Direct Access Links to Each NCEE Middle School Math WS&C Set

1. Pieces of String      [Includes the NYC modifications]
2. Dart Board            [Includes the NYC modifications]
3. Science Fair
4. Cubes                 [Includes the NYC modifications]
5. Points and Segments   [Includes the NYC modifications]
6. Locker Lunacy
7. Who is Best?
8. Probability Booth
9. Logic Puzzle
10. Scaling Project
11. A New Look at a Budget
12. Candle Life

The Work Sample & Commentary Sets for NCEE Middle School Math

1. Pieces of String
• Task:"A length of string that is 180 cm long  is cut into 3 pieces.  The second piece is 25% longer than the first, and the third piece is 25% shorter than the first.  How long is each piece?"
• Method:  The student guessed the length of first piece as 100 cm.  Then the length of the other two pieces would be 75 cm and 125 cm, and the total length of the string would be 300 cm.  The student then observed that 100, the guessed length of the first piece, is 1/3 of 300.  Hence the actual length of the first piece must be 1/3 of 180, or 60 cm, and the length of the other two pieces must be 45 cm and 75 cm.
• Comments:
1. This problem is easily solved by setting up and solving a linear equation.  If Let x equals the length of the first piece piece, then  x + (x + .25x) + (x - .25x) = 3x  = 180.   There's an NSPS comment that recognizes this fact, but the authors appear pleased that "the work does not show a standard approach that a student of algebra might take."
2. Notice that the guessing strategy only works with carefully chosen numbers.   Constructivist math educators frequently offer such contrived problems to show that real math isn't really necessary.
• NYC Version:  The NYC authors modified this problem, changing the total length to 160 cm, the second piece percent to 40% longer, and the third piece percent to 20% shorter.  The NYC authors omitted the NCEE method and offered the following two alternatives:
1. NYC Method 1:  The student solved the equation  x +1.4x + .8x = 160.  The numbers work out nicely, yielding lengths 50 cm, 70 cm , and 40 cm.
2. NYC Method 2:   This student avoided working with decimal by saying there are 10 pieces in the first (big) piece, 14 pieces in the second (big) piece, and 8 pieces in the third (big) piece.  This yields a total of 32 (small) pieces in the 160 cm total length.   The student divided 160 by 32 to get 5 cm as the length of each small piece and then multiplied by 10, 14, and 8 to get the length of the three big pieces.
• NYC Comments:
1. This NYC variation is still easily solved using "guess and check."   The obvious initial guess (50) works.  The success of Method 2 depends on the nice numbers.
2. Method 1 does demonstrate important middle school math: using properties of equality to solve a linear equation.   NCEE math doesn't include a similar demonstration.
2. Dart Board
• Task:  "Design a dart board that has four regions with the following features:
Score value      Probability %
 100 points          10%
  50 points          20%
  25 points          30%
  10 points          40%
The dart board may be any shape (circle, square, etc.) and must have an area between 1,000 sq. cm and 3,000 sq. cm   Assume the probability is proportional to the area of the region."
• Method:
• The student drew 4 concentric circles, indicating that the area of the inner circle and three donut-shaped regions, moving progressively outward, should be 10%, 20%, 30%, and 40% of the total area contained in the outer circle.
• Using A = (3.14) r²  as the formula for the area of a circle, the student used guess and check to choose 20 cm. as the radius of the outer circle.  Equivalently, the student chose 1,256 sq. cm as the total area.  The student then found:
1. The radius of the inner circle by solving (3.14) r² = 125.6.
2. The outer radius of the 20% region by solving (3.14) r²  = 3 x (125.6).
3. The outer radius of the 30% region by solving (3.14) r²  = 6 x (125.6).
• Comments:
1. There's no mention of the calculation methods.  At the middle school level it's safe to assume that a calculator was used.
