About Homework (with bonus lesson on area of a circle)

In the midst of extreme busyness, I left a short, hurried post nearly four weeks ago about how I recently became the temporary legal guardian of a sixth grader. Now it is spring break and I am even busier, but I have managed to carve out half an hour to write. I do not remember being so busy with the demands of a child when I was raising my own kids.

Attitudes of School Officials

Although I have teaching experience in elementary and even preschool, I spent most of my thirty-five-year career in secondary school settings. Suddenly, I have been spending a lot of time interacting with an elementary school. The classroom teacher and the school secretary do not know I am a teacher. The way they choose to interact with me is quite interesting, if somewhat condescending. I can only surmise that school people, intentionally or not, treat parents as if parents are basically ignorant. I apologize if this observation offends some. I completely understand that some parents are difficult, but it was disconcerting to have the teacher or secretary dispute with me almost from the get-go as if I knew nothing about the child.

Homework

Homework? What homework? In spite of the (shall we say) whining that kids these days are overly burdened with homework, I am just not seeing it. My ward brings very little homework home. Alfie Kohn, an educator with whom I am generally on the same page, has been crusading against homework for years. Just a few weeks ago, he sounded similar alarms, making it sound as if any and all homework is bad, bad, bad.

Funny thing I actually agree with most of his points. I absolutely detest homework as busy work. I remember when my own third-grade child came home with an assignment to write out the sevens ten times. His teacher knew that he could recite the times tables on demand, so the assignment was a complete waste of his time. In an effort to salvage some usefulness, he decided to type it in order to he could use the assignment as an excuse to practice the ten-key pad on the right side of his keyboard. His teacher gave him an “F” and scolded him for shortcutting the assignment. “You could have written it once,” she said, “and then simply copied and pasted.”

Nevertheless, homework does have a useful place, especially when used to generate fodder for idea generation during class discussion, as when the student measures the circumference, diameter and radius of ten round things. Or perhaps the student writes up a report of an experiment done in class in order to prepare to discuss the findings with the rest of the class the next day. The homework my ward brought home was generally useless, but certainly not time consuming. The teacher says she is only allowed to give 20 minutes of language arts and twenty minutes of math per day. The teacher does not even expect sixth graders to write their last name on their papers.

Homework as Practice

Normally, if a student has acquired the concept, it does not take a lot of practice to reinforce it. In my experience, homework as practice often means the student has not acquired the concept. The younger the student, the more control the teacher has over acquisition (but it must always be remembered that the teacher does not have total control). Anyway, homework as practice at the elementary level is worse than useless if the student has not acquired the underlying concept. Such students spend the twenty minutes reinforcing the wrong learning. It would also be helpful if elementary math teachers actually had what Liping Ma calls the “profound understanding of fundamental mathematics.”

Area of a Circle

My ward brought home a worksheet to practice finding the area of a circle. From her point of view, it was nothing but plug and chug. She asked me what “pi” meant. We spent a little time developing the concept of pi. Then we cut up a paper plate into pie slivers and arranged them, point up point down, into a sort of parallelogram with scalloped edges. I asked her what shape it reminded her of. Not surprisingly, she answered, “Rectangle.”

I asked her how to find the area of a rectangle. “Length times width” she replied.

“Right. So what part of the circle is the width of this rectangle?”

“The radius.”

“Right. What part of the circle is the length?”

She thought a bit and offered, “Half the circumference?”

“Exactly so. Then instead of length times width, what can we write?”

I began writing “A = r,” whereupon she shouted, “times 1/2C.”

We continued along these lines until we had written A = r X ½ X ∏ X 2 X r. Then r times r equals r squared. ½ times 2 equals 1. When she was done substituting, she had A = ∏r^2.

“Okay,” I said, “Look at your worksheet.”

To her amazement, the formula for the area of a circle was exactly what she had written. She had figured it out herself. That is the sort of success that builds genuine self-esteem. My disappointment came when I described this experience to her math teacher. He had no idea what I was talking about.

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.

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.

Place Value Part 3: The Bake Sale

Place value is such a fundamental concept that we should ensure the students recognize place value and its significance wherever it occurs. An activity I call “The Bake Sale” highlights place value in the operation of division. I will present just one example. Of course, teachers can have as many examples as groups within the classroom. The groups should not be too large, not more than three of four students per group.

