One is one ... or is it?

Summary

The video script explores the concept of units in mathematics and everyday life, emphasizing the flexibility of what we consider as 'one.' It explains how units can be both composed, like a dozen eggs, and partitioned, like slices of bread. The script uses relatable examples to illustrate the importance of units in our number system, showing how they can be manipulated to represent different quantities. It concludes by connecting these ideas to the broader mathematical principles of place value, fractions, and the variable nature of the number 'one' in different contexts.

Takeaways

• 🍎 The concept of 'one' is relative and can change based on the unit of measurement.
• 🥚 'A dozen eggs is' is correct because 'a dozen' is a composed unit representing 12 eggs as one entity.
• 🛒 Units can be composed by grouping smaller units together (e.g., 12 eggs make a dozen) or partitioned by dividing larger units (e.g., a loaf of bread into slices).
• 🔢 Whole numbers and decimals are based on the idea of place value, which is dependent on the unit of measurement.
• 🎴 Composed units are created by combining smaller units (e.g., a deck of cards, a pair of shoes), while partitioned units are parts of a whole (e.g., a slice of pizza, a chocolate square).
• 🍞 The grocery store example illustrates the flexibility of units, where a bag of apples or a loaf of bread can be considered as 'one' for the purpose of purchase.
• 📦 When units are nested, such as a box of toaster pastries containing packs of two, it demonstrates that 'one' can represent multiple levels of composition.
• 🍕 Sharing a pizza slice shows that partitioning can occur at different levels, changing what 'one' represents in a practical context.
• 🧮 In mathematics, the value of 'one' is not static; it can represent a single item, a group, or a larger collection depending on the unit system used.
• 📘 Understanding the flexibility of units is crucial for grasping mathematical concepts like place value, fractions, and the representation of numbers in different systems.

Q & A

Which phrase is grammatically correct: 'A dozen eggs is' or 'A dozen eggs are'?

The grammatically correct phrase is 'A dozen eggs is' because 'a dozen' is considered a singular unit.

What is the significance of the story about buying a bag of apples in the script?

The story illustrates the concept of units in counting, emphasizing that 'one' can represent different quantities depending on the context, such as a single apple versus a bag of apples.

What does the speaker mean by 'whole number place value' and 'decimal place value'?

The speaker refers to the way numbers are structured in our counting system, where the value of a digit changes based on its position, whether it's in the whole number or decimal part.

How does the speaker define 'composing units' and 'partitioning units'?

Composing units is the process of combining smaller units to form a larger one, such as 12 eggs making a dozen. Partitioning units is the opposite, where a larger unit is divided into smaller parts, like a loaf of bread into slices.

What is an example of a composed unit mentioned in the script?

A deck of cards is an example of a composed unit, as it is made up of multiple individual cards grouped together.

Can you explain the concept of partitioned units using an example from the script?

A partitioned unit is a larger unit that is divided into smaller parts. An example from the script is a chocolate bar divided into squares, where each square is a partitioned unit.

Why does the speaker say that 'one isn't always one' in the context of mathematics?

The speaker points out that in mathematics, the concept of 'one' can represent different quantities depending on the unit. For instance, 'one' could mean one item, a dozen, or even a hundred, depending on the context.

How does the speaker use the example of toaster pastries to explain units?

The speaker uses toaster pastries to show how units can be composed of other composed units. A box of toaster pastries contains multiple packs, which in turn contain individual pastries, demonstrating different levels of units.

What is the mathematical significance of the number 10 according to the script?

The number 10 is significant because it represents a transition from single units to groups of units, where 'one' in the tens place signifies a group of ten ones.

How does the concept of units relate to the idea of fractions discussed in the script?

The concept of units relates to fractions because both involve dividing a whole into parts. Just as a unit can be composed or partitioned, fractions represent parts of a whole, emphasizing the flexibility of the 'one' unit.

What does the speaker imply about the certainty in mathematics when they say 'one isn't always one'?

