On the tenth night, Robert found himself in the middle of a blizzard admiring the different kinds of snowflakes. In this dream, he began to learn about continuing fractions, and new explorations of how equations are connected to area and perimeter and how 3-D figures are formed by using nets.
9th Night
1) Which set of numbers are represented by the first row? second row?
2) How many red-shirted numbers are there compared to white-shirted numbers?
3) Which set of numbers are represented by the third, fourth, fifth, sixth, and seventh rows?
4) What did the number devil write on the ceiling of Robert's room? What is a series?
5) The first two numbers in the series was 1/2 + 1/3, taking the next four terms in the series and adding them together, predict what the sum will be. What computation strategy was used to predict the answer?
10th Night
6) How many sides and or points did the snowflakes have? What geometric shape do they resemble?
7) How does the computer that Robert refers to in his dream similar to the technology used in our mathematics class?
8) Which number do all the numbers "wobble" around? Identify the mathematical term for this number.
9) Take 1.618033989. Subtract 0.5. Double the result. Square the new result. What is the final result? Does the result make sense? Explain your reasoning.
10) Use the figures on page 203 to prove the formula true. Hint: White spaces are closed shapes. What types of figures does this formula work for?
11) Identify the figures that result from folding the nets on pages 204-205.
12) Dots plus spaces minus lines equals two. Translate to mathematical symbols. What types of figures does this formula work for?
When answering questions in a post, be sure to answer them in complete sentences. In order to avoid echoing someone else's comment, each student will be required to answer one of the questions above. Utilize more of the questions, for more points. In class on Monday, students will share their posts and respond to at least one of someone's response to a question. When answering a question, remember to expand your thinking and think of mathematical examples that will make your response richer.
Your task this holiday break is to choose one of the following questions from the 9th OR 10th night and post an intelligent response. The deadline to post for this blog is Sunday, January 3rd by 2pm no later. WRITE OUT YOUR POSTS ON PAPER AND BRING THEM TO CLASS. We will continue our discussion on Monday. Use the bookmark to help guide your comments. Then continue reading through Chapter 11 and 12 which will be due by January 10th.
Let's get reacquainted with important parts of the rubric. Follow this rubric to achieve the highest scores possible.
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Ideas & Content
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The student expresses some original ideas. The majority of ideas are related to the subject matter.
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The student has many original ideas and expresses them clearly. The great majority of ideas are related to the subject matter. Student makes connections to real-world situations and prior mathematical concepts learned.
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Commenting
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Student offers constructive comments and/or asks questions that provide assistance in reaching the solution of the problem.
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Student offers constructive comments and/or asks questions of at least two classmates that clarify the problem solving process and/or provides assistance in reaching the solution of the problem with examples and counter examples.
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Writing Quality
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Posts show above average writing style.
The content demonstrates that the student reads moderately, and attempts to synthesize information and form new meaning.
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Posts are well written, and are characterized by elements of a strong writing style. The content demonstrates that the student is well read, synthesizes learned content and constructs new meaning.
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Mathematical
Communication
Clarity of
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There is a clear
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There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made.
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Mathematical
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Representation
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or required)
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There is appropriate
use of accurate mathematical
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Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem.
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Writing Conventions
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There is one spelling or
grammatical error in the explanation or comments.
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There are no spelling or grammatical
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The first row is representing ordinary numbers. It goes 1,2,3,4,5,6,7,8 and on. On page 173 it said,"He seemed tremendously pleased with a line of perfectly ordinary number." The second line represents odd numbers. It goes 1,3,5,7,9,11,13,15 and on. On page 174 it said ,"I see said Robert. The odd numbers"
ReplyDeleteI agree with Fatoumata's response and I would like to add on. The chart on page 178 shows 7 rows of different types of numbers. In my point of view, the numbers in the first row are consecutive positive integers. This is because integers have no fractions or decimals, and also they numbers go in order and do not skip anything. I agree that the second row shows odd numbers. An odd number is a number that divided by 2, has a remainder of 1. For example, 15 is in the second row. 15 divided by 2 is 7 remainder 1. Or, you can think of 7.5. Either way, you cannot get a whole number that has no remainder.
DeleteI agree with you but you should be more specific even though the number devil didn't as positive integers since 0.5 and 1/2 are ordinary numbers.just like factorials you multiple with the positive integers preceding the number next to the factorial including that number.
DeleteDear Fatoumata,
DeleteI like how your response is straight to the point. However, I think you can add more and write more about what you know about ordinary and odd numbers.
I agree with you Fatoumata but you should elaborate more on your answer by including more details from the book and you should also restate your answer. Also you should explain your example.
DeleteI agree with the comments above me because I think you should be more elaborate and try seeing why odd numbers are after ordinary numbers.