The scenario: They are getting ready for a bake sale. They have a platter of cookies and they want to make sure they will have enough cellophane bags to package the cookies. In today's example, the platter has 173 cookies and they will be packing 6 cookies to a bag. I use beans for cookies and little squares of paper for the bags. So the students would start with 173 precounted beans.

The first concept I want them to see is division as repeated subtraction. They are to remove 6 beans at a time, just as if they were really packing cookies, and place them on a square of paper. As they do so they place a tally mark. Very young children would have a specially designed “worksheet” for recoding each “bag.” For example, a page of squares that the students color as they “pack” each “bag.” When they are through, the number of squares with beans and the number of tally marks or colored squares on the worksheet should be the same.

Older students will want to cut to the chase and simply perform the long division. But one purpose of this activity is to help students see the math behind the procedure, and besides in real life, they really would be subtracting 6 cookies at a time, repeatedly, until there were no longer enough cookies to pack a bag.

They should have 28 bags with 5 cookies left over. Some older students already know that the “real” answer is 28 and 5/6, or maybe 28.83 or ... depending on what decisions they make. Some will be sure that the answer is 29 because they learned to round somewhere along the way. Some of them may believe an answer with a remainder (as in 28 R5) is juvenile, and not as good an answer as some of the other possibilities. Students must always be reminded that math is the servant, not the master.

Later in the activity students will see that the “juvenile” answer is the most useful answer.

Once they have determined the answer, it is time to revisit the standard algorithm with a variation. Rewrite the division problem like this:




The green lines show the place value columns. In a class discussion, we establish that a 2 goes above the 7, not because 6 goes into 17 twice, but because the 7 is in the tens’ place, 6 is going into 170 (17 tens) 20 times. The 2 is really a twenty. Students need to be reminded continually what the numerals really signify as they complete calculations. Otherwise, students are merely manipulating abstract, meaningless symbols.

Because we are writing the division problem with Arabic numerals, naturally each digit and its columns represent a place value. Since 6 roundly goes into 170 twenty times, meaning we can show 20 repeated subtractions in one step, we write a 20, not a 2, over the 173. Since we have filled 20 bags at once with 6 cookies per bag, we have removed or subtracted 20 x 6, or 120 cookies from the platter. We show this very concrete action by subtracting 120 from 173, leaving 53 cookies on the platter. We remove enough cookies to fill eight more bags, that is 48 cookies, leaving 5 cookies on the platter, not enough to fill a bag. We needed 28 bags.

Although not “wrong,” 28 and 5/6, 28.833, 28.83 or 29 have no practical utility in this scenario. Students will have an easier time evaluating the reasonableness of an answer if they are encouraged to keep the context and the numbers together. When they round to 29, they are saying 29 what? 29 bags. By the end of the activity, it should be clear that 5/6 of a bag is not helpful and that typical rounding serves no useful purpose. I require students to write their answers in complete English sentences. The answer to this problem is not “28,” or even “28 bags,” but something like “we needed 28 bags to pack the cookies.”

The finished problem would look like this:






The format looks a little different than the standard algorithm, but the significance of place value is preserved. This type of format did not have a name when I first started using it, or perhaps I mistakenly thought at the time that it was an innovation of my own. I was little surprised when the format began appearing in textbooks as “scaffolding.”

Incidentally, at every opportunity we should insist that students read numerals correctly. Simply reading numerals correctly can prevent confusion. “And” marks the spot between “wholes” and “parts.” Although the answers with fractional parts served no real purpose in this activity, of course there are other contexts where the fractional part is important. In any case, some of the other possible answers would be read “twenty eight and five-sixths,” “twenty eight and eighty three hundredths.” I would use “twenty eight point eight three” only for dictation purposes, not for mathematical purposes.

Place Value Part 2: Base Ten for Young Students

One of the most fundamental mathematical concepts, yet one of the most poorly understood, is place value. The typical primary school lesson presents only a superficial, nominal understanding of place value. Students learn merely to correctly name the place-value columns, or identify the digit in a given column, but they often do not understand the significance of the column names.


In Part 1, The Chocolate Factory, I introduced a middle school activity for rebuilding often weak base ten foundational concepts. The activity extends understanding to place value in other bases. In Part 2, I will introduce activities suitable for much younger children. Young children can construct the meaning of base ten place value through many activities and games.

There is some evidence from Jean Piaget's work as illustrated in the video, that base ten is conceptually out of reach for very young children. If there is demand, I will present some activities that help young children explore “Two Land” and “Three Land.” Years ago I field tested a unit called “The Land of Hand” which of course would be “Five Land” in the terminology of the video.




