Place Value Part 5: Applications of Place Value

So far we have completed four parts of the place value series:

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.

Fractions

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 column. 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).

Base Clocks

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

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.

Conclusion

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.

To summarize:
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.

Resources

O'Block Books sells manipulatives.

So does Creative Teaching Press.

Base ten packing set from Digi-Block.

"Great Source" for calendar math.

When a President Speaks: 6 Reasons to Object to Objectors

I remember President Kennedy urging us kids to be physically fit, and the national president's fitness program that went with it. Anybody else out there earn a Presidential Fitness Award while they were in school? In fact, the program has followed us into adulthood.

Another president is planning to give a speech to school children urging fitness of another kind, educational fitness.

During this special address, the president will speak directly to the nation’s children and youth about persisting and succeeding in school. The president will challenge students to work hard, set educational goals, and take responsibility for their learning.

Why the furor?


Because of the breathtaking opposition.

Unbelievable. Maybe there is no grand tradition, but presidents have addressed remarks to schoolchildren from time to time. Ronald Reagan in 1986, George H.W. Bush in 1991, George W. Bush in 2001.

My problems with all this hullabaloo:

First, the objections are premature. It is silly to object to figments of the imagination. Wait till the president actually says something objectionable in the speech before objecting to it.

Second, the objections break the Golden Rule. In the nutshell, the right is worried that the president may voice a tenet or two of liberalism. I have trouble believing they would object to a Republican president voicing a tenet or two of conservatism. I am not letting the left off easy; they will violate the Good-for-Goose-Good-for-Gander principle when it suits them as often as the right does. As Shakespeare might say, “A pox on both their houses.”

Third, the objections are largely ad hominem. The objections are not criticizing the speech on its merits (probably because the speech has yet to be broadcast). Ad hominem is one of the defining marks of lack of critical thinking. What a lousy role model to set before our kids.

Fourth, the objections are misplaced. Edweek reports that White House efforts to quell the furor have been ineffective.

But the planned 15- to 20-minute noontime speech—and, especially, a menu of classroom activities (for younger and older students) suggested by the White House in connection with it—continued to draw denunciations...

Especially?! I looked at the “menu of classroom activities.” The White House's companion lesson ideas for elementary students and secondary students have no leading questions and emphasize strategies for comprehension. The secondary lesson plans ask students to create a specific action plan for meeting their goals. Maybe it is about time we adults directly ask students what they want, and then find specific ways to help them, instead of creating burdens for students in the name of reform.

Fifth, the objections are politically-motivated disturbance in the guise of concern for our children.

Finally, sixth, and perhaps most importantly, whatever happened to free speech?

Amendment I
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech (bold added) , or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.

What have we come to when a subset of the public maintain foul language is protected speech, and a subset of the public agitate to pre-censure the freedom of the President of the United States to encourage students to study hard and stay in school? Even stranger is that both subsets very likely contain many of the same members.

The White House has a video of the President's speech here.

Place Value Part 4: Geometry of Place Value

So far we have completed three parts of the place value series:

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 demonstrates the role of place value in long division.

Today, Part 4: Geometry of Place Value will explore place value within a quadratic equation. We will further show that each monomial can be modeled geometrically.


Review

The expression, 5x² + 6x + 3, appeared in The Chocolate Factory, as a summary of the chocolate packing activity. Five cases and six boxes were packed with three leftover chocolates. Where x stood for the number of chocolates per box, five cases and six boxes could represent different absolute numbers of chocolates. If x =10, or 10 chocolates per box, then ((5 times 100) + (6 times 10) + 3) chocolates, or 563 chocolates came down the conveyor belt, I Love Lucy style. In fact, this episode of I Love Lucy was the inspiration for the math activity.

If the chocolates are packed in boxes of five then the 563 means ((5 times 25) + (6 times 5) + 3) or 158 chocolates came down the conveyor belt. So a quadratic equation can be thought of as an expression of place value in any base. In fact, a polynomial of any degree can be seen as an expression of place value. Missing terms are represented by zeros. So 2x⁶ + 5x⁵+ 3x² + 7x + 2 would be 2,500,372_{base x}.

Now we can see where the analogy to place value breaks down. If x = 6, then a term like 7x would be “illegal.” Once six “boxes” had been packed, those six boxes would immediately be packed into one “case,” so in base 6 the last three terms would properly be 4x² + x + 2, either way, the last three terms represent 152 “chocolates.” Obviously I have just been speaking to adults initiated into the joys of algebra, not children.

