there are 6 brownies that are to be shared amongst 4 friends fairly. How much should each person receive?

Like many situations in life, there are multiple ways to share these brownies fairly. Let's represent a few of these methods.

Using Partitive Division to Partition the Brownies into Fourths

Since there are 4 people to share the 6 brownies, we can divide the 6 brownies into fourths and then pass them out, one-by-one:

After splitting the brownies into quarters, you'll notice that each of the four people should receive 6 quarters. Hence why we can represent this fraction as quotient as the improper fraction, 6/4.

Sharing Full Brownies Equally, Then Partitioning Remainder

Some people might find it useful to share whole brownies equally and then when there are not enough whole brownies to share fairly, consider partitioning up the remaining brownies appropriately:

In the case with 6 brownies, we can share one brownie each and then partition the remaining brownies into four halves to share. This method would be equivalent to the mixed fraction of 1 and 1/2 brownies per person.

Partition Brownies Into Four Equal Portions

This method involves considering all 6 brownies and where cuts must be made to create four equal groupings:

This method would be another representation of the mixed fraction, 1 and 1/2 brownies for each person.

For each of the methods above, we could also consider the use of a number line to support the partitioning of the brownies to distribute a fair amount to each person. Here are a few examples:

Fraction as Operator

Another common construct is using a fraction as operator where a fraction is used to act as a function to:

scale a quantity larger;

scale a quantity smaller; or,

find equivalent fractional quantities.

Using a fraction as operator to scale a quantity larger or smaller feels pretty intuitive.

Consider a $100 investment.

Over time, this investment could increase in value or decrease in value.

If the investment decreases to $50, we could say that the investment "halved" from its initial value of $100.

In other words:

1/2 of $100 = $50

It is also possible that the $100 investment increases in value to $150.

In that case, we could say:

3/2 of $100 = $150

In both of the above cases, the fraction is still a part-whole relationship, however the intent is to use the fraction for scaling.

While it seems natural in both cases to think of the operation as simply multiplication, we can approach both cases through multiplication and/or division.

For example, in the second case, we could arrive at the equivalent result of $150 through each of these approaches:

3/2 of $100

$100 multiply by 3, divide by 2

$100 divide by 2, multiply by 3

Thus, the intent of the fraction 3/2 was to use as a function for operating.

Exploring Fractions Developmentally

Now that we have an understanding of the different fraction constructs and when we might bump into those situations, many are likely wondering how students will develop their understanding of fractions.

As mentioned earlier, students will likely bump into fractions through experiences in the world around them. For example, seeing that half of a brownie tray is missing, one fourth of a glass of milk remains, or each of three children will receive one third of the share of chocolates.

Exploring part-whole fractions beginning with unit fractions through counting can be helpful for building an understanding of the relationship of the part to the whole. This can be done through mathematical modelling using area models such as a bar model to encourage representing, comparing and ordering of fractions.

How Many Rolos: Adding and Subtracting Fractions

Let's have a look at another problem. We mind as well keep the context as food, because that is just yummy.

There are 7 pieces in every full roll of Rolo chocolate.

Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces.

How many full rolls of Rolo are there?

If we use concrete manipulatives or visuals, like we will in this instance, it is much easier to see how to add and subtract fractions. In this case, we'll show a few different representations including how using a number line can be helpful:

So in this particular example, you can see that 5 pieces plus 4 pieces will yield 9 pieces, which is more than a full 7 piece roll; 9/7 or 1 and 2/7.

Gimme a Break: From Adding and Subtracting to Multiplying Fractions

If you're a 3 act math fan, you might be familiar with my Gimme a Break 3 act math task involving the use of fractions as operators. I should note that while there are some useful pieces in that task, it has really evolved over time and I just haven't had the time to update the post yet. Here's some of the pieces I've added along the way.

The task begins with an opportunity for students to notice/wonder after watching a video of me opening a KitKat bar.

Later in the task, the following video is shown asking students to consider what operation could be taking place:

Since the video is open ended, I'm always hoping that all students will be able to share something mathematical about the situation. For example, we might see some responses like these:

However, the new learning I'm hoping to spend some time on is the multiplication of a fraction by a whole number. In this case, a student might think of this situation as 4 groups of one quarter of a whole bar:

Alternatively, we could think of this situation as one quarter groups of four whole KitKat bars:

In either case, the use of arrays and/or area models can be very useful when trying to help build an understanding of multiplication with fractions. In the following example, we extend this idea to the multiplication of two fractions:

While I find that multiplication of fractions seems to be the one operator I hear the least noise about, my gut tells me that it is more about being able to remember the algorithm and not due to any conceptual understanding. In my experience, many students can manage to remember to "multiply the tops, multiply the bottoms" without necessarily having any real conceptual understanding to fall back on in times of doubt.

Dividing Whole Numbers by Fractions and Fractions by Fractions

On the other hand, division of fractions seems to be a sore spots for many students (and adults) because the algorithm is a bit more complicated and less intuitive. Pair that with a very low level of conceptual understanding as to why the algorithm actually works and you'll end up with confusion and frustration for many. The gap between the pure memorization of steps and the conceptual understanding of what dividing fractions really means is often so large that many educators simply follow suit and focus purely on the algorithm. While I believe it is so important to build the conceptual understanding behind a complex idea like dividing fractions, I know that the task can be quite difficult and many teachers aren't so sure how to approach it. When we rush to an algorithm before constructing conceptual understanding, students (and adults) are often very difficult to engage in the heavy thinking required to connect the dots. This leaves many teachers believing that successful use of the algorithm is enough (or all they have time for) and they just move on to the next topic.

