MathNotations

And the seasons they go round and round
And the painted ponies go up and down
We're captive on the carousel of time
We can't return we can only look behind
From where we came
And go round and round and round
In the circle game...

Oh, how I love Joni Mitchell's lyrics made famous by the inimitable Buffy Sainte-marie. Oh, how The Circle Game lyrics above describe my feelings about the state of U.S. math education. I feel I've been on this carousel forever. But I do believe that all is not hopeless. I do see promise out there despite all the forces resisting the changes needed to improve our system of education.

Our math teachers already get it! They get that more emphasis should be placed on making math meaningful via applications to the real-world, stressing understanding of concepts and the logic behind procedures, reaching diverse learning styles using multiple representations and technology, preparing their students for the next high-stakes assessment, trying to ensure that no child is ... They've been hearing this in one form or another forever. BUT WHAT THEY NEED IS A CRYSTAL CLEAR DELINEATION OF ACTUAL CONTENT THAT MUST BE COVERED IN THAT GRADE OR THAT COURSE.

The vague, jargon-filled, overly general standards which have been foisted on our professional staff for the past 20 years is frustrating our teachers to the point of demoralization. THIS IS NOT ABOUT THE MATH WARS. THIS IS NOT AN IDEOLOGICAL DEBATE. JUST TELL OUR MATH TEACHERS WHAT MUST BE COVERED AND LET THEM DO THEIR JOB!

BY "WHAT MUST BE COVERED" I AM INCLUDING THE SKILLS, PROCEDURES AND ESSENTIAL CONCEPTS OF MATHEMATICS. NONE OF THIS CONSTRAINS TEACHER STYLE OR CREATIVITY. BUT WITHOUT THIS STRUCTURE THERE IS ONLY THE CHAOS THAT CURRENTLY EXISTS. AND IF YOU DON'T THINK THERE IS CHAOS OUT THERE, TALK TO THE PROFESSIONALS WHO HAVE TO DO THIS JOB EVERY DAY.


UPDATES...

Results of MathNotation's Third Online Math Contest

The Common Core State Standards Initiative

NCTM's latest response to the Core Standards Movement - the forthcoming Focus in High School Mathematics

Validation Committee selected for draft of Core Standards

The results of the latest round of ADP's Algebra 2 and Algebra 1 end of course exams

It will take several posts to cover all of this...


RESOURCES FOR YOU

MODEL PROBLEMS TO DEVELOP HIGHER-ORDER THINKING AND CONCEPTUAL UNDERSTANDING

Consider using the following as Warm-Ups to sharpen minds before the lesson and to provide frequent exposure to standardized test questions (SAT, ACT, State Assessments, etc.). I hope these problems serve as models for you to develop your own. I strongly urge you to include similar questions on tests/quizzes so that students will take these 5-minute classroom openers seriously.

I've provided answers and solutions/strategies for some of the questions below. The rest should emerge from the comments.

MODEL QUESTION #1:

For how many even integers, N, is N² less than than 100?

Answer: 9

Solution/Strategies:
Always circle keywords or phrases. Here the keywords/phrases include
"even integers"
N²
"less than".

This question is certainly tied to the topic of solving the quadratic inequality, N² "<" 100 either by taking square roots with absolute values or by factoring. Of course, we know from experience, when confronted with this type of question on a standardized test, even our top students will test values like N = 2, 4, 6, ... However, the test maker is determining if the student remembers that integers can be negative as well and, of course, ZERO is both even and an integer! Thus, the values of N are -8,-6,-4,-2,0,2,4,6, and 8.


MODEL QUESTION #2

If 99 is the mean of 100 consecutive even integers, what is the greatest of these 100 numbers?

ANSWER: 198
Solution/Strategies:
There are several key ideas and reasoning needed here:

(1) A sequence of consecutive even integers (or odd for that matter) is a special case of an arithmetic sequence.

(2) BIG IDEA: For an arithmetic sequence, the mean equals the median! Thus, the terms of the sequence will include 98 and 100. (Demonstrate this reasoning with a simpler list like 2,4,6,8 whose median is 5).

(3) The list of 100 even consecutive integers can be broken into two sequences each containing 50 terms. The larger of these starts with 100. Thus we are looking for the 50th consecutive even integer in a sequence whose first term is 100.

(4) The student who has learned the formula (and remembers it!) for the nth term of an arithmetic sequence may choose to use it: a(n) = a(1) + (n-1)d. Here, n = 50 (we're looking for the 50th term!), a(1) = 100, d = 2 and a(100) is the term we are looking for.
Thus, a(50) = 100 + (50-1)(2) = 198.

However, stronger students intuitively find the greatest term, in effect inventing the formula above for themselves via their number sense. Thus, if 100 is the first term, then there are 49 more terms, so add 49x2 to 100.



MODEL QUESTION #3: A SAMPLE OPEN-ENDED QUESTION FOR ALGEBRA II

If n is a positive integer, let A denote the difference between the square of the nth positive even integer and the square of the (n-1)st positive even integer. Similarly, let B denote the difference between the square of the nth positive odd integer and the square of the (n-1)st positive odd integer. Show that A-B is independent of n, i.e., show that A-B is a constant.