The speaker implies that while mathematics is often seen as absolute, the concept of 'one' is flexible and context-dependent, challenging the idea of absolute certainty by showing the relativity of numerical units.

Outlines

00:00

🥚 The Concept of Units in Mathematics

This paragraph discusses the significance of units in mathematics and everyday life. It starts with a reflection on the correct usage of 'is' versus 'are' when referring to a dozen eggs, leading to a broader exploration of the concept of 'one'. The author uses personal anecdotes, such as buying a bag of apples and a loaf of bread, to illustrate how the idea of 'one' can change based on the unit of measurement. The paragraph explains that our number system relies on the flexibility to redefine what constitutes 'one', whether through composition (like a dozen eggs) or partition (like slices of bread). It emphasizes that once a new unit is established, it can be treated as a single entity, which is fundamental in understanding whole numbers, decimals, and fractions. The narrative also touches on how these concepts apply to more complex mathematical operations and the importance of recognizing that the number 'one' can represent different quantities depending on the context.

Keywords

💡Whole Number Place Value

Whole number place value refers to the position of a digit within a number, which determines its value. In the script, this concept is used to explain how our number system is built on the idea of what constitutes 'one'. For example, the number 10 is not just a single entity but represents one group of ten, emphasizing that 'one' can be redefined as a unit of a larger quantity.

💡Decimal Place Value

Decimal place value is the system of expressing numbers using base-10, where each digit after the decimal point represents a fraction of a whole. The script uses this concept to illustrate that 'one' can represent different quantities, such as 1, 0.1, or 0.01, depending on its position relative to the decimal point.

💡Fractions

Fractions represent a part of a whole, expressed as a ratio of two integers. In the script, fractions are mentioned as a way to understand that 'one' can be divided into smaller parts, such as halves or quarters, which are still considered units within the context of the whole.

💡Compose

To compose, in the context of the script, means to form a larger unit by combining smaller units. This is exemplified by a dozen eggs, where 12 individual eggs are grouped together to form a new unit called a dozen, demonstrating the flexibility of what can be considered 'one'.

💡Partition

Partitioning is the act of dividing a whole into smaller parts. The script uses the example of a loaf of bread being cut into slices, where each slice represents a partitioned unit. This concept shows that 'one' can be broken down into smaller units, changing its value relative to the whole.

💡Unit

A unit, in the script, is a single entity within a given system of measurement or counting. The concept is central to understanding how 'one' can change based on the context, such as a dozen eggs, a box of toaster pastries, or a slice of pizza, each representing a different unit of measurement.

💡Place Value

Place value is the numerical value given to a digit based on its position in a number. The script discusses how place value is crucial for understanding how 'one' can represent different quantities, such as in the case of the number 10, where the '1' represents one group of ten rather than a single entity.

💡Composed Units

Composed units are created by combining multiple smaller units into a larger one. The script uses examples like a dozen eggs and a deck of cards to illustrate how composed units are formed and how they can be treated as a single entity for the purpose of counting or measurement.

💡Partitioned Units

Partitioned units are formed by dividing a larger unit into smaller parts. The script explains this with examples like a loaf of bread being divided into slices or a chocolate bar into squares, showing how a single unit can be partitioned into smaller, manageable units.

💡Mathematical Certainty

Mathematical certainty implies that mathematical operations and values are fixed and unchanging. The script challenges this notion by arguing that even the concept of 'one' can vary depending on the unit, suggesting that mathematics can be more flexible and context-dependent than it might initially appear.

💡Flexibility of Units

The flexibility of units is the idea that units can be redefined or recontextualized based on the situation. The script emphasizes this by discussing how 'one' can mean different things in different units, such as a dozen eggs versus a single egg, highlighting the adaptability of mathematical concepts.

Highlights

The debate between 'A dozen eggs is' and 'A dozen eggs are' highlights the importance of units in language and mathematics.

The concept of 'one' is flexible and can change based on the unit of measurement being used.

Whole number place value, decimal place value, and fractions are all dependent on the idea of changeable units.