Let's look at the Gimme a Break context in terms of division of fractions. With manipulatives on the table such as pattern blocks or relational rods, I might ask students:

What would 1 ÷ 1/4 "look like" in terms of the KitKat task?

Can you show a neighbour what that looks like to you?

Hopefully students come up with some representation to model what a whole KitKat bar would look like an a quarter of a KitKat bar:

Then, I would ask them to determine how many quarter pieces there are in a whole KitKat bar. With this in mind, students should be able to "see" and manipulate their representation on the table to come up with an answer of 4.

Then, I might consider asking students to represent a couple more similar situations using whole numbers divided by a certain number of quarter pieces to build some confidence.

For example, 2 divided by 1/4:

When ready, students might be ready to start stepping outside of whole numbers to situations like 3/4 divided by 2/4:

Then, once students are really comfortable dividing fractions with common denominators, you might start moving to situations where the fractions have uncommon denominators.

Note that context is always very important for students to build a conceptual understanding. If we are trying to explain abstraction through abstraction (i.e.: explaining a rule algebraically based on another algebraic rule), then it is unlikely to stick. However, if we, the educators, spend the time necessary to understand mathematics conceptually, it will be a huge pay off for the understanding of our students. Only once students are able to visualize what mathematics "looks like" does it make sense for us to move towards more efficient methods like algorithms or more abstract problems where context is stripped away.

Throughout this post we have tackled the idea of fraction constructs by using a similar approach to how I introduce most big ideas in math class: using math tasks that are contextual, visual and concrete in order to build a conceptual understanding.

This post is by no means the "fractions rulebook", but rather a journal outlining my own journey to understanding fractions conceptually. As a secondary mathematics teacher, much of what our curriculum is built on is the assumed understanding of topics like fractions. However, the more I reflect on my own mathematical understanding, the more I realize that I never truly had a conceptual understanding of most concepts that I was teaching.

Let's avoid the rush to the algorithm and slow down to let kids truly experience and understand mathematics. I'd appreciate any pieces I should add to this post in the comments section.

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The Progression of Fractions | Understanding Fractions Conceptually

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Fractions are a beast of a concept that causes struggles for many adults and students alike. While we all come to school with some intuition to help us with thinking fractionally and proportionally, the complexity quickly begins to increase as we move from concrete, to visual, to symbolic and from identifying, to comparing, to manipulating.

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, Secondary Math Teacher and Instructional Coach

Published on

OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking. Webinar for Ontario Teachers Federation delivered December 2016.

1. 1. Making Connections Between Proportional Thinking and Fractional Thinking tapintoteenminds.com/OTF Ontario Teachers' Federation CONNECT

2. 2. @MathletePearcewww.tapintoteenminds.com playkh.complaykh.comGo to:

3. 3. @MathletePearcewww.tapintoteenminds.com wiz623

4. 4. @MathletePearcewww.tapintoteenminds.com Tap, then wait. wiz623

5. 5. @MathletePearcewww.tapintoteenminds.com Tap wiz623

6. 6. @MathletePearcewww.tapintoteenminds.com VISION

7. 7. @MathletePearcewww.tapintoteenminds.com Visual and Concrete Contextual VISION

8. 8. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual VISION

9. 9. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual CONFIDENCE

10. 10. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual PERCEPTION

11. 11. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual ACHIEVEMENT

12. 12. @MathletePearcewww.tapintoteenminds.com kylep.ca/patfrac

13. 13. @MathletePearcewww.tapintoteenminds.com Paying Attention to Fractions "The research suggests that explicit and precise changes to learning and teaching practices can have a substantial impact on children's understanding of fractions and future mathematical success. Instructional decisions have a signiﬁcant bearing upon students' ability to understand the concept of fractions, including the ability to represent fractions appropriately, compare the relative magnitude of two fractions, and complete calculations accurately." (Bruce, Chang, Flynn & Yearley, 2013, p. 32)

14. 14. @MathletePearcewww.tapintoteenminds.com "No area of elementary school mathematics is as mathematically rich, cognitively complicated, and difficult to teach as fractions, ratios, and proportionality. These ideas all express mathematical relationships: fractions and ratios are 'relational' numbers. They are the ﬁrst place in which students encounter numerals like ' 3/4 ' that represent relationships between two discrete or continuous quantities, rather than a single discrete ('three apples') or continuous quantity ('4 inches of rope')." Why Is Understanding Fractions Important? (Litwiller & Bright, 2002, p. 3)

15. 15. @MathletePearcewww.tapintoteenminds.com What is a Fraction? A fraction is a number. While fractional notation is typically used to represent quantities that are not whole, it is possible for all quantities to be represented as a fraction. 0 1 2 1 23 4 1 12 5 12 4 These simple descriptions do not appropriately communicate the complex constructs that lie within this big idea.

16. 16. @MathletePearcewww.tapintoteenminds.com "The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms." kylep.ca/patprop Connecting Fractions and Proportional Reasoning

17. 17. @MathletePearcewww.tapintoteenminds.com "The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms." 3 4 Relationship Connecting Fractions and Proportional Reasoning kylep.ca/patprop

18. 18. @MathletePearcewww.tapintoteenminds.com ...when you think of 8 as two fours or four twos rather than thinking of it as one more than seven. ...when you think of how a speed of 50 km/h is the same as a speed of 25 km/30 min. Connecting Fractions and Proportional Reasoning kylep.ca/patprop

19. 19. @MathletePearcewww.tapintoteenminds.com Students begin working towards proportional reasoning even before arriving at school when they make comparisons: this bag is heavier than that one Connecting Fractions and Proportional Reasoning I'm "this much" shorter than you are there are 5 more red candies than green While these early comparisons are likely absolute (i.e.: additive) in nature, we are one step closer to relating these quantities relatively (i.e.: multiplicatively).