MODEL QUESTION #4: GEOMETRY

If two of the sides of a triangle have lengths 2 and 1000, how many integer values are possible for the length of the third side?


MODEL QUESTION #5: GEOMETRY

There are eight distinct points on a circle. Let M denote the number of distinct chords which can be drawn using these points as endpoints. Let N denote the number of distinct hexagons which can be drawn using these points as vertices. What is the ratio of M to N?

Answer: 1
Solution/Strategies: The student with a knowledge of combinations doesn't need to be creative here but a useful conceptual method is the following:
Each hexagon is determined by choosing 6 of the 8 points (and connecting them in a clockwise fashion for example). For each such selection of 6 points, there is a uniquely determined chord formed by the 2 remaining points. Similarly, for each chord formed by choosing 2 points, there is a uniquely determined hexagon. Thus the number of hexagons is in 1:1 ratio with the number of chords.

MODEL QUESTION #6: GEOMETRY AND THE ARITHMETIC OF PERCENTS

If we do not change the angle measures but increase the length of each side of a parallelogram by 60%, by what per cent is the area increased?

(A) 36% (B) 60% (C) 120% (D) 156% (E) 256%

Posted by Dave Marain at 6:42 AM 0 comments

Labels: core curriculum standards, national math curriculum, reasoning, SAT strategies, SAT-type problems, update, warmup

Additional SAT/Contest/Challenges

Challenge 1:

HOW MANY DIGITS OF 1000¹⁰⁰⁰ - 1 WILL BE EQUAL TO 9 WHEN THIS EXPRESSION IS EXPANDED?

Challenge 2:

HOW MANY 5-DIGIT POSITIVE INTEGERS HAVE A SUM OF DIGITS EQUAL TO 43?

Challenge 3:

Jorge can run a 6-minute mile while Alex can run a 5-minute mile. If they start at the same time, how much less distance, in miles, will Jorge run in 10 minutes?

(Yes, you can respond with answers and solutions to these in the comments!)
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Tired of hearing about THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST! ? IF I RECEIVE 10 MORE REGISTRATIONS, I MAY JUST STOP!
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The Common Core State Standards Initiative
First look here for a quick overview and here for an index to the latest draft of the standards. Of course, this blog only discusses the mathematics part of the document.

Overview

The Common Core State Standards Initiative is a joint effort by the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) in partnership with Achieve, ACT and the College Board. Governors and state commissioners of education from across the country committed to joining a state-led process to develop a common core of state standards in English-language arts and mathematics for grades K-12.

These standards will be research and evidence-based, internationally benchmarked, aligned with college and work expectations and include rigorous content and skills. The NGA Center and CCSSO are coordinating the process to develop these standards and have created an expert validation committee to provide an independent review of the common core state standards, as well as the grade-by-grade standards.

• Core Concepts and Core Skills
• 11 Core Standards including the new "Mathematical Practice":
  
• Mathematical Practice
• Number
• Quantity
• Expressions
• Equations
• Functions
• Modeling
• Shape
• Coordinates
• Probability
• Statistics

• Each standard is broken into Core Concepts and Skills, provides research-based evidence and many illustrative examples to clarify the language
  
• Alignment of these standards to those of 5 representative states: California, Florida, Georgia, Massachusetts and Minnesota
• Standards reduce the number of Core Concepts and Skills in accordance with many recommendations to pare down the number of required topics to allow for greater depth

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs, which can be graphed in the plane. Equations can be combined into systems to be solved simultaneously.

An equation can be solved by successively transforming it into one or more simpler equations. The process is governed by deductions based on the properties of equality. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, stimulating the formation of expanded number systems (integers, rational numbers, real numbers and complex numbers).

A formula is a type of equation. The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b₁ + b₂)/2) h, can be solved for h using the same deductive process.

Inequalities can be solved in much the same way as equations. Many, but not all, of the properties of equality extend to the solution of inequalities.

Connections to Functions, Coordinates, and Modeling. Equations in two variables may define functions. Asking when two functions have the same value leads to an equation; graphing the two functions allows for the approximate solution of the equation. Equations of lines involve coordinates, and converting verbal descriptions to equations is an essential skill in modeling.

1. Exceptionally clear and definitive document
2. Influenced by NCTM (Curriculum Focal Points), Achieve, College Board, ACT
3. Illustrative examples are of high quality
4. Will serve as a basis for states' revisions of current standards hopefully creating more consistency than currently exists
5. Leaving curriculum to local districts and states was a politically necessary decision, however, in my opinion, developing a reasonably consistent curriculum by grade level and/or course across districts and states from these standards may prove to be difficult and may again lead to considerable disparity. Hopefully, this will be self-correcting when standardized assessments are created as is currently being done with the End of Course Tests from Achieve
   

Posted by Dave Marain at 6:01 AM 4 comments

Labels: achieve, core curriculum standards, SAT-type problems, update