New ACT Math Test Content Part 1: The Factorial
Combinatorics has been added to the ACT math test—and 36U is here to get you ready!
Don’t worry. Combinatorics may sound intimidating, but it’s really about fancy, advanced counting techniques that can be a lot of fun and save lots of time. Let’s get right to it.
Factorials
First, let’s work on a possibly new concept with some new notation—the factorial.
6 factorial, written 6!, means 6 • 5 • 4 • 3 • 2 • 1.
And, again:
5! = 5 • 4 • 3 • 2 • 1
To simplify, 5! = 5 • 4 • 3 • 2 • 1 = 120.
Let’s put the factorial to work…
Example 1: In how many different ways can you arrange the letters of the word OVERCAST?
____ ____ ____ ____ ____ ____ ____ ____
You have 8 options for the first letter, 7 options for the 2nd, 6 options for the 3rd, and so on.
Mathematically, that looks like this:
_8 options •_7 options •_6 options •_5 options •_4 options •_3 options •_2 options •_1 option
or
8! –> 40,320
There are 40,320 different ways the letters of OVERCAST can be arranged.
Easy so far. Let’s move on…
Example 2: In how many different ways can you arrange the letters of the word CANYON?
This is a slightly trickier item because CANYON has 2 N’s! Fortunately, if you understood our first example, this isn’t much more difficult.
Step 1: There are six different spots to place the letters in the word CANYON:
_6 options •_5 options •_4 options •_3 options •_2 options •_1 option = 6!
There are 6! or 720 different ways of arranging C-A-N-Y-O-N, but…You have to account for repeated options because there are 2 Ns. Here’s how:
Take your total number of possible arrangements (6!) and divide by 2! to account for N appearing twice.
6!/2! –> (6 • 5 • 4 • 3 • 2 • 1)/(2 • 1)
–> 360
There are 360 different ways the letters of the word CANYON can be arranged!
That’s your introduction to new counting techniques (combinatorics) that are being tested on the ACT.
Take time to brush up on your combination and permutations, too. For more instruction and practice on these topics, check out our online program.
-Dr. Kendal Shipley, 36U
10/13/17
THE Problem I Missed on the ACT Math Test
The original title of this post was supposed to be “The Math Problem I Had to Skip and Come Back To,” but it didn’t work out that way. I formulated the first title after I had to skip THE problem the first time.
I ended up missing the problem!
Recently, ACT Inc put out a new “Official ACT Prep Guide” and, as I always do, I sat down to take the tests (timed, of course) and to categorize the new items. This time, I set the timer and began working.
I was halfway through at the 19-minute mark and had no troubles. I noticed the problems were taking me a little more time than I wanted, but I only had a phone calculator and was only willing to use a small notepad to write something down if absolutely necessary.
No problems so far.
And then I hit #51:
(This item is owned by ACT Inc and printed in the first practice test in The Official ACT Practice Guide 2016-2017.)
I made a mental list of all possible perfect square sums from students 1 through 18: 4, 9, 16, 25. I started working and realized I couldn’t see the “quick way.”
Yes, almost all ACT Math items have a “quick way.” So, contrary to what I think most students should do at this point (which is to give it their best guess and move on), I skipped it.
I made my way through the rest of the test. The dumb “how many quarters are there?” problem. The Venn diagram problem. The Law of Sines. A probability distribution (That’s new!). And I headed back to #51…
I had about 4 minutes left at this point and decided to continue with the the brute force approach (another critical error?).
I eliminated choices A and C, because 1 + 16 and 1 + 9 don’t give perfect squares.
A little more deduction told me 17 had to be be paired with 8 to give the perfect square 25. Student 17 had to be paired with another student to give a perfect square. There wasn’t a student who could pair with 17 to reach the perfect square 36 (17 + 18 = 35). There was no way to pair student 17 with another student and get a number less than 17, so the only option was to add to 25. That made student 17 pair with student 8.
I was left with choices B and E.
I knew that 1 had to be paired with 3 or 15 (I had known that since the first pass through.). I scribbled. I paired. I scribbled. I paired.
And I still didn’t get it. I was running out of time! I chose…3.
Wrong choice!
