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1) After a quick read-through, it seems like this paragraph is describing meals in some traditional sub-Saharan Africa households. Now I skim the paragraph for clues. There are a few things in this paragraph I underline right away. In sentences Q, R, S and U they mention “the starch”. What is the starch? They don’t mention it in the first sentence. They must explain what it is somewhere! They introduce the starch in sentence T. That immediately tell us that T must come before Q, R, S and U, so T is the second sentence. it seems to make sense after the first sentence- it gives a general description of typical meals.

I underline “Diners then use” in sentence S because it tells us that something happens right before sentence S. Let’s find what it is. U talks about taking starch from a communal plate. S can come right after U. First the people take some starch, then they use it as a scoop. I draw an arrow from U to S to show that U comes right before S.

The other thing I underline right away is “another purpose as well” in sentence R. This tells us that R is a connecting sentence that will come between two purposes for “the starch”. Let’s look for those purposes.

One of those purposes is using the starch as a scoop, from U and S. Sentence Q is the other purpose: the starch cools the spicy food. All we need to know is which purpose comes first in the paragraph.

Right now there are two possible orders: TQRUS and TUSRQ. Trying both, we see TUSRQ works better because T and U both talk about the main course/ main dish, and together they work as a good overall description of the meal. In TQRUS, U and S don’t follow as nicely from R. R has no mention of the main course, so Q fits better after R because it is more specific to the starch. U is more general and has more connection to T.

Lets read through and make sure everything fits. The paragraph describes what is served at this type of meal, how the main dish is scooped up with a starch, and how the starch serves another purpose of cooling down the spicy food. Makes sense!

This paragraph fits a common pattern of sentences going from general to more specific.

2) After the first quick read, this paragraph appears to be about bluegrass music. I also see in sentence S they mention a variation on this type of music called “newgrass”. Whenever you see phrases like “this” something, “that” something or “these” somethings, you should underline them. They must’ve introduced what they’re talking about before. The same goes for “he”, “she”, “they” and when you see a shortened version of someone’s name. If you see just a last name, there’s going to be a sentence earlier in the paragraph with the full name. In this paragraph, I underline “that style” in Q, “Monroe” in T, and “these two meanings” and “this style of music” in U.

I also underline “on the other hand” in S since it indicates contrast.

I also see some repeated phrases. I see “traditional” and “non-traditional” three times, so I underline all of them. I also see “non-amplified strings” in R and “non-amplified stringed instruments” in T, so I underline them.

Now let’s start connecting. “These two meanings” in U is a strong connection to the first sentence, which gives two meanings of the word “bluegrass”, so U probably is the next sentence. “This style of music” in U must refer to bluegrass music. U also introduces Bill Monroe.

T gives use details of Monroe’s style, and we can connect the mention of “Monroe” back to “Bill Monroe” in U. T also mentions the style uses “non-amplified stringed instruments”. R gets more specific about those “non-amplified strings”. At this point UTR is a good bet for the order of the first three sentences.

Q says “you won’t hear that style at… traditional bluegrass festivals”. What is “that style”? S mentions that when the instruments or music are “non-traditional“, the style is called “newgrass”. It makes sense that they wouldn’t have a non-traditional style at a traditional festival, so the style is “newgrass”. So S must come before Q, and our order is UTRSQ!

Let’s read through in that order to make sure everything makes sense. The passage talks about the origin of bluegrass, give details about the style, and then talks about a non-traditional variation called “newgrass”. Seems to make sense!

So how should you prepare for scrambled paragraphs? I would read and work through the Barron’s section on it and practice on the practice exams you do. However, beyond that it should not be the main focus of your studying. Some paragraphs are so tricky that you can’t expect to get them all. Getting 3 out of 5 scrambled paragraphs is pretty good for most students. I have been dong this for a while and still sometimes run into scrambled paragraphs that I have trouble solving. In addition, scrambled paragraphs are only worth a fifth of the points on the verbal section. Reading comprehension is worth 30 points, and many students don’t pay enough attention to it. One exception: you may want to pay a good amount of attention to scrambled paragraphs if you are pretty good on everything else and you are trying to get into Stuy or one of the higher cutoff schools. It is possible that one or two correct scrambled paragraphs could make the difference there!

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]]>Relevant question from SHSAT handbook:

2012-2013 Handbook Form A

57) If , what is the value of in terms of ?

A)

B)

C)

D)

E)

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]]>When we have two if/then statements, sometimes we can connect them. For example, suppose these two statements are true:

If I go to Roma’s restaurant, then I get pizza. If I get pizza, then I get a soda.

This is a chain of events: when I go to Roma’s, I get pizza, and when I get pizza I also get a soda. So whenever I go to Roma’s I end up getting a soda! We can get a new if/then by skipping over the middle step: If I go to Roma’s restaurant, then I get a soda. This must be true!

