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# Why do new rules cause students to forget and misapply older rules?

I teach at a community college. I have taught everything from arithmetic to linear algebra. I have also taught at 4-year schools, but at present, I'm devoting my energies to the problem of helping remedial algebra students to succeed.

I have noticed a pattern in my remedial classes. It's disturbing. You see, my students first learn the concept of "combining alike terms" example:

$3x^2y - 5x^2y = -2x^2y$

I present this topic in a number of ways including manipulatives and concrete examples. We then apply this concept in a variety of contexts including systems of linear equations, and in word problems. Basically, it appears that they are getting quite good at working with variables.

But, later when we study exponent rules such as:

$x^ax^b=x^{a+b}$

$(x^a)^b=x^{ab}$

$x^{-a}=\frac{1}{x^a}$, for $x \neq 0$

etc.

these new rules seem to displace and muddy the older rules in the minds of the students. A week ago they would have considered:

$2x + 5x = 7x$

to be "easy" and every single student (even the weakest) had mastered this type of problem (signed numbers and fractions could be another matter...but still) Yet, after teaching the exponent rules I notice students doing things like this:

$2x + 5x = 7x^2$

to me this indicates a fundamental disconnect in terms of how mathematics works, I know they are aware of the older "rules" but it is as if they expect each problem to have different set of rules. This has happened all three times that I have taught this course, despite my effort to teach it in a different way each time.

I find that this type of error is much more common in remedial classes. Why is that? I have also taught elementary school algebra and I simply never saw mistakes like this. Not, at least, with the frequency I'm finding them now, even among responsible students who are clearly intelligent people as evidenced by their work in other subjects and pursuits, students who are putting in large amounts of time studying, who take notes etc. And these erors are hard to fix, explanations don't seem help much.

What is going on here? Is there a name for this?

I honestly wonder what it is I've taught them in the past two months if each new concept displaces and corrupts the old concepts.

Eventually the students will master the new rules but I get the feeling many of them are working much harder than they should be to do so. It's like they are doing something that's more like memorizing a complex gymnastics routine than mathematics. Others become very frustrated, to them it must seeming like I'm just making up random stuff as I go along to vex them.

But I know mathematics makes sense. That's why I love it. How can I help them to see this?

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Instead of rules one might make them notice that they are facts; the rule of exponents is not something you told them, it is a fact of nature. – Mariano Suárez-Alvarez Nov 17 '12 at 18:00
I don't know if that distinction would make sense to them. I have tried replacing x with numbers and showing them that $2x+5x=7x^2$ is simply not true for most values of x. But this seemed to be even more confusing. That fact that it is true for the solutions to that quadratic is just too complicated to even mention. The kind of "proof" that I find very helpful and convincing seems to sound like gibberish to them. I try to avoid talking too quickly or putting too many symbols on the board at once... – futurebird Nov 17 '12 at 18:09
What I'm really looking for hear is a name for this kind of destabilization. Something I can look-up in math education research journals. Maybe there is a way to avoid it. – futurebird Nov 17 '12 at 18:13
but you can turn their confusion about some numbers satisfying that equation into theproblem of finding which do, and thereby getting some motivation for the problem of solving the quadratic. A motivation they'll feel close to them. – Mariano Suárez-Alvarez Nov 17 '12 at 19:19
@Josh: The problem is that with students like these, principles don’t stick, because they aren’t understood as principles in the first place. To these students the subject is just a maze of complicated rules with no underlying system. Some of them even want it to be that way: they want rules to follow and actively resist the idea that understanding some underlying principles will make life easier. Rules are seen as easy, principles as hard. – Brian M. Scott Nov 17 '12 at 23:40

One approach for dealing with this problem: make a three-part worksheet. Part one is "stuff using the formulas we just learned today." Part two is "review of old stuff." Part three is "mixed problems."

In essence, when you teach the first concept, the student isn't learning it as an if-then statement: "when you see this, then do this." The student is just learning "do this," because that's the only sort of problem the student sees. So when you introduce the second concept, really the student has twice as many new things to learn - how to solve the second type of problem, and how to distinguish the first type of problem from the second type of problem. Of course there's no getting around this. You have to introduce something first. But that's my theory for why the second thing you introduce makes things slower.

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One article that immediately came to mind, and which I think you'd find very helpful for your purposes, is Memory and Mathematical Understanding (link to article abstract).

Also perhaps relevant is Working Memory and Mathematics. It is relatively recent (published in 2010) and has a long list of references, to explore further.

You might want to check out the Mathematical Association of America "special interest group"'s SIGMAA RUME website and this auxiliary site. (RUME = Research in Undergraduate Mathematics Education.) You will find additional resources and references on undergraduate math education, and perhaps connect with others sincerely interested in post-secondary teaching.

That said, I really think much of what you are observing is a "cognitive" phenomenon: not necessarily a phenomenon of "what is being learned" (i.e., mathematics), but more appropriately addressed in the domains of cognitive & educational psychology. Certainly, math cognition is, in many respects, qualitatively different than (or a more specific domain of) cognition in general. So I do not mean to imply that this question doesn't belong here. I suspect that there are a lot of people here with lots of experience teaching, and many of whom are also interested in improving students' comprehension of mathematics.

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Thanks, the link to 'Memory and mathematical understanding'. seems to be pointing to the wrong place? – futurebird Nov 17 '12 at 18:17
Updated with correct link. It's in pdf, which is always handy! (I think I lost the link when I added the first link). – amWhy Nov 17 '12 at 18:19
Thanks. I'm going to think about this angle, though I'm not certain it's a memory problem. These students seem to have normal memories, some of them might have better memories than I do. When I ask they are aware that last week 2x+3x was equal to 5x... it's more like they no longer think this is valid because of the new rules... – futurebird Nov 17 '12 at 18:35
The article discusses "working memory" among other things. There may be some "cognitive competition" going on between working memory (that which is most recently learned, or that which with one is currently trying to grapple) and longer-term memory (being that which was previously learned). And it could be that "circuits" are crossing/clashing. – amWhy Nov 17 '12 at 20:03
The links were corrected. In my original post, "Memory and Mathematical Understanding" was actually linked to "Working Memory and Mathematics," the former article being , I suspect, more relevant to the question at hand. – amWhy Nov 18 '12 at 14:28

Here is a guess:

It is possible, that as remedial students, they have an expectation that the problems will be difficult. I.e. they are expecting to put a lot of mental strain and effort into the solution.

This instinct may be appropriate in some situations, e.g. when they are learning the problem. However, in other situations it may make them discard the simpler (correct) solution and employ a more complex (incorrect) solution, simply because it seems to be more complex, and therefore meets the level of mental effort that they are expecting to experience.

If this hypothesis is correct, then perhaps additional practice on problems that they are already comfortable with will help recalibrate their "maths-is-hard" mental parameter?

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If the needed rule is not the rule being directly studied, the student must first select the rule from their entire toolbox of rules that could be used. This is much harder than trying to apply the one rule that will probably be needed to solve all problems in the current problem set.

To put it another way, when students are studying a specific topic and they try to solve exercises in that topic's section, they are effectively given a really big hint that the problem probably has something to do with that topic. If any other topics are also needed, they must refer to all previous topics to consider the set of possible applicable rules.

You can see this phenomenon at work by simply suggesting the rule, like combining alike terms, when the student gets stuck. Often they can then immediately see that this rule can be applied. Explanation: They got their hint back.

Once you move out of the laboratory and into the real world, what seemed easy may now be much harder!

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This is mostly speculative.

### Consider doing exercises on when not to use a 'rule'.

I like to give an example from machine learning: if you design an AI to answer a particular question, and then give it a lot of training data for which the answer is "yes", then what the AI will learn is that every question should be given a "yes" answer. You have to give it enough examples where the answer is "no" in order to learn properly.

