Wednesday, December 20, 2006

Sample Place Value Questions from TEA

Here are a few questions from the state-wide math assessment that show what a possible place value question could involve and possible ways to solve them. In no way are possible questions limited to the style, format, and/or level of difficulty shown here.

Sample 1
According to a report published in 1999, the
population of Dallas was 1,063,292. What does
the 6 in this number represent?

A Six thousand
B* Sixty thousand
C Sixty-three thousand
D Six hundred thousand


Strategy to solve this question:
Write out the place value chart and put the number in it.


Sample 2

Some of the greatest long-jump distances by
Olympic athletes are listed in the table below.

















According to this table, in which year was the
greatest long-jump distance recorded?
F* 1968
G 1976
H 1988
J 1992


Strategy for solving this problem:
Stack the distances and determine which is largest.
8.90
8.35
8.72
8.67
All 4 distances are the same in the ones place. The first place a difference is seen is in the tenths place. Since all the digits in the tenths places are different, the greatest long-jump distance is 8.90.

Place Value of Whole Numbers and Decimals

Place Value
The value of a digit as determined by its position in a number
Example:


The Texas Education Agency (TEA) requires the following be taught in all 5th grade public school classrooms:
(5.1) Number, operation, and quantitative reasoning. The student uses place value to represent whole numbers and decimals.
The student is expected to:
(A) use place value to read, write, compare, and order whole numbers through the 999,999,999,999; and
(B) use place value to read, write, compare, and order decimals through the thousandths place.


The Difference Between Place and Value:
The place of a digit in a number is where it is placed in the place value chart. The value of a digit in a number is what it’s worth in the entire number.
For example, in the first number above, the place of the digit 6 is the “hundred thousands place”. The value of the 6 is 600,000.
In the second number above the place of the 5 is the hundredths place and it’s value is five hundredths – or 5 pieces out of a hundred.



In the decimal chart above, 0.05 or five hundredths would be shown by 5 individual squares being colored red. This should not be confused with 0.5 or five tenths which is five entire rows OR columns being shaded. (See below)











If you were to compare 0.5 (five tenths) and 0.50 (fifty hundredths) you would see that they are equal. 5 out of 10 rows is half of the whole, and 50 out of 100 is also half of the whole.



Required skills:


  • Read whole numbers.
    Read whole numbers through 999,999,999,999 – or nine hundred ninety-nine billion, nine hundred ninety-nine million, nine hundred ninety-nine thousand, nine hundred ninety-nine.

Each whole –number period of the place value chart (ones, thousands, millions, billions etc.) is separated from the others by a comma. When reading a number, read from left to right one period at a time. For example, in the number above, it is read “one million, six hundred twenty-three thousand, fifty-one”.



  • Write whole numbers

Write whole numbers from left to right as they are read. For example, in “one million, six hundred twenty-three thousand, fifty-one” a person should use the place value chart to be sure the digits are in the correct places.



  • Compare and Order Whole Numbers

The most effective way to compare and order whole numbers is to stack the numbers on top of each other being sure to line up the decimals. (in the case where no decimal is visible, know that it always falls to the right of the ones place) After stacking the numbers begin at the left most digits and compare. As soon as the digits (above and below) are different, you know which number is bigger.
For example, to compare 121,112,010,212 and 121,122,010,212 you would stack them then compare. The digit in red is the first place value that is different thus making 121,122,010,212 the larger number.
121,112,010,212.
121,212,010,212.
Using this method, you could take any list of numbers and find the largest, smallest, and/or order them from least to greatest or greatest to least.


DECIMALS

In understanding decimals, it is vital to know that ALL decimal numbers represent part of one whole. All decimals have a fractional equivalent. Therefore, a longer number does not always mean a larger amount. For example 1 is a greater amount than 0.123456789 because 0.123456789 is part of a whole. In thinking about decimal place value, you will notice that all of the places on the decimal side match places on the whole number side EXCEPT the ones place. It is not possible to have a “oneths” place because it is impossible to cut something into one piece. As soon as something is cut (equal parts or not) it immediately becomes at least two parts.



  • Read decimal numbers.
    Read decimal numbers through 0.009– or nine thousandths.
    When reading a number, read from left to right as if the number were a whole number then call the place of the last digit. . For example, in the second number above, 0.053, is pronounced “fifty-three thousandths”

  • Write decimal numbers.

    Write decimal numbers from left to right as they are read. For example, in “seven hundred sixty-four thousandths you write seven hundred sixty-four then count three places for the thousandths place. A person should use the place value chart to be sure the digits are in the correct places.

  • Compare and Order Decimal Numbers

The most effective way to compare and order decimal numbers is to stack the numbers on top of each other being sure to line up the decimals. After stacking the numbers begin at the left most digits and compare. As soon as the digits (above and below) are different, you know which number is bigger.
For example, to compare 0.112 and 0.2, you would stack them then compare. The digit in red is the first place value that is different thus making 0.2 the larger number.
0.112
0.2
Using this method, you could take any list of numbers and find the largest, smallest, and/or order them from least to greatest or greatest to least.
*note this again illustrates that the longer number is not necessarily the larger number.

Tuesday, December 19, 2006

Math Background

The content and skills found in the following pages presupposes that students have basic knowledge of place value, addition, subtraction, multiplication, division, fractions, geometry, probability, and problem solving through 4th grade according to the Texas Essential Knowledge and Skills requirements.
http://www.tea.state.tx.us/

The purpose of this blog is to serve as a math notebook for students to refer to when studying for a test, doing homework, tutoring, or practice. It may be helpful to parents or tutors when working with their child on such assignments. Please feel free to ask questions, make (polite) comments, or help others. I cannot promise how often I will check and respond myself.