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### Course: 6th grade (Illustrative Mathematics)>Unit 4

Lesson 5: Lesson 15: Volume of prisms

# Volume with cubes of fractional side lengths

Before, we've found volume by seeing how many cubes with 1-unit side lengths would fit into an object. Find out what happens when we find volume with smaller cubes.

## What is volume?

Volume is the amount of $3$-dimensional space an object occupies. We measure volume in cubic units.
For example, the rectangular prism below has a volume of $24$ cubic units because it is made up of $24$ unit cubes.
We can also find the volume of a rectangular prism by multiplying the side lengths.
$4\phantom{\rule{0.167em}{0ex}}\text{units}\cdot 2\phantom{\rule{0.167em}{0ex}}\text{units}\cdot 3\phantom{\rule{0.167em}{0ex}}\text{units}=24\phantom{\rule{0.167em}{0ex}}\text{cubic units}$
That works well when we can fill the prism completely with unit cubes. How could we find volume when a prism has fractional side lengths and spaces too small to fill with unit cubes?

## Filling a unit cube with smaller cubes

Let's try starting with smaller cubes.
1.1
This is a cubic centimeter because each of its sides is $1\phantom{\rule{0.167em}{0ex}}\text{cm}$ long.
How many dice with edge lengths of $\frac{1}{2}\phantom{\rule{0.167em}{0ex}}\text{cm}$ do we need to fill the cubic centimeter?
dice

1.2
What is the volume, in cubic centimeters, of a die with edge lengths of $\frac{1}{2}\phantom{\rule{0.167em}{0ex}}\text{cm}$?
cubic centimeters

## Filling a rectangular prism with smaller cubes

Let's consider the following rectangular prism.
2.1
What is the volume of the prism?
${\text{cm}}^{3}$

2.2
Label how many dice with edge lengths of $\frac{1}{2}\phantom{\rule{0.167em}{0ex}}\text{cm}$ would fit across the length, width, and height of the same prism.
Click each dot on the image to select an answer.

2.3
Based on the numbers of dice you found above, how many dice with edge lengths of $\frac{1}{2}\phantom{\rule{0.167em}{0ex}}\text{cm}$ would it take to fill the prism?
dice

Before continuing, take a minute to tell a friend how you know the number of dice it would take to fill the prism.
2.4
How does the volume of the prism, in cubic centimeters, relate to the number of dice with edge lengths of $\frac{1}{2}\phantom{\rule{0.167em}{0ex}}\text{cm}$ it takes to fill the prism?
The volume of the prism, in cubic centimeters, is
times the number of dice it takes to fill it.

Why is the number of dice different from the volume?
2.5
Suppose we fill the following prism with dice with $\frac{1}{2}\phantom{\rule{0.167em}{0ex}}\text{cm}$ side lengths.
What is the product of the number of dice and the volume per die, and what does that product represent?

## Finding volume with cubes with fractional side lengths

Now we know $2$ different ways to find the volume of a prism with whole number side lengths:
1. Find the number of cubes of some size that would fit it and multiply by the volume per cube.
2. Multiply the side lengths.
Either method gives us the same volume. Do you have another method? Tell us about it below in the comments.

## Prisms with fractional side lengths

Suppose we fill the following prism with cubes with side lengths of $\frac{1}{4}\phantom{\rule{0.167em}{0ex}}\text{cm}$.
How would you find the number of cubes that fill the prism?
3.1
How many cubes with side lengths of $\frac{1}{4}\phantom{\rule{0.167em}{0ex}}\text{cm}$ does it take to fill the prism?
cubes

3.2
What is the volume, in cubic centimeters, of a cube with side lengths of $\frac{1}{4}\phantom{\rule{0.167em}{0ex}}\text{cm}$?
${\text{cm}}^{3}$

3.3
What is the volume of the prism?
${\text{cm}}^{3}$

3.4
How is the number of cubes with side lengths of $\frac{1}{4}\phantom{\rule{0.167em}{0ex}}\text{cm}$ related to the volume, in cubic centimeters, of the prism?
The number of cubes is
times the cubic centimeters of volume of the prism.

3.5
Evaluate $1\frac{1}{4}\phantom{\rule{0.167em}{0ex}}\text{cm}×2\frac{1}{2}\phantom{\rule{0.167em}{0ex}}\text{cm}×3\phantom{\rule{0.167em}{0ex}}\text{cm}$ (the product of the dimensions of the prism).
${\text{cm}}^{3}$

## Summary

Both strategies of finding volume work with rectangular prisms with fractional side lengths, too! Describe those $2$ strategies to a friend.
Do you have another way of finding volume when the rectangular prism has fractional side lengths? Tell us about it in the comments.

## Want to join the conversation?

• This can be confusing at time, so here's some tips!
Anything to the power of 3, is cubed, so it's multiplied by itself 3 times.

Anytime you have mixed numbers, just turn everything into a fraction (remember to simplify!) and multiply by the least common multiple (LCM)!

Area can get annoying, but think of it like your bedroom! By measuring two of your walls by your floor, you can see how many cubes could fill your room! (2 x 3 x 10 = 60)
• wow, thanks so much! this explanation made the mixed numbers way easier :0 i think i'll still need help from my classmates or something though lol those are hard
• if you understand it
(1 vote)
• L x W x H = volume
• thanks
• 😭this is actually hard but I might just have to go to over videos
• Its not hard wit yo dum ahhhhhhhh
• OMG this is so confusing i only got like 3 questions right
• allowd talking
(1 vote)
• Cap thats not true
• THis is a little bit confusing.
• yeah this is so confusing
• I am kinda confused
• Are you kidding me this is so hard
• I don't get this at all to the point were I guess
• To find the volume in cubic centimeters, of a cube with fractional side lengths, simply cube the denominator of the fraction:
the volume of a cubic centimeter, in 1/2 centimeter side length cubes, would be =8 because we multiply the denominator (2) times itself 3 times for length, width, height.

So, the volume of a 1/2 centimeter cube will be 1/8 of the 1 cube centimeter.

The same will go for the volume of a cube centimeter if we calculate in 1/4 centimeter cubes: multiply the denominator of the fraction times three to get the length, width, height, like this: 4 x 4 x 4 = 64.

So, the volume of a 1/4 centimeter cube will be 1/64 of the 1 cube centimeter.

Example:

We want to know how many 1/2 centimeter cubes we need to find the volume of a rectangle that is 3(length) x 5(width) x 1(height), so we multiply 3 and 5 and 1 by 2 to work with 1/2 centimeter cubes.

Now we have 6(length) x 10(width) x 2(height), so we multiply: 6 x 10 = 60 and x 2 = 120.

The result will be x8 the original number of the volume in cube centimeters, and likewise, the original number in cube centimeters will be 1/8 of the 1/2 cube centimeter result.

I really hope this was helpful, good luck!👍🥳