August 2014 Geometry Regents, Part 1 (Multiple Choice)

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I just spent a few hours grading the Geometry Regents exams at my old school. It was described as "doable". I think that the students who came to my tutoring classes the past few days will be happy ... assuming that they were all working out their own problems and not just copying my answers.

A few observations: the proof was pretty straightforward, with a choice of AAS or ASA, as long as you backed up the one you used; the locus question was very unusual in that you had to give an equation of the circle locus and the line was neither vertical nor horizontal; the construction question had you bisecting multiple times (and using your ruler and drawing the curves later didn't count). These issues with the Open-Ended problems with be discussed tomorrow (I hope).

Here are the multiple-choice questions from Part I. Keep in mind, that I have to type all of these, so the rest of the test may not show up on my blog as quickly as you may like. Questions are always welcome. Likewise, because I've been asked to hurry with this, there are no diagrams included. They may get added at a later time.

1. A rectangular prism is shown in the diagram below. [DIAGRAM IS BOX-SHAPED WITH DIAGONALS ON TOP AND BOTTOM DRAWN.]
Which pair of line segments would always be both congruent and parallel?

(4) DB and HF are parallel. Of the other three choices, two pair intersect and one pair is skew.

2. In parallelogram QRST, diagonal QS is drawn. Which statement must always be true?

(3) Triangle STQ is congruent to QRS. The diagonal of a parallel creates two congruent triangles by SSS. (Proof left as an exercise to the reader.)

3. In the diagram below of circle O, diameter AB and chord CD intersect at E. [DIAGRAM IS A CIRCLE AS DESCRIBED.]
If AB is perpendicular to CD, which statement is always true?

(4) Arc CB is congruent to arc BD. (Also arcs AC and AD are congruent, in case you were wondering.)

4. What is an equation of the line that passes through (-9, 12) and is perpendicular to the line whose equation is y = 1/3 x + 6?

(2) y = -3x - 15. If it is Perpendicular then the slopes are negative reciprocals. The slope of the new line must be -3. (Eliminate two choices.) Plugging in (-9, 12) into choice (2) shows that it is a point on that line. Likewise, substitution into equation (4) shows that it is not correct.

5. In the diagram below, under what transformation is triangles X'Y'Z' the image of triangle XYZ? [DIAGRAM SHOWS A CO-ORDINATE PLANE WITH A TRIANGLE IN QUADRANT I AND A CONGRUENT, BUT *ROTATED* TRIANGLE IN QUADRANT IV.]

(3) It's a rotation of 90 degrees clockwise.

6. What is the solution of the system of equations y - x = 5 and y = x² + 5?

(1) You can check very quickly that (0, 5) is a solution. From the choices, the other solution has to be (1, 6) or (-1, 6). Check (1, 6); it works.

7. In the diagram below, parallelogram ABCD has vertices A(1, 3), B(5, 7), C(10, 7) and D(6, 3). Diagonals AC and BD intersect at e.

What are the coordinates of point E?

(3) (5.5, 5) The diagonals of a parallelogram bisect each other, so the coordinates of point E is the midpoint of either diagonal.

8. Right triangle ABC is shown in the graph below. [DIAGRAM SHOWS A CO-ORDINATE PLANE WITH RIGHT TRIANGLE ABC IN QUADRANT I.]
After a reflection over the y-axis, the image of triangle ABC is triangle A'B'C'. Which statement is not true?

(4) AC will not be parallel to A'C' after the reflection. A'C' will have a positive slope.

9. What is an equation of circle O shown in the graph below? [DIAGRAM SHOWS A CO-ORDINATE PLANE WITH A CIRCLE CENTERED AT (-2,4) WITH RADIUS 4.]

(4) Flip the signs, square the radius.

10. In the diagram below of right triangle ABC, an altitude is drawn to the hypotenuse AB.

Which proportion would always represent a correct relationship of the segments?

(3) The Right-Triangle Altitude Theorem. Altitude z is used twice in the proportion.

