### Includes the  New York City Modifications

By Bill Quirk  (wgquirk@wgquirk.com)

Here you will find a compact version of the NCEE  "Work Sample & Commentary" sets (WS&C sets) for elementary school math.  We also cover the New York City modifications to these sets.  In brief, the New York City authors added one set, changed some work samples for the first two sets, and chose to omit four NCEE sets.  As you read below, please note that the NYC authors regularly missed opportunities to improve the NCEE's product.

NCEE elementary school math is exemplified by the TERC K-5 math program, Investigations in Number, Data, and Space.  Widely used in New York City, TERC openly claims a "constructivist approach," and uses the phrase "constructivist math" to identify their curriculum.   For more about TERC math, click on TERC Hands-On Math: The Truth is in The Details.

The NCEE claims that the student work samples found below are "genuine student work," demonstrating "standard-setting performances" at the end of the fourth grade.  Apparently there's one more year to learn elementary school math.  Unfortunately, the Chapter 3 middle school work samples offer no evidence of genuine K-5 math by the end of the 8th grade.

### Direct Access Links to Each WS&C Set

1. Sharing 25                 [Includes the NYC modifications]
2. Arithmetic                 [Includes the NYC modifications]
3. 3-D to 2-D
4. Patterns
5. Pumpkin Activity
6. Height Measurement Statistics
7. Two Dice Sums
8. Creatures
9. Se Hace un Triangulo?
10. The Great Fish Dilemma     [Not in the NYC version]
11. How Many Handshakes?
12. Tangram Dispute
13. Feverish Freddy
14. Counting on Frank
15. School Uniforms Project    [Not in the NYC version]
16. Catapult Investigation     [Not in the NYC version]
17. The Never Ending Four
18. Dream House Project
19. Constructing a Polyhedron  [Not in the NYC version]
20. Making a Cube              [Only in the NYC version]

### The Work Sample & Commentary Sets for NCEE Elementary School Math

1. Sharing 25
• Task: "Share 25" as equally as possible in three different concrete situations.
1. How can four friends share 25 balloons "as equally as possible?"
2. How can four friends share \$25 "as equally as possible?"
3. How can four friends share 25 cookies "as equally as possible?"
• Method:  Answers are shown for one student.  Computation details are missing.
1. Constructivist math students are expected to remember many "number fact equivalents," such as 25 ÷ 4 can be expressed (for the three contexts) as 6 R1,  \$6.25,  and  6 1/4.
2. Constructivists wants students to develop number sense and the "meaning" of operations  by working constantly  with "familiar numbers." The "landmark numbers" are small whole numbers, multiples of 10, and multiples of 25.  The  "familiar fractions" are proper fractions with denominator equal to 2, 3, 4, 5, 6, 8, 10 or 12.  Yes, 7, 9, and 11 are unfamiliar denominators.
• Although constructivist math educators expect students to remember many "familiar number" facts, they reject the importance of remembering single-digit number facts, such as 7 x 8 = 56, and 56 factors as 7 x 8.  Automatic recall in the forward direction is necessary for later mastery of multidigit computation, operations with fractions, and comparing fractions.  Automatic recall in the reverse (factoring) direction is necessary for simplifying fractions.  It also orients the mind to learn about the next level of factoring found in algebra.  But these are all non-calculator skills,  and that explains the attitude of the NCTM and NCEE.  They believe that such skills are now obsolete.
• New York City Variation:  Answers are shown for two students.  Computation details are missing.
1. Physical objects can't always be divided, but this point isn't mentioned. The balloon problem could have been used to make that point clear.  Instead both NYC students found a way around the problem of the extra balloon.  One threw it "up into the air,"  and the other made "this one bigger so they can all play with it together."
2. One NYC student drew a picture (an arrow pointing to drawing of cookie) showing the extra cookie equally divided into 4 pie-shaped sections. Inserting the picture in the sentence, the student wrote "There is one left.  I  (picture inserted) this cookie into 4 shares as there are 4 people."  The fraction 1/ is never mentioned.
2. Arithmetic
1. This is the longest set.  With the exception of one work sample (computing 13 x 14) found below in the  Dream House set, all student work demonstrating non-calculator multidigit computation is found here. .
2. We are assuming that these work samples demonstrate non-calculator work, but the NCEE instructions never rule out the use of a calculator, and many details are omitted.  Since constructivist math educators encourage the regular use of calculators, our assumption may be wrong.
