Post 2a: Are You Ready? Rational Number Operations: Multiplying and Dividing Fractions

(by Chris B. with help from Sage the intrepid student dog)

A massive chasm yawns before Sage and the kids as they enter the room. Looking to his left and right, Sage notices several blocky items strewn on that look like sections of a bridge. However, they are all different sizes. As Sage approaches one, he notices that they have some fractions floating in the air around the blocks. Sage realizes he can interact with the floating math just like in the last room. He wonders if by changing the fractions, he can get the bridge back together...

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You and Sage probably have a pretty good handle on Integers from the last couple of posts. The next step is to review how to apply operations to Rational Numbers. Rational numbers expand from Integers to include all the partial values in between the integers. This would include positive and negative fractions and decimals. This post is going to focus on working with fractions specifically. To learn more about rational operations, many web tools can help such as this one.

Sage is a big fan of fractions because dinner is measured up in fractions of a cup. Yum!

Unlike Integer Operations, Fractional Operations are easiest to think about using Multiplication and Division first as it is the most straightforward. The next posts cover Addition and Subtraction as they can be a little more complicated, requiring a "Common Denominator". The final post in this string will cover exponents.

Remember, a Fraction is made up of two parts. The "Numerator" is the number on top of the Fraction. The "Denominator" is the number on the bottom of the fraction.

Multiplying Rational Numbers

Recall from your classwork that if you multiply two fractions, simply multiply the numerators together to form a new numerator and then multiply the denominators together to form a new denominator.

Sage finds some math work on a wall mounted touchscreen TV showing an example for how to multiply fractions:

Sage sniffs at the stack of blocks. As he nears the glowing blocks, a math problem pops up floating in space. The math problem reads:

5/11 x 23/15

Sage thinks back to the interactive TV Demo and wonders if he can recreate this math work there. He walk back into the room and see that the touchscreen TV has a new blank worksheet with the new math problem pre-populated. Hints appear on the side. Help Sage derive the correct answer!

Sage worked through the problem and ended up with 115/165. Is that what you got? Sage walks over to the pile of blocks and sees that there are 6 blocks with 6 different numbers. Sage finds a 5, 15, 23, 33, 115 and 165. Sage mouths the 165 and places it on a plate in front of the chasm, and it just sits there. Then he confidently takes the 115 and places it on top. Again, nothing happens. Sage is confused. "What did I miss?" Sage wonders.

"Wait," Sage thinks "I didn't remove the common factor! Both 115 and 165 are divisible by 5. After dividing the numerator and denominator by 5, Sage ends up with 23/33. Sage knocks the incorrect blocks out of the way and places the 33 on the plate and it latches into place as if drawn to the plate by a magnet. Then Sage grabs the 23 block and feels it pulling him to the plate. Sage releases the block just above the 33 and it snaps into place. Woohoo! The Blocks flash and all 6 blocks grow into large tiles that begin to fill in the missing flooring over the chasm. Sage tentatively paws at the new floor, and it feels supported, but it doesn't feel strong enough to stand on.

"What now?" Sage wonders, and just then the 2nd pile of blocks begins to glow and a new tab on the TV activates.

Dividing Rational Numbers

Dividing rational numbers is the next easiest rational operation when considering fractions. To divide rational numbers, simply multiply first rational number by the reciprocal of the second rational number.

"The what-rical?" Sage wonders.

Reciprocals are when you take a fraction and write it upside down, swapping the numerator and denominator. So, the reciprocal of 3/4 is 4/3. The reciprocal of 5/22 is 22/5.

"I can do that!" Sage exclaims with a bark. "See?"

Sage returns to the wall mounted touchscreen TV to see this:

Sage notices a new problem hovers over the chasm.

5/11 ÷ 10/12

Can you help Sage solve this?

Sage came up with 25/66. Is that what you got? Again, Sage looks at the stack of blocks which now include 6,11, 25, 60, 66, and 110. Sage mouths the 66 and begins walking towards the plate in front of the chasm. Sage realizes that just like the first attempt, this block does not seem to be attracted to the plate. The dog puts the block down and thinks.

"Did I do something wrong?" Sage wonders. Just then Sage remembers the clues in the scratchpad. Upon closer inspection, Sage sees a bunch of X's right from the first step.

"Hey, what do you expect. I'm a dog! Like I can see red!" Sage exclaims!

"Wait, I forgot the reciprocal in the first step!" Sage races back through the math after performing the reciprocal in the first step. Sage ends up with 6/11. The dog confidently grabs the 11 block and it hums in Sage's mouth with eager anticipation. The block snaps to the plate! Sage mouths the 6, and again he can feel the energy in the block. The 6 ends up on top of the 11 and another flash of brilliance happens as the blocks turn into more tiles to cover even more of the chasm!

Sage notices that the other kids don't seem to be even acknowledging what Sage is doing. He wonders if they are scared of math...

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Catch your algebra mistakes before they cost you

The scratchpad in this guide runs on moment.of.math — a free Chrome extension that checks your algebra line by line, on any problem. It can generate new variations problems, so you can get as much practice as you need. On many homework sites, it automatically detects the math on the page, so you don't have to copy anything over. Try it on tonight's homework.

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