(aside: for the AI enthusiasts out there, I'm sure there are designs that the above doesn't apply to. I'm just trying to make a good example)

Of course, a human will realize that they're supposed to be learning some sort of pattern, and so they'll try and guess the pattern on their own. They'll get some feedback that their guess is wrong in the form "you did this problem wrong", but it will come slowly.

For students who are having trouble working out the pattern, seeing examples that don't fit the pattern may be far more informative than more examples that do. A variety of examples will surely help as well.

### Consider explaining how to recognize when to use the rule

Rather than trying to learn the pattern through osmosis, it may help to explicitly describe what features they should be looking for in an expression that allow a rule to be used.

Explain how to break down expressions into terms and factors and such, and to see that one rule might be applied to combine terms of a sum (or to split one term into several), or another rule might be applied to combine the factors of a product (or to split one factor into several).

A full blown lesson on mathematical grammar might even be useful. I've always been curious if learning how to make parse trees out of an expression could help students.

On a related note, I think a lot of people (both on the learning and teaching sides) have trouble even recognizing the existence of this skill and how important it is -- students who focus on learning how to use tools without ever realizing they're supposed to be learning how to select tools (or maybe even how to pack a toolbox!), and teachers who think the latter skills will come automatically to anyone who learns the former.

I don't know how to teach it though, especially to a student to doesn't acknowledge it, or worse, rejects it because they think they need to focus on how to use tools. :(

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I have similar experience in teaching high school algebra and trigonometry class. We stream students into three levels of math: honor, intermediate and elementary. The elementary level is where I observe similar problem.

After many years of teaching I am starting to suspect that some topics are not teachable (if that is a word) for some students. Is this a polite thing to say these days?