What Did I Do Wrong?
I’m not really sure where I went wrong. Is there a key that unlocks this problem quickly? I don’t know.
Eventually, I did figure that 1 had to be paired with 15, and this is how I did it…
Step 1: Start with the big numbers.
Pair student 18 with student 7, which is the only option for reaching a perfect square. Likewise, student 17 has to be paired with student 8. Student 16 has to be paired with student 9. Of the nine pairs, we have 3 pairs figured quickly!
Student 15 can be paired with student 10 (sum of 25) or student 1 (sum of 16).
Student 14 can be paired with student 11 (sum of 25) or student 2 (sum of 16).
Student 13 can be paired with student 12 (sum of 25) or student 3 (sum of 16).
Student 12 can be paired with student 13 (sum of 25) or student 4 (sum of 16).
Student 11 can be paired with student 14 (sum of 25) or student 5 (sum of 16).
Student 10 can be paired with student 15 (sum of 25) or student 6 (sum of 16).
Student 9 is already paired with student 16 (see above).
Student 8 is already paired with student 17 (see above).
Student 7 is already paired with student 18 (see above).
Student 6 can be paired with student 10 (sum of 16) or student 3 (sum of 9).
Student 5 can be paired with student 11 (sum of 16) or student 4 (sum of 9).
Student 4 can be paired with student 12 (sum of 16) or student 5 (sum of 9).
Student 3 can be paired with student 13 (sum of 16) or student 6 (sum of 9) or student 1 (sum of 4).
Student 2 can be paired with student 14 (sum of 16), not student 7 (already paired with student 18), not student 2, because he is student 2. We have our fourth pair! Student 2 is paired with student 14!
Student 1 can be paired with student 15 (sum of 16) not student 8 (already paired with student 17) or student 3 (sum of 4).
Step 2: Student 2 is the key.
Student 2 is paired with student 14. From there, work backwards from student to student eliminating possible choices until student 1’s pair is identified.
Go to the Student 14 line. Student 14 is paired with student 2, so that leaves student 11.
Student 11 can’t be paired with student 14. Student 11 is paired with student 5.
Student 5 is paired with student 11, so that leaves student 4.
Student 4 must be paired with student 12.
Student 12 is paired with 4. That leaves student 13.
Student 13 is paired with student 3. The leaves students 1 and 6.
Student 6 is paired with student 10, because student 3 is paired with student 3.
Student 10 is paired with student 6. That leaves student 15.
Student 15 must be paired with student 1!!!
Is there a better way?
For a 60 second per item pace, ACT is asking a lot in this problem.
Do you see a more efficient way to work this problem? If so, tell me how in the comments below.
Kendal Shipley, Ed.D.
8/29/2017
36U Blog Posts
The Six Trickiest Math Problems We Found on the Most Recent ACT
If you learned what your teachers taught in your math classes, then you should do great on the ACT math test. However, like most standardized tests, the ACT will throw a few items your way that look a little different than the items your teacher assigned.
So, here are the tricky items we found on the latest publicly-released ACT. There are six of them. Have questions or comments? Let us know! Please note: these aren’t necessarily the most difficult items on the math test, though some are, but they are the ones that we consider the slyest.
All items shown below are from the 2016-2017 Preparing-for-the-ACT Guide. They are the property of ACT Inc., not 36 University.
Tricky ACT Math Item #6
What’s Tricky About It?
This item is asking for an application of an exponents rule, but in a way that isn’t usually stressed in math classrooms. Usually, students are taught to distribute exponents to the bases:
(ab)x –> axbx
But this problem asks students to apply the same concept, but in reverse.
How You Should Work It
The problem asks for x•y.
Step 1) Use the information from the problem (x = ab and y =cb) to substitute for x and y.
x•y –> abcb
Step 2) Apply the exponent rule (ab)x –> axbx in reverse. In other words, both bases have the same exponent, so the exponent can be placed outside parentheses:
abcb –> (ac)b
The answer is H.
Tricky ACT Math Item #5
What’s Tricky About It?
Monthly payment. P dollars. Short-term loan. Annual interest rate. This item is wordy, and the vocabulary is a little tough. On top of that, that equation is intimidating.
