3.1 Designing Another Spinner

Focus Question:
How can we use a probability simulation to make predictions?

Part 1: Looking At Spinners

Part 2: Analyzing a Spinner

Kalvin decides to use the spinner shown below:

1. What is the theoretical probability that the spinner will land on each time?
2. Based on the theoretical probabilities, if Kalvin spins the spinner 64 times, about how many times can he expect to go to bed at each time? Explain. Why do you think I chose 64 times?

Part 3: Testing the Spinner

1. Go to the following website: Spinner Site (you will probably need to use Firefox or Safari)
2. Click on "Change Spinner"
3. Design your spinner.
4. What was the experimental probability that the spinner landed on each time?
5. How many times did you "spin" the spinner? How did you decide that you had enough trials?
6. How do your experimental results compare to your theoretical probabilities?

Part 4: Designing a New Spinner

Pretend that you are Kalvin's father.  Design a new spinner where it is "most likely" that Kalvin will go to bed at 9:00. Your spinner must have at least three different times and four different sections.

1. Test your spinner to be sure that it meets the criteria.
2. Take a screen shot and post it on your blog along with the theoretical probability of Kalvin going to bed at each time.

Scratching Spots:

• How might we design a simulation to test the probability of winning this contest?
• Try to use one of the probability tools to design a simulation for this situation. Be prepared to share your idea with others.

1. Based on your simulation, what is the experimental probability of winning this game?
2. Build a sample space to try and determine the theoretical probability of winning this game?
3. Is it worth it to spend $1000 at Tawanda's Toys to try and win video games?  Explain.

Homework:

• Design a Spinner Challenge Task
• ACE (13,14)
  
  
• Complete these Khan Academy activities:
  
  Probability Space
  Probability 1

Analyzing Compound Events

Focus Questions:

• How can you decide if a game is fair or not?
• How can we determine the theoretical probability of a compound event?

New Vocabulary:

• Fair Game - a game where the winning and losing are "equally likely"
• Sample Space - a list of all of the possible outcomes for an event.
• Tree Diagram - One type of "sample space" that is useful for events that have more than two parts.
• Compound Event - An event that has two or more parts or actions

Part 1: The Coin Game

Here are the rules of this game:

• You will toss three coins.
• If all three of the coins match, you get a point (and some money).
• If all three of the coins do NOT match, I get a point.
• Everybody gets one turn.
• At the end, the team (students or teacher) with the most points wins.

Part 2: Analzying the Game

1. What is the "experimental probability" of having all three coins match?
2. How many outcomes are there in this game?
3. Draw a sample space to determine all of the outcomes. (which type of sample space should you draw?)
4. Is this a "fair game"? Explain why or why not.

Part 3: The Bonus Prize

You are a contestant on Gee Whiz Everybody Wins and have made it to the final "Bonus Round".  In the Bonus Round you get to choose one of two games to play.

Game #1

• How to Play - Choose a block from Bag #1 THEN Choose from Bag #2
• How to Win - If you correctly predict the color of block that you will choose from each bag, you win.

Game #2

• How to Play - Choose a block from Bag #1 AND Bag #2 at the same time.
• How to Win - If you correctly predict the color combination that you will choose, you win.

Which game should you play?  Justify your decision with experimental and theoretical probabilities.

Part 4: Playing The Game

Task A:

• Run each experiment 18 times.  Record the results in your table.  Please remember to treat the bags kindly as they need to be used in multiple classes.
• After you have completed your experiment, fill out the forms below to contribute to class data.
• Game #1
• Game #2
• Here are the results from our class data: 

Part 5: Analyzing the Experiment

Answer the following questions in your squarebook:

1. What is the "experimental probability" of choosing each outcome for Game #1? Game #2?
2. What is the "theoretical probability" of choosing each outcome for Game #1? Game #2?
3. Are all of the outcomes "equally likely" for Game #1? Game #2?
4. How do the "experimental probabilities" and the"theoretical probabilities" compare for Game #1? Game #2?
5. Which Game should you choose to play? What should you predict if you play this game?