20. 20. @MathletePearcewww.tapintoteenminds.com Fraction Constructs Let's Explore Four Common Fraction Constructs

21. 21. @MathletePearcewww.tapintoteenminds.com playkh.complaykh.comGo to:

22. 22. @MathletePearcewww.tapintoteenminds.com wiz623

23. 23. @MathletePearcewww.tapintoteenminds.com Tap, then wait. wiz623

24. 24. @MathletePearcewww.tapintoteenminds.com Tap wiz623

25. 25. @MathletePearcewww.tapintoteenminds.com Carton Counting

26. 26. @MathletePearcewww.tapintoteenminds.com Carton Counting

27. 27. @MathletePearcewww.tapintoteenminds.com How many eggs are there? Carton Counting

28. 28. @MathletePearcewww.tapintoteenminds.com How many cartons are there? Carton Counting

29. 29. @MathletePearcewww.tapintoteenminds.com How many cartons are there? 1 1 2 Carton Counting

30. 30. @MathletePearcewww.tapintoteenminds.com How many cartons are there? 1 1 2 2 1 21 Carton Counting

31. 31. @MathletePearcewww.tapintoteenminds.com How many cartons are there? 1 1 2 2 1 21 6 12 18 24Eggs Cartons Carton Counting

32. 32. @MathletePearcewww.tapintoteenminds.com How many cartons are there? Carton Counting

33. 33. @MathletePearcewww.tapintoteenminds.com How many cartons are there? 1 2 3 2 6 12 18 24Eggs Cartons 2 2 4 2 Carton Counting

34. 34. @MathletePearcewww.tapintoteenminds.com Part-Whole Relationship How many cartons are there? 1 2 3 2 6 12 18 24Eggs Cartons 2 2 4 2

35. 35. @MathletePearcewww.tapintoteenminds.com HOT "Hot or Not"

36. 36. @MathletePearcewww.tapintoteenminds.com HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? "Hot or Not"

37. 37. @MathletePearcewww.tapintoteenminds.com HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? "Hot or Not"

38. 38. @MathletePearcewww.tapintoteenminds.com HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? 1 6 "Hot or Not"

39. 39. @MathletePearcewww.tapintoteenminds.com "Hot or Not" HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? 1 6 "Hot" Chips

40. 40. @MathletePearcewww.tapintoteenminds.com "Hot or Not" HOT What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? 1 6 "Hot" Chips "Not Hot" Chips

41. 41. @MathletePearcewww.tapintoteenminds.com What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? HOT 1 6 "Hot" Chips "Not Hot" Chips "Hot" "Not Hot"

42. 42. @MathletePearcewww.tapintoteenminds.com What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette? 1 6 "Hot" Chips "Not Hot" Chips HOT "Hot" "Not Hot" Part-Part Relationship

43. 43. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Baking Brownies

44. 44. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Baking Brownies

45. 45. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Baking Brownies

46. 46. @MathletePearcewww.tapintoteenminds.com Partitioning Brownies into Fourths How can 4 friends share 6 brownies fairly? Multiple Representations: Baking Brownies

47. 47. @MathletePearcewww.tapintoteenminds.com Partitioning Brownies into Fourths How can 4 friends share 6 brownies fairly? Multiple Representations: 6 brownies 4 people Baking Brownies

48. 48. @MathletePearcewww.tapintoteenminds.com Partitioning Brownies into Fourths How can 4 friends share 6 brownies fairly? Multiple Representations: 6 brownies 4 people = 6 4 Baking Brownies

49. 49. @MathletePearcewww.tapintoteenminds.com Each person would get 6 one- fourth pieces of brownies. Partitioning Brownies into Fourths How can 4 friends share 6 brownies fairly? Multiple Representations: 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 6 brownies 4 people = 6 4 Baking Brownies

50. 50. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally Baking Brownies

51. 51. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally Baking Brownies

52. 52. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. Baking Brownies

53. 53. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. Baking Brownies

54. 54. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. 1 2 1 2 1 2 1 21 1 1 1 Baking Brownies

55. 55. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. 1 2 1 2 1 2 1 21 1 1 1 6 brownies 4 people = 1 2 1 Baking Brownies

56. 56. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Sharing full brownies equally, then partitioning the remainder. 1 2 1 2 1 2 1 21 1 1 1 6 brownies 4 people Each person would get 1 full and 1 half piece of brownies. = 1 2 1 Baking Brownies

57. 57. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Partition the brownies into four equal-size portions Baking Brownies

58. 58. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Partition the brownies into four equal-size portions Baking Brownies

59. 59. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Partition the brownies into four equal-size portions Baking Brownies

60. 60. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Partition the brownies into four equal-size portions 1 2 1 2 1 2 1 21 1 1 1 6 brownies 4 people = 1 2 1 Baking Brownies

61. 61. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Baking Brownies Partition the brownies into four equal-size portions 1 2 1 2 1 2 1 21 1 1 1 6 brownies 4 people Each person would get 1 full and 1 half piece of brownies. = 1 2 1

62. 62. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Using a number line 1 2 1 2 1 2 1 21 1 1 1 6 brownies 4 people Each person would get 1 full and 1 half piece of brownies. = 1 2 1 1 20 1 2 3 4 5 6 3 2 5 2 7 2 9 2 11 2

63. 63. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: 1 2 1 2 1 2 1 2 6 brownies 4 people Each person would get 1 full and 1 half piece of brownies. = 1 2 1 1 20 1 2 3 4 5 6 3 2 5 2 7 2 9 2 11 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Using a number line

64. 64. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: 6 brownies 4 people Each person would get 1 full and 1 half piece of brownies. = 1 2 1 2 40 1 2 3 4 5 6 6 4 10 4 14 4 18 4 22 4 Using a number line 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4

65. 65. @MathletePearcewww.tapintoteenminds.com How can 4 friends share 6 brownies fairly? Multiple Representations: Fraction as Quotient Partition the brownies into four equal-size portions 1 2 1 2 1 2 1 21 1 1 1 6 brownies 4 people Each person would get 1 full and 1 half piece of brownies. = 1 2 1

66. 66. @MathletePearcewww.tapintoteenminds.com How Many Rolos?

