Common Number Patterns

7/29/2019 Common Number Patterns

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Common Number PatternsNumbers can have interesting patterns.

Here we list the most common patterns and how they are made.

Arithmetic Sequences

An Arithmetic Sequence is made by adding some value each time.

Examples:

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.The pattern is continued by adding 3 to the last number each time.

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.

The pattern is continued by adding 5 to the last number each time.

The value added each time is called the "common difference"

What is the common difference in this example?

19, 27, 35, 43, ...

Answer: The common difference is 8

The common difference could also be negative, like this:

25, 23, 21, 19, 17, 15, ...

This common difference is -2

The pattern is continued by subtracting 2 each time.

Geometric SequencesA Geometric Sequence is made by multiplying by some value each time.

Examples:

2, 4, 8, 16, 32, 64, 128, 256, ...


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This sequence has a factor of 2 between each number.

The pattern is continued by multiplying the last number by 2 each time.

3, 9, 27, 81, 243, 729, 2187, ...This sequence has a factor of 3 between each number.

The pattern is continued by multiplying the last number by 3 each time.

Special Sequences

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, ...

This Triangular Number 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:

Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, ...

The next number is made by squaring where it is in the pattern.

The second number is 2 squared (22

or 22)The seventh number is 7 squared (72 or 77) etc

Cube Numbers

1, 8, 27, 64, 125, 216, 343, 512, 729, ...

The next number is made by cubing where it is in the pattern.The second number is 2 cubed (2

3or 222)

The seventh number is 7 cubed (73 or 777) etc

Fibonacci Numbers

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0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number is found by adding the two numbers before it together.

The 2 is found by adding the two numbers in front of it (1+1)The 21 is found by adding the two numbers in front of it (8+13)The next number in the sequence above would be 55 (21+34)

Can you figure out the next few numbers?

Integer Sequences I: The Fibonacci

Sequence

Introductory Remarks

This lesson could be subtitled "Guess my rule." I write a sequence of four or five numbers on

the board. I then ask the students what number comes next. After the students have correctlyadded two or three numbers, I ask what rule they are using to figure out the next number in

the sequence. In determining the rule, I remind them that the rule must work for every

number in the sequence.

I have taught this lesson to second and third graders.

Supplies

In addition to a chalkboard, you will need pieces of paper, about a quarter of a sheet, enoughto give each student three pieces.

Lesson Plan

1. Write the numbers 1 3 5 7 on the chalkboard.2. Ask the students what number comes next. Usually a student will correctly guess 9.3. Ask for the next number in the sequence. Ask the student who answers how she or he

knew that was correct. Students will offer explanations such as "You're skipping anumber every time." If they don't bring it up themselves, point out that these are the

odd numbers.

4. Write the numbers 1 4 7 10 on the board. Ask for the next number (13). Ask for thenumber after that (16). Ask the students to explain the pattern.

5. Write the numbers 1 2 4 7 11 on the board. Ask for the next number. It may take afew guesses for the students to come up with the correct answer of 16. Ask for thenext number (22). Ask the students to explain the pattern.

It may take several minutes for the students to figure out this pattern. Often they willsay, "You're skipping two numbers." I respond by referring to the sequence and

asking whether I am skipping two numbers between 1 and 2. I then ask what is


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happening between 2 and 4. As we proceed along the sequence a few students will

guess the rule. They usually express it as "skipping one, then you skip two, then youskip three."

It is important when discussing these explanations to take all suggested rulesseriously, and to try to apply them to the pattern. I avoid labelling ideas as "right" and

"wrong." Sometimes a student will come up with a new rule that fits the pattern. I tell

the student that it is a good rule, but it is not the one I had in mind.

6. Write the numbers 1 3 6 10 on the board. Ask the students what comes next. Afterthey discover the next two numbers, 15 and 21, ask them to explain the rule. They

usually figure this out pretty quickly, since it is the same as the rule for the precedingsequence.

7. Draw three dots on the board, with one dot on top, and two dots in a row below sothat all three form a triangle. Cover the lower two dots, and tell the students that hereis one, then uncover the dots and ask them to count. Draw a row of three dots below

the row of two, and ask the students to count the total number of dots. Add a row offour dots, then a row of five dots, counting each time. You are building a triangulararray. Tell the students that the numbers 1 3 6 10 15 . . . are sometimes called the

triangular numbers.8. Now it's time for the challenge. Write the numbers 1 1 2 3 5 on the board. Ask the

students what comes next. You should get wildly varying responses. To involve all of

the students, pass out squares of paper, and have each student write his or her guess inlarge digits so you can see it. Have the students hold their guesses up in the air. I read

the guesses out loud. It generally comes out, "I see a 5, and a seven, and another 7,and an 8, and there's a 10, and more sevens . . ."