On top of that, many students won’t know how to handle multiplying a by 2.
How You Should Work It
Step 1) Multiply a by 2.
Here’s the new value for p:
Step 2) Factor out the 2.
Don’t make this too complicated: use the distributive property in reverse.
and because multiplying the numerator by 2 is the same as multiplying the fraction by 2:
Step 3) Compare to the previous value for p.
In the problem, the value for p was:
When a is multiplied by 2, p is also multiplied by 2. The answer is D.
Tricky ACT Math Item #4
What’s Tricky About It?
Reasoning through this item, especially with its fractions, is very difficult. Only the top students may find that a navigable path.
How You Should Work It
The easiest way to work the item is to set up an equation that mirrors the situation.
Let x = volume of the container in cups
Step 1) Set up the equation.
The problem says they took 1/8 of the container, added 10 cups, and ended with the container 3/4 full. Or, maybe this is simpler: One-eighth of the container plus 10 cups equals three-fourths of the container.
As an equation, that description looks like this:
Step 2) Solve the equation for x, the volume of the container.
Subtract (1/8)*x from both sides.
Multiply both sides of the equation by 8/5.
x = 16 cups
The answer is J.
It’s also a good practice to check your solution with the information given in the problem. Have fun!
Tricky ACT Math Item #3
What’s Tricky About It?
Students are often more comfortable working from point A to point B along a path they’ve trod several times before. This item invites students to reason to try to find the relationship between x and z, but that reasoning is difficult.
How You Should Work It
There is a variable common in both ratios—y. Rewrite both ratios so that y has the same value in both.
Step 1) Rewrite both ratios.
The variable y corresponds with the 2 in the first ratio and a 3 in the second ratio. Rewrite both ratios so that y has a value of 6 (least common multiple).
and
Step 2) Make the comparison between x and z.
Rewriting both ratios as fractions with a y value of 6 allows us to compare x and z.
If the ratio of x to y is 15 to 6 and the ratio of y to z is 6 to 4, then the ratio of x to z is 15 to 4.
The answer is E.
Tricky ACT Math Item #2
What’s Tricky About It?
This item, like a few others in this list, is likely to have students spending an inordinate amount of time with a trial-and-error method trying reason their way to the answer. That is a tough road!
How You Should Work It
Students need a Venn Diagram in their math tool chest. Unfortunately, many of them may not have used a Venn Diagram since the 8th grade.
Set up a Venn Diagram for all 120 students.
Step 1) Survey question one tells us that 55 students have neither skied cross-country or downhill.
Step 2) Use survey questions two and three to find the number of students who have skied both cross-country and downhill.
Survey question one states that 65 students have either skied cross-country or downhill. Survey question two states that 28 have skied downhill and 45 have skied cross-country. That’s 73 total, 8 more than the 65 from Q1. That means some of them have done both!
Take the surplus of 8 and place them in the overlap for students who have skied both downhill and country-country.
Step 3) Use survey questions 2 and 3 to finish out the Venn Diagram.
Twenty-eight students have skied downhill.
Forty-five students have skied cross-country.
Step 4) Check your Venn Diagram with the original information.
After your check, you’ll see that indeed 8 of the students had skied both cross-country and downhill.
The answer is E.
Tricky ACT Math Item #1
What’s Tricky About It?
There are two difficulties here:
1) Rate problems are standard fare in algebra class, but this item isn’t worked like most of those textbook problems are worked.
2) The wording is strange. Who ever asks about how many cans of food a dog eats in “’3 + d days?’”
How You Should Work It
Step 1) Recognize the rate at which the dog is eating.
Seven cans in three days means the dog is eating 7/3 of a can per day. This means that you can take the rate (7/3 of can per day) and multiply by the number of days to get the total amount of dog food consumed.
Try this: After 3 days the dog has eaten (7/3) * 3 = 7 cans of food, just like the problem stated.
Step 2) Substitute to find the solution.
How many cans does the dog eat in 3 + d days? Break this into two separate parts. In 3 days, the dog eats 7 cans, just like the problem told us. In d days, the dog eats (7/3)*d cans, just like we figured in Step 1. This means the dog eats 7 + (7/3)*d cans.
