## Thursday, September 24, 2009

### More Challenges/SAT Practice, Core Curriculum Standards, Reminders, Comments...

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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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.

HIGHLIGHTS

• 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
Example of a Standard (Standard 5)

Equations | see evidence
An equation is a statement that two expressions are equal. Solutions to an equation are the values of the variables in it that make it true. If the equation is true for all values of the variables, then we call it an identity; identities are often discovered by manipulating one expression into another.

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.

Core Concepts
Students understand that:
1. An equation is a statement that two expressions are equal.
see examples

2. The solutions of an equation are the values of the variables that make the resulting numerical statement true.
see examples

3. The steps in solving an equation are guided by understanding and justified by logical reasoning.
see examples

4. Equations not solvable in one number system may have solutions in a larger number system.
see examples

Core Skills
Students can and do:
1. Understand a problem and formulate an equation to solve it.
see examples

2. Solve equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.
see examples

3. Rearrange formulas to isolate a quantity of interest.
see examples

4. Solve systems of equations.
see examples

5. Solve linear inequalities in one variable and graph the solution set on a number line.
see examples

6. Graph the solution set of a linear inequality in two variables on the coordinate plane.
see examples

FUNDAMENTAL ASSUMPTIONS AND CONSIDERATIONS

Very Important!
(Click on image to see a clearer view)

INITIAL MATHNOTATIONS REACTIONS

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

Emily W said...

Following are my answers to the questions.

Challenge 1: 1000 digits
Challenge 2: 15
challenge 3: one-third mile

Dave Marain said...

Emily,
I agree with your 2nd and 3rd answers but consider the following pattern:

1000^1 -1 = 999 (three 9's)
1000^2 - 1 = 999999 (six 9's)
etc....

There are many other ways to look at #1 using laws of exponents but the best strategy for these types is usually to "Make the problem simpler and look for a pattern!"

Keep commenting!

Emily said...

Oops!
For the challenge 1, I thought it was 10^1000 - 1...

Okay, so I think the correct answer to challenge 1 is,
3000 9's.

Thanks.

Dave Marain said...

You got it, Emily!