Parts of Things

Some of the dolls were doing their homework at the dining room table.

Jolena thought Billy and Emil looked unhappy. "What's the matter?" she asked.

"We're learning about fractions (FRAK-shuns) in school," Billy replied. "I just don't get it."

"I'm having trouble, too," Emil sighed. "I don't understand why we have fractions."

"I understand a half," Mariah said. "That's a fraction. A half is like when you share a cookie with a friend. One of you cuts the cookie into two pieces, and the other one gets to choose which piece to take."

"Yes," Jolena agreed. "That's so the doll who cuts the cookie will be careful to make both the same size, because otherwise the other doll will pick the larger piece. Two halves are two pieces of one cookie that are exactly the same size."

"I think that's a great idea!" Emil exclaimed. "It's fair that way. If the doll who cut the cookie got to pick which piece to take, the other doll would get the smaller piece for sure!"

Jolena thought for a moment. "Fractions help us tell how much of something to use," she said. "I know a little about fractions. I need to know that for cooking. Come into the kitchen, and I'll show you what I know. Maybe it will help you with your homework!"

Jolena led the other dolls into the kitchen, where she pulled out some measuring spoons and some measuring cups. Those are spoons and cups she uses to measure how much to use of something, so she can follow a recipe. Because the countertop had lots of spots, she placed the spoons and cups on the small cutting board. She was afraid that Emil might not see things as clearly as the other dolls, even with his special glasses, and Jolena wanted to make it easier for him to see them.

"These are measuring spoons," Jolena said, showing the other dolls the spoons. "Each one has a fraction on it except the one teaspoon measure."

The dolls all looked at the spoons. Three of the spoons each had two numbers on them. Jolena picked up the spoon that had "1/8" on it. "This is a one eighth teaspoon," she said. "That means you would have to fill it eight times to have as much as one teaspoon."

Then she picked up the spoon that had "1/4" on it. "This is one fourth teaspoon. You need four of this one to be one teaspoon, and this," Jolena said, picking up the spoon that had "1/2" on it, "is a half teaspoon. There are two of these in a teaspoon."
Billy picked it up and looked at it. Then he looked at the other dolls.

"I wonder why we don't we call it a 'one second' teaspoon," Billy said after he placed it carefully back on the cutting board.

"I don't know," Jolena replied. "Maybe it's because we use seconds for time."

"A one second teaspoon wouldn't last very long," Emil said, and they all laughed.

Jolena pointed to the largest spoon. "If the recipe says to add one teaspoon, I use this one, but if I need an eighth of a teaspoon of something, I use this one," Jolena explained, pointing to the smallest measuring spoon. 

"I have measuring cups in fractions, too. These blue cups are for dry things, like flour and sugar. They are 1/4, 1/3, 1/2 and a whole cup. If you fill one of these to the top and no more, you will have an exact amount." 

"For measuring liquids, like water or milk," Jolena went on, "I use these glass measuring cups. You can see through them, so you can pour in a liquid until it reaches the amount you need. You can use the same cup for different amounts. The cup is marked with the same fractions as the set of blue cups, but it's all in one cup."
"I see the glass cup is bigger than the mark at the top," Emil said.
"Yes," Jolena agreed. "Liquids spill easily, so the cup has extra room in it, but you only fill it just to the line."

"The one eighth teaspoon is smaller than the one fourth teaspoon," Billy noticed, "but eight is a bigger number than four."

"I think," Mariah said, "it's because when you cut something into more parts, like eight, each part is smaller."

"That's right," Jolena agreed. "If you had to share a cookie with five friends, you would need to cut it into smaller pieces, if everyone were going to have a piece."

"So the "1" on each spoon means how many parts you get with it," Billy asked, "and the other number is how many parts are in the whole thing?"

Emil nodded. "I remember the number on the top, is the numerator (NOO-mer-ay-ter), but I've forgotten what the number on the bottom is called."
Billy couldn't remember, either.

"It's the denominator (dee-NAW-min-ay-ter)," Mariah said.

Jolena agreed. "Sometimes the number on top in the recipe is more than one. It might be three eighths (3/8), for instance. That means you use this spoon three times. That's usually for something you need in small amounts, like salt or baking powder."

"I wonder if you could save time," Mariah said, "if you used the one fourth (1/4) teaspoon measure once and the one eighth (1/8) teaspoon measure once. Wouldn't that be the same as three eighths (3/8)?"

