3.1+Walking+Together+0809

 21st March 2009 MS Using distributive and the commutative properties How do I use distributive and the commutative properties in everyday life?

Distributive property: A mathematical property used to rewrite expressions involving addition and multiplication. The distributive property states that for any three quantities //a, b,// and //c, a(b+c)=ab+ac//. If the expression is written as a factor multiplied by a sum, you can use the distributive property to //multiply// the factor by each term in the sum. 4(5+//x//) = 4(5)+4(//x//)=20+4//x// If an expression is written as a sum of terms and the terms have a common factor, you can use the distributive property to rewrite the expression as the common factor multiplied by a sum. This processing is called //factoring//. 20+4//x// = 4(5) +4(//x//) = 4(5+//x//) Commutative property of addition: A mathematical property that staes that the order in which quantities are added does not matter. For example, 5+7 = 7+5 and 2//x//+4 = 4+2//x//. The commutative property of addition is sometimes called the //rearrangement property of addition.// Commutative property of multiplication: A mathematical property that states that the order in which quantities are multiplied does not matter. For example 5x7 = 7x5 and 2//x// (4) = (4)2//x//. The commutative property of multiplication is sometimes called the //rearrangement property of multiplication.//

Problem 3.1 A Leanne: //x// (16x1) A Gilberto: //x// (7x2) A Alana: x (5.5x11) A Total: x (16) + x (14) + x (60.5) 16 + 14 + 60.5 = $90.50 A total//: x// (90.50). C. ** 1. Use the distributive property and commutative properties to show that the two expressions you wrote for the total amount the team will raise are equivalent. ** A Total: //x// (16) + //x// (14) + //x// (60.5) = //x// (90.50) A Total: //x// (14) + //x// (60.5) + //x// (16) = //x// (90.50) I know they are equivalent because when I made another table the result was the same.
 * 1) ** For each student, write an equation for the amount of money the student will raise if he or she walks //x// miles. Then write an equation for the total amount the three-person team will raise if they walk //x// miles. **
 * 1) ** Alana asked each of her 11 sponsors to pledge $5 in addition to an amount per mile, so the team will raise $55 regardless of how far they walk. **
 * 2) ** Excluding $55, how much will their team raise per mile? **
 * 2. Use your answer from part 1 to help you write a different equation for the total amount the team will raise if they walk //x// miles. **
 * 2. Verify that the expressions are equivalent by making and comparing tables or graphs. **
 * || No. of Sponsors || Amount per mile || Total per person ||
 * Leanna || 16 || $1 || $16 ||
 * Gilberto || 7 || $2 || $14 ||
 * Alana || 11 || $5.50 || $60.50 ||
 * Total || || || $90.50 ||

Follow-Up 3.1 These are equivalent because the commutative property of addition applies. 8x is the simplified version of 3x + 5x. These are not equivalent because 3x + 5 can not be simplified further. The 5 in this expression doesn’t have any x. These are not equivalent because the simplified version of the distributive property would be 4 (x) + 4 (7), not 4x + 7. These are equivalent because 5 times x equals 5x, and 5 times 2 equals 10. Therefore 5x + 10 and 5 (x + 2) are equivalent. These are equivalent because 4 (3 + 2x) simplified becomes 8x + 12. When 4 is multiplied by 3 and 4 is multiplied by 2x the expressions remain equivalent. No, these are not equivalent because 4x + x = 5x. 5x + 2x = 7x. 7x is not equivalent to 8x. Yes, these are equivalent because these two are simply reversed by the commutative property of addition. These are not equivalent because the commutative properties can not flip the amount of x’s from one number to the other. These are equivalent because the commutative property of addition is being used. Both of these expressions add up to 12x. These are equivalent because using the distributive property we can find out that 3 times 2 equals 6 and 2 times t equals 2t. If you put them together you get 6 + 2t. These are equivalent because of the distributive property. 2 times L equals 2L and 2 times 2 becomes 4. So, 2L + 4 + 2W is the same as 2L + 2W + 4. Yes, it is equivalent because of the distributive property. 2 times L is 2L. 2 times is 2W and 2 times 2 is 4. Therefore 2L + 2W + 4 = 2L + 2W + 4.
 * In questions 1-12, tell whether the expressions are equivalent, and explain your reasoning. **
 * 1. 3x + 5x and 8x. **
 * 2. 3//x// + 5 and 8//x// **
 * 3. 4(x + 7) and 4x + 7 **
 * 4. 5(x + 2) and 5x + 10 **
 * 5. 12 + 8x and 4 (3 + 2x) **
 * 6. 4x + x + 2x and 8x **
 * 7. 7 + 5x and 5x + 7 **
 * 8. 3x + 8 and 8x + 3 **
 * 9. 5x + 3x + 4x and 4x + 5x + 3x **
 * 10. 6 + 2t and 2 (t + 3) **
 * 11. 2(L+2) + 2W and 2L + 2W +4 **
 * 12. 2L + 2W + 4 and 2 (L + W + 2) **