2. There's no mention of approximating pi.  Pi equals 3.14 for these students.
3. Decimals outside the range .00 to .99 are not mentioned in NCEE middle school math.
4. This is considered a problem in probability, but it's only about using the formula for the area of a circle.
5. The problem is not well posed.  That is, the conditions given don't lead to one correct solution.  The NCEE notes that an acceptable solution would be "a choice of dart board as a 100 cm x 20 cm rectangle, divided along the length in 10, 20, 30, and 40 cm segments."  Thus, a 4th grade solution is acceptable.
• NYC Version: The NYC authors omitted the NCEE method and offered the following substitutes:
1. NYC Method 1: The student drew a 10 by 10 grid with 100 component 10 sq. cm boxes.  The student then used four different colors to mark ten 100 point boxes, twenty 50 point boxes, thirty 25 point boxes, and forty 10 point boxes.
2. NYC Method 2:  Similar to NYC Method 1.  Used 10 congruent rhombi.
• NYC Comment:  These are 4th grade solutions.
3. Science Fair
• Task:  A rectangular floor must be divided into 3 rectangular rooms for a science fair.  Room 1 is for a school with 1000 students. Room 2 is for a school with 600 students.  Room 3 is for a school with 400 students.  Express the size of each room,  first as a fraction of the total floor, and then as a percent of the total floor.  If the total floor costs $300, find the cost of each room.
• Method:  The student wrote three fractions with denominator 2000, simplified to 1/2, 3/10, and 1/5, and then expressed these fractions as 50%, 30%, and 20%.  The student recognized the corresponding decimal equivalents and multiplied each decimal by 300 to get the cost per room.
• Comments:
1. This is an appropriate problem at the end of the 4th grade, not at the end of the 8th grade.
2. The NCEE says that this work sample illustrates standard-setting performance for computing accurately with arithmetic operations on rational numbers.But there's no illustration of the addition, subtraction, multiplication, or division of fractions.  Simplifying fractions doesn't qualify as operating with fractions.
• Such overreaching claims are typical in the NSPS.
4. Cubes
• Task:  Find the volume and surface area of a cube.  Then determine if a claim is true.
1. "What is the volume of a cube whose edges each measure 3 centimeters?"
2. "What is the surface area of a cube whose edges each measure 3 centimeters?"
3. "A student named Eddie says,  'No matter what size the cube is, the number you get when you calculate its surface area is always twice as big as the number you get when you calculate its volume.' Is Eddie correct?  Show how you know."
• Method:
1. The student computed the volume as 3 x 3 x 3  = 27 and the surface area as 6 x (3 x 3) = 54.
2. To test Eddie's claim the student tried three other edge lengths, but didn't find another cube example with surface area double the volume.  The NCEE says "the students gave several examples to counter Eddie's claim.  Not only is Eddie's claim not true for all cubes, it is false for most cubes."
• Comments:
1. Tasks 1 and 2 are basic 5th grade math.
2. Note the constructivist reasoning method for task 3.  The students tried several cases, while looking for a pattern.  But no pattern was found and that ended the discussion.  These students don't know another way to reason.
3. No one mentioned the algebraic formulation of Eddie's conjecture.  It's equivalent to saying that the equation  2x³ = 6x²  is true for all x greater than zero.  But,  if x is greater than zero,  both sides can be divided by 2x² to yield the answer 3.
4. The students and the NCEE didn't notice that Eddie said "always,"  or perhaps they don't know that one counterexample is sufficient.   It's an example of how they have redefined mathematical logic.
5. Once the students determines that the claim isn't generally true, the focus should shift to finding other patterns involving volume and surface area.   No other patterns are discussed anywhere in NCEE math.
• NYC Version:  Three students tested Eddie's claim by trying seven other edge lengths.  More drawings and written explanations, but no new ideas.
5. Points and Segments
• Task: "How many segments are needed to connect 5 points?  6 points?  8 points?  10 points?  30 points?   100 points?  n points?"
