Karteikarten
Math Module 1
Karten in diesem Stapel (12)
- 1Star PolygonA Star Polygon is formed by plotting n dots that are equally spaced (on a circle) and connecting every kth point with a straight line. The n star polygon where every kth point is connected is named or {n/k}.
- 2A Star Polygon is formed by plotting n dots and we will work with 8 dots, so n = 8.Step 1: Visualize a circle and place n (8 in this example) equally spaced dots around it.
Step 2: Connect every kth point with a straight line (use a ruler or any sort of straight edge, like an index card or a thick piece of paper) to make {8/k}. Every dot must be used. - 3Identical Star PolygonsStar Polygons are identical when the n’s are the same and the bottom values added together will give you the top!
Star Polygons are called Regular if you do NOT need to lift up your pen/pencil/mouse while creating them.
-Regular Star Polygons are made in one cycle.
-Compound Star Polygons because they are created with more than one cycle aka you have to lift your pen/pencil/mouse more than once!
A Regular Star Polygon is denoted by {n/k}, where n and k are relatively prime (they share no factors except 1) and k ≥ 2. A Compound Star Polygon has common factors besides 1. - 4Sequenceis a list of numbers having a first number, a second number, a third number, .... These numbers are called terms. We use the three dots (ellipsis) in the sequence to indicate it goes on in this pattern forever.
3, 7, 11, 15, 19, 23, … - 5Arithmetic SequenceAn Arithmetic Sequence is made by adding the same value each time. The value being added is called the common difference.
In Example 3 we see an arithmetic sequence and it has the common difference of 6.
→ 3, 9, 15, 21, 27, 33
(You are adding 6 to get to the next term.) - 6Geometric SequenceA Geometric Sequence is made by multiplying the same value each time. The value being multiplied is called the common ratio.
In Examples 1 and 4 we see a geometric sequence and it has the common ratio of 2.
→ 1, 2, 4, 8, 16, 32 and → 5, 10, 20, 40, 80, 160
(You are multiplying by 2 to get to the next term.) - 7Method of Successive DifferencesThe method of successive differences involves repeated subtraction until the difference is a constant value. Once you have a row of constant values, we work backward by adding until the desired term of the original sequence is obtained.
- 8Fibonacci Sequence1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … . Each term in the Fibonacci sequence is the sum of the previous two. After the first two terms (both 1) in the sequence, each term is obtained by adding the two previous terms.
- 9Recursion Formula.If Fn represents the Fibonacci number in the nth position in the sequence (for example: F1 = 1 F2 = 1 F3 = 2), then using the recursion formula we obtain: Fn= Fn-1 +Fn-2 for n ≥ 3.
Example 2) 987
We can use our formula, Fn= Fn-1 +Fn-2 , to come up with the answer!
F18= F17 +F16 Now we plug in what we are given:
2,584 = 1,597 + F16
Here we have that the 16th Fibonacci number added to 1,597 gives us 2,584.
To get F16 by itself we subtract 1,597 from both sides of the equation:
2,584 - 1,597 = F16
987 = F16 - 10Fibonacci Numbers climbing stepsThese are Fibonacci numbers!
# of stairs: 1 2 3 4 5 n
# of ways to climb: 1:F2 2:F3 3:F4 5:F5 8:F6 ?
# of stairs # of ways to climb
0 F1= 1
1 F2= 1
2 F3= 2
3 F4= 3
4 F5= 5
5 F6= 8
6 F7= 13
7 F8= 21
8 F9= 34
n Fn+1= # of ways to climb n steps!
If we have 9 stairs then the answer of how many ways to climb will be the 10th Fibonacci number. If we have 20 stairs then the answer would be the 21st Fibonacci number. (c) n stairs? Fn+1 ways to climb. - 11Number Patterns-fibonacci sequenceRemember we use Fn to represent the Fibonacci numbers. The Fibonacci numbers go odd, odd, even, odd, odd, even, odd, … Every THIRD number is even. So, if the n (in Fn) is a multiple of 3, the number is even and if it is not a multiple of 3, then it is odd.
- 12tEvery square is composed of two consecutive triangles.
Triangular numbers are: 1, 3, 6, 10, 15, 21, 28 . . . and
1 + 3 = 4 ← the 2nd Square number
1 + 3 = 4 ← the 2nd Square number
T1 + T2 = S2
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- Karteikarten am Math Module 1 , erstellt von Deborah Wilson am 06/03/2025.
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