Thursday Nov 1 Lesson
Presentation
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Mathematics
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10th Grade
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Practice Problem
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Hard
Catherine Hosack
FREE Resource
13 Slides • 33 Questions
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This builds on what we learned about conditional statements in the first couple days of this unit.
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Fill in the Blanks
First, a quick Warmup. Is this true or false?
A straw man fallacy is when someone oversimplifies an argument, and then attacks the oversimplied idea instead of the original idea.
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The questions are designed so that if you read the examples carefully, you will get most of the following questions correct. It's expected to get some wrong but again: this Quizizz was designed so that even if you are still learning, you will get them correct if you read carefully and work thoughtfully.
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Statement: If you wear long pants, then it is winter. (This is false, BTW.)
As we learned earlier in this unit, the "P" part of the sentence is after the IF (in this case: "you wear long pants"). The "Q" is after the THEN (in this case: "it is winter.")
The contrapositive of the statement is formatted like:
IF NOT (insert Q here), THEN NOT (insert P here)
You won't always use the word "NOT." Sometimes it's "never," "don't," "isn't," and other "negative" words.
So the contrapositive is: If it is not winter, then you don't wear long pants. (Also false.)
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Multiple Choice
Practice: Write the contrapositive of "If it is Saturday, then I watch movies."
If it is not Saturday, then I don't watch movies.
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Multiple Choice
Practice: Write the contrapositive of "If it is Monday, then tomorrow is Tuesday."
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Multiple Choice
Practice: Write the contrapositive of "If you are a carnivore, then you eat meat."
If you are not a carnivore, then you do not eat meat
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If a statement STARTS with "negative" statements, then its contrapositive DOESN'T have the negative statements.
Example: If you aren't green, then you aren't blue. (This is false BTW.)
The "P" is "you aren't green." The "Q" is "you aren't blue."
Contrapositive: If you ARE blue, then you ARE green. (Also false.)
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Multiple Choice
Select the contrapositive of: "If you aren't a rabbit, then you don't have ears."
If you aren't a mammal, then you aren't a rabbit.
If you are a rabbit, then you have ears.
If you don't have ears, then you aren't a rabbit.
If you have ears, then you are a rabbit.
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Multiple Choice
Select the contrapositive of "If I am not sleepy, then I don't sleep."
If I sleep, then I am sleepy.
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Multiple Choice
Select the contrapositive of "If I am not Jane, then I am not your sister."
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Two examples of conditional statements and their contrapositives:
If the timer runs out, then it dings. (True.)
If the timer doesn't ding, then the timer hasn't run out. (Also true.)
If the air doesn't smell like apples, then it's not Fall. (False.)
If it's Fall, then the air smells like apples. (Also false.)
See: each pair has the same truth value. (Either both true, or both false.)
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Multiple Choice
If a statement is true, then its contrapositive...
will be true
will be false
will be either
will be both
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Multiple Choice
If a statement is false, then its contrapositive...
will be true
will be false
will be either
will be both
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An advertisement says "if you don't hire our law firm, you won't win your case." If this statement appears on a well-designed, legit-looking advertisement, then it will be easy for people to subconsciously believe this statement (even if it's an arrogant thing to say.)
But if you test its contrapositive, you will notice that the contrapositive is clearly a false statement.
Contrapositive: "If you win your case, you hired our law firm."
This is clearly false, because lots of court cases are won every day by lots of different law firms, not just that one.
So by determining that the contrapositive is false, you know the original statement is also false.
(Reminder: you know this, because you know that a statement and its contrapositive share the same truth value.)
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Multiple Choice
An advertisement says "If you are not planning your retirement, then you must not care about your future."
Bill thinks this might be true, but the double negatives in the sentence are a bit confusing to think about, so he writes the contrapositive of the statement. What is the contrapositive?
If you are planning your retirement, then you are caring about your future.
If you care about your future, then you are not planning your retirement.
If you care about your future, then you are planning your retirement.
If you do not care about your future, then you are planning your retirement.
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Multiple Choice
What is one reason why "If you care about your future, then you are planning your retirement" could be a FALSE statement? Choose a valid counter example.
Because some people might care a lot about their future, but they get overwhelmed thinking about retirement plans so they just avoid it
Because retirement is an economics subject
Because the future doesn't exist
Because some people might care a lot about their future, but they are actually Martians
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Open Ended
What's one more reason you can think of, that someone might care a lot about their future but they don't think about retirement? There are lots of reasons so feel free to be creative as long as it's realistic.
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Multiple Choice
From the counterexamples on the previous two slides, we know that "if you care about your future, then you are planning your retirement" is a FALSE statement.
The original statement was "if you aren't planning your retirement, then you don't care about your future."
Since its counterpositive is false, you now know that the original statement must also be:
False
True
Could be either one
Both
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Multiple Choice
To summarize the previous few slides: If you see a confusing conditional statement (an if-then statement) and want to investigate whether it's true or not, you can write its contrapositive. If its contrapositive is true, then the original statement must be true, and vice versa. When might this be helpful?
Researching ads to buy a car
Determining whether a company is being truthful or not in their public statements
When organizations run campaigns to convince the public to vote a certain way
All three
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Open Ended
Do you think it's important to know different strategies for determining if something is true or not? Why?
