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Mathematics Test for Conceptual Understanding

How well do you understand math concepts? See if you can explain the following 13 statements:

1) “You can’t divide by zero.” Explain why not, (even though, of course, you can multiply by zero.)

2) “Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.

3) You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.

4) Place these numbers in order of largest to smallest: .00156, 1/60, .0015, .001, .002

5) “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.

6) A catering company rents out tables for big parties. 8 people can sit around a table. A school is giving a party for parents, siblings, students and teachers. The guest list totals 243. How many tables should the school rent?

7) Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.

8) Construct a mathematical equation that describes the mathematical relationship between feet and yards. HINT: all you need as parts of the equation are F, Y, =, and 3.

9) As you know, PEMDAS is shorthand for the order of operations for evaluating complex expressions (Parentheses, then Exponents, etc.). The order of operations is a convention. X(A + B) = XA + XB is the distributive property. It is a law. What is the difference between a convention and a law, then? Give another example of each.

10) Why were imaginary numbers invented? [EXTRA CREDIT for 12th graders: Why was the calculus invented?]

11) What’s the difference between an “accurate” answer and “an appropriately precise” answer? (HINT: when is the answer on your calculator inappropriate?)

12) True or false: .99999.... = 1

13) Explain why a negative times a negative is a positive.

I came across this list in Grant Wiggins blog.

Note by Calvin Lin
2 years, 5 months ago

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(1)Proving that you cannot divide by 0: First I will prove that you cannot divide a \(\textbf{non-zero}\) number by 0.Assume that proving by 0 is possible.Let \(a\) be any non-zero real number.Then \(\frac{a}{0}\) results in a number \(b\) which can be written as: \[\frac{a}{0}=b\\a=b\times 0\] Anything times 0 is 0,so \(b\) times 0=0.But then the equation becomes \(a=0\).But we have already fixed that a is non-zero.Therefore,we have arrived at a contradiction.Therefore we cannot divide non-zero numbers by 0. Second part of proof:Proving that 0/0 is undefined: Let 0/0 be equal to \(c\) Then: \[\frac{0}{0}=c\\0=c\times 0\] But LOOK:\(c\) can be any number;it can be \(0,1,\frac{1}{2},\pi\;etc\) \(\textbf{ANY NUMBER}\).But a division can only have 1 answer.So it is proved that \(\large{\textbf{YOU CANNOT DIVIDE BY 0}}\)

(4)1/60,0.00156,0.0015,0.002,0.001

(5)\(\sqrt{2}\times\sqrt{3}\),which cannot be expressed as repeated addition.In general \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) cannot be expressed as repeated addition if both \(a\) and \(b\) are not perfect squares. \(\textbf{Explanation:}\)You can express \(a_1\times a_2 \times a_3\times\dotsm a_n\) as repeated addition if any one of the numbers from \(a_1\) to \(a_n\) is an integer.But this is not necessarily true.

(6)You cannot order 30 tables because 3 people will still be left standing so you have to order 31 tables.

(7)For example,if a teacher assigns a final grade by using the mean of a students with marks:19/20,18/20,20/20,19.5/20 and one really bad 10/20 (Which does happen) the student's final grade will be really low because of that 10/20.In such cases,it is better to use other types of averages.

(8)\(F=\frac{Y}{3}\)

(10)To find solutions to those equation which had no real solutions.

(12)True \(\textbf{Proof:}\)Let: \[0.999\dots=x\\9.9999\dots=10x\\10x-x=9.999\dots-0.999\dots\\9x=9\\x=\frac{9}{9}=\boxed{1}\] (13)\[0=(-1)\times0\\=(-1)\times[1+(-1)]\\=(-1)(1)+(-1)(-1)\\=-1+(-1)(-1)=0\\-1+(-1)(-1)=0+1\\\boxed{(-1)(-1)=1}\] So \((-x)\times(-y)=(-1)\times(-1)\times x\times y=1\times xy=xy\) Abdur Rehman Zahid · 2 years, 2 months ago

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12) let us suppose that there are n number of 9

we can write it in this way

\[\frac{9999.....}{10^{n}}\]

\[= \frac{10^{n} - 1}{10^{n}}\]

\[ = 1 - \frac{1}{10^{n}}\]

thus as n will go on increase \[ \frac{1}{10^{n}}\] go on decrease (tends to 0)

thus \[ = 1\] Megh Choksi · 2 years, 5 months ago

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13)

x any be real numbers

\( - x . - y + ( -x . y) = -x( - y + y) =0\)

thus negative . negative is equal to positive

sir is this right? Megh Choksi · 2 years, 5 months ago

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10)

to find the root of

\( x^{2} + 1 = 0\) Megh Choksi · 2 years, 5 months ago

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@Megh Choksi actually it was created by a teacher to prevent his students from taking root of -1 x root of -1 is equal to 1 that is how 'i' was created. Uthkarsh Salian · 2 years, 4 months ago

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I think #2 is a bit too sweeping of a statement. It sort of reminds me how Richard Feynman [jokingly] proposed a GUT, which is to round up and state all equations of physics in the form \(X=0\) and then multiply them all to get \(\prod { X } =0\). Yes, it'd be a GUT, but it wouldn't be very illuminating, wouldn't simplify anything, and wouldn't help us learn anything new.