3. The student is regularly asked to describe multiple methods. This allows the student to explain the method, provide answers, but omit the complete details.  This presentation approach is very useful when you have a superficial knowledge of the methods used.
4. The distributive law is used multiple times below, but it's never explicitly recognized in NCEE math.  The NCEE sometimes comments that the student "broke the number apart."
5. When we express 87 as (80 + 7) or as (8 x 10) + 7, we are expanding 87 using its place value definition.  Such expansions are used multiple times below, but place value isn't explicitly mentioned here or elsewhere in NCEE math.  Similar to applications of the distributive law, place value expansions are vaguely described as "breaking numbers apart."
6. Handling the need to carry (compose relative to place value) and borrow (decompose relative to place value) is an essential requirement of any computational method.   It goes to the heart of our ingenious system for representing numbers in terms of powers of 10.  But carrying, borrowing, or the general term, regrouping, are never mentioned in NCEE math.  This is accomplished by choosing numbers so that  neither carrying nor borrowing is required, or by transforming the the problem into an equivalent problem which requires neither, or by simply providing the answer, with no mention of the carrying or borrowing details.
7. Properties of equality, such as substitution, are used frequently in this WS&C set, but the NCEE doesn't list specific properties of equality.  They simply say that the student should understand that "an equality relationship between two quantities remains the same as long as the same change is made to both quantities."   It's up to the student to decide what they mean by "the same change."
8. The standard algorithms for multidigit computation are not demonstrated  here or elsewhere in NCEE math.  The standard algorithms are efficient, accurate, general, and work the same way in every case.  Their power is best understood with more difficult computations, such as 8,756 x 334.26.  Once mastered, the standard methods can be carried out automatically, without conscious thought.  This frees the mind to learn about the generalization of these ideas found in algebra.   But this is no problem for the NCEE.   They also omit genuine algebra.
• Task 1Compute 63 x 46.
• Method:  The student wrote "I knew that I could add 63  46 times but it would take too much time so I decided to do it with doubling."  The student then computed 63 + 63, 126 + 126, 252 + 252, 504 + 504, and 1,008 + 1,008  to reach the intermediate total of 2,016.  Then, realizing that 2,016 corresponded to 32  63's and doubling again would go too far, the student changed strategies to take advantage of  previously calculated sums.  The student continued by computing  2,016 + 126, 2,142 + 504, and 2,646 + 252 to get the final total of 2,898.  All along the way, the student kept track of how many 63's corresponded to each intermediate total.
1. This method certainly demonstrates impressive thinking, but it requires significant conscious effort to carefully calculate the eight sums and write down the "keeping track" details.  Additionally, it's a special case method, limited to small numbers, with details that vary depending on the specific numbers involved.
2. The need for separate notes to keep track is a major inefficient characteristic of the non-standard alternatives to the standard algorithms of multidigit computation.
3. The NSPS authors say this sample "was produced before the class had received any instruction about two digit by two digit multiplication."   Approaching the end of elementary school, that instruction is overdue.  The good ideas presented don't require continued demonstration with numbers requiring so many steps.
• Task 2.1Compute 522 - 367 in two different ways.
• Method 1:  The student added up from 367 as follows: 3 + 30 + 100 + 22 = 155.  A note says: "the student used landmark numbers to go from 367 to 370 to 400 to 500."
• Method 2:  The student added 43 to both 367 and 522 to "make the bottom number easier to subtract," and then subtracted 410 from 565.
1. Both methods avoid borrowing.  That's the idea.
2. Why add 43?  Consider 367 and note that 4 + 6 = 10 and 3 + 7 = 10.  This method usually leads to an equivalent subtraction that avoids borrowing.  But one computation is replaced by three, and carrying may be required for calculating the two sums.
• This borrowing avoidance strategy doesn't always work.  Try 578 - 385.   The magic number is now 25, the subtraction transforms to 603 - 410,  and the need for borrowing raises its ugly head.  This is a characteristic of constructivist math programs.  Their special case methods often don't work if you change the numbers.
• With a little experience (familiarity),  the new "bottom number" can be quickly written down, without thinking about carrying.  See the pattern?
3. The idea of transforming a problem into an equivalent, easier problem is a good strategy, but the transformation here significantly increases the effort.
• Task 2.2Compute 87 x 9 in two different ways.
• Method 1: The student wrote  "I used 8 x 9 as a stepping stone and I tacked on a zero."  After writing down 720, the student wrote 7 x 9 = 63, placed the 63, right-justified under 720, drew a lines under the 63, and then wrote 783 under the line.  The NSPS authors observed that the student "broke the number apart."
