Unit 3 Consolidation Day

Main Objective:

The main goal for today is to consolidate our understanding of ratios, proportions, and percentages in order to be prepared for our unit test on Wednesday.

Part 1: Do Now

Try to answer each of the questions below in your squarebook.

When you are finished, try to think about how each of the two questions are connected to each other in terms of how you work with percentages.

Part 2:  Preparing for your assessment

One of the main problems that students encounter when preparing for a math assessment is that they engage in only PASSIVE studying when what they need to be doing is ACTIVE studying.  Below is a list of things that would be considered PASSIVE and ACTIVE studying techniques.

Passive Studying Techniques (20% of your time)

• Reading your notes
• Reading blog posts
• Reading your textbook
• Watching YouTube videos

Active Studying Techniques (80% of your time)

• Doing the Review Package handed out in class
• Working through the MathXL For School Study Plan
• Redoing the questions from your Mid-Unit Assessment
• Working through ACE questions

What you MUST do before next class:

• Complete your preparation for the Unit Test on Wednesday
• Here are the answers to the Review Package.

Working With Percentages

Focus Question:

How can we use proportions and the constant of proportionality to solve problems involving percentages?

Part 1:  What is a Percentage?

So...remember that a percentage is just another type of special ratio.  The key characteristics of a percentage are:

• It is a part to whole ratio.
• The "whole" is always equal to 100.

Part 2: How to Solve Percentage Problems?

There are a couple of different ways to solve problems involving percentages.  We can set-up a pair of part to whole ratios, we can use a table to organize our thinking or we can construct equations in the form y=kx where "k" represents the percentage written in the form of a decimal.

For the problem below, try to set-up a table to organize your thinking.  Be really thoughtful about how each value is connected to the other values. (This is where the percentage work comes in).  In case you are not familiar with the language, here are some definitions to help you out:

• Purchase Price - the amount that Carla paid to get the car.
• Markup - the amount that Carla has increased the price of the car by in order to make a profit.  In this case, the markup is equal to 10% of the purchase price.
• Selling Price - the amount that Carla is going to sell the car for.  This is usually equal to the purchase price + markup.

Now, try to write an equation for each of the following relationships.  Remember that "k" is the percentage as a decimal.

• The markup based on the purchase price.
• The selling price based on the purchase price.
• The selling price based on the markup.

What you MUST do before next class:

• Finish TOFU for your Mid-Unit Assessment
• ACE (1-6) (11 OR 12) p. 71

More Constant of Proportionality

Objective:

The main focus for today was to consolidate our learning from the last couple of classes in terms of how to find the constant of proportionality from various different representations.

In case you missed class, here are the notes that we took during the first part of class:


For the final hour of class, we worked in our groups through a booklet that combined a lot of different representations of the constant of proportionality (k).  Here are the solutions to the problems in the booklet.

Here is what you MUST do before next class:

• Complete the work in the booklet
• We have a quiz next class that focuses on unit rates, best buys, and the constant of proportionality

The Constant of Proportionality

Focus Questions:

• What is the relationship between the constant of proportionality and a unit rate?
• How do we determine the constant of proportionality from a table, equation, or graph?
• How do we represent proportional relationships using an equation?

Do Now:

Which pizzeria is the best buy?  How do you know?

Who is the fastest runner?  How do you know?

Part 1:  Connecting Unit Rates and the Constant of Proportionality

The constant of proportionality is another name used to describe the constant ratio that exists between two different quantities.  In all cases, this is the same as the ratio expressed in lowest terms. In most cases, this is the same as the "unit rate".

Look back on your work from last class, "The Four Pizzerias" and determine the equation to find the Total Cost at each pizzeria.

Notice that each equation has a constant multiplier that we had previously identified as the unit rate.  This is also known as the "constant of proportionality"

Part 2:  Determining and Using the Constant of Proportionality

As a group, work through the "Constant of Proportionality" assignment that you will be given in class.  If you get stuck, look through your notes or ask for help.

What you MUST do for next class:

• Complete the "Constant of Proportionality" assignment handed out in class.
• Complete the "Constant of Proportionality" assignment on Math XL For School

Comparing Unit Rates in Different Forms

Objective:

I can compare unit rates using tables, graphs, words, and equations.

I can create rate tables, line graphs, and equations to represent proportional relationships.

Part 1:  Comparing Unit Rates

In this part of the lesson, you will compare the unit rates of four different pizzerias.  The catch is that each pizzeria has represented their pricing structure in a different way.

Part 2: Representing Unit Rates

Now that you know how to find the unit rate in each of the four different representations, your goal is to convert each pizzeria's pricing structure into the different representations.  To do this, you can follow the instructions on this handout.

What you MUST do before next class:

• Relax and have a great Thanksgiving weekend.

2.1 and 2.2 Scaling and Using Unit Rates

Lesson Focus: The main goal for this lesson is to look at how to create and use unit rates as a strategy for comparing ratios.

Part 1: Comparing Rates

As a follow-up to our work last class, try to use one of the "comparing ratios" strategies from last class.  If you need a reminder, here is the list that we generated:

• Create a common numerator
• Create a common denominator
• Convert into percentages

Part 2: Creating and Comparing Unit Rates

One way to compare two ratios (or rates) is to make a unit rate by scaling each ratio to find the cost of one pizza.  If you want some additional practice with finding unit rates, you can watch the video below.


Now try to find the unit rate for each of the pizza specials shown below:

When you are ready, try to determine which jar of peanut butter is the "better buy".  Remember to give a reason for your final decision.

Here is what you MUST do before next class:

• ACE (11) p. 54
• "Ratios, Rates, and Unit Rates" on Math XL For School.

1.2 to 1.4 Comparing and Scaling Ratios

Lesson Objectives:

• Students will be able to create and use a table to scale ratios up and down.
• Students will be able to use a variety of strategies to compare ratios.

Part 1: The Language of Ratios

Remember that ratios are about comparing two quantities using the language of "There are ____ _____ for every ____ _____".

Look at the image below and see how many ratio descriptions you can come up with:

Part 2: Scaling Ratios

Take a look at your ratio statements.  Do you have any statements that are really describing the same relationship, but using different numbers?  For example:

• There are 2 pigs for every 1 sheep.
• There are 4 pigs for every 2 sheep.

These two statements are really the same but the second set of numbers has been "scaled" up by a factor of 2.  So what is "scaling"?

Scaling is the process of multiplying or dividing one ratio by a particular "scale factor" to create an equivalent ratio.

Work through the task below.  Be sure to clearly show your "scale factor".

Part 3: Comparing Ratios

While scaling produces equivalent ratios, other situations require that we compare ratios.  Complete the task below.  If you get stuck, try some of the following strategies:

• Rewrite each ratio so that the "cups of concentrate" is the same.
• Rewrite each ratio so that the "cups of water" is the same.
• Convert each ratio into a part:whole fraction.
• Convert each ratio into a percentage.

Here is what you MUST do before next class:

• ACE (10, 17, 18, 22, 23)
• Remember to set these up as a table when possible to make your work more organized.