DeleteThe third row had prima donna numbers. All of the numbers in the third row are only divisible by 1 and themselves. An example is 31. It only has 2 factors, 1 and 31. This makes 31 and all the other numbers in the 3rd row prima donnas. For the fourth row, they are all Bonacci numbers. The rule for Bonacci numbers, as we read in Chapter 6, is that to get the next number in the sequence, you add the 2 numbers before it. The 6th Bonacci number is 8. To get 8, you add the two prior numbers, 5 and 3, to get your sum. Row 5 of the chart represents the triangle numbers. To get triangle numbers, you simply add 1 to whatever you added to get the last number. So, the first triangular number is 1. Then you add 2, giving you 3. Then to 3 you add 3 to get 6. The 2nd addend always increases by 1. The 6th row represents hopping numbers, or exponents. In this case, it is 2 to the second power, 2 to the third power, two the fourth power, etc. 2 squared is 4. and 2 cubed is 8. So, it goes on in that pattern. Finally, the 7th row to me has a multiplication rule and the 2nd factor is always increasing by 1. For example, 1 times 2 is 2. Then, you add 1 to the 2nd factor. 2 times 3 is 6. I think of it of the triangular numbers, but with multiplication.
ReplyDeleteWhen you said that the pattern is triangular but only with multiplication doesn't that mean the addition could also work on it as well.
DeleteI like how you used your previous knowledge from the previous chapters and included that withim your response.
DeleteI will answer the 9th night first. The 3rd row or the green row is represented by are all prime numbers as you see they have only 2 factors one and itself.3 examples are 2, 19, and 47.All are prime numbers.Does 2,19 or 47 have 3 or more factors.You can test them out on page 178.The blues are all Fibonacci numbers.The way you can find is by using the previous rules given in the book.You add the the addend and the Fibonacci number before it.An example is 5.2+3 (Fibonacci numbers) or 5+3 equals 8 the next Fibonacci number. Or you can know if you remembered that the first two numbers of the Fibonacci sequence is 1.The next row are triangle numbers. A way is to use the algebraic equation is xn = n(n+1)/2 so if you want to find the 10th number of the sequence.So,you'll substitute 10 so x10 is equal to 10(10+1)/2 so 11 times 10 is 110 divided by 2 is 55 and you can check it with the third row count 10 spaces and you will find 55.The following are square numbers or hopping numbers
ReplyDelete2 square is 4,2 cubed is and 2x2 two times is 16.The last row are factorial products. 1 factorial is 1 1. The positive integers preceding it is being multiplied.2! is 2x1 is 2.5! is 5x4x3x2x1 which is 120.You can think of it as what Ndeye says triangle numbers but instead of addition it is multiplication but not with other factorial numbers but just an integer(positive) before it except 0 but people agreed 0! is 1 so don't get confused. .
Dear Ibrahim,
DeleteI feel that your response is very well written. However, I would like to see if you can explain what you mean using less words.
But,I only wrote so much to show why I am correct.
Deletei agree with Ibrahim because he wrote this much just to break it down to everyone so that they would understand
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ReplyDelete2) How many red-shirted numbers are there compared to white-shirted numbers?
ReplyDeleteThere are 13 white shirted numbers. While there are 25 red shirted numbers.The difference between the shirts is that the white shirted shirts go by one. What this means is that the next number in the pattern is the sum of the number before and one. However, the red shirted numbers go by two. So,the next number in the pattern is the sum of the number before and two. This example of white shirts and red shirts is related to the daily math classroom. This is because this example shows patterns, which are constantly in math problems.
I agree with you Vianny,but you should give more visual information like equations to strengthen your answer.
DeleteI agree with you Vianny but I would like to give an example from the book. "That's obvious," said Robert. "Every other number is odd, so there are half as many reds as there are whites."
DeleteI would like to add on to what Mia said, "See? Each ordinary number from one on. Can you show me one single red without a partner? So there is an infinite quantity of ordinary numbers and an infinite quantity of odd numbers." Robert thought for a while. "So if I divide an infinite quantity in half I get two infinite quantities. But then the whole is the same size as the half." This shows that the odd numbers and ordinary numbers are two quantities that go on forever and are the same size as the whole.
DeleteI agree with your answer Vianny and I like that it was explained thoroughly. But, you could have shown where in the text you saw that so others can verify your answer.
ReplyDelete4) What did the number devil write on the ceiling of Robert's room? What is a series?
ReplyDeleteA series is a sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series. While some sequences are simply random values, other sequences have a definite pattern that is used to arrive at the sequence's terms. The number devil had wrote a series of numbers on a number line in Robert room. In the book it states " But by then the number devil had drawn the first series on the ceiling with his stick."
Do you think you could've given an example? An can you compare the Number Devil's definition of a series compared to the one you just gave me. This will help clarify your explanation.
