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ACT Math Posts

ACT Math Practice: How to Graph Trigonometry Equations

In order to graph trig functions, you need to understand some Trig basics. If you haven’t already, check out this quick refresher on SOHCAHTOA.

Sine, cosine, and tangent are the three main trig identities. They are usually graphed and expressed in degrees, but you may also see them expressed in radians. There are 2π radians in one circle. Each point on a circle corresponds to a certain number of radians. To convert degrees to radians, simply multiply by π/180.

Sine and cosine both have standard graphs that you need to memorize. The standard equation for sine looks like this:  y = sin x. The “period” of the wave is how long it takes the curve to reach its beginning point again. The coefficient in front of “sin” (here 1), is called the amplitude. It effects how high and how low the wave reaches vertically. If that coefficient changes, then the height changes. For example, y = 3 sin x, would show a curve that reaches +3 on the y-axis and extends down to -3 on the y-axis.

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Measurement Concepts to Ace the ACT Math Test

The ACT Math Test tests three distinct categories of measurement concepts: Perimeter, Area, and Volume. There are a number of need-to-know formulas you should have memorized as you start working on free ACT practice tests like the one offered at actstudent.org. Come Test Day, you’ll get better ACT Math scores if you can mentally recall them quickly and confidently. Remember that unlike the SAT, these formulas won’t be given to you on Test Day, so make sure to know them cold to ace the ACT!

Perimeter:  The perimeter is the distance around any shape. For a triangle, the perimeter will be the sum of the sides. For a rectangle, the formula is P = l + l + w + w, or P = 2l + 2w. For a square, this becomes P = 4s. For other quadrilaterals, you need to know the length of each side in order to find the perimeter, unless you are given more information about the comparative lengths of the sides.

For a circle, the perimeter is equal to the circumference: C = 2πr.

Try this ACT math question for practice!

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How to Get Better Scores on the ACT with Approximation & Estimation

When you encounter a question on the ACT Math Test that you don’t know how to solve, what do you do? Even if your Math skills are strong, you’re likely to see at least a couple of challenging questions on the ACT Math Test. Sometimes doing the Math is not always the best way to get to the correct answer – a strategy like Picking Numbers or Backsolving can be preferable. You can also use Approximation & Estimation on difficult test questions; this is especially helpful when you’re running out of time, or for questions where there is no applicable formula. Let’s look at how we can use this in our Test Prep to get better ACT scores on Test Day.

Figure ABCDE was drawn on a grid of unit squares, with each vertex at the intersection of two grid lines. What is the area of the figure in square units?

A            19

B            23

C            25

D            27

E            33

 

Since there is no formula to solve for the area of this 5-sides figure, we know that Approximation will be our best strategy here. The way to find the area of this figure is to divide it into sections, first counting the full squares, then the half squares, then the pieces of square.

Whole squares (blue) = 19 full squares

Half squares (green) 2 = 1 full square

Grey pieces = 1 full square

Pink pieces = 1 full square

Purple pieces = 1 full square

Adding together all the areas gives a total of 23 square units. The answer is (B).

When you approximate, let the answer choices be your guide. If they are very close together, then you will need to be more accurate in your approximation. However, if the answer choices are markedly far apart, then by all means round to the nearest integer. Let’s look at a more challenging question:

In the table to the left, the amount of hardcover and paperback copies sold in a given year for novels L, M, N, P, and Q is provided. For which novel is the ratio of hardcover copies sold to the paperback copies sold the greatest for that year?

 

To solve this question it is MUCH faster to approximate by rounding each number to make the comparison easier. L is 800,000/1,300,000, or 8/13. Following the same logic of approximating and estimation: M is 3/6. N is 8/10. P is 3/7. Q is 10/25.

The question asks for the greatest ratio. Notice how 3/6, 3/7 and 10/25 are all around 50% or less. That leaves only 8/13 and 8/10. When the numerator stays the same and the denominator shrinks, we know we are getting a larger fraction of the total. Therefore, 8/10 is greater, so the answer is Title N.

For more practice looking at questions where Approximation & Estimation is an appropriate strategy, click on “Create Game” in the Grockit ACT lobby and make a custom game using only the skill tag “Approximation & Estimation.”

Find 2011 and 2012 ACT test dates here!

ACT Practice: How to Get Every Question on Factors, Multiples, and Divisors Correct

Your ACT Math Test will almost certainly feature questions on factors, multiples, and divisors. Out of 60 questions, you can expect these concepts to play a role in at least 2 or 3 questions. This is everything you’ll need to know to get those questions correct on Test Day!

