collegemathforelementaryeducation.files.wordpress.com · Web viewProject 2: Number Patterns, p. 21 Name:_____ Math 21 4 Project 2: Number Patterns Each problem is worth 5 points Do

Project 2: Number Patterns, p. 1

Name:_______________________________________________

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

Do 20 for a total of 100 points, or do all 24 for a possible 110 points!

This page is where your professor will put your points for each question. Each question is worth a total of 5 points.

1. 6. 11. 16. 21.

2. 7. 12. 17. 22.

3. 8. 13. 18. 23.

4. 9. 14. 19. 24.

5. 10. 15. 20.

Project Grade (total of all questions attempted): _______________

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!


How to write up this project: You can print out the project, write on it by hand, then scan or take pictures

of each page. Remember that all the pictures/scans must be together in one file. If you get stuck on how to do this, directions are in our files tab.

You can use the word version of the project, type out parts of your answers into the word doc and handwrite other parts. You can then take pictures of the handwritten parts and paste them into the word doc in the right places.

You can write the whole thing out on separate paper if you really want to. (You do not have to copy out each question.) But it’s a bit harder to stay organized and make sure you answer every part that way.

Remember: put your name on the first page! Name your file with your name or you initials in it somewhere, so I know it’s

yours!

Tips for doing well on this project:1. If you are confused, asked for help from your professor or your classmates right

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

To ask for help from your professor, 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, or describe what you did. “Professor, on #3, I tried multiplying by 2 and then I tried…but I’m still stuck.”

2. Look up definitions/vocabulary words in the textbook or on the internet. That’s not cheating, it’s research! LOOK UP what the word product means! LOOK UP perfect numbers!

3. 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).

4. Do more than 20 problems – try all of them! That way, if you don’t do so well on a few, you have points on others!

5.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.

Math 214: Project 21. a.) Goldbach’s conjecture states that every even number greater than two is the

sum of two prime numbers. This is in section 2.1 Make your own three 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 is true.

Example 1:

Example 2:

Example 3:

b.)There is a conjecture called “Goldbach’s weak conjecture” 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.

Step 1: 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? Don’t guess! Do a web search!) →

three primes (make a list of primes) →

Step 2: Show your own two examples that show an odd number greater than 5 that is expressed as the sum of three primes .

Example 1:

Example 2:

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

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

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




2. 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. Caution: in the first part, (2¿¿P−1) ,¿all of the P−1is in the exponent. In the second part, (2P−1 ), only the P is in the exponent.

You should get the first perfect number, 6.

c.) Show how you know 6 is perfect by showing what its proper divisors add up to (see sec. 2.1).

d.)Find the second perfect number using the formula (2¿¿P−1)(2P−1)¿ and the next prime number after P=2.

TIP: do a web search for perfect numbers to see if your answer is correct!

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


f.) What happens when you try a number that is not prime? Use the formula (2¿¿P−1)(2P−1)¿ and P=4. Show your work.

g.)Show that the number you got in the previous part is not perfect by showing what its proper divisors add up to. Tip: use divisibility rules to help you see what goes into the number. Write down both the number that divides in, and the result. Both are divisors.

h.)Find the third perfect number using the formula (2¿¿P−1)(2P−1)¿ and the next prime number for P. Show your work.

TIP: do a web search for perfect numbers to check if you are correct.


3. In third or fourth grade, children can begin to explore block patterns like the one below, in pre-algebraic reasoning. We have added the algebra to it, as at the end of section 2.2. 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 = 6 L2 = 11 L3 =_______ L4 =___________

b.)Use the pattern of successive differences to find the number of blocks in next L’s (see the end of section 2.2).

differencesgo here →

1 6 11

5


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

d.)Write a formula using y and x, for the area (number of blocks), using the starting number and the difference. This is at the end of sec. 2.2

e.)Show that your formula gives you the correct result for x = 3.

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

y

x

L Number, x

Number of blocks (area), y

0 11 62 11345

https://www.desmos.com/calculator


4. More block patterns! a.)Create a block pattern for a classmate to solve. It could be the first letter of

your name, getting bigger and bigger, or just a fun pattern. Caution: the same amount must be added each time! Draw the first three of your shape (each one bigger than the last) by shading in the blocks in this grid:

Under each one, write the number of blocks in it. Remember, the number of blocks must increase by the same amount each time!

b.)Solve a classmate’s block pattern:Classmate’s name: _______________Number of blocks in the first three they drew: _______, _________, ________

What is the successive difference each time?

(If the difference is not the same each time, use someone else’s pattern!)

Write the formula for the number of blocks in their pattern:

Show how you can use the formula you wrote to find the next number of blocks.


5. Third grade area activityThis is an activity from Mindset Mathematics, by Jo Boaler, Jen Munson and Cathy Williams. Third graders are given square tiles, and asked to create rectangles from them:

a.)Draw all the possible rectangles you can make that have an area of 30 (have 30 square tiles inside). The first two rectangles have been drawn for you.

