with Kim Rodgers Distributive Property for Addition/SubtractionFor the last couple of weeks we have been working on the distributive property for addition which is stated as: a (b + c) = (a x b) + (a x c) We started by using actual number problems to prove how this property works before introducing the property algebraically. After using base ten blocks the first week, we worked on problems in our notebooks last week. Those who finished all the problems began helping their classmates. The students took turns “teaching” the class how they solved each problem. We then moved on, trying to express the problems we had been working on in an algebraic way. We took the problem 3 x 46. After splitting 46 into tens and ones (40 + 6), we assigned a letter to each number in the problem: 3 x (40 + 6) = (3 x 40) + (3 x 6) a = 3 b = 40 c = 6 This led us into the property itself, which we added onto our Algebra Toolkit. We discussed why it might be called the distributive property, with a consensus that “a” has to be distributed to “b” and “c”. This week we took what we learned about the distributive property for addition and asked if we could do the same thing using subtraction. We took one problem we had worked with when using addition (6 x 39) and asked how we could express 39 as a difference of two numbers instead of an addition. We worked through it like this: After doing one more together we mapped out the steps and the students tackled more problems on their own. I really saw a difference from the week before. They had the concept of distribution down. The only problem was they kept reverting back to addition. We laughed as it kept happening over and over, which led us into a great discussion about why it’s good to SHOW YOUR WORK! I have been emphasizing this over the last several weeks because it’s very difficult to see where the problem lies if all you have is an answer. Because they were showing their work I could see where they were having problems, and most of the time it was a mistake that had nothing to do with the concept of the distributive property. They added instead of subtracted or they wrote a number down wrong, etc. It was clear to them by the end of class yesterday that it only helps them when they show their work. Wrapping up our work on the distributive property for subtraction, we expressed it algebraically and put it on our toolkit as well: a (b - c) = (a x b) - (a x c) Next week we will begin discussing how algebra is used in the real world. They are welcome to do some research in this area by looking into how people might use algebra in their jobs. They began discussing this a little bit at the end of class and seemed excited by it. Looking forward to seeing what they come up with! with Leigh Ann Yoder Routing and DeadlockToday was all about routing and deadlock. When you have a lot of people using one resource, such as cars using roads, there is a possibility of deadlock. Many students have experienced this in large cities. Routing and deadlock are also problems in many types of networks, including computer systems. Engineers spend a lot of time figuring out how to solve these problems, and how to design networks that make the problems easier to solve. The students learned these concepts by playing a game called The Orange Game. They had to develop their own strategies for avoiding deadlock and quickly learned that working cooperatively is imperative to avoid problems. Toss 'n' SortWe then played a second game called Toss ‘n’ Sort, which again mimicked routing and deadlock in networks. This time each student was like a “packet” of information, which had a destination. The “packet” had to travel along the edges of a graph to reach its required node. Of course there were rules regarding movement. We played several variations and discussed which methods were most successful. The students were stumped on the last challenge, but towards the end of the class they were close to solving the puzzle. I have asked them all to continue working on this for homework. Next week, we will try one more time! We ran out of time, so next week we will play another game demonstrating Message Routing. Recommended Reading |
Today we started class by finishing up our NaNo TV interviews. We then announced the winner of the Sequence and Structure Challenge using the Langston Hugh's short story, "Thank You, Ma'am." The boys, Team WriMo, took home the victory as their re-arrangement of the story still maintained its original meaning. Nice work, boys. Without further ado, we updated our in-class progress chart for the final time and started our novel-sharing. Each student read aloud the first two pages of his/her novel. I was so impressed with every story! I told the students my expectations were very high going into September, but they managed to surpass them! I am truly proud of these talented young people! For the last 30-minutes of class, we broke into two workshop groups. Students traded novels and made constructive notes and comments on the first two pages of their partners' novels. | We then did what all good novelists do -- we celebrated with food and drink! |
Homework
Lastly, here's what students should bring next week. We will be continuing our work on editing for our last class.
Re-type your first two pages. This time you should incorporate any changes your partner suggested (if they finished in class, otherwise, make no changes). Pages should be double-spaced, not single spaced, and Times New Roman, 12 pt. Please let me know if there are any questions about the homework!
Thank you and see you next week!
Foundations of Philosophy (ages 9-11)
The Little Prince
Chapter IV
First we considered the Turkish Astronomer who made his presentation "in Turkish costume, and so nobody would believe him."
1. How do we determine whether a person is qualified? What biases might get in the way?
Then, with the help of Aristotle, we considered friendship.
The pilot points out that, "When you tell them about a new friend, [grown-ups] never ask questions about what really matters. They never ask: 'What does his voice sound like?' 'What games does he like best?' 'Does he collect butterflies?' They ask: 'How old is he?' 'How many brothers does he have?' 'How much does he weigh?' 'How much money does his father make?'"