The equals sign, \(=\), is actually used in a variety of context, as for example, in an identity, where both sides are always the same, or in an assignment, where a variable is given a specific value, or in an definition, as for example a Log function is defined to be the inverse of an exponential, etc. Not all equals are equal. One who programs computers would be familiar with these kinds of issues. To say that "solving problems typically requires finding equivalent statements that simplify the problem", suggesting that problem solving is a chain of equal quantities from unsolved to simple solved is just grossly oversimplifying the practice of doing mathematics. Michael Mendrin · 2 years, 5 months ago

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I'm trying to actually solve this. I saw the answer for #1. It says that if you consider 1/0 to infinity, 1/0 * 0 = 1, if you cancel 0 or 1/0 * 0 = infinity * 0 = 0. This leads to ambiguity of saying (any number)/0 is infinity. Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #2 No idea. Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #3 \[ \frac{x}{\frac{a}{b}} = x * \frac{b}{a}\] \[ \frac{x}{\frac{a}{b}} * \frac{b}{b} = \frac{xb}{a}\] \[ \frac{xb}{\frac{ab}{b}} = \frac{xb}{a}\] \[ \frac{xb}{a} = \frac{xb}{a}\] Joeie Christian Santana · 2 years, 5 months ago

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Comment deleted Oct 17, 2014

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@Joeie Christian Santana In #4 1/60, .002, .00156, .0015, .001, Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #5 Multiplying by fractions. 3 * 1/3 cannot be interpreted by repeated addition. Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana can't we say it as adding 1/3 three times Will Jain · 2 years, 5 months ago

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@Joeie Christian Santana In #6 31. The excess people should be given tables for themselves, too. Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #7 Averaging lowers your grade, if you get consistent near-the-highest grades and one really low one. If you get 90% 95% 92% and 60%, the average would be around 84%, which is far from the high scores. Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #8 To be typed. Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #9 In my opinion, convention is a way to put things in a specific set of rules. Example, synthetic division. A law is a rule that's been proven to be true. Example is (). (What am I even saying?) Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #10 My hunch would be to describe the answer when you take the square root of a negative number, but other than that, I can't think of anything else. Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana Close. What is the reason that we will need to take the square root of a negative number? Calvin Lin Staff · 2 years, 5 months ago

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@Calvin Lin Solving quadratic equations. That is the simplest use of imaginary numbers that I can imagine. (no pun intended) Shenal Kotuwewatta · 2 years, 5 months ago

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@Joeie Christian Santana In #11 I think that an accurate answer is the answer shown on the calculator, and an appropriately precise answer is the answer rounded to the given number of significant numbers. Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #12 True. 0.999... is equal to 1 because infinite nines won't end, and you can't add 0.0.........1 (infinite zeroes after the decimal point with a 1 on the end.) Joeie Christian Santana · 2 years, 5 months ago

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@Joeie Christian Santana In #13 Let a, b be the element of positive real numbers. Assume -a * -b = -ab. -ab/-b = -a. a = -a. A contradiction. Therefore, -a * -b = ab. :v Joeie Christian Santana · 2 years, 5 months ago

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For #11: your calculator answer is inappropriate when it answers 58008 Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 2) An equal sign is the representation of equivalence. Say that x=y and y=y, the x is a y in disguise.

...that's deep Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 3) \(\dfrac{a}{b}=a*\dfrac{1}{b}\). Multiply both sides by b

\(\dfrac{ab}{b}=a*\dfrac{b}{b}\Rightarrow a=a\) Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 4). 1/60, 0.002, 0.00156, 0.0015, 0.001 Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 5) I don't got nutin on 5.... Best I got is that x*x can't be written as addition. Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro Maybe Multiplying by 0 or by negative numbers Dinesh Chavan · 2 years, 5 months ago

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@Dinesh Chavan multiplying by 0 is like adding it 0 times and multiplying by a negative is like adding a negative x amount of times. Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 6) 30, because guarnteed three people won't show.