This is a 1 by 30 rectangle, with 30 square tiles inside it.

This is a 2 by 15 rectangle, with 30 square tiles inside it.

Draw all the other rectangles you can make with 30 tiles:

b.)What is the commutative property of multiplication? Do a web search (Google it) if you don’t know!

c.) How would the rectangles you drew above help children visualize the commutative property of multiplication?


d.)Now let’s imagine we change the activity! Would students have to draw more rectangles if you gave them 21 tiles or if you gave them 23 tiles? Explain you know.

If I gave them 21 tiles, they could make ______ rectangles with them.

If I gave them 23 tiles, they could make ______ rectangles with them.

Which one has more rectangles that the kids could make? Why?

6. Two numbers add together to get 12. 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 12 Multiply to…0 + 12 = 12 0 x 12 = 0

b.)What patterns do you see? What is the greatest product? Smallest product? (Look up the word product on the internet if you have forgotten what it means.)


c.) How do the patterns show the students the commutative property of both

addition and multiplication?

d.) If you were adding two numbers to get 16, what 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 another number, what is their greatest product? The smallest product? Describe the pattern in words. Do not use numbers here, just talk about the pattern.


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

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

consecutive numbers Remember you can do a web search (Google “consecutive numbers meaning”) to see what consecutive numbers are! (Caution: it does not say “consecutive even” or “consecutive odd,” just “consecutive.”)

Meaning of the word sum

will always be

a multiple of

how many numbers you added up

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

Example 1:Does it work?




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

8. Repeating Patterns.


a.) In the pattern LEARNEDLEARNEDLEARNED…. what will the 105th letter be? Hint: this is in section 2.3.

Start with a smaller problem first! What will the 14th letter be? The 20th? Show how you know, using the table:

What type of number is always in the last row (hint: 7, ____, ____, ….)

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

What will the 98th letter be? Explain how you know using the pattern.

What will the 100th 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:

Write your conjecture (prediction) for the last digit of 647.Fibonacci Number Patterns – for this and the next problems, find a list of Fibonacci numbers on the web or in the textbook (sec. 2.4) and write them out, here, to at least F 19. Note: F1 is 1, not zero.

1: L 8: 2: E 9: 3: A 10: 4: R5: N6: E7: D


F1 = 1 F10

F2 = 1 F11

F3 = 2 F12

F4 = 3 F13

F5 F14

F6 F16

F7 F17

F8 F18

F9 F19

9. Even and odd Fibonacci Numbersa.)Which of the Fibonacci numbers are even? What pattern do you see in the

subscripts of the even ones?

b.)Will the 30th Fibonacci number be even or odd? Explain how you know, using the pattern of the subscripts that 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 of the subscripts you found. LOOKING UP or writing out up to the 100th Fibonacci number does not count as using the pattern!


10. 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:


11. 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:


12. 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?13. Fruit Puzzles – Find the number represented by each fruit and use that to

get the answer to the last equation. Think about which of these problem-solving strategies you can use for each one:

make a table guess and check (also called trial and error) work backwards describe and solve the problem algebraically

a.)Puzzle 1 each fruit is a whole (positive) number. Show your work.

× × = 27× × × = 24× × × = 96 + × = __?___

Puzzle 2 each fruit is a whole (positive) number. Show your work.

+ + = 9+ + = 5

– = 2/ + = _?____


Puzzle 3 each fruit is a whole (positive) number. Show your work.

× × = + = 6/ = – 1

+ + = ___?____

b.)Which of these problem solving strategies did you end up using? (You can circle the ones you used.) make a table guess and check (also called trial and error) work backwards describe and solve the problem algebraically

How did you use them? Did your strategy change as you went along? Why or why not? This answer should be at least four sentences.

Patterns in Variables – before you do the next problems watch the short video: https://www.youtube.com/watch?v=0u9Cj6yGywk&feature=youtu.be

https://www.youtube.com/watch?v=0u9Cj6yGywk&feature=youtu.be



14. For the rectangles belowa.)Write the length, width and area of each rectangle under it (the first have

been done for you), and draw the next rectangle.

b.)Describe in words how each new width is related to the previous width:

Describe in words how each new length is related to the previous length:

c.) Another pattern is how each width is related to each length. How much more is the length than the width, for each rectangle?

If the width is 17, what will the length be?

d.) If the width is n, what will the length be? (hint: whatever you did to 17, do that same operation to n instead)

e.)Using the variable n, write a formula for the area. Area = L x W = (_____ )(______)Hint: the width is n. The length is what you wrote in part d.

