Vaneisms
What strategy should be used to show how to factor quadratics with a>1?
"Splitting the Middle Term" 3x^2+7x-20 = 3x^2 - 5x + 12x - 20. Then explicitly group the terms after splitting the middle term. (3x^2-5x)(12x-20) Then factor each group and proceed like normal. x(3x-5)+4(3x-5) = (x+4)(3x-5)
What is the origin of "billion"?
"Bi" means 2, so there are two sets of zeros past the thousands (1,000,000,000)
What is the origin of the word "percent"?
"percent" comes from two Latin words "per centum." Later, it was abbreviated per cent. (with a final period), and eventually it was combined into one word. It means, "for each 100"
What is the meaning of the word "Polynomial?"
"poly" = many "nomine" = names law of SAMEness applies, only things which are the same may be added together.
What is the origin of "trillion"?
"tri" means 3, so there are three sets of zeros past the thousands (1,000,000,000,000)
How should we teach the midpoint formula between two points?
(avg of x's, avg of y's)
What points should we make when dividing monomials?
- Answer should have no negative exponents, even though the answer key uses them. - When moving variables to simplify, compare the exponents on both the numerator and the denominator. Whichever is the larger integer tells you on which side, numerator or denominator, the variable will be in the answer keeping the exponent positive. Start with the larger exponent (even if both are negative) and subtract the exponents from the other side. That will be your final exponent on that variable. Exponents can never be negative, but can be zero.
What are the special cases with absolute value equations?
- Where there is no solution, because the absolute value has to be smaller than a negative number, such as in |x+3|<-5 - When the answer is all real numbers. Such as an abolute value needing to be greater than a negative number |x+3| > -5 = all real numbers.
Is 0 a prime or composite number?
0 cannot be prime because it has infinite factors - any number(s) times zero It also cannot be composite because it would violate the "Fundamental Theorem of Arithmetic" - it also would not have a unique prime factorization.
Is 1 a prime or a composite number?
1 cannot be prime due to the "Fundamental Theorem of Arithmetic" that states that every composite number has a unique prime factorization. If one is prime, then every number has an infinite number of prime factorizations. 1 cannot be composite either, because it has only one factor.
When multiplying polynomials larger than two binomials...
Once they distribute, sse multiple lines and line up matching variables to make it easy to simplify. This uses the law of sameness. Ex: (2x+3)(X^2-4x+5) 2x^3-8x^2+10x+3x^2-12x+15 Line them up like an addition problem: 2x^3-8x^2+10x 3x^2-12x+15 2x^3 - 5x^2 - 2x + 15
What nmenonic should be used for Order of Operations.
PEMDAS, but there should be a perenthesis around the P, an exponent on the E, and both "MD" and "AS" should be circled with an arrow. This is to help students know that they do multiplication and division as they come across them. Same with addition and subtraction.
When teaching multiplying binomials...
PK 3393. Teach FOIL acronym to connect with their school teaching methods
How to point out place value to younger students.
Point out on an example number the 100's, 10's, 1's, 1/10's, and 1/100's. 10^2=100's 10^1=10's 10^0=1's 10^-1=1/10's 10^-2=1/100's
What should you know how to do before teaching completing the square?
Should be able to use it to derive the quadratic formula. Tell Algebra II students, could be a bonus question on a Completing This Square test.
How to teach the circle/angle formulas?
Show how they are related by drawing all three cases (see the Vaneisms p14 for the pictures) m<=1/2(a+/-b)
How can we solve simultaneous equations?
1. "Addition" method also works with subtraction (you can subtract one from the other if the student is good with subtraction of integers) 2. Students will initially prefer substitution method, but encourage them to become proficient with the elimination method as most experienced students prefer it 3. in the elimination method, unlike many textbooks, Mathnasium problems often have messy fractions. It may be quicker to redo the elimination again to eliminate the other variable rather than do the messy fraction work to calculate the other variable
What wording should be used for graph transformations?
