collegemathforelementaryeducation.files.wordpress.com… · Web viewNames of group members: Math 21. 4 . Project. 2: Number Patterns. Each problem is worth 5 points. Do 20 for a

Project 2: Number Patterns, p. 1

Name:_______________________________________________

Names of group members:

Math 214 Project 2: Number PatternsEach problem is worth 5 points

Do 20 for a total of 100 points, or do all 22 for a possible 110 points! Project Grade

Points earned for each question (each is worth a total of 5 points):1. 6. 11. 16. 21.

2. 7. 12. 17. 22.

3. 8. 13. 18. Brought printout of project to class

4. 9. 14. 19. Total:

5. 10. 15. 20.

The scoop: The more you work, the higher your grade can be! It is not about how smart you are, and not about how good you are at math now -- it is about how much time and effort you are willing to spend each week, outside of class, to think about the problems, so that you can become good at math! This work gives you the experience of sustained thinking about patterns and problem solving that you want for your own students. And you will become better at math because of your effort!


1. Tips: If you are confused, asked for help right away! Talk over answers with your

group, with a classmate or the professor.

I am not bothered if you email me ([email protected]) or text me (917-676-9865) about math. Math is what I LOVE to explain. Ask me! Be as specific as you can about what you tried before you got stuck.

o DON’T: “Professor, can you help with #3, I don’t get it!” o DO: “professor, on #3, I tried multiplying by 2 and then I tried,

…but I’m still stuck. Here’s a picture of my work.”

Look up definitions/vocabulary words in the textbook or on the internet. That’s not cheating, it’s research!

2. To get full credit on a problem, do all problem parts — generally, the final, concluding parts of a problem are worth more than the initial parts (for example, part d may be worth more than parts a to c).

3. Each person in the group must submit their own project, written in their own words. Computer copies of projects from other group members, or identical language in explanations will not be accepted.

mailto:[email protected]


Math 214: Project 2

1. a.) Goldbach’s conjecture states that every even number greater than two is the sum of two prime numbers. Make your own example (different from your group) that shows this is true. Share your example with your group and help check that each of your group members is correct!

b.)What does conjecture mean? (Do a web search or look in the textbook!) Why is the above called a conjecture? Answer both parts of this question!

c.) There is a conjecture called “Goldbach’s weak conjecture” (yes, really!) that every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). Find two examples of this.

Step1: Understand the problem. Show your word by word translation.

Every odd number (make a list) →

greater than 5 (fix the above list!) →

can be expressed as →

the sum of (what does sum mean?) →

three primes (make a list of primes) →

Pick your own odd number (different from your group) and show the conjecture is true for that number. Share your work with your group.

2. The prime versus composite challenge!

https://en.wikipedia.org/wiki/Prime_number

https://en.wikipedia.org/wiki/Odd_number


Only ONE of the numbers below is prime! Each group member should pick one of these to try, and report back to the group if it is prime or composite. If it is composite, say what the factors are. In the table below, summarize the findings from your group. You must fill out the whole table to get full credit, which might mean you have to figure out more than one of these on your own, depending on how big your group is.

Number Prime or composite? The factors of the number1369

1887

1171

4477

3031

1067


3. Block Patternsa.)Use the pattern to draw the third figure.

Figure 0 figure 1 figure 2 figure 3 (draw it)

b.)Write the total number of blocks in each figure. Use the pattern of successive differences to find the number of blocks in the 7th figure: n = 0 1 2 3 4 5 6 7

1, 5, _____, ______, ______, _______, ______, _______

c.) Write an algebraic formula for the number of blocks that will be in the nth figure. Use the variable, n (see sec. 2.1, p. 7, for help).

d.)Test that your formula is correct by showing that it works for n = 3, to give you the correct number of blocks in the third figure.

4. Repeating Patterns. a.) If you raise 5 to the 47th power, 547, what will the last digit be? Since you

cannot go that high on your calculator, try smaller powers of 5. Show several examples, then write your conjecture (prediction) for the last digit of 547.

Prediction:b.) In the pattern KITTYKITTYKITTYKITTY…. What will the 102nd letter be?


Start with a smaller problem first! What will the 10th letter be? The 15th? Show how you know, using the table:

What type of number is always in the last row (hint: 5, 10, ____, ____, ….)

What letter is always in the last row of the table?

What will the 100th letter be? Explain how you know.

