Pre Algebra Period 2

Statistics Unit

There are many ways to ORGANIZE and REPRESENT data.
The whole point is to make the data more UNDERSTANDABLE.

There are 3 ways to express the CENTRAL TENDENCY of data:
Mean or average (add up all the values and divide by the number of items)
Mode or the most often seen value
Median or middle of data from least to greatest (if even number of items, average the middle two)

The Range can be expressed in two ways:
Showing the lowest value to the highest value
Calculating the difference between those two values (highest - lowest)

Tally Tables help you put COUNT the data and begin to ORGANIZE it.
You can tally numeric data (like grades) or non-numeric data (like names for the giraffe)

Frequency Tables take the tallies and SUMMARIZE them in an easy to read table.
Can be numeric or non-numeric (like the Tally Table)

Line Plots are graphs that make the Frequency Table summary more VISUAL.
It's really easy to see the MODE in the Line Plot.
For numeric data, you can determine the MEDIAN easily because it's in order from
LEAST TO GREATEST
Can be numeric or non-numeric (Like Tally and Frequency Tallies)

Stem and Leaf Plots are graphs that CLUSTER the data in INTERVALS.
This helps make the graph more UNDERSTANDABLE because you can see PATTERNS.
Data must be NUMERIC.
You can determine the MEDIAN easily because it's in order from
LEAST TO GREATEST
You can also see the MODE but not as quickly as the Line Plot because it will be in an interval with other data items. On the other hand, if you're looking for the interval that has the most data (not the individual item), the Stem and Leaf will show the "Interval Mode" very quickly.
You can read every data item on the Stem and Leaf.

Scatter Plots are graphs that plot TWO DIFFERENT TYPES OF DATA against each other, one one the x axis and one on the y axis to determine if there is a CORRELATION between the two types of data. ("co" meaning together and "relation" meaning a pattern)
If the two types of data have a correlation, you can draw a LINE OF BEST FIT through the center of the data points.
POSITIVE CORRELATION: the data moves in the SAME DIRECTION
(the more you study, the higher your grade....or the less you study, the lower your grade)
The line of best fit looks like it is going UP from left to right
NEGATIVE CORRELATION: the data moves in OPPOSITE DIRECTIONS.
(the higher the temperature, the lower the number of people using their heat....or the lower the temperature, the higher the number of people using their heat)
The line of best fit looks like it is going DOWN from left to right
NO CORRELATION: the two types of data has no relationship and so the data points are scattered everywhere with no pattern
(temperature changes should have no relationship with how well a student does)

TODAY:
LINE GRAPHS: p. 98-99
Data that takes place OVER TIME is represented well on a Line Graph.
For example, a student's math grade over the course of a year, or over middle school, or even over a single month. The length of the time interval depends on what you're trying to show.
If you have more than one set of data over the same time period,
you can make MULTIPLE LINES on the graph.
For example, a student's math grade, along with his science grade.

BAR GRAPHS:
Data over time can also be shown as a Bar Graph, but Bar Graphs are especially useful to show data that compares either non-numeric data or comparing multiple types of the same data at the same time.
Non-numeric data: Really a Line Plot but without the "X's" Allows you to represent very large data sets easily because you can label the y axis (vertical axis) in any way that you need to.
Numeric data: Going back to a student's grades on the Line Graph...Say you wanted to compare multiple students' math grades and science grades in 8th grade, you could make a DOUBLE BAR GRAPH with one bar color representing math grades and another bar color representing their science grades

HISTOGRAMS
Isn't this just a Bar Graph???
Yes! It's a specific type of bar graph that shows the FREQUENCY of data items

Isn't that what a Line Plot and a Stem and Leaf Plot show???
Yes!
If you made Bar Graph out of a Line Plot, it would be called a HISTOGRAM
or
If you turn a Stem and Leaf Plot on its side, outline the data listed in the leaves and then erase the data numbers and instead color in the bars, you will also have a HISTOGRAM.

What you lose:
The actual data items (you'll still know 12 students scored in the 80's, but not exactly what score in the 80's that each of them received)
You cannot determine the exact MODE or MEDIAN, but can make an analysis of the intervals.
What you gain:
For large amount of data items, a HISTOGRAM is much less cluttered than either a Stem and Leaf or Line Plot, making the patterns easier to understand.
For large amount of data items, because you can adjust the labels on the y axis, you can fit all the data in in whatever

Math 6 Honors Periods 6 & 7 (Tuesday )

Multiplying or Dividing by a Power of Ten 3-7

We have learned that in a decimal or a whole number each place value is ten times the place value to its right.

