Primary Understanding and Use of Place Value

Eleanor Brzozowski             Hanson Park School
                               5411 W. Fullerton Avenue
                               Chicago IL 60639
                               312-534-3100

Objectives:

    1. To give students an awareness of some of the peoples that have 
contributed to the development of modern arithmetic.  (Number systems) 
    2. To help students gain a better understanding of our base 10 number system 
and its application to real life problems. 
    3. To increase a student's ability to add and subtract two numbers having as 
many as three digits and involving renaming in the ones place and in the tens 
place. 

Materials Needed:

    Place Value board               Dice
    Base ten blocks                 Activity sheets
    Chip-trading boards             Number Wheel Response Cards (Optional)
    Chips (4 colors)                Multicultural materials

Strategy:

    1. Discuss with the children some of the number systems used by other 
peoples such as the Romans, Egyptians and Mayas.  Roman Numerals contain a 
symbol for one (I), five (V), ten (X), fifty (L), one hundred (C), five hundred 
(D) and one thousand (M).  It lacks a symbol for zero and place value.  Explain 
that Egyptian numerals are base ten as are our Hindu-Arabic numerals and are 
additive.  The Egyptian numerials are usually written from right to left, do not 
use place value and the symbols used to write their numbers are called 
hieroglyphics.  Mayan numerals are base 20, with place value and a zero place-
holder.  The increasing powers of 20 go up vertically instead of horizontally. 

    2. Put two Roman numerals on the board such as CX (110) and XLV (45).  Ask 
children, "How did the Romans add?"  As they see the problems encountered in 
doing this, lead the discussion to an explanation of our base 10 system and its 
advantages, the 10 symbols used are called digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) 
and how the groupings by tens form a place value system.  It is, therefore, 
possible to represent any number using only the ten basic symbols (digits).  The 
position of each digit in the numeral determines the value of that digit.  This 
makes the task of doing arithmetic operations much easier than in systems like 
the Romans or Egyptians. 

    3. Using a place-value chart showing ones, tens, hundreds and thousands, 
model with base ten blocks four problems -- the addition of two 3 digit numbers 
with and without regrouping and two subtraction problems up to 3 digits with and 
without regrouping.  At the same time of modeling and solving the problems, have 
a student at the board showing abstractly what is being done as it is being 
modeled.  This will enable the student to transfer what he sees concretely to 
the abstract.  Be sure to stress that the largest digit each place value can 
have is a nine and that exchanges must be made for numbers larger than nine. 

    4. Give students experience in renaming or making exchanges by playing "Chip 
Trading".  Give each student a playing board made from 8 x 11 inch manila paper 
marked in four vertical columns.  At the top of the left-hand column, paste a 
small rectangular piece of black construction paper, red at the top of the next 
column, then blue and white.  Cut out circles of these same colors and use for 
chips -- white representing ones, blue representing tens, red representing 
hundreds, and black thousands. 
       Children play in groups of two to four.  Each player throws the dice in 
turn and always takes the number of white chips corresponding to the sum of the  
numbers shown on the dice.  Using 10 as the number (or base) for making 
exchanges, whenever a player has 10 or more chips of one color, a trade must be 
made.  (10 whites are traded for 1 blue; 10 blue for 1 red; 10 red for 1 black).  
The first player to obtain a red chip (when playing to 100) or a black chip 
(when playing to 1000) is the winner.

        Variation:  Use other numbers as rates of exchanges such as 2, 3, 5, 8, 
etc.. 

    5. Provide students with appropriate activity sheets for reinforcing 
addition and subtraction skills presented in lesson.  

Performance Assessment:

    Children will be given three problems, one orally and two written, to assess 
how well the objectives of the lesson were met. 
    A. Oral:
       Students will write two 3-digit numbers from clues given orally by the 
teacher and then find the sum.  Example:  In the first number, the ones' digit 
is 5.  The tens' digit is l more than the ones' digit.  The hundreds' digit is 2 
more than the tens' digit.  The second number has 4 ones.  The tens' digit 
equals the number of boys in the class.  The hundreds' digit is the number of 
feet a chicken has. 
    B. Written:
       (1) Write a question that can be answered using the information in the 
story.  Then find the answer. 
       Mary swam 567 meters at the indoor pool today.  Yesterday she swam 498 
meters.
       (2) Write a story problem that you can solve using this number 
sentence:  459 + 64 = ? 

Additional Activities:

       1. Cooperative Problem Solving -- In groups of four, give each group an 
envelope with four papers in it.  Each paper states the problem with a different 
clue on it.  The group discusses the problem and the four clues.  The first 
group that solves the problem and can explain how the solution was obtained 
wins.  Example:  Problem: How many seeds are in the sunflower?  Clue 1: The 
number uses the digits 2, 7, and 8, not in that order.  Clue 2:  The 2 is not in 
the ones' place.  Each digit is used once.  Clue 3:  The 7 is not in the 
hundreds' place.  Clue 4:  The 2 is not in the hundreds' place. 
       2. Rolling-the-Cube -- On a piece of paper, students make three short 
lines and directly below that three more short lines in preparation for adding 
two 3 digit numbers.  A cube or die is rolled.  The number appearing on the cube 
is put on one of the six blanks.  The cube or die is rolled again and the second 
number is placed in one of the remaining blanks.  The process is repeated until 
all six blanks are filled in.  The two numbers are then added, and the student 
with the largest sum is the winner. 
       3. Find-Hide-Show -- Give students a "Number Wheel Response Card" (Open 
Court Publishing Co.).  State a number to make on their cards.  After they find 
the number, they "hide" the card by their chest and when the teacher says 
"show", they hold up their card.  Teachers can easily identify students having 
an understanding of place value and those needing additional help. 


Extension:

    To provide students with experiences in regrouping larger numbers, make a 
three dimensional mat called the "Daily Depositor" with places up to one 
million.  Have the class predict whether they can accumulate one million dollars 
by the year's end if each day the class receives the number of $100 bills for 
the date of the month.  For example, on the third of the month $300 is put into 
the Depositor and on the 27th of the month $2700 is added to the Depositor.  The 
accumulation occurs throughout the years so there is experience with large 
numbers and regrouping (hundreds in the early months and thousands in the later 
months).  Use play money from a game such as Life.