What will the 102nd letter be? Explain how you know.

c.) If you raise 7 to the 102nd power, what will the last digit be? Start with a smaller problem first! Find the last digit of 71, 72, etc. Look for a pattern and create a table.

Answer:

Letter 1: K

6: K

2: I3: T4: T5: Y


Fibonacci Number Patterns – for this and the next problems, find a list of Fibonacci numbers on the web and write them out, here, to at least F19.

F1

F2

F3

5. Even and odd Fibonacci Numbersa.)Which of the Fibonacci numbers are even? Which are odd? What pattern do

you see?

b.)Will the 30th Fibonacci number be even or odd? How can you tell?

c.) Will the 100th Fibonacci number be even or odd? Explain how you can tell.


6. Find the pattern when every other Fibonacci number is added, starting with the first:

F1 = 1 = ___1______

F1 + F3 = 1 + 2 = _________

F1 + F3 + F5 = 1 + 2 + 5 = _________

F1 + F3 + F5 + F7 = 1 + 2 + 5 + 13 = _________

a. Complete the blanks, above, and then three more rows of the table in the space above, including numbers and subscripts.

b. Explain the pattern in words. Caution: it is not enough to say that the result is a Fibonacci number. WHICH Fibonacci number do you get in relation to the numbers you just added? See 2.3, p. 6.

c. Use the pattern to predict the sum 1 + 2 + 5 + 13 + ... + 1597 =______. Your answer should show that you know how to get the answer using the pattern, without having to actually add up all the numbers! Hint: which subscript does 1597 have? Which subscript will your answer have? How do you know?


7. Find the pattern when the squares of the Fibonacci numbers are added:a. Complete the table for the first six rows: The squares of Fibonacci Numbers Sum

Pattern (F1)2 = 12 = 1 = 1 x 1(F1)2 + (F2)2 = 12 + 12 = 1 + 1 = 2 = 1 x

2

(F1)2 + (F2)2 + (F3)2 = 12 + 12 + 22 = 1 + 1 + 4 = 6 = 2 x ?

= = __ = _x __

= = __ = _ _x __

= = __ = __ x ___

Complete the blanks and the next three rows. Hint: look for two special numbers that multiply to get the sum.

b. Explain the pattern of the answers in words. Hint: relate the two multiplied numbers to the numbers you just added.

c. Use the pattern to predict the sum when 12 + 12 + 22 + 32 + 52 + ... + 2332 is added. Your answer should show that you know how to get the answer using the pattern, without having to actually add up all the numbers! Hint: which subscript does 233 have? Which subscript will the two numbers in your answer have? How do you know?


8. Fibonacci and Lucas numbersa.)Given four consecutive Fibonacci numbers (this means four Fibonacci

numbers in a row, for example, 1, 1, 2, 3) if you square the middle two and then subtract the smaller result from the larger result, the result is equal to the ________ of the smallest and largest of all four Fibonacci numbers. Fill in the blank and show several examples that fit this pattern.

Understand the problem:Given four consecutive Fibonacci numbers → 1, 1, 2, 3 square the middle two → 1, 12, 22, 3

1 4and then subtract the smaller result from the larger result → 4–1 = 3

the result is equal to the ________ of the smallest and largest → look at the smallest and largest in the sequence 1, 1, 2, 3 versus the answer of 3. What can you do to 1 and 3 to get 3?

Another example:Given four consecutive Fibonacci numbers → 2, 3, 5, 8 square the middle two → 2, 32, 52, 8Fill in the answers: and then subtract the smaller result from the larger result. Fill in the answers:

the result is equal to the ________ of the smallest and largest → look at 2 and 8 and see how they relate to your answer and describe in words.

Your own example:

b.)Another sequence that is constructed in a similar way to the Fibonacci sequence is the Lucas Sequence: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, …. Find the next number in the Lucas sequence.

c.) If you take four consecutive Lucas numbers, square the middle two and then subtract the smaller result from the larger result, do you get the ____ of the smallest and largest, the same way as the Fibonacci sequence? Show several examples, then state your conclusion.

SEVERAL examples means at least three examples!

