Tip #50 to Get a Top ACT Math Score (page 3)
Learning mantras is like learning martial arts. Practice until they become part of you, until you follow them naturally: When you see a proportion, you cross–multiply; when you see a linear pair, you fill in the angles. … This will fundamentally change you as a math student. In fact, after ACT prep, many students begin to like math, they realize that they "get" it, and it stops being intimidating and becomes easy. I've even seen kids overcome serious math phobia with the mantras. Your ACT score and probably even your math class grades will go way up.
Here are the rest of the ACT math mantras. Check the box next to each one when you have mastered it.
- Skill 23. "Use the Diagram" to estimate an answer. When a diagram is not drawn to scale, redraw it. And when a picture is described but not shown, draw it! When estimating an answer, translate fractions, √, or π into decimals.
- Skill 24. When you see a right triangle, try a2 + b2 = c2.
- Skill 25. When you see a 30°, 45°, or 60° angle in a right triangle, try using the pecial right triangles.
- Skill 26. Similar triangles have sides that are proportional.
- Skill 27. Translate word problems from English to math.
- Skill 28. Word problems are no–problems; translate English to math, and translate fractions to decimals.
- Skill 29. When something can be factored, FOILed, reduced, or simplified, do it. When you have two equations in a question, ask how they relate. And convert fractions to decimals.
- Skill 30. Memorize the laws of exponents.
- Skill 32. For the equation y = mx + b, the m is the slope and b is the y intercept.
- Skill 33. When you see an arrangement question, draw a blank for each position, fill in the # of possibilities to fill each position, and multiply. When an arrangement question mentions a "team of two," or specifically points out repeats, divide your result by 2.
- Skill 34. SohCahToa!
- Skill 35. When trig seems tough, "Use the Answers" or "Make It Real."
- Skill 36. When you see the word "probability," use the equation Probability .
- Skill 37. For questions like (x + 4)(x – 3) = 0, just "Use the Answers" or set each parenthesis equal to zero and solve for x.
- Skill 38. For the equation y = ax2 + bx + c, the a tells whether the U shaped graph opens up or down, and the c is the y intercept. For the equation y = (x – h)2 + k, the h and k give the coordinates of the vertex of the graph (h, k). The vertex is the highest or lowest point of the graph and is therefore also called the maximum or minimum point.
- Skill 39. The equation for a circle is (x – h)2 + (y – k)2 = r2, where (h, k) is the center and r is the radius of the circle.
- Skill 40. If one side of a triangle is the diameter of a circle, and the opposite vertex is on the circle, then the triangle is right, with its right angle opposite the diameter.
- Skill 41. When you see absolute value on the ACT, "Use the Answers" or "Make It Real," and remember that absolute value means, "Ditch the negative sign."
- Skill 42. An arithmetic sequence is a sequence of numbers where a certain number is added to each term to arrive at the next, like 3, 7, 11, 15, 19; and a geometric sequence is a sequence of numbers where a certain number is multiplied by each term to arrive at the next, like 3, –6, 12, –24, 48.
- Skill 43. For a Fahrenheit/Celsius conversion question, when you are given degrees Celsius, just plug in and simplify; but when you are given degrees Fahrenheit, you can either do the algebra or "Use the Answers."
- Skill 44. "Careless errors are bad mmmkay," so underline all vocabulary words and remember to distribute the negative.
- Skill 47. A log is just a fancy way of writing exponents. For example, log5 25 = 2 means 52 = 25.
- Skill 48. The key to complex number questions is to treat i like a normal variable, and then in the final step, replace i2 with –1.
Let's look at this question:
Solution: This question would be ranked "medium" or even "hard," but if you just know what "median" means, it's totally easy! Median is the middle number in a list of numbers. To find the median, rewrite the list in order: 79, 80, 82, 85, 90, 92, 94. Then cross out a number on each side until you get to the one in the middle. (If there are two middle numbers, then the median is the average of the two.) Here the middle number is 85.
