The Importance of Organization for Word Problems

red and white roller coaster on railings

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Let’s start with this challenging question:

Molly worked at an amusement park over the summer.Every two weeks she was paid according to the following schedule:at the end of the first two weeks she received $160.At the end of each subsequent two week period she received $1, plus an additional amount equal to the sum of all the payments she had received in the previous weeks.How much money was Molly paid during full 10 weeks of summer?

A. 644
B. 1288
C. 1770
D. 2575
E. 3229

To start, let’s create a table just to lay out the general information first:

After 2 weeks —> $160

After 4 weeks —-> $1 + (all previous)

After 6 weeks —-> $1 + (all previous)

After 8 weeks —-> $1 + (all previous)

After 10 weeks —-> $1 + (all previous)

Then fill in what the “all previous” would mean:

After 2 weeks —> $160

After 4 weeks —-> $1 + $160 = $161

After 6 weeks —-> $1 + ($160 + $161) = $161 + $161

After 8 weeks —-> $1 + ($160 + $161 + $161 + $161) = $161 + $161 + $161 + $161

After 10 weeks —> $1 + (all previous) = eight $161’s

At this point we can see that “all previous” is going to equal the initial $160 + seven $161 and then we’re adding $1 to it, so it’s really just $161 x 8 = $1288

We have an interesting sequence:

After 2 weeks —> $160

After 4 weeks —-> one $161

After 6 weeks —-> two $161

After 8 weeks —-> four $161

After 10 weeks —-> eight $161

Basically the “number” of $161’s just doubled from 4 weeks to 10 weeks.

Now to add them all up, again, we’re just going to count up all of the number of $161’s.

We have our initial $160 + 15*($161) = $2575. The correct answer is (D).

You will see over and over again that it’s not the Math calculations in Word Problems that will trip you up, but the inability to be super organized on your scratch paper. 🙂

Interpreting Evidence Correctly on the ACT and SAT

In all three types of ACT Science passages, you will need to be able to understand how evidence is presented and be asked to answer questions based on the evidence. These are a bit like Reading questions.

To interpret evidence correctly, you need to focus on the results. Draw logical conclusions based on the presented data, and actively read the information in the accompanying passage. It’s important to know the difference between direct and inverse variation.

To answer Interpreting Evidence questions correctly, ask yourself these 3 questions: – What is the evidence presented? – Whose position is supported by the evidence? – What does the evidence suggest?

Interpreting evidence for Data Representation and Research Summaries ends up requiring more Data Analysis skills, since data is a large component of those passages. For Conflicting Viewpoints, interpreting evidence is a matter of keeping the two theories (two scientists) straight. Watch this video for a strategy on how to do this!

This video describes how to answer “Inference” questions on the SAT and ACT Reading Test. You may wonder what this has to do with ACT Science, but sometimes interpreting evidence is a LOT less scientific than it sounds, and a LOT more like reading comprehension. Remember that you can only make an inference BASED on something directly stated. Incorrect answer choices are frequently “out of scope.”

 

Grammar Guide: the usage of “those”

Confused about how “those” works on the SAT, ACT, GRE, or GMAT? Let’s take a look at a question from Manhattan GMAT that uses this word!

Salt deposits and moisture threaten to destroy the Mohenjo-Daro excavation in Pakistan, the site of an ancient civilization that flourished at the same time as the civilizations in the Nile delta and the river valleys of Tigris and Euphrates.

A) that flourished at the same time as the civilizations
B) that had flourished at the same time as had the civilizations
C) that flourished at the same time those had
D) flourishing at the same time as those did
E) flourishing at the same time as those were

In general, “those” is the plural of “that.” It’s in a group called “determiners” along with “this”, “that”, “these”, “those,” “here” and there that are technically pronouns but often function more like adjectives.

For C, D, and E, “those” is meant to mean something like “the ones” and could be used as a pronoun to replace a plural antecedent if there was one in the sentence, and if the construction was parallel.

The correct answer is (A).

What to Expect on the ACT Science Test

The ACT Science section can cause a lot of unnecessary worry among test-takers. However you can still receive a strong score even if you aren’t a budding Albert Einstein. Careful reading and note-taking (the same skills you use for Reading Comp!) are enough to answer most questions. Remember – the answer has to be based on the information in the paragraphs and/or tables. You just have to know where to look!

The ACT Science Test will always be the fourth test you’ll take. It will have 7 passages and you’ll have 35 minutes to complete them. That’s about 5 minutes per passage so moving confidently through this test is essential! It takes practice to gain confidence in interpreting data and understanding the meaning of unfamiliar vocabulary. Luckily, you already have all of the skills necessary to do this from your high school Science classes.

Experiments of some type are described in most of the passages. A group of scientists will be studying some type of phenomenon. They will usually conduct two experiments to see how certain factors affect the phenomenon. Often a graph, table or diagram will accompany the description of the experiments to show the results. Here are some two important steps to help you analyze the experiments:

1. Identify the Purpose & Method

Make sure to underline the Purpose & Method for each Experiment as you read (don’t wait until you finish reading everything or you’ll be too confused and overwhelmed). The Purpose tells you why the scientists are conducting the experiment. What are they trying to find out? Look for verbs like “to study…” or “to examine…” in the first explanatory paragraph. That is often where the description of the Purpose can be found. The Method for each experiment will be described in the following paragraphs. Make sure to make note what is similar and what is different between the two Experiments. Sometimes the scientists will change one factor or more factors between the experiments to see if the results change.