• Method 2:  The student wrote "What I did is split 80 in half and multiplied it by nine."  In a vertical column the student wrote 40 x 9 = 360, then another 40 x 9 = 360 directly below, then  7 x 9 = 63 directly below that, and finally 783 for the sum: 360 + 360 + 63.
1. Note the non-standard student language ("stepping stone", "tacked on", and "split in half").  The NCEE makes no comment.
2. The NCEE  uses "broke the numbers apart" as a blanket phrase to cover the two key ideas illustrated here:
• 87 is expanded, relative to place value, as (80 + 7).
• (80 + 7) x 9 is rewritten as (80 x 9) + (7 x 9), using the distributive law.
• 80 x 9 is rewritten as (40 + 40) x 9 = (40 x 9) + (40 x 9) using the distributive law.  The fact that that substitution is also involved is never mentioned by anyone.
• Task 3: Demonstrate multiple ways to multiply 62 x 85.
• Task 4Four children are to "fairly" share 3 bags, with each containing 52 of M & M's.  How much does each child get?
• Method: The student first drew a four column table, identifying the columns with 1, 2, 3, and 4 as column headings.  The student then said "3 times 52 equals ... 150 ...152, 154, 156.  And that equals 156.  So since there's four children, I split the 156 and I said, 20, 20, 20, 20."  The student wrote one 20 in each column and said  "I added all the 20's up.  20, 40, 60, 80. . . . Then I said 85, 90, 95, 100."  The student wrote one 5 in each column and said "10, 20, 30, 40.  These are tens."  The student wrote one 10 in each column.  The student then counted from 141 to 156, placing a 1 in one column for each number counted, moving left to right from columns 1 through 4 and starting again at column 1 until all 16 1's were placed.  At the end of this process each column contained 20, 5, 10, 1 , 1, 1, 1.  The student then chose one column and added (in the following sequence) 20 + 10 + 5 + 1 + 1 + 1 + 1 to get 39.
1. For twice as much fun, try 312 divided by 8.
2. Students at the level should know how to divide a 3-digit number by a 1-digit number.
3. The NYC version omits the preceding method and substitutes the following two methods:
• NYC Method 1: The student added 52 + 52 + 52  and then divided 156 by 4 to get 39.  The computational details were omitted.
• NYC Method 2: The student added 52 + 52 to get 104, and then added 52 to 104 to get 156.  The student then wrote the column:
•
156 ÷ 4 =
100 ÷ 4 = 25
50 ÷ 4 = 121/2
6 ÷ 4 = 11/2
The student then wrote the column:
25
121/2
11/2
____
39
1. NYC Method 2 converts to familiar divisions, allowing the student to utilize remembered number fact equivalents.  The two computations, 100 ÷ 4 and 50 ÷ 4, involve the "landmark numbers" 100 and 50, and the third computation, 6 ÷ 4, is remembered as an  equivalent form of the "familiar fraction" 3/2.
• The NYC authors say: "the student broke 156 into 100, 50, and 6 and divided each part by 4."  This explanation hides the use of the distributive law to expand 156 ÷ 4 as (100 ÷ 4) + (50 ÷ 4) + (6 ÷ 4).
2. Students in NYC TERC classrooms work constantly with the numbers in NYC Method 2.  They are expected to remember the four equivalent values  (25,  121/2 , 11/2 , and 1) for the four familiar expressions (100 ÷ 4, 50 ÷ 4, 6 ÷ 4, and 1/2 + 1/2 ).   [If you find this last statement hard to believe, click on 50 ÷ 4  to see an illustration in TERC math.]
3. The NYC authors make no mention of the incorrect use of the "=" sign following 156 ÷ 4 in the first column for Method 2.  See  Adding It Up (page 270) for a related comment.
3. 3-D to 2-D
• Task 1: Given a picture of a 3-D "cube building," draw the front view.
• Method: The student was shown a 14 cube "building," consisting of two 2 by 3 cube layers, topped by the remaining two cubes positioned diagonally opposite in corners.  The student drew the correct front view and then belabored the obvious by providing a lengthy written defense of the drawing.
• Task 2: Given pictures of three 3-D "cube buildings," draw the front, side, and top views.
• Method: The student first "built these shapes with inter-locking cubes" and then drew the three 2-D views.
• Comment: Too much busywork, with pictures, plastic cubes, drawings, and written opinions.