Delete6) How many sides and or points did the snowflakes have? What geometric shape do they resemble?
ReplyDeleteIn the book it states " Most had six sides or points, and when Robert looked closer he saw that certain patterns tended to return: hexagonal stars in a hexagonal star, points branching off into smaller and smaller points..." this example states how the snowflake has 6 sides or points also they make hexagonal star.
I agree with Vianny. The red and white numbered shirts there was difference cause the white numbered shirts were counting by one which means that the next number in that pattern is before and one. The red numbered shirts are counted by odd numbers because in page 175 the numbers in the white its counted by one and the red numbers is counted by two and they are odd. This is related to in my daily math classroom. These examples shows a pattern which in in a constant math problem.
ReplyDeleteKiara, I think you could have answered another question. It seems like you just restated Vianny's answer.
DeleteKiara, I like how you answered the question but i think you could've written a different example so it doesn't seem like your saying the same thing as Vianny.
DeleteI agree with the previous comments, Kiara could give examples on how it relates to your daily math classroom.
Delete9th night question 1: The first and second row of all the numbers show the odd numbers that are from 1-13 and all the first 1-13 numbers. So the first row shows people the first 13 numbers which it goes as 1,2,3,4,5,6,7,8,9,10,11,12,13. The second row then goes as all the odd numbers but only the first 13 of them. So that row will look like this,1,3,5,7,9,11,13,15,17,19,21,23,25. In the book it states "I see said Robert. The odd numbers. Right. Now I want you to guess how many of them there are compared with their white shirted comrades along the wall." So this shows us the odd numbers and how they work with the first row. They are similar and they have the same amount of numbers on them.
ReplyDeleteCould you be more elaborate when you say they have the same amount of numbers?
Delete10th night question 6: The snowflakes on page 193 are hexagonal. They are shaped like hexagons and also have the same amount of sides as them. The snowflakes have six sides to them. On page 193 if you count the sides of the snowflakes that are coming down then you can count that they each equal to a total of six sides. Then if you count the sides that are on a hexagon then you will also find that there are six sides to the shape. These are the sides and shape that the snowflakes resemble in the story.
ReplyDeleteQuestion 1: The first row are ordinary numbers, they go as the number 1,2,3,4,5,6,7,8,9,10 and it keeps going. The second row are odd numbers, they go as the numbers 1,3,5,7,9,11,13 and it keeps going. To support my answer, on page 174 the second table only has odd numbers and it says "I see, said Robert. The Odd Numbers."
ReplyDeleteQuestion #4: On the page 180 the number devil had drawn the first series on the ceiling. They were fractions. A series is the value you get when you add up all the terms of a sequence. A sequence is an ordered list of numbers, the numbers in the list are called "elements" or "terms".
ReplyDeleteQuestion #6: On page 192 it says "Most had six sides or points, and when Robert looked closer he saw that certain patterns tended to return: hexagonal stars in a hexagonal star, points bramching off into smaller and smaller points..."
ReplyDeleteThe number devil wrote the numbers 0,1/2,and 1 in order. He describes what a series is by writing these numbers on the ceiling. If you look without reading 1/2 is half of one. Meaning two halves make showing how adding more halves makes a series of numbers such as 1/2,1,1 1/2,2,2 1/2. If you get my point the number is just trying toexplain how series originate from one number. He specifically describes describes what a serie is on page 180
ReplyDeleteI think snow flakes have more than 6 to 7 shapes because if you take away all of the decorations and filter them it looks like as if it were made up of triangles and hexagons. If you count around the sides you see 6 triangle like points but,m if you look into those triangles there are more triangles. To add on I even think there is an infinity amount of triangles because you can ,more triangle out of other triangle just not equal triangles. Their angles would be very different from each other.
ReplyDeleteCould you use evidence to clarify your observations? But good thinking!
DeleteOn the first row are ordinary numbers and on the second row are odd numbers. Actually the first and second row have the same number of quantities that go on forever and equal the whole. FOr example on page 175 it states, Each ordinary number from one on. Can you show me one single red without a partner? So there is an infinite quantity of ordinary numbers and an infinite quantity of odd numbers." Robert thought for a while. "So if I divide an infinite quantity in half I get two infinite quantities. But then the whole is the same size as the half." This shows that the odd numbers and ordinary numbers are two quantities that go on forever and are the same size as the whole.
ReplyDeleteThe snowflakes had 6 sides and the resembled a hexagon. FOr example on page 191-192 it states, " Everyone of the large soft flakes
ReplyDeletewas unique. Most had 6 sides or points, and when Robert looked closer he saw that certain patterns tended to return; hexagonal stars in a hexagonal star, points branching off into smaller and smaller points...." This shows this is a hexagonal shape. TO add on, hexagons have 6 sides.