What? No Fourth Dimension

Obviously you can use standard base ten blocks to model quadratic equations. If we assume that x represents base 10, for 4x³ + 3x² + 7x + 2, we would use 4 large cubes, 3 flats, 2 rods, and 2 small cubes to model the expression. However, if all I meant by geometry was geometric solids, I would not have meant much. The geometry is more interesting, and becomes clearer when you look at a set of blocks in a different base, say, base 5, the small cube looks the same as a base ten small cube, but the rod is five cubes long, the flat is a square of 25 cubes, the large cube is 5 flats stacked or 125 cubes.

So the rod of any set of base blocks determines the base of the set. If we look at the large cube of any base, we see that any one of the 12 edges shows x¹, the first dimension, any one of the six faces shows x², the second dimension, and the whole cube shows x³, all three dimensions. Now we run smack into a physical limitation of manipulatives; not one can show more than three dimensions. The power of math is that math is the language of imagination. We can imagine a fourth dimension, x⁴ and beyond, even if we cannot model it. How fun is that?

Interestingly, we can also show x⁰ on the cube. Remember for any x, x⁰ = 1. Cubes of varying bases are all different sizes, or volume. The x¹, x², and x³ is different on each cube. But since x⁰ = 1 for any base, it stands to reason that x⁰ or 1 would have an identical appearance no matter the size of the cube. In fact, it does. You can find x⁰ at any vertex, that is to say, the corner shows 1, the zero-th dimension, if you will. In fact, the vertex is a geometric point, described as having no length, width or height.

Multiplication with Base Ten Blocks

So far we have spent a great deal of time establishing that x² means x times x, and that we can show x times x geometrically, by using a flat from a base block set. The flat has a square shape which we would expect from an expression like x-squared. But let's consider a rectangle shape. Now we are not multiplying the same number by itself, x times x, the very definition of squaring, “the product obtained when a number or quantity is multiplied by itself”.

With a rectangle, we are multiplying two different numbers, x times y (or length times width, the formula for the area of a rectangle). Using the cubes from a base blocks set, we can model 5 x 3.






Math educators call this type of diagram a multiplication array. Now lets try 13 x 11.





To show the factor 13 along the top, I used a rod and three small cubes. The factor 11 is along the side with a rod and one small cube. One rod times one rod equals one flat (square, and you expected a square, right?), one rod times three small cubes equals three rods or three lengths. Then, one small cube times one rod equals one length, and one small cube times three small cubes equals three small cubes.

Combining like terms, that is, similar objects, together, we have one flat (10² or 100), four rods ((3 times 10) + (1 times 10)) or 40, and three small cubes (1 times 3, or 3) for a total of 143 which I could express as(1 x 10²) + (4 x 10) + (3 x 1) .

What if we wanted to multiply (x+3)(x+1). I am using the magenta to stand for x, a number we don't know, also called a variable.




The product is 1x² + 4x +3, and geometrically, the product is the picture of a quadratic equation showing both its factors above and to the left of the crossbars. I recommend manipulatives that elucidate the geometry of quadratic equations, available, for example, the Montessori Binomial Cube and Creative Publications Algebra Lab Gear. Remember we have shown that the magenta rod could stand for any value, that is, for any base.

We can show three factors and therefore three dimensions with the same model by standing a rod and/or stacking small cubes vertically in the corner where the crossbars intersect. If I were to stack four small cubes in that intersection, I would be modeling (4)(x+3)(x+1) or by multiplying the x-factors first, (4)(x² + 4x +3). You could think of it as stacking four layers of the x-factor product. In fact, the formula for volume is height times base, or height layers of the base.

In terms of base blocks, the product would be modeled with 4 flats, 16 rods, and 12 small cubes. If we are working in base ten, we would have 400 + 160 + 12. We can exchange 10 of the rods for a flat, and 10 of the small cubes for a rod, ending up with 5 flats, 7 rods, and 2 small cubes or 572.

If I were to stand a rod in the intersection, I am modeling 10 layers of 143 or 1430. In the upper left hand corner of the product there would be ten flats stacked which I can exchange for a large cube worth 10x10x10. Completing any other exchanges, the product would consist of 1 large cube, 4 flats, 3 rods and 0 small cubes. If I were to stand a magenta rod in that intersection. I would be modeling (x)(x+3)(x+1). The product would have 1 magenta cube, 4 magenta flats, 3 magenta rods and 0 small cubes or x³ + 4x² + 3x.

The Candle Problem: How to Damage Motivation

Herbert Kohl says we are missing the boat, motivation wise, in an open letter to Arne Duncan, Secretary of Education.