Correct answer: H
Name the skill(s) that you can use, and then solve each question.
- Which of the following is equivalent to 2(3x4)3 ?
- What is the length of the hypotenuse of a right triangle with legs that are 7 and 24 inches long, respectively?
- Kayla purchased a new phone at 31% off its original price. If she paid $34 for the phone, which of the following is an equation that could be used to find x, the original price of the phone?
- 34 = 1.31x
- 31 = x – 0.34x
- 31 = x – 0.66x
- 34 = x – 0.31x
- x = x – 0.31
- What are the values for x that satisfy the equation x2 – 5x – 14 = 0 ?
- –2 and 9
- –2 and –9
- 2 and –7
- –2 and 7
- In the standard (x, y) coordinate plane, the midpoint of is (–3, 5). If point M is located at (2, 3), what are the coordinates of point N ?
- (–8, 7)
- (7, 1)
- (–6, 15)
- (0, 0)
- For right triangle ΔABC, the sides measure 6, 8, and 10 inches. Which of the following equations could be used to solve for smallest angle, x, which is opposite the smallest side?
- sin x° = 0.6
- sin x° = 0.8
- tan x° = 0.8
- sin x° = 0.75
- tan x° = 0.6
- If m < 0, which of the following is greatest?
- 3 m
- 6 m
- 9 m
- It cannot be determined from the information given.
- If triangle MNO is similar to triangle PQR shown below, which of the following could be the measures of the three sides of triangle MNO ?
- If 4 times a number p is subtracted from 20, the result is negative. Which of the following gives the possible value(s) for p ?
- 0 only
- 5 only
- 20 only
- All p < 5
- All p > 5
- Which of the following could be the equation of the graph below?
- y = x – 1
- y = x2 –1
- y = (x – 1)2
- y = (x +1)2
- y = x2
- The inequality 3(3x – 2) < 6 – (5x – 2) is equivalent to
- x < 7
- x < 5
- x < 3
- x < 1
- x < –1
- formula for the surface area of a rectangular solid can be SA = 2lw + 2lh + 2wh where l represents the length of the base, w represents the width of the base, and h represents the height of the solid. By halving each of the dimensions (l, w, and h) of a certain geometric solid, the surface area will be multiplied by what number?
- The dartboard shown below has a small circle, with radius 3.5, inside a larger circle, with radius 5. When Matty randomly throws a dart at the board, which of the following is closest to the probability that it will land in the shaded region?
- If the width of a little league baseball must satisfy the inequality |w – 3| ≥ 0.12, what is the widest that a ball can be?
- E Exponents (Skill 30) and avoiding careless errors (Skill 44). 2(3x4)3 = 2(27x12) = 54x12. Careless Error Buster: Remember to apply the exponent to the 3 as well as the x4, and remember to multiply by 2 only after USING the exponent.
- J When you have a right triangle, try a2 + b2 = c2 (Skill 24). The legs are 7 and 24, so (7)2 + (24)2 = c2, and 49 + 576 = c2. So 625 = c2, and 25 = c. Remember to plug in the legs (the shorter sides) for a and b and the hypotenuse (the longest side, always opposite the right angle) for c.
- D Translation (Skill 27). $34 equals the original price minus 31% of the original price, so 34 = x – 0.31x.
- K Zero Times Anything Equals Zero (Skill 37). You can just "Use the Answers" on this type of question; try the choices and use the process of elimination, eliminating the ones that don't work. That's a quick, reliable, and easy way to get this one. Or you can factor x2 – 5x – 14 = 0 to (x – 7)(x + 2) - 0. Since zero times anything is zero, either x – 7 = 0 or x + 2 = 0. Solve each of these to get x = 7 or –2.