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Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x–p) 2 =q that has the same solutions. ©m w260 H1z4x qK ju 5t 3ab 7S Xokf vtIw SaLr lev 4L3L TC B.P n wAClnl F CrDiEguh8t 8s9 vr2e RsYePr RvkePdb.W I SMsa 7dTe l kw 7i 3t Ohm zI an2f ViDnbidt7e N 7A zlpgTeFb6r5a L e2z. Let's look at an example: 4 - 7 i and 4 + 7 i . ©H z2 C0U1x2w VK4uNtvaL WSQoTf xtyw haDr6e 1 IL mLhC t. L Y uA 7l 4l 9 UrUi3gKh8t Gsz crXeOs0eurtv YeJd D.9 W yM Haxdde s 9wxi pt8h6 PI3n XfYitngi 1t neq EA LlAg Pe5b NryaZ s2 K.H Worksheet by Kuta Software LLC Algebra 2: Complex Numbers ENTIRE UNIT Mini Formative Assessments, Conjugates and Quotients of Complex Numbers Guided Notes and HW, Quadratics Revisited - Complex Numbers Algebra 2 Unit 1 Bundle, Divide Complex Numbers using Conjugates Guided Notes, Algebra 2 - Conjugates of Complex Numbers, Complex Conjugates Theorem Doodle Graphic Organizer, Multiply A Complex Number and its Conjugate: 24 Task Cards, Complex Conjugate - Find the Conjugate, Moduli, and Quotients of Complex Numbers, Math Operations with Complex Numbers Mystery Drawing, Complex Numbers No Prep FUN Notes Doodle Pages also for Distance Learning, Algebra 2 FUN Notes Doodle Bundle also for Distance Learning Packets, Simplifying Complex Numbers Color Worksheet, Simplifying Complex Numbers Coloring Activity, Operations With Complex Numbers Cut & Paste Activity {Algebra 2}, Scavenger Hunt {School/Home/Stations} - Dividing Complex Numbers, Complex Numbers Digital Drag & Drop Activity - Distance Learning, Algebra 2 Curriculum - Texas TEKS Aligned, Complex Numbers Review or Quiz x 2｜2 Similar Versions!｜Math Google Forms. And I want to emphasize. Example: Displaying top 8 worksheets found for - Conjugate. Given a complex number z = a + b i {\displaystyle z=a+bi} (where a and b are real numbers), the complex conjugate of z {\displaystyle z} , often denoted as z ¯ {\displaystyle {\overline {z}}} , is equal to a − b i . Displaying top 8 worksheets found for - Conjugate To Be. W, Teach your students the techniques to determine tricky limits.Two techniques that are often found on the AP Exam are using the conjugate to rationalize a numerator and simplifying complex fractions. Write the problem in fractional form. A way todo thisisto utilizethe fact that(A+B)(A−B)=A2−B2 in order to eliminatesquare roots via squaring. It's the perfect, Your Algebra students will love learning about Complex Numbers with this fun format.Topics Include:Adding and subtractingMultiplying Dividing (rationalizing)Finding conjugatesFinding the absolute value (modulus)Graphing complex numbers. Simplify a … Change your answers in Q1 to exact x + yi form. 5. Power of a Complex Number. Then plot all eight complex numbers in the same complex plane. b. Hopefully yours will find it enjoyable and helpful as well :-) In some classes I ran this activity in class while in others I assigned it at home as a homework. Some of the worksheets for this concept are How to conjugate french verbs present tense, Conjugating verbs to go work, Conjugate acid base pairs name chem work 19 2, Conjugate of complex numbers 1, Bronsted, How to conjugate portuguese verbs, How to conjugate french verbs imperfect tense, … In general, the complex conjugate of a complex number is simply that number with the sign of the imaginary part reversed, that is, if z = x + i y, then the complex conjugate is z ∗ = x − i y. Then plot all eight complex numbers in the same complex plane. Adding and subtracting complex numbers. Students love to doodle while learning and it has been shown th, 25 carefully thought out problems on simplifying complex numbers. Simplify a bi a bi 2. $\overline{z}$ = 25 and p + q = 7 where $\overline{z}$ is the complex conjugate of z. ©f i2 N0O12F EKunt la i ZS3onf MtMwtaQrUeC 0LWLoCX.o F hA jl jln DrDiag ght sc fr 1ersve1r2vte od P.a G XMXaCdde 9 9waiht5hB 1I2nAfUizn ZibtMeV fA Sl Agesb 7rfa G G2D.Z Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Operations with Complex Numbers Date_____ Period____ Simplify. 29 min. F Worksheet by Kuta Software LLC Math for College Readiness Name_____ Date_____ Period____ ©2 A2G0f1t4 r pK MuStEad YS5oHfXt Lw ia br8eW QLFLrC x. Question 2. It has the same real component x, but the imaginary component is negated. There are 20 problems total, separated into two columns. Dividing Radicals. Plus model problems explained step by step We learn the theorem and illustrate how it can be used for finding a polynomial's zeros. ... 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For example, the complex conjugate of $$3 + 4i$$ is $$3 − 4i$$. Complex Numbers in Polar Form. nth Root of a Complex Number. Types of angles worksheet. Algebra II Complex Conjugate Worksheet Name: _ Find all zeros (real and imaginary) of each polynomial function using the Square Roots of a Complex Number. Each sheet contains a short summary in addition to the differentiated questions. 28 min. 2. Write the conjugate of each. a) 6 i b) 11 i c) 24 i 4. EXAMPLE 6 Find 8 −1 . Polynomials Complex Conjugate Root Theorem Worksheet 2 Answer each of the following without using a calculator and using the boxes provided for your answers. Also included in: Maze - BUNDLE IMAGINARY NUMBERS (Complex Numbers). Rez = Rez z = z; Imz = 0 Imz = Imz z = z; Rez = 0 z +z = 2Rez z = z z z = 2iImz zz = jzj2 jzj = jzj 1=z = z=jzj2; z ̸= 0 argz = argz; z ̸= 0 z = 1=z; jzj = 1 z1 +z2 = z1 +z2 z1z2 = z1z2 z1=z2 = z1=z2; z2 ̸= 0 z1 +:::+zn = z1 +:::+zn z1::: zn = z1::: zn 2. REF: 010714b 2 ANS: 3 REF: 089325siii 3 ANS: 2 REF: 088620siii 4 ANS: 4 If 1 i is one solution, the other is 1 i. To download/print, click on pop-out icon or print icon to worksheet to print or download. ... Sal discusses the usefulness of the conjugate of the complex number is this video. This lesson covers how to find a conjugate of a complex number as well as how to simplify expressions including division by a complex number. Complementary and supplementary worksheet. Our comprehensive worksheet numbers are free to download, easy to use, and very flexible. The real complex numbers lie on the x–axis, which is then called the real axis, while the imaginary numbers lie on the 7 + 2i 3 – 5i The complex conjugate of 3 – 5i is 3 + 5i. “Relearning” h, Heroes are not made overnight, they are developed with practice. Guided notes and practice problems on dividing complex numbers. Great for use in the classroom when first learning the topic, or as homework or revision sheets. This resource works well as independent prac, In this activity, students will practice operations with complex numbers. 2. Conjugate - Displaying top 8 worksheets found for this concept.. Discussion: In Class Exercise – 02. Then plot all eight complex numbers in the same complex plane.4. Students simpli You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. Write the expression as a complex number in standard form. Some of the worksheets for this concept are Conjugate of complex numbers 1, Complex conjugate root theorem, 2 5 and 2 6b complex zeros and the fundamental theorem of, Complex zeros and the fundamental theorem of algebra, Irrational and imaginary root theorems, 1pf1 complex analysis, Complex numbers in polar form demoivres theorem, Complex numbers and … (x) 0 x 08 x 5 g = x 4 − 4 2 − 2 − 2 i is a zero − 4 + 3 The following diagram explains complex conjugate pairs. These complex worksheet numbers are randomly created and never repeated, so you have an infinite inventory of quality complex worksheet numbers to use in class or at home. Not only does, Students will simplify 18 algebraic expressions with complex numbers/imaginary numbers including adding, subtracting, multiplying and dividing complex numbers (includes rationalizing the denominator by multiplying by the conjugate) (Algebra 2 Curriculum) Worksheet Q1: Find the complex conjugate of − 7 − and the sum of this number with its complex conjugate. Quick and easy warm-up or exit ticket. A complex conjugate is formed by changing the sign between two terms in a complex number. There is a variety of problems from basic operations, rationalizing, conjugating, distributing, and negative radicals. Show all of your working. The answer to a puzzle will appear at the bottom of each worksheet once the solutions are found.Complex Numbe, This maze is part of : Maze - BUNDLE COMPLEX NUMBERSThis product is a good review of checking for understanding how to Divide Complex Numbers and Rationalizing the Denominator - Binomial/Binomial ONLY. Area and perimeter worksheets. First, he shows both graphically and algebraically adding a complex number and its conjugate. In other words if we find, or are given, one complex root, then we can state that its complex conjugate is also a root. Multiply the next few complex number pairs and try to recognize a pattern 21) 3 2 3 2 ii 22) ii 4 5 4 5 23) ii7 3 7 3 24) Look carefully at questions 21-23. 1. Notation: w= c+ di, w¯ = c−di. 15 min. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Complex Conjugates. Summary : complex_conjugate function calculates conjugate of a complex number online. Then plot all eight complex numbers in the same complex plane. Free worksheet(pdf) and answer key on Dividing Complex Numbers. Students will simplify 20 dividing radical expressions problems WITHOUT variables in this independent practice riddles worksheet. When b=0, z is real, when a=0, we say that z is pure imaginary. Also included in: Complex Numbers Bundle｜8 Self-Grading Math Google Forms｜Editable! Enjoy these free printable sheets focusing on the complex and imaginary numbers typically covered unit in algebra 2 each worksheet has model problems worked out step by step practice problems as well as challenge questions at the sheets end. There are two versions of the puzzle included. Conjugate - Displaying top 8 worksheets found for this concept.. 2. Multiply the next few complex number pairs and try to recognize a pattern 21) 3 2 3 2 ii 22) ii 4 5 4 5 23) ii7 3 7 3 24) Look carefully at questions 21-23. Multiplying by the Conjugate Sometimes it is useful to eliminate square roots from a fractional expression. Complex numbers is vital in high school math. A.REI.B.4: Complex Conjugate Root Theorem Answer Section 1 ANS: 3 The product of the roots equals . The complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude, but the complex value is opposite in sign. We also work through some typical exam style questions. These notes go along step-by-step with a presentation sold separately. The real part of the resultant number = 5 and the imaginary part of the resultant number = 6i. Scroll down the page for more examples and solutions for dividing complex numbers. G Worksheet by Kuta Software LLC Math for College Readiness Name_____ Date_____ Period____ This right here is the conjugate. Maze - Operations on Complex Numbers - Division of Binomial by Binomial, Maze - BUNDLE IMAGINARY NUMBERS (Complex Numbers), Learning Limits - 5: Rationalizing a Numerator and Complex Fractions, Herowork - Conjugates - Wilma Rudolph Mystery Pic & Weekday Joke. (iii) 3i... (2) If z = x + iy , find the following in rectangular form. Use the conjugate root theorem and the … Some of the worksheets for this concept are How to conjugate french verbs present tense, Conjugating verbs to go work, Conjugate acid base pairs name chem work 19 2, Conjugate of complex numbers 1, Bronsted, How to conjugate portuguese verbs, How to conjugate french verbs imperfect tense, Rationalizing imaginary denominators. When you access the file you, This five problem assessment helps you to quickly determine students' understanding of the term "i" within complex numbers (and that it's equal to square root of negative 1) as well as i squared and how to write complex conjugates. Write the conjugate of each. For example, the complex conjugate of 3 + 4 i is 3 − 4 i. There are 20 problems total, separated into two columns. Complementary and supplementary word problems worksheet. 4. 2. Complex number any number that can be written in the form where … Simplifying Expressions with Complex Numbers. Introduction to Complex Numbers ... i2 = –1 To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the complex number in the denominator of the fraction. Use basic rules like combining the like terms together, FOIL method, multiplying the top and bottom by the complex conjugate of the denominator to simplify complex-valued expressions. Sum of two complex numbers a + bi and c + di is given as: (a + bi) + (c + di) = (a + c) + (b + d)i. Complex conjugation negates the imaginary component, so as a transformation of the plane C all points are reflected in the real axis (that is, points above and below the real axis are exchanged). I implement. Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials. Subset, Universal Set, Power Set & Intervals. Write your answers in polar form. Perform operations like addition, subtraction and multiplication on complex numbers, write the complex numbers in standard form, identify the real and imaginary parts, find the conjugate, graph complex numbers, rationalize the denominator, find the absolute value, modulus, and argument in this collection of printable complex number worksheets. 19 min. Answer key included. For division, students must be able to rationalize the denominator, which includes multiplying, Students will practice adding, subtracting, multiplying, and dividing complex numbers with this coloring activity. Conjugate Showing top 8 worksheets in the category - Conjugate . My kids loved this activity. Algebra 2: Complex Numbers ENTIRE UNIT Mini Formative Assessments, Also included in: Quadratics Revisited - Complex Numbers Algebra 2 Unit 1 Bundle, Also included in: Algebra 2 Credit Recovery, Also included in: Algebra 1 Mystery Drawings Volume 1, Also included in: Algebra 2 FUN Notes Doodle Bundle also for Distance Learning Packets, Also included in: Algebra 2 Activities Growing Bundle, Also included in: Algebra 2 Curriculum - Texas TEKS Aligned. Complex Numbers Test It covers the following questions: 1. For example, the complex conjugate of $$3 + 4i$$ is $$3 − 4i$$. The following notation is used for the real and imaginary parts of a complex number z. (x) x x 3 x x f = 5 4 + 6 3 − 1 2 − 6 + 8 is a zero − 2 2. About This Quiz & Worksheet About This Quiz & Worksheet Test your ability to solve complex conjugate problems and find the product of these mathematical expressions in this … 2. They need to do their Herowork! A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers,... Complex Conjugates. Some of the worksheets for this concept are Conjugating verbs to go work, Conjugate acid base pairs name chem work 19 2, How to conjugate french verbs present tense, How to conjugate portuguese verbs, Conjugate of complex numbers 1, Rationalizing imaginary denominators, Dividing radical. These complex numbers are known as complex conjugates. Explain why complex conjugates do not have an “i” term Challenge Problems 1. Equal, Conjugate, Opposite and Reciprocal Complex Numbers. If = + then ̅(pronounced zed bar), is given by = − , and this is called the complex conjugate of z. Divide Complex Numbers Dividing by Complex Numbers For the complex number , identify the real number and the imaginary number. But 7 minus 5i is also the conjugate of 7 plus 5i, for obvious reasons. 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# Budget built MiniMax 1100 Ultralight