The first row is representing ordinary numbers. It goes 1,2,3,4,5,6,7,8 and on. On page 173 it said,"He seemed tremendously pleased with a line of perfectly ordinary number." The second line represents odd numbers. It goes 1,3,5,7,9,11,13,15 and on. On page 174 it said ,"I see said Robert. The odd numbers"
ReplyDeleteyou should include more evidence than just the odd numbers for example on page 175 it states each ordinary number from on has its own odd number from one on
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ReplyDelete1) Which set of numbers are represented by the first row? second row?
ReplyDeleteAs shown on page 174, the numbers that are shown in the first row count up by, 1, then 2, 3, and 4, and so forth...
And in the second adds by 2 starting with 1 for Ex: 1 + 2 = 3, 3 + 2 = 5, 5 + 2 = 7, and so on...
don't forget to conclude evidence to inform /persuade the reader about your statement otherwise it wont stand (meaning supporting
Delete9th night#33) Which set of numbers are represented by the third, fourth, fifth, sixth, and seventh rows?
ReplyDeleteThe set of numbers listed in the frist row are ordinary numbers which in outher words are called natural numbers. natural numbers are A natural number is a number that occurs commonly and in nature as in The whole numbers from 1 upwards: 1, 2, 3, and so on Or from 0 upwards in some fields of mathematics: 0, 1, 2, 3 and so on there are No negative numbers and no fractions.In the second row we have odd numbers which are the color red odd numbers are Any integer that cannot be divided exactly by 2 is an odd number.
The last digit is 1, 3, 5, 7 or 9. in the third row we have prima donnas which are green they are usally or today as we know are called prime numbers prime numbers are A prime number (or prime integer, often simply called a "prime" for short) is a positive integer that has no positive integer divisors other than 1 and itself. More concisely, a prime number is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. But in outher words Prima donnas are special divas of operas, and the comparison is being made between prime numbers and the importance of divas.as stated in page 55-66 ,In the fourth row which is the color blue they are the bonaccis which are (Fibonacci numbers) a series of numbers in which each number is the sum of the two preceding numbers. The simplest is the series 1, 1, 2, 3, 5, 8, etc The next number is found by adding up the two numbers before it. (1+1)•Similarly, the 3 is found by adding the two numbers before it (1+2),•And the 5 is (2+3).0, 1, 1, 2, 3, 5, 8, 13, 21, 34.as we read in chapter 6 of number devil that to get the next number in the sequence, you add the 2 numbers before it.page 108 till the end of the chapter 6. another way to slove for the fibbonichi numbers is hopping Robert and the number devil state on page 110 for example you can take the bonicci number 4 which is three then you can make it hop to 3 the exponents of 2 which is 9 which you is alike to the coconuts and the rutabagas had mentioned in page 76,93.the 5th row which is orange are triangular numbers which are The triangular number is a figurate number that can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one.but in outher words this is theThis is the Triangular Number Sequence; 1,3, 6, 10, 15, 21, 28, 36, 45
This sequence is generated from a pattern of dots which form a triangle. by adding another row of dots and counting all the dots we can find the next number of the sequence. in the 6th row (black numbers) the numbers they represent are hopping numbers rising a number to a power. A visual description of raising numbers to powers. and then we have hopping backwards which is taking the square root of something and squaring like Raising a Number to a Higher Power: Approximating higher power uses Squaring Numbers (many kinds), Multiplying 2 Numbers exponential expression, you find the new power by multiplying the two powers together. for another example you hopped from 2 to 4 to 6 to 8. The pink colored numbers or seventh row is representing vroom!(factorials)which is Vroom! is the visual description of the speed at which factorial numbers grows. 4! is shorthand for 4 x 3 x 2 x 1 factorial Symbol
The factorial function (symbol: !) means to multiply a series of descending natural numbers.
10th night #10 Use the figures on page 203 to prove the formula true. Hint: White spaces are closed shapes. What types of figures does this formula work for? If you go back to page 203 the formula (d + s - l = 1)worked for each of the figures if you add the number of dots=d and spaces=s together and subtract the number of lines=l the total would be 1 so to prove this right I am going to provide you with 2 examples #1 we will provide you with an five pointed star inside a five sided figure if you count the dots it would equal 10 so this is when you start your equation 10=d(dots)then count the spaces inside the shape which equal 11=s(spaces) then add the d+l= plug in the numbers 10 +11=21 now count the lines which would equal 20 then subtract 21-20 which would equal 1 and you would end up with an equation like this 10+11-20=1.#2 but it has to be an flat figure (plane figure)made with straight lines . for the figure on the right( 1 figure) it has 7dots then 2 spaces so 7+2 =9-lines which is 8 so 9-8=1
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