6: The Story of Fractions
- Page ID
- 186351
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Long ago, people started noticing that whole numbers just weren’t enough. Imagine trying to divide a pizza between three friends or pour half a can of paint—you can’t describe those situations using just whole numbers! So, people invented fractions to solve real-world problems, “fractions were invented because it was not convenient to describe many problem situations using only whole numbers” (Musser et al., 2014). Around 1600 B.C.E., the ancient Egyptians were among the first to use fractions in a big way. They used them to track land, food, and time. Their system was based on unit fractions—fractions like ½ or ⅓, always with a top number of 1. Even when they needed something like 2/3, they would break it down into a sum of unit fractions. Their methods were different from ours, but their need was the same: to describe parts of a whole. (Burton, 1984)
Just like kids today naturally split snacks or toys into fair shares, ancient people had to figure out how to share and divide things too. Young children often show early understanding of halves, thirds, and fourths even before they learn formal math. This makes fractions one of the oldest—and most human—mathematical ideas. (Feike et al., 2018). As math developed, so did the way we write and use fractions. Our modern notation (like ¾) is a tool created to make life easier. But understanding what a fraction means—a number that represents part of a whole—is way more important than just writing it correctly. (Burton, 1984)
Why does this matter for your future students? Because fractions are a bridge. Once students understand them, they can move on to decimals, percentages, algebra, and even higher math. The California Math Framework (2013) emphasizes that building a strong number sense, including a deep understanding of fractions, sets the stage for success in middle and high school math. Yes, fractions can be tricky. In fact, they're considered one of the most challenging topics in math. But here’s the twist: that difficulty means it’s even more important to build meaning first, not just teach rules. Understanding the story behind fractions—the real-life needs that inspired them—can help both teachers and students feel more confident.
In the end, fractions show us something powerful: Math was created to solve real problems, and it keeps evolving to help us understand the world better. And that’s something your students deserve to see from the start.
Chapter Learning Objectives
- SLO 6.1 Students will model and compare fractions using visual, verbal, and symbolic representations, and explain how these models support conceptual understanding and simplification strategies.
- SLO 6.2 Students will analyze and compute sums and differences of fractions, applying addition and subtraction properties and using estimation techniques to verify the reasonableness of results of real-world problems.
- SLO 6.3 Students will use visual models and real-life contexts to represent and compute multiplication and division of fractions, justifying the methods they choose and interpreting the results.
- SLO 6.4 Students will evaluate the application of mental math and estimation techniques for operations with fractions, including rounding, range estimation, and using compatible numbers to simplify calculations.
Essential Questions
- How do conceptual and visual representations of fractions deepen understanding and support meaningful real-world application—especially in teaching and parenting contexts?
- What misconceptions do students commonly have about fractions, and how can we design instruction to address them?
Learning That Transfers
Students will connect conceptual understanding of fraction operations to real-world contexts, using multiple representations and strategies to explain and model these ideas for others—especially young learners.
One Part Love One Part Math
Morgan's A Fraction Fiesta
It was cold in December, the sky turning gray, When Morgan arrived for Tamale Day.
The house smelled of chili, of masa and steam— It was time to cook up Abuelito’s dream!
“Morgan,” said Grandpa, “today’s just for you, We’ll make tamales—red, green, and blue!
(Well not blue... but sweet ones with raisins and spice!) You'll measure and mix and cut things just right.”
He laid out the masa, a warm golden mound, In a bowl so big it could spin ‘round and ‘round.
“We’ll need to divide this in parts of a whole—Each type of tamale has its own role.”
“One-half of the masa is spicy and bold, With green chile filling, just like I told.
One-fourth will be sweet, with cinnamon flair, The rest will have beans and some cheese we can share.”
Morgan looked close, her mind in a spin— “How do you know that these parts all fit in?” Grandpa just grinned and pulled out a tray, And showed how the pieces could fit the whole way.
She spread the masa with careful precision, Laughed when she made her very first division.
“Half for the green, one-fourth for the sweet...This fraction thing? It's actually neat!”
Next came the toppings—“Now don’t go too fast! We’ve got 24 tamales to wrap up and cast.
If there are six people, all hungry and loud, How many tamales for each of the crowd?”
Morgan thought hard, then gave him a grin—“That’s four each!” she beamed. “Let the feast begin!”
“But wait!” said Grandpa, “Let’s count them again, What if five more cousins arrive at ten?”
Morgan blinked twice, “Oh my... that’s eleven! Now 24 tamales must stretch like heaven!”
She estimated quickly, rounding with care—“Two for each, maybe three if we share!”
They made a new batch, dividing once more, Using thirds, halves, and even three-fourths galore.
As they worked, she added and mixed with pride, Explaining her math with Grandpa by her side.
“When I take 1/2 and add 1/4 in the pot, I get 3/4, and that’s quite a lot!”
They multiplied fillings—“Three kinds by eight!” And divided them fairly upon every plate.
They laughed and they learned, just as families do, Mixing culture and numbers into something brand new.
And when the last tamal was done—The family gathered, everyone!
They ate and they smiled, with stories to say, And Morgan knew math in a whole different way.
So now when she sees a pie or a stew, She thinks of the math in the cooking they do.
Fractions aren’t just for numbers or tests—They live in our homes, in our hearts, in our quests.