Lets look at this problem in symbols. Let R = “I go to Roma’s restaurant”, P = “I get pizza”, and S = “I get a soda”. Then our statements are:

and

R leads to P and P leads to S, so R actually leads to S. The Chain Rule lets us get the new if/then:

Let’s try using the chain rule on a few examples:

1) If Ron goes to the theater, then Sandra goes to the mall. If Sandra goes to the mall, then Anna goes to a concert.

2) If you don’t get good grades, then you can’t go to the party. If you don’t study then you don’t get good grades.

3)

An important note: you can only use the chain rule on two statements if the “then” part of one of the statements is the “if” part of the other. For example, we can use the chain rule on “If Jane goes to the cafeteria, then Bob goes to the cafeteria” and “If it is Tuesday, then Jane goes to the cafeteria” because “Jane goes to the cafeteria” if the “then” part of the second if/then and the “if” part of the first. We can’t use the chain rule with “If an alien is from planet Neon, then they eat florks” and “If an alien has a flying saucer, then they eat florks” because the part they share, “they eat florks”, is the “then” part of both if/thens.

In symbols, if our 3 statements are A, B and C, our statements need to look like and to use the chain rule.

The chain rule says that if we have two if/thens like and , we know must also be true. If this were an SHSAT, might show up as the correct answer. However, there are other possible correct answers.

Since is true, its contrapositive must also be true. The contrapositive would be . So using the chain rule and then taking the contrapositive would be another possible correct answer.

The other possible correct answers you might see are just the contrapositives of the original statements.

Let’s take two if/thens and write out all the possible answers:

If it is not snowing, then Abby goes running. If Abby goes running, then she does not play hockey.

a) Write both statements symbolically.

b) Get the contrapositive of both if/thens.

c) If possible, use the chain rule on the statements.

d) Get the contrapositive of the if/then we got from the chain rule.

Now for a few SHSAT-style questions:

4) Suppose the following statements are true: If it is Tuesday, then Ann does laundry. If Ann does laundry, then she walks the dog. Which of the following must also be true?

a) If Ann walks the dog, then it is Tuesday.

b) If Ann does not walk the dog, then it is not Tuesday.

c) If Ann does not do laundry, then she does not walk the dog.

d) If it is not Tuesday, then Ann does not walk the dog.

e) Ann likes to get all her chores done at once.

5) Suppose the following statements are true: When Daniel goes fishing, he has a sandwich for lunch. Daniel does not have a sandwich for lunch if he does not have juice. Which of the following must also be true?

a) Daniel only eats fish sandwiches.

b) If Daniel does not go fishing, then he does not have a sandwich for lunch.

c) If Daniel does not have a sandwich for lunch, then he does not have juice.

d) If Daniel does not go fishing, then he does not have juice.

e) If Daniel goes fishing, then he has juice.

6) Suppose the following statements are true: If Kayla joins the track team, then she joins the art club. If Kayla joins the debate team, then she does not join the art club. Which of the following must also be true?

A) Kayla does not join the track team if she joins the debate team

B) Kayla joins the track team if she joins the art club

C) If Kayla does not join the debate team, then she joins the track team

D) Kayla is the busiest kid in school

E) If Kayla does not join the debate team, then she does not join the art club

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]]>Relevant questions from the SHSAT handbook:

2012-2013 Handbook Form A

68) If , what is the value of in terms of ?

F)

G)

H)

J)

K)

2012-2013 Handbook Form B

88) If , what is the value of in terms of ?

F)

G)

H)

J)

K)

96)

For any value of and , what is the value of in the equation above?

F)

G)

H)

J)

K)

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]]>q) Suppose the following statement is true: If it is Monday, then Bob goes to the park. Which of the following **must** also be true?

A) If it is not Monday, then Bob does not go to the park

B) If Bob goes to the park, then it is Monday

C) Bob’s favorite place is the park

D) If Bob doesn’t go to the park, then it is not Monday

E) Bob doesn’t like going to the beach

We call this type of question an if/ then question. Barron’s has a pretty solid section on these questions, but since this is one of the harder question types I find it helpful to go over them in a little more depth. You should definitely check out this section of Regents Prep, and if you have the time and patience to go really deep, this section can help make you a Logical Reasoning wiz!

Note: At least in my edition of the Barron’s book, one or two questions in the if/then section have more than one correct answer. Don’t let that confuse you!

Now lets dive into the example:

It is helpful to write down if/ then statements in shorthand. The statement “If it is Monday, then Bob goes to the park” is made up of two sub-statements:

- It is Monday
- Bob goes to the park

Let’s let the letter M stand for “it is monday” and P stand for Bob goes to the park. The symbol stands for “If… then…”.

So we can write “If it is Monday, then Bob goes to the park” in symbols as . You say this aloud as “If M then P”. Sometimes I like to say something like “If Monday, then Park”.

So what does this if/ then statement tell us? Well, we know if it’s Monday, then Bob must go to the park.

What if it’s not Monday? You might think “If it’s not Monday, then Bob doesn’t go to the park”. That’s even an answer choice!

“If it is Monday, then Bob goes to the park” only tells us about what happens when it’s Monday! It doesn’t say anything about what happens when it’s not Monday. Choice A is wrong because we don’t have any information about what happens when it’s not Monday! Maybe Bob goes to the park, maybe he doesn’t; we have no idea.