Test your ACT math skills with this ACT math practice question!

Factors

A factor, or divisor, of a number is a positive integer that evenly divides into that number. For example, 4 is a factor of 12 because 12/4 is an integer, 3.

All integers have 1 and themselves as factors. For example there are six factors for the number 12: 1, 2, 3, 4, 6, 12.

The GCF, or greatest common factor, is the biggest factor that two numbers have in common. Think of GCF questions as: what is the biggest number that divides evenly into both?

What is the GCF of 36 and 108?

The answer is 36, because 36 is a factor of itself, and 108 is a multiple of 36.

A prime number is a special kind of integer that has exactly two factors: itself and 1.

Remember that 1 is NOT prime, because it only has one factor, itself. 2 is a special number because it is the smallest prime number and the only even prime number.

The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Multiples

A multiple is a number that is evenly divisible by another integer. For example, 6 is a multiple of 3. 10 Is a multiple of 5. The multiple is always larger than the other integer. The other integer would be considered a factor of the multiple. Every number is a multiple of itself.

For example, for the number 12, possible multiples are 12, 24, 36, 48, 60, etc.

The LCM, or lowest common multiple, is the smallest number for which both given numbers are factors.

What is the LCM of 12 and 9?

9 = 3 x 3, and 12 = 3 x 4, so the LCM must be 3 x 3 x 4 = 36. By saying that the LCM of 12 and 9 is 36, you are saying that 36 is the SMALLEST number that both 12 and 9 will divide into evenly. Notice how the LCM is not necessarily just the product of the two integers.

Divisibility

Here are the Rules of Divisibility to memorize for the ACT!

A number is divisible by 2 if it’s an even number.

A number is divisible by 3 if the sum of the digits is divisible by 3.

A number is divisible by 4 if the last two digits are divisible by 4.

A number is divisible by 5 if the last digit is either 0 or 5.

A number is divisible by 6 if it’s divisible by BOTH 2 and 3.

A number is divisible by 9 if the sum of the digits is divisible by 9.

For more practice with these rules, click on Create Game in the ACT lobby and create a custom game using these skill tags at the difficulty level of your choice! You can also message any of the Grockit ACT tutors to set up a private tutoring session for more help. Find out more by clicking the Tutoring tab in the Grockit lobby.

How to Analyze Information on the ACT Science and ACT Math Tests

Data Analysis is heavily tested on the ACT Science Test and occasionally on the ACT Math Test, especially for Data Interpretation and Research Summaries questions. You will often be presented with a graph or a table and required to answer relevant questions. (After all, you’ll need to analyze a lot of information in college, so it makes sense that the best schools want ACT students to start now!) Make sure to take notes on your ACT Science practice tests and on the various ACT online resources as you study. It will help you improve your ACT score if you review your ACT Science and ACT Math practice questions in a set right after you complete them. Try to apply this strategy to your ACT Test Prep for better ACT scores on Test Day!

1. Analyze the data first. Don’t jump to the question and then go back to the data to look for an answer. Read all of the labels on the presented graph or table. What is in each column? What is in each row? What is the range of values? Does the data have a direct or indirect relationship? Do the lines have positive or negative slopes? Where was there the most change or growth? Where was there the least?  With this kind of discipline, you will already understand what is being presented. The question becomes almost an afterthought, and you’ll know exactly where to look for the correct information.

2. Ask yourself: what data do I need to solve? Graphs and tables often give us extraneous information. You will not need everything to solve. Chances are the correct answer hinges on just 1 or  2 numbers from the data. Most ACT students get ACT Science questions and ACT Math questions involving data analysis wrong because they do not know what data to use to solve, not because the Math itself is difficult. You will likely only be performing calculations as simple and easy as addition, subtraction, multiplication and division.

3. Approximate when possible. Let the answer choices be your guide. If they are very close together, then you will need to be more accurate in your approximation. However, if the answer choices are markedly far apart, then by all means round to the nearest integer.

Let’s apply these test day tips to a sample data table from Grockit’s question bank:

Analyze: When we analyze this chart, we can see that it represents figures for 5 books. In general, all of the books sold more in paperback then they did in hardcover.

In the table to the left, the amount of hardcover and paperback copies sold in a given year for novels L, M, N, P, and Q is provided. For which novel is the ratio of hardcover copies sold to the paperback copies sold the greatest for that year?

What data do I need?: We’ll need to divide the number in the 1st columns by the number in the 2nd columns to express the “ratio.”