4. Patterns
• Method:  The student gave examples of  "linear patterns" and "non-linear patterns."  A "linear pattern" example is   {9, 1, 9, 1, 9, 1, 9, 1, ....}  A "non-linear pattern" example is {1, 4, 9, 16, 25, 36, .....}.
• Notes from the NCEE:
1. What do they mean by "linear" and "non-linear?"  One NCEE note tells us the student "shows how one quantity determines another in a linear ('repeating') pattern."  In the next sentence they say "a pattern can be linear, i.e. 'may or may not repeat itself.' "  Next we are told that "a pattern can 'grow or lower itself,' i.e., be non-linear."   These are the only "definitions" offered for "linear" and "non-linear."
2. The NCEE points to "pattern may or may not repeat itself" and "shapes that keep repeating" as examples that "parts of the work provide evidence of appropriate use of mathematical terms and vocabulary."
1. The "definition" of "linear pattern" should link to the concept of a straight line. The "linear pattern" examples offered here should be described as repeating sequences. The "non-linear" examples offered should be described as increasing sequences.
2. Notice that {4, 8, 12, 16, ...} isn't a linear pattern because it's not "repeating."  On the other hand, maybe it is linear because a linear pattern "may not repeat itself."
3. If non-linear ("grow or lower") really means "increasing or decreasing," and non-linear also means "not linear," then any pattern (sequence?) can be classified as repeating, increasing, or decreasing.   Then, since there are sequences that don't fall into one of these three classes, it follows that such sequences that don't qualify as patterns.  But  {1, 1, 2, 1, 2, 3, 1, 2, 3, 4, .... } looks like a pattern to us.   We must have misunderstood.  Perhaps they're not using 2-valued logic.
5. Pumpkin Activity
• Task: Estimate the height, diameter, circumference, weight, and number of seeds for your group's pumpkin.  Then measure and count the the number of seeds.
• Method:  Eight groups of four students carried out this task.  The estimates (guesses) were recorded for each student.  One set of measurements was recorded for each group. Weighing was accomplished by weighing a student, first holding the pumpkin, and then without the pumpkin. The "diameter" was measured by placing two rulers, vertically and "on the sides," with another ruler balanced on the top used to determine the measurement. The "circumference" was measured by "wrapping a string around the pumpkin" and then measuring the string.  The "height" was determined via a vertical ruler on "the side" and a horizontal ruler on "the top".   Seed counts, ranging from 233 to 824, were recorded for the eight groups, but the counting method wasn't mentioned.  The students developed charts and tables to record each group's estimates and to show how the eight pumpkins compared for weight, "diameter," "circumference," "height," and number of seeds.
1. The terms "diameter," "circumference," and "height" are used for an irregular object.  But a pumpkin doesn't have a well defined diameter, circumference, or height.  This attempt to be "real world" leads to misconceptions, such as the need here to distinguish between diameter and height.
2. One student estimated the "diameter" as 6 1/2  inches and the "circumference" as 7 1/2  inches. Another student estimated the "diameter" as 7 inches and the "circumference" as 9 inches.  There's no NCEE comment.
3. There's nothing about the key idea that circumference divided by diameter is a constant.   On the other hand, this can't be demonstrated with a pumpkin.
4. A note claims that this example demonstrates statistical knowledge, but the students counted every seed in all eight pumpkins.  Keeping the students busy was more important then the time-saving concept of a sample.
6. Height Measurement Statistics
• Task: "Collect everyone's height measurement in inches, make a line plot with the data, and write about what you noticed about the data."
• Method:  The students developed a line plot of 24 measured heights. They found the range, median and identified an outlier.  They also identified "bumps" where "the line plot rose," and and "holes" where "no one is that height."
1. Bumps and holes?  Students are being mislead into thinking that such non-standard descriptors are of some statistical importance.
2. More time-consuming busywork.  Why not give them the data?   They could then find the range, median, and outliers for multiple data sets in less time.  With the repetition of multiple examples, they might remember the ideas.  There might be time to also cover the arithmetical mean and mode(s).
7. Two Dice Sums
• Task List all possible combinations for two dice, list all possible sums for two dice, and list all possible probabilities for two dice sums.  Then play a game involving two dice sums.
• Example:  The combinations (1,4), (4,1), (2,3), and (3,2) are the only combinations for the sum 5.
• Note:  This statement of the task and the example are not found in the NSPS notes. The actual notes say  "think of a way to figure out all the combinations (sums) for dice rolled together." This  apparent equating of the term "combinations" with the term "sums" is confusing and has been avoided here to improve clarity for the reader.
