FTCE Prep Grade 5-9
Competency 1-4
Solutions to Practice Questions #1 - 15:
1. What would be the total cost of a suit for $295.99 and a pair of shoes for
$69.95 including 6.5% sales tax?
Solution: First find the cost of suit and shoes  $295.99 + $69.95 or $365.94
Second figure the tax at 6.5% on $365.94  365.94 • .065 = $23.79
Last = add tax to cost of items  $365.94 + $23.79 = $389.73 or A
2. Which is the least appropriate strategy to emphasize when teaching
problem solving?
The basic problem solving strategies are:
Find a Pattern
Make a Table
Work Backwards
Guess and Check
Draw a Picture
Make a List
Write a Number Sentence
Use Logical Reasoning/Write Simpler Version
This means B is the least appropriate strategy!
3. A student had 60 days to appeal the results of an exam. If the results
were received on March 23, what was the last day that the student could
appeal?
There are 31 days in March and May and 30 days in April and June.
If you count 8 days left in March to 30 days in April (30+8 = 38) you then need
(60-38) 22 days in May. This means B or May 22 is the last day
4. The diagram below would be least appropriate for illustrating which of
the following?
The diagram illustrates 28 blocks and 3 blocks. This can mean 28 block plus
3 blocks or 31 blocks – 3 blocks or 7(columns) x 4 (rows) + 3 blocks. It does
NOT show 28 blocks X blocks. Answer choice is C.
5. According to Piaget, at which developmental level would a child be able
to learn formal algebra?
Answer is D Concrete Operational. Piaget levels are:
Sensorimotor stage (Infancy). In this period (which has 6 stages), intelligence is
demonstrated through motor activity without the use of symbols. Knowledge of the world is
limited (but developing) because its based on physical interactions / experiences. Children
acquire object permanence at about 7 months of age (memory). Physical development
(mobility) allows the child to begin developing new intellectual abilities. Some symbolic
(language) abilities are developed at the end of this stage.
FTCE Prep Grade 5-9
Competency 1-4
Pre-operational stage (Toddler and Early Childhood). In this period (which has two
substages), intelligence is demonstrated through the use of symbols, language use
matures, and memory and imagination are developed, but thinking is done in a nonlogical,
nonreversable manner. Egocentric thinking predominates
Concrete operational stage (Elementary and early adolescence). In this stage
(characterized by 7 types of conservation: number, length, liquid, mass, weight, area,
volume), intelligence is demonstrated through logical and systematic manipulation of
symbols related to concrete objects. Operational thinking develops (mental actions that
are reversible). Egocentric thought diminishes.
Formal operational stage (Adolescence and adulthood). In this stage, intelligence is
demonstrated through the logical use of symbols related to abstract concepts
[emphasis added by Moursund]. Early in the period there is a return to egocentric thought.
Only 35% of high school graduates in industrialized countries obtain formal operations;
many people do not think formally during adulthood.
6. Sandra has $34.00, Carl has $42.00. How much more does Carl have
than Sandra? Which would be the best method for finding the answer?
How much more than indicates subtraction. The answer is B.
7. Choose the least appropriate set of manipulatives for a sixth grade
class.
Choice A contains conics section models – which is used for Algebra 2…not
for 6th grade.
8. Which statement is incorrect?
While we would like for every child to have their own workstation, Choice C is
incorrect because the teacher can use one computer for demonstration
purposes. A computer is excellent for drill and practice (A) and to
teach/reinforce problem solving (B) and for data analysis (D).
9. Given XY = YZ and AYX  AYZ . Prove AYZ  AYX . Which property
justifies step 3? STEP 3: AY = AY
The answer is REFLEXIVE which is defined as a = a.
The symmetric property is if a = b then b = a.

 b = c, then a = c
The transitive property is if a = b and
The Identity property of addition is a + 0 = a.
10. Given l1 || l2 prove b  e
1) vertical angle theorem
b  d
d  e
b  e

2) alternate interior angle theorem
3) symmetric axiom of equality
Which step is incorrectly justified?
Vertical angle theorem says If two angles are vertical angles, then they have
equal measures.
Alternate angle theorem says When a transversal intersects two parallel lines,
the alternate angles are equal.
Symmetric axiom says if a = b, then b = a
The answer choice C or 3 since this represents the transitive property (if a = b
and b = c hen a = c)
FTCE Prep Grade 5-9
Competency 1-4
11. Which is a postulate?
A postulate (or an axiom) is a proposition that is not proved or demonstrated
but considered to be either self-evident. Therefore, its truth is taken for
granted, and serves as a starting point for deducing and inferring other
(theory dependent) truths.
A, B & C can be proven. D is accepted as truth.
12. When you begin by assuming the conclusion of a theorem is false, then
show that through a sequence of logically correct steps you contradict
an accepted fact, this is known as?
Inductive reasoning, also known as induction or inductive logic, is a type of reasoning
that involves moving from a set of specific facts to a general conclusion.
A direct proof is a way of showing the truth or falsehood of a given statement by a
straightforward combination of established facts.
An indirect proof is a type of proof in which a statement to be proved is assumed false and if
the assumption leads to an impossibility, then the statement assumed false has been proved
to be true.
An exhaustive proof requires all possible values of the expression to be evaluated and thus
convinced that the proof is concrete.
Answer is C Indirect Proof
13. Given that n is a positive even integer, 5n + 4 will always be divisible
by:
If n is even, then 5n is also even. When you add 4 to an even number (5n) it
is an even number. Therefore 5n + 4 will always be an even number thus
divisible by 2. Answer is D.
14. Given similar polygons with corresponding sides of lengths 9 and 15,
find the perimeter of the smaller polygon if the perimeter of the larger
polygon is 150 units.
Similar polygons are polygons for which all corresponding angles are
congruent and all corresponding sides are proportional. This means the
perimeters are directly proportional to the lengths of their sides.
9
x

This means
 15x = 9 • 150  15x = 1350  x = 90
15 150
Answer Choice is C
15. A student turns in a paper with this type of error:
7 + 
16 ÷ 8 x 2 = 8
8 – 3 x 3 + 4 = -5
In order to remediate this error the teacher should?
First you need to figure out the error the student is making.
In the first problem, the student is multiplying first, then dividing, then adding
7 + 16 ÷ 8 x 2  7 + 16 ÷ 16  7 + 1  8
In the second problem, the student is multiplying first, then adding 4 to 9, then
combining 8 and -13.
8 – 3 x 3 + 4  8 – 9 + 4  8 – 13  -5
This indicates a lack of understanding of the order of operations—that you
add or subtract, multiply or divide from Left to Right. Correct Answer is C.