The answer is K.
Solution Page
The Locker Problem
We want thank all who attempted the January 2017 Math Challenge. We hope you enjoyed and learned from the problem.
For those who missed it, here was the challenge problem:
Imagine 100 lockers numbered 1 to 100 with 100 students lined up in front of those 100 lockers:
The first student opens every locker.
The second student closes every 2nd locker.
The 3rd student changes every 3rd locker; if it’s closed, she opens it; if it’s open, she closes it.
The 4th student changes every fourth locker.
The 5th student changes every 5th locker.
That same pattern continues for all 100 students.
Here’s the question: “Which lockers are left open after all 100 students have walked the row of lockers?”
The Solution
As many of you found, the perfect square lockers (#s 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100) are the only lockers left open. Cool, huh?
We hope you realized that lockers are only touched by students who are factors of that locker number, i.e. locker #5 is only touched by students 1 and 5. Student 1 opens it and student 5 closes it. In fact, because factors come in pairs, the first student factor will open it and the corresponding factor student closes it. At first, maybe you thought every locker would be closed because factors come in pairs. But there was a twist…
Here are a couple of ways you could have gotten there:
Method 1: Solve a Simpler Problem
Start with just 20 lockers and try to find a pattern.
We used a code: O = Open, C = Closed.
Here’s what the lockers look like after the first student walks through. They are all open.
After the second student walks the row of lockers, the odd-numbered lockers are left open and the even-numbered lockers are closed:
Here’s how the lockers look after the third student changes every 3rd locker:
And here’s how it looks after the first 20 students have walked the row of lockers. Note: after student 20 has gone, the first 20 lockers aren’t touched again.
Of the first 20 lockers, locker #s 1, 4, 9, and 16 are left open. Those are perfect squares. You can extend that pattern to identify the remaining open lockers.
Lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are left open!
Method 2: Who Touches Which Lockers
Identifying which students touch which lockers is a little less of a brute-force approach and would likely have gotten you to the solution a little more quickly.
Here’s what I mean:
Consider locker #1. The only student who touches locker #1 is student #1. Student 1 opens the locker, and since no one else touches it, it will be left open at the end.
Consider locker #2. Student 1 opens the locker, and student 2 closes it. No one else touches the locker, so it will be closed.
Consider locker #10. Students 1 opens the locker. Student 2 closes it. Students 3 and 4 skip right by it. Student 5 opens it. Students 6, 7, 8, and 9 skip right by it. And student 10 closes it. Locker #10 will be closed.
Mental Milestone 1: After looking at several lockers, you should notice that lockers are only changed by student numbers that are factors of the locker number. In other words, locker 12 is changed by students 1, 2, 3, 4, 6, and 12.
Mental Milestone 2: You should also have noticed that factors always come in pairs. This means that for every student who opens a locker, there is another student who closes it. For locker #12, student 1 opens it, but student 12 closes it later. Student 2 opens it, but student 6 closes it later. Student 3 opens it, but student 4 closes it later.
By this logic, every locker would be closed.
But there are exceptions!
Consider locker #25. Student 1 opens it. Student 5 closes it. Student 25 opens it. The locker will be left open, but why? In this case, the factors do not come in pairs. One and 25 are a pair, but five times five is also 25. Five only counts as one factor. This causes the open-close pattern to be thrown off. Locker #25 is left open.
Mental Milestone 3: When factors don’t come in pairs, the locker will be left open. And factors don’t come in pairs when numbers are multiplied by themselves. Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100,…) are the only numbers whose factors don’t come in pairs because one set of factors, the square root, is multiplied by itself. This means that only perfect square lockers will be left open.
Locker #s 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are left open!
The Winner
Congratulations to Sydney H, our January 2017 ACT Challenge winner! Her solution was simple, precise, and just as important, correct. In fact, I like her solution better than the explanations I provided above. Here’s what she had to say:
The Winnings!
