Math Study Guide for the SHSAT

General Information

The Specialized High School Admissions Test (SHSAT) Math section is a paper-and-pencil test that contains 57 questions, the same number as the English Language Arts (ELA) section. Most of them are typical four-answer choice questions. However, five questions require you to grid in your answer on a small chart. There is a complete explanation of this process at the end of this study guide.

The time limit for the Math section is up to you. You will have a total of three hours to complete the entire test and you will need to determine how much of that is spent on the Math section and how much time you devote to the ELA section.

You must know terms and basic procedures, and you should be familiar with the symbols used in math. However, since this is a test to determine if you would benefit from enhanced math instruction at one of the specialty high schools, many of the questions require you to combine basic math skills to solve complex problems that have multiple steps.

Definitions and explanations for terms and symbols and most formulas will not be given in the test booklet. You should practice using basic math formulas to solve problems so they are memorized and easy for you to use.

The content assessed by the SHSAT math section is taken from the New York State Next Generation Mathematics Learning Standards. If you take the test as an eighth-grader, you are responsible for all skills taught through the seventh grade.

This study guide, our practice questions, and our flashcards are geared toward the SHSAT that is taken by students in eighth grade. If you are a new New York City resident or wish to retake the SHSAT, you may take the test as a ninth-grader or rising ninth-grader. In that case, you will also need to review the eighth-grade skills listed in the link above.

No calculator is allowed during this SHSAT section, and you are highly encouraged to take notes and do calculations using scratch paper (which will be provided).

Numeration

Most math involves numeration, which is simply using and manipulating numbers in a variety of situations. The following are some seventh-grade numeration concepts you’ll need to be fluent with.

Integers

Integers are the set of numbers that comprise both positive and negative whole numbers and zero. Positive integers are all numbers greater than zero, such as \(1\), \(10\), and \(567\). In contrast, negative integers are less than zero and denoted with a minus sign, like \(-1\), \(-10\), or \(-567\).

Zero, neither positive nor negative, serves as the neutral point between positive and negative integers. Understanding integers offers a foundation for arithmetic operations and applications in real-world scenarios, from representing financial transactions to measuring temperatures and distances.

Expression on a Number Line

Understanding expressions on a number line is important for visualizing mathematical concepts involving integers. The number line serves as a graphical representation of numbers, with positive integers positioned to the right of zero and negative integers to the left.

For instance, \(3\) lies three units to the right of \(0\) on the number line, while \(-2\) is two units to the left. This visual aid facilitates comprehension of arithmetic operations like addition and subtraction, as well as expressions such as \(5+(-2)\) or \((-3)-7\). Mastering expressions on a number line enhances mathematical fluency and problem-solving abilities by providing a tangible representation of abstract integer concepts. Here is a typical number line:

Additive Inverses

Additive inverses are pairs of numbers that, when combined, result in zero. They serve as the cornerstone for various mathematical operations, particularly subtraction, and they play a pivotal role in balancing equations and solving equations using algebra. The additive inverse of a number, \(a\), is its opposite counterpart, which would be represented as \(-a\). For instance, \(3\) and \(-3\) are additive inverses since \(3+(-3)=0\).

Beyond mathematical abstractions, additive inverses have practical applications in numerous real-world scenarios, particularly those involving money.

Imagine you have a bank account with a balance of \(\$100\). You decide to withdraw \(\$100\) from your account. Mathematically, this withdrawal can be represented as \(-100\), indicating a decrease in your account balance. Now, the concept of additive inverses comes into play. To find the new balance after the withdrawal, you add the withdrawal amount to your initial balance:

Initial balance:\(+100\) Withdrawal: \(-100\)

Adding these together, you get:

\[+100+(-100)=0\]

As you can see, the withdrawal \((-100)\) cancels out the initial balance \((+100)\), resulting in a final balance of zero. This means that after withdrawing \(\$100\), you have no more money in your account.

Rules of Operations

The Distributive Property

The distributive property is a mathematical concept that describes how multiplication operations can be distributed over addition or subtraction operations within an expression. It plays a basic role in simplifying expressions, solving equations, and understanding algebraic concepts.

The distributive property states that for any three numbers, \(a\), \(b\), and \(c\), the following equality holds:

\[a \times (b+c)=a \times b+a \times c\]

Similarly:

\[a \times (b-c)=a \times b-a \times c\]

Order of Operations

The order of operations, often expressed using the acronym PEMDAS, is a set of rules used to clarify the sequence in which mathematical expressions should be performed in a problem involving multiple operations. Performing operations in the wrong order will produce a wrong answer.