Discussion in 'Hangar Flying' started by Little Scrapper, Mar 10, 2018.

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1. Mar 10, 2018

### Little Scrapper

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Cancelled work today and stayed home with my son who is pretty sick. He took a nap so I decided to rummage through all my airplane plans and organize them.

Came upon these MiniMax plans. Pretty fun to page through these. It's amazing how simple something like this is. Why is it people aren't building this in droves?

So you know, a guy starts thinking.

Most here who have followed my Cassutt thread have seen my shop in the background of the photos as well as some of my tools. You wouldn't believe how many people make comments to me about how they can't build because they don't have a shop like that and a few other excuses. I'm very greatful for my shop but the photos don't show just how small and simple it really is. Still, the excuses come in droves. A shop for most builders is pretty rare, it's just not a big requirement. Many seem to forget that a great deal of successful builders didn't spend thousands of dollars and months of labor to build a shop. I'm guilty of doing this but it's not required. A kitchen table, a cheap bench and a $5 light is really all anyone needs. Cost. I'm sure most of you have heard from people about the constant bickering on affordable aviation. It's as if everyone has to have a$80,000 airplane or they refuse to participate. Looking at these plans is there really anything wrong with a simple wood 2 stroke powered ultralight? Again, paging through these plans I can't imagine why people don't just build this little guy and go have fun.

So you know, I did some more thinking.

Airventure is 20 weeks away. I have about 800' of 1/4" capstrip and a sheet of 1/16" mahogany in my basement. Definitely have nails. I have a few spars. I have a Spruce/Doug fir company 30 minutes from me. Airventure is a hundred miles away. My Cassutt gets worked on only when at I'm at my shop, which is not at my house it requires a trip. So none of this really intrudes on my Cassutt time. I have no shop at all left in my home, it's all moved out and in my shop. The only tools I have are in my service truck. Pretty sure I can build this at home with the bare minimum of tooling.

I'm thinking I could build 2 airplanes at once. The Cassutt in my shop and a Minimax 1100 Ultralight at home at night when my wife is working out. Cassutt is a long term project for fun.

So today I decided to rapidly build a MiniMax 1100. Here's a thread about building a simple airplane with minimal tools, minimal money, and for people who can't afford a pilots liscense......it's an ultralight. There's no excuse for anyone! And for those who don't want to wait a few years like my Cassutt or a lot of other airplanes will take well, I plan on flying this before summer is over. I don't think I can build it in time for Airventure because 20 weeks just doesn't seem possible does it? I've never built a wood airplane so let's just say I have no experience at all. Everyone knows my passion is metal and welding, so I'm completely out of my realm here. I still think I can build this and fly it in under 7 months. I could be wrong, but I think I could do this.

As a recap, this is a fun low stress thread about building a little open cockpit airplane made of wood on a budget and in a short period of time in a small space starting on a kitchen table. There's no excuse not to fly and I'm gonna show that on this thread.

I'm off to find a board for a wing jig. In order for me to get this done in a few months I'll need to not only build my rib jig this weekend but actually build a rib by Sunday so I can do ribs next week.

Because I have a project thread already I decided I'm going to make automatic monthly donations here to the forum to pay for the extra photos. It's the responsible thing to do.

If anyone wants to participate here's how you can help.

1.) If anyone has ever built or is building a MiniMax please post some helpful tips to expedite the build process.

2.) If anyone here has built one or has just studied the Ultralight MiniMax any advice on budget wheels, fuel tank or parts etc is welcomed. This will save me some research time which is greatly needed.

3.) I need to find a little Rotax and do it on a budget so we can say "here's how you can do it on a budget". Any advice or tips here is greatly appreciated!

4.) Please keep comments fun, exciting and encouraging because this thread is for those who say aviation isn't for the average person. Let's prove them wrong.

Time to get busy, summer will be here before we know it.

Scrap.