Add the number 8 to the sequence, and ask the students to write a guess for the next

number (13).

Have the students guess one more number (21).

Ask the students if they can figure out the rule. I've had a number of third graders and

an occasional second grader get this one. To get the next number in the sequence, you

add the previous two numbers, i.e. 1+1=2, 1+2=3, 2+3=5, and so on. This is called theFibonacci sequence.

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Integer Sequences II: Combinatorics

Supplies

Each student will need a piece of scratch paper--a half sheet will do--and a pencil.

Warm-up: Guess my rule
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1. Write the number sequence 1 3 5 7 on the board. Ask the students what numbercomes next. Add 9 to the sequence. Ask for the next number. Add 11 to the sequence.Ask if the students know the special name for the sequence. If they don't know, tell

them these are the odd numbers.2. Write the number sequence 1 2 4 8 on the board below the odd numbers. Ask for the

next number (16), and the next (32). Ask the students to explain the rule thatgenerates the sequence (doubling the previous number.) Tell them these are the binary

numbers.

3. Write the number sequence 1 3 6 10 under the binary numbers. Ask for the nextnumber (15), and the next (21). Ask the students to figure out the rule. (Add 2, then 3,

then 4, and so on, to the previous number.) Tell the students that these are called the

triangular numbers. If they haven't seen this sequence before, draw a triangular arrayof dots on the board with 1 dot in the first row, 2 dots in the second row, 3 dots in the

third row, and so on. Show that the number of dots above any particular row is 1, 3, 6. . .

4. Write the number sequence 1 4 9 16 on the board. Ask the students to guess the nextnumber (25), and the next (36). Ask them to figure out the rule. Two explanations are

possible: Adding successive odd numbers, 3, 5, 7 . . . to the previous number, or

squaring 1, 2, 3, 4 . . . Tell students these are thesquare numbers.

The handshake puzzle

Pose the following problem: A baseball team is practicing for the first time. In order to get to

know each other, the players shake hands. Each player shakes hands with every other playerexactly once. How many total handshakes are there?

Have the students guess the answer, and write their guesses on the sheets of scratch paper.

Tell them to put the paper aside.

Now work it out inductively. Begin by calling two students to the front of the room andhaving them shake hands. Draw a table on the board. Label the left column "number of

players" and the right column "handshakes." Write 2 and 1 in the left and right columnsrespectively.

Have three students come to the front of the room and shake hands with each other. Count the

handshakes (3) and add the numbers to the table.

Continue with four students, five students, and six students. You may need to impose a bit oforder on the handshakes to make sure they get counted correctly. You should get 6, 10 and

15.

Ask the students if they recognize the numbers in the right-hand column. They should

identify them as the triangular numbers.

Use the rule for generating triangular numbers to work out the seventh (21), eighth (28), and

ninth (36) numbers in the sequence. Have the students compare their guesses to the answeryou worked out with them.

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Pascal's Triangle

Introductory remarks

This is one of my most popular lessons.

Pascal's triangle, named after the seventeeth-century French mathematician Blaise Pascal,turns up in a number of mathematical problems. Those who remember their high school

algebra will recognize the numbers as binomial coefficients. If you are familiar with

probability theory, you may know that these numbers give the probability of tossing m heads(or tails), in n tosses.

Lesson Plan

1. Draw the following street map on the chalkboard: At the top in the middle is the maincorner, or starting point. The two boundary streets run at 45-degree angles down fromthe starting point, one to the right, and one to the left. Inside these boundaries draw asquare grid, fairly widely spaced, of lines parallel to, and ending at, these boundary

streets. If this description confuses you, take a piece of graph paper, rotate it 45

degrees so that one corner is at the top, and copy a 6-by-6 square of it. I space the"streets" about 6 inches apart.

2. Pose the problem. Starting at the top of the grid, you want to know how manydifferent paths there are from the starting point to any corner, subject to the followingconditions. All paths must follow streets, and you are only permitted to move

downward, either to the right, or to the left.3. Begin with the first row of intersections below the starting point. Show that there is

only one path to each of these intersections. Write the number 1 on each intersection.