67. 67. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every roll of Rolo chocolate.

68. 68. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there?

69. 69. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? +

70. 70. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? + =

71. 71. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? + =1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7

72. 72. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? + =1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 5 pieces 7 pieces = 9 pieces 7 pieces + 4 pieces 7 pieces

73. 73. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 5 pieces 7 pieces = 9 pieces 7 pieces + 4 pieces 7 pieces 1 70 1 2 3 7 5 7 6 7 9 7 11 7 2 7 4 7 8 7 10 7 12 7 13 7 +

74. 74. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 5 pieces 7 pieces = 9 pieces 7 pieces + 4 pieces 7 pieces 1 70 1 2 3 7 5 7 6 7 9 7 11 7 2 7 4 7 8 7 10 7 12 7 13 7

75. 75. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 7 5 pieces 7 pieces = 9 pieces 7 pieces + 4 pieces 7 pieces 1 70 1 2 3 7 5 7 6 7 9 7 11 7 2 7 4 7 8 7 10 7 12 7 13 7

76. 76. @MathletePearcewww.tapintoteenminds.com How Many Rolos? There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? 1 7 1 7 5 7 = 2 7 + 4 7 1 70 1 2 3 7 5 7 6 7 9 7 11 7 2 7 4 7 8 7 10 7 12 7 13 7 1 There is one full roll of Rolos and an additional 2 pieces out of a roll of 7. 1

77. 77. @MathletePearcewww.tapintoteenminds.com Fractions As Operators There are 7 pieces in every full roll of Rolo chocolate. Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces. How many full rolls of Rolo are there? 1 7 1 7 5 7 = 2 7 + 4 7 1 70 1 2 3 7 5 7 6 7 9 7 11 7 2 7 4 7 8 7 10 7 12 7 13 7 1 There is one full roll of Rolos and an additional 2 pieces out of a roll of 7. 1

78. 78. @MathletePearcewww.tapintoteenminds.com Fraction Constructs

79. 79. @MathletePearcewww.tapintoteenminds.com What is a Fraction? A fraction is a number. While fractional notation is typically used to represent quantities that are not whole, it is possible for all quantities to be represented as a fraction. 0 1 2 1 23 4 1 12 5 12 4 These simple descriptions do not appropriately communicate the complex constructs that lie within this big idea.

80. 80. What Do You... Notice? Wonder?

81. 81. What Operation Could It Be?

82. 82. =? What Operation Could It Be?

83. 83. =? What Operation Could It Be?

84. 84. =? 4 4 or 1 1 4 What Operation Could It Be?

85. 85. = 4 4 or 1 1 4 What Operation Could It Be? 1 4 1 4 1 4

86. 86. = 4 4 or 1 1 4 What Operation Could It Be? 3 4+

87. 87. 4 4 or 1 1 4 What Operation Could It Be? =4x

88. 88. 4 4 or 1 1 4 What Operation Could It Be? "1/4 groups of 4" =4x

89. 89. 1 4 4 What Operation Could It Be? "1/4 groups of 4" x

90. 90. What Operation Could It Be? 1 4 4 "1/4 groups of 4" x

91. 91. 0 0 What Operation Could It Be? 1 4 4 "1/4 groups of 4" x

92. 92. 0 0 What Operation Could It Be? 1 4 4 "1/4 groups of 4" x 1 4 2 4 3 4 1 1 2 3 4

93. 93. 0 0 What Operation Could It Be? 1 4 4 "1/4 groups of 4" x 1 4 2 4 3 4 1 1 2 3 4

94. 94. 0 0 What Operation Could It Be? 1 44 "4 groups of 1/4" x 1 4 2 4 3 4 1 1 2 3 4

95. 95. What Operation Could It Be? 4 4 or 1 1 4 4 "1/4 groups of 4" = x 1 44 "4 groups of 1/4" xor

96. 96. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 = ? x

97. 97. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 = ? x

98. 98. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 x = ?

99. 99. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 x = ?

100. 100. @MathletePearcewww.tapintoteenminds.com 2 3 x 3 4 = 6 12 x

101. 101. @MathletePearcewww.tapintoteenminds.com = 6 12 ? ?

102. 102. @MathletePearcewww.tapintoteenminds.com = 6 12 ? ?

103. 103. @MathletePearcewww.tapintoteenminds.com = 6 12 ? ?

