In our first meeting of fall 2021, Sarah and Amy led through the group through an exploration of the ways of cutting a shape into pieces.
Warm-up Discussion: How are you creative?
- I don’t follow rules. Sometimes I don’t know how I know something. It’s not always logical.
- I don’t think in a straightforward way. Sometimes it’s a complicated way that gets me to an answer mathematically.
- Cooking. I look at recipes, but I don’t usually follow them.
- I like to see how things come apart and go together.
- I think in concentric circles, overlapping circles. Sometimes, this makes it hard to focus.
- Installing floors in my apartment.
- Coming up with a CAMI project! (but I’m a perfectionist and haven’t found the right topic yet…)
- Putting together professional development, writing articles. I like writing because it gives me a chance to express myself when I sometimes have trouble finding the words when speaking.
- Having pictures in my head of abstract concepts. Taking concepts from math and turn them into visual explanations.
Which One Doesn’t Belong?
Sarah and Amy led through the group through a discussion of these images.
This activity is in the form of a Which One Doesn’t Belong?
Which one doesn’t belong?
- I think that the top right doesn’t belong because it is the only one where all the segments have the same height as the overall shape.
- I think the bottom right doesn’t belong because all the shapes aren’t congruent.
- I think the bottom left doesn’t belong because it has triangles.
- I think the top left doesn’t belong because each of the small shapes are the same shape as the complete blue shape.
What do they have in common?
- Each shape starts with a square and breaks it up into four equal areas. (Though we don’t have enough evidence to say that the lines are exactly breaking up the squares in equal ways.)
What rules can we make about fourths based on these pictures? Are these rules always true?
- If you add four fourths together, you will get one whole
- Two fourths make a half
- Fourths can be different shapes
- If you cut all the fourths in half, you get eighths
- if the fourths were cake, they would each be worth the same number of calories.
Work in breakout rooms
What if... You start with a different shape? You make thirds? Or some other fraction? You want all the pieces to be the same? You want all the pieces to be different? You want to build a picture we already have into a fraction with a different denominator? You create fractions by building instead of partitioning? Another possible direction... Play with this interactive graph at Desmos: https://www.desmos.com/calculator/wwg2shmxcd What questions or ideas does it inspire for you?
How many cuts can I use to cut a square into fourths?
How many different ways are there to cut a square into fourths with four cuts?
Sarah’s Strategy: Start by cutting into 4 parts, then work on making the parts equal.
Maggie’s conjecture: If you start with a polygon with more and more sides, you can make 4ths with infinite cuts.
How many ways are there to split a shape into pieces, such that all pieces of the shape are touching each other? (Imagine neighboring countries.)
The group used circles instead of squares.
The group made a table to show the number of families for each number of areas:
Conjecture: There are only two families of cuts for fractions larger than fifths, such that each country touches every other country.
Attendance: Amy, Benny, Eric, Maggie, Maya, Patricia, Sarah, Sophie