Last edited: Mar 10, 2018
2. Mar 10, 2018

### cdlwingnut

#### Well-Known MemberHBA Supporter

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Good luck scrap. You will find working with wood very fun. My ragwing is very similar and im guessing will come in under $3000 to build CobraCar11 and Little Scrapper like this. 3. Mar 10, 2018 ### rbrochey ### rbrochey #### Well-Known Member Joined: Oct 17, 2010 Messages: 1,503 Likes Received: 610 Location: Gallup, New Mexico USA I think that is a terrific idea!!!! And you're right, more people should be building them! If I'm not mistaken, Lynn built a nice mini max... CobraCar11 and Little Scrapper like this. 4. Mar 10, 2018 ### Chris In Marshfield ### Chris In Marshfield #### Well-Known MemberHBA Supporter Joined: Jan 29, 2008 Messages: 1,229 Likes Received: 1,006 Location: Germantown, WI USA Nice! I have an LMA wood plane I’m dying to build some time. I may build it at the lake cabin so I have something to do when not fishing. Little Scrapper likes this. 5. Mar 10, 2018 ### Pops ### Pops #### Well-Known Member Joined: Jan 1, 2013 Messages: 7,202 Likes Received: 6,100 Location: USA. Friend of mine built 4 Mini-Max's in a very small upstairs spare bedroom. Put 1K hrs on one with a 32 hp 1/2 VW engine. Ran the car VW distributor powered by a 12 volt, 5 amp model airplane starting battery from Tower Hobbies. Would fly several hundred cross country flight by stopping every 2 hrs and refuel and charged the battery at the airport. Burned 1.7 gph and cruised at 75 mph. Then he put a small 2 cylinder Cont-GPU on the last MM just to see if it would fly. BTW-- Its a legal Ultralight by about 2 pounds. He uses a Hummelbird canopy on the MM's and hot wires foam and tape to streamline the wing struts. https://www.youtube.com/watch?v=cj0hdymXfOg&t=29s https://www.youtube.com/watch?v=ppR5Boo7cn4 CobraCar11 and Little Scrapper like this. 6. Mar 10, 2018 ### Little Scrapper ### Little Scrapper #### Well-Known MemberHBA Supporter Joined: Jan 3, 2014 Messages: 5,352 Likes Received: 3,280 Location: Wisconsin I may have made a mistake. The plans I have are the 1100 but after reading pops post I discovered actual Ultralight from Minimax is the 1030 or 103 model not the 1100. If anyone knows of the wing is the same let me know or I'll need to print a set of 103 plans on Monday. In the mean time can anyone she'd some light on the Rotax 277 vs the Hirth F33 singles. I have some Rotax experience but not enough to matter. 7. Mar 10, 2018 ### fly2kads ### fly2kads #### Well-Known MemberHBA Supporter Joined: Jan 2, 2010 Messages: 1,461 Likes Received: 537 Location: Justin, TX I thought it was the various options that put the 1100 over the weight and/or speed limits of Part 103, and that with the F33 and just the basics, it would come in under spec? I applaud the effort, one way or the other! Little Scrapper likes this. 8. Mar 10, 2018 ### TFF ### TFF #### Well-Known Member Joined: Apr 28, 2010 Messages: 11,702 Likes Received: 3,311 Location: Memphis, TN One of my old as dirt buddies, retired, at the airport was at the airport 6 days a week messing with his planes; at the time he had about 8 planes stuffed in his hangar and then projects. His wife said, you have to be home at least Saturday and Sunday. He built two Piets at the same time, when home. I would 1/2 v-dub it. Little Scrapper likes this. 9. Mar 10, 2018 ### don january ### don january #### Well-Known Member Joined: Feb 11, 2015 Messages: 2,660 Likes Received: 1,086 Location: Midwest Scrap: I can't answer much engine questions but I will give a bit of insight of things you will need and you may already have. I have found a Shopsmith is priceless when building an all wood aircraft but your three basics are Table saw-drill press- disk sander and don't forget some soft tipped clamps. Wood doesn't like that metal squeezing on it. As you build your Cassutt you probably don't worry to much about banging your tubes around a bit but I found wood is more dent happy. You must be ready for a much more dusty work area so daily sweeping is part of your routine and I can say try to set up your work area away from your furnace and laundry area the best you can. Spring cleaning becomes a daily thing even upstairs in my house I have to dust some of the wife's pretties. There are many things you will come up against and the good thing is there is alot of help here on the forum and I will say unlike the Cassutt you wont be able to cut the tack loose if the fit wasn't where you wanted it. I say "once glued once screwed" I'm pulling for you Scrap and like you I have been getting the parts together for my PDQ. 10. Mar 10, 2018 ### lake_harley ### lake_harley #### Well-Known Member Joined: Jan 5, 2007 Messages: 954 Likes Received: 300 Like Robert mentioned in a post above, I did build a MiniMAX. It's basically an 1100, but I did use the plans for the 1030 wing and tail. The 1030 wing is slightly different, mostly in the sizes of anti-drag members used. I also used 1.0MM ply for the leading edge instead of 1.5MM. A Legal Eagle wing is almost a copy of the MiniMAX wing and uses .8MM ply leading edges. Also after talking with a MiniMAX builder who told me he used 1.0MM, I felt quite confident to follow suit. Pops' friend in the earlier post with video links was the builder I talked with about the 1.0MM leading edge ply. He made weight with the 2A042 surplus engine that out weighs a 277 by about 10 or 15#! The 1030 plans also show the "light tail". It's a bit more time consuming to build than the standard tail, but with a light engine like a 277 you would need to be concerned about a tail heavy plane with the standard version. My plane was on the border of tail heavy, and that was with the light tail. With that said I'll mention my "hybrid" MiniMAX came in at 246# empty. I built from scratch using carefully selected Northern White Pine from a local lumber yard and aircraft grade ply from Wicks. I ended up with about$4500 in my completed plane and that included the 277 that I freshened up, and bought new instruments. I was pleased!

Something I'd do differently if I were building another MiniMAX is to use 1.0 or possibly even .8MM ply to cut the rib gussets. Every bit of weight savings adds up in the end and it doesn't cost anything to save weight where it's only a matter of being careful to not oversize anything that goes into the plane. All of the wood that I re-sawed to the required dimensions to build the plane were done with a dial caliper in hand to set the table saw. To me, 1/4" is .250" and not .280 or .300" like one might do using just a tape measure or ruler

The plans are VERY GOOD and pretty easy to follow, even for a 1st time builder like I was. I did seek advice on several occasions and will have to admit many of my questions were related to thinking I would so something slightly different than shown on the plans. In hindsight I'd say stick to the plans straight though the build and things will go faster. I don't really feel I'm leading you astray with the suggestions I made in the first two paragraphs though.

If you were closer, I'd loan you my rib jig and get you off to a faster start by saving you the time to build one!

BTW, my MiniMAX has flown but only by an experienced pilot friend of mine. He said it flew nicely, had no bad habits and landed without drama. I hope to fly it too, but I need more stick time trying to land a Kitfox II of which I own 50%. That seems to be a damned cantankerous plane to land, but I want to get proficient on it before it's just me up there.

I'll also mention the great support that available on the East Tennessee Lonesome Buzzard website. It's all about building and flying a MiniMAX. http://www.lonesomebuzzards.com/cgi-bin/forum/Blah.pl/Blah.pl?b-cc/

Happy building!

Lynn

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11. Mar 10, 2018

### Beragoobruce

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Plus one on the Lonesome Buzzards forum - it's a great resource.

I built, and am flying, an Eros, but all the Max family are essentially very similar. Here's a link to my build thread in the Buzzards board:
http://www.lonesomebuzzards.com/cgi-bin/forum/Blah.pl?m-1409444380/ Also includes a link on making a propeller, if you really want to save money. Note that the first few photos were done from photobucket, & the miserable sods have stopped showing those now. Later build pics were uploaded from my camera, so appear ok.

You will surely enjoy building one of these great little planes, and they are great fun to fly: very controllable, highly responsive, and leave you with a huuuge smile after every flight.

I look forward to your build blog.

Bruce

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12. Mar 10, 2018

### FritzW

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If your really, truly, no s#it, honest injun serious about quick building a Mini/Hi Max I'll donate the 1/8" ply parts of a quick build "plug and play" MiniMax wing:
http://www.homebuiltairplanes.com/forums/showthread.php?t=20101

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13. Mar 10, 2018

### Little Scrapper

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Fritz, that's a gosh darn great offer. I looked at the pages in your link and what an awesome job. Top notch all the way.

I'm going to have to respectfully decline. Please don't be offended and let me explain.

This could easily backfire on me. I have to be realistic. While I certainly think my plan is solid often times the real world steps in and things look a bit different. If I took that from you it's a larger burden than I'm willing to carry if things don't go as planned. I really hope you understand. And I think there's sort of a second reason as well. I need to bleed a little bit and go through the full process 100%. It may sound a bit off or possibly stupid but I really want to show the guy who thinks he can't fly affordably and is out of time that it's quite possible to do this. I'm still very much involved in my Cassutt in fact I might be buying a 0-200 tomorrow if things work out so this little ultralight is just a kind a thing I'm trying to do at home with free time at home.

Again, I'm humbled by your offer, I have the upmost respect for even considering that.

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14. Mar 10, 2018

### Little Scrapper

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just a quick notes to the other folks who responded. Thank you.

I visited the Lonesome Buzzards site, pretty cool indeed. And thanks for explaining the model differences and the improvements.

Question. I have a large amount of 1/16" Mahogany ply for gussets, should I just use that or just get the finish birch for the ribs? These minimaxes really seem to like the birch plywood.

2nd question. I see a lot of people using alternate wood. Where in the plans does it say you can substitute Spruce with N White pine? Obviously it works and it's common, I'm just curious how and why this came about. I guess I'm not even sure I know where to locate NW Pine.

Thanks!

15. Mar 10, 2018

### Victor Bravo

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FWIW, first decide if you genuinely need to make Part 103 weight. Most of the 103 air "vehicles" flying around as ultralights may be a pound or two over weight. The letter of the law is one thing and reality is another. If you have a Minimax that looks, sounds, and is not N numbered like an UL, with a 5 gallon can as the fuel tank, and it is operating in areas where UL's operate, buzzing around with other UL's, you will find that the FAA is not driving around with scales in their car looking to bust you on being overweight.

If you are conservative with the aircraft and you are not caught on video flying in a risky or scofflaw manner, the FAA has a lot of other things to do instead of running around enforcing strict 103 rules.