• Method:  The student listed all possible pairs and then entered the sum for each of the 36 pairs in a 6 x 6 grid provided by the teacher. The student then counted the number of distinct occurrences for each possible sum and wrote the eleven requested fractions (example: 4/36 for the sum 5). Fractions were not simplified.  Students observed that 7 was the most likely outcome in the game, with probability 6/36.  This fraction wasn't simplified to 1/6.
• The Game: Each student is to draw eleven circles, placing each circle under one of the eleven column headings for the eleven possible sums, 2 through 12.  For example, the student might draw five circles under the sum 7, three under 6, and three under 8.  Each time the student rolls the dice, all the student's circles corresponding to the two dice sum are filled with an "X".  The first student who so marks all eleven circles is the winner.  Each student was asked to "write the reason why you put the circles  where you did."
1. A note states "The activities were spread out over two days."   There must be time to fill out more tables and charts.  It's necessary to carefully record every case for two dice combinations and two dice sums, and then there are the lengthy written justifications for circle placement choices in the game.  The students are kept busy.
2. A note tells us that the game comes from constructivist math educator, Marilyn Burns.   By emphasizing "the winner" and subjective arguments defending the choices for placing the eleven circles, this game misleads students to think that probability has major significance for just one game.
8. Creatures
• TaskHow many characters can you make by combining 4 (different) heads with 4 (different) bodies.  Fully stated, this task called for drawing "four Halloween characters (a ghost, a witch, a skeleton, and a pumpkin-head scarecrow)."   The students were then instructed "to cut the figures into four head and four bodies and staple each set into small 'flip books.' "  The teacher then asked  "How many characters could you possibly come up with by combining the different parts in different ways?  Show and explain in detail all the combinations you could make."
• Method:  The NCEE says "the student used two approaches to solve the problem: a 'flip book'/systematic listing and a student-created multiplicative formula."  The student developed 16 drawings, showing each of the four heads combined with each of the four bodies.   The drawings are accompanied by a lengthy written explanation.   The NCEE used bracketed insertions to clarify part of the student's statement "the student made connections among different conceptual approaches: 'I know that my [systematic listing] solution is correct because there are four monsters, so that means there are four bodies [and four heads] and four times four equal sixteen.' "
1. The "student-created multiplicative formula" should eventually lead to the recognition of the Fundamental Counting Principle (FCP) and advanced counting techniques, including permutations and combinations.  But that's never done at any level in NCEE math.
• The FCP: If one event can occur in N ways, and a second event can occur in M ways, then both events can occur in N x M ways.
• An application of the FCP quickly produces 36 as the number of combinations for the preceding Two Dice Sums WS&C set.
2. Why can't the students be asked "How many different ways can each of the four letters A, B, C, and D be combined with each of the four numbers, 1, 2, 3, and 4?   If it must be "hands-on," why not use manipulatives already available in the classroom?
3. The NYC version includes more samples of student work, but no new ideas.
9. Se hace un triangulo?
• Task"Suppose you were given a string that is sixteen inches long.  If you cut or fold it in any two places, will it always make a triangle?"
• Method:  Using "trail and error" and "string, scissors, and a ruler," the student discovered "if the two sides add up to more than the bottom side, it will make a triangle."
• Comment:  Constructivist math is hands-on concrete math.  This can't be done more simply with a pencil, paper, and a ruler.
10. The Great Fish Dilemma  [Not in the New York City version]
• Task"How many different ways can you put nine fish in two bowls?"
• Method 1:  The first student made a two-column chart, with "Bowl 1" and "Bowl 2" as column headings, and then wrote 10 rows.  The first row indicated 0 in Bowl 1 and 9 in Bowl 2.  The second row indicated 1 in Bowl 1 and 8 in Bowl 2.  Continuing in this fashion, the last row indicated 9 in Bowl 1 and 0 in Bowl 2.  The student wrote "I found 10 possibilities," and then wrote why some possibilities were better than others for the comfort of the fish.
• Method 2:  The second student "used cubes and blocks, and then drew pictures."  A note says "the student used cubes and blocks to make decisions."  This student modified the problem by declaring that "two of the fish are Simese (sic) Fighting Fish, and the rest are Neons.  The Simese (sic) Fighting Fish can't be together ever or they kill each other." The student then drew eight pictures, each showing two bowls, always with one "S" in each bowl.  The first two-bowl picture shows the letter "N" repeated 7 times in bowl 1.  The second two-bowl picture shows the letter "N" repeated 6 times in bowl 1 and one "N" in bowl 2.  Continuing in this fashion, the last picture shows the letter "N" repeated seven times in bowl 2, with just the letter "S" in bowl 1.  This student found 8 possibilities.