PEMDAS stands for:

Parentheses—Perform any operations within parentheses first, from left to right.

Exponents—Evaluate any exponents (powers and roots) from left to right.

Multiplication and Division—Perform multiplication and division operations in the order they appear from left to right.

Addition and Subtraction—Perform addition and subtraction operations in the order they appear from left to right.

A good way to remember PEMDAS is with the phrase, “Please excuse my dear Aunt Sally.”

Let’s consider the expression.:

\[2+3\times(4-1)^2\]

Start by evaluating the expression within the parentheses:

\[4-1=3\]

Next, evaluate the exponent:

\[3^{2}=9\]

Then, perform the multiplication (there is no division):

\[3 \times 9=27\]

Finally, complete the addition (there is no subtraction):

\[2+27=29\]

Signed Numbers

Multiplication and division of signed numbers, which include positive and negative integers, follow specific properties that help simplify calculations and ensure accurate results. These properties are fundamental in arithmetic and algebra, facilitating problem-solving in various mathematical contexts.

Multiplication of Signed Numbers

There are two rules that come into play when multiplying numbers that have signs (i.e., \(+\) and \(-\)) in front of them:

• When multiplying two numbers with the same sign (both positive or both negative), the result is always positive. Here is an example:

\[(+3)\times(+4)=+12\]

• Multiplying two numbers with different signs (one positive and one negative) results in a negative product. Look at this example:

\[(-5)\times(+2)=-10\]

Note: When there is no sign in front of a number, a positive sign (\(+\)) is assumed.

Division of Signed Numbers

As with multiplication, when dividing numbers with signs in front of them, there are two rules to know:

• When dividing two numbers with the same sign (both positive or both negative), the quotient is positive. Here is an example:

\[(+8)\div(+2)=+4\]

• Dividing two numbers with different signs (one positive and one negative) yields a negative quotient. Look at this example:

\[(-12)\div(+3)=-4\]

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction or the quotient of two integers. All integers are rational numbers. For example, \(3\) can be expressed as \(\frac{3}{1}\)). However, not all rational numbers are integers.

Adding and Subtracting Rational Numbers

Adding and subtracting can be slightly more complex procedures when done with rational numbers than they are with integers. The process involves combining or comparing quantities that can be expressed as fractions or decimals. Understanding these rules is necessary for quickly performing arithmetic operations with rational numbers.

Addition of Rational Numbers

When adding rational numbers, follow these steps:

Find a common denominator—If the denominators of the fractions are different, find the least common multiple (LCM) to make the denominators the same.

Add the numerators—Once the denominators are the same, add the numerators while keeping the denominator unchanged.

Simplify, if necessary—After performing the necessary mathematical operation, reduce the resulting fraction to its simplest form, if necessary.

Here is an example:

\[\frac{1}{3}+\frac{2}{3}\] \[=\frac{1+2}{3}\] \[=\frac{3}{3}\] \[=1\]

Subtraction of Rational Numbers

Subtracting rational numbers follows similar steps to addition:

Find a common denominator—As with addition, ensure that the fractions have the same denominator by finding the LCM.

Subtract the numerators—Once the denominators are equal, subtract the numerators while keeping the denominator unchanged.

Simplify, if necessary—Reduce the resulting fraction to its simplest form, if necessary.

Here is an example:

\[\frac{5}{4}-\frac{1}{4}\] \[=\frac{5-1}{4}\] \[=\frac{4}{4}\] \[=1\]

Multiplying and Dividing Rational Numbers

Multiplying and dividing rational numbers are essential operations in mathematics, particularly when dealing with fractions and decimals. When multiplying or dividing rational numbers, it’s important to understand the specific rules and procedures involved to ensure accurate calculations and interpretations of the results. As with adding and subtracting, multiplying and dividing rational numbers can involve more complex processes than they do with integers.

Multiplying Rational Numbers

Rational numbers, which are represented as fractions, are typically quite easy to multiply together. The product of two rational numbers is also a rational number. To compute the product of two rational numbers, you simply multiply the numerators together and the denominators together.