4. Move on to the next row of three intersections. Ask the students how many pathsthere are to the leftmost intersection in that row. Ask them to describe the path bytelling you which direction to take at each intersection. Starting at the top, you movedown to the left, which takes you to the leftmost intersection in row two. Then you

move to the left again, taking you to the intersection you are trying to reach. This path

is therefore "left-left."5. Have the students describe the paths leading to the middle intersection in row three.

The two legal paths are "left-right" and "right-left." Next, have them find the only

path to the rightmost intersection, "right-right."6. Proceed to the fourth row, with four intersections. This is where the puzzle starts to

get interesting. By this time, most students will realize that there is only one legal pathto any intersection on the right or left boundary. Don't tell them how many paths to

look for in the middle. As they describe paths, write them down using the letters L

and R, for example LLR for "left-left-right." Have the students continue to look forpaths until both you and they are satisfied that they have found them all. (You, of

course know that the three legal paths for the second intersection in the row are LLR,LRL, and RLL.) The students may use symmetry to conclude that the number of paths

to the third intersection in the row is equal to the number of paths to the second

intersection, or they may need to list them (LRR, RLR, and RRL.)7. Continue with the next row. The numbers of paths to the fifth row intersections are 1

4 6 4 1.


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8. Summarize by writing in the form of a triangle the numbers discovered so far:------------1-----------------------1---1-----------

--------1---2---1---------------1---3---3---1-------

----1---4---6---4---1-----

Ask the students if they see a pattern that will generate the next row of numbers. Moststudents quickly figure out that the first two numbers in the row are 1 5, and the last

two are 5 1. Some students can guess the rule: add the two numbers above to get thenext number. Thus 1+1=2, 2+1=3, 1+3=4, 3+3=6. Applying this rule, the middle

numbers of the sixth row are 10 10. If students do not figure the rule out in a few

minutes, I tell them.

9. Tell the students this pattern is called Pascal's Triangle. Tell them it comes up in anumber of mathematical contexts, the two most common areas being the probability

of coin tosses, and the expansion of binomials in high school algebra.

Follow-Up Activity: Probability of Coin Tosses

Supplies

Each student will need four pennies.

Lesson Plan

1. Ask the students if they know what "odds" are. If they don't, explain that odds arenumbers that give the relative likelihood of events. For example, when tossing one

penny, the chances of getting heads or tails are equally likely. Thus the odds are 1:1.2. Test the prediction by having each student toss one coin. Count the numbers of heads

and tails. Explain to the students that experimental results do not always match the

theoretical probability.3. Now ask the students to list the possible outcomes of tossing two coins. (They should

come up with two heads, two tails, and one head/one tail.) Explain that order mattershere, so that HT and TH add together to double the odds. Thus the odds for HH

HT/TH TT are 1:2:1.

4. Test the prediction by having each student toss two coins.5. Have the students figure out all the possible outcomes of tossing three coins. (They

are HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT.) Write down the theoreticalodds 1:3:3:1. Test the prediction by having the students toss three coins.

6. Figure out the possible outcomes for tossing four coins. (HHHH, HHHT, HHTH,HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT,TTTH, TTTT.) The theoretical odds are 1:4:6:4:1. Test the prediction by tossing four

coins.

7. Write the odds in triangular format:----1:1----

---1:2:1-----1:3:3:1--

-1:4:6:4:1-


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Ask the students where they have seen this before. They will recognize Pascal's

triangle from the street map puzzle.

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Generalized Fibonacci Sequences and the

Golden Mean

Supplies

Students will each need a calculator for this activity.

Lesson Plan

1. Write the number sequence 1 1 2 3 5 on the board. Ask the students what comes next.If they don't get it in three or four guesses, tell them the next number is 8. Have them

guess the next number (13), and the next (21). Ask them if they can figure out the rulethat tells how to generate the next number in the sequence. (The rule is to add the

previous two numbers.)

2. Have the students calculate about 12 more terms of the sequence. They may do thismentally, on paper, or with a calculator.

3. Have the students divide the next term by the next-to-the-last term of the sequence. Itwill be approximately 1.618. Write this number on the board.

4. Start a new sequence by choosing any two numbers between 1 and 20. It does notmatter if the second number is smaller than the first. Apply the Fibonacci rule ofadding the previous two numbers until you have a sequence of 20 numbers.

5. Have the students divide the last number in the sequence by the next-to-the-lastnumber. Again the result will be approximately 1.618. Explain that this number iscalled the "golden ratio" or the "golden mean." According to the ancient Greeks, arectangle with its sides in this ratio was the most beautiful rectangle a person could

draw.

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Data Compression

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Common Number Patterns