- How do you describe a friend?
The pilot says grown-ups will believe the little prince existed if you say, "The planet he came from is Asteroid B-612." He says grown-ups won't believe he existed if you say, "The proof of the little prince's existence is that he was delightful, that he laughed, and that he wanted a sheep."
- What do you think proves that the little prince existed?
- Are numbers and people equally real?
Chapter VIII
The Little Prince says that he misunderstood the flower. “The fact is that I didn’t know how to understand anything. I ought to have judged by deeds and not words.”
How do you know a person is your friend?
For homework, students should read through Chapter XIX. They should also write a paragraph either about Aristotle's criteria for friendship (do they agree or disagree with Aristotle, and why?), or reality (Are plants and stones equally real? Are centaurs and square circles equally real?).
Philosophy for Children (ages 12-14)
Secrets and Trust
(We discussed this and I referred her to "Harry Stottlemeier's Discovery," Professor Lipman's first Philosophy for Children novella, the novella with which we began our Mosaic P4C journey. In it, Professor Lipman devotes much time discussing children's rights.)
A second student responded to her comment by poignantly observing that she felt too old for her own age group, but too young for an adult age group. The other students all related to this comment. I said that it must be challenging to navigate such an "in-between" time. They said it was indeed challenging. I assured them that this awkward time will pass, and, if they stay disciplined and diligent, and open and receptive to life, they will find their place in the overall scheme of things.
We then moved on to the notion of secrets. In the novella "Lisa," we started Chapter 4. It begins with Millie sharing secrets with her pet Peruvian guinea pig, Pablo. One student began the discussion by saying we keep secrets because we are afraid of being judged. Another then said that the test for a good friend is if they can keep your secrets. We just applied the critical thinking skill of establishing criteria for the idea of friendship, which we spent quite a bit of time on earlier in the term!
The dialogue continued: The students agreed that secrets are based upon trust, and trust is based upon reciprocity (a concept we discussed in Chapter 2.) Therefore, secrets are based upon reciprocity; i.e., you trust someone with your secrets only if they trust you with theirs first. So we astutely asked, "Who goes first?" To which we replied, "You can trust someone even without expecting trust in return. If you want the relationship, it's worth taking the risk."
Critiques
- Was the rhyme forced? In many of these cases, we reconsidered the problem end rhyme, thinking of alternatives.
- Did the line scan? There were times when a syllable or two needed to be added or cut.
- Did all of the words make sense in terms of the vocabulary of the poem? There were some that were too elevated or not in keeping with the general tone.
- Could additional words (a, the, and, for instance) be dropped to tighten the line? Poetry should be as tightly constructed as possible.
There were also a couple of stories, both of which lead us into a discussion of the rules of the particular worlds. One was a fanfic (Dr. Who), which prompted a discussion of that particular genre - easy because characters and world rules are established, but can be restrictive because of that. Fan fiction can be good writing practice, though, just as student artists paint already existing works to work on their technique.
The Study of Networks
Engineers spend a lot of time figuring out the most cost effective way to link objects in a network and computers play an important roll in solving these real world problems.
A simple example is air travel. Airlines use airports as hubs, and there are many ways to travel from Destination A to Destination B, but travelers often want the most direct route. Airlines also want to use the most direct routes since fuel and workforce costs add up quickly. Next week we will talk more about how to take the safest route as well as avoid crashes!
The students worked on "The Muddy City Problem". This activity helped them to discover direct and efficient routes by linking a network of houses.
After working through the problem I defined some computer science specific terms: Graph, Node, Vertices, Edge, Network and Tree. Students should understand the difference between a statistical graph and a computer science graph. After mastering these terms we defined a Minimum Spanning Tree.
Students then went on to create and solve their own Muddy City Problems. They were also asked to define rules for solving the Muddy City Problem.
Finally, I explained two famous algorithms for solving Minimum Spanning Trees -- Prim’s Algorithm and Kruskal’s Algorithm. Of course this lesson would not be complete without the famous Traveling Salesman Problem. Ask your student about it and if it has been solved!
November 11
Question 1: Ethics
Cyril says that “stealing is stealing even if you've got wings.” Jane argues that no one scolds the birds, and then they go on to eat “quite as many plums as were good for them.” However, when Jane sees the man, she says, “We had some of your plums; we thought it wasn’t stealing, but now I am not so sure.” Later in the day, they feel that it is right to take the necessities of life from the clergyman, until they find themselves locked on the roof. Then, they begin once again to think that their actions are wrong.
Do you think they were wrong to eat the clergyman’s food? How do you know when something is wrong?
What kind of knowing is this?
Was Robert wrong when he lassoed the baker’s boy? What about when he put him on the roof of the cowshed?
Was it wrong to go to the fair as a giant to make money?