** Monique Spruijt ** ** April 22, 2009 ** ** Math 8D **
 * __ Through the Looking Glass – Lewis Carroll __**
 * Going by the pen name, Lewis Carroll, Charles Lutwidge Dodgson was born on January 27th, 1832 and died on the 14th of January in 1898. Dodgson was an English mathematician, logician and writer. His most famous writings are ‘Alice in Wonderland’, its sequel ‘Through the Looking Glass’ and poems such as ‘The Jabberwocky’, which are all considered to be within the genre of literary nonsense. His gift for word play, logic, and fantasy has delighted audiences all over the world from young children to the literary elite and has influenced artists, such as Walt Disney, too. Apart from writing, his talent as a mathematician won him the Christ Church Mathematical Lectureship, which he continued to hold for the next 26 years. **
 * In his early adulthood, Charles Ludwidge Dodgson was about six feel tall. He was slender and quite attractive with his curly brown hair and blue-gray eyes. He was described as somewhat asymmetrical, carrying himself awkwardly, which may have come about from a knee injury. As a young boy, Lewis acquired a stammer in his speaking and around the same time he suffered a fever which left him deaf in one ear. **
 * In the first book, ‘Alice in Wonderland’ the theme is a pack of cards, whereas ‘Through the Looking Glass’ is based on a game of chess. Chess and symmetry are the two major mathematical themes in this book. **
 * The symmetry in ‘Through the Looking Glass’ is very obvious when you reach the ‘Tweedledee and Tweedledum’ section. Tweedledee and Tweedledum are mirror images of each other or enantiomorphs. Tweedledee and Tweedledum are always finishing each others sentences or arguing against each other. Another example of directional nonsense is when Alice must approach the Red Queen. She may only approach the queen by walking backwards. Cake being handed around first and then sliced is another example as well as the king’s messengers who are ‘One to come’ and ‘One to go’. In addition Lewis Carroll applied this mathematical nonsense to his real life. He sometimes wrote letters from the last word to the first and liked playing music tapes backwards. **
 * Chess in this book, a very significant theme, is played with fields as squares. Most of the main characters met in the story represent a chess piece, with Alice herself being a pawn. The Looking Glass world is divided into sections by brooks or streams, with the crossing of each brook usually signifying a notable change in scene or action of the story. The brooks represent the divisions between the squares on a chessboard and, as Alice crosses them, it signifies the advancing of her piece one square. Even though the sequence of moves (white then red) is not always followed it relates back to the book’s mirror image theme. **
 * John Tenniel, the illustrator of the ‘Alice in Wonderland’ series was not given instructions about what to draw but rather had to choose what to draw and then try to guess what Lewis Carroll was saying and what he meant by it. **
 * Martin Gardener is a popular American math and sciences writer who has published over 70 books. One of those books, ‘Annotated Alice’, has helped me understand what the mathematical mysteries are in Lewis Carroll’s book ‘Through the Looking Glass’ and has given me a lot of interesting information. He is best known for his column ‘Mathematical Games’ in the Scientific American. **
 * ‘Through the Looking Glass’ has quite a lot of mathematics in it and I hope you have learned as much as I have in this project. **