Conclusion: yes or no?9. Perfect Numbers


Euclid (born approx. 300 BCE) discovered that the first four perfect numbers are generated by the formula (2¿¿ P−1)(2P−1)¿, where P is prime. a.)The second part of this formula, (2P−1), is a special kind of prime number,

invented by a French monk, called a _____________ prime (sec. 2.1).

b.)Show how you would use this formula (2¿¿P−1)(2P−1)¿with P = 2. You should get the first perfect number, 6. Caution: in the first part, (2¿¿P−1) ,¿all of the P-1 is in the exponent. In the second part, (2P−1 ), only the P is in the exponent.

c.) Show how you know 6 is perfect by showing what its proper divisors add up to.

d.)Find the second perfect number using the formula (2¿¿P−1)(2P−1)¿and the next prime number for P. (Do not use P = 4, since 4 is not prime). TIP: when you are done, you might do a web search for perfect numbers to see if you are right.

e.)Show how you know this new number is perfect by showing what the proper divisors add up to.

f.) Find the third perfect number using the formula (2¿¿P−1)(2P−1)¿ and the next prime number for P. Show your work, not just your answer.

g.)Use a prime factor tree to find all the prime factors of your new perfect number. This will help you find the proper divisors.

h.)Show how you know this new number is perfect by showing what the proper divisors add up to.



10. The Staircase: How the triangular numbers connect to Gauss’s methodThe triangular numbers can also be drawn like an increasing staircase within a rectangle. The rectangle makes an exact copy of the staircase, upside-down.

T1 = 1 T2 = 3 T3 = 6 T4 = 10 T5 = ?=1+2 =1+2+3 =1+2+3+4

a.)Draw T5 in a similar way. b.)What is the size of the rectangle around T2? The rectangle is ___ by ___, with

area ___.c.) What is the size of the rectangle around T3? The rectangle is ___ by ___, with

area ___.d.)What is the size of the rectangle around T4? The rectangle is ___ by ___, with

area ___.

e.)Complete the table:Triangular number

The same as adding the numbers….

Each staircase takes up half of the rectangle.

T2= 3 = 1+2 ½ of a 2x3 rectangle =(2x3)= ½ (6)=3

T3 = 6 =1+2+3 ½ of a 3x4 rectangle =(3x4)=½(12)=6

T4 = 10 =1+2+3+4½ of a ___x ___ =

T5 = =½ of a ___x ___ =

T10 = =

T100 = =

Tn = Fill in the blanks, using n instead of numbers.½ of a _______ x _______(tip: see part f)

f.) The formula for a triangular number Tn ¿ n(n+1)2 is the same as the formula in

the last box, but written in a different way. Write both formulas, and use circles and arrows to show where you see the same elements.


To help you with the next several questions, write a list of the first seven square and triangular numbers, here, with subscript notation:S1 = 1 S2 = 4 S3 = _____ S4 = _____ S5 = _____ S6 = _____ S7 = _____T1 = 1 T2 = 3 T3 = _____ T4 = _____ T5 = _____ T6 = _____ T7 = _____

11. Pentagonal numbersa.)The formula for Pentagonal numbers is Pn=

n (3n−1 )2

. Use the formula to find P5, the fifth Pentagonal number. Show your work.

b.)Pentagonal numbers can be drawn as below. Write the next pentagonal number and draw the corresponding picture. Make sure your picture matches what you found in part a!

P1 = 1 P2 =5 P3 = 12 P4 = 22 P5 = _______c.) Show that the P2, P3 , P4 and P5 can be split into two shapes, a triangular

number and a square number. Hint: the square is the bottom of the house and the triangle is the roof! Circle each part! The first one has been done for you.

P2 =5 P3 = 12 P4 = 22 P5

d.)Complete the table below.Pentagonal number

The same as adding the triangular and square numbers….

P2= 5 T1 + S2P3 = 12P4 = 22


P5 = e.)Write the general formula, with the correct subscripts, using the variable, n.

Pn = T? + S?


12. Patterns in figurate numbersa.)Subtract the third square number minus the third triangular number, S3 – T3.

What number do you get? (Remember to use the list on the previous page!)

b.)Subtract S4 – T4. What number do you get?

c.) Complete the table, below. In the last rows, add your own examples. Problem solving strategy: create a table and look for a pattern.

Subscript

Notation

Numbers Result Type of figurate number you get, with subscript

S3 – T3S4 – T4S5 – T5

d.)How is the subscript in the answer always related to the subscripts you started with? Is it the same? Different by a certain amount?

e.)Use your observations to complete the following: Sn – Tn= ______ Use a subscript in your answer, and the variable, n.