104. 104. @MathletePearcewww.tapintoteenminds.com = 6 12 1 2

105. 105. 2 31 ÷ Dividing Fractions

106. 106. 1 2 3÷ Dividing Fractions

107. 107. 1 2 3÷ Dividing Fractions

108. 108. 1 2 3÷ Dividing Fractions

109. 109. 1 2 3÷ Dividing Fractions

110. 110. 1 2 3÷ 1 1 2 Dividing Fractions

111. 111. 1 2 3÷ 1 1 2 = 1 1 2 Dividing Fractions

112. 112. 1 2 3÷ Dividing Fractions ÷

113. 113. 1 2 3÷ Dividing Fractions ÷

114. 114. Dividing Fractions 2 3÷ ÷ 1

115. 115. Dividing Fractions 2 3÷ 3 3 ÷

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135. 135. @MathletePearcewww.tapintoteenminds.com kylep.ca/patfrac

136. 136. @MathletePearcewww.tapintoteenminds.com Interconnected Visual and Concrete Contextual VISION

137. 137. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com

138. 138. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding

139. 139. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding Connect to Prior Knowledge CONNECTIONS

140. 140. Visual and Concrete Contextual TASKS VISION @MathletePearcewww.tapintoteenminds.com INQUIRY Construct Conceptual Understanding Connect to Prior Knowledge CONNECTIONS CONSOLIDATE Move Towards Procedural Fluency

141. 141. Two Groups of Students Struggle with unfamiliar problems Good at Math Not Good at Math @MathletePearcewww.tapintoteenminds.com

142. 142. Two Groups of Students Good at Math Not Good at Math Struggle with unfamiliar problems @MathletePearcewww.tapintoteenminds.com

143. 143. 3 Act Math Tasks iTunes U kylep.ca/curiousmath1p @MathletePearcewww.tapintoteenminds.com

144. 144. 3 Act Math Tasks tapintoteenminds.com/3act-math @MathletePearcewww.tapintoteenminds.com On The Web

145. 145. "Gimme a Break" kylep.ca/kitkatmath @MathletePearcewww.tapintoteenminds.com

146. 146. @MathletePearcewww.tapintoteenminds.com Visual and Concrete Contextual TASKS CONSOLIDATE INQUIRY Move Towards Procedural Fluency Connect to Prior Knowledge CONNECTIONS Construct Conceptual Understanding VISION

147. 147. Making Connections Between Proportional Thinking and Fractional Thinking tapintoteenminds.com/OTF Ontario Teachers' Federation CONNECT

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OTF Connect - Making Connections Between Proportional Thinking and Fractional Thinking

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What are fractions?

A fraction is part of a whole. It's less than 1 whole thing, but more than 0. We use fractions all the time in real life. Have you ever ordered a quarter-pound burger? Or noticed that your gas tank is half full? Both of these are fractions of the whole amount-a whole pound of meat, or a whole tank of gas.

Fractions look a little like division expressions, but they aren't problems to be solved. They are a way of expressing an amount. Like numbers, fractions tell you how much you have of something.

Click through the slideshow to learn how fractions work.

Writing fractions

Every fraction has two parts: a top number and a bottom number. In math terms, these are called the numerator and the denominator. Don't worry too much about remembering those names. As long as you remember what each number means, you can understand any fraction.

As you saw in the slideshow, the bottom number, or denominator, is the number of parts a whole is divided into. In our pizza example, we said each slice was 1/8 of the pizza. The denominator was 8, since the pizza was divided into 8 slices.

The top number, or numerator, refers to a certain number of those parts. It lets us know how much we're talking about. Since we're talking about one slice of pizza, our numerator is 1.

Let's look at another example. What if we divided the same pizza into 12 slices instead of 8? If we took one slice, that would be 1/12 of the pizza- 1 slice out of the 12 total slices. No matter what fraction you're trying to write, you always write it the same way-with the number of parts on the bottom, and the parts you're referring to on top.

Try This!

Write these images as fractions.

Reading fractions

In the example above, if you had a pizza with eight slices, each slice would be 1/8 of the pizza. You'd read that like this: one-eighth.

When we read or talk about fractions, we use special numbers called ordinal numbers. A good way to remember this is that many of them are the same numbers you use when you're putting things in order: third, fourth, fifth, and so on.

You might know some of these numbers already. For example, when you tell your boss you'll be at work in half an hour, you're saying that you'll get there in 1/2 of an hour. If you're helping a friend bake a cake and she asks you for a third of a cup of sugar, you might know to hand her the measuring cup that says 1/3.

Here are some of the most commonly used fractions:

A good rule to remember is that most ordinal numbers end in " th." So, 1/20 is one-twentieth. 1/35 is one-thirty-fifth. 1/54 is one-fifty-fourth.

What about fractions that don't have a 1 on top? Read these as if you were counting. So if 1/5 is one-fifth, then 2/5 is two-fifths, and 3/5 is three-fifths. The top number will always be a "normal" number like the ones you use to count, and the bottom number is always an ordinal number.

Try This!

Write fractions to match the text.

Mixed numbers

Sometimes you might see a fraction next to a whole number. We call this a mixed number. We'll talk more about mixed numbers in the next lesson. For now, we'll concentrate on learning how to read them. Let's take a look at this example:

is a mixed number. If we say we have 2 1/2 pizzas, it means we have 2 whole pizzas and 1/2 of another pizza. You can read 2 1/2 like this: two-and-a-half.

Let's try another example. What if you pour 1 whole cup of tea, then fill only 2/3 of another cup? You could write that situation like this:

You'd read 1 2/3 like this: one and two-thirds. Remember, the whole number is always first.

Try This!

Write the correct mixed number next to each picture.

Whole fractions

So far, you've learned that a fraction is a part of a whole. For example, 3/4 means you have three parts out of four parts total. But what if you had a fraction like this?

8/8

In this example, we have eight parts out of eight parts total. If the top number and the bottom number of a fraction are the same, then the fraction is equal to 1. That's because you have every part of the fraction, or one whole thing. This is sometimes known as a whole fraction.

So if you had eight slices of pizza out of eight total, you'd have one whole pizza.

Let's look at another example: 8/8 and 2/2. While these fractions may look different, they're actually just two ways of saying the same thing. Because these are whole fractions, 8/8 and 2/2 are both equal to 1. And because they're both equal to 1, they're also equal to each other.