1. "vertical shift up/down #" - graph works the way you expect (+ moves up and - moves down) 2. "horizontal shift left/right #" - works opposite the way you expect (- moves right and + moves left) 3. y=-|x| is "reflection across the x-axis" 4. y=2|x| is "vertical stretch by a factor of 2" (again, works the way you expect). This is NOT "getting narrower;" it is getting TALLER 5. y=(1/2)|x| is "vertical compression/shrink by a factor of 1/2" 6. y=|2x| is "horizontal compression by a factor of 1/2" (again works opposite the way you expect). This IS getting narrower
How should we graph inequalities in a plane?
1. Convert to slope intercept form. 2. Put ruler on paper in preparation of drawing line. 3. Check to see if the line is included (=) or not. If included, draw a solid line. If not, draw a dashed line (much easier to draw correctly versus erasing) 4. if greater than, shaded region above the line. If less than, shade area below the line.
How should we teach absolute value inequalities?
1. Convert to two absolute value equation and solve to find the critical points. 2. Determine if the critical points are included or not. 3. Use the mnemonic "greatOR" [two disconnected parts=outside of critical points] versus "less thAND" [one part, between the critical points]
What are the ways we use to solve proportions?
1. Cross multiplying, which always works. 2. Equivalent fractions. If you can see a relationship in the equivalent fractions, can save a lot of unpleasant division.
What is the easier way to find the interior angle of a regular polygon?
1. Find the exterior angle. (360/n) 2. Interior angle equals supplement of exterior angle
How can you find square roots mentally?
1. Narrow it to between two squares of tens 2. Figure out if the numbers is closer to the lower square or the upper square (this gets you the first digits of the square root) 2. Use the list below to figure out the one's place 1^2 -> 1 2^2 -> 4 3^2 -> 9 4^2 -> 6 5^2 -> 25 6^2 -> 6 7^2 -> 9 8^2 -> 4 9^2 -> 1
How should you teach GCF and LCM?
1. Prime Factorization: Factor tree down until you reach all prime numbers, then write them exponentially. 2. Cake method (Or "upside down division") - Make sure you can demonstrate.
How can you generate a Pythagorean triple from an even number?
1. Square the number 2. Take the whole number on either side (+/-1) 3. Double original number 6 = 36 = 35, 37. 12. = 12, 35, 37
How can you generate a Pythagorean triple from an odd number
1. Square the number 2. Cut the result in half 3. Take the whole number on either side (+/- 1/2) 5 = 25, 12 1/2 = 12 & 13 = 5, 12, 13
What should students do when finding fractions between two other numbers or fractions?
1. Use equivalent fractions to get common denominators. 2. while student's initial thought will be to keep doubling the numerator and denominator to make room between the two fractions, it's a lot quicker and easier to multiply both numerator and denominator by 10 or 100 to make more room
What to know when teaching slant (oblique) asymptote?
1. When a slant asymptote exists 2. How to calculate them (long or synthetic division) 3. What the remainder of the problem represents
How to square numbers that end in 5?
1. Write down 25 for the last two digits 2. Write down the leading digit times one more for the first digits
When finding the square of an ugly decimal, what should be done to prevent an error?
1. convert to fraction 2. find square root of fraction 3. reconvert to decimal ex: 0.0064 -> 64/10000 -> 8/100 -> 0.08
When working with uniform motion word problems...
1. make a drawing 2. general rule: if going in opposite directions, add the speeds. If going in same direction, subtract the speeds.
What are some strategies for teaching word problems?
1. wholes and parts 2. encourage drawings
What are the divisibility rules for 2, 4, and 8?
2: is an even number 4: can be divided by two twice OR last two digits are divisible by 4 8: can be divided by two thrice OR last three digits are divisible by 8
Which Pythagorean triples should you memorize?
3, 4, 5 and 5, 12, 13
What is the divisibility rule for 3 and 9?
3: The sum of the digits is divisible by 3 9: The sum of the digits is divisible by 9
What are the divisibility rules for 5, 10, and 100?
5: The last digit is 0 or 5 10: The last digit is 0 100: The last two digits are 0
What is the hard way to find the interior angle of a regular polygon?
<= [(n-2)180]÷n
What do you make clear when teaching rational expressions?
A rational function NEVER crosses a vertical asympotote, but CAN cross a horizontal asymptote.
What does a "score" of something mean?
A score is twenty of ANYTHING, not just years
How should we teach absolute value equations?
Always isolate the absolute value. For example, 2|x+3|=8 becomes |x+3|=4
Triangle Congruencey Theorems - what is the that case won't work?
Angle-Side-Side. Tell students to remember that they don't want to be "one of those." You should also be able to demonstrate this "ambiguous case" with pencils as manipulatives.
What do you ask students when they are studying logarithms?
Ask them what a logarithm is, and the response should be "an exponent", if not have a conversation with the student about this.
How should you explain x to the power of 0?
Begin with something they know, such as (x^2)(x^3)=x^5 (x^2)(x^0)=x^2; since multiplying by x^0 does not change the number, x^0 must equal one. Since x can be anything, any number to the zero power is one. (x^2)(x^-2)=x^0=1 Ask for examples of pairs of numbers that multiply to 1, such as 2 and 1/2. These numbers must be reciprocals. So negative exponent means take a positive exponent and then take the reciprocal
How can you find the total number of factors of a composite number?
By using Prime Factorization and the Fundamental Counting Principle. For example, 120=(2^3)* 3 * 5. So there are four possibilities for exponent of 2 (0, 1, 2, 3), two possibilities for exponent of 3 (0, 1), and 2 possibilities for exponent of 5 (0, 1). So total number of factors is 4 * 2 * 2 = 16 total factors.
What should you say and do when a student is wrong?
Circle the answer and say "we'll talk about that one" to avoid embarassment.
How should you explain negative exponents?
Clarify to a student that since (x^2)(x^-2)=(x^0), then x^2 and x^-2 must be reciprocals.
How to teach partial fraction decomposition?
Clearing the fractions method. Multiplying the entire equation by the factored form of the original denominator and then selecting convenient values to find A,B, etc. *Understand this well before teaching*
What is the simplest way to convert to scientific notation?
Count how many places the leading digit was moved from the ones place. To the left is positive power of 10; to the right is negative power of 10. It's easier to count the digits than the spaces where the decimal point would move.
What should you do when learning to borrow with one or more "middle zeroes"?
Cross out the instructions in the PK and teach as shown in the Vaneisms. (go to the first place you can make a number and borrow from it.)
How should we describe the horizontal asymptote?
Describe as a battle. 1. If numerator has larger degree, numerator "wins" and there is NO horiztonal asymptote. 2. If denominator has larger degree, denominator "wins" and the horizontal asymptoteis y=0 3. If there is a tie between the degrees, than y= numerator/denominator of the coefficents with the largets degrees.
What is the divisibility rule for 7?
Double the final digit of the number and subtract that double from the remaining digits. If the final result is divisible by 7, then so is the number. This can be repeated if the number is too large.
What is the Mathnasium trick for dividing by 5?
Double the number and then divide by 10
When teaching the discriminant (b^2-4ac), what should you always do?
Draw the three versions of a parabola to show the number of real solutions When the discriminant is greater than 0: 2 real solutions When the discriminant is equal to 0: 1 real solution (double root at that point) When the discriminant is less than 0: no real solutions (Or 2 complex solutions for Alg II and up)
How to show students angle pair relationships?
Draw two parallel lines cut by a transversal. Draw one with an exaggerated angle. If the angles look the same, they are the same; if they do not look the same, they are supplementary. This will help with vertical angles.
What emphasis should you make when discussing fractional parts?
Emphasize how many pieces you have of a fraction. "What is ONE fourth of 20. What is THREE fourths of twenty?" "TWO thirds of some number is 40, so what is ONE third of the number?" "If 20 is ONE third of the number, what is the number?"
What should you do when encountering sign errors on linear equations?
Encourage the student to find a way to avoid having a negative coefficient on the variable. Often the sign error comes from dividing by a negative number, so show them how to get to a positive coefficient of the variable as quickly as possible without division.
How to solve an exponential equation?
Get rid of the exponent, take the log of both sides (base same as the experssion with the exponent)
How to solve a logarithmic equation?
Get rid of the log, "exponentiate" both sides (base the same as the base of the log)
When students are working on "reasoning in groups," what word should you use consistently?
Group and Groups
When a student is working on significant fraction work in other PK's, what should you begin having them do with division?
Have the student write their remainders as fractions.
What is special about higher math PKs?
If degree of difficulty of PK goes past what is typically taught in US high schools, or even local high schools, ask CD if student should end the PK.
For remainders with complex numbers...
If n>4, divide the exponent by 4 and find the remainder: remainder = 1 >> i remainder = 2 >> -1 remainder = 3 >> -i remainder = 4 >> 1
What is the prime directive at Mathnasium?
If you don't know how to teach something, ASK!! Do NOT fake it
What is the relationship between the GCF and the LCM?
If you multiply the GCF of two number by the LCM of two numbers, you will get the two numbers multiplied together.
What is the origin of the dollar sign?
It is unclear, but some believe that the dollar sign started as the initials of the United States - "U" superimposed on the "S." The bottom of the "U" would overlap the "S" so it was easier to just draw two vertical lines.
What is the origin of "million"?
Million comes from the Latin word for 1,000 - "mille." But instead of meaning 1,000, it means one thousand thousands.
When PK's in algebra have equation steps horizontally instead of vertically, what should be done?
Model the correct format by redoing the example vertically on the back of the previous page.
What is the Mathnasium trick for multiplying by 5?
Multiply by 10 and then cut in half
When multiplying integers, how should we describe sign changes?
Multiplying by a positive number keeps the sign the same, but multiplying by a negative will switch the sign.
What should you show students when they are learning to divide decimals?
Show students why the modern method of moving the decimals works by converting the division problem into a fraction and then multiplying the numerator and denominator by 10 to make an equivalent fraction and clear the decimals. Then convert back to a division problem, now with no decimals.
What should you say when discussing the percent sign?
Show the students the wall art that represents the transformation of the percent sign contains the "1" and the two "zeroes" from the number 100.
What should we use to teach the Quadratic Formula?
Sing to the tune of "Pop Goes the Weasel" "The quadratic formula's negative b, Plus or minus the square root, Of b squared minus 4ac All over 2a."
What should students do when adding denominators?
Students must use best common denominator (LCM). If they do not use the best denominator, they must redo the problem even if the answer was correct. Erase their answer and have them start over. Teach them to do multiples of larger denominator until they find one that can be divided by the smaller.
How do you determine which way a rational function goes, as it apporaches a vertical asymptote?
Test a point close to either side. Substitute these into a factored form and check the sign of each term, and calculate sign of overall function.
What should you show students about the distance formula between two points?
That it is a restatement of the Pythagorean Theorem (draw the right triangle in the grid to show them).
What is strategy should you use to teach borrowing in subtraction when learning it for the first time?
The Borrowing Game: Give the student dimes and a ten sided die. Ask them to move the number of cents equal to what they roll to another side of the table. When they realize they cannot do less than 10 cents, offer to give them 10 pennies for a dime. After the trade, ask them how much money they have (It will be the same). Then tell them to move the pennies. Repeat until out of pennies. This should reinforce that they will have to trade or borrow.
How do you explain percent change?
The amount of the change, divided by what it was before the change. Ex: $90 to $60. = $30 (amount of change)/$90 = 1/3 = 33% discount.
What information can you share with students about the early Roman calendar
The month names used to make sense. An early Roman King added two months at the beginning of the year, pushing them out of the numbered position. September: 7+2=9 October: 8+2=10 November: 9+2=11 December: 10+2=12
What is the origin of the word "mile?"
The word "mile" comes from the Latin word "mille," which means "thousand" (think millennium or millipede). A mile is roughly the distance a Roman legion could march in 1000 paces (a pace is 2 steps)
What is special about answer keys for higher math PKs?
They are always suspect! **You can use "Let's do that one together to double check," to avoid embarassment, in case the answer key is wrong and their answer is right.**
When factoring quadratics with a>1: what should you do if there is one positive and one negative key number?
They should put the negative number first when they split the middle term. However, as an example, show them what happens if you don't - you either have to change the sign of the last term as you effectively factor out a -1 as you are grouping or you realize it later when the terms left inside the parentheses in the next-to-last step do not match. It is key to show students that you do get the same answer either way.
Should you use the word "right...?"
This is almost always wrong. It is a leading term that tells the student that they should have already known this.
What are the oddball cases with compound inequalities?
This is in PK 3412. Somewhere in the second half of the PK, teach the Oddball cases: 1. No solution "and" problems. Example: x<-2 AND x>7 2. "and" or "or" problems where both inqualities go the same way. Example: x<4 OR x<=8; x>-4 AND x>6
How does one graph quadratics in vertex form?
Use graph transformations. Know these steps: Find vertex using vertical shift (outside the parent function) and horizontal shift (inside the parent function). Make sure the student learns the basic pattern for quadratic where a=1. Over one up one, over two up four, over three up nine. If a is negative, go down, not up. If a is not 1, just multiply the basic pattern by a.
Once student understands completing the square approach, what other thing can we teach?
Use the short form of completing the square for circles, ellipses, etc. (Don't show full quadratic)
When working with mixture word problems...
Use the table method. If you do not know this method, do not work with a student on mixture problems until you learn it.
What aids can you use for teaching slope?
Use your arms to test students on their general knowledge of the slopes of lines. 1. For simple version, ask student to pick the right slope that matches your arm. Choices are "positive, negative, zero, or undefined". 2. For more complex version, choices are "+2, +1, +1/2, 0, - 1, -2, or undefined"+
What is a shortcut for finding the equation of a line that is perpendicular?
Using standard form, Ax+By = C, swap the coefficients A&B and change one of the signs, making sure a>=0. Then plug "a" in for x and "b" in for y and calculate the new C. Use caution in teaching this method. Make sure students can do the y=mx+b and/or point-slope method first since they may not be given a line in standard form very often. Even after teaching these easy methods, require student to do half of each page using these more typical methods.
How should you teach types of quadrilaterals?
Using the quadrilateral family tree
How to find the other endpoint of a line segment, given one endpoint and the midpoint
Whatever you do to get from one end to the midpoint, do it again to get to the other endpoint.
What is a shortcut for finding the equation of a line that is parallel ?
When in standard form, Ax+By = C, find the equation of a parallel line, Ax+By stays the same. Simply plug "a" in for x and "b" in for y and calculate the new C. Use caution in teaching this method. Make sure students can do the y=mx+b and/or point-slope method first since they may not be given a line in standard form very often. Even after teaching these easy methods, require student to do half of each page using these more typical methods.
When encountering a linear equation with fractions, what skill should you be able to demonstrate?
You should know how to clear fractions from a linear equation. *If you don't know how, do PK 701. This is absolutely essential if you are going to work on "Work" problems.
What should we use to teach Graph Transformations?
Zane's packet "Transformation Boot Camp" Beginning with the first pages, the student calculates the points and learns the names of transformations. For the final three pages, the student writes the transformations using the formal verbiage and has them checked by instructor, then graphs the function without plugging in values to find points. Student can look back at earlier pages. Always use this formal language when referring to the transformations. If the teacher uses more colloquial verbiage, such as shrink versus compression, use both. Remind student that these same transformations can be used with any parent function and will almost certainly show up again
Imaginary numbers
i^1 = 1 i^2 = -1 i^3 = -i i^4 = 1 i^5 = i i^6 = -1 i^7 = -i i^8 = 1
How to help students struggling with coin word problems
most students struggle with the value equation. Remind them how they would find the value of a pile of coins
When working with consecutive integer word problems...
the PK 3401 does not include any problems that do not have a solution. At some point in the second half of the PK, ask students to find three consecutive odd integers with a sum of 60. There is no solution to this problem. Nice introduction to extraneous solutions.
What are the divisibility rules for 20 and 50?
the last two digits of the number can be divided by 20 or 50
What is the divisibility rule for 6?
the number is divisible by 2 and 3
How does one graph quadratics in standard form?
use graph transformations. Find the vertex. Make sure the student learns the basic pattern for quadratic where a=1. Over one up one, over two up four, over three up nine. If a is negative, go down, not up. If a is not 1, just multiply the basic pattern by a.
When working with work word problems...
use the general work equation, which reflects parts adding to one job. T/x + T/x = 1 Always clear fractions to simplify the equation