Sydney won the 36U Winter Care Package:
3 months access to 36U ACT Prep
36U Tee (Long-sleeve)
36U toboggan
$15 Starbucks gift card
More 36U Resources:
36U Blog Posts
36U ACT Tips
36U ACT Prep Program
The Locker Problem – January 2017 Challenge
Number properties are rarely reviewed, but they are sometimes tested on the ACT. So, we’ve decided to share a fun, straightforward, and hopefully enlightening item that will have you thinking about number properties. Here’s your January 2017 ACT Math Challenge:
Imagine 100 lockers numbered 1 to 100 with 100 students lined up in front of those 100 lockers:
The first student opens every locker.
The second student closes every 2nd locker.
The 3rd student changes every 3rd locker; if it’s closed, she opens it; if it’s open, she closes it.
The 4th student changes every fourth locker.
The 5th student changes every 5th locker.
That same pattern continues for all 100 students.
Here’s the question: “Which lockers are left open after all 100 students have walked the row of lockers?”
How Do I Enter?
Take a pic of your solution, with your work included, and post as a reply to any of our Locker Problem social media posts or send to lockerchallenge@36university.com. Submissions must be posted by midnight eastern time on January 31st. Impress us with your approach to solving this problem!
What Will I Win?
The winner will receive the 36U Winter Care Package:
3 months access to 36U ACT Prep
36U Tee (Long-sleeve)
36U toboggan
$15 Starbucks gift card
How Will 36U Choose a Winner?
At 36U, we value simple, precise solutions. We will draw a winner on February 1st from among entries with correct answers and easy-to-understand explanations. (or: whose solutions are correct and whose approach is easily understood.)
More 36U Resources:
36U Blog Posts
36U ACT Tips
36U ACT Prep Program
Side-by-Side Comparison of the Updated ACT and the New SAT
Wow! The SAT has changed. Instead of 10 smaller sections, now there are 4 bigger tests, just like the ACT. The SAT no longer penalizes for guessing. And the essay is now optional. It has even begun incorporating science items in the English and reading tests. To top it off, question formatting is very similar, too. Let’s get right to it…
(This post contains screenshots of items released by ACT Inc. and the CollegeBoard (makers of the SAT). The items are not the property of 36 University.)
Test Order and Structure: ACT vs SAT
Both tests have 4 main tests and an optional essay.
|
ACT |
SAT |
| Section 1 |
English (75 questions in 45 min) |
Reading (52 questions in 65 min) |
| Section 2 |
Math (60 questions in 60 min) |
Writing/Language (44 questions in 35 min) |
| Section 3 |
Reading (40 questions in 35 min) |
Math (20 questions in 25 min). Last 5 are grid-in. |
| Section 4 |
Science (40 questions in 35 min) |
Math (38 questions in 55 min). Last 8 are grid-in. |
| Section 5 (Optional) |
Writing (Optional Essay in 40 min) |
Essay (Optional in 50 minutes) |
| Total Time |
2 hrs 55 min or 3 hrs 35 min w/ Essay |
3 hrs or 3 hrs 50 min w/ Essay |
Scoring
|
ACT |
SAT |
| Max Score |
36 |
1600 |
| Details |
Each test has max score of 36.
Composite is average of four scores. |
SAT Reading max score: 800
SAT Reading derived from Reading & Writing/Language.
SAT Math max score: 800
SAT Math score derived from 2 math sections.
Composite is sum of Reading and Math scores. |
There is no penalty for wrong answers on either test.
Writing scores are kept separate from the other scores on both tests.
Subject Comparisons
ACT English vs SAT Writing/Language
The English tests on both the ACT and SAT have a very similar structure. The SAT English incorporates science graphs occasionally and seems to require a slightly more extensive vocabulary.
ACT Math vs SAT Math
|
ACT |
SAT |
| Setup |
60 questions in 60 minutes |
Part 1 (no calculator): 20 questions in 20 minutes (15 MC, 15 Free Response)
Part 2 (calculator): 38 questions in 55 minutes (30 MC, 8 Free Response) |
| Pace |
60 seconds per item |
Part 1: 60 seconds per item
Part 2: ~87 seconds per item |
| Question Format |
ACT Math has 5 answer choices. |
SAT Math has 4 answer choices on multiple choice questions.
