Exploring the power of Place Value
Let's start our journey in a rather unorthodox way...
Consider six magic cards all with some numbers between 1 and 60 on them. (a printable download is available in resources)
Ask a student to choose a number between 1 and 60.
Then have the student separate all the cards that have that number.
Ask for all the cards that have the student's number.
Then, using just the cards with the number, you, the Math Magician, reveal the number the student selected.
The real bonus of this activity (after the secret is revealed to you and then by you to students in your class) is that students can take cards home and impress their family members!
This is place value for base 2.
This is how the magic cards work. Look at the number in the top left corner of each card. Add those numbers together from the cards selected and you will be able to guess the chosen number correctly.
Why is it so important to memorize basic math facts of addition, subtraction, multiplication and division?
Why is it important to be familiar with the concept of place value?
Why, especially in light of the proliferation of so many devices that will do the calculations for you, is it crucial?
Because, I would argue, it is fundamental to understanding how numbers work. Succeeding in math in the upper grades depends on feeling competent in manipulating these basic math facts!
It is all about counting. From the beginning human beings have wanted to count things. People want to count almost anything and everything. In sports we count things like times at bat, average yards per down, time it takes to run 26.2 miles, and so on. We count the number of hot dogs a person can consume in a minute, and we count the number of rings around Saturn.
Throughout history people have developed a variety of systems to help them count things. One of these systems is familiar to many as Roman Numerals. This system will reveal the power of place value by presenting a negative example.
Consider for example: IV, which is equivalent to 4, added to IX, which is 9. Of course individuals could remember that the answer is XIII just like we commit the math fact 4 + 9 = 13 to memory. But what about something like MDC + XL + CX?
By contrast here is a simple system that is used in computers. The binary, or base 2, system would look something like this for counting from 1 to 16:
An example of a base 2 application of place value would look like the following:
Would be 128 + 8 + 4 + 1 that is equivalent to 141 base 10
Consider adding two numbers in base 2
Just as the largest numeral possible in a place in our base ten system is 9, and when a column adds to greater than 9, one is added to the next column and the ones column starts over again at zero (plus any excess over 10), In base 2, 1 is the largest numeral possible in a column and when the numbers in a column add up to greater than 1 then zero comes down and the one is “carried” to the next column.
Fairly easy when dealing with 1 plus 1 in base 2, but consider the following base 2 problem:
Taking this route to discuss place value hopefully helps you appreciate place value in our base 10 system, and the importance for students to learn basic math facts. Because of place value a student who has mastered addition and multiplication facts, and their subtraction and division inverses, can learn to solve any simple calculation.
Consider these examples:
This may seem a bit daunting to students who are used to working with problems involving 3 digit numbers or less. But if they know their math facts just show them that all that is needed is to take the problem one column at a time, and the solution ( 999,999,999 ) is easily calculated.
The reality is that one hundred twenty-three million, four hundred fifty-six thousand, seven hundred eighty-nine plus eight hundred seventy-six million, five hundred forty-three thousand, two hundred ten is not generally a math fact that most people carry around in their head. The amazing thing about our base 10 system is that we only need to deal with the basic math facts and take them one at a time. One hundred million plus eight hundred million equals nine hundred million. Because of place value we can reduce that calculation to the basic math fact of 1 + 8 = 9 and so long as we keep results in the right columns place value will handle the rest