11. Quadrilateral ABCD is graphed on the set of axes below. [DIAGRAM SHOWS A CO-ORDINATE PLANE WITH DIAMOND-SHAPED QUADRILATERAL IN QUADRANTS I AND iv.]
Which quadrilateral best classifies ABCD?

(3) It's a rhombus, but not a square.

12. Circle O is represented by the equation (x + 3)² + (y - 5)² = 48. The coordinates of the center and the length of the radius of circle O are...

(1) Second time the circle equation is used in this exam. Flip the signs and simplify the square root of 48. (Not that you have to, there's only one possibility in the choices.)

13. In the diagram below of circle O, chord AB is parallel to chord CE.

A correct justification for mAC = mBD in circle O is

(1) Parallel chords intercept congruent arcs.

14. What is theslope o fa line perpendicular to the line whose equation is 3x - 7y + 14 = 0?

(2) Perpendicular again. The slope of the given line is 3/7. The slope is the inverse reciprocal, which is -7/3.

15. Line segment AB has endpoint A located at the origin. Line segment AB is longest when the coordinates of B are ...

(2) The question essentially asks, which of the four choices is the farthest from the origin. You might also notice that if you add the absolute value of x and the absolute value of y, in each case, the sum is 10. However, that isn't how you find distance. The distance formula involves squaring numbers, so it shouldn't be a surprise that (2, -8) is the farthest point.

16. In triangle FGH, m<F = m<H, GF = x + 40, HF = 3x - 20 and GH = 2x + 20. The length of GH is ...

(3) Because angles F and H are congruent, it is an isosceles triangle with legs GF = GH. So x + 40 = 2x + 20. (Note: HF does not matter in this.) Subtract x from both sides, and subtract 20 from both sides, and you find that x = 20. THAT IS NOT THE ANSWER. Plug 20 in for x, and 2(20) + 20 = 60.

17. In the diagram of quadrilateral ABCD, diagonals AEC and BED are perpendicular at E. [DIAGRAM SHOWS A KITE-SHAPE QUADRILATERAL WITH TWO DIAGONALS.]
Which statement is always true based on the given information?

(4) If the lines are perpendicular, then four right angles are formed, which are all 90 degrees, and therefore all congruent to each other.

18. Which set of numbers could represent the lengths of the sides of a right triangle?

(4) {8, 15, 17} is the only Pythagorean Triple listed. You should know it just from seeing it. If you didn't know it, you had to check each using the Pythagorean Theorem, although you should have realized that {7, 7, 12} couldn't be correct because the hypotenuse of that isosceles triangle would have to be 7(root 2).

19. In quadrilateral ABCD, the diagonals bisect its angles. If the diagonals are not congruent, quadrilateral ABCD must be a

(3) It's a rhombus, but not a square. I think I wrote that once before.

20. Line m and point P are shown in the graph below. [DIAGRAM SHOWS A CO-ORDINATE PLANE WITH A STRAIGHT LINE WITH A POSITIVE SLOPE AND A P BELOW IT IN QUADRANT IV.]
Which equation represents the line passing through P and parallel to line m?

(2) The line has a slope of 2, so the parallel line also has a slope of 2. The equation was given in point-slope form: y - y₀ = m(x - x₀).

21. Which compound statement is true?

(1) A disjunction (OR) only needs one part to be true. A square has four sides.

22. In triangle CAT, m<C = 65, m<A = 40 and B is a point on side CA, such that TB is perpendicular to CA. Which line segment is shortest?

(2) In any triangle, the side opposite the smallest angle is the smallest side, but there's a complication here. The triangle is cut into two smaller triangles, and because CAT is NOT a right triangle (it's 65-40-75), we can't apply the Right Triangle Altitude Theorem, even if that would have been of some use. Because TB is an altitude, it creates two right angles, so we can find the size of the smaller angles that T has been cut into, 50 and 25. The side opposite the 25-degree angle is going to be the smallest of the four segments listed.