13. If you subtract the squares of two consecutive triangular numbers, what kind of number do you get? Put the problem solving strategies together: use a table, look for a pattern, and use similar problems.

a.)Complete the table below.Subscript Notation

Numbers Result Rewrite the number as a base and exponent Hint: the result is

not a square number, but instead, a different power.

(T3)2 – (T2)2 62 – 32 27(T4)2 – (T3)2

(T5)2 – (T4)2

b.)Explain in words how you can predict the base and exponent of the answer.

c.) Write a general formula using n. Put the correct subscripts on each T. (T )2 – (T )2 = _________

14. Pattern Blocks


a.) I made a design using pattern blocks. On the left you see the original blocks, and on the right, I traced it onto pattern block paper.

If my design is one whole, what fraction of it are two of the blue rhombuses? Use the triangles to figure it out! (Tip: watch the 2.5 videos first.)Write the fraction using the original numerator and denominator, then simplify the fraction, if possible.

b.)Using the pattern paper on the next page to create your own design. Make sure it has at least one of each kind of shape (yellow hexagon, green triangle, red trapezoid, blue rhombus). Then pose a question to give to one of your classmates, like the one in part a, above, choosing any shape you like (for example, you could ask, “If my design is one whole, what fraction of it are three hexagons?”) Be sure you know the answer to your own question!

c.) Answer another student’s question.

Your project should have a copy of your question and answer, and a copy of the other student’s question and answer.


15. Babylonian Fractions To write fractions in Babylonian, you must convert our fractions into 60ths. For example, to write ½ as a Babylonian fraction, you must write the Babylonian for 30, since ½ = 30/60. However, some Babylonian fractions had to be written using 60ths and 3600ths!

a.)The Babylonian fraction for 18 would have been written as 760+303600 !

Show how you know that it is true that 18=760

+ 303600 by adding the two

fractions (be sure to get a common denominator) and reducing to show that you get 18 .

b.) 29 can’t be written as an exact fraction over 60. Explain why not.

c.) 29 would have been written as the sum of 1360 and what other fraction over

3,600? That is, 29=1360

+ ?3600 . Find the missing number and show how you

know that you are correct.

d.) 49 would have been written as the sum of what two fractions? That is, 49= ?60

+ ?3600 . Find both missing numbers, then show how you know that you

are correct.


16. In about the third grade, children can work on creating block patterns like L’s and staircases to explore the patterns they find. a.)Draw the 5th L-shape and fill in the correct number of blocks for the 4th and 5th

shapes. Careful – notice how much higher and how much longer the L gets each time!

L1=1 L2=4 L3=7 L4=? L5=?

b.)Use the pattern of successive differences to find the number of blocks in next L’s.

c.) If you graph the number of blocks, will you have a line or a curve? Complete the table and draw the graph.

If your graphing skills are a little rusty, check the online grapher at https://www.desmos.com/calculator

1 4 7

3

L Number, x

Number of blocks, y

1 12 43 745

y

x


d.)Conclusion: I have a line/curve (circle the correct one) because…. (think of how many rows of differences you had and look at section 2.4 p. 2 & 3)

17. a.) In this staircase, children start with 1 block, then add blocks enough so that they can step up three blocks and right two each time. Fill in the correct number of blocks and draw the 4th staircase.

S1=1 S2=9 S3=23 S4=?b.)Use the pattern of differences to find the number of blocks in next staircases.

c.) If you graph the number of blocks, will you have a line or a curve? Complete the table and draw the graph.

d.)Conclusion: I have a line/curve (circle the correct one) because…. (think of how many rows of differences you had and look at section 2.4 p. 2 & 3)

1 9 22

8

Staircase Number, x

Number of blocks, y

1 12 93 2245

y

x



18. Multiplying Fractions: Partial Products and Areaa.)Multiply 412×3

12 using partial products: (4+12 ¿×(3+ 1

2)

Show all your work.

b.) Show how you can find 412×312 using area. Caution: this grid is not quite the

right size. Mark the correct rectangle on the grid. Each box is divided into quarters. Clearly label each partial product on your rectangle.

c.) Multiply 323×212 using partial products.

d.)Show how you can find 323×212 using area. Caution: this grid is not quite the

right size. Mark the correct rectangle on the grid. Each box is divided into sixths. Clearly label each partial product on your rectangle.


