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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.

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 from me or your classmates right away!

Caution: when you are giving help, give hints and ideas, not whole answers.

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.

DON’T: “Professor, can you help with #3, I don’t get it!”

DO: Take a picture of your work so far and send it to me, or describe what you did. “Professor, on #3, I tried multiplying by 2 and then I tried…but I’m still stuck.”

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

mailto:[email protected]


Math 214: Project 2

1. Third grade area activityIn this activity from Mindset Mathematics, by Jo Boaler, Jen Munson and Cathy Williams, third graders are given 24 square tiles:

a.)Draw all the possible rectangles you can make that have an area of 24 (have 24 square tiles inside).

b.)How do your answers relate to multiplication?

c.) How would the answers help the students begin to understand the distributive property?

d.)Would this activity take a longer or shorter time if you gave the students 36 squares instead of 24 squares? Explain.


2. Two numbers add together to get ten. What might you get when you multiply those two numbers? This puzzle has been given to third graders, who work on it in groups for a whole class period (also from Mindset Mathematics). Imagine you are a third grader, exploring this topic, instead of thinking there must be one right answer, as grown-ups have been taught!

a.)Make a table of your answers:Two numbers that add to ten Multiply to…

b.)What patterns do you see? Is there a greatest product? a smallest product? (Look up the word product if you have forgotten what it means.)

c.) How do the patterns show the students the commutative property?

d.) If you were adding two numbers to get 12, where do you think the greatest product would be? The smallest product? (This would be a whole new exploration for third graders, not a quick question like this one!)

e.)Generally, if you have two numbers that add to get n, what is their greatest product? The smallest product? (Not for third graders! Hint: what did you do to the 12 to get the greatest and smallest products? Do the same thing to the n! Don’t worry about simplifying or rewriting it!)

3. a.) Goldbach’s conjecture states that every even number greater than two is the sum of two prime numbers. Make your own two examples (different from your


group if you are face to face, or from what your other classmates post if you are online) that shows this true.

Example 1:

Example 2:

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 greater than 5 (make a list of odd numbers larger than 5) →

can be expressed as translates to (circle the correct one) → A. ½ B. = C. D.

the sum of (what does sum mean?) →

three primes (make a list of primes) →

Pick your own odd number (different from what other people have) and show the conjecture is true for that number. Then show a second example, with the same number or a different number (your choice). Share your work with your group or classmates so they can help you check that you are correct.

Example 1:

Example 2:

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

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


4. Perfect NumbersEuclid (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 (after P=2). TIP: when you are done, 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 (not P=4, 4 is not prime). Show your work, not just your answer. TIP: when you are done, do a web search for perfect numbers to see if you are right.

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.

Tree:

The prime factors are:The proper divisors are:


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

5. In third or fourth grade, children can begin to explore block patterns like the one below. a.)Draw the next L-shape and fill in the correct number of blocks for L3 and L4.

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

L0=1 L1=5 L2=9 L3=_______ L4=___________b.)Use the pattern of successive differences to find the number of blocks in next

L’s.

c.) Not for third graders! Complete the table and graph the number of blocks. If your graphing skills are rusty, check the online grapher at https://www.desmos.com/calculator.

d.) Write a formula using y and x, for the area (number of blocks), using the starting number and the difference.

e.)Show that your formula gives you the correct result for x = 4.

1 5 9

4

y

x

L Number, x

Number of blocks (area), y

0 11 52 934


f.) Do you have a line or a curve? Explain how you can tell without looking at the graph.

6. Create your own letter or shape pattern!a.)Using a grid like the one below, create your own letter or shape that

increases by the same amount each time. Draw at least four of your shape, in increasing sizes, like the L shapes on the previous page. Do not show what is being added each time – that’s for another student to figure out!

b.)Make a copy (or a picture or a pdf) of your pattern and give it to another student in the class (if you are in the online class, post it to “files”).

c.) Find another students’ pattern. Describe the pattern and say how much is being added each time. (You do not need to copy the picture.)

Student’s name: Type of shape being made:

Amount of blocks in their first shape: Amount being added each time:

The formula that describes their pattern, using n:

Show that the formula works to get you the number of blocks (area) of the 5th

shape:


7. Repeating Patterns. a.) In the pattern LEARNSLEARNSLEARNS…. what will the 105th letter be?

Start with a smaller problem first! What will the 18th letter be? The 20th? Show how you know, using the table:

What type of number is always in the last row (hint: 6, ____, ____, ….)

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

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

What will the 105th letter be? Explain how you know.

b.) If you raise 6 to the 47th power, 647, what will the last digit be? Since you cannot go that high on your calculator, try smaller powers of 6. Show several examples, then write your conjecture (prediction) for the last digit of 647.

1: L 7: 2: E 8: 3: A 9: 4: R5: N6: S


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 F 19.

F1

F2

F3

F4

F5

8. 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? Explain how you know, using the pattern you found. LOOKING UP or writing out up to the 30th Fibonacci number does not count as using the pattern!

c.) Will the 100th Fibonacci number be even or odd? Explain how you can tell, using the pattern you found. LOOKING UP or writing out up to the 100th Fibonacci number does not count as using the pattern!


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

F1 = 1 = ___1______ = F2

F1 + F3 = 1 + 2 = ___3______ = F4

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

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. Look for a pattern in how the answers are related to the numbers being added.TIP:F1 + F3 + F5 = 1 + 2 + 5 = _________ = F ?How are these related to this?NOT how each old answer is related to the new answer.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?

c. Use the pattern to predict the sum 1 + 2 + 5 + 13 + ... + 1597 =______.