23. In the diagram of triangle ABC below, DE || BC, AD = 3, DB = 2, and DE = 6.

What is the length of BC?

(2) Because the lines are parallel, the triangles are similar and the sides are proportional. 3 : 6 :: (3 + 2) : x, or 3x = 30. So x = 10.

24. In triangle ABC, an exterior angle at C measures 50 degrees. If m<A > 30, which inequality must be true?

(1) It must be less than 20 because of the Exterior Angle Theorem.

25. Which graph represents the graph of the equation (x - 1)² + y² = 4?

(2) Third time they are asking about the equation of a circle.

26. The equations of lines k, p, and m are given below:

k: x + 2y = 6
p: 6x + 3y = 12
m: -x + 2y = 10
Which statement is true?

(1) Lines p and m have perpendicular slopes (-2 and 1/2).

27. Peach Street and Cherry Street are parallel. Apple Street intersects them, as shown in the diagram below:

If M<1 = 2x + 36 and M<2 = 7x - 9, what is M<1?

(4) The two angles are supplementary, so 2x + 36 + 7x - 9 = 180. (They are NOT congruent, equal to each other.)
Therefore 9x + 27 = 180, 9x = 153, and x = 17. Again, that is NOT the answer.
Substitute 17 for x: 2(17) + 36 = 34 + 36 = 70 degrees.

28. A regular pyramid has a height of 12 centimeters and a square base. If the volume of the pyramid is 256 cubic centimeters, how many centimeters are in the length of one side of its base?

(1) 256 * 3 = 768. 768 / 12 = 64. The square root of 64 is 8.

That's it for the multiple-choice. I hope you did well.

As always, if there are any typos in the above, please holler at me at your earliest convenience so I can adjust things.

Math Can Be Painful

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(C)Copyright 2014, C. Burke.

The puns can be deadly.

That's my attempt at constructing the Gowanus Canal. It's a trendy area now, even though it still stinks. (Literally, the smell is nauseating.) However, the boats aren't really as big as they are on the map. Nor do they float on their sides, as seen in this aerial view.

Math Isn't Everything. It Isn't Even the Only Thing.

Math isn't everything. It isn't even the only thing.

That may sound shocking coming from me, but theoretical math (you know, a lot of that Algebra stuff) needs to be applied to real life using real life conditions, which sometimes (most times?) take you out of the scope of any classroom problem. This is why some people don't think they're using algebra (when, in fact, they are), or why they think it isn't really practical.

A quick example of what I mean: there's a joke floating around that goes something like this: only in a math class can you buy 36 oranges and 25 apples and not be thought crazy.

Here's a different example (not a joke). Each weekend in August, the Miller family barbecues six hot dogs. Buns come in packages of eight. Over four weekends, how many packages of buns should they buy?

Don't scroll down until you're ready for the answer.

The "correct" answer is four packages.

Now, wait a minute, you protest! The cook 24 franks, they need 24 buns, and you get 24 buns in three packages of eight.

That is certainly true. On the second weekend, you still have 2 buns leftover from the prior weekend. This brings the next question -- the real world question -- who gets the stale buns? Probably the shy, quiet one who doesn't speak up for himself. Or the youngest one who doesn't know any better. Most likely, Mom, who who sacrifice for her children, giving them the food from her mouth if need be, assuming she wanted two hot dogs to begin with. (Take better care of Mom, she's been good to you!)

In the real world, even if the bread hasn't reached it's expiration date, those leftovers still won't be as fresh as new rolls will be. Moreover, consider the fourth weekend. All of the bread is leftover, and no one gets a fresh roll. If you're not on a really tight budget, buy new bread each week.

What do you do with the extra bread? Feed the birds. Make breadcrumbs. Have a really funky looking sandwich on Monday.

What do I know? I'm a math teacher, not a cook.

Devil in the Details

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(C)Copyright 2014, C. Burke.

Not just improper, but infernal!