Today I am going to concentrate on base ten, or “Ten Land.”
1. Morning Circle
Many kindergarten and first grade teachers have a regular morning circle time when they gather the children and go through a structured routine of talking about the calendar, the season, birthdays and other topics using a set of visual materials that are permanently on display. The two main math components are the calendar and the base ten pocket chart. The periodicity of the calendar lends itself to a number of activities for building number sense. The base ten pocket chart is decribed below.

The teacher prepares a display of three horizontal pockets with transparent envelopes on the front of each pocket. On the side is a cup full of Popsicle sticks and a stack of cards numbered with the digits from 0 to 9. Pocket charts can also be purchased from various vendors. Every morning the teacher takes one Popsicle stick and places it in the far right pocket (as you face the display). Each day the teacher replaces the card in the envelope to reflect the number of sticks in the pocket.

On the tenth day, the teacher places the tenth stick in the pocket and then makes a show of pointing out there are ten sticks. The teacher then bundles up the ten sticks with a rubber band and places the bundle in the middle pocket. The pocket envelopes should now show (empty, 1, 0) representing 1 bundle of ten sticks and 0 single sticks. The teacher goes through the Popsicle stick routine every day.

On the hundredth day, a celebration day in many schools, the teacher gathers the 10 bundles, ties them together with a piece of yarn and places the whole bundle in the far left pocket and changes the display to show (1,0,0) representing 1 packet of 10 bundles, 0 bundles of 10 sticks, and 0 single sticks. The teacher continues the routine until the last day of school at which point the display should show something like (1, 8, 5).



2. Trading Activities and Games

Playing games is a natural way for children to acquire all sorts of different aspects of number sense. Years ago I checked a book out of the library that was chock full of wonderful tutoring games. The book has long since gone out of print but no matter. I found the author, Peggy Kaye's website. Here is my version of a game she calls "Fifty Wins."

The teacher creates two boards on heavy card stock, one for each player. Each player also has a die. I recommend using extra large die if you can find them. Each player also has a collection of 50+ beans, pennies, or other counters. My own modification involves using the board at first, then doing away with the board and playing with pennies and dimes.




Each child casts their die in turn, and draws the number of counters that matches the number of dots on their die, placing one counter in each of the small squares of which there are nine. Upon accumulating the tenth counter, they transfer ten counters to one of the five big squares. The first person to get fifty counters wins. Children learn there can never be more than nine in the one's place, and that the ten's place is precisely groups of ten. If three big squares are filled and none of the little squares, they can see very clearly 3 (groups of ten) 0 or 30.

A modification I have made is to use poker chips for counters. I change the design of the board so that the nine little squares become a long rectangle outlined in one color (say blue) and the big squares are outlined in another color (say red). Then as the child accumulates 10 blue chips, the child exchanges the 10 blue chips for one red chip and places it in one of the red squares. The poker chip modification leads quite naturally in the penny-dime modification I mentioned earlier. I have also used the same poker chips with the same color signification for "The Chocolate Factory" activity, blue for leftovers, red for boxes, white for cases.

Another modification of mine which may be considered a weakening of the game is the use of a die to generate numbers. The original game uses a spinner where some of the fields say “Win 10.” At the beginning the child will dutifully count out ten beans and place them one by one in the small squares, only to have to transfer the entire group of ten to a big square. Very soon the child counts out the ten beans and straightway places them in a big square. The opportunity to realize a group of ten in one turn is lost when die are used, but I suppose you could use a set of two dice. I like the die because the child does not have to read words or numerals. With die, the child has only to match, by one-to-one correspondence, the beans to the die spots. There is no need to reference numerals at all, so the game stays squarely focused on number and avoids number/numeral conflation.


“Make Fifty” is just one example of what is known as a “trading activity.” Cuisenaire rods also work well for trading activities. Every ten cubes makes one rod. Any base-ten block set goes one step further where every ten rods makes one flat, and every ten flats makes one cube. Many base ten block worksheets can be adapted to active lessons.

All manipulatives have limitations and some researchers are concerned about the limitations of base ten blocks. Nevertheless, with a good mix of activities, the teacher can address the differing learning styles of each student.