• Method: The students looked for a pattern.  They sketched and counted  for cases 2, 3, 4, 5, 6, 7, 8, 9, 10, and 30.     Finally, one student said "then I found this  ((n ÷ 2) - .5) x n  as the formula and it worked."  The NCEE commented: "the student's phrase, "Then I found this," leaves the reader to wonder, 'How'."
• NYC Version:  The NYC authors revised the method, omitting the "leaves the reader to wonder" statement.   They want to explain "How" the student discovered the formula.  For cases 5, 6, and 8, the NYC student "noticed" that the number of points times the number of segments equals "double the answer."
• Comments:
1. This problem is a variation of the "Handshake Problem" found in the NCEE elementary school math examples. Here it's points and segments.  There it was bubbles and lines.   Both problems required finding the number of combinations of 5 elements taken 2 at a time.  The formula given above yields the number of combinations of n elements taken 2 at a time.
2. The NCEE never acknowledges that the student "discovered" an important general formula.  More generally, advanced counting methods, such as permutations and combinations, aren't covered in NCEE math.  Why not?  These topics are quite accessible for middle school students.  All that's required is prior mastery of fractions.  Ooops!  That may be the problem.    But the "wonder how" comment gives us a hint that the NCEE may recognize the limitations of pattern recognition and discovery learning.   Perhaps they are conceding the stretch necessary to sell the idea that the student really discovered the formula for the number of combinations of n objects taken 2 at a time.   If so, they also know that no one will buy student discovery of more general counting formulas.
6. Locker Lunacy
• Task:  "Analyze a scenario in which school lockers are alternately opened and closed in a particular manner."  Lockers, numbered 1 to 100, are all opened.  Then lockers with numbers divisible by 2 are all closed.  Then lockers with numbers divisible by 3 are "reversed" (closed if already opened, or opened if currently closed).  If this "reverse" pattern continues for N = 4, 5, ... 100, which lockers will remain open?
• Method:  The NCEE explains that  "Students notice experimentally that those lockers whose numbers are perfect squares remain open at the end.  It is only on reflection (asking 'Why?') that they realize the relationship between the perfect squares and their odd number of factors."
• Comment:   NCEE notes refer  to "a long while of experimenting" and "the student created a large table of values."  Much busywork and more pattern recognition, but calculators save time.
7. Who is Best?
• Task:  "Determine which of three golfers is the 'best' chipper."  If three golfers each chips 10 golf balls onto a green, decide "who is closest and most consistent" based on a statistical analysis of "distance from the pin" data.  Measured in inches, here's the data:
1. Rick:    40, 60, 100, 120, 312, 320, 152, 105, 95, 46
2. Mike:    52, 76, 184, 288, 230, 120, 64, 60, 88, 188
3. Sarah:  84, 99, 130, 135, 200, 165, 120, 129, 135, 152
• Method: For each of these three sets of ten measured distances, the student found the lower extreme, lower quartile, median, upper quartile, and upper extreme.  The student then developed three box plots.  Sarah was voted "most consistent" because her measurements had the smallest inter-quartile range (upper quartile minus lower quartile). In spite of an extended discussion, no golfer was voted "who is closest."
• Comments:
1. The consistency part is fine for 7th grade math.
2. The failure to award "closest" is puzzling.  Two of Rick's chips, at 40 and 46 inches, are closer than any chip for  the other two golfers.  Looks like a winner!   But Rick's chips were considered flukes.  He was penalized for lack of consistency.  But wasn't that the other award?
8. Probability Booth
• Task: "Design an activity that uses multiple probability events and that would attract lots of players."
• Method:  The student's activity consisted of first predicting a coin toss and then rolling an 11 or 12,  using a pair of dice.  The student wrote "the probability of winning is 1 in 24 because it's 1 in 2 (¹/₂) to win the coin toss and 1 in 12  (¹/₁₂)  for the roll of the dice."  The student proposed to "charge $1 to play and give $10 to a winner."  The student wrote "Out of every 24 players one would win.  My profit out of 24 players would be $14." The NCEE complimented the student for "realizing that his ¹/₂₄ probability implied that he could expect one winner per 24 games played."