N-RN.2 (Real Number System) - Rationalizing the Denominator

This is the third and final column on Common Core Standard N-RN.2. If you missed the first two parts, Part 1 dealt with evaluating expressions with rational exponents, and Part 2 showed how to simplify using factor trees and how to add and subtract radicals. The last piece of this standard (and since I'm only dealing with part ".2", I could really call it a "substandard" if I wanted to, mockingly) is to "Simplify radical expressions by rationalizing the denominator (Algebra 1 - EE.2)".

Previously, we mentioned that you can multiply two radical numbers by multiplying their radicands. We also factored radical numbers in order to simplify them. Let's talk about division. When you divide, you multiply by the reciprocal; that is, you can create a fraction of the two numbers without relying on your early education "gazintas". (You remember, "2 gazinta 6 three times".) Likewise, when you take the square root of a fraction, you are actually dividing one radical number by another.

So if you wanted the square root of 1/4, you would take the square root of the numerator (radical 1 is 1) over the square root of the denominator (radical 4 is 2). The result would be 1/2.

But suppose we wanted the square root of 1/2? Again, we can split it up into the square root of 1 (which is 1) over the square root of 2.

Here's where we run into a problem because there are rules from fractions. One of them is that there cannot be any radicals in denominator. You have to get rid of them.

We haven't discussed this before, but there's really only one simply way to get rid of a square root sign: square the number. We need to multiply the denominator by radical 2. We are allowed to do this because it's a fraction and we won't change the value of the fraction at all as long as we multiply the numerator by the same amount as the denominator. The fractions (square root of 2 over square root of 2 looks scary to evaluate until you remember that any number, even an irrational, divided by itself is one, with the exception of zero. If you multiply a fraction by 1, it doesn't change its value, even if it looks different. The result is that the radical is gone from the denominator and has moved into the numerator, which is allowed.

One more example. Try it yourself before scrolling down and looking at the image. What is the square root of 4/5?
Take the square root of each number. Rationalize the denominator. What's left in the numerator? What's left in the denominator?

Okay, check your work.

That's it for this standard. Time to move on to part 3, coming soon.

N-RN.2 (Real Number System) - Dealing With Radicals

In a recent post, we explored evaluating expressions with rational exponents in them, but there's more to Common Core Standard N-RN.2. Don't worry, some of it's easier to deal with than what we already tackled.

These are the items listed below the standard, at least according to the IXL website, where I found the list:

• Simplify radical expressions (Algebra 1 - EE.1)
• Simplify radical expressions by rationalizing the denominator (Algebra 1 - EE.2)
• Multiply radical expressions (Algebra 1 - EE.3)
• Add and subtract radical expressions (Algebra 1 - EE.4)
• Simplify radical expressions using the distributive property (Algebra 1 - EE.5)

Is there anything else to deal with? I don't know. Dealing with rational exponents isn't in this list, and yet I think that they might be encountered before Algebra 2. FYI, the notation "EE" stands for Expressions and Equations, which allows me to once again state that "Expressions don't have equal signs and are evaluated, and Equations do have equal signs and are solved."

Simplifying radical expressions is not a difficult task -- as long as you know that it does NOT mean pushing buttons on your calculator and coming up with an approximate decimal equivalent to 8 or 12 or 15 decimal places. Simplifying a radical is similar to reducing a fraction to its lowest terms. It makes it easier to deal with for computations (particularly adding and subtracting, when the radicals have to be "like terms") and comparisons. If the only thing you're planning to do with a radical number is square it, then, yes, simplifying it is a bigger waste of time than converting an improper fraction into a mixed number when it's only going to be used for slope.

There is a very straightforward method of simplifying square roots, but it seems to mystify some of my students who, apparently, never grasped the concept of what a square root (or a perfect square) was in the first place. They memorize steps, but uncertainty about the order causes them to mess up at the very end, removing radical signs from irrational numbers or leaving them in after taking a square root. (For example, they'll write that the square root of nine = the square root of three, instead of three.)