Stuff to Avoid
Worksheets
Generally speaking, worksheets should be avoided. Nevertheless, I like to design special worksheets as data recording instruments for math labs utilizing base-ten blocks and Cuisenaire rods. Students can learn a lot of math without writing numerals. In fact, a foundation of math reasoning skills without reliance on numerals helps children acquire the concept of the difference between numbers and culturally-determined symbols for numbers such as Arabic numerals. Schools “accidentally on purpose” teach children to confuse number and symbol. Cuisenaire has a few such worksheets along this idea, but I have some problems with the worksheet design. Maybe I'll collect my math lab worksheets into some kind of cohesive with comprehensive directions for using them with children and make them available.

Computer-Based Materials

Too many of the computer-based materials, animated mathematics and virtual manipulatives, though so appealing to adults, often have a magical quality to young children. Regrouping happens before their very eyes but they do not understand the mathematical concept and mechanism. They do not get from the computer what I call the psychology of numbers, or how numbers behave. It is just a lot of cool special effects without specific mathematical concept acquisition benefit.

Calculators

Despite the National Council of Teachers of Mathematics (NCTM) claims to the contrary, calculator studies with the youngest students show no advantage in the development of children's number sense. In 2002, I conducted a major survey of research, research critiques, case studies, and editorials. I periodically asked NCTM to provide me a list of what they characterized as supporting research, but they never did. I found no basis for NCTM's assertion that research backed their recommendation for calculator use in the earliest grades. I found that calculator usage need not hinder the development of math reasoning skills, but it may in fact do so. Teachers report that children become overly dependent on the calculator and have difficulty learning to evaluate the reasonableness of their answers. They trust the calculator more than themselves.

Links

The following is a list of links of base ten lessons. They are presented as is. Many exemplify what I believe are the main weaknesses of most base ten teaching.
Lesson plans reviewed by teachers:
Crayola Tally Sticks:
Applet: but better off using concrete manipulatives.
An Unreviewed Collection of various resources:
A favorite resource for getting teaching ideas:
Vendor:

Lesson Plan: The Chocolate Factory or Place Value in Algebraic Thinking

Because students typically have fuzzy notions of place value, they may be able to correctly name the place-value columns, but they often do not understand the significance of the names. For example, they cannot give a mathematical explanation of why regrouping works. One reason may be that they rarely receive mathematical explanations.

The explanations may certainly be chock full of numbers yet without having the least connection to the way the numbers work. A good example is the standard method for finding 10% of a number: just move the decimal place one digit to the left. The method is nothing but a trick, and our children learn to mistake performance of tricks for understanding of math.

Students need help in constructing mathematical explanations. In an activity I call “The Chocolate Factory,” students pack chocolates in boxes, then in cases, while keeping a tally. At the end of the activity, students will be able to trade and regroup in order to add or subtract.

I usually use beans instead of chocolate because it is less messy and less tempting. I explain that the students are working for Hershey Chocolate Company packing chocolates as chocolate pieces roll down the conveyor belt a la a famous “I Love Lucy” episode. The number of chocolates in each group is simulated by drawing a card from a shuffled deck with no picture cards. A specially made set of number cards with spots but no numerals would be better. Students pack the pieces into boxes of ten pieces each, then pack the boxes into cases of ten boxes each, keeping a running tally in a table on the blackboard.

Draw	Cases	Boxes	Leftovers
1			/////
2			////////
Result		/	///
3			//////
Result		/	/////////
4			//
Result		//	/
	etc. until, say,
Result	/////	//////	///

Each pair of students shares a set-up: 100 beans, a container capable of holding ten beans to represent boxes, and a larger container to hold ten “boxes.” The teacher explains that the rule of the game is that a “box” can only hold ten beans. Once a box is filled, they begin filling another box, and so on until they have ten boxes. Ten boxes are then packed into a case.

The teacher shuffles the cards and holds the deck face down. The teacher uses any suitable method to select a student to pick a card. The student takes a card from the deck (a five-spot in the example) and shows it to the class. Each pair counts out five beans and puts them in a “box.” The teacher records the five as tally marks in the “leftovers” column. Another student picks a card (an eight spot). The students count out eight beans and the teacher records the tally in the “leftover” column The students use the beans to fill a box, pointing out that they have one full box and three leftovers. The teacher records the result with one tally mark in the “box” column, and three tally marks in the “leftover” column.