For example, computing the product of \(\frac{3}{5}\) and \(\frac{4}{7}\) goes as follows:

\[\frac{3}{5} \times \frac{4}{7} = \frac{3 \times 4}{5 \times 7} = \frac{12}{35}\]

If you are multiplying an integer with a rational number, you must first change the integer into a fraction by dividing it by \(1\).

For example, multiplying \(8\) by \(\frac{3}{5}\), we have:

\[8 \times \frac{3}{5} = \frac{8}{1} \times \frac{3}{5}\] \[= \frac{8 \times 3}{1 \times 5} = \frac{24}{5}\]

Dividing Rational Numbers

Rational numbers, represented as fractions, can be divided by any nonzero rational number. Understanding the principles of dividing rational numbers is vital for accurate calculations and interpreting the results effectively. There are three important rules when dividing rational numbers:

• The divisor cannot be zero—The divisor is the number that is dividing the other number, the dividend. Dividing by zero produces an undefined result and leads to mathematical inconsistencies.

• Every nonzero rational number can be used as a divisor—Even fractions (except those with a denominator of zero) can be divided by another nonzero fraction to yield a rational number as the result, or quotient.

• All quotients are rational numbers—When dividing one rational number by another nonzero rational number, the quotient is always a rational number. This property highlights the closed nature of dividing rational numbers.

Consider dividing the rational number \(\frac{3}{4}\) by \(\frac{2}{3}\):

\[\frac{3}{4} \div \frac{2}{3}\] \[= \frac{3}{4} \times \frac{3}{2}\] \[= \frac{9}{8}\]

In this example, both the dividend \((\frac{3}{4})\) and the divisor \((\frac{2}{3})\) are rational numbers, and the quotient \((\frac{9}{8})\) is also a rational number.

Rule for Signed Rational Numbers

When dealing with signed rational numbers (fractions), a key rule to remember is that the negative of the number can be obtained by negating the numerator, the denominator, or the number as a whole. This rule is expressed as follows:

If \(p\) and \(q\) are integers, then:

\[-\bigg(\frac{p}{q}\bigg) = \frac{-p}{q}=\frac{p}{-q}\]

These notations are all equal::

\[-\bigg(\frac{12}{3}\bigg)\] \[-\frac{12}{3}\] \[\frac{-12}{3}\] \[\frac{12}{-3}\]

Matching to Real-World Situations

Multiplying and dividing rational numbers are mathematical operations that can be applied to various real-world situations. Understanding these operations not only helps in solving mathematical problems but also allows us to interpret and analyze practical scenarios.

Multiplication

Cooking with a recipe is a common real-world situation that you will see used in problems. The problem might tell you the amount of an ingredient and ask you to multiply it. For instance, you may be told a recipe calls for \(\frac{3}{4}\) cup of flour, but you need to double it.

To find out how much flour you need for the doubled recipe, you use multiplication:

\[\frac{3}{4} \times 2 = \frac{3 \times 2}{4}=\frac{6}{4}=\frac{3}{2}\]

So, for the doubled recipe, you would need \(\frac{3}{2}\) cups of flour.

Division

Conversely, if you have the same recipe but need to cut it in half, you simply divide the given amount by \(2\):

\[\frac{3}{4} \div 2 = \frac{3}{4\times 2}=\frac{3}{8}\]

So, for the halved recipe, you would need \(\frac{3}{8}\) cup of flour.

Converting Fractions to Decimals

In some situations, it might be easier to work with a decimal than a fraction. In such cases it is helpful to be able to convert fractions to decimals, usually using long division. Let’s try an example.

Convert \(\frac{3}{8}\) to a decimal.

Solution

If needed, you may need to first convert a mixed number to an improper fraction. In this case, that isn’t necessary. So, we can skip to the next step, dividing the numerator \((3)\) by the denominator \((8)\):

\[\begin{array}{r} \phantom{0}0.375 \\[-3pt] 8 \overline{)30\phantom{0}} \\[-3pt] \underline{\phantom{0}24\phantom{0}} \\[-3pt] \phantom{00}60 \\[-3pt] \underline{\phantom{00}56} \\[-3pt] \phantom{000}40 \\[-3pt] \underline{\phantom{00}40} \\[-3pt] \phantom{0000}0 \end{array}\]

The result of the long division is the fraction converted to a decimal, so:

\[\frac{3}{8}=0.375\]

Solving Word Problems with Rational Numbers

Word problems involving rational numbers require applying mathematical operations to real-life situations. Whether it’s dividing quantities, calculating proportions, or determining rates, rational numbers offer a precise way to represent and solve these problems. Let’s look at example word problems involving rational numbers.