Approximate: Round each number to make the comparison easier. L is 800,000/1,300,000, or 8/13. Following the same logic of approximating and estimation, M is 3/6. N is 8/10. P is 3/7. Q is 10/25.

The question asks for the greatest ratio. Notice how 3/6, 3/7 and 10/25 are all around 50% or less. That leaves only 8/13 and 8/10. When the numerator stays the same and the denominator shrinks, we know we are getting a larger fraction of the total. Therefore, 8/10 is greater, so the answer is Title N.

For more ACT practice create a custom game in the Grockit lobby to focus in on ACT Science and ACT Math questions only, or contact one of the Grockit tutors to set up a 1-on-1 private tutoring lesson. Hours can be purchased on the Tutoring tab in the Grockit lobby and each lesson can be fully customized to your needs. Happy Grockiting!

Find out how Grockit can predict your performance on the SAT.

 

ACT Pre-Algebra Cheat Sheet

Need a quick refresh on the basics of Pre-Algebra tested on the ACT? Print out this handy cheat sheet and review it often to quickly improve your ACT Math foundation. The more you practice, the better your scores will be!

Basic definitions:

A variable is a symbol representing a numerical quantity. Variables are represented by letters in the alphabet such as x, a, b, y, etc. The number that the variable represents is called a value.

A constant is a symbol that represents a definite quantity (such as pi).

A term is a product (multiplication) with an unspecified number of factors, made up of either variables or constants. Terms that have the same factors which differ only in their numerical coefficients are called similar terms. For example, 5y and 9y are similar terms.

An algebraic expression is a mathematical statement which often uses constants and variables. For example: 75x + 12

Because of the distributive property, similar terms can be combined into one term. The new term has the same factors as the similar terms, but its coefficient is the sum (addition) of the coefficients; this is commonly known as combining like terms. 3xy + 2xy = 5xy

Some concepts to know:

Test your ACT skills with this ACT practice math question. Read more »

Top 5 Tested Intermediate Algebra Concepts on the ACT

1. QuadraticsQuadratic equations have three terms and are in the form ax² + bx + c. An example of a quadratic is x² – 5x + 6. To find the factors of this equation, we must set up our set of two parentheses: (        )(        )

The first term in both parentheses must be x, since x multiplied by x is the only way to get x². Then we look at the coefficient of the second term, -5. It’s important to include the sign in front of the integer as part of the coefficient. One of the rules of quadratic equations is that the second terms in the two factors must add together to equal the middle term’s coefficient. So we need to think of two numbers that add together to give us -5.

Already, we can think of many combinations: -6 and 1, -2 and -3, -200 and 105. So which pair is it? Now we have to look at the integer that’s the third term of the quadratic. Here it’s  + 6. Another rule of quadratic equations is that the third term of the quadratic equation will equal the product of the second terms in the two factors. So not only do we need the two numbers to add together to equal -5, but we need them to multiply together to equal + 6. Therefore the factors must be: (x – 2) (x – 3). The “roots” or the “solutions” for this quadratic would be 2 and 3.

2.  Systems of Equations – The ACT will often present you with two or more equations with multiple variables. Remember the “n equations with n variables rule.” If there are 2 variables in an equation (for example, x and y), then there must be 2 equations that each contain those variables in order to solve. The two common ways to solve are Substitution and Combination. Jordan Schonig reviews each method in detail here.

3. Functions –It’s helpful to think of (x, f(x)) as another way of writing (x, y). For many function questions, you can Pick Numbers for the variables to solve! Read more »

When is the ACT offered in 2012?

The ACT is offered 6 times a year in most states. Before you schedule your ACT test you should be aware of the college application deadlines and what it means for rolling admissions and early action deadlines. You should give yourself enough time for ACT test prep to give you the best advantage in achieving the highest ACT score possible.

2012 ACT test dates:

Exam Date:          Register By:            Late Registration:

*2/11/2012             1/13/2012                   1/20/2012

4/14/2012              3/09/2012                   3/23/2012

6/09/2012              5/04/2012                   5/18/2012

9/08/2012                  TBD                            TBD

10/27/2012                TBD                            TBD

12/08/2012                TBD                            TBD

Test your ACT skills by answering this ACT reading question. Good luck!

* Note: There will be no ACT test offered in New York for the February test date.