Try This!

Write the correct whole fraction under each picture.

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Comparing fractions

In Introduction to Fractions, we learned that fractions are a way of showing part of something. Fractions are useful, since they let us tell exactly how much we have of something. Some fractions are larger than others. For example, which is larger: 6/8 of a pizza or 7/8 of a pizza?

In this image, we can see that 7/8 is larger. The illustration makes it easy to compare these fractions. But how could we have done it without the pictures?

Click through the slideshow to learn how to compare fractions.

As you saw, if two or more fractions have the same denominator, you can compare them by looking at their numerators. As you can see below, 3/4 is larger than 1/4. The larger the numerator, the larger the fraction.

Comparing fractions with different denominators

On the previous page, we compared fractions that have the same bottom numbers, or denominators . But you know that fractions can have any number as a denominator. What happens when you need to compare fractions with different bottom numbers?

For example, which of these is larger: 2/3 or 1/5? It's difficult to tell just by looking at them. After all, 2 is larger than 1, but the denominators aren't the same.

If you look at the picture, though, the difference is clear: 2/3 is larger than 1/5. With an illustration, it was easy to compare these fractions, but how could we have done it without the picture?

Click through the slideshow to learn how to compare fractions with different denominators.

Reducing fractions

Which of these is larger: 4/8 or 1/2?

If you did the math or even just looked at the picture, you might have been able to tell that they're equal. In other words, 4/8 and 1/2 mean the same thing, even though they're written differently.

If 4/8 means the same thing as 1/2, why not just call it that? One-half is easier to say than four-eighths, and for most people it's also easier to understand. After all, when you eat out with a friend, you split the bill in half, not in eighths.

If you write 4/8 as 1/2, you're reducing it. When we reduce a fraction, we're writing it in a simpler form. Reduced fractions are always equal to the original fraction.

We already reduced 4/8 to 1/2. If you look at the examples below, you can see that other numbers can be reduced to 1/2 as well. These fractions are all equal.
5/10 = 1/2

11/22 = 1/2

36/72 = 1/2
These fractions have all been reduced to a simpler form as well.
4/12 = 1/3

14/21 = 2/3

35/50 = 7/10
Click through the slideshow to learn how to reduce fractions by dividing.

Irreducible fractions

Not all fractions can be reduced. Some are already as simple as they can be. For example, you can't reduce 1/2 because there's no number other than 1 that both 1 and 2 can be divided by. (For that reason, you can't reduce any fraction that has a numerator of 1.)

Some fractions that have larger numbers can't be reduced either. For instance, 17/36 can't be reduced because there's no number that both 17 and 36 can be divided by. If you can't find any common multiples for the numbers in a fraction, chances are it's irreducible .

Try This!

Reduce each fraction to its simplest form.

Mixed numbers and improper fractions

In the previous lesson, you learned about mixed numbers. A mixed number has both a fraction and a whole number. An example is 1 2/3. You'd read 1 2/3 like this: one and two-thirds.

Another way to write this would be 5/3, or five-thirds. These two numbers look different, but they're actually the same. 5/3 is an improper fraction. This just means the numerator is larger than the denominator.

There are times when you may prefer to use an improper fraction instead of a mixed number. It's easy to change a mixed number into an improper fraction. Let's learn how:

Try This!

Try converting these mixed numbers into improper fractions.

Converting improper fractions into mixed numbers

Improper fractions are useful for math problems that use fractions, as you'll learn later. However, they're also more difficult to read and understand than mixed numbers. For example, it's a lot easier to picture 2 4/7 in your head than 18/7.

Click through the slideshow to learn how to change an improper fraction into a mixed number.

Try This!

Try converting these improper fractions into mixed numbers.

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Fractions: Comparing and Reducing Fractions

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en/fractions/introduction-to-fractions/content/ In Introduction to Fractions, we learned that fractions are a way of showing part of something. Fractions are useful, since they let us tell exactly how much we have of something. Some fractions are larger than others. For example, which is larger: 6/8 of a pizza or 7/8 of a pizza?

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Converting fractions, decimals, and percents

When we talk, we often use different words to express the same thing. For example, we could describe the same car as tiny or little or small. All of these words mean the car is not big. Fractions, decimals, and percents are like the words tiny, little, and small. They're all just different ways of expressing parts of a whole.

In this image, each measuring cup has the same amount of juice in it. But we've expressed this amount in three ways: as a fraction, as a percent, and as a decimal. Since they're expressing the same amount, we know that 1/2, 50%, and 0.5 are equal to each other. Any time we see 1/2, we'll know it can also mean 50% or 0.5.

Sometimes it's useful to convert one kind of number into another. For example, it's much easier to add 1/4 and 0.5 if you turn 0.5 into a fraction. Learning how to convert fractions, decimals, and percents will also help you as you learn more advanced math.

Fractions and decimals

Every fraction can also be written as a decimal, and vice versa. You may not do this very often, but converting decimals and fractions can help you in math. For example, it's easier to subtract 1/6 from 0.52 if you turn 1/6 into a decimal first.

Converting a fraction into a decimal

Let's convert a fraction into a decimal. We'll be using a math skill you've already learned: long division. To refresh your memory on this skill, you can review our Long Division lesson.

Click through the slideshow to learn how to convert a fraction into a decimal.

Try This!

Convert each of these fractions into a decimal.

Converting a decimal into a fraction

Now we'll do it in reverse. Let's convert a decimal into a fraction.

Click through the slideshow to see how to convert a decimal into a fraction.

Reducing a fraction may seem unnecessary when you're converting a decimal. But it's important if you're going to use the fraction in a math problem. If you're adding two fractions, you may even need to reduce or change both fractions so they have a common denominator.