SAT Math also has some Grid-In answer items. |
| Basic Items |
 |
 |
| Same Writers? |
 |
 |
| Grid-Ins |
Not applicable. |
***Students grid-in answers on answer document. |
| Notes |
Calculator is allowed on all items. |
Calculator is allowed on Part 2, not Part 1. |
The math tests are not as similar as the English tests.
ACT Reading vs SAT Reading
|
ACT |
SAT |
| Setup |
40 questions divided over 4 passages
35 minutes |
52 questions divided over 5 passages
65 minutes |
| Pace |
52.5 seconds per item |
75 seconds per item |
| Passages |
4 passages:
Prose Fiction
Social Science
Humanities
Natural Science |
5 passages:
U.S. or World Lit
U.S. Constitution or Global Conversation
Social Science
2 Science passages |
| Comparison Items #1 |
 |
 |
| Comparison Items #2 |
 |
 |
| Notes |
|
The SAT has begun adding items that require
students to locate justification in the text.
|
| Notes |
|
SAT Reading has science graphs included
in the 2 science passages.
|
As with the English tests, the Reading tests are very similar. On the SAT English tests, students will work with 2 science passages. On the ACT, the science is in a section on its own.
ACT Science in the SAT Writing and Reading
|
ACT |
SAT |
| Setup |
40 questions divided over 6 or 7 scenarios
35 minutes |
N/A |
| Pace |
52.5 seconds per item |
N/A |
| Question Format |
The ACT Science test requires students to interpret
information in tables and graphs.
|
The tables and graphs are incorporated in the
SAT English and Reading tests.
|
| Basic Items |
Questions require students to evaluate
experiments and research.
|
The tables and graphs are incorporated in the
SAT English and Reading tests. |
ACT Writing vs SAT Essay
|
ACT |
SAT |
| Setup |
1 essay in 40 minutes |
1 essay in 50 minutes |
| Format |
1 Central Question.
3 Perspectives Provided.
Students are asked to:
-state and develop their own position.
-analyze and evaluate the 3 perspectives
in light of their stated position. |
Students are given a relatively lengthy
passage to evaluate. Instead of stating
their own position, they are asked to dissect,
analyze, and evaluate the passage. |
| Examples |
You can find an example of an ACT Writing
test prompt provided by ACT Inc here:
Sample Prompt |
The Collegeboard has two sample prompts
available on its website:
Prompt 1
Prompt 2 |
Final Word
I don’t consider one test more difficult than the other, and the new SAT is new enough that I haven’t completed enough practice tests to make a complete assessment. At this point, maybe the most obvious difference between the tests is that 1/4 of the ACT score is derived from math items and 1/2 of an SAT score comes from math items.
Kendal Shipley, Ed.D.
6/3/2016
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The Story of Lil Carl and the Arithmetic Series
Lil Carl bounced into kindergarten on a warm spring morning– energetic, but bored. Not one thing Ms. Artwaller had thrown at him that year had come close to testing his abilities. Exasperated Ms. Artwaller had had enough of Lil Carl disrupting class and decided to put an end to his shenanigans. “Find the sum of the first 100 whole numbers,” she said. At long last, Ms. Artwaller was finally able to teach the class without Lil Carl doing Lil Carl things…
After just a few moments Lil Carl was back: “5050. You did want me to add whole numbers 1 to 100, right?”
(This story is based on a legendary tale of one of the greatest mathematicians of the 18th and 19th centuries, Carl Friedrich Gauss. Read more here.)
How Did He Do That?
Even though the numbers are sequentially 1, 2, 3, and so on, you don’t have to add them in that order. In fact, the addition is much quicker if you change the arrangement. Add 1 and 100; add 2 and 99; continue with that pattern.
Step 1: Wrap the second half of the series below the first half, and then add.
Now, we have a bunch of 101s. In fact, we have 50 of them. So, instead of adding 100 terms, we can now multiply!
Step 2: Multiply!
We have fifty 101s now, so multiply:
50 • 101 –> 5050
The sum of 1 + 2 + 3 + … + 100 is 5050. We folded the series in half, so that we added the first half of the series with the second half of the series. This eventually allowed us to multiply to get the sum of the series.
What about Other Arithmetic Series?
Suppose we wanted to add 5 + 12 + 19 + … + 75. (Hint: The difference between consecutive terms is 7.)