Also, understand that many of todays UL engines are lighter than the old UL engines. You may well find that a Polini or Simonini paramotor engine that makes 27 or 30HP is lighter than the old 277. So you MAY be able to make UL weight without resorting to the extremes that others have had to in the past.

If you already have any sort of pilot license you probably don't need to make 103 weight... unless reason for this entire exercise includes a true 103 airplane that does not need a license.

IMHO whatever you do, try not to put those tiny little dinner plate size wheels and tires on the airplane. Larger wheels protect against damaging the airplane when operating off of dirt, sod, etc. There is a safety issue under all that as well, you can do a nose-over much more easily with the tiny wheels, leaving you upside down soaked with gasoline and a hot exhaust pipe. Goes from comical to dead serious in a heartbeat.

As mentioned by other posters, sawdust, sanding dust and scrap wood is a much bigger safety hazard than metal. So a dust collector hooked up to your power tools is a good investment.

Another way to save significant weight if you are shooting for Part 103 is to use the Oratex covering or the Stewart Systems method, both of which are lighter than Stits, Poly-Fiber, Ceconite, Blue River, etc. The choice of covering alone can be the difference in meeting 103 weight. White Oratex with no large areas of trim color is probably worth a few pounds versus a beautiful American flag or aerobatic starburst paint scheme. Also takes less time. I think Oratex may be lighter than Stewart, not 100% sure.

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16. Mar 10, 2018

### Dana

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If you want to go 2-stroke you might consider a rebuilt Cuyuna engine instead of a Rotax. The Cuyunas got a bad reputation back in the day but I believe that was because nobody understood 2-strokes and Rotax would have fared no better. Anyway, Cuyunas (and their parts) are a lot cheaper than any of the other options. My Ultrastar had a Cuyuna and it was trouble free for the time I had it, about 200 hours of flying.

But if I were building a Minimax I'd be inclined to go with a half VW. I had one in my Fisher, of course, and my friend has a Minimax (N numbered) with a Global half VW... been sitting for a few years, we're hoping to get it flying this spring.

Dana

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17. Mar 10, 2018

### lake_harley

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Providing I clicked the correct buttons my comments should be below in red.

Lynn

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18. Mar 10, 2018

### Wayne

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Scrap -
Don't be reluctant to call Dave Cooper at Mini Max if needed, he is a great guy and has encyclopedic knowledge. He is the real deal when it comes to these aircraft. Before my Dad decided to fund the Cruzer I was going to build a Hi-Max and have spent hours out there in Niles. I met a CFI there and flew about 20 hours out of Niles, MI and South Bend. The TEAM factory is a hangar full of spruce and partially built Mini Max's.

Dave told me that with my years of model airplane building (balsa sticks, tissue, dope anyone?) he thought 300-400 hours. That seems low but they are simple aircraft.

Dave will be thrilled to see this thread - he's a good dude and those guys are doing this for the love of it - not to get rich.

Little Scrapper likes this.
19. Mar 10, 2018

### Little Scrapper

#### Well-Known MemberHBA Supporter

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Thank you.

Just to hit a few points and answer some questions.

The purpose of this thread is to produce a cheap fun flying airplane from start to finish in a few months and fly it before summer ends. I think I can do it. I'm probably doing this for the wrong reasons but I want to personally answer for those that say aviation is not affordable. When I'm done with this project anybody can come to this thread and say, ok, here's something I can do with minimal tools, minimal experience, minimal time and minimal input costs. I kinda felt that was a good idea. So, sticking with that theme it made sense to me to pick a well proven design that's readily available and has a great track record.

It also means sticking to the plans and non customizing because that just adds time. It means an ultralight that's a true ultralight and a airplane that's bone stock and to the plans with no reverse engineering of any type. Just read the plans and build it.

On engines. While I've had rotax 2 stroke experience I'm smart enough to know I'm still lacking in knowledge with any sort of real depth. I just now that the engine I pick needs to be part of the company recommended ones and that alone will save me time.

Materials. These plans are interesting the more you read them, they are actually quite good. Last night was the first time I really sat down and studied them and it's starting to make sense. Attached is a photo that talks about material.

RS 0-50 (raw stock) can be any pine if I'm reading this correctly. And of course this pine needs to meet aircraft standards as it concerns grain, direction, run-out, etc. I'll call TEAM to verify.

Fortunately I can get started today on my kitchen table and build a dirt simple, cheap, no frills, rib jig and start making ribs immediately because I have an absolute ton of 1/4" x 1/4" spruce cap strip.

Just a note. Because my plumbing shop where my Cassutt is is not my home I'll be doing all wood cutting at home.....in my lawn....out in the open. Kinda old school. I don't want any woodworking dust in my plumbing shop at all and I certainly don't want it mixed in with customer inventory or my metal fab stuff. Remember, this is a at home project with basic tools and no "shop" to speak of. After this is complete my home life resumes

Appreciate people chiming in on this. Should be interesting going forward.

20. Mar 10, 2018

### Little Scrapper

#### Well-Known MemberHBA Supporter

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One thing I forget to mention. Forum member Kevin N sent my son a Minimax either a tail kit or rudder kit that I have in a box. I think tail kit. I've been sitting on it for over a year and forget all about it. I want my son to be just a bit older and more mature. That's a real bonus. I'll be digging that out today.


text:
# Thomas Company recorded operating data for its shoe division for the year. Sales....................

Thomas Company recorded operating data for its shoe division for the year.

Sales.......................$1,500,000 Contribution margin.......300,000 Controllable fixed costs..180,000 Average total operating assets:$600,000

How much is the departmental margin for the year?

a) 20%

b) 50%

c) $300,00 d)$120,000

e) none of the above

Answer is Option D Explanation: The departmental margin is the difference between the value of contribution margin and controllable fixed costs. The formula that can be derived for departmental margin is: Departmental Margin = Contribution Margin -...


text:
## d00gie_h0w5er_ Group Title ***HELP*** A car accelerates uniformly from +10.0 m/s to +40.0 m/s over a distance of 125 m. How long did it take to go that distance? Show all your work, including the equation used, given and unknown quantities, and any algebra required. Make sure your answer has the correct number of significant figures. one year ago one year ago

1. ivancsc1996

$v _{f}^{2}=v _{i}^{2}+2ax=1600\frac{ m ^{2} }{ s ^{2} }=100\frac{ m ^{2} }{ s ^{2} }+2a125m \rightarrow a=\frac{ 1600\frac{ m ^{2} }{ s ^{2} }-100\frac{ m ^{2} }{ s ^{2} } }{ 2 \times 125m }$Since $a=\frac{ v _{f}-v _{i} }{ t }$Then$t=\frac{ (40\frac{ m }{ s }-10\frac{ m }{ s })125m2 }{ 1600\frac{ m ^{^{2}} }{ s ^{2} }-100\frac{ m ^{2} }{ s ^{2} } }=5s$


text:
# estimate of the error variance Blackwood, New Jersey

Therefore, the brand B thermometer should yield more precise future predictions than the brand A thermometer. However, one can use other estimators for σ 2 {\displaystyle \sigma ^{2}} which are proportional to S n − 1 2 {\displaystyle S_{n-1}^{2}} , and an appropriate choice can always give Like the variance, MSE has the same units of measurement as the square of the quantity being estimated. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Skip to Content Eberly College of Science STAT 501 Regression Methods Home » Lesson 1: Simple Linear Regression 1.4

The result for S n − 1 2 {\displaystyle S_{n-1}^{2}} follows easily from the χ n − 1 2 {\displaystyle \chi _{n-1}^{2}} variance that is 2 n − 2 {\displaystyle 2n-2} Make all the statements true Meaning of S. Near Earth vs Newtonian gravitational potential Physically locating the server Why does argv include the program name? This also is a known, computed quantity, and it varies by sample and by out-of-sample test space.