By the end of this lesson, you should be able to do the following:

• Determine whether a game is "fair".
• Describe how a "compound event" is different from a "simple event"
• Determine the "sample space" for a compound event using a probability tool such as a tree diagram or table

Homework:

1. ACE (9,11,12) (26) (*38)
2. Mid-Unit Summative on Investigations 1 and 2 next class.

Choosing Marbles

Focus Question: 

How do we determine the theoretical probability of some slightly more complex situations.

New Vocabulary:

• AND - All conditions described must be met.
• OR - Only one of the conditions described need to be met.
• NOT - The conditions listed must NOT occur.

Part 1: More Kalvin

For this part of the lesson you are going to do a quick formative assessment to see how well you understand the main focus questions from last week's lessons on probability.

You will do the assessment independently and can use calculators if you think that you need them.

Part 2: Simple Events

In this part of the lesson you will watch a video describing some of the basics around finding the theoretical probability of an outcome for a given event.  We have already talked about this a little bit, so this should be mostly a reinforcement of prior knowledge.

When you have finished watching the video, make sure that you complete the "On Your Own" questions.  Show your teacher when you have finished the questions.

Part 3: Choosing Marbles - Determining Theoretical Probabilities

The main part of this lesson is for you to now practice applying the concept of theoretical probability.  This is also in your textbook on page 30.

By the end of this problem you should be able to do the following:

• Determine the theoretical probability for a series of outcomes that contain the "OR", "AND", and "NOT" conditions.
• Develop a probability situation based on theoretical probabilities.

Homework:

1. ACE (p. 36) (6-7) (18-25)

Choosing Cereal

Focus Questions: 

• How does collecting more data helps you to predict the outcome of a probability situation?
• How can we use the results of a probability experiment to determine the "relative frequency" of an outcome?
• How does running a probability experiment help you determine how many possible outcomes there are as well as the likelihood of each outcome?

Kalvin would prefer to eat Cocoa Blast cereal for breakfast everyday.  His parents would prefer that he eats Health Nut Flakes.  In order to decide, his mother allows Kalvin to design a probability experiment.

Option 1: Flipping Coins

Kalvin will flip a coin.  If the results are heads, he will eat Cocoa Blast.  If the results are tails, he will eat Health Nut Flakes.

1. To run this experiment we will use our integer counters.  With your partner, designate one color to be "Heads" and the other color to be "Tails".  Make a note of your legend at the top of the table.
2. After you have completed your experiment, fill out this online form to contribute your data to our class data.
3. Here are the class results:
   
   
   RED = HEADS and BLUE = TAILS

Option 2: Tossing Cups

Kalvin will toss a cup.  If the cup lands on either end, he will eat Cocoa Blast.  If the cup lands on its side, he will eat Health Nut Flakes.

• Run your experiment 20 times.  Record the results in your table.  Please remember to treat the cups kindly as they need to be used in multiple classes.
• After you have completed your experiment, fill out this online form to contribute to the class data.  Here are the results of the class data:

Option 3: Matching Coins

Kalvin will toss two coins.  If the coins match, he will eat Cocoa Blast.  If the coins do NOT match he will eat Health Nut Flakes.

1. Run your experiment 30 times.  Record the results in your table.  Remember to use the "shake and drop" strategy.  
2. After you have completed your experiment, fill out this online form to contribute to the class data.  Here are the results of the class data:

At the end of this Problem you should be able to do the following:

• Describe how the accuracy of a probability experiment is influenced by the amount of data collected.
• Describe the difference between the probability of each outcome for an event and the relative frequency of each outcome for a probability experiment.
• Use the results of a probability experiment in order to predict the likelihood of each outcome.
• Extend the results of a probability experiment in order to predict the results of a similar probability experiment.

Here is what you MUST do before next class:

• Complete all of the questions on the handout.
• ACE (3, 6, 8, 10)
• Khan Academy: Simple Probability and Experimental Probability

Introduction to Probability - Block G

Our next unit is my personal favorite - Probability.  This is a unit where we get to explore "how likely" is something to happen.  This is a great skill to have in anybody's decision making skill set.