1. This WS&C set includes pictures of different types of fish and both students mention different types, but the fish are considered identical for the math of this problem.  Put another way, both students simply counted the different possiblities for the total number of fish in  Bowl 1.
2. But a "real world" owner of fish is likely to distinguish one fish from another, so the correct answer, using the FCP, is 29 for the first student and 28 for the second student.
• Considering how this problem should be correctly solved, 3 or 4 fish would be a better example at the elementary school level.   The 9 fish case should wait until exponents and the FCP are both formally introduced.   Neither are ever discussed in NCEE math.
3. The NCEE is pleased with the second student's creative modification of the problem.  They don't recognize the concept of a well-posed problem with precise conditions that lead to one right answer.
4. This problem was apparently contributed by a Vermont source.  A note says that "students could choose to include their work in the Vermont statewide portfolio assessment."   One wonders if the 29 solution would be considered wrong in Vermont.
11. How Many Handshakes?
• Task: How many ways can five people shake hands with each other, shaking every hand just once.
• Method:  Students drew "bubbles" and connected each bubble with one line drawn to each other bubble.  They then counted the lines.  Some students noticed a pattern:  the first person can shake four hands, then the second person can only shake three hands not previously shaken, then the third person can only shake two hands not previously shaken, and the fourth person can only shake one hand not previously shaken.  So the answer is 4 + 3 + 2 + 1.
1. The preceding statement of the method isn't found in the NSPS document.  Although work samples are shown for several students, they all say "number 1 shakes 4 people, number 2 shakes 3 people, etc." or something similar.  Of course each person shook 4 hands.  At each point we are interesting in counting the hands that haven't already been shaken.  It's not clear that any of these students really understood this.
2. More generally, the answer for N people is found by summing  (N-1) + (N-2) + + + 1.  The interest then should shift to an efficient way to compute such a sum.  But, although examples other than 5 are shown, the students always carried out the full extended addition.
3. The FCP can be applied  here.  There are 5 ways to choose the first hand (first event) and 4 ways to choose the second hand (second event).  So there are 20 ways to choose a pair of hands.  But this counts each pair twice, so 20 must be divided by 2 to yield the answer 10.
4. For more about the teaching possibilities associated with the handshake problem, click on Adding It Up  The discussion, beginning in the middle of page 107, begins by noting  "this problem appears often in the literature on problem solving in school mathematics, probably because it can be solved in so many ways."
• At the bottom of page 108, read about busywork when Carl Friedrich Gauss (1777 - 1855) was a schoolboy.  His teacher attempted to occupy the class by asking the students to compute the sum of the first N whole numbers, for the case N = 100.   How Gauss quickly produced the solution and the simplicity of his idea (see below) should fascinate young children.  But this beautiful opportunity to teach about a math "discovery" was missed.  Perhaps the NSPS authors don't know about the most famous schoolboy discovery in the history of mathematics.
•  Gauss recognized that he could quickly compute twice the desired sum:

•  1.   (      1 +    2 +     3 +     4 +     5   +  +  . . . .  + 100)  = S
2.  + (100 +  99 +   98 +   97 +   96   +  +  . . . .. +     1)  = S
3.      101 + 101 +101 + 101 +  101  +  +  . . .  . + 101    =  100 x 101  = 2S
Therefore  S = (100 x 101)  ÷  2 = 50 x 101 = 5050
5. The "Handshake Problem" reappears as one of the WS&C sets for NCEE middle school math.  There it's called "Points and Segments."  Points correspond to bubbles and segments correspond to lines.
12. Tangram Dispute
• TaskTracy's "whole" parallelogram, formed from two tangram triangles,  is compared to Terri's "small" tangram parallelogram.  Terri claims that her parallelogram is "half as big" as Tracy's.  Tracy thinks that Terri's is smaller.  They have a dispute.
• Method:  The student physically demonstrated (via tangram manipulative pieces from multiple tangram sets) that the "small parallelogram is equivalent to one out of four equal parts of the whole parallelogram."
1. Concrete experience relating plastic triangles to plastic parallelograms.
2. The NCEE suggests that this example involves considerable content about fractions.  The only evidence is the student's statement that "Tracy's shape was 1/ not 1/ of Terri's shape."
13. Feverish Freddy
• Task: The appraised value of lot A is \$88,000.  Find the value of 6 other lots relative to the value of lot A
• Note: The student is given a diagram showing 7 lots.   Two lots are shaped as hexagons, 2 as trapezoids, 2 as parallelograms, and one as a triangle.
