Baseline Assessment: Mathematical Proof
Name ________________________________________
Grade_________________
1. Take an odd number and an even number and multiply them together. Their
product is always an even number.
Provide a justification for this fact, explaining as clearly as you can.
Baseline Assessment: Mathematical Proof
2. Consider the statement that (n 1)2  n 2  2n 1. Four students have provided
explanations below.
a. Which of the following students have proven this statement?

b. Whose explanation is best? Why?
Baseline Assessment: Mathematical Proof
3. Write down exactly what you would have to do to prove that the following
sentences are false.
a. All High School students are lazy.
b. Some Major League Baseball Players have taken steroids.
c. If the sun is shining, then it is at least 70 degrees outside.
Baseline Assessment: Mathematical Proof
Recording Student Responses
1. This question is intended for you to assess what your students consider to be
an appropriate level of justification. The “theorem” itself is intentionally not
high school level mathematics, because we don’t want students to struggle
with whether or not the statement is true.
With such a simple fact it’s unlikely that too many students will justify the
fact with an appeal to authority. (Who/What would they appeal to? An
elementary school teacher?) But some of them might check a few simple
examples, like 3x4 and 5x2, whereas others will try more extreme examples
using 0, 1, negative numbers, numbers of widely varying magnitude, etc.
Keep a tally of responses in the following table.
Appeal to Authority
Justification with
Simple Examples
Justification with
Simple and Complex
Examples
Attempted General
Argument
Valid General
Argument
2. This problem gives you more insight into what students consider to be valid
mathematical proof.
Dave checks two simple examples and concludes the statement is true. This
is not a valid proof.
Sherman uses an area model diagram to construct a square whose sides are
each (n+1) units long. The diagram clearly shows how the areas inside add
Baseline Assessment: Mathematical Proof
up to the required expression. This is a valid proof, despite the fact that the
square is not drawn to scale. Your students’ reactions to this proof could be
quite interesting and revealing. Many students are visual learners and might
prefer this type of proof, especially if they are familiar with area models.
Other students will think a picture cannot represent a proof.
Veronica gives a valid proof of the statement using algebra. Students who
have done two-column proofs in a geometry course might be drawn to this
proof since it uses that framework. However, the proof would be just as valid
without the reasons listed on the side – those reasons would be implied at
the high school level.
Allison gives a fancy version of “Justification by Example,” injecting some
geometric reasoning into the mix for good measure. In the end she only
verified that the function values (i.e. the y-coordinates of the parabolas)
agree at one single point, which is not enough to conclude they are identical.
(Consider y  x 2 and y  x 2 for x  0 .)
Tally your students’ responses to (a) below, and summarize their answers to
(b).



Dave
Sherman
Veronica
Allison
What types of arguments did your students think were best, and why?
Baseline Assessment: Mathematical Proof
3. Correct answers are as follows:
a.
You need to find at least one high school student who is not lazy.
b.
c.
You would have to show that no major league baseball player has taken
steroids.
You need to demonstrate that it can be sunny outside but cooler than 70
degrees. (This should be possible for students who have lived in
Minnesota during January.)
Tally your students’ responses in the following table.
Correct
(a)
(b)
(c)
Incorrect