The grocery store anecdote illustrates the practical application of units in daily life.

The idea that 'one' can represent different quantities is fundamental to understanding mathematics.

There are two ways to change units: composing and partitioning.

Composed units are created by grouping smaller units together, such as a dozen eggs.

Partitioned units are created by dividing a larger unit into smaller parts, like slices of bread.

Once a new unit is established, it can be treated as a single entity for further composition or partitioning.

Examples of composed units include a deck of cards, a pair of shoes, and a jazz quartet.

Examples of partitioned units include a square of chocolate, a section of an orange, and a slice of pizza.

The toaster pastries example shows how composed units can be further composed into sets.

Sharing a slice of pizza involves partitioning a partitioned unit into smaller pieces.

In mathematics, the concept of 'one' is not always a single entity, as it can represent a group or unit.

The number 10 is an example of a unit that can represent one group of ten individual units.

The flexibility of 'one' in mathematics allows for the understanding of larger numbers like 100 as composed of smaller units.

The concept of units is crucial for understanding place value and the structure of the number system.

Transcripts

Which is correct: "A dozen eggs is?" Or "A dozen eggs are?"

I remember being in elementary school,

and my teachers making a big deal about the unit.

And I never really got that,

until one day, I was in the grocery store,

and I wanted to buy an apple, but I couldn't buy one apple.

I had to buy a whole bag of apples.

So I did. I bought one bag of apples,

I took it home, I took one apple out of the bag, and I cut it up.

And then I ate one slice.

One bag, one apple, one slice.

Which of these is the real "one"?

Well, they all are of course,

and that's what my elementary teachers were trying to tell me.

Because this is the important idea behind whole number place value,

decimal place value and fractions.

Our whole number system depends

on being able to change what we count as "one".

Our whole number system depends on being able to change units.

There are two ways to change units.

We can compose, and we can partition.

When we compose units,

we take a bunch of things, we put them together to make a bigger thing,

like a dozen eggs.

We take 12 eggs, put them together to make a group,

and we call that group a dozen.

A dozen eggs is a composed unit.

Other examples of composed units

include a deck of cards, a pair of shoes, a jazz quartet

and of course, Barbie and Ken make a couple.

But think about a loaf of bread.

That's not a composed unit,

because we don't get a bunch of slices from a bunch of different bakeries

and put them together to make a loaf.

No, we start with a loaf of bread

and we cut it into smaller pieces called slices,

so each slice of bread is a partitioned unit.

Other examples of partitioned units

include a square of a chocolate bar, a section of an orange

and a slice of pizza.

The important thing about units is that once we've made a new unit,

we can treat it just like we did the old unit.

We can compose composed units,

and we can partition partitioned units.

Think about toaster pastries.

They come in packs of two,

and then those packs get put together in sets of four

to make a box.

So when I buy one box of toaster pastries,

am I buying one thing,

four things, or eight things?

It depends on the unit.

One box, four packs, eight pastries.

And when I share a slice of pizza with a friend,

we have to cut "it" into two smaller pieces.

So a box of toaster pastries is composed of composed units,

and when I split a slice of pizza,

I'm partitioning a partitioned unit.

But what does that have to do with math?

In math, everything is certain.

Two plus two equals four,

and one is just one.

But that's not really right.

One isn't always one.

Here's why: we start counting at one,

and we count up to nine: 1, 2, 3, 4, 5, 6, 7, 8, 9,

and then we get to 10, and in order to write 10,

we write a one

and a zero.

That one means that we have one group,

and the zero helps us remember

that it means one group, not one thing.

But 10, just like one,

just like a dozen eggs,

just like an egg,

10 is a unit.

And 10 tens make 100.

So when I think about 100,

it's like the box of toaster pastries.

Is 100 one thing,

10 things

or 100 things?

And that depends on what "one" is,

it depends on what the unit is.

So think about all the times in math when you write the number one.

No matter what place that one is in,

no matter how many things that one represents,

one is.