First of all, let's start by playing a game.  The rules are very simple:

• Predict what you will pick out of the bag.
• If you are correct, you win
• Questions???

Next, let's get a sense of what you will be doing in this unit.  Take a look at the "I Can..." statements.  On the scale above each statement, place a mark to indicate how well you think you can do each of the unit objectives.  Remember that this is the beginning of the unit so I am not expecting you to be an expert.

Now...let's get some basic probability language down.  Take a look at this video and complete the note-taking sheet that you have been provided with.

Here are some key words that you should be able to use next class:

• Event
• Outcome
• Certain
• Impossible
• Likely
• Unlikely
• Theoretical Probability
• Experimental Probability

What does it mean if outcomes are "Equally Likely"?

If two or more outcomes are "equally likely", it means that they have an equal chance of occurring.  Think about the game we just played.  Each block had an equally likely chance of being picked, but that didn't necessarily mean that each color had an "equally likely" chance of being picked.

Take a look at the spinners below and see if you can decide which one has "equally likely" outcomes.  Be prepared to justify your answer:

 

Now let's look at a variety of other events.  For each event, list all of the possible outcomes (if this hasn't already been done for you) and then decide whether the outcomes are all "equally likely" to occur.

As you can see, this can get a little bit tricky.

Now let's Practice

I have included a link to a Khan Academy activity that addresses the same things that are discussed in the video above.  You need to complete this before next class: Understanding Probability

Before next class you MUST have completed the following:

• Self assess your current level of understanding of each "I can..." statement
• Watch the "What is Probability" video and complete the note-taker page.
• Pass the first level of the "Understanding Probability" exercise on Khan Academy
• ACE (11-18) p. 19
• ACE (26-29) p. 23

Introduction to Probability - Block C and E

Our next unit is my personal favorite - Probability.  This is a unit where we get to explore "how likely" is something to happen.  This is a great skill to have in anybody's decision making skill set.

First of all, let's get a sense of what you will be doing in this unit.  Take a look at the "I Can..." statements.  On the scale above each statement, place a mark to indicate how well you think you can do each of the unit objectives.  Remember that this is the beginning of the unit so I am not expecting you to be an expert.

Now...let's get some basic probability language down.  Take a look at this video and complete the note-taking sheet that you have been provided with.

Here are some key words that you should be able to use next class:

• Event
• Outcome
• Certain
• Impossible
• Likely
• Unlikely
• Theoretical Probability
• Experimental Probability

Now let's Practice

I have included a link to a Khan Academy activity that addresses the same things that are discussed in the video above.  You need to complete this before next class: Understanding Probability

Before next class you MUST have completed the following:

• Self assess your current level of understanding of each "I can..." statement
• Watch the "What is Probability" video and complete the note-taker page.
• Pass the first level of the "Understanding Probability" video on Khan Academy

3.1 Introduction to Finding Solutions

Here is what you MUST do before next class:

• Complete ACE (1, 3, 4) p. 69
• Watch the following YouTube Video:

Here is what you CAN do before next class:

• Watch the following YouTube video that reviews equation solving skills from 6th grade:

Mid-Unit Assessment Day

The main goal for today was to give everybody a chance to demonstrate their learning so far this unit on our mid-unit assessment.

Part 1:  Do Now

• Use DESMOS to make a graph that shows the price plan for each of the three different DJ's.  Think carefully about how to set the values for the x and y axes.
• Write a question that you could answer by looking at the graph.

Part 2: Mid-Unit Assessment

As you do the assessment remember to think about the connection between tables, graphs, and equations as well as how to determine the rate of change and the starting value using each representation.

Part 3: Swipe, Split, Flip

This is a fun logic game that we will play today.  If you need a review of the objective and the rules, watch the slideshow below:

What you MUST do before next class:

• Complete all Swipe, Split, Flip challenges.  Remember to do the following:
• Draw a picture of your "solution".
• Briefly record your steps.
• Glue the pages in your workbook.

