Algebra: Chapter 10-5 and 10-6, p 445 and 451
Addition and Subtraction: Unlike Denominators (Chapter 10-5)
This lesson is TOUGH and takes time to do right. We have to bring all of our tools to the problem. Know how to FOIL, recognize binomial squares, and use the BOX method of FACTORING ACCURATELY.
Unlike Lesson 4, we have differing denominator terms and we must find a common denominator of both rational expressions.
Given that p, q, r and s are rational numbers or expressions, the easiest way for addition and subtraction is with this equation: (p÷q) + (r÷s) converts to become (ps + rq) ÷ qs for addition or (ps – rq) ÷ qs for subtraction.
Sometimes, we need to find the LCM (least common multiple of the denominators. These steps will help:
Factor each expression
For the product using each factor the greatest number of times it occurs.
Just go slow and be methodical. SHOW ALL THE WORK and you’ll be fine, try shortcuts and you’ll regret it.
Purplemath has a good explanation and plenty of examples here too. Just make sure to scroll down to the middle of the page and the next page as well.
Solving Rational Expressions (Chapter 10-6)
We are now solving rational equations, they have an equal sign. With rational expressions on both side, we can sometimes structure these as ratios or proportions. We can solve proportions by criss-crossing, if a ÷ b = c ÷ d, then ad = bc. We just plug in the numerators and denominators where apprpriate and work it out.
Remember too, that a quadratic equation has 0, 1 or at most 2 roots or answers. Sometimes, one of the solutions can be considered extraneous or invalid. Usually, it results in the denominator of an expression being = 0. In Algebra I, we don’t know how to handle that but eventually in higher math, you will
Math-8
Chapter 10-1, p 486
Stem and Leaf Plots
Stem and leaf plots are a handy and quick way to show data trends. In a stem and leaf plot, the greatest place value common to all the data values is usually used for the STEMS. The next greates place value forms the LEAVES. First, you find the stems (the trunk) and then the leaves. A back to back stem and leaf plot is used to compare 2 sets of data. In this type of stem and leaf plot, the leaves for one set of data are on one side of the stem and the leaves for the other set of data are on the other side of the stem. Two keys to the data are needed. Here are some examples!
Chapter 10-2, p 490
Measures of Variation
Definitions:
RANGE: The range of a set of numbers is the difference between the least and the greatest number in the set.
MEDIAN: The middle number in a given data set. IF there are 2 numbers (that is the number sequence has an even number of numbers), then the median is the average of the 2 middle numbers. Add the 2 numbers together and divide by 2!
QUARTILE: The division of a data set into 4 equal parts. There are 2 quartiles we will worry about, the lower quartile (LQ) and the upper quartile (UQ)
INTERQUARTILE RANGE: The interquartile range (IQ) is the range of the middle half of a set of numbers. In equation form, the IQ =UQ – LQ
Chapter 10-3, p 495
Displaying Data – Box and Whisker Plots
A box and whisker plot summarizes data using the median, the upper and lower quartiles and the highest and lowest extreme values. See this link to do a few examples. It’s easier to see than to explain in words.
Data that are more than 1.5 times the interquartile range from the quartiles are called outliers. Essentially, these are points that are off the charts and don’t belong on the box and whisker plot.