Graphing Solutions to Equations and Inequalities

Topics for Today:

Today we started a unit on graphing by discussing what a graph is, and also the difference between solutions of an equation and solutions of an inequality.  We also solved simple inequalities using addition and subtraction.

Our graphing focused on number lines.  We will be working with the coordinate system (all four quadrants) in the upcoming days.  

Vocabulary:

coordinate, graph, inequality, solution of an inequality

Sections Covered in Textbook:

2-8: Graphing and Writing Inequalities (pages 107-111)

Resources & Tutorials:

1) What is an inequality?

2) How do you write an inequality from a number line graph?

3) How do you solve an inequality using subtraction?

4) Graphing Solutions to Equations and Inequalities Class notes

Scale and Scale Drawings

Topics for Today:

Today we worked on scale drawings.  A scale drawing is an enlarged or reduced drawing of an object that is similar (in proportion to) the actual object.  Maps and floor plans are examples of scale drawings.  

image credit: https://windexchange.energy.gov/maps-data/128

Vocabulary:

scale, scale drawing

Sections Covered in Textbook:

5-7: Maps and Scale Drawings (pages 275-281)

Resources & Tutorials:

1)  What is a scale drawing?

2)  Scale Drawing and Scale Factor Class notes

Similar Figures

Topics for Today:

We used our understanding today in the context of Geometry.  We discussed similar polygons, and know now that similar polygons have the same shape but a different size.  Furthermore, similar polygons have sides that are in proportion to each other.  

Vocabulary:

polygon, similar polygons

Sections Covered in Textbook:

5-6: Using Similar Figures (pages 267-274)

Resources & Tutorials:

1) What are similar figures?

2) How do you find missing measurements of similar figures using a proportion?

3) Using Similar Figures Class notes

Percent of Change

Topics for Today:

We continued the topic of percents in the context of percent of change. We can have a positive percent of change (representing an increase) or a negative rate of change (representing a decrease). To determine the percent of change, we have to compare how much something changed to its original quantity. Percent of change is relative to the original value.

Vocabulary:

percent of change, percent of increase, percent of decrease

Sections Covered in Textbook:

6-8: Finding Percent of Change (pages 327-332)

Resources & Tutorials:

1) What is the percent of change?
2) How do you find percent of change?
3) How do you determine percent of increase or decrease?
4) Percent of Change Class notes

Proportions and Percent Equations

Topics for Today:

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.

Vocabulary:

percent, percent proportion

Sections Covered in Textbook:

6-5: Solving Percent Problems Using Proportions (pages 311-317)

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) Proportions and Percent Equations Class notes

Solving Proportions

Topics for Today:

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.

We also used the means-extremes (cross products) property to solve proportions.

Vocabulary:

proportion, cross products

Sections Covered in Textbook:

5-5: Using Proportional Reasoning (pages 260-265)

Resources & Tutorials:

1) What is a proportion?
2) How to solve a proportion by using cross products?
3) Solving Proportions Class notes

Solving Two-Step Equations

Topics for Today:

Our discussion about solving equations moved on to solving 2-step equations.  The order of operations still figures in to this process, although since we are undoing operations, we go in reverse.

Sections Covered in Textbook:

2-6: Solving Two-Step Equations (pages 98-101)

Resources & Tutorials:

1) How do you solve a 2-step equation?
2) Math Antics - How to solve 2-step equations.

3) Two-Step Equations Class Notes

Solving Equations by Multiplying or Dividing

Topics for Today:

We continued our discussion today about inverse operations are, and also further defined the term solution.  We will continue to follow the same processes to solve equations, even as they become more complicated, with more operations and steps.  Because we have to "undo" operations to isolate our variable (get it by itself), we must follow the order of operations, but IN REVERSE!

Vocabulary:

division property of equality, multiplication property of equality, solution

Sections Covered in Textbook:

2-4: Solving Equations by Multiplying or Dividing (pages 88-92)

Resources & Tutorials:

1) Math Antics - How to Solve a 1-step Equation.
2) How to solve an equation using multiplication.
3) How to solve an equation using division.
4) How to solve an equation when multiplying fractions.
5) Solving Equations by Multiplying or Dividing Class notes

Solving Equations Using Addition and Subtraction

Topics for Today:

We began our journey into solving algebraic equations today.  We also discussed what inverse operations are in the context of solving equations.  Students are reminded that all they are accountable for all the mathematics that came before this class, as well as any new learned material.  Mathematics is a cumulative subject, and the skills built in the past will continue to be used to solve new problems.  That means we will continue to integrate fraction and decimal operations into our problem solving.  Equation operations will include all types of real numbers.

Vocabulary:

addition property of equality, subtraction property of equality, inverse operations

Sections Covered in Textbook:

2-3: Solving Equations by Adding or Subtracting

Resources & Tutorials:

1) How to solve an equation using addition.
2) How to solve an equation using subtraction.
3) How to solve a decimal equation using subtraction.
4) Solving One-Step Equations class notes

Variables and Algebraic Expressions

Topics for Today:

Our lesson focused on some of the language of Algebra and how words can be translated into algebraic expressions.

Image Credit: https://www.pinterest.com/pin/651473902320727044/

Vocabulary:  variable, algebraic expression, evaluate, equation, open sentence, substitution property of equality

Sections Covered in Textbook:

2-1: Evaluating and Writing Algebraic Expressions (pages 71-76)

Resources & Tutorials:

1) What are numerical and algebraic expressions?
2) Translating mathematical expressions
3) Variables and Algebraic Expressions Class Notes

Discount, Markup, and Sales Tax

Topics for Today:

Percentages have many practical and every day uses.  Today we discussed how percents are used in retail situations.  Most items purchased (toys, food, clothing) are subject to sales taxes in most areas of the United States, and the tax is a percentage based upon the sale price of an item.  Merchants who sell items purchase these items for one price, and then sell them for a higher price called a markup.  Sometimes items go on sale and are sold for less than their original price, and we call this a discount.  We used our skills learned previously about how to find percent of a number to calculate discounts, markup, and sales tax.  

Vocabulary:

discount, markup, sales tax

Sections Covered in Textbook:

Not in Textbook; Lesson Pulled from Other Sources

Resources & Tutorials:

1) How do you figure out how much something is marked down?

2) How do you figure out sales tax?

3) Discount, Markup and Sales Tax Notes

Finding Percent of a Number

Topics for Today:

Today we worked on finding percent of a whole.  We can accomplish this many ways.  First, we must recognize that when we find a percent or fraction OF a number, that really means that we must multiply.  

Vocabulary:

Sections Covered in Textbook:

6-4: Finding Percent of a Number (pages 305-310)

Resources & Tutorials:

1) How do you use an equation to find percent of a whole?

2) Percents Mental Math

3) Finding Percent of a Number Class Notes

Percents: Large and Small

Topics for Today:

Today we reviewed what percents are - percents are ratios that compare a number to 100.  Percents can be large (bigger than 100%) and very small (less than 1%).  We handle these large and small percentages the exact same way we do other percents, and we can convert among fractions, decimals, and percents by using the strategies we've discussed.  

Vocabulary:

percent

Sections Covered in Textbook:

6-3: Percents Greater than 100 or Less Than 1 (pages 301-304)

Resources & Tutorials:

1) What is a percent?

2) How do you turn a percent into a decimal?

3) How do you turn a percent into a fraction?

4) Percents Large and Small class notes