• Method:  The students used "pattern blocks" to find the size (relative to size of lot A) of the other 6 lots. They found them to be 1/6, 1/3, 1/2, or the same as the size of lot A.  They used a calculator to find 1/6, 1/3, and 1/2 of 88,000 and to compute totals.
1. It's assumed, but never stated, that the value of the lot depends only on the size of the lot.
2. The student ignored cents because "I didn't think it was important."   The NCEE says "the work provides evidence of making sense of what to do with the cents that were left over."
3. The student uses letters to represent lots, and writes equations such as 3F = A, commenting that 3 F lots equal one A.   The student means that 3 times the area of lot F equals the area of lot A, but this is never stated in this way.   The NCEE makes no comment about this lack of precision in language.  They say that the student demonstrates "Function and Algebra" concepts by using "letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations with concrete materials, i.e demonstrates the use of a beginning concept of variable."
14. Counting on Frank
• TaskTest Three claims made in a book.   (The book is Counting on Frank)
• Claim 1: It took 11 hours and 45 minutes to completely fill a bathroom with water from two running faucets.
• Method:
1. Used the school's sink to determine that one faucet produces one cubic foot in 40 seconds.
2. Deduced that two faucets produce one cubic foot in 20 seconds.
3. Assumed that a typical bathroom measures 8 x 8 x 8  (unit = feet).
4. Used a calculator and a formula provided by the teacher ( L x W x H), to calculate:
• (8 x 8) x 8 = 64 x 8 = 512  (cubic feet)
• 20 x 512 = 10, 240 (seconds)
• 10,240 ÷ 60 = 170 (minutes)
• 170 ÷ 60 is "almost 3 hours"
5. Argued for the correctness of the process and concluded "I have proved him wrong."
• Comment: Why assume that a typical bathroom faucet produces water at the same rate as the school faucet?  Could water be also draining out, via the tub, shower, toilet, door, or windows?  There's no statement to the contrary, and this is promoted as "real world" math.  Finally, why assume such a small bathroom?  Today's bathrooms can be very large.    With some different assumptions, you might conclude that the claim could be true.
• Claim 2: If 15 peas were dropped on the kitchen floor every day for 8 years, then the peas would reach the level of the kitchen table top.
• Claim 3: An average ball point point will draw a line 2,300 yards long, before running out of ink
• Method for Claims 2 and 3:  Similar to the method of claim 1, the student used estimates, assumptions, "experiments," personal opinions, and a calculator to "prove him wrong" two more times.
• Comment: Students shouldn't be allowed to go away thinking that such subjective reasoning constitutes a mathematical proof.
15. School Uniform Project [Not in the New York City version]
• Task: "The teacher asked students to come up with a question for a data study that they would then carry out."
• Method:  The work for one student is shown.
• This student conducted a survey, asking students if they would like school uniforms.  Three possible color schemes were offered.  The student chose a sample size of 100 students, dividing that into 25 students from each of four different grades.  The student conducted the survey and recorded the results in two tables, one table for a yes/no vote about having uniforms and the other recording the choice of color scheme.  These two tables were also presented as bar graphs.
• Four out of 100 said no to uniforms, leaving 96 to choose a color.  The student wrote 23, 37, and 40 as the three totals for the choice of color, and then expressed these as fractions with denominator 96 (23/96, 37/96, 40/96).
• Comment:   The presentation of this project is spread out over 6 pages in the NSPS Volume 1 book.  There are extensive author notes about the complex set of activities carried out by the student.  But no one noticed that 23 + 37 + 40 = 100, not 96, and the sum of the three factions is 100/96, not 1.   Apparently this problem was modified to recognize the student's right to say "no."  Now it's politically correct, but not mathematically correct.
16. Catapult Investigation [Not in the New York City version]
• Task: "Design and carry out a test that determines the optimum setting for shooting a wet sponge the furthest."
• Note: The teacher gave this instruction "after each student had built an individual catapult according to the teacher's design."
• Method:
• The "student used a calculator (as verified by the teacher) to compute averages" of three shots, for each of 9 catapult settings.  The student listed the results in a  table and "graphed" mean shot length vs. catapult setting using graph paper.