Write the answer, then explain how you know 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?Your explanation:


10. 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 SumPattern (F1)2 = 12 = 1 = 1 = 1 1 = F1 F2 (F1)2 + (F2)2 = 12 + 12 = 1 + 1 = 2 = 1 2 = F2 F3

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

= = ____= _ __ =

= = ____= _ _ ___ =

= = ____= __ ____ =

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. TIP: Look for a pattern that goes across(F1)2 + (F2)2 = 12 + 12 = 1 + 1 = 2 = 1 2 = F2 F3Look for how these are related to these.NOT how each old answer is related to the new answer.

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?Your answer: 12 + 12 + 22 + 32 + 52 + ... + 2332 = _____________Your explanation:


11. 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

___ , ____ write the squaresand then subtract the smaller result from the larger result → 4–1 = 3

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? 1 ____ 3 = 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 two numbers 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?12. The Staircase: How the triangular numbers connect to Gauss’s method

The 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 = ___ + ___ + ___ + ___ + ____Rectangle: Rectangle: Rectangle: Rectangle:2 by 3 3 by 4 ___ by ____ _____ by ____

a.)Fill in the blanks and draw T5.

b.)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 ___ rectangle =

T5 = =½ of a ___x ___ rectangle =

T10 =Caution: this is 10, not 6!

=

T100 = This is 100, not 11!

=

Now imitate the pattern you see, above, using n instead of numbers. Hint: each second number, above, is always how much more than the first number?

Tn = ½ of a _______ x _______

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

the last box.


Tn ¿ n(n+1)2is the same as ½ of a _______ x _______

Fill in the blanks and use colors or circles and arrows to show where you see the same elements.

To help with the next questions, write a list of the first seven square and triangular numbers, here, with subscript notation. Look online or in the textbook.

S1 = 1 S2 = 4 S3 = _____ S4 = _____ S5 = _____ S6 = _____ S7 = _____

T1 = 1 T2 = 3 T3 = _____ T4 = _____ T5 = _____ T6 = _____ T7 = _____

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

What number do you get?

b.) Is your answer in part a) a square number or a triangular number? Which one?

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.


14. If you subtract the squares of two consecutive triangular numbers, what kind of figurate 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 answer as a base and exponent Hint: the result is not a square number, but instead, a different power.

(T2)2 – (T1)2 32 – 12 9 1 = 8(T3)2 – (T2)2 62 – 32

(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 = _________


15. 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 to look like pentagons, or like houses! 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 = 22P5 =

e.)Write the general formula, with the correct subscripts, using the variable, n.Pn = T? + S?


16. Fraction Puzzlesa.)What fraction of this rectangle is shaded white?

What fraction of the same rectangle is shaded black? First write the fraction over the same denominator as before, then write as a reduced fraction.

b.)What fraction of this square is shaded black? Write as an unreduced fraction, then reduce.

c.) In the diagram, outline a rectangle that shows 1/3 white (hint: find a fraction that reduces to 1/3).

d.)Outline and label rectangles that show 1/2 black,

3/4 grey and 3/8 white.

e.)Outline and label any shape (not a rectangle!) that shows 3/5 black.


17. 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, or copy a picture of it.

b.)Based on your folding, what whole number does the answer to 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? (add in the next fraction after 1/8).

Add 12+14+ 18+ 1? + 1? (add in the next fraction after the above).

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

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

½

½ ¼


f.) Does this pattern agree with what you found in part b? Explain.


18. 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. Note: neither fraction can have a numerator larger than 59, since that’s as high as you can go in base 60.


19. Multiplying Fractions: Partial Products and Areaa.)Multiply 4 12×3 using partial products and the distributive property. It may be

easier to write as: 3 ×(4+ 12)

Show all your work using fractions, not decimals.

b.) Show how you can find 4 12 3 using area. Caution: this grid is not quite the right size. DRAW the correct rectangle on the grid. Use the ruler to help you.

Label each partial product on the rectangle, above.

c.) Multiply 4 12×312 using partial products and the distributive property:

(4+ 12)×(3+ 1

2)

Show all your work using fractions, not decimals.


d.)Show how you can find 4 12 312 using area. Caution: this grid is not quite the

right size. DRAW the correct rectangle on the grid. Use the ruler to help you.

Label each partial product on the rectangle, above.


20. Coloring Multiples in Pascal’s Trianglea. 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?

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?

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, in the middle of the Pascal’s triangle?

There is the start of a shape in the bottom row. What do you think this shape will be? Why?

d. Color all the multiples of 6 (tip: use divisibility rules):What divisibility rule do you use to color in multiples of 6?

Is the same kind of shape still made by the multiples of 6 that you found with the multiples of 5, 3 and 2? Why do you think this happens?

21. Binomial Multiplication Patterns in Pascal’s Triangle

a.)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, but it should be your best guess at the algebraic answer.

22. Conjecture: “When you add any consecutive numbers together, the sum will always be a multiple of however many numbers you added up.”

a.)Show your work understanding the problem by writing the meaning next to each part

consecutive numbers (Caution: the above does not say “consecutive even” or “consecutive odd,” just “consecutive.”)

the sum


will always be

a multiple of

however many numbers you added up

b.)Show at least four examples, being sure to choose all kinds of different amounts of consecutive integers.

c.) Based on your examples, does it look like the conjecture is true, or only true sometimes? Explain.

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