Chocolate Fractions

Here in the Upper Mississippi River Valley we are under the cover of snow, and the temperatures are frigid. After a bit of snowplay who doesn't want to come in for a little warming up with a cup of cocoa. Following on the theme of using food for math manipulatives, here is one of my students' favorite math lessons.

(This lesson can be done with a piece of construction paper cut into the appropriate shape, but how much more fun is the tactile experience of chocolate, and therefore more memorable.)

Each student needs his or her own bar of Hershey's chocolate (standard size, without almonds). I have my students draw each stage and label the picture accordingly.

One whole bar. 1/1 Note three rows (top to bottom) and four columns (left to right).

Then break the bar in half between the second and third row. One half. 1/2 Ask how many halves make up one bar. Go back to the whole bar to note 2/2 make one whole.

Break the halves in half between the columns. One fourth aka one quarter. 1/4 Ask how many fourths make up one whole bar. Go back to the whole bar to note 4/4 make one whole.

(For older students point out 1/2 times 1/2 = 1/4. Note that the multiplication sign can be translated as the word "of" in English. Half of half is one quarter (or one fourth). Also point out that 1/2 divided by two represents the same thing. This leads naturally to the discussion of how the saying, "Ours is not to reason why, just invert and multiply." pertains to dividing with fractions. Show how multiplying by 1/2 is essentially the same as dividing by two.)

Now break all the fourths into twelfths. At this point we will be rearranging and drawing the different combinations.

Put the twelfths back together to resemble the original bar, but leave a small gap between the rows. This shows three thirds. Draw a third lengthwise. Rearrange to show that there are the same number of pieces in drawing the thirds in columns (this means stacking four twelfths on top of each other). 1/3 Ask how many thirds make one whole bar. Go back to the whole bar to write 3/3 equals one whole.

Rearrange the whole bar and group the pieces in groups of two. This shows 1/6. Ask how many of these fit into the whole bar. Go back to the whole to write 6/6 equals one whole.

Have the student count how many separate pieces of chocolate there are. Draw the piece and label it 1/12. Ask how many of these pieces are in whole bar. Go back to the whole bar to write 12/12 make one whole bar.
Now go through all the other groups to see how many twelfths are in half, one fourth, one third, and one sixth. Do this for all of the fractions. This will show the equivalent fractions.

Regroup the pieces to represent three fourths, five twelfths, seven twelfths, ten twelfths, and eleven twelfths.

http://www.amazon.com/Hersheys-Milk-Chocolate-Fractions-Book/dp/0439135192
Have the student write down all of the equivalent fractions on a number line.

___________________________________________________________________________
0 1/2 2/2

1/3 2/3 3/3

1/4 2/4 3/4 4 /4

1/6 2/6 3/6 4/6 5/6 6/6

1/12 2/12 3/12 4/12 6/12 8/12 9/12 10/12 12/12


Future post: Adding and subtracting fractions with chocolate fractions.

Zero's House

Shiv who is nine was working on completing factor houses with me. I pointed out that 1 is the only number with a one story house as 1x1 is the only factor - it is neither prime nor composite, unique. Then I showed him how the primes are two story houses. Take 2 as an example -- 1x2 and 2x1. Then we looked at some factor houses for other numbers.

We noticed that some houses had an odd number of stories. "Which numbers are those?" I asked. "Ahhh, they are, umm...perfect squares," he answered. I put my finger on the house for the number 24. "Wow! Look at this house. It's really tall compared to the other houses, isn't it?" I queried. Shiv started to write the facts in the stories, then he looked up and said, "It's not as big as zero's house."

His brother who is going into the eighth grade stopped working on his math and looked over at Shiv's paper. He then looked at me quizzically. I returned the look when it hit me, "Ah, yes, Shiv. Zero's house is infinitely tall. Anything multiplied by zero equals zero. God's house, we could say, because it's infinite. Good one, Shiv."

The older brother continued to look at me a little bewildered by his younger brother as I went on. "I have never heard anyone mention that before, Shiv. I am going to write about what you have told me. I can't wait to tell my math mentor. He will like how you are thinking about zero's house."

Mathletes

People often ask me what types of students I have in my private tutoring practice. Most folks assume that I work primarily with children who need remediation. The truth is that I mostly work with children who are in the top of their class. When I tell people this they are genuinely surprised. Then I go on to explain my theory based on the following incident:

I had a perfect tutoring moment today - watching a child get the "A-ha" look and a sense of mastery. I turned to the student's younger brother and said, "It's like Olympic swimming. We are shaving seconds off the time it takes for him to do these problems. He won't have to work so hard to get the answers." The student has gained some insight into the nature of what he is solving, and I get to witness the discovery.

I see myself like an Olympic coach. When an athlete is competing at the upper levels, he or she needs a coach to encourage and foster excellence. Top students benefit from one on one tutoring in similar ways. A tutor can point out habits both physical and mental that hinder a student from performing at his or her peak level.

Many students who are high achievers are busy outside of the classroom. Some are involved in sports year round, others are musicians or dancers. Children who are busy with other activities need to work more efficiently. Tutoring can help students see their particular foibles more quickly. Classroom teachers do not have the luxury of working one on one with students to address individual needs.

Innate ability can be honed to a finer degree. Strengths can be identified, and weak areas can be overcome or worked around. Most students benefit from personalized strategies as much as from the extra practice they receive.

Among my students there is no shame attached to being tutored. This benefits all types of students. The ones who come for remediation see the advanced students coming in for help which helps to alleviate some of the possible labeling that might occur on their part or on the part of others. I point out to all of my students the similarities in the mistakes that I identify. They work to their ability and strive to attain their own personal best.

Platonic Solids - Regular Polyhedra

At the last meeting of the Mighty Math Club, everyone enjoyed creating models of the five Platonic Solids using gumdrops, mini marshmallows, and nearly one thousand toothpicks. Some kiddos decided to build other objects, including a gorgeous replica of the Sears Tower and a biplane, but most had tremendous success with all but the dodecahedron. (That's the most wobbly of the five given the lack of triangles in the construction. This brings up some other topics for discussion along the lines of stability in building with triangles, having braces on bookshelves, etc.)

Convex regular polyhedra (poly-many; hedra-faces) are composed of faces, edges, and angles that are all congruent. These regular polyhedra are called Platonic solids. There are only five Platonic Solids - tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The name of each figure is derived from the number of its faces: respectively 4, 6, 8, 12 and 20.

Building models helps children to see the component parts and to understand the concepts of dimensionality and regularity.

The toothpicks are the edges. Just count the number used.
Each gumdrop or marshmallow is a vertex. Count them to find the number of vertices.
Paper models show the faces more clearly. Use the link below to find nets (two-dimensional paper outlines).

Euler's Rule:

If V is the number of vertices, F the number of faces, and E the number of edges of a convex regular polyhedron then the following equation is valid: V + F = E + 2

This link show how to put the models together.
http://www.youtube.com/watch?v=5QgIJOy7T7Y

Here's a link to make the Platonic Solids out of paper.
http://www.otrnet.com.au/IntegratedMathsModules/H04/H04_Platonic_Solid_Nets.pdf

This link leans in the mystical direction:
http://themathlab.com/wonders/godsdice/dicepatn.htm

Blame it on my friend

A place to put all the tidbits that I send piecemeal to folks. What a brilliant idea! I guess the fact that I figured a blog is a conceit I never thought I'd succumb to is yet another hat I will have to eat.

Welcome to Mathknotter where I will post my math ideas along with other trivia from my adventures as a homeschooling math tutor.