Jk, 31. Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 7) teachers shouldn't take the mean of the grades because there is a lot of effort put into work. Many a time, the student's hard work is overshadowed by one careless error. I'm the sweatiest try hard you'll ever meet. I know how it feels to get a B on something you worked so hard for.

Also, the mean might account for outliers. Say that a student gets a 20/20 on every quiz and one 2/20 (yes, I've seen his happen to many students in my chem class) and one student gets a 20/20 on every quiz but gets two 19/19. If each student gets 1 dropped quiz, then the one who had a lower total before the drop will end up with a higher average. Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 8) \(F=Y/3; Y=3F; 3=Y/F; 3^3=Y^3/F^3\) Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 9) idk my vacabilary=0.....punn intended. Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro Numbers were invented to quantify things. Because the ancient Romans were too lazy to say that "bill, Sally, joe, Bob, Dan, sean, Dylan... Heather, Sally are present" they just said 15 people are present.

in actuallity, numbers aren't invented. Numbers are to math as letters are to language. One doesn't exist without the other. Numbers were made to prove, without a doubt, why something happens or doesn't.

Yoda, John muradeli, Michael Mendrin, and Brian Charlesworth discuss something similar to this AND MUCH MORE in the comment section here

Calc was also created for these purposes, just for physics stuff. Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 12) \(0.\overline{9}\neq 1\) because \((0.\overline{9})^\infty=0\) while \(1^\infty=1\) if two values are equal, then putting those values to any power should keep them equal. Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro 13) I tried to argue this, but it became a cyclic arguement. So all that I can say is that if u tell some one "I do not not want this" its like saying that "I want this".

(I know in mathematical logic this is flawed because it technically doesn't mean u like it, it just means that u don't not want it.) Trevor Arashiro · 2 years, 5 months ago

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@Trevor Arashiro When you figure this out (as in a proper formal proof), you should add it to Trivial Inequality.

Hint: What is a definition of positive number that doesn't involve 0? Calvin Lin Staff · 2 years, 5 months ago

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@Trevor Arashiro Lol. Joeie Christian Santana · 2 years, 5 months ago

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12). True Nashita Rahman · 2 years, 4 months ago

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6) No. Of tables required= 31 Nashita Rahman · 2 years, 4 months ago

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1) we cannot divide by zero because the result is undefined where as if we multiply by zero then the result itself is zero. Nashita Rahman · 2 years, 4 months ago

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help me learn here:

@no.1- zero implies "infinite space where infinite number possible can fit." You can't possibly divide something or anything with an infinite space where infinite number possible could exist.

Am I making sense in my statement? Les Narvasa · 2 years, 4 months ago

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The first statement says that "You can't divide by zero." You can simply say

\(\frac{x}{0}\)

is impossible because...

\(y \times 0 = x\) ,

in which where \( x \neq 0\). No such real number can stand for \(x\). Jeremy Bansil · 2 years, 5 months ago

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@Jeremy Bansil In which somehow, any number, such as \(y\), multiplied to \(0\) is ALWAYS equal to \(0\). Jeremy Bansil · 2 years, 5 months ago

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1)

Using binary operation on set of complex numbers , when we divide any complex number by zero it does not belongs to its set

Using binary operation on set of complex numbers , when we multiply any complex number by zero it belongs to its set

therefore we can multiply any number with zero but can't divide

IS this right? Megh Choksi · 2 years, 5 months ago

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I disagree with #1.

Depends on how you define "dividing," you can actually get a "defined" result.

In the Riemann Sphere,

s

s

For this set of arithmetic, one may yield, for any nonzero \(a\), \(\frac{a}{0}=\) complex infinity. John Muradeli · 2 years, 5 months ago

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Multiplication is repeated addition only for integers, we cannot define it for real numbers where we cannot express it as such,., in fact if multiplication was really just repeated addition,,

then x^2=x+x+x+x......x (x times for all real no.) and on differentiating, we get 2x=1+1+1+1+... (x times) = x in which case we get 2x=x for all real number which is false Mvs Saketh · 2 years, 5 months ago

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for number 12: its true using geometric infinite series. Paul Ryan Longhas · 2 years, 5 months ago

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1 we cannot divide by 0 but multiply because in division we not only have 1 but 2 conditions. in multiply it creates no problem because the answer is always 0 and we don;t need to do discussion over that , but in division we have to find the nearest multiple of any positive or negative no. for eg. - 6 . but even if we multiply 0 *10000000000 we know we would never reach to 6. and hence it just results to be ∞

Palash Som · 2 years, 5 months ago

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@Palash Som how did you increase the font size? Megh Choksi · 2 years, 5 months ago

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12) 0.999.... = 1 It is only a matter of notation. No mystery at all. Samarth M.O. · 2 years, 5 months ago

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