There is even a special name for this wrong answer choice: It’s called the **inverse**!

Before we can understand the inverse, we have to understand **negation**: the opposite of a statement is called the **negation**. Negations are often created by adding a not. For example, the negation of “It is Monday” would be “It is not Monday”. The symbol for “not” is . So if stand for “It is Monday”, then “It is not Monday” would be . You say as “not M”.

Now you try:

q) What is the negation of “Bob goes to the park”?

A twist:

q) What is the negation of “It is not raining”?

So what does negation have to do with the inverse? Well, to get the inverse of an if/ then, you replace the if and the then parts with their negations. For example, the inverse of “If it is Monday, then Bob goes to the park” would be “If it is not Monday, then Bob does not go to the park”. Symbolically, the inverse of is .

Let’s practice getting the inverse of a couple if/ then statements:

q) What is the inverse of “If it is warm out, then Jane goes running”?

q) What is the inverse of “If it is snowing, then the pool is not open”?

Remember the inverse of an if/then is not necessarily true and is a wrong answer choice. If we are given “If it is Monday, then Bob goes to the park”, we don’t know what happens if it’s not Monday. We can’t say “If it is not Monday, then Bob does not go to the park”. That would be a wrong answer choice.

What if we knew that Bob went to the park? Does that mean it has to be Monday? Can we choose choice B, “If Bob goes to the park, then it is Monday”?

We can’t say for sure that if Bob went to the park, it must be Monday. We don’t know what happens when it’s not Monday! It’s possible he goes to the park when it’s not Monday also, so just because he went to the park doesn’t necessarily mean it must be Monday.

Choice B is also a special type of wrong answer. It is called the **converse**. To get the converse, you **switch the if and the then**. In this example, we started with “If it is Monday, then Bob goes to the park”, so the converse would be “If Bob goes to the park, then it is Monday”. In symbols, we started with , and the converse is .

Let’s practice finding the converse.

q) “If Samuel L. Jackson is in a movie, then the movie is good”

q) “If it is Saturday, then school is closed”

So we know that the inverse and the converse are often wrong answers. Let’s talk about one more scenario. We are starting with “If it is Monday, then Bob goes to the park”. What if we know that Bob didn’t go to the park today? Then it can’t be Monday, because on Monday Bob goes to the park! So we can say “If Bob does not go to the park, then it is not Monday”. This is called the **contrapositive** of the original statement, and it is the right answer! We see choice D is the contrapositive (choices C and E we don’t have information about- they’re kind of nonsense).

We can think of the if part of the if/ then as the cause and the then part as the effect. So we can restate all if/ then’s as “If the cause happens, then the effect happens”. The contrapositive is saying “If the effect didn’t happen, then the cause couldn’t have happened”. The cause forces the effect to happen.

We often have to figure out the contrapositive in these questions. To make the contrapositive, you switch the if and the then and negate them both. So the contrapositive of “If it is Monday, the Bob goes to the park” is “If Bob does not go to the park, then it is not Monday”. In symbols, the contrapositive of is .

Try making contrapositives on your own:

q) If you work, then you get paid.

q) All purple people are not ticklish.

q) If you don’t study for the test, you won’t pass

Now let’s try a couple SHSAT style if/then questions.

q) Suppose the following statement is true: If Alison plays tennis, then Ryan does not play football. Which of the following must also be true?

a) If Ryan does not play football, then Alison plays tennis

b) Alison likes tennis more than football

c) If Ryan plays football, then Alison does not play tennis

d) If Ryan plays tennis, then Alison plays football

e) If Alison does not play tennis, then Ryan plays football

q) Suppose the following statement is true: Joan does not wake up early if school is not open. Which of the following must also be true?

a) If school is open, then Joan wakes up early.

b) Joan is a heavy sleeper.

c) If Joan wakes up early, then school is open.

d) If Joan does not wake up early, then school is not open.

e) If school is open, then Joan does not wake up early.

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]]>Links:

Regents Prep lessons and practice

Take special notice of this hint from the Regents Prep ratios lesson: When working with ratios such as 2 : 3 : 5, use 2x, 3x, and 5x** **to write an equation!

Khan Academy lessons and practice

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]]>http://www.regentsprep.org/Regents/math/ALGEBRA/AD2/indexAD2.htm

There are a couple types of tricky problems related to averages that often show up on the SHSATs:

Finding the average from a frequency table:

Practice 1:

http://www.bbc.co.uk/bitesize/ks3/maths/handling_data/measures_average/revision/7/

Practice 2- don’t worry about the second half of the page, which is more advanced:

https://www.mathsisfun.com/data/mean-frequency-table.html

Missing numbers in averages:

Lesson/ practice 1:

https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-data-statistics/cc-6th-mean-median-challenge/v/using-mean-to-find-missing-value

Practice 2:

http://www.softschools.com/math/topics/determining_a_missing_data_value_given_the_mean/

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