19. Infinite fractionsa.)Cut the bottom off a piece of paper so that you have a long rectangular strip.

Fold it in half, lengthwise. Open it back up and write the fraction 12 on one of the two halves you have created, and shade that half with a pencil or highlighter.

Now fold the paper back in half, and fold that in half again. Open it back up. Write the fraction 14 on a piece next to the 12, and shade with a pencil or highlighter.

Continue folding and writing in this way until you have 1/32. If you can’t fold, approximately judge the amount. Include this paper in with your project, glued or stapled to this page.

b.)Based on your folding, what does 12+14+…+ 1

32 get closer and closer to, but never reach?

c.) Add 12+14 , by first getting a common denominator.

Add 12+14+ 18 by first getting a common denominator.

Add 12+14+ 18+ 1? (+ the next fraction).

d.)What pattern do you see in the fractions you are adding in part c?

½

½ ¼


e.)What pattern do you see in your answers in part c?

f.) Does this pattern agree with what you found in part b? Explain.20. Coloring Multiples in Pascal’s Triangle

a. Color all the multiples of 2:What divisibility rule do you use to color in multiples of 2?

What kind of shape (a triangle, rectangle, square?) is made by your colored-in numbers?

You should see at least three sizes of shapes in your coloring. How far across does each shape go (what is the width of the shape)? The first one is filled in for you.

Largest shape: 7 wide

Medium shape ______ wide Smallest shape ______ wide

b. Color all the multiples of 3 (tip: use divisibility rules):

What divisibility rule do you use to color in multiples of 3?


What kind of shape is made by your colored-in numbers?

What is the width of the shape?Medium shape ______ wide

Large shape ______ wide

c. Color all the multiples of 5 (tip: use divisibility rules):What divisibility rule do you use to color in multiples of 5?

What kind of shape is made by your colored-in numbers?

What is the width of the shape?

d. Complete the table:Multiples Width of smallest

shapeWidth of medium shape

Width of largest shape

235

How does the width of each shape relate to the multiple you are coloring? Explain.

e. What width do you think you would get if you colored in the multiples of 7?


Check to see if you are correct: go to http://www.shodor.org/interactivate/activities/ColoringMultiples/Click “increase depth” to make the triangle at least 7 rows bigger. Set the multiple to 7 and click “auto color.” Was your idea correct? Explain how you know.

21. Number Patterns in Pascal’s Trianglea.)What type of figurate numbers are found in the third diagonal (colored in

yellow) that starts 1, 3, 6…?

b.)The powers of 11 are 110 = 1, 111 = 11, 112 = ______, 113 = _____, 114 = _____.

Where do you see the powers of 11 in Pascal’s triangle? Hint: look at the first row, 1 then the second row, 1,1….

Does the pattern keep going on after 114? Explain.

c.) Find the sum of each row in Pascal’s triangle, up to row 9. Rows 0 to 2 have been completed for you.

row 0 sum = 1

row 1 sum = 2row 2 sum = 4row 3 sum = _______row 4row 5row 6row 7

http://www.shodor.org/interactivate/activities/ColoringMultiples/


Explain the pattern of the sums answers, in words. Yes, you can see what happens as you look down the answers, but instead, look across: how does the row number relate to the sum?

d.)Use your answer to the previous part to find the sum of row 20, without writing out the row.

22. Binomial Multiplication Patterns in Pascal’s Trianglea.)What is (x+1)0? (Hint: anything to the 0 power is ___)

b.)What is (x+1)1?

c.) What is (x+1)2? Caution: it is not x2+12, but is gotten by multiplying (x + 1)(x + 1).

d.)Where can the coefficients and constants of each answer be found in Pascal’s triangle? The coefficient is the number in front of x, and the constant is the added on number. For x + 1, the coefficient in front of the x is a 1, so we have 1x + 1.

Where do you see 1 1 in the triangle?

Where do you see the coefficients and constants for your answer to part c?

e.)What is (x+1)3? Hint: multiply the result of part c by (x+1), then combine like terms.

Check your answer using Pascal’s triangle. Show how you know you are correct.


f.) Make a conjecture as to what (x+1)4 will equal, using Pascal’s triangle. You do not have to multiply it all out to check!

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collegemathforelementaryeducation.files.wordpress.com… · Web viewNames of group members: Math 21. 4 . Project. 2: Number Patterns. Each problem is worth 5 points. Do 20 for a