Part 1: The Chocolate Factory which covered the regrouping or trading aspect of place value and explored regrouping in base ten and other bases.
Part 2: Base Ten for Young Students which introduced several games and trading activities to help young children acquire a solid foundation in place value.
Part 3: The Bake Sale demonstrated the role of place value in long division.
Part 4: Geometry of Place Value demonstrated the dimensions of a cube in terms of place value and explored the geometric representation of the quadratic equation.
In the conclusion of the series, Part 5, Applications of Place Value, I will show you a sampling of interesting applications of place value to some real-life situations.
It is sometimes easy to add and subtract fractions by putting fractions in place-value-type columns. The usual addition and subtraction with regrouping requires students to conceptually pack or unpack “boxes” (See Part 1: The Chocolate Factory). What if I want to add 3 5/7 + 2 4/7?
The denominator, 7, means that the “whole” has been divided into seven equal pieces. Notice that instead of packing in boxes, the student creates “wholes.” With nine-sevenths, there are enough pieces to make one whole with 2 pieces or two-sevenths left over. There is that word, “leftover” again. In The Chocolate Factory, “leftover” was used to label the ones. Here we are using it to label the pieces, helping students to transfer and integrate concepts.
If students need to “borrow” (the current term is “exchange”) in order to subtract fractions, then they would have to cut up a whole into the necessary number of equal pieces. In the case of 4 2/5 – 1 3/5, the student creates two place-value columns, the place value of one being the “wholes” and the place value of the other being the “pieces.”
With fractions of unlike denominators, the procedure is essentially the same with the preliminary step of finding common denominators.
The Olympics and Time
The odometer of a car is obviously a place value representation, but have you noticed that the clock at the bottom of an Olympic race is formatted in terms of place value with colons marking the separation between hours, minutes, seconds, and hundredths of a second? As the race progresses, the clock gathers hundredths into seconds, then seconds into minutes, minutes into hours. The expression of a runner's time might be 2:14:52:27.
Thinking of time in terms of place value columns simplifies addition and subtraction.
Here's a (sort of) real life problem that occurred in my house recently. My son was invited to watch Lord of the Rings at a friend's house. If the DVD starts playing at 4:17:48 pm and the movie lasts 3 hours 44 minutes and 25 seconds, will my son be able to catch the 8:15 bus home?
Starting with the seconds column, 48 seconds plus 25 seconds equals 73 seconds which is 60 plus 13 seconds. Since 60 seconds equals one minute, cross off the “60” and carry the one to the top of the minutes column. One minute plus 17 minutes plus 44 minutes equals 62 minutes which is 60 plus 2 minutes. Since 60 minutes equals one hour, cross off the “60” and carry the one to the top of the hours column. One plus 4 plus 3 equals 8. The solution: The DVD will end at 8:02:13 pm. The answer: The DVD will end at 8:03 pm, in time to make the bus. “Packing” seconds into minutes and minutes into hours is just like packing chocolates into boxes and boxes into cases.
Subtraction is a matter of unpacking then. Suppose we want to work backwards. If the DVD must end by 8:10 pm to make the bus, what is that latest time it can start. The problem is 8:10:00 minus 3:44:25.
We must subtract 25 seconds from zero seconds. So we just unpack one minute or dump a minute into the seconds columns. Since we unpacked one of the 10 minutes, there are 9 minutes in the minutes column and 60 seconds in the seconds column which still represents the original ten minutes but in a slightly different form. Now it is easy to subtract 25 seconds from 60 seconds. Now we have to subtract 44 minutes from 9 minutes. Simple. Just unpack one hour into minutes, leaving 7 hours and giving 69 minutes. Now it is easy to subtract the 44 minutes. The solution: the latest the DVD can start playing is 4:25:35 pm. The answer: the latest the DVD can start playing is 4:25 pm.
(An small digression: I have twice made a distinction between the solution and the answer with the movie problems. I did not make this distinction with the fraction problems. The fraction problems did not require the distinction because they were context-free problems.
The movie problems were word problems or story problems. They have context. I teach students to find a solution, then interpret the solution to find the best or most reasonable answer to the question. Home clocks are not like Olympic clocks, and rounding by the rules does not always produce the best answer. In the case of the second movie problem, rounding to 4:26 pm might cause my son to either miss the bus or run like the dickens to catch it.
In real life, people cannot stop when they have found the solution. They must then apply the solution to the situation, or context, in the most reasonable way. We are failing to teach our children critical thinking when we allow them, even encourage them to conflate solution with answer).
Number base clocks are a manipulative that use the student's internalized understanding of clock time as a peg to hang the concept of bases. A base-7 clock means that a rotation of 7 “minutes” equals one base-7 “hour.” Using a variety of base clocks as aids, students can become quite adept at adding and subtracting in different bases. It is even possible to extend the skill to multiplication and division in bases as well.
Calendars can also be used to teach in place value especially if you confine yourself to hours, days, weeks, years because the months are not uniform “packages” of days or weeks. In calendar math, the number 14 would be 1 week, 4 days. The number 305 would be 3 months, 0 weeks and 5 days.
You can use the calendar to explore interesting patterns numerically as Michael Naylor does. Although he does not specifically target place value, the algebraic pattern partly follows from the place value of a calendar.
Fast box addition (Grades 6-8)
Have a student choose a 2 x 2 box and demonstrate how you are able to quickly give the total. Tell your students the secret: Add 4 to the first number and multiply the result by 4. Have the class test the result on several boxes.
The secret is algebra; if the first number is x, the other numbers are x + 1, x + 7 and x + 8. The total is 4x + 16, which is the same as 4(x + 4).
Have your students outline a 3 x 3 box and ask them which is greater – the sum of all of the numbers or 9 times the center number? Relabel the center number as x and write all the other numbers in terms of x as shown here:
When adding all of those terms, the constants cancel (–8 + 8 = 0, –7 + 7 = 0, etc.) so all that is left 9x. The total of all nine numbers, then, is 9 times the center number.
One teacher considers the calendar as probably the one math essential for her kindergarten class even though she may not necessarily focus on place value. Her website lists activities and links.
What we see today is that although regular number bases maintain column values in terms of powers of the base, place value is way more flexible. Each place, or column, can be whatever it needs to be as long as it is in terms of groups of the preceding place.
Fractions: Wholes, parts
Olympic Time: Hours, Minutes, Seconds, Hundredths of a Second
Calendars: Years, Months, Weeks, Days
Gettysburg Time: Score, Days
You can have more fun working in multiple places with
Historical Time: Millenia, Centuries, Decades, Years, Months, Weeks, Days, Hours, Minutes, Seconds, Hundredths of a Second. That's eleven places. Once the students understand grouping and ungrouping, they do not worry about the number of places. The more places, the more fun.
Maybe you can think of some more ways we use place value everyday.
O'Block Books sells manipulatives.
So does Creative Teaching Press.
Base ten packing set from Digi-Block.
"Great Source" for calendar math.