The standard error of the estimate is a measure of the accuracy of predictions. MR1639875. ^ Wackerly, Dennis; Mendenhall, William; Scheaffer, Richard L. (2008). Formulas for a sample comparable to the ones for a population are shown below. I got lost when $\sigma^2$ is calculated.

regression"? Why did it take 10,000 years to discover the Bajoran wormhole? See also stats.stackexchange.com/questions/5135/… –conjugateprior Sep 8 '14 at 13:11 add a comment| 3 Answers 3 active oldest votes up vote 2 down vote accepted Looking at ISL's parent book, ESL (Elements For an unbiased estimator, the MSE is the variance of the estimator.

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. There are, however, some scenarios where mean squared error can serve as a good approximation to a loss function occurring naturally in an application.[6] Like variance, mean squared error has the The denominator is the sample size reduced by the number of model parameters estimated from the same data, (n-p) for p regressors or (n-p-1) if an intercept is used.[3] For more The usual estimator for the mean is the sample average X ¯ = 1 n ∑ i = 1 n X i {\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}} which has an expected

If the estimator is derived from a sample statistic and is used to estimate some population statistic, then the expectation is with respect to the sampling distribution of the sample statistic. Probability and Statistics (2nd ed.). You measure the temperature in Celsius and Fahrenheit using each brand of thermometer on ten different days. Examples Mean Suppose we have a random sample of size n from a population, X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} .

That being said, the MSE could be a function of unknown parameters, in which case any estimator of the MSE based on estimates of these parameters would be a function of Substitute $\frac{RSS}{N-2}$ into the equation for SE$(\hat{\beta_1})^2$ and you will get the values in ISL. Mathematical Statistics with Applications (7 ed.). Please try the request again.

Contents 1 Definition and basic properties 1.1 Predictor 1.2 Estimator 1.2.1 Proof of variance and bias relationship 2 Regression 3 Examples 3.1 Mean 3.2 Variance 3.3 Gaussian distribution 4 Interpretation 5 Introduction to the Theory of Statistics (3rd ed.). Belmont, CA, USA: Thomson Higher Education. Your cache administrator is webmaster.

This definition for a known, computed quantity differs from the above definition for the computed MSE of a predictor in that a different denominator is used. The numerator is the sum of squared differences between the actual scores and the predicted scores. The mean square error: $MSE=\frac{\sum_{i=1}^{n}(y_i-\hat{y}_i)^2}{n-2}$ estimates σ2, the common variance of the many subpopulations. On the other hand, predictions of the Fahrenheit temperatures using the brand A thermometer can deviate quite a bit from the actual observed Fahrenheit temperature.

Mean squared error From Wikipedia, the free encyclopedia Jump to: navigation, search "Mean squared deviation" redirects here. See also James–Stein estimator Hodges' estimator Mean percentage error Mean square weighted deviation Mean squared displacement Mean squared prediction error Minimum mean squared error estimator Mean square quantization error Mean square And, the denominator divides the sum by n-2, not n-1, because in using $$\hat{y}_i$$ to estimate μY, we effectively estimate two parameters — the population intercept β0 and the population slope The MSE can be written as the sum of the variance of the estimator and the squared bias of the estimator, providing a useful way to calculate the MSE and implying

Values of MSE may be used for comparative purposes. New York: Springer. Note that, although the MSE (as defined in the present article) is not an unbiased estimator of the error variance, it is consistent, given the consistency of the predictor. more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed

Estimator The MSE of an estimator θ ^ {\displaystyle {\hat {\theta }}} with respect to an unknown parameter θ {\displaystyle \theta } is defined as MSE  ( θ ^ ) Addison-Wesley. ^ Berger, James O. (1985). "2.4.2 Certain Standard Loss Functions". What is alluded to by "In general, σ2 is not known, but can be estimated from the data. MR0804611. ^ Sergio Bermejo, Joan Cabestany (2001) "Oriented principal component analysis for large margin classifiers", Neural Networks, 14 (10), 1447–1461.

The estimate of σ2 shows up indirectly on Minitab's "fitted line plot." For example, for the student height and weight data (student_height_weight.txt), the quantity emphasized in the box, S = 8.64137, What is the best way to upgrade gear in Diablo 3? Two or more statistical models may be compared using their MSEs as a measure of how well they explain a given set of observations: An unbiased estimator (estimated from a statistical Previous Page | Next Page | Top of Page Copyright © 2009 by SAS Institute Inc., Cary, NC, USA.

Applications Minimizing MSE is a key criterion in selecting estimators: see minimum mean-square error. MSE is a risk function, corresponding to the expected value of the squared error loss or quadratic loss. Note that, although the MSE (as defined in the present article) is not an unbiased estimator of the error variance, it is consistent, given the consistency of the predictor. The system returned: (22) Invalid argument The remote host or network may be down.

self-study multiple-regression residuals terminology share|improve this question edited Nov 17 '13 at 18:57 gung 74.1k19160309 asked Nov 17 '13 at 18:02 yasar 12315 1 What exactly do you mean by Is that how you are using the term, or do you mean a model w/ >1 predictor variable but only 1 response variable? –gung Nov 17 '13 at 18:47 Theory of Point Estimation (2nd ed.). If we use the brand B estimated line to predict the Fahrenheit temperature, our prediction should never really be too far off from the actual observed Fahrenheit temperature.


text:
Thesis Open Access

# Feedback Regulated Star Formation in Cool Core Clusters of Galaxies

Tremblay, Grant Russell

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The classical "cooling flow" model historically associated with "cool core" clusters of galaxies fails in the absence of an external, non-gravitational heating mechanism needed to offset catastrophic radiative losses of the X-ray bright intracluster medium (ICM). Numerous proposed solutions exist, including feedback from active galactic nuclei (AGN), which may elegantly calibrate fundamental relationships such as the coupled co-evolution of black holes and the stellar component of their host galaxies. AGN feedback cannot completely offset cooling at all times, however, as the brightest cluster galaxies (BCGs) in cool core clusters harbor extensive warm (∼104 K) and cold (10 < T < 104 K) gas reservoirs whose physical properties are regulated by ongoing star formation and an unknown, non-stellar heating mechanism.
\n\n
We present a doctoral thesis broadly related to these issues, particularly as they pertain to cooling flows, the triggering of AGN activity, and the associated energetic feedback that may play a critical role in heating the ambient environment on tens to hundreds of kiloparsec scales. We begin with a summary of the relevant background material, and in Chapter 2 we present a multiwavelength study of effervescent AGN heating in the cool core cluster Abell 2597. Previously unpublished Chandra X-ray data show the central regions of the hot intracluster medium (ICM) to be highly anisotropic on the scale of the BCG, permeated by a network of kpc-scale X-ray cavities, the largest of which is cospatial in projection with extended 330 MHz radio emission. We present spectral maps of projected, modeled gas properties fit to the X-ray data. The X-ray temperature map reveals two discrete, "hard-edged'' structures, including a ∼15 kpc "cold filament'' and an arc of hot gas which in projection borders the inner edge of the large X-ray cavity. We interpret the latter in the context of the effervescent AGN heating model, in which cavity enthalpy is thermalized as the ambient keV gas rushes to refill the wake of the buoyant bubble. The hot arc revealed in the temperature map may be one of the first instances in which ICM/ISM heating by AGN feedback is directly observed. The ∼15 kpc soft excess filament, part of which is cospatial with extended 1.3 GHz radio emission, may be associated with dredge-up of low entropy gas by the propagating radio source. Results from our study of the hot X-ray gas are framed in the context of inferred young stellar component ages associated with the central emission line nebula in the BCG. We find that inferred ages of the young stellar component are both younger and older than the inferred ages of the X-ray cavities, suggesting that low levels of star formation have managed to persist amid the AGN feedback-driven excavation of the X-ray cavity network.
\n\n
In Chapter 3 we present Hubble Space Telescope far-ultraviolet imaging of seven BCGs in cool core clusters selected on the basis of elevated star formation rates. We find that even at low levels, star formation provides a dominant contribution to the ionizing photon reservoir required to power the observed luminosities of the emission line nebula. Weak, compact radio sources are observed in each of these seven BCGs. The combination of higher SFR and lower radio power is consistent with a scenario wherein a low state of AGN feedback allows for increased residual condensation from the ambient X-ray atmosphere, accounting for the elevated star formation rates.
\n\n
In Chapter 4 we present a comparison study of episodic star formation and AGN activity in the giant radio galaxy 3C 236, which is not associated with a cluster. We find that an episodic AGN/starburst connection can be fostered by a non-steady transport of gas to the nucleus. These results are then compared with Abell 2597, enabling a better understanding of the roles that may be played by cooling flows vs. mergers and hot vs. cold accretion modes in depositing the gaseous reservoirs that fuel both star formation and AGN activity.
\n\n
In Chapter 5 we broaden the context of the thesis with a search for high redshift Fanaroff-Riley class I radio galaxies, which may act as observable "beacons" for assembling protoclusters. Probing the epoch of cluster assembly will be critical to a better understanding of the evolution of the cool core phenomenon and the history of cluster entropy regulation in general. The relative inability of X-ray cluster selection techniques to extend to these redshifts necessitates alternative detection methods, one of which we describe in this thesis.
\n\n
Finally, in Chapter 6 we discuss the main conclusions of this thesis, which can be summarized as follows: (1) AGN feedback is real, and likely plays a dominant role in regulating the pathway of entropy loss from hot ambient medium to cold gas to star formation; (2) AGN feedback does not establish an impassable "entropy floor" below which gas cannot cool; and (3) star formation plays an important role in determining the temperature and ionization of the warm (∼104 K) and cold (10 < T < 104 K) gas phases in brightest cluster galaxies.
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text:
Quotient Rule Of Exponents Calculator