Step 1: Fold the series in half.
Step 2: Find the number of terms after the series is folded in half.
a. From the first term of 5 to the last term of 75 is a jump of 70.
b. The jump between each term is 7, so there are 10 total jumps.
c. There are 11 terms in the original series of 5 + 12 + 19 + … + 75. (10 jumps and the first term)
d. After the series is folded in half, there will be five full pairs and an odd term in the middle without a pair. We will call that 5 ½ pairs.
Step 3: Multiply!
The pairs add to 80. There are 5.5 pairs.
80 • 5.5 –> 440
Arithmetic Series Formula
Arithmetic series can always be added with the method shared above. If had rather use a formula, however, here it is:
where n = # of terms in the series, t1 is the first term in the series, and tn is the last term in the series.
Please note: The formula mirrors the method used above. The first and last terms are added (t1 + tn) and then multiplied by half of the number of terms in the series (n/2).
How Will This Look on the ACT?
Try this example from the 2014-2015 Preparing-for-the-ACT Guide (property of ACT Inc). If needed, start by writing out the terms in the sequence.
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#1 Skill Needed to Score Well on the ACT May Be Reading Comprehension
Often, I hear from parents of students who have recently received ACT scores between 13 and 17. They are concerned that the low ACT score is going to limit their student’s college options and don’t know what skills to begin addressing.
Standardized tests don’t always accurately reflect what a student really knows or his capabilities, and sometimes students just have an off day on the day of the test.
However, more often than not, low ACT scores are symptomatic of one prevailing problem: poor reading comprehension. A student who has weak reasoning and math skills may score poorly on the math and science tests. A student who doesn’t understand the basics of grammar and punctuation will likely score poorly in English. But the student who doesn’t read well will be penalized on all four subject tests. As strange as it may sound, many times low ACT scores can be traced back to a student’s ability to read and comprehend.
(All of the released ACT items below are from 2015-2016 Preparing-for-the-ACT Guide and are the property of ACT Inc., not 36 University.)
The English Test and Reading Comprehension
Examine the two items below. By my count, they are representative of 24 of the 75 English items from the 2015-2016 Guide, and they require strong reading comprehension skills! As you can see, a struggling reader will struggle with the English test.
(If you missed our recent post The Juggling Act Required by the ACT, click here.)
The Math Test and Reading Comprehension
Maybe it’s not surprising that the English test requires strong reading skills, but the math test is text-heavy also. In fact, 39 of the 60 items from the 2015-2016 Preparing-for-the-ACT Guide required students to read at least 3 lines of text. Not all math items are like the three shown below, but many of them are.
The Reading Test and Reading Comprehension
Of course, the reading test is all about reading comprehension, and it’s one-fourth of an ACT score! Sure, a fair number of the 40 items ask students to interpret meaning from what is explicitly stated in the text. However, a fair number also ask students to reason implicitly to determine inferences, main idea, compare and contrast, etc. It comes as no surprise that strong reading skills are required to score well on the reading test.
The Science Test and Reading Comprehension
To understand the importance of reading skills on the science test, you must first understand the test’s format. The science test consists of six or seven separate scenarios. One of those scenarios is comprised of only text, usually four or five paragraph’s worth. It is like a mini version of a reading scenario. All of the other scenarios require a significant amount of reading, too. In the 2015-2016 Preparing-for-the-ACT Guide, the other five passages accounted for another 19 paragraphs, almost 4 paragraphs per scenario.
To give you an idea of a typical ACT science passage, we have provided a sample passage from the same ACT publication we’ve used for this entire post:
Summary
As you can see, reading comprehension skills are needed on every section of the ACT. Scores in the 13 – 17 range are often a result of insufficient reading skills and not from a lack of content knowledge.
What Can I Do?
Be intentional! In your casual reading—through a news article, or your favorite blog, or that new novel—stop occasionally and ask yourself some very specific questions:
“What was the main idea of that paragraph?” (Try to summarize in 5 words or less.)
“Were the paragraph’s sentences presented in a logical sequence?”
“What was the main idea of the passage?”
“Were the paragraphs within the passage placed in a logical sequence?”