2.3 More Comparing Relationships

Focus question:

How can we decide if a table, graph, or equation represents a linear relationship?  What does it mean when two linear relationships intersect?

Part 1: Do Now

1. Have your two screenshots from last class ready on your computer.  We will have one person share their "triangle graph" and one person share their "non-triangle graph".  Be prepared to share your thinking.
2. Complete the "Linear Relationships Matching Activity".  For each story you need to do the following:

• As you read the story try to decide what the "rate of change" and "starting position" are.
• Determine which table goes with the story.
• Determine which graph goes with the story.
• Write an equation in y = mx + b format using the rate of change and starting value for the story.

Part 2:  Vocabulary

This lesson introduces some even more formal math vocabulary that mathematicians use when describing linear relationships.  These words are:

• y-intercept
• coefficient
• solution

Read through your textbook with your table group from page 32-33 and complete the note table.  Glue this into your notebook.

Part 3: Comparing Costs

In this part of the lesson you are going to continue the work from last class where we are going to look at the pricing plans for three different T-shirt companies and try to decide when each company represents the better value.  Remember that when looking at a linear relationship the two most important things to identify are the STARTING VALUE and the RATE OF CHANGE.

Here is what you MUST do before next class:

• Complete 2.3 More Comparing Relationships Notes (this should have been completed in class)
• Complete 2.3 Comparing Costs (this also should have been completed in class)
• ACE (4,5,6,7)....remember that Desmos is your friend when it comes to analyzing relationships.

2.1 and 2.2 Comparing Relationships

Focus Question: 
How can we use tables, graphs, and equations to compare linear relationships?

Part 1:
In this lesson, we are going to focus on how to use different representations in order to determine the distance were two brothers meet in a race.

Here is the set-up:

Henri challenges his older brother Emile to a race.  Because Emile’s walking rate is faster, Emile gives Henri a 45-meter head start.  Emile wants the race to end in a tie.

What distance should the race be so that the race ends in a tie?

In order to better understand how we can use the various representations to help us tackle this problem, work through the attached activity.

Here is what you MUST do before next class:

• Finish parts 2 & 3 of the attached activity (Block E only)
• ACE (2,3) p. 38
• Desmos HW: 
• Discover three equations whose graphs will intersect to form a triangle.  Take a screenshot of your graph and equations.
• Discover three equations whose graphs will NOT intersect to form a triangle.  Take a screenshot of your graph and equations.

1.2 and 1.3 Looking at Linear Relationships

Here is what you MUST do before next class:

• Finish the handouts given in class.  There are four of them and they should all be glued into your notebook.
• ACE (6,24)

1.1 Walking Marathons

Focus Question:

How can we write an equation to represent our own constant walking rate?

Part 1:

In this part of the lesson, we will conduct a mini-experiment in order to determine your average walking rate over a distance of 10 meters.  You will then use this walking rate to determine how far you can walk in a given amount of time.  Here is what you will need to do:

• Designate one person in your group to be the timer.  This person will be responsible for starting and reading out times.
• After each of your three trials, you will need to record your time in the table provided.
• After you have completed all three trials, you will need to find the average, or mean, of your three trials.  You can round your final answer to the nearest half-second.

Part 2:

In this part of the lesson, you will determine your walking unit rate, and then use that rate to complete a rate table, make an equation, and draw a line graph as multiple ways of representing your walking rate.

Here is what you MUST do before next class:

• Finish the "1.1 Walking Rates" handout.  This should be glued into your squarebook.
• Complete ACE (1,2) (16,18) starting on page 16.

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.

Accentuate the Negative Assessment Day

The main goal for today is for you to show all of the "positive" growth that you have made during this unit.

After you have completed your assessment, here are a couple of things that you need to do:

• Complete the "Comparing and Scaling Readiness Test" that is on MathXL for school.
• Complete the "Learning Behaviors Reflection" for this unit: