Math Knowledge Now

finding the perimeter of a scalene triangle

2010-04-13T19:37:00.000-07:00

we have to use the altitude to get a right triangle to get to the Pythagorean theorem.

decimal number addition

2010-04-13T19:35:00.000-07:00

we must line up the decimals when we add.

solving multi step problems

2010-04-13T19:31:00.000-07:00

here are a few tips before i go over the examples.

here are the examples explained in great detail.

rounding decimal numbers

2010-04-13T19:28:00.000-07:00

this is a step by step approach to rounding.

percents

2010-04-13T19:21:00.000-07:00

in this example we go from a drawing to a fraction to a percent. we use the concept of ratios. this is an introductory idea. eventually we see we must divide the fraction to get a decimal, then move the decimal to the right 2 places to get the percent value.

Summer Tutoring Program Explained

2010-03-31T18:07:00.000-07:00

I am offering a tutoring program this summer.

In San Clemente the rate is $60 per hour.

Not in San CLemente the rate is $70 per hour.

The dates I will be tutoring are:

June 21st - July 30th (6 weeks) and Aug 16th - Aug 27th (2 weeks)

Monday - Friday from 10:30 - 5:30

I can tutor any subject. I am prepared with a focused curriculum in the following subjects:

==Basic Skills Practice

==Mathematical Modeling ( word problems and rectangular modeling)

==Singapore 3

==Singapore 4

==Singapore 5

==Saxon Alg 1/2

==Saxon Alg

==Geometry

==Algebra II

==Elementary Statistics

==Learning How to Learn Math (Note Taking, understanding the processes and mechanics of mathematics, organization skills, which questions to aks and when to ask them)

Obviously these are not the only concepts I am available for tutoring.

Summer tutoring is a fantastic way to reinforce prior learned concepts, introduce new concepts, relate concepts to each other, and receive "one on one instruction".

Please contact me via email for any tutoring request:

mrblack@vandammeacademy.com or mrblacksmath@live.com

coin problem

2010-03-25T19:10:00.000-07:00

surface area of a triangular prism

2010-03-25T19:09:00.000-07:00

increase in percent #2

2010-03-25T19:07:00.000-07:00

another coin problem

2010-03-25T19:05:00.000-07:00

evaluate variable expressions

2010-03-25T19:02:00.000-07:00

we substitute values for the variable and honor the order of operations.

Pythagorean theorem and a diagonal

2010-03-25T18:11:00.000-07:00

a 6 year old's quiz

2010-03-18T19:23:00.000-07:00

decimals to fractions and the thousandths place

2010-03-18T19:18:00.000-07:00

conversions of measurement

2010-03-18T19:13:00.000-07:00

conversions of measurement are a 4 step procedure

increase in percent

2010-03-18T19:08:00.000-07:00

rational equations

2010-03-18T19:04:00.000-07:00

in the below examples we see a rational equation is an equation made of ratios. the key to solving these equations is to utilize the distributive property. multiply both sides of the equation by the least common multiple of the denominator. cancel where appropriate. now you have an equation. isolate the variable.

multiply scientific notation

2010-03-16T07:15:00.001-07:00

similarity

2010-03-14T17:28:00.000-07:00

inverting matrices with a 6 year old.

2010-03-05T06:48:00.000-08:00

please excuse the poor quality of this photo. I didn't have my camera with me , i just had my phone. this is a photo from the material my 6 year old student is learning after school.

My introduction as a SDB Fellow

2010-02-02T09:47:00.001-08:00

Class of 2009 Induction Comments

Delivered at SDB Annual Conference, Friday, January 22, 2010

Michael Black oversees the math program at VanDamme Academy, a relatively new private K-8 school in Aliso Viejo, California, including the math instruction for grades 4 through 8. He is a graduate of Texas A&M and UC Santa Barbara, where he earned an MA in Pure Mathematics. Michael has traveled widely, often in pursuit of the perfect wave. He has taught in both large public schools and modest-sized independent schools. He appears to have found his niche at the Academy. His nominating student, currently an 8^th grader, confirms this good match. “The ordinary math teacher has 30 pupils and teaches all of them the same thing with no special attention towards any of them. Mr. Black is different. A student in Mr. Black’s classroom does not have to worry about getting lost in the crowd.” The student goes on: “Mr. Black finds a way to relate the logic he teaches us in geometry to real life.” For Michael, the mission is “to help free our school community of people who say, ‘I hate math.’” In his words, “I enjoy the responsibility of finding creative ways to break through mental barriers to learning, of challenging students at their own levels of ability, and make them feel recognized and appreciated as unique individuals.” Michael goes on: “Being able to adapt, creatively and compassionately, to the individual child is one of the greatest pleasures of teaching.” Responsibility and satisfaction – these are strong bookends for any professional. Nice work, Michael, and welcome as an SDB fellow.

What is satisfying about being a teacher?

2010-01-24T18:39:00.001-08:00

I just returned from the Johns Hopkins Center for Talented Youth annual seminar. A student nominated me for a Fellowship. I was awarded it this past weekend. I was prompted to send in a Resume and a letter concerning what satisfies me about teaching. Below is what I wrote:

Nothing is more satisfying than being a teacher. Seeing my students grow in maturity, intelligence, and depth of understanding—and consequently seeing my community benefit from the addition of thoughtful, responsible, educated young adults—gives me profound satisfaction.

For many, math is regarded as intimidating, tedious, a chore. It is my goal to help free our community of people who say, “I hate math.” I have the pleasure of teaching students at every stage of development, from simple arithmetic to algebra to geometry to pre-calculus. It is my goal to ensure that at each step in the progression they have not just learned but mastered the concepts, and that they can advance with confidence and clarity.

When I teach my students long division, they do not simply churn through an algorithm. Long division is a step in a mathematical sequence, from counting (which is addition), to multiplication (which is repeated addition), to subtraction (which is addition of negative numbers), to division (which is keeping track of repeated subtraction). The students do not view long division as a process that exists in a mathematical void; they see its connection back to addition. Their depth of understanding allows them to grasp later concepts such as the division of polynomials and the reason behind the order of operations. This approach is one example of a broader theme: I am satisfied only when my students truly understand.

The task is a creative one. It cannot be achieved by cranking students through some preconceived program at some predetermined pace. I enjoy the responsibility of finding creative ways to break through mental barriers to learning, of challenging students at their own levels ability, and of making them feel recognized and appreciated as unique individuals.

When one brilliant 5-year-old proved himself a prodigy in geography and zoology but formed an early aversion to mathematics, I taught him, for example, hexagons by reference to a bee’s honeycomb and addition through summing the number of countries in the world. When presented a 6-year-old with a seemingly infinite capacity and appetite for math, I nourished his ability, marveled as I witnessed him grasp the connection between fractions, division, and decimals (a year-long curriculum for most) in the span of about a single class period, and was pained at the thought of how many such students must exist, neglected. When after repeatedly asking the kids to put on their “thinking caps,” I brought in my own hand-sewn, colorful, patterned cap to give playful emphasis to the point, and was greeted by cries of, “I want one!”—I indulged. Being able to adapt, creatively and compassionately, to the individual child is one of the greatest pleasures of teaching.

Helping the child to build a solid conceptual structure and recognizing him as an individual, I empower the child to believe in himself. I am in a position to help provide students with a solid understanding of math, a generalized feeling of competency, and the sense that they have an important place in this world. What could be more satisfying?

Singapore Word problem

2009-06-13T17:10:00.000-07:00

Rectangular modeling with money.

Word Problems!

2009-06-10T19:03:00.000-07:00

In these examples rectangular modeling is used.

Uniform motion problem.

2009-06-05T17:27:00.000-07:00

Rectangular modeling with money.

2009-06-05T17:24:00.000-07:00

Rectangular modeling word problem.

2009-06-05T17:20:00.000-07:00

Success here occurs by creating a model that has the lightest weight as the base size rectangle.

Volume of a Sphere

2009-06-05T17:18:00.000-07:00

Area of a regular Polygon

2009-06-04T16:35:00.001-07:00

We rely on the definition of an apothem and we notice that any regular polygon can be thought of as a collection of isosceles triangles. In the below example I utilize the definition of Tangent.

Fish Tank Volume word problem

2009-06-04T16:33:00.000-07:00

Graphing a line in slope intercept form.

2009-06-02T17:48:00.000-07:00

Isolate variable in denominator

2009-06-02T17:44:00.000-07:00

The trick here is to multiply both sides of the equation by the least common multiple of the denominators. We MUST honor the distributive property.

Linear system

2009-06-02T17:41:00.000-07:00

When we are asked to find a point of intersection we are actually being asked to solve a linear system. We could have chosen to graph this system or we could have chosen to use substitution. I chose to use elimination. I made this choice based on the fact the equations were presented in standard form.

Simplify like terms

2009-06-02T17:37:00.000-07:00

This example is an effort to explain why and how to simplifying like terms work.

Unit multipliers

2009-06-02T17:33:00.000-07:00

Here we must honor the fact that cubic meters is the same as (meter)(meter)(meter) and cubic inches is (inch)(inch)(inch). We must also know the conversions for meters to centimeters and centimeters to inches.

Finding an angle measurement

2009-06-02T17:29:00.000-07:00

The key to being successful with this example is to look at the two triangles inside the large triangle. We must recall the measure of a straight angle is 180 degrees and so is the sum of angles in a triangle.

Isolate Variable

2009-05-30T08:32:00.000-07:00

Isolate a variable by:
1) Changing all "-" signs to "+ the opposite"
2) Use the distributive property
3) Identify like terms
4) Combine like terms
5)Add the opposite
6)Divide since division is the opposite of multiply

Long Division

2009-05-30T08:25:00.000-07:00

Long division requires a repeated process and a knowledge of relevant vocabulary.

Showing a triangle is a right triangle by using algebra.

2009-05-28T16:30:00.000-07:00

Rectangular modeling problem involving "fractions of a remainder"

2009-05-28T16:24:00.000-07:00

The trick to these "fraction of a remainder" rectangular modeling problems is to choose the correct number of squares to make up your original rectangle. Always choose the least common multiple of your denominators.

Sales commission

2009-05-28T16:19:00.000-07:00

Volume in cubic miles

2009-05-28T16:14:00.000-07:00

We must think of the irregularly shaped object as a collection of known geometric shapes. Remember the formulas for are of a rectangle and a semicircle. We need to recall that the formula for Volume is area of base multiplied by height.

Multiplication and Division of mixed units

2009-05-27T18:47:00.001-07:00

Here we need to have a firm understanding of how many smaller units make up the larger unit. I choose to convert all units to the smaller of the two, then preform the required arithmetic , then convert back to the mixed units.

Geometry Test Review.

2009-05-23T10:27:00.000-07:00

Her I post various explanations to support the solutions to the problems on the Review for the Final exam in a neighboring school district.

3D tesselation

2009-05-22T18:43:00.000-07:00

In this example my 10 year old students take a line segment and divide it into 3 equal pieces to form a triangle. They take that triangle and form a triangular prism , they take that triangular prism and replicate the process many times to produce the polyhedron shown.

Isolate variable with mixed numbers

2009-05-22T18:41:00.000-07:00

Container word problem

2009-05-22T18:35:00.000-07:00

Here we recall "times" means multiply and we let a variable represent the unknown quantity.

Finding a point with certian constraints.

2009-05-21T17:08:00.000-07:00

Simplifying an expression invoving exponents.

2009-05-21T16:04:00.000-07:00

Value of a fraction

2009-05-21T16:02:00.000-07:00

Irregularly shaped object volume

2009-05-21T16:01:00.001-07:00

Consecutive Integers

2009-05-21T15:59:00.001-07:00

Translating words to mathematical symbols is the key here.

Word problems GALORE!

2009-05-21T15:55:00.000-07:00

Basic skills practice: ROUNDING

2009-05-20T19:07:00.001-07:00

Word problem city!

2009-05-20T18:58:00.000-07:00

Here we see word problems of varying degrees of challenge. Email me if the explanations are not clear enough.

Word problems with rectangular modeling

2009-05-19T15:21:00.000-07:00

Rectangular modeling is relied upon here.

Rectangular modeling is used here also.

Rounding the Sum to one decimal place

2009-05-19T15:18:00.000-07:00

The concept of rounding and place value is reviewed here.

Inverse Function

2009-05-18T17:35:00.000-07:00

In this example you must exchange the place of the x and y variable and then isolate the y.

Basic Fraction skill review

2009-05-18T17:26:00.000-07:00

We need to be familiar with the Least Common Multiple, and reducing fractions to be successful with adding and subtracting fractions. The following pictures illustrate examples in the aforementioned concepts.

Geometric Proofs

2009-05-18T17:02:00.001-07:00

To be successful with proofs one must think logically. Do NOT try to use a characteristic of the conclusion. Too often developing mathematics students try to assert something they are actually trying to prove. I find thinking about every characteristic of the conclusion can help with your discovery of the needed definitions, postulates , properties, or previously proven theorems you will use in your proof.

A word problem involving Ratios and books.

2009-05-16T07:05:00.000-07:00

The key to understanding this problem that was asked to the 9 year olds is a proper model. We use a rectangular modeling set up. In rectangular modeling we always use the smallest unit as the squares that make up the larger rectangles. In this case: Lina's original amount of books. Now, we must focus our attention on the fact that AFTER 24 books are purchased the amount Lina has is twice that of Suling. Suling had 2 rectangles in our original model. Hence Lina MUST have 4 rectangles in our new model as 4 is twice 2. we step back and look at the model. We notice that 3 small rectangles represent the 24 books. Hence the division.

PERCENTAGE

2009-05-15T21:49:00.000-07:00

In this example there are alternative methods utilized. Here we must focus on the English language. What is described by the word "is"? What is described by the word "of"? What is associated with the word "percentage"? The answers to these questions are what guide us in the solution.

pythagorean triplets

2009-05-15T21:01:00.001-07:00

multiples are very important here.

Simplifying a Rational Expression

2009-05-14T16:35:00.000-07:00

We honor mathematical properties here.

Consecutive Even Integers

2009-05-14T16:31:00.000-07:00

Basic Skill: Fractions. Adding and Subtracting fractions with unlike denominators.

2009-05-14T16:17:00.000-07:00

When adding and subtracting fractions with unlike denominators we have to utilize the least common multiple. We also rely heavily on the fact that when you multiply by 1 you do not effect the value of the other factor.

Calculating the Volume of a Pyramid

2009-05-14T16:14:00.000-07:00

The formula is 1/3 of the area of the base of the pyramid multiplied by the height or the pyramid.

Graphing an exponential equation with out a graphing calculator

2009-05-13T19:19:00.000-07:00

Here I demonstrate how our grandparents did it.

Calculating Volume involving liters and centimeters.

2009-05-13T19:11:00.000-07:00

A rectangular solid with a base measuring 96 square cm has 1.2 liters of liquid in it. What is the height of the liquid.

We must remember that 1 cubic centimeter is equal to 1 millimeter. 1000 cubic centimeters equals 1 liter.

Motion problem with equal Distance

2009-05-13T19:06:00.000-07:00

Substitution and observing the distance between the school and the house is the same regardless of whether you are going to or coming from school is the key in this example.

Volume Displacement

2009-05-13T19:01:00.000-07:00

Here we think of the irregularly shaped rock with volume 300 cubic centimeters as a rectangular solid with the same volume.

Evaluate Variable Expression

2009-05-13T18:56:00.000-07:00

Care must be taken when substituting your values for your variables.

Isolate Variable Example

2009-05-13T18:52:00.001-07:00

I refrain from explaining in great detail given my prior post.

Polynomial Division

2009-05-13T18:36:00.000-07:00

It is critically important when dividing polynomials to honor the euclidean algorithm. You can't think in terms of "how many times does the divisor go into the dividend?" You must instead repeatedly ask the question "what do I multiply the first term in my divisor by to get the first term in my dividend?" Once this question is answered you MUST honor the distributive property and handle your subtraction with care.

Basic skills Review and explanations involving zeros.

2009-05-13T18:21:00.000-07:00

Basic arithmetic skills are reviewed and explained.

Finding the X and Y intercepts of a Linear Equation

2009-05-12T18:31:00.000-07:00

This concepts becomes clear when we realize what is happening to our coordinates when we are on the respective axis. Substitution yields the solution.

Calculating the volume of an object by observing a change in liquid height

2009-05-12T18:27:00.000-07:00

Here the water level was first observed with the rock in the container, then it was observed after the rock was removed.

Calculating Volume by way of displacement

2009-05-12T18:20:00.000-07:00

In this example I chose the approach of looking at the volume of the liquid in the tank before the rock was put in , then placing the rock in the tank and observing the new height of the liquid. I use this new height to calculate the new volume inside the tank. The difference in the two volumes is the volume of the rock. This is not the most efficient approach, but as an instructor I believe it is the most effective at illustrating the need for subtraction.

Measurement Word problems and Arithmetic

2009-05-12T18:11:00.000-07:00

Here are a few basic word problems involving measurement. The key is to always convert your units to the smaller measure then do the required arithmetic then convert the units back to the larger unit making sure to have the remainders show up as the smaller unit.

Isolate a variable

2009-05-12T17:56:00.000-07:00

"Isolating a variable" is the way to describe the process we do when finding the value for a variable in an equation. All operations in mathematics can be paired with their inverse operation. When thinking of the solution to x+2=5 most student's brain is saying "what number do I add to 2 to get a 5?" This is all fine and dandy until you get to a more complicated problem. That is why certain language of the discipline must be introduced. Words and phrases such as: inverse operation, multiplication property of equality, addition property of equality, and combining like terms. With this language in place we can now think of x+2=5 as actually saying "what number do i get when I subtract 2 from 5?"

Classifying Triangles

2009-05-12T17:47:00.000-07:00

Here I identify the various words for classifying a triangle. Every triangle can have two words associated with it. One word describes some information about the triangle's side lengths (scalene, equilateral, or isosceles) the other word describes some information about the triangle's angle measures (acute, right, or obtuse).

Simplifying Rational Expressions

2009-05-12T17:40:00.000-07:00

In this post I sum two rational expressions. I use the Least Common Multiple. Although there are Polynomials for denominators I treat them as if they were numbers. Polynomials behave like numbers.

Uniform Motion

2009-05-11T22:24:00.000-07:00

Here we investigate the classic train station type problem.

Dividing fractions

2009-05-11T22:20:00.000-07:00

Here is an example involving division of two mixed numbers. We first change the mixed numbers to improper fraction then we honor the definition of division.

Elementary Volume

2009-05-11T22:13:00.000-07:00

In this post we investigate elementary volume calculations.

The volume of an object can be thought about as how many cubes can fit in it in some cases in other cases it can be thought of as how much liquid can fit in it.

Elementary Measurement

2009-05-11T21:34:00.000-07:00

The definition of measure I will be using today is

measure: a method to describe an amount.

some examples:

Johny's diving board was 5 meters.

Julia is 4 feet 3 1/2 inches tall.

Paige is 9 years old.

Sidney had 1/2 liter of juice this morning.

The following pictures are some problems concerning elementary measure.