- A Midpoint formula (Skill 17). Midpoint is really just the average of the x's and the average of the y's. Just plug in what you know and solve for the variable:
- F Trig (Skill 34). When a picture is described, draw it. That always helps. Use SohCahToa; the smallest side is 6 and the hypotenuse is 10, so sin
- A Weird Number Behavior (Skill 46) and "Make It Real" (Skill 18). Without our Skills many kids would get this one wrong. I almost did, just now, as I was writing this solution. It's tempting to say, "Oh, easy, 9m must be the biggest." But remember our Weird Number Behavior Skill, the higher the digit of a negative number, the smaller the number actually is. Let's "Make It Real," and you'll see what I mean. Let's say m = –3. Now –3 is greater than 9(–3), which equals –27.
- G Similar triangles (Skill 26) and special right triangles (Skill 25). As soon as you see a right triangle with a 30 or 60 degree angle, consider special right triangles. A 30–60–90 triangle has sides that fit the pattern x, x , 2x. So since triangle PQR has a 60 degree angle and a side of 3, it must have sides of 3, 3 6. And triangle MNO, since it is similar, will have sides that are proportional to 3, 3, 6. Choice G works since 6, 6 and 12 is proportional (double) to 3, 3, 6
- E Translation (Skill 27). The question translates to: 20 – 4p = a negative. You could also say 20 – 4p < 0. Solve this to get
- G y = x2 + bx + c (Skill 38). For the equation y = x2 + bx + c, the sign in front of the x2 term tells us if the U– shaped graph opens up or down, and the c term tells us the y intercept of the graph. The graph shown in the question opens up, so it should not have a negative in front of the x2; and it has a negative y intercept, so c should be negative. It also must have an x2 since it is a parabola (a U–shaped graph). That eliminates choices F, J, and K. Choices G and H are left. Choice G is correct since the –1 outside the parenthesis represents the y intercept, whereas the –1 inside the parenthesis relates to a shift of the x values, not y values. If that last part is confusing, you could also just graph the two choices on your calculator and see which one matches the diagram shown in the question.
- D Algebra (Skill 2) and avoid careless errors (Skill 44). Simplify the inequality:
- G Make It Real" (Skill 18) and area/volume (Skill 19). Let's say l = 4, h = 6, and w = 8; then the surface area of the solid is SA = 2lw + 2lh + 2wh = 2(4)(8) + 2(4)(6) + 2(8)(6) = 64 + 48 + 96 = 208. When we halve each dimension, we get l = 2, h = 3, and w = 4, so the new surface area is SA = 2lw + 2lh + 2wh = 2(2)(4) + 2(2)(3) + 2(4)(3) = 16 + 12 + 24 = 52. So the new surface area is onequarter the original; the original times 0.25 equals the new surface area. Without "Make It Real" many people would incorrectly pick choice J, since we halved the dimensions.
- C Probability (Skill 36), donut area (Skill 20), or use the diagram (Skill 23). Probability = , so the probability of hitting the shaded region is
- J Absolute value (Skill 41) and "Use the Answers" (Skill 1). You could solve the inequality statement or just try the choices and see which is the greatest one that works. Choices F, G, H and J work, but choice J is the largest one that works.
In other words, . When you see a proportion, cross–multiply, and solve to get x = –8 and y = 7. (–8, 7). You could also get this question, of course, by "Using the Answers," just trying the choices to see which one works.
Remember to flip the > sign when you divide by a negative! As is so often the case, you could also just "Use the Answers" to find which one works. Choices A, B, and D do not work. Choice C works, but 20 is not the only solution; when you test choice E, with any number more than 5, it also works, so choice E is correct.
3(3x – 2) < 6 – (5x – 2)
9x – 6 < 6 – 5x + 2
14x < 14
x < 1
Careless Error Buster: Remember to distribute the negative sign.
Once again, you could also just "Use the Answers" and see which choice works.
You could also use the diagram and realize that the shaded region being half the area of the larger circle is the only answer choice that makes sense with the diagram.
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