The class worked to develop a criteria for right and wrong that can be applied in many situations, rather than simply stating an opinion of the ethics of each individual case. One budding philosopher articulated that it is the knowledge and willingness of the participants in each event that determine whether something is right or wrong. Another student of philosophy developed the term prehenceive: a preexisting law, something that has been set in stone for as long as anyone can remember; it is almost a sixth sense, something that you know you can or can't do. Similar ideas can be found in the work of Plato.
Question 2: Identity
Was Robert still Robert when he was a giant? Why or why not?
What is your criterion for identity? Has it changed?
Homework: Write your criterion for identity and bring it next week. What makes you you? Is your identity independent of the world around you, or does it depend on the world around you?
Next week we will review our criteria for identity before we discuss chapters 15 and 16 from The Wonderful Wizard of Oz.
November 18
You might enjoy trying these at home:
Reality
Look at an object placed on the table. Describe the (a) size, (b) shape, and (c) color without looking at your neighbors answers.
A.
B.
C.
Now list them on the chalkboard. Are the answers all the same? Discuss.
How you do know what you know?
Do you know 2 + 2 = 4? How do you know?
a. It feels right when I look at it.
b. I learned arithmetic in the first grade.
c. I realize it can be proven using universally accepted mathematical principles.
a. Because I can clearly see its redness.
b. Because everyone calls this color “red.”
c. Because I can see that it is red – and it is red.
Next week we will begin discussing The Little Prince. Students should read through section VIII, and write down one or two different issues in the book that strike them as philosophical. We will discuss these in class.
The Professor Returns!
We received another letter from Professor Arbegla this week! She informed us that she has a trick that makes multiplying numbers outside the times table easier. The best way to describe it was to use an example. Students chose one of the three following problems to try and solve mentally -- without paper and pencil: 4 x 26 6 x 59 9 x 87 Each student took a turn sharing how they arrived at an answer. I was so impressed by all the creative strategies they used to figure it out. We then used base ten blocks to test Professor Arbegla’s new strategy. Using the blocks to warm up, students practiced making two digit numbers. Next, we discussed the meaning of 3 x 46. Students made three sets of 46 with a partner and exchanged blocks to make 138. The question was then asked, “How do you write 46 using tens and ones?” (40 + 6 was the answer.) So, we decided another way to say 3 x 46 was to say that we have three 40's and three 6's. The students were able to build that and find the same answer of 138. We learned that the strategy of splitting up the two digit number and sharing the factor equally was called the distributive property over addition. The students practiced regrouping the blocks as I wrote the following on the board --> --> --> --> --> --> | 3 x 46 = 3 x (40+6) 46 + 46 + 46 = (3 x 40) + (3 x 6) 138 = 120 + 18 138 = 138 The partners were then challenged to solve 4 x 38 using the base ten blocks and be prepared to share their strategy. Each pair solved the problem and took turns standing at the board explaining which strategy they used. I loved hearing how their minds worked, and how the partners shared the process together. Next week we will learn how to write the distributive property algebraically, which always seems to throw them for a loop at first. We’ll take it slow and have some fun! |
Making Mummies
We also molded tinfoil into the shape of mummy bodies and wrapped these in dry "linen", careful to place special jewels inside the wrapping just as the ancient Egyptians did to protect the mummy on its journey to the afterworld. After our mummies were wrapped, we applied wet plaster strips to finish the mummification process. When these dry, we will be able to paint the gold mask on our mummy's face so that the gods will recognize the mummy when it arrives in the afterlife! Students each hope to create a sarcophagus for their mummy when we meet next in class. | Students covered glass jars with gold paper and used air dry clay to mold the heads of gods to create their own canopic jars. We will paint these once they dry to complete our designs. |
Writing Ancient Numbers
The students also enjoyed learning about the Ancient Egyptian number system, and wrote page after page of numbers. The Ancient Egyptians did not use place value - but rather a simple system of adding symbols together. Here are their number symbols:
Parallel Processing and a Sorting Network
Speaking of efficiency, our computers are NEVER fast enough! Computer scientists are constantly searching for new and improved ways of solving problems quicker. Although our Quicksort from last week seemed quite quick, when working with millions of numbers it could actually turn into a dreadfully slow process.
This week was all about Parallel Processing. We started our lesson with a life size demonstration of a Sorting Network. In a sorting network the students could see that 12 comparisons could actually be done in the time of 5, if we used multiple processors to do the job.
After everyone had a chance to walk through the human network, they were all given a challenge to design their own sorting network. I believe all of the students were successful at this!
Parallel vs. Serial Processing
We wrapped up class with a friendly competition. The students were divided into groups and given the task of adding 32 four-digit numbers. The goal was to divide the task to speed up the process. However, accuracy was important, so they were encouraged to develop a check system. The girls finished first, but the boys came closer to the actual answer.
Next week we will continue learning about Networks, which were introduced this week.
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