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Multiple Choice
A TV show says "if you aren't watching only this show, then you aren't watching good TV." What is this statement's contrapositive, and is it true or false?
The contrapositive is "if you are watching good TV, then you are watching only this show." It is FALSE.
The contrapositive is "if you are watching good TV, then you are watching only this show." It is TRUE.
The contrapositive is "If you are watching this show, then you are watching good TV." It is FALSE.
The contrapositive is "If you are watching this show, then you are watching good TV." It is TRUE.
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Multiple Choice
Choose the counterexample that proves the contraposiive statement "if you are watching good TV, then you are watching only this show" to be false.
You could watch Breaking Bad instead, which is universally known to be an excellent TV show.
You could watch bad TV
You can turn the TV off
You could watch only that show forever
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Multiple Choice
Since the counterexample is FALSE, then the original statement ("if you aren't watching only this show, then you aren't watching good TV") must be...
False
True
Either one
Both
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When first learning a concept, the examples are written in a way that help people learn the concept.
So when conditional statements (if-then statements) are used in real life, they can be a lot more complicated. But learning with careful example is still helpful in developing your brain towards more complicated logical thinking.
So yeah, let's practice some more.
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Multiple Choice
"If you don't visit Virginia you won't have the time of your life!" What is its contrapositive, is it true or false, and why?
If you have the time of your life, you are visiting Virginia. False. Because you could have the time of your life somewhere else.
If you have the time of your life, you are visiting Virginia. True. Because Virginia is the only place where good things happen.
If you have the time of your life, you aren't in Virginia. False. Because there are other places.
If you have the time of your life, you aren't in Virginia. True. Because you can't have fun in Virginia.
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Multiple Choice
Since its contrapositive is false, is "If you don't visit Virginia you won't have the time of your life!" true or false?
False
True
Both
Neither
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Multiple Choice
What is the contrapositive of "if you don't take this vitamin, then you won't live to be 100"
If you live to be 100, then you take this vitamin
If you live to be 100, then you don't take this vitamin
If you don't live to be a 100, then you took the vitamin.
If you don't live to be 100, then you didn't take the vitamin.
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Open Ended
Do you think the correct contrapositive ("If you live to be 100, then you take this vitamin") is true or false? Why?
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Open Ended
Based on what you said on the previous Slide, do you think the original statement "If you don't take this vitamin, then you won't live to be 100" is true or false?
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Multiple Choice
What is the contrapositive of "If you buy my ice cream, you'll be happy?"
If you aren't happy, you didn't buy my ice cream
If you buy my ice cream, you won't be happy
If you aren't happy, you bought my ice cream
Ice cream is weird
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Multiple Choice
What is the contrapositive of "If you don't follow this advice, then you won't reach your dreams"
If you reach your dreams, you followed this advice
If you reach your dreams, you didn't follow this advice
If you follow this advice, you'll reach your dreams
Advice and dreams are related
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Multiple Choice
Select a counterexample that proves "If you are sleepy, you didn't get enough sleep" to be false.
Sometimes there are medical conditions that cause people to be sleepy all the time
Sleep is overrated
If you are sleepy, you need to sleep
Sometimes there are people who never need to sleep
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Multiple Choice
If "If A happens, then B happens" is true, then "If B doesn't happen, then A doesn't happen" is ...
True
False
Both
Neither
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Open Ended
In this free response box:
1 - Write a *TRUE* IF-THEN statement (make up your own.)
2 - Write its contrapositive.
3 - Answer this question: is the contrapositive statement true or false?
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Open Ended
In this free response box:
1 - Write a FALSE IF-THEN statement (make up your own.)
2 - Write its contrapositive.
3 - Provide a counterexample to prove that the statements are false.
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Multiple Choice
Write this statement as an IF-THEN statement.
I will make my bed when I get home.
If I get home, then I will make my bed.
If I make my bed, I will get home.
I will get home eventually.
I won't make my bed when I don't get home.
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Multiple Choice
Write this statement as an IF-THEN statement.
I'll wash the car if the weather is nice.
If the weather is nice, then I'll wash the car.
I'll wash the car even if the weather isn't nice.
If the weather isn't nice, then I won't wash the car.
Some people say that public transportation is better than private ownership of cars.
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Multiple Choice
Write this statement as an if-then statement.
I am going to feed my dog because I care about him.
If I care about my dog, then I will feed him.
If dogs eat, they will eat dog food.
If my dog is home, then I will feed him.
I'm going to pay someone to feed my dog.
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Open Ended
Make up your own statement and then write it as an IF-THEN statement. Use the previous few slides to help you get ideas of how to write your original statement.
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REMINDER: if you are absent on any day, you are responsible for looking in Classroom, reading the Agenda, and doing that day's work. The test for this unit will be on Tuesday no matter what so you need to do your part to catch up if you are absent. (If you have extended sick days then talk to me about those.)
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If you have time before the bell, then you should now practice the Quizizz from yesterday to get better at it.
It is linked on the next Slide.
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https://quizizz.com/join?gc=71372519
This builds on what we learned about conditional statements in the first couple days of this unit.
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