Now the mantra is high expectations and high standards. Yet, with all that zeal to produce measurable learning outcomes we have lost sight of the essential motivations to learn that moved my students. Recently I asked a number of elementary school students what they were learning about and the reactions were consistently, “We are learning how to do good on the tests.” They did not say they were learning to read.

Mr. Kohl sees a fundamental contradiction between what we say we want and what we are doing to get it.

It is hard for me to understand how educators can claim that they are creating high standards when the substance and content of learning is reduced to the mechanical task of getting a correct answer on a manufactured test.

What, for Mr. Kohl, motivates learning, at least for learning to read?

...reading is a tool, an instrument that is used for pleasure and for the acquisition of knowledge and information about the way the world works. The mastery of complex reading skills develops as students grapple with ideas, learn to understand plot and character, and develop and articulate opinions on literature.

Nowhere does Mr. Kohl mention extrinsic rewards. Teachers have observed, and Robert Slavin's research has confirmed the dissipating effect of extrinsic rewards.

Robert Slavin's position--that extrinsic rewards promote student motivation and learning--may be valid within the context of a "facts-and-skills" curriculum. However, extrinsic rewards are unnecessary when schools offer engaging learning activities; programs addressing social, ethical, and cognitive development; and a supportive environment.

Not only do extrinsic rewards fail to motivate, except in limited cases, but research has also found that extrinsic rewards actually sabotage motivation.



So what's with the ubiquitous classroom token economies? Why must teachers have jars ofmarbleson their desks? Are we deliberately sacrificing long-term learning benefits for short-term classroom management? How about pay-for-performance or merit pay? First. And foundationally, EVERYONE deserves to be paid FAIRLY. “Getting the issue of money off the table,” as Dan Pink says.

If our society want to motivate the highest performance from teachers, then give them:

Autonomy
Mastery
Purpose

NOT merit pay.

Merit pay is inherently unfair. The bug-a-boo with merit pay is that teachers have so little control over the factors that impact student achievement. What do we say, for example, about the student who actually scored worse after his first year with me only to leapfrog three grades the second year with me.  Should I have lost pay the first year?  I was still the same great teacher.  I had no idea his alcoholic uncle moved in with him and his mom that first year. What do you do if you are a great teacher in an environment where just about everything seems to be conspiring against the kids? And what if you are lucky enough to teach in a school where kids have all kinds of advantages and their scores show it regardless of who is their teacher? Policy-makers have not figured out any equitable mechanism for awarding merit pay.

Western Education has the Wrong Mindset

Science educators know full well that school textbooks lag at least a generation behind the times. Teachers who do not take the initiative to independently keep up and supplement the textbook with current information are teaching possibly out-of-date stuff. Sad to say, the vast majority of teachers teach the book, especially at the lower grades where the lifetime foundations for critical thinking are laid.

Hans Rosling, a professor of public health, in a presentation to the US State Department, marvels that the Western world is a generation behind in its understanding of the global situation, especially regarding the developing world.

My problem is that the worldview of my students corresponds to the reality in the world the year their teachers were born.

In Dr. Rosling's words, “Their mindset does not match the data set.”

We have a world that cannot be looked upon as divided.
...snip...
The world is converging.

We have completely misunderstood the HIV “epidemic.”

There is no such thing as an HIV epidemic in Africa...It's not war...It's not economy...Don't make it Africa. Don't make it a race issue. Make it a local issue and do (appropriate) preventative approaches.

Dr. Rosling has made his data presentation software available for free at Gapminder.

I Love Eureka! Physics

Here is a complete episode guide.

It is pretty expensive to purchase the entire series. Here is one source.

Here are a handful of the first episodes:

Episode 1-Inertia




Episode 2-Mass



Episode 3-Speed



Episode 4-Acceleration Part 1



Episode 5-Acceleration Part 2



Here is a video player.

Enjoy.

The New School House Rock

Give a listen to Georgia teacher, Crystal Huau Mills, her students and friends performing their version of Grammar: the Musical, entitled Grammar Jammer, available on DVD.
Crustal Huau Mills wrote the lyrics and her friend, Bryan Shaw, put them to music. When they were all done, they had thirteen songs.

The teacher, who is played by Crystal, falls into a dream world where her class and some of her co-workers are transformed. Many of the normal classroom objects come to life to help her reinforce the underlying lesson behind each song. The clock, the flag, the globe, a crayon, the computer, the ruler, her class pet, a goldfish as well as the dictionary all spring to life to help her teach the class.