• Comments:
1. It appears that both the student and the NCEE believe that exactly one win will occur every 24 games.
2. Used here, but not explicitly recognized by the NCEE:  If A and B are independent events, with probabilities p and q, then the probability that both events occur is p x q.
3. The NCTM and NCEE claim that they emphasize probability for all the K-12 years, but this is as difficult as it gets in the 38 math-related WS&C sets found in the three volumes of the NCEE's New Standards Performance Standards.
9. Logic Puzzle
• Task: Given 7 equations involving 10 variables,  where each variable represents a unique digit between 0 and 9, find the value of all 10 variables.
• The Equations:
1. g + g + g = d
2. j + e = j
3. g² = d
4. b + g = d
5. f - b = c
6. i/h = a (h > a)
7. a x c = a
• Method:  The student recognized zero as the additive identify and one as the multiplicative identity.   Here's a taste: j + e =  j, therefore e = 0.   Since 3g = d, g² = d, and e already occupies the zero slot, it follows that g = 3 and d = 9.   You get the idea.
• Comment:  The student solved a system of equations, by using substitution, properties of equality, and knowledge of identity elements.  If done correctly, this problem qualifies as genuine algebra.  Unfortunately, because of the "unique digit between 0 and 9" constraint, it can be solved using "guess and check."
10. Scaling Project
• Task: "Design and build a scale model enlarging or shrinking an everyday object using a ratio of 1:10 (or specify a different ratio) for each dimension."
• Method:  The student built a 10 to 1 enlargement of a LEGO piece.  The materials used included rectangular "tagboard," toilet paper rolls, glue, and red spray paint. The student applied standard volume formulas for rectangular prisms and cylinders.  The student assumed that pi = 3.14.
• Comment: A less time consuming variation may be an acceptable 6th grade activity.
11. A New Look on a Budget
• Task:  "Determine the cost of  redecorating your room.  You must carpet the room, paint two coats, and use wallpaper in some way.  Draw to scale, on graph paper, each wall, including windows and doors."
• Method:
• The student "measured the walls in yards and inches," and then converted "yards to inches and found  the total inches."  Then "I multiplied on my calculator the length and height in inches to find the square inches of each wall.  I divided the area by 144 to find the square feet."  The student carried out similar calculations for the floor, ceiling, and woodwork.  The student divided by 9 to convert to square yards for the floor.
• Given coverage information, the student used the calculator to determine how much paint, wallpaper, and carpet to buy, and then calculated total costs for all materials and supplies.
• The student developed all the required drawings.
• Comment:  This is 4th grade math. With the "real world' aspects and required drawings, there's too much time-consuming busywork.
12. Candle Life
• Task: "Determine the relationship between the volume of a container and the length of time a candle will continue to burn when covered by it."  Experiment with several containers.  Graph the data and "choose the line that best fits your data."
• Method: The student:
1. Measured the volume of 12 different glass containers by filling them with water and then pouring the water into a measuring cup.
2. Placed each container over the burning candle and used a stop watch to determine the time the candle would continue to burn.  This was done 3 times for each container.
3. Selected the "intermediate value" from each of the 3 trials and used a coordinate plane to graph container volume vs. flame extinguish time.
4. "Drew a line of 'best fit' and identified two points off of it."
5. Found the equation of the straight line joining these points.
• Comment: Finding the equation of a straight line, when given the coordinates of two points on the line,  is a very important middle school topic.  Too bad more time wasn't devoted to equations of straight lines and less time to pseudo science.

Next?   Please click on one of the following links:

Chapter 1:  A Summary View of NCEE Math
Chapter 2:  How the NCEE Limits Elementary School Math
Chapter 4:  How the NCEE Limits High School Math

 Understanding NCTM Math "Standards" and NCTM "Math Reform"  [Home Page]