• The "graphing" uses a 2-D coordinate grid, but in a non-standard manner.  The x-axis is marked off in 5 inch increments, as 0, 5, 10, . . . 105.  The y-axis is marked off with the nine catapult settings, A1, A2, A3, B1, B2, B3, C1, C2, C3.   The nine (x, y) coordinates, (0, A1), (5, A2), (10, A3), (15, B1), . . . (35, C3), all fall on the line y = x.  The student "graphed" the 9 average three-shot averages by placing a period at the (x, y) coordinate where the y-coordinate equals the catapult setting, and the x-coordinate equals the average distance for that setting.  For example, there's a period at the coordinate (A3, 30) to indicate that the average distance for catapult A3 was 30 inches.
1. The non-standard graphing method is a bad idea at any time, but can be particularly misleading for students in elementary school.  The 2-D coordinate grid should only be used to represent the standard coordinate plane, where two perpendicular number lines provide a system of coordinates for each point in the plane.  The pair (x, y) should fall on the line y = x if and only if x and y are both real numbers, and x = y.
2. Why did the teacher need to verify that the student could correctly use a calculator to add three number and divide the sum by 3?   It's not explained.
17. The Never Ending Four
• TaskInvestigate and explain why the following iterative process eventually leads to four:
• NSPS Note:  The teacher asked students to complete a large scale mathematics project chosen from among five kinds. This student chose to do a "Pure Mathematics Investigation."
• The Process:
1. Choose any  whole number.
2. Spell the number.
3. Count the number of letters used to spell the number.
4. Continue to repeat these three steps, each time using the count of step 3 as the number in step 1.
• Note: This explanation of the process is not found in NSPS.  The idea is explained though an example.
• Method: Multiple demonstrations showing that it works.
1. The student attempted to give a verbal explanation as to why this works, but finally pointed to a table for the first 30 whole numbers.  Here's the explanation: The number of letters is always less than or equal to the value of the number for whole numbers greater than or equal to 4, and the number of letters needed to spell any whole number is greater than or equal to 4, except for one, two, and six. It takes 3 letters to spell each of these, and "three" contains 5 letters.  It takes 4 letters to spell five.
2. The NCEE classifies this set as demonstrating "Functions and Algebra" because "the student built (again and again) the non-linear pattern generated by this 'trick' question."
3. This NCEE also classifies this set as demonstrating "Statistics and Probability" because "the student collects and organizes data to answer a question."
4. This is a parlor trick, not "pure math."
18. Dream House Project
• Task: Given a maximum budget of \$100,000, a cost of \$75 per square foot for "regular room", and cost of \$150 per square foot for "special rooms," design a dream house.  There are several other "requirements" associated with the task.  These include:
1. "Rooms and hallways must have reasonable areas."
2. "The overall design must be convenient and practical."
3. "The number of sides to your floor plan should be limited, to avoid a sprawling, awkward design."
There are several instructions describing how to draw the floor plan, record calculations, and draw views.  Students are to meet "once in a small peer response group and once with an editing partner."  They "were allowed to use calculators to check calculations and for multiplication with multipliers of three or more digits."
• Method: We have omitted the "real world" forms and written defenses of "design decisions."  As for the math in the method, the student multiplied two 2-digit numbers to calculate the area for multiple rectangular rooms, multiplied by cost per square foot to determine the cost per room, and then added room costs to get the total cost.  Calculators were usually used for computations, but one student carried out a two digit multiplication as follows:
• "If one of my rooms is 13 ft x 14 ft, I first do 3 x 4 = 12.  So I write the 2 and add the 10 to the 4 x 10 which is 40, but if I add a 10 to the 40, it'll be 50 so I write the 5 on the left side of 2, it's 52.  Then I do 3 x 10 equals to 30.   I write the 3 under the 5 on top from 52.  Then I do 10 x 10 and I write the 1 on the left side of 3.   Then I add the 52 and 130 together and I got the answer of 182 square feet."
1. Note the very loose conditions.
2. The NCEE notes: "students completed this project over the course of four weeks."  All that time.  What a shame.
19. Constructing a Polygon [Not in the New York City version]
• Task: Select a polyhedron from choices on a poster.  Use a compass, rubber bands, etc. to build the polyhedron of your choice.
• Method:  The student chose to build a snub icosadodecahedron, which is a "polyhedron consisting of 80 triangles and 12 pentagons."   The NCEE informs us tht the student "spent several hours beyond the allotted time for this project, including before and after school and during lunch recesses."
• Comment:  This student will be able to recognize a pentagon.
20. Making a Cube [Only in the New York City version]
• NYC TaskDraw a pattern for making a cube
• NYC Method:  The student provided answers, with drawings of two correct patterns and a lengthy written explanation about patterns that "don't work."