Try This!

Convert these decimals into fractions. Be sure to reduce each fraction to its simplest form!

Percents and decimals

Knowing how to convert percents and decimals will help you calculate things like sales tax and discounts. To learn how, check out our Percentages in Real Life lesson.

Converting a percent into a decimal

Converting a percent into a decimal is surprisingly easy. It only takes a few simple steps.

Click through the slideshow to learn how to convert a percent into a decimal.

Why does this work?

Converting percents into decimals is so easy that you may feel like you've missed something. But don't worry-it really is that simple! Here's why the method we showed you works.

When we turn a percent into a decimal, we're actually doing two steps. First, we convert our percent into a fraction. Since all percents are out of 100, we just put the percent over 100, like this:

78% = 78/100

In the second step, we convert 78/100 into a decimal. You already know this means we'll divide the numerator by the denominator, like this:

78 ÷ 100 = 0.78

So why didn't we show you these steps in the slideshow? Because you can get the answer without them. You know that all percents are out of 100, so you can skip making the percent into a fraction. You have to divide the percent by 100 to get a decimal, but there's a quick way to do that. Just move the decimal point two spaces to the left! This way, you can get the same answer with just one easy step.

Try This!

Convert these percents into decimals.

Converting a decimal into a percent

Now we'll reverse what you just learned. Let's convert a decimal into a percent.

Click through the slideshow to see how to convert a decimal into a percent.

Try This!

Calculate these decimals as percents.

Percents and fractions

Knowing how to write percents as fractions and vice versa can help you in your everyday life. For example, let's say you earned a grade of 80% on a test. You can convert 80% into a fraction to find out how many of your answers were correct. When your teacher grades the test, she may do the opposite. If a student got 8 out of 10 questions right, the teacher can convert 8/10 to a percent to give the student a grade.

Converting a percent into a fraction

When you're converting a percent into a fraction, it helps to remember that percents are always out of 100. You can practice with percents in our Introduction to Percentages lesson.

Click through the slideshow to learn how to convert a percent into a fraction.

Try This!

Write these percentages as fractions. Make sure to reduce each fraction to its simplest form.

Converting a fraction into a percent

Converting a fraction uses two of the skills you just learned: writing a fraction as a decimal, and writing a decimal as a percent. Let's see how we can use these skills to convert a fraction into a percent.

Click through the slideshow to learn how to convert a fraction into a percent.

Try This!

Convert these fractions into percents.

Assessment

Want even more practice? Try out a short assessment to test your skills by clicking the link below:
Start Assessment

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Fractions: Converting Percentages, Decimals, and Fractions

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en/fractions/multiplying-and-dividing-fractions/content/ When we talk, we often use different words to express the same thing. For example, we could describe the same car as tiny or little or small. All of these words mean the car is not big. Fractions, decimals, and percents are like the words tiny, little, and small.

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Multiplying fractions

A fraction is a part of a whole. In the last lesson, you learned how to add and subtract fractions. But that's not the only kind of math you can do with fractions. There are times when it will be useful to multiply fractions too.

Click through the slideshow to learn how to write a multiplication problem with fractions.

Try This!

Try setting up the multiplication problem below. Don't worry about solving it yet!

A recipe calls for 2/3 of a cup of milk. You want to cut the recipe in half.

Note: Although our example says the correct answer is 2/3 x 1/2, remember, with multiplying order does not matter. 1/2 x 2/3 would also be correct.

Solving multiplication problems with fractions

Now that we know how to set up multiplication problems with fractions, let's practice solving a few. If you feel comfortable multiplying whole numbers, you're ready to multiply fractions.

Click through slideshow to learn how to multiply two fractions.

Try This!

Try solving the multiplication problems below.

Multiplying a fraction and a whole number

Multiplying a fraction and a whole number is similar to multiplying two fractions. There's just one extra step: Before you can multiply, you'll need to turn the whole number into a fraction. This slideshow will show you how to do it.

Click through the slideshow to learn how to multiply a fraction and a whole number.

Try This!

Try solving the multiplication problems below.

Dividing fractions

Over the last few pages, you've learned how to multiply fractions. You might have guessed that you can divide fractions too. You divide fractions to see how many parts of something are in something else. For example, if you wanted to know how many fourths of an inch are in four inches, you could divide 4 by 1/4.

Let's try another example. Imagine a recipe calls for 3 cups of flour, but your measuring cup only holds 1/3, or one-third, of a cup. How many thirds of a cup should you add?

We'll need to find out how many thirds of a cup are in three cups. In other words, we'll need to divide three by one-third.

We'd write the problem like this:

Try This!

Try setting up these division problems with fractions. Don't worry about solving them yet!

A recipe calls for 3/4 of a cup of water. You only have a 1/8 measuring cup.

Solving division problems with fractions

Now that we know how to write division problems, let's practice by solving a few. Dividing fractions is a lot like multiplying. It just requires one extra step. If you can multiply fractions, you can divide them too!

Click through the slideshow to learn how to divide a whole number by a fraction.

Try This!

Try solving these division problems. Don't worry about reducing the answer for now.

Dividing two fractions

We just learned how to divide a whole number by a fraction. You can use the same method to divide two fractions.

Click through the slideshow to learn how to divide with two fractions.

Try This!

Try solving these division problems. Don't worry about reducing the answer for now.

Multiplying and dividing mixed numbers

How would you solve a problem like this?

As you learned in the previous lesson, whenever you're solving a problem with a mixed number you'll need to convert it into an improper fraction first. Then you can multiply or divide as usual.