The dolls looked at the measuring spoons.
"The 1/4 teaspoon measure looks like twice as much as the 1/8 teaspoon measure," Emil pointed out.
"Hmmmm," said Jolena. "Now that you mention it, there are four of the 1/4 teaspoons in each whole teaspoon, but there are eight of the 1/8 teaspoons in a whole teaspoon, so the 1/4 teaspoon measure would be the same as two of the 1/8 teaspoon measures. I think you're right, Mariah!" 

Then Jolena thought about it.

"If I had to put down the 1/8 teaspoon measure," Jolena said, "and then had to go find the 1/4 teaspoon measure in the measuring-spoon box, I don't think I would be saving any time."

"My head is starting to spin," Billy said. "I'm getting confused."

Just then, Mandy came into the room with a basket of costumes from the last photo shoot. She was planning to fold them, so they could be put away.

"Maybe Mandy can explain it better," Mariah said. "She's good at explaining things."
All the dolls looked at Mandy.

Mandy put down the laundry basket, seeing all the dolls looking at her. "What?" she asked.

"We're trying to explain fractions to the boys," Jolena said. "Maybe you can help."
"I'll try," Mandy said, putting down the basket with the costumes. "Go get the Legos, Billy! I'll use them to show you how to understand fractions."

Billy left and came back with the big box of Legos. Mandy went through the Legos until she found what she was looking for.

Mandy held out a piece that had eight little bumps on it. "Lets say," she said, "these little bumps are how many parts you can divide the piece into. How many bumps does this one have?" she asked.

The dolls counted the bumps. "Eight!" they all said together.
Mandy put the piece with eight bumps down on the floor.
"How about this one?" Mandy asked, holding out a smaller piece.

The dolls all looked. "Four!" they all said together. Mandy nodded and put the piece down on the floor.
"Now this one?" Mandy took out an even smaller piece.

"How many bumps does this one have?" Mandy asked.
"Two," they all said.

"Now," Mandy said, "let's use these for denominators."

"That's the number that's on the bottom," Billy said.
"Right!" Mandy agreed. Then she took a piece with just one bump. "Now let's say this piece with one bump is the numerator," she went on. She placed it above the piece with eight bumps. "What do we have?"

The dolls all looked. They thought. They used their little imaginary brains.
"That's my 1/8 teaspoon!" Jolena cried suddenly, and Mandy nodded.

"OK," Mandy went on, taking away the piece with one bump and replacing it with a piece with four bumps. She put it next to the piece with eight bumps. "Now what do we have?"

"Four eighths?" Emil asked. "There are four bumps in the numerator," he said, pointing to the smaller piece, "and eight bumps in the denominator."

"That's right," Mandy agreed.
"You know," Billy began, "it looks like half to me."

"Why do you say that," Mandy asked.
"Well," Billy began, "the smaller piece looks like half the size of the larger piece."

Mandy picked up the smaller piece and fastened it to the larger piece. 

Then she put it down again. "You're right," she said, "because four eighths (4/8) is the same as one half (1/2). Can you make one half with these other two?"

The dolls looked through the pieces on the floor. They put several down on top of the other two pieces.

They talked about it together. Finally, they came up with these, and Mandy said they were right.

"So how do you know that four eighths is the same as one half?" Billy asked.
Mandy asked if anyone knew the answer. Then she waited.
Finally, Mariah said, "I think when you put the smaller piece on top of the larger piece, and you can still see the same number of bumps as are on the smaller piece, it's one half, like if we cut a cookie in half."

"That's right," Mandy agreed."Now, you should be able to put these pieces together and make all kinds of fractions out of them."
"This is fun," Emil said. 
The dolls played with the Legos until they had made as many fractions out of them as they could with the Legos they had.

"I think we need more Legos," Billy said.
While the dolls helped Mandy fold the clothes in the laundry basket, they talked about how they could use fractions.

Emil said, "A football game has a first half and a second half."
"I would need a fourth of a yard of fabric for a dress," Mariah said.
"I can do a half twist on my skis," Jolena said. "When I do, I land facing backwards."
"I needed three fourths of a ball of yarn for the sweater Billy is folding," Mandy said.
"If we cut that cookie into enough equal pieces for each of us to have one," Billy suggested, "we would each have one fifth of the cookie.
"If...," Mariah pointed out, "...if we could cut the cookie into five pieces that are all the same size."
"I think a half twist would be easier," Jolena said with a sigh.


Cast--
Mandy: Götz Happy Kidz Katie 2015
Jolena: Götz Happy Kidz Lena in Aspen
Mariah: Götz Happy Kidz Mariah, "Chosen" from My Doll Best Friend
Billy: Götz Happy Kidz Lily at London
Emil: Götz Happy Kidz Emilia

Message to young readers: Remember that these are dolls. It is not good for real children to get up on tables or countertops, especially wearing shoes.

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