Problem:  Starting with the letter A = 26 cents, assign each letter of the alphabet a money value going in consecutive descending order, down to Z = 1 cent.  Which day of the week is a $1.00 word?

Math Topic/Concept:  Number Sense

Classroom Use: (Introductory)

Classroom use comments*:  After solving.  Question different strategies of student answers.  How did you do it?  Did you start with a certain day, why?  Did anyone start with a different day, why?

Grade:  6-8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  6

Standard:  6.B.3a

Applied? (1-4):  1

Source:  Historical connections in Mathematics  Vol.1  (Wilbert Reimer)
                ISBN  1-881431-35-5

Answer:  Thursday

Strategies Listed: Make a table

Solution:  t = 7  h = 19  u = 6  r = 9  s = 8  d = 23  a = 26  y = 2
           7 + 19 + 6 + 9 + 8 + 23 + 26 + 2

Extensions or related problems*:  Which American made automobile is a $1.00 word?
Find as many words as you can that are worth exactly $1.00.
Supply a list of words to find the value of.

Intended rubric or assessment method:  Table correct.  Did they show the addition.  Did they include all their work up to the point where they got the solution.

Write-up submitted by:   Jenni Robinson and Denise Mann

Problem:  Jonna brought home a strip of wood 120 cm in length and 2 cm in width.  She instructed her daughter, Diana, to make her a rectangular picture frame and to use all of the wood.  She wanted the frame to be made as shown below.
 What are several sizes of frames Diana could make?
 What frame would give Diana the largest enclosed area?

    Frame Style:

 

Math Topic/Concept:  Area and perimeter.

Materials: Pencil, paper, graph paper, calculators (optional)

Classroom Use: (Developmental/Evaluation)

Classroom use comments*:  It will likely take some discussion to decide that the outside perimeter and the insider perimeter of the design is not 120.  The design is not the strongest design, but has been chosen so the problem is easier to solve.  Students may need to spend time thinking about how the boards need to be cut so that the rectangular frame can be made.

Grade:  4 - 6

Grade Cluster: (LateElem/MS-Jr.High)

Illinois Goal:  Goal 7 and Goal 6

Standard: 7A3b, 6B3a, 6B3b, 6C3a, 6C3b

Applied? (1-4):  2

Source:  original

Answer:  There are several rectangles that can be made. The rectangle with an outside perimeter of 32 cm and an inside perimeter of 28 cm gives the largest area for the picture.  This area is 576 cm2.

Strategies Listed:  Making an organized list.

Solution:  See answer.  It is important to note that the area on the inside of the frame is desired.  The width of the wood matters here as well.

Extensions or related problems*:  The length or the width of the board can be varied.  Different ways to make the frame could also be discussed.

Intended rubric or assessment method:  The following rubric could be used.
Exceeds:  Student finds a pattern to the length and width combinations and lists possible combinations. Clear explanation and/ or work to show understanding.  Finds the combination that makes the largest area and explains why it is the largest. May use decimals or fractions in the pattern and identify an infinite number of possibilities.

Meets: Student inds a pattern to the length and width combinations and lists combinations with perhaps a few errors. Finds the frame dimensions that make the largest area but does not provide justification.  Otherwise, explains work fairly well.

Does not meet: Student may find some patterns or sizes for the frame. Work contains major errors. May not find the rectangle with the largest area. Does not explain or gives poor explanations.

Write-up submitted by:  Melfried Olson

Problem:  Suppose you have a set of double-nine dominoes.  Take all the doubles from the set except the 0-0 domino and you’ll have these dominoes:
 
 
 
 

Now arrange these dominoes on the grid below so that the total number of “dots” along each row, column, and diagonal is equal to 30.

 
 
 
 
 
 
 

Math Topic/Concept:  Algebra, Magic Squares

Materials:  Grid, Dominoes (paper or actual)

Classroom Use: (Introductory)

Classroom use comments*:  Do you understand the total.  Did you recheck all your work before turning in?  How did you reach 30?   (2 + 7 + 6 = 15) (15  x 2 = 30)     or     (4 + 14 + 12 = 30)

Grade:  6-8

Grade Cluster: (MS-Jr.High)

Illinois Goal:  6

Standard:  6.B.3a

Applied? (1-4):  2

Source:  Awesome Math Problems (Creative Publications)   Grade 8  ISBN  0-7622-1287-X

Answer:                2              7                    6
 

                             9               5                    1
 

                             4              3                     8

Strategies Listed:    Guess and check

Solution:  Teacher should check individual answers, because rows or columns could be flipped and still get 30.

Extensions or related problems*:  Magic Square problems.

Intended rubric or assessment method:  10 point problem.  1 point for each row, column, and diagonal that adds correctly to 30.  2 points for trying.

Write-up submitted by:  Denise Mann and Jenni Robinson

Problem:  In Duluth Minnesota the temperature at 6 am on Jan. 1 was –30 degrees F.  During the next 8 hours the temperature rose 38 degrees.  Then during the next 12 hours the temp dropped 12 degrees.  Finally the next 4 hours it rose 15 degrees.  What was the temperature at 6 am on Jan. 2?

Math Topic/Concept:  Integers

Materials:  Optional:  number line or thermometer

Classroom Use: (Introductory)

Classroom use comments*:  Discuss different strategies students used.

Grade:  7

Grade Cluster: (MS-Jr.High)

Illinois Goal:  6 (7 if use extension)

Standard:  6.B.3a     (7.A.3b)

Applied? (1-4):  3

Source:  Connected math  (Dale Seymour)

Answer:  11 degrees F

Strategies Listed:  Addition and Subtraction Computation

Solution:  -30 + 38 +-12 +15 = 11  or  -30 +38 = 8 – 12 = -4 +15 = 11

Other solution methods (if any)*:  Move with arrows up and down on a vertical number line to the end solution.

Extensions or related problems*:  What would final temperature be in Celsius?

Intended rubric or assessment method:  worksheet with addition and subtraction of more than 2 integers.

Write-up submitted by:  Jenni Robinson and Denise Mann

Problem:  Travis has 19 toothpicks.  How many ways can he arrange them into three
  piles so that each pile contains an odd number of toothpicks.

Math Topic/Concept:  Problem Solving, Patterns

Materials:  Toothpicks

Classroom Use: (Introductory)

Classroom use comments*:  Problem Solving Unit

Grade:  7

Grade Cluster: (MS-Jr.High)

Illinois Goal:  6

Standard:  6B3

Applied? (1-4): 1

Source:  Mathematics Teaching in the Middle School (Feb. 1999)

Answer:  10 ways

Strategies Listed: Make a Table, Organized List, Act Out

Solution:
Pile 1   Pile 2   Pile 3
  1         1         17
  1         3         15
  1         5         13
  1         7         11
  1         9          9
  3         3         13
  3         5         11
  3         7          9
  5         5          9
  5         7          7

Other solution methods (if any)*:  Show the solution with pictures.

Extensions or related problems*:  Use more or less toothpicks. Separate into greater
     than 3 piles. Make the piles have an even number.

Write-up submitted by:  Denise Mann and Jenni Robinson
 
 

James R. Olsen, Western Illinois University
E-mail: jr-olsen@wiu.edu
updated Aug. 20, 2001