MATH 5370
Problem Solving Concepts in K-8 Mathematics

View the course syllabus.
View problems worked or assigned in class.
View links to the supplemental readings.
Visit my math ed web resources page.
View student interviewing tips.
View the sample interview write-up.
Read tips on writing papers on math problems.

NOTE: Technical information such as prerequisites, text materials, course format and assignments, and other details can be found in the syllabus, a copy of which is provided in a link at the top of this page.

"College-level" problems

• Session 1: Poisons --- complete, justified solution for both classic Poison (with any number of counters) and Super Poison (generalizing the number of counters one may take at a time to 1,2,3; etc.).
• Session 2: Rhinos by Pairs --- with a careful explanation of the definitions of the variables, and the consequent chain of reasoning necessary to write equations.
• Session 3: Pentominoes --- with a focus on rigorous justification of the completeness of the list; also include generalizations.
• Session 5: Pick's Formula --- with proof.
• Session 6: Squares Problem 3 (p. 7).
• Session 10: The Stamps Problem --- be careful to distinguish clearly conjecture from proven conclusion in discussing Question 3.
• Session 11: Triangle centers --- be sure to include reasoning for the answers to Questions 2 and 3.
• Session 12: Geoboard Eighths --- include a line-by-line analysis of the reasoning in the given response (i.e., don't just say "this line is true" (or false), say why), in answering the questions on p. 16.
• Session 13: Triangles Around Cones --- justify the equation relating cone angle to the sum of the vertex angles.
• Session 14: The Shepherds Problem --- a coherent narrative (not "1. [answer] 2. [answer] etc.") addressing the questions on p. 19.
• Session 15: Linear Modeling --- answer the last question, and discuss the salient features of each problem as regards linear models.

Links for specific class meetings

• Session 1: slides used in class
• Session 3: Lesson anatomy chart
• Session 5: Proofs of Pick's Theorem (be sure to develop your own argument if you write about this problem in your two-problem paper)
  • mine
  • an informal proof
  • short formal proof
  • medium formal proof
  • long formal proof
  • another one from a math ed site
• Session 6: Squares problem 3 -- a combinatorial approach
• Session 7: Modified versions of the Peas and Bagels problems

  From 2011:

  1. If Calvin ate 16 peas on Monday, and he ate twice as many peas on Tuesday, how many more peas did Calvin eat on Tuesday than on Monday?
  2. Calvin ate 16 peas on Monday and 32 peas on Tuesday. If he continues this pattern, how many peas will he eat on Wednesday? on Thursday?
  3. Calvin ate some peas on Monday and then twice as many peas on Tuesday. He ate 135 peas in all. How many peas did Calvin eat on Monday?
     The solution could be found in various ways such as working backward, using algebra, drawing a picture and guess and check. It differs from the original comparison problem by using division instead of subtraction. In the original problem, students have two known quantities to compare, while in the modified version, students are working from a known total to split proportionally for each day’s feeding. The modified version would be appropriate for 4th-5th grades.
  4. If 3 bagels are shared equally among 5 people, how many bagels will each one get?
  5. If 3 dozen bagels are shared equally among 5 people, how many bagels will each one get?

  From 2007:

  • Group 1:
    • Peas: On Monday, the Blazers scored 8 points. On Tuesday, the Blazers scored 20 points. How many more points did the Blazers score on Tuesday than on Monday?
    • In January, my chicken laid 50 eggs. In February, she laid 2 dozen eggs. In which month did she lay more eggs? How many more eggs did she lay in this month?
    • Bagels: If 3 dozen bagels are shared equally among six people, how many bagels will each person get?
    • I'm having a party with 11 guests and I want all of us to get 2 slices of pizza. If a pizza is cut into 8 slices, how many pizzas will I need to order?
  • Group 2:
    • Peas: Calvin ate 16 peas on Monday. On Tuesday he ate 32. On which day did he eat more?
    • Calvin ate 16 peas on Monday. On Tuesday he ate 5 peas more than on Monday. How many peas did he eat on Tuesday?
    • Bagels: If 3 bagels are shared equally among 5 people, how many bagels will each one get?
    • Five people ate 3/5 dozen bagels each. How many dozen bagels did they start with?
  • Group 3:
    • Peas: If Calvin ate 32 peas on Monday and on Tuesday he ate 16 fewer peas, how many peas did he eat on Tuesday?
    • Each pea pod has 3 peas. On Monday Calvin ate 5 pea pods and on Tuesday he ate 7 pea pods. How many more peas did Calvin eat on Tuesday than on Monday?
    • Bagels: If 36 bagels are shared equally among five people, how many bagels will each person get?
    • If 3 dozen bagels are to be shared equally among 5 people but 18 of the bagels are moldy, how many dozens of bagels will each person get? What fraction of the whole is this?
  • Group 4:
    • Peas: If Calvin ate 5 peas at lunch and 10 peas for dinner, how many more peas did he eat at dinner?
    • If Calvin ate 1/2 cup of peas on Monday and 3/4 cup on Tuesday, how much more did he eat on Tuesday than on Monday?
    • Bagels: If there are 12 bagels and 4 friends shared them equally, how many bagels did each friend have?
  
  Two area tasks to compare
  A summary handout on cognitive demand
• Session 11:
  • GeoGebra applet 1
  • GeoGebra applet 2
  • This page, with an encyclopedia of over 3000 different centers of a triangle, includes interactive applets where you can click and drag triangle vertices to see the effects of changing triangles. The first four centers listed are the four we investigated in class.
  • On a different topic: Virtual diffy box generator
• Session 13: Here's my clumsy attempt at a proof of the triangles on cones formula.
• Session 14: NetLogo Shepherds problem (see the documentation, and then click on "Run Shepherds in your browser")
• Session 15: spreadsheet with data for the 3 linear modeling problems