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Q
uantitative
R
easoning &
P
roblem
S
olving
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© 2014 Pacific Crest
The Practice
9. Ask “What if?”
as a way to identify what you understand
Description
When/How
to Apply
After completing
each problem,
1) Identify variables that can change 2) Explore a range of values for variables 3)
Check for limitations 4) Check for any special situations
In mathematics, there is a strong tendency to do a whole bunch of practice problems for each new idea
encountered. A different approach based on many years of thoughtful practice indicates that learners
can test their understanding with just three to four problems, once they have addressed the Critical
Thinking Questions. Once the first few problems have been solved, we can revisit them to perform a
“What if?” analysis. Making changes to these first few problems and understanding how the changes
affect our process and the solutions is much more productive than constantly moving to new problems.
Examples
Multiplication and Division of Fractions
Through applying “What if?” to a previous example,
we can create more difficult problems. We can change components of this mathematical problem
from half a pizza to, what if we have three halves (3/2) and then distribute an equal portion to four
people (one fourth of three halves: 1/4 x 3/2)? How much does each person get then? How does
the original multiplication process change? We can see that we just substitute and recalculate:
Instead of
1 1 1
2 2 4
=
we now have
1 3 3
4 2 8
=
We see now that we can take any fraction multiplication problem and solve it, where the two
fractions can take on any values.
Solving a linear equation
Let’s start simple: you have $5 for buying candy bars at the movies
and each candy bar costs $1.25. How many candy bars can you buy? The answer is obviously
4 (4 x $1.25 = $5). What we did was calculate $5 = $1.25(
x
) where
x
represents the number of
candy bars.
Now let’s ask, “What if...we have $12 and candy bars cost $2.50? How many could we buy
then?” We see that we can change nearly all the information involved in this scenario and still
solve the problem:
12 2.50( )
12 120 24 44
2.50 25 5 5
x
x
=
= = = =
The answer is
4 candy bars
because
we can only buy whole candy bars.
The Practice
10. Validate
to make sure your results are 100% correct
Description
When/How to Apply
Every time you solve a problem (see Activity 4.4).
The solution to a problem is often a calculation or derivation. Knowing that your calculations are accurate
and validated is important for building trust in your understanding.
Examples
Long Division
In long division, we multiply the answer (quotient) by the divisor to determine if
we arrive at the original dividend as our answer. If so, we have used multiplication to validate
the process of division.
Identifying Prime Factors
In identifying the prime factors of a numbers, we simply take the
potential prime factors we find, multiply them together, and determine whether or not the product
is our original number.