Whole Numbers

You have \(\$500\) in your bank account. On Tuesday, you withdraw $150 to buy groceries and pay bills. On Friday, you receive your paycheck for \(375\) and deposit it in the bank account. What is your account’s balance at the end of the week?

Solution

Problems involving money are common examples of situations where you’ll need to set up equations using signed numbers. Withdrawals are written with a negative sign, while deposits are written with a positive number.

You know your starting balance is \(\$500\) (i.e., \(+\$500\)).

The amount withdrawn is \(\$150\), which is written as \(-\$150\).

The amount deposited is \(\$375\), which is written as \(+\$375\).

Now, let’s create an equation with all three numbers:

\[500 - 150 + 375 = 725\]

So, at the end of the week, you have \(\$725\) in your bank account.

Fractions

There is often a use for fractions in solving word problems. Here are a couple examples:

Example 1:

Susan is making a cake for her mother’s birthday dinner. The recipe she is using says it will make a cake big enough for four people and calls for \(\frac{2}{3}\) cup of butter, \(1\) cup of sugar, \(\frac{5}{3}\) cups of flour, and \(\frac{1}{2}\) cup of chocolate. However, Susan’s cousins are now coming to the dinner, and there will be eight people there. How much of each ingredient will Susan need to make enough cake for everyone?

Since there will be eight people at the dinner and the recipe is for four people, we simply need to double all the ingredient amounts:

\[\frac{2}{3} \times 2 = \frac{4}{3}\] \[1 \times 2 = 2\] \[\frac{5}{3} \times 2 = \frac{10}{3}\] \[\frac{1}{2} \times 2 =1\]

So, Susan will need \(\frac{4}{3}\) cups of butter, \(2\) cups of sugar, \(\frac{10}{3}\) cups of flour, and \(1\) cup of chocolate.

Example 2:

You have \(\frac{3}{4}\) of a cake remaining from your birthday party. The next day, you eat \(\frac{1}{2}\) of what’s left. What fraction of the original cake remains?

Solution

This will be a multi-step problem. Since you ate half of what was left of the cake, you have to find what \(\frac{1}{2}\) of \(\frac{3}{4}\) is:

\[= \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}\]

Now, you had \(\frac{3}{4}\) of a cake to start with, which means \(\frac{1}{4}\) was initially eaten. So, add that to the amount eaten on the next day and subtract that total from the whole:

\[= \frac{1}{4} + \frac{3}{8} = \frac{2}{8} + \frac{3}{8} = \frac{5}{8}\] \[= \frac{8}{8} - \frac{5}{8} = \frac{3}{8}\]

So, \(\frac{3}{8}\) of the cake remains.

Decimals

You are on a business trip and must drive across the state to reach your client. You drive at a speed of \(55\) miles per hour for \(2.5\) hours. How far did you travel?

Solution

Your speed is \(55\) miles per hour.

The time you drove was \(2.5\) hours.

We can set up a simple formula to figure out the distance you traveled:

\[d = s \times t\]

where \(d\) is the distance traveled, \(s\) is speed, and \(t\) is time. Using the given numbers in this formula, we get:

\[55 \times 2.5= 137.5\]

So, you traveled \(137.5\) miles.

Conversion between Forms

A recipe calls for \(\frac{3}{4}\) cup of flour. There are 16 tablespoons in one cup. How many tablespoons is this?

Solution

\[1 \,\text{c} = 16 \,\text{tbsp}\]

So:

\[\frac{3}{4} \text{c} = \frac{3}{4} \times 16 \,\text{tbsp} =12\, \text{tbsp}\]

Therefore, the recipe calls for \(12\) tablespoons.

Evaluation of Answer

After performing calculations involving rational numbers, it’s essential to evaluate the answer to determine if it’s reasonable within the context of the problem. If you ever get an answer that seems wrong at first glance, your first step should be reviewing your calculations and identifying any errors that could have led to the incorrect result.

Consider a question in which you are asked to calculate the profit of a business. You are told the business had expenses of \(\$5\text{,}000\) (a negative number) and revenue of \(\$7\text{,}000\) (a positive number), the profit should be \(\$2\text{,}000\). If your calculation yielded a profit of \(\$-2\text{,}000\), that wouldn’t make sense. In this case, the most likely mistake is that you reversed the signs of expenses and revenue.