Width: 1Length: 4Area: 1 x 4 =

Width: 3Length: 6Area: 3 x 6 = 18

Width: 5

Length: ____

Area: _______

Width: ________

Length: ________

Area: ___________

Width: _________

Length: ________

Area: ___________


15. 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 = ___ + ___ + ___ + ___ + ____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)2

is 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 = 9 S4 = _____ S5 = _____ S6 = _____ S7 = _____

T1 = 1 T2 = 3 T3 = 6 T4 = _____ T5 = _____ T6 = _____ T7 = _____

16. 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.


17. 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?


18. Fraction patterns

Fraction of an object = thenumber of parts youhavethetotal number of equal parts

For example, the shaded part of this shape shows the fraction 28

You can see why if divide the shape into equal parts, each the same sized triangle:

Now you can see we have 8 equal parts, with two shaded.28=2 parts shaded8equal parts

We can also move the two shaded parts together

…to show that this is the same as ¼ (1 square part shaded out of 4 equal squares)

a.)What fraction of the object are the three shaded triangles? Show how you know, by dividing the object into equal parts.

b.)What fraction of the object are the shaded triangles in this star? Show how you know, by dividing the object into equal parts. For credit, you must show the fraction and the drawing with equal parts shown. Do not reduce!

1 3 5 7

2 4 6 8

1 2

3 4


c.) Now combine pairs of triangles together to make squares (two triangles combine to make one square).

How many filled-in squares do you now have?

Draw a picture of the filled in squares and empty squares:(There is more than one correct way to draw this.)

What fraction does your new picture show? Use number of shaded squarestotal number of squares

d.)Combine all the squares in part c together. Draw the new picture, showing equal larger squares. What new fraction does this show? Use

1large shaded squaretotalnumber of large squares

e.)What three equivalent fractions did you find in parts b, c, and d? Write the three fractions as equal to each other.


19. Infinite fractionsImagine you have a number line:

The fraction ½ is halfway between 0 and 1 on this number line. We can also show this by folding in half a long strip of paper that goes from 0 to 1.

The dashed line shows where we folded the paper and the shaded part shows ½. If we fold the paper back up and fold it in half again, we get quarters, four equal parts, each worth ¼.

a.)Shade up to 12 and then shade 14 more than that on the paper. This shows 12+ 14 .

Based on your picture (without having to get a common denominator), 12+

14=¿ ______

b.) If you fold the paper back up and fold the whole thing in half again, what new

fractions does this show? (Hint: everything above gets folded in half again.)

c.) Shade 12+ 14+¿ your new fraction in your picture, above. What answer does

this show?

|0

|1

|2

12

14

24

34


d.)Based on your folding, what whole number does the answer to 12+14+…+ 1

32 get closer and closer to, but never reach?

e.)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).

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

What pattern do you see in your answers in part c?

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


20. 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. Then reduce your result to show that you get 18 .

b.) 49 can’t be written as an exact fraction over 60. Explain why not.

c.) 49 would have been written as the sum of 2660 and what other fraction over

3,600? That is, 49=2660

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

know that you are correct.

d.) 89 would have been written as the sum of what two fractions? That is, 89= ?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.


21. Multiplying Fractions: Partial Products and Areaa.)Multiply 5 12 ×3 using partial products and the distributive property. It may

be easier to write as: 3 ×(5+ 12) Tip: see examples 6 and 7 in section

2.6.Show all your work using fractions, not decimals.

b.) Show how you can find 5 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.

c.) Label each partial product on the rectangle, above. That means, show where you can find all three parts of your answer from part a.


c.) Multiply 5 12 ×312 using partial products and the distributive property:

(5+ 12)×(3+1

2)

Show all your work using fractions, not decimals.

d.)Show how you can find 5 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.

e.)Label each partial product on the rectangle, above. That means, show where you can find all four parts of your answer from part c.


22. 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? (See the end of section 2.1.)

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?

d. Color all the multiples of 15 Fill in the blanks: Multiples of 15 are the numbers that are

multiples of ____ and

multiples of _____

so, these are numbers that end in:

and the digits add up to:

Caution: all of the numbers you color in should be divisible by 15!

Is the same kind of shape still made by the multiples of 15 that you found with the multiples of 5, 3 and 2?


23. 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.

Explain how you know you are correct using Pascal’s triangle.

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 instead, say how you know you are correct using Pascal’s triangle.


24. In Pascal’s triangle, some of the rows start with a prime number after the number 1. a.)For example, the row

1 5 10 10 5 1starts with the prime number 5 after the number 1.

How are the numbers in the middle of this row related to the number 5?

b.)The next row that starts with a prime is the row 1 7 21 35 35 21 7 1

How are the numbers in the middle of this row related to the number 7?

c.) Find two more rows that start with a prime. Write out the rows:

Is the pattern still the same? Explain.

d.)Find two rows that do not start with a prime.

Is the pattern still the same? Explain.

e.)State your conclusion about this pattern:

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collegemathforelementaryeducation.files.wordpress.com · Web viewProject 2: Number Patterns, p. 21 Name:_____ Math 21 4 Project 2: Number Patterns Each problem is worth 5 points Do