Proportions and Percent Equations

Today We Discussed:

We expanded our discussion of proportions to include the percent proportion.  We deconstructed the word per-cent to mean "out of 100".  We can solve percent problems using a proportion or using a percent equation with the percent expressed as a decimal.

This unit will continue to explore proportions and percents, and we'll take some time to review the conversions of fractions to decimals to percents and vice-versa.  Students will be encouraged to memorize the decimal equivalents of common fractions as a time-saver.  Normally I am not a big fan of memorization, unless it serves a useful purpose - memorizing common concepts (like divisibility rules, the quadratic formula, and common numbers) can be a big time-saver leaving more time for higher order problem solving.

As we continue to explore the relationships among decimals, fractions, and percents, I plan to expand our discussion to other proportions, like circles.  We will investigate some probability in this unit as well.

Vocabulary: percent proportion, percent equation

Sections Covered in Textbook:

4-3: Proportions and Percent Equations (pages 197-202)

Resources & Tutorials:

1) What is a percent proportion?
2) How do you use a proportion to find a whole?
3) How do you use a proportion to find what percent a part is of a whole?
4) How do you use a proportion to find part of a whole?
5) What is a percent equation?
6) More Percent Equation Links

Ratios and Proportions

Today We Discussed:

We began our unit on solving and applying proportions today, and introduced/reviewed some important vocabulary, beginning with ratios.  A ratio is just a comparison of numbers by division.  Students have seen ratios ever since they began working with fractions.  When we talk about rates, we create a ratio of two numbers that have different units.  We have already seen rates this year, when dealing with uniform motion -  rate of speed (comparing a distance with how much time elapses). 

We also used conversion factors to convert rates.  A conversion factor is a rate that is equal to 1 (multiplicative identity states we can multiply by 1 and not change the identity of our number).   For example, a unit conversion would be 60 seconds per minute since 1 minute=60 seconds.

Finally, we used the means-extremes (cross products) property to solve proportions. 

Vocabulary: ratio, rate, unit rate, conversion factor, unit analysis, dimensional analysis, proportion, cross products

Sections Covered in Textbook:

4-1: Ratio and Proportion (pages 182-187)

Resources & Tutorials:

1) What is a ratio?
2) What are rates and unit rates?
3) What is dimensional or unit analysis?
4) What is a proportion?
5) How to solve a proportion by using cross products?

6) Ratio and Proportion Notes

Solving Absolute Value Inequalities

Today We Discussed:

We continued our discussion about absolute value but moved on to inequalities.  Just like absolute value equations, we must consider TWO cases for absolute value inequalities - the positive case and the negative case.  Furthermore, we have to analyze which direction our solutions go based upon whether we are dealing with a greater than absolute value inequality or a less than absolute value inequality.

• Greater than absolute value inequalities function like OR compound inequalities.
• Less than absolute value inequalities function like AND compound inequalities.  

Graphic Credit: http://andymath.com/solving-absolute-value-inequalities/

Sections Covered in Textbook:

3-6: Absolute Value Equations and Inequalities (pages 167-171)

Resources & Tutorials:

1) How do you figure out if you have an AND or OR compound inequality?
2) How to solve an AND absolute value equation.
3) Introduction to Absolute Value Inequalities.
   (Use navigation on the left for more types of examples.)

Solving Absolute Value Equations

Today We Discussed:

We moved into the next section in our book and discussed absolute value equations.  We will tackle absolute value inequalities on Thursday and Monday.

First, we reviewed absolute value and what it means - a number's positive distance from zero.  Absolute value equations add a small level of complexity because when we take the absolute value of a quantity, it will always be positive.   We can have an expression inside the absolute value bars be either positive OR negative, so we can end up with two solutions for the variable in these cases.

We must also analyze whether or not our absolute value equation makes sense.  In most cases, we will get two solutions, but there will be times when no solutions will be possible.  We need to make sure our equation is logical. 

Take for example the equation |x -2| = -3.   

There will never be a case when we take the absolute value of an expression that will result in a solution that is less than 0.  By its very definition, absolute value is always positive.  

For each of these absolute value equations, we will need to consider TWO cases for each solution set:  the positive case and the negative case. We will need to solve TWO equations to get the complete solution for the variable.

Sections Covered in Textbook:

3-6: Absolute Value Equations and Inequalities (pages 167-171)
(We will only cover equations today!)

Resources & Tutorials:

1) Four steps to solve absolute value equations. 
2) Introduction to absolute value equations.
3) Chili Math - Solving Absolute Value Equations (Not a video)

Solving Compound Inequalities

Topics for Today:

Our discussion about inequalities has moved on to compound inequalities.  We discussed the word "compound" and related it to compound words and compound sentences.  There are two types of compound inequalities:

• inequalities using OR 
• inequalities using AND. 

For the OR types, only part of the inequality needs to be true for the entire compound statement to be true.  For the AND types, we must have both parts true at the same time.  OR inequalities can be related to the UNION of two sets, and AND types represent the INTERSECTION (where both criteria are true at the same time).   Venn Diagrams (circle diagrams) are often used as pictorial representations of our sets.  

Graphic Credit:  

https://rstudio-pubs-static.s3.amazonaws.com/282515_ddb4251ce953466fb8055bb4fd4837b9.html

Vocabulary: compound inequality

Sections Covered in Textbook:

3-5: Compound Inequalities (pages 161-165)

Resources & Tutorials:

1) What is a compound inequality?
2) How do you solve an OR compound inequality and graph it?
3) How do you solve an AND compound inequality and graph it?
4) How do you solve an AND compound inequality rewriting it as two?
5) What is a Venn Diagram?
6) Schoolhouse Rock - Conjunctions

Solving Multi-Step Inequalities

Topics for Today:

We continue to build on our problem-solving skills with solving inequalities.  Today we moved on to more complicated inequalities that involve several steps.  Again, we approach these problems just like solving equations, with the first step being to identify the variable.  Once we identify the variable, we need to plan for how we "undo" operations performed on the variable with the goal of getting the variable by itself.  To accomplish this goal, we perform the order of operations (PEMDAS) in reverse.  *Students must always keep in mind that when multiplying or dividing an inequality by a negative number, they must reverse (flip) the inequality sign to keep the truth of the inequality.*

Sections Covered in Textbook:

Solving Multi-Step Inequalities (pages 153-159)

Resources & Tutorials:

1) How do you solve a multi-step inequality?
2) How do you solve an inequality with variables on both sides?
3) How to solve multi-step inequalities.

Solving Inequalities Using Multiplication and Division

Topics for Today:

Our discussion about solving inequalities moved to solving by using the multiplication and division properties of inequality.  We solve inequalities using the same steps and procedures as solving equations, but there is one notable exception.  For cases when we either multiply or divide both sides of our inequality by a negative number, we must switch the inequality sign to preserve the truth of the inequality.  To illustrate why this works, we did a little exploration with simple inequalities in class to help understand why the "truth" of an inequality changes.

Vocabulary: multiplication property of inequality, division property of inequality

We considered the following examples in class:

Sections Covered in Textbook:

3-3: Solving Inequalities Using Multiplication & Division
       (pages 146-151)

Resources & Tutorials:

1) What is the division property of inequality?
2) What is the multiplication property of inequality?
3)  Solving inequalities using multiplication and division
4) Virtual Nerd Page with more tutorials.