10 ∙ 1 = 10 10 ∙ 10 = 100 10 ∙ 100 = 1000

10 ∙ 0.1 = 1 10 ∙ 0.01 = 0.1 10 ∙ 0.001 = 0.01

Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.

Multiplying by ten moves the decimal point one place to the right

10 ∙ 762 = 7620 = 7620

10 ∙ 4.931 = 49.31

At the beginning of this chapter you learned about powers of ten

10⁴ = 10 ∙10 ∙ 10 ∙10 = 10,000

We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.

2.64874 ∙10⁴ = 2.63874 = 26,387.4

Notice that we have moved the decimal point four places to the right.

Rule

To multiply a number by the nth power of ten, move the decimal point n places to the right.

Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of a number greater than or equal to 1, but less than 10, AND a power of ten are said to be written in scientific notation.

To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the umber of places we moved the decimal point.

4,592,000,000 in scientific notation

First move the decimal point to the left to get a number between 1 and 10

4,592,000,000 4.592

Since the decimal point was moved 9 places, we multiply 4.592 by 10⁹ to express the number in scientific notation

4.592 x 10⁹

When we move a decimal point to the left, we are actually dividing by a power of ten.

Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros

3.1 ÷ 10⁴ = 0003.1 = 0.00031

Rule

To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.

Math 6 Honors Periods 1, 6 & 7 ( Yosemite Week in review)

Comparing Decimals 3-4

We have used number lines to compare whole numbers. Number lines can be used to show comparisons of decimals. As with whole numbers, a larger number is graphed to the right of a smaller number.

In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.

Compare the following:

1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68.

2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 .

3. 0.83 and 0.833

To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833

0.83 = 0.830 Then compare

0.830 and 0.833 since 0 <3 Then 0.830 < 0.833.

Write in order from least to greatest

4.164, 4.16, 4.163, 4.1

First, express each number to the same number of decimal places Then compare. 4.164, 4.160, 4.163, 4.100

The order of the numbers from least to greatest is

4.1, 4.16, 4.163, 4.164

Rounding 3-5

A method for rounding may be stated as follows: Find the place to which you wish to round, mark it with an underline ___ Look at the digit to the right. If the digit to the right is 5 or greater, add 1 to the marked digit. If the digit to the right is less than 5, leave the marked digit unchanged. Replace each digit to the right of the marked place with a 0

Round 32,567 to (a) the nearest ten thousand, (b) the nearest thousand, (c) the nearest hundred, and (d) the nearest ten

(a)32, 567: since 2 is less than 5, we leave the 3 unchanged, and replace 2, 5, 6, and 7 with zeros

30,000

(b) 32,567: since the digit to the right of 2 is 5, we add a 1 to 2 and get 3 and we replace 5, 6, and 7 with zeros

33,000

(c)32,567: since 6 is greater than 5, we add 1 to 5 and replace 6 and 7 with zeros

32,600

(d) 32,567: since 7 is greater than 5, we add 1 to 6 and replace 7 with a zero

32,570

A similar method of rounding can be used with decimals. The difference between the two methods is that when rounding decimals, we do not have to replace the dropped digits with zeros.

Round 4.8637 to (a) the nearest thousandth, (b) the nearest hundredth, (c) the nearest tenth, and (d) the nearest unit

a. 4.8637: Since 7 is greater than 5, we add 1 to 3 --get 4 & drop the 7

4.864

b. 4.8637: Since 3 is less than 5, we leave 6 unchanged and drop 3 & 7

4.86

c. 4.8637: Since 6 is greater than 5, we add 1 to 8 and drop 6,3, &7

4.9

d. 4.8637: Since 8 is greater than 5, we add 1 to 4 and drop 8, 6, 3, & 7

5

Adding and Subtracting Decimals 3-6

Decimals may be added or subtracted using the same rules as whole numbers

Write the given numbers one above the other with the decimal points in line.

Annex zeros to get the same number of decimal places and then add or subtract as if the numbers were whole numbers.

Place a decimal point in the number for the sum or difference in position under the decimal points in the given numbers.

Add 6.47 + 340.8 + 73.523

STEP 1	STEP 2	STEP 3
0006.47	006.470	006.470
0340.8	340.800	340.800
+ 073.523	+ 73.52	+ 73.523
	420 793	420.793

The use of rounded numbers to get an approximate answer is called estimation. We use estimates to check actual answers. Use estimates as a habit to check if your answer is reasonable. To find an estimate, first round the highest place value of the smallest number, then compare.

Add 8.574 + 81.03 + 59.432. Then estimate to check your answer. What is the highest place value of the smallest number? 9 + 81 + 59 = 149