It is important to give students experience with “Cases, Boxes and Leftovers” before renaming these columns “100’s, 10’s and 1’s.” Another advantage to using the column names, “cases, boxes, leftovers” is that the activity can be recycled later for teaching any base. I have found it is more helpful to rename the “ones” place “leftovers”. Then it is easy to explain that there are leftovers when the amount is insufficient to fill a box. Thus, there will never be 10 leftovers, because 10 will fill a box, thereby adding 1 to the tally in the “Boxes” column. Converting the final tally in the table to numerals yields 563. Students readily understand that as they accumulate 10 boxes, they transfer those boxes as 1 case and put a tally mark in the “Case” column. It is often at this very point the light bulbs go on, and students see the why carrying works for the first time.

Then we repeat the activity, but the cards now simulate consumed chocolate (yum). From the deck, a student draws, say, an eight-spot to stand for eating eight pieces. Students will naturally want to open a box to accomplish this. As they take a box, they erase a tally mark and dump the 10 chocolates (beans) with the leftovers, and record ten more tally marks for a total of 13 tally marks in the leftover column. They continue subtracting in this way. This activity is very similar to most trading activities, but seems to be more effective at building the concept of place value because we avoid giving the columns numerical names at the outset.

With older students we simultaneously keep a record of this computation in the standard algorithm. Again, students often understand regrouping for the first time. We expand and repeat the activity with other groupings which I have carefully planned in advance. I tell students that they have done such a good job that now they work for a more expensive chocolate company, perhaps Ghirardelli, where chocolates are packed in boxes of 5 pieces, and cases of 5 boxes. I give the students 158 chocolates, knowing full well they will again end up with 5 cases, 6 boxes, and 3 leftovers, the same tally as for the Hershey exercise.

In the ensuing class discussion, we talk about why the first 563 (10 to a box) has more chocolate pieces than the second 563 (5 to a box). Students discover that neatly lining up their addition and subtraction columns is not merely for neatness sake, but because the columns have real meaning. Students find they can work just as readily in other bases as long as they remember the basis (pun intended) of the groupings. (It is also valuable to finish the base five regouping so that there is one crate of 5 cases, one case of 5 boxes, one box of 5 pieces, and 3 leftovers, or 1113 in base five, and why "563 base five" is technically illegal).

If working in base ten, I prefer to name the columns from right to left “leftovers, 10, 100, 1000,” etc. As the Chocolate Factory activity illustrates, the “ones” are “ones” only because there are not enough of them to make a “ten”. They are the ungrouped leftovers, whether in base ten or any other base. In fact, students get very comfortable with working in a variety of bases and discover that for any base (b), the column names will be (from right to left) “(leftovers), (b), (b x b), (b x b x b), and so on. For example, they would name the base 7 columns “(leftovers), (7), (7 x 7), (7 x 7 x 7), and so on.

I like using the parenthesis early on so students become familiar with the idea of parenthesis holding a number just as cupped hands hold an apple, and that a number can have different appearances, and all are still equivalent. In practice, I often go beyond leftovers, boxes, and cases, and extend the activity to crates, trucks and warehouses. Just like the the song from School House Rock says, “Don't you worry 'bout the big numbers, they're just bigger, that's all.”

Later the columns can be renamed with exponents, 10^2, 10^1, 10^0 or 7^2, 7^1, 7^0. Then it is a small step to b^2, b^1, b^0 (where b stands for "base"), then x^2, x^1, x^0. Students often practice writing in expanded notation without ever grasping real significance of what they are doing. In algebra, many polynomial expressions are really bases in disguise. For example, the base 10 tally and the base 7 tally were both 563. Algebraically, both would be expressed as 5x^2 + 6x + 3, where x is the base. The algebraic expression is nothing more than expanded notation. If x is 10, then the expanded notation is (5 x 10^2) + (6 x 10) + (3 x 1). If x = 7, then the expanded notation is (5 x 7^2) + (6 x 7) + (3 x 1). Rewriting numbers as polynomial expressions often makes calculations in different bases much easier, and The Chocolate Factory activity enhances such algebraic understanding.

There is a vendor, Digi-Block, who sells a manipulative that would be ideal for the base 10 chocolate factory. The set has pieces, boxes and cases. Each box holds exactly 10 pieces, and each case holds exactly 10 boxes. The big advantage is that students are prevented from overpacking or underpacking. I have usually had to rely on materials I have scrounged: beans, little cough syrup cups for boxes, and little containers to hold 10 cough syrup cups. If you are looking for basic base ten blocks, Nasco probably has the most complete assortment anywhere.