Quotient Rule Calculator is a free online tool that displays the derivative of a function. BYJU’S online quotient rule calculator tool makes the calculation faster, and it displays the derivative of a given function in a fraction of seconds. How to Use the Quotient Rule Calculator? The procedure to use the quotient rule calculator is as follows:

1. Estimated Reading Time: 30 secs

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2019-05-28 · Example. Use the quotient rule for exponents to simplify the expression. x 4 x 3 \frac {x^4} {x^3} x 3 x 4 . The base of the expression in the numerator is x x x, and the base of the expression in the denominator is x x x, which means that the bases are the same, so we can use the quotient rule for exponents.

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To divide these two exponents with a base of ten, we will use the quotient rule. This rule states that when you are dividing two exponents with the …

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Example. Use the quotient rule for exponents to simplify the expression. x 4 x 3 \frac {x^4} {x^3} x 3 x 4 . The base of the expression in the numerator is x x x, and the base of the expression in the denominator is x x x, which means that the bases are the same, so we can use the quotient rule for exponents. We’ll subtract the exponent in

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Posted in: 10 laws of exponents

Exponents Calculator - Symbolab tip www.symbolab.com. Simplify exponential expressions using algebraic rules step-by-step. \square! \square! . Get step-by-step solutions from expert tutors as fast as 15-30 minutes. Your first 5 questions are on us!2^X\Cdot2^Y‧(X^{12})/(X^{-2)} - Exponents Calculator

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Posted in: Quotient rule calculator with steps

Basic exponent laws and rules When exponents that share the same base are multiplied, the exponents are added. a n × a m = a (n+m) EX: 2 2 × 2 4 = 4 × 16 = 64 2 2 × 2 4 = 2 (2 + 4) = 2 6 = 64 When an exponent is negative, the negative sign is removed by reciprocating the base and raising it to the positive exponent. a (-n) = 1 a n

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Quotient rule of differentiation Calculator online with solution and steps. Detailed step by step solutions to your Quotient rule of differentiation problems online with our math solver and calculator. Solved exercises of Quotient rule of differentiation.

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Law of Exponents: Product and Quotient Rule DRAFT. 21 hours ago by. michaellwilliams. 7th - 8th grade . Mathematics. Played 0 times. 0 likes. 0% average accuracy. 0. Save. Edit. Edit. Print; Share; Edit; Delete; Report an issue; Live modes. Start a live quiz . Classic . Students progress at their own pace and you see a leaderboard and live results. Instructor-paced BETA . Control …

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Law of Exponents: Power of a Quotient Rule ( (a/b) m = (a m /b m )) The quotient rule states that two powers with the same base can be divided by subtracting the exponents. Follow this simple rule to adeptly and quickly solve exponent problems using the power of a quotient rule.

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Laws of Exponents: Quotient RuleIn this video tutorial, I will teach you how to simplify expressions using the quotient rule.If you want to test your skill,

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In order to multiply powers with the same base, we add their exponents. Preview this quiz on Quizizz. Quiz. Law of Exponents: Product and Quotient Rule. DRAFT. 7th - 8th grade . Played 0 times. 0% average accuracy. Mathematics. 4 minutes ago by. eddington_gambiza_22039. 0. Save. Edit. Edit. Law of Exponents: Product and Quotient Rule DRAFT. 4 minutes ago by. …

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Quotient of powers Calculator online with solution and steps. Detailed step by step solutions to your Quotient of powers problems online with our math solver and calculator. Solved exercises of Quotient of powers. Calculators Topics Solving Methods Go Premium. ENG • ESP. Topics Login. Tap to take a pic of the problem. calculators. Quotient Of Powers. Quotient of powers …

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This is my way of providing free tutoring for the students in my class and for students anywhere in the world. Every video is a short clip that shows exactly how to solve math problems step by step. The problems are done in real time and in front of a regular classroom. These videos are intended to help you learn how to solve math problems, review how to solve a math problems, study for …

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New Popular Law

What is the quotient rule of exponents?

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as displaystyle m>n m > n.

What is the law of exponents?

Explaining Law of exponents with crystal-clear examples, this chart helps them drive home the concept. The product rule is: when you multiply two powers with the same base, add the exponents. Train 8th grade students to rewrite each exponential expression as a single exponent with this set of pdf worksheets.

How to teach quotient rule to students?

Also, help them develop substantial skills in finding the value of the unknown exponent and MCQ. Upgrade your skills in solving problems involving quotient rule by practicing these printable worksheets. The rule states that you can divide two powers with the same base by subtracting the exponents.

What is the product rule for exponents?

The product rule is: when you multiply two powers with the same base, add the exponents. Train 8th grade students to rewrite each exponential expression as a single exponent with this set of pdf worksheets. Also, help them develop substantial skills in finding the value of the unknown exponent and MCQ.


text:
## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$\text{a) a solution }\\\text{b) a solution }\\\text{c) a solution }$
$\bf{\text{Solution Outline:}}$ Use the position of the number on the number line to evaluate if it satisfies the given inequality, $c\ge-7 .$ $\bf{\text{Solution Details:}}$ The inequality $c\ge-7$ is satisfied by all numbers that appear on/on the right of $-7$ in the number line. a) In the number line, $0$ appears on the $\text{ right }$ of $-7 .$ Hence, it is $\text{ a solution .}$ b) In the number line, $-5\dfrac{4}{5}$ appears on the $\text{ right }$ of $-7 .$ Hence, it is $\text{ a solution .}$ C) In the number line, $1\dfrac{1}{3}$ appears on the $\text{ right }$ of $-7 .$ Hence, it is $\text{ a solution .}$ Therefore, \begin{array}{l}\require{cancel} \text{a) a solution }\\\text{b) a solution }\\\text{c) a solution } .\end{array}