The simplest method involves finding the largest perfect square which is a factor of the radicand (i.e., the number under the radical sign). If it isn't the largest perfect square, then the radical hasn't been fully simplified. An example:

One problem my students face is not understanding the concept of a perfect square, so instead of 25 and 2, then use 5 and 10. After that, they're stuck, or they just decide, for example, that the square root of 5 is the same as 5 without the radical sign.

Because of this, I tried a different approach, using factor trees. They remembered doing them in middle school, and actually liked using them again. (You see, your teacher was right! You are using them again!) The example looked something like this instead:

After they have the prime factorization under the radical, I have them circle the pairs of numbers, cross them out and write one factor outside the radical. This has two downsides to it: first, if the number has a lot of factors, there will be a lot of extra work (but at least they will know, for certain, that they simplified their answer); second, if they don't complete the problem, they basically just drew a factor tree, which looks kinds childish and silly from a high school student.

Multiplying, Adding and Subtracting Radicals

Multiplying two radicals is as simple as multiplying two fractions. Just multiply the numbers under the radicand. For instance, radical 7 times radical 10 equals radical 70. If the number can be simplified, do it, according to the rules above. Obviously, if you square a radical, such as radical 6 times radical 6, the radical symbol goes away. In this case, you get radical 36, which is just 6.

As mentioned above, if you want to add or subtract radicals, they have to be alike. You can't add or subtract the following the way they are:

They aren't alike. It's like two to add 5² + 4² and getting 9². (In other words, you don't.)

But if you simplify the radicals, how to combine them becomes much clearer:

Finally, there is Division, but I'll save that for another column because of the standard, above, Simplify radical expressions by rationalizing the denominator.

How I Started Today's Geometry Regents Review

Students always ask me the same thing: What do I got to know?

Finally, I created a proper response:

I used a meme generator for this. It's a variation of one I saw some time back by a player of boardgames and RPGs which said "Own All the Things". Whether anyone else has done this one before or not, I don't know. (NOOOOOO! I don't know ALL the THINGS!!!!) But I do know I posted it on Facebook, and a friend share it a few hours later, and then a mutual friend shared it from the other friend! So it's spreading.

And I want this one a poster. I could probably make one because there's no copyright notice on it.

Geoboard

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(C)Copyright 2014, C. Burke.

I understand the feeling: sitting around a math classroom, not doing much of anything.

ObMath: Find the perimeter and area of each geometric figure (in units and square units, respectively). Are these numbers rational or irrational?

N-RN.2 (Real Number System) - More Rational Exponents

Continuing my post from Tuesday, on the Real Number system and rational exponents, let's move on to Standard N-RN.2, which reads Rewrite expressions involving radicals and rational exponents using the properties of exponents.

There's a lot to consider under this standard, so I'll continue with rational exponents, i.e., fractions. What if we wanted to evaluate an expression like this?

We need to recognize that the radical 5 is the same as 5^1/2, so

The rules for exponents say to multiply the 1/2 and the 4, giving us 5² or 25.

We can take this further. Suppose we had

The cube root is the same as 1/3 power. So

We can evaluate 6³ as 6 * 6 * 6 = 216.

One more example: How would we handle

The fourth root becomes the 1/4 power.

Now we can get a little fancy with and deal with the multiplication of two fractions:

One final note: The answer doesn't always have to be a rational number. You may exchange one rational power for another, one root for a different one. Consider:

Problems could contain any combination of roots and improper fractions, which may or may not have a simple rational answer. But keep the calculator handy just in case you need to know the sixth root of 117,649. Showing your work, of course.

Putting Down Roots

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(C)Copyright 2014, C. Burke.

They're always trying to put down those radicals!

I haven't hand-drawn one in a while. This one was touched up on the PC, instead of going over the original with a Sharpie.