Intentionally evaluating and critiquing your reading material will make you a better reader and writer – and an improved ACT score will be a by-product of your new skills!
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Three Types of Probability Items to Expect on the ACT Math Test
You walk into the testing site and sit down, nervous but confident. You know you’ve prepared and you’re ready to give the ACT one last shot. You are sure this will result in your best your score yet. The English test is no problem. You open the math test and begin working, but something has changed…
Modifications to the ACT Math Test
Maybe you haven’t heard, but ACT announced test-takers could look forward to “the inclusion of additional statistics and probability items in the mathematics test” (Source: ACT FAQ). ACT hasn’t released volumes of materials on the changes, but the last two released exams contained in the Preparing-for-the-ACT Guides are consistent with the announced changes and provide the best clues we have on the amendments. We have divided the following items into three categories. The items are provided to give you examples of the ways you can expect ACT to test probability. These are screenshots from the ACT’s booklets, not the property of 36 University.
Item Type 1: Basic Probability Items
These items require you to apply the basic probability formula:
P(A) = (# of events corresponding to A/ total # of possible events)
Item Type 2: Probability from Graphs
These items require you to utilize your graph reading skills and basic probability concepts.
Item Type 3: Probability of Multiple Events
Multiple events probability items ask you take one additional step. Often, that means you’ll just need to multiply the probabilities of each event. Here are examples:
Wrapping It Up
The good news is these items aren’t that complicated and are likely to replace more difficult items. These changes may mean it’s time for you to brush up on basic probability, especially since ACT may begin testing these concepts in more challenging ways.
Do you want to see these items worked? Check out our follow-up post. For more practice, check out the online Sample Items published by ACT. Specifically, try Set 3 Item #2.
Recent & Related Posts:
Post Follow-up: Working the New ACT Math Probability Items
Origami: Sharpen Visual-Spatial Skills & Boost Your ACT Math Score
Your Week-Before-the-ACT Game Plan (Part 4 of 4: Science)
Your Week-Before-the-ACT Game Plan (Part 3 of 4: Reading)
Your Week-Before-the-ACT Game Plan (Part 2 of 4: Math)
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36U Blog List
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Origami: Sharpen Visual-Spatial Skills & Boost Your ACT Math Score
Learning and test scores shouldn’t always be about test prep. It’s more fun when a good test score is a natural result of time spent learning…
For as long as I can remember, I’ve been fascinated by turning sheets of paper into dragons, frogs, and flowers. Fortunately, access to origami instruction has changed drastically since the days I used to forage through book after book of origami illustrations in the school library. Oh, the convenience of the Internet!
Origami and Visual-Spatial Skills
Engineers, architects, software user experience designers, and myriad other professionals excel because of their visual-spatial skills. Paper-folding may seem like a strange way to develop skills for such high-tech professions. When working with origami, you’ll practice some of these skills:
- Diagram Interpretation
- 2-d and 3-d Rotation Visualization
- Design Manipulation
How Can I Get Started?
Start by picking a simple design and trying to fold it. Peruse a few websites (Option 1, Option 2, Option 3) until a design catches your eye. Along the way, make sure to digest key terms like mountain fold, valley fold, and reverse fold. Become acquainted with the concept of building a base.
From there, you can progress toward more difficult designs. Instructions for creating the dolphin from the video above can be found here.
Can Working with Origami Help My ACT Score?
Visual-Spatial skills such as mental rotation, spatial perception, and spatial visualization have been linked to higher ACT Math scores (Sorby and Baartman, 2000). You’ll see those same skills tested in these recently released ACT Math items:
Reflection over the y-axis?
Lines of Symmetry?
Interpreting a diagram:
Ratio of areas?
Try it. Share it.
I get a real kick out of completing a new design. I hope you do, too. If you’re intrigued by origami, try it and share your creation on our Facebook page. You’ll be having fun and practicing critical skills!
-Kendal Shipley, Ed.D.
Recent Posts:
Your Week-Before-the-ACT Game Plan (Part 4 of 4: Science)
Your Week-Before-the-ACT Game Plan (Part 3 of 4: Reading)
Your Week-Before-the-ACT Game Plan (Part 2 of 4: Math)
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36U Blog List
