Friday, April 3, 2015

Math problems of the week: Common Core-inspired introductory geometry problems

The opening exercises of New York State Common Core Geometry Curriculum



The opening exercises of Weeks Adkins A Course in Geometry (1970 edition)

The following exercises provide situations in which deductions can be made. Consider carefully the way in which a supporting argument may be developed.

1. Two coins of the United States have a total value of 30 cents. Can you deduce the names of the coins?

2. Three coins of the United States have a total value of 60 cents. Can you deduce the names of the coins?

3. Three coins of the United States have a combined value greater than 70 cents and less than 80 cents. Can you deduce the names of the coins, if no two of them have the same value?

4. The sum of two whole numbers is an even number. One of the numbers is even. Can a deduction be made about the other number?

5. The product of two whole numbers is an even number. One of the numbers is even. Can a deduction be made about the other number?

6. Can a deduction be made about Tom from the statements that follow?
(a) All dogs have two ears. Tom is a dog.
(a) All dogs have two ears. Tom has two ears.
(a) All dogs have two ears. Tom has no ears.

7. In a class of 30 students everyone must take at least one of the languages French, Latin. If 20 are enrolled in French and 17 are enrolled in Latin, how many students are enrolled in both?

8. If x,y are whole numbers and x-y is an odd number, can a deduction be made about x + y?

9. A, B, C, D are four towns. A route runs through each pair of towns and no route runs through more than two of the towns. What deduction can be made about (a) the number of routes through each town, (b) the total number of routes?


Extra Credit

1. Are the Weeks Adkins exercises, above, irrelevant to 21st century mathematics?

2. The Common Core Math Standards include this one:
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples…
(I've omitted the "critique the reasoning of others" part of this goal.)

Try to find a Common Core-inspired problem in which:

A. the primary challenge, as with the 1970s Weeks Adkins problems, is in the construction of viable arguments
B. the logical challenges are similar to those of the Weeks Adkins problems.

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