Using canceling to simplify problems

Sometimes you might have to solve problems like this:

Both of these fractions include large numbers. You could multiply these fractions the same way as any other fractions. However, large numbers like this can be difficult to understand. Can you picture 21/50, or twenty-one fiftieths,in your head?

21/50 x 25/14 = 525/700

Even the answer looks complicated. It's 525/700, or five hundred twenty-five seven-hundredths. What a mouthful!

If you don't like working with large numbers, you can simplify a problem like this by using a method called canceling. When you cancel the fractions in a problem, you're reducing them both at the same time.

Canceling may seem complicated at first, but we'll show you how to do it step by step. Let's take another look at the example we just saw.

Step 1

First, look at the numerator of the first fraction and the denominator of the second. We want to see if they can be divided by the same number.

In our example, it looks like both 21 and 14 can be divided by 7.

Step 2

Next, we'll divide 21 and 14 by 7. First, we'll divide our top number on the left: 21.

Then we'll divide the bottom number on the right: 14.

We'll write the answers to each problem next to the numbers we divided. Since 21 ÷ 7 equals 3, we'll write 3 where the 21 was. 14 ÷ 7 equals 2, so we'll write 2 where the 14 was. We can cross out, or cancel, the numbers we started with.

Our problem looks a lot simpler now, doesn't it?

Step 3

Let's look at the other numbers in the fraction. This time we'll look at the denominator of the first fraction and the numerator of the second. Can they be divided by the same number?

Notice they can both be divided by 25! You might have also noticed they can both be divided by 5. We could use 5 too, but generally when you are canceling, you want to look for the biggest number both numbers can be divided by. This way you won't have to reduce the fraction again at the end.

Step 4

Next, we'll cancel just like we did in step 2.
We'll divide our bottom number on the left: 50.

Then we'll divide the top number on the right: 25.

We'll write the answers to each problem next to the numbers we divided.

Step 5

Now that we've canceled the original fractions, we can multiply our new fractions like we normally would. As always, multiply the numerators first:

Then multiply the denominators:

Step 6

Finally, let's double check our work. 525/700 would have been our answer if we had solved the problem without canceling. If we divide both 525 and 700 by 175, we can see that 525/700 is equal to 3/4.

We could also say that we're reducing 525/700 to 3/4. Remember, canceling is just another way of reducing fractions before solving a problem. You'll get the same answer, no matter when you reduce them.

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Fractions: Multiplying and Dividing Fractions

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en/fractions/adding-and-subtracting-fractions/content/ A fraction is a part of a whole. In the last lesson, you learned how to add and subtract fractions. But that's not the only kind of math you can do with fractions. There are times when it will be useful to multiply fractions too.

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Find out how to add and subtract fractions with matching and different denominators in this video, activity and KS2 Maths Bitesize Guide.

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Learn how to simplify fractions when the numerator and denominator can be divided by the same number in this Bitesize KS2 maths guide.

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Revise how to multiply fractions and learn how to calculate proportions in this BBC Bitesize KS2 maths guide.

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Draft Grade 5 Math Overview and Scope and Sequence

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Grade 4  / Unit 5 / Fractions / Priority Document

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Draft Grade 4 Math Overview and Scope and Sequence

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Grade 4  / Unit 5 / Fractions / Priority Document

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Comparing & Ordering Fractions for Beginners

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Compare Fractions with the Same Denominator. Grade 3

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Compare Fractions with the Same Numerator. Grade 3

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Ordering Fractions | How to Order Fractions with Unlike Denominators

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Comparing and Ordering Fractions with Different Denominators (fraction strips)

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Practice fractions with our popular math games. Learn about equivalent fractions, adding and subtracting fractions, multiplying fractions and more.

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Fractions: Parts of a Set Pancakes | Game | Education.com

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Get your flapjacks! Kids flip pancakes to represent fractions in this fun-filled diner game.

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In this galactic space fractions game, students compare like denominators to help Floyd get to planet Zorb. Students are asked to fill enough fuel cells to match the numbers represented in the fraction. Then they compare the fractions to determine which one is bigger or smaller. Comparing fractions is a key element of the math curriculum in fourth grade, in particular.

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Comparing Fractions with Common Denominators Game | Game | Education.com

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In this galactic space fractions game, students compare like denominators to help Floyd get to planet Zorb. Students are asked to fill enough fuel cells to match the numbers represented in the fraction. Then they compare the fractions to determine which one is bigger or smaller.

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Can you find and match all of the equivalent fractions? In this interactive online game, Match: Equivalent Fractions, students will boost their fractions knowledge and gain confidence identifying equivalent fractions. With several rounds of play, learners will get plenty of practice with this important third grade math skill.

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Match: Equivalent Fractions | Game | Education.com

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Can you find and match all of the equivalent fractions? In this interactive online game, Match: Equivalent Fractions, students will boost their fractions knowledge and gain confidence identifying equivalent fractions. With several rounds of play, learners will get plenty of practice with this important third grade math skill.

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Kids help Roly swing from vine to vine as they practice adding jungle fractions on a number line! This interactive math game is designed to help third-grade students understand and visualize fractions as parts of a whole by seeing them on a number line. With colorful animation and amusing jungle sounds, practicing adding fractions has never been more entertaining!

Grade

Subject

View aligned standards

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Jungle Fractions on a Number Line | Game | Education.com

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Kids see how fractions get added by swinging across a jungle-themed number line.

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Least Common Multiples vs. Greatest Common Factors (LCM vs. GCF) | Math with Mr. J

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Multiples vs. Factors | What are Multiples and Factors? | Math with Mr. J

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Greatest Common Factor and Least Common Multiple | GCF and LCM | Math with Mr. J