2. Understand the Factors

Factors, also known as variables, are important elements of the experiments. These are often things like temperature, pH, pressure, time, distance, etc. Depending on a way the variable is being used in each experiment, it can be called either dependent or independent. Independent variables are those factors that are controlled by the scientists. Did the scientists increase the heat in the experiment? Did they add or remove pressure? If the scientists were the ones controlling the variable, it is independent. Dependent variables are what the scientists observed changing. Let’s say that when the scientists increased the heat in our hypothetical experiment, the time also increased.

The ACT: All things coordinate geometry

Some of the most challenging Math questions on the ACT involve Coordinate Geometry, so it’s important you have a solid grasp on the formulas and concepts tested. The most basic concept you’ll encounter is inequalities. To graph a range of values, you’ll need to draw a number line and plot the beginning and end of the range using an open or closed circle and a solid line.

Which of the following graphs represents the solution to the set of the inequality |x| > 3 on the real number line?

For a question, like this we’d need to solve |x| > 3 before we can graph it. Remember that the ACT loves to make seemingly easy problems more difficult by combining them with other topics (like absolute value). We have to remove the absolute value symbol by splitting the inequality in two, and we have to remember to flip the sign for the inequality that becomes negative.

x > 3  and x < – 3 become our answer to |x| > 3.

Remember that when we graph this solution set, the circles must be open because the symbols don’t have the line that indicates “or equal to”.

A few more formulas to know:

Slope = Rise / Run = Change in y / Change in x

As long as you know any two points on a line, you can find the slope. Remember that parallel lines have the same slope, and perpendicular lines have negative reciprocal slopes. You’ll also need to know how to recognize the graphs for linear and non-linear equations.

y = mx + b

This is called slope-intercept form. An equation in this form will always make a straight line on a graph (notice how neither x nor y have an exponent). In this form, b is the y-intercept (the point on the y-axis where the line crosses) and m is the slope.

y = ax^2 + bx + c

This is the standard equation for a parabola. In this equation c represents the y-intercept. A standard equation in which a variable is squared will never make a straight line.

The standard equation of a circle is (x – h)^2 + (y – k)^2 = r^2 where (h, k) is the center point of the circle and r is the radius. For example, on test day you might see a circle plotted on a graph, let’s say its center is (0,4) and the diameter is 8. All we’d have to do to find the equation of that circle is plug in r = 4 and (h,k) = (0,4) into our standard equation:

(x – h)^2 + (y – k)^2 = r^2
(x – 0)^2 + (y – 4)^2 = 4^2
(x)^2 + (y – 4)^2 = 16

Check out some awesome video explanations using problems with these formulas on Learnist!

Learnist: Point of View in ACT Science questions

The key to better scores on the ACT Science Test Conflicting Viewpoints passages is to hunt down each author’s point of view. As you read each passage, look closely for keywords that help you identify the author’s opinions.

If there are multiple paragraphs, remember that the scientist or student usually uses the first few sentences to introduce his topic and start a discussion of the main idea. The final paragraph wraps up the discussion and reinforces the Main Idea. If you are having trouble finding what the overall point of view is for the passage, go back to the very beginning and the very end.

Don’t feel like you need a big background in Science to get Point of View or other Science questions correct. These are very close to Reading Test questions! Use these three tips:

  • Don’t Be Confused by the Extraneous Information
  • Cut Through the Scientific Jargon
  • Never Leave an Answer Blank

Try a few practice questions on this ACT Science: Point of View learnboard!

ACT Science: How to Compare Two Experiments

One type of ACT Science passage, Research Summaries, will present descriptions of one or more related experiments and will require you to answer questions about one or both experiments. To compare them accurately, answer the following questions as you read. Make sure to underline, circle, or otherwise mark up the passages as you read!

1. How is each experiment set up? Make sure you understand the method for each experiment. What tools/processes/chemicals are used?

2. What does the data show? Pay close attention to the results of each experiment. Usually this is presented in tables or charts. How do the different variables relate to one another? Draw arrows on the tables to show the trends. Keep the following definitions in mind:

Independent variables: These are factors that are controlled by the scientists. Did the scientists increase the heat in the experiment? Did they add or remove pressure? If the scientists have control over the variable, it is independent.

Dependent variables: These are factors that the scientists observe changing. This is what the look for and how they record data — but they don’t control it.

Direct variation: This is when two things change in the same way over time. If Column A increases and Column B increases at the same time, we can say that the two vary directly.

Indirect variation: If when Column A increases, Column B decreases, there is an indirect (also called inverse) variation between the two elements.

3. How do the experiments differ? There will be certain elements common to both experiments, and one or more elements will change from Experiment 1 to Experiment 2. Circle the new information. Then focus on the results – do the variables interact similarly or differently in the 2nd experiment? Is the range of data greater or smaller?