Register to take the ACT at act.org

ACT Science: Strengthen Hypothesis

Science questions on the ACT may ask you if a piece of evidence, always supplied in the reading passage or in a chart, graph or table, supports and strengthens the hypothesis put forth in the passage. Answering strengthen hypothesis questions require you to identify a hypothesis and understand how data can strengthen or weaken that argument. Although this question type can be time consuming to answer, getting the proper test prep at Grockit will help you master strengthen hypothesis questions. So will these tips and strategies:

  • Read the passage before looking at the questions. Reading the questions first, especially if there’s a hypothesis strengthening question, may mislead you when reading the passage and examining any charts, graphs or tables.
  • Read the first paragraph very closely. It’s very likely that the passage’s hypothesis will be explicitly stated in the first paragraph, maybe even the first sentence. Make sure you understand what it is.
  • Read for the main idea. For any reading passage, you should be reading for the main idea, or purpose. If you do this, you should be able to identify the passage’s hypothesis.
  • Don’t read for the details. It’s a waste of time memorizing details for strengthen hypothesis questions. You’ll be asked about a specific detail, and at that point go back in the passage or to the correct chart, graph or table and find the information you need.
  • Skip questions you don’t know. Since you’re ACT score is based off of how many questions you answer correctly, move on if a strengthen hypothesis has you stumped.

Try this ACT science question for more practice!

Now let’s use these tips/strategies on a sample question:

Stars often form in large groups.  For instance, an “open cluster” such as the Pleiades may contain hundreds of stars that were formed at approximately the same time (and are thus the same age). Even larger “globular clusters” may contain a million or more stars of approximately the same age.  By observing clusters of increasing age, astronomers are able to see how the brightness and temperatures of stars change as they age.
Shortly after a group of stars forms, a plot of their brightness versus their color shows that they form a diagonal line called the “Main Sequence.” The bluest stars, are the brightest, and the reddest stars are the dimmest.  (Star colors are divided into a series of “spectral classes” – O, B, A, F, G, K.  A star’s brightness is expressed as an “absolute visual magnitude,” with the dimmest stars having the largest magnitudes.)
After millions of years, the bluest stars in the cluster begin to swell into “giant” or “supergiant” stars, and they become redder.  Thus, they no longer lie on the Main Sequence line.   As the cluster gets older, the same process happens to redder and redder stars.

Astronomers observing a nearby galaxy have measured the position of the “turnoff” in 25 globular clusters.  If the turnoff occurs at the same spectral class in different globular clusters, those clusters must be of approximately the same age.  Furthermore, if most of the clusters in a galaxy are of similar ages, the galaxy itself may be of that age.  Figure 2 shows the number of clusters with turnoffs in each spectral class.


Does Figure 2 support the hypothesis that most of the galaxy’s globular clusters are of similar ages?

  1. No, Figure 2 shows that the clusters in the galaxy have different turnoffs, and thus different ages
  2. No, Figure 2 shows that the turnoffs of the clusters vary from A to K in spectral class, and these clusters thus vary widely in age
  3. Yes, Figure 2 shows that most of the clusters in the galaxy have a turnoff in spectral class G, and are thus similar in age
  4. Yes, Figure 2 shows that all of the clusters in the galaxy have a turnoff, and are thus similar in age

There are several things you need to know to figure out if Figure 2 supports the passage’s hypothesis. First, you need to know the hypothesis, which is “Stars often form in large groups.” Now, we have to examine Figure 2 to figure out what information it’s relaying. The y-axis shows an increase in the number of clusters as we move up; the x-axis labels the different spectral class of “turnoff” for the star. What we then see is that Figure 2 shows the highest number of clusters in the G class of turnoff, meaning that the scientists’ hypothesis is correct. If we examine the answer choices, we can eliminate A and B since they misinterpret the question. C looks correct since the chart shows most of the clusters in spectral class G and the passage told us this meant they were of similar age. And D doesn’t seem to mention any points mentioned in the passage or chart. We’re left with C as the correct answer.

By following the tips and strategies above, you’ll be able to solve any strengthen hypothesis question on the ACT . Just make sure you get plenty of test prep, going to Grockit to fully prepare yourself for test day. You may be surprised at how much fun you’ll have preparing for the ACT .

Need more help with ACT science questions? Get personalized tutoring in live online sessions with an experienced expert instructor.

ACT Math: The Basics of Lines & Angles

Need better scores in Geometry? This quick review will tell you everything you need to know about Geometry basics! These are the essential building blocks for ACT Math success.

An angle is formed by two lines or line segments which intersect at one point. The point of intersection is called the vertex. Angles are measured in either degrees or radians.

Challenge yourself with adaptive solo practice sessions on Grockit.

A circle has 360 degrees total. You might see in your online studying questions involving radians, and some test questions will ask you to convert radians to degrees. To convert from degrees to radians, multiply by π/180.  To convert from radians to degrees multiply by 180/π.

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