N-RN.1 (Real Number System) - The Meaning of Rational Exponents

While I'm trying to update every day in August, I might as well start taking a closer look at the Common Core standards, which have now been in place for one year in high school Algebra 1 classes in New York. The first standard I find is N-RN.1. The first N stands for Number and Quantity. Unfortunately, the RN stands for Real Number System and not Registered Nurse because the latter would be helpful when you got sick of all this! I could have helped that telling that joke -- I chose to tell it any way, if only because I had to look up what the letters meant, particularly the first "N".

The entire standard reads as follows: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

It then has two components, one for Algebra 1, the other for Algebra 2. The Algebra 1 component is Evaluate integers raised to rational exponents (Algebra 1 - V.9). Don't ask me about the "V.9", I've done enough searching for this column.

I used to start teaching each year with Order of Operations, something that all the students should have seen before, and yet seemed to forget about. They know an acronym, such as PEMDAS, but don't know what it means. Oh, they know what the six letters mean, but they don't get the concept. And even when they can explain the concept, when push comes to shove and the pressures on (and they're taking a quiz), you find them calculating from left to right as if they hadn't learned anything. I had to change that when we started welcoming calculators into the lesson (even before we started requiring them). The calculators were down the work for them, so they didn't have to learn it, right? Wrong. I just adapted the problems. I started added more operations within fractions and adding exponents, forcing them to pay attention to what they put into their calculator. For instance, you need explicit parentheses to group things in a calculator because the numerator and denominator are implicitly grouped.

But Common Core changes that. They have to learn it much earlier, so they're ready for Algebra by the time they get to high school. Yeah, right. I'm still doing it. But wait, there's more.

After parentheses, come the exponents. Some students know the concept of exponents, and some just press buttons on the calculator. They know (or they'll learn!) that they are multiplying the factor some number of times. What they haven't seen before is a fraction as an exponent. What do you do when you see a fraction? Hide under the desk, usually. Wait for it to be over. I'm not exaggerating much.

First, I have to teach the concept that exponent 1/2 means take the square root, as stated in the gem. And then see how well they know they're square roots, either with or without a calculator. (I try to get them to learn up to 256, with some success.)

Second, I have to teach them that exponent 1/3 means take the cube root. This usually entails explaining what a cube root is, and where to find it on the calculator. Maybe reviewing what "cubing" means, and possibly through in Volume = length X width X height somewhere, just so I can spiral back to it when that comes around again. Depending on the results, I could try to conquer the concepts of fourth and fifth roots. Seriously, a practice webpage asked me (729)^(1/6).

At this point, I'm asking: are they getting it, or are they pushing buttons? I don't mind the button pushing if they understand the concept, because then they'll start recognizing patterns and will be smarter about their button pushing.

Okay, so the next step is the real doozy: explaining exponents of 2/3 or 3/5. The students have to deconstruct the fraction. That is, they need to know that 2/3 = (2)X(1/3), so they need both to square it and then take the cube root. And then I'll suggest that you take the cube root first, so that they're dealing with smaller numbers. Sometimes this makes sense to them.

Finally, there's the kicker: improper fractions. If fractions are Dr. Frankenstein, improper fractions are his monster. They're a whole new level of scary, and the first thing they want to do is turn them into mixed numbers. Or decimals. No! Wait! Stop!

Taking the (5/2) power of 16 is as simple as taking the fifth power of the square root of 16. Okay, read that again like you're a ninth grader.

No, it's not that difficult to do once the concept is learned, but it's something they haven't seen before, and they're learning it earlier. I can't remember exactly when the first time I was required to find a third or fourth root. I was probably using logs to do it. But then, I didn't have the calculators that they have today on their phones. They'll have the answers at their fingertips, but I'll make them show the work so I know they get the idea behind it.

Hear Yourself Talk

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(C)Copyright 2014, C. Burke.

Is ''Hear Yourself Talk'' = ''Talking To Yourself'' ?

"Speak up more?" Isn't it enough that they're awake and in class? (Or, at least, in class?)

Reviewing that exam, I could have used an actual negative correlation question, but they were too wordy to fit in a panel.
And, to be honest, I never would have copied it onto the board with chalk. (Maybe with a projector, but not chalk.)