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Math 11 Foundations Reasoning Exam
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Bradley gathered the following evidence.
4(44) = 176
5(44) = 220
6(44) = 264
Which conjecture, if any, is Bradley most likely to make from this evidence?
a. When you multiply a one-digit number by 44, the first and last digits of the product form a
number that is four times the original number.
b. When you multiply a two-digit number by 44, the first and last digits of the product form a
number that is twice the original number.
c. When you multiply a one-digit number by 44, the sum of the digits in the product is equal
to the original number.
d. None of the above conjectures can be made from this evidence.
____
2. Which conjecture, if any, could you make about the sum of two odd integers
and one even integer?
a.
b.
c.
d.
____
The sum will be an even integer.
The sum will be an odd integer.
The sum will be negative.
It is not possible to make a conjecture.
3. Make a conjecture as to which line segment is longer, A or B.
a. I conjecture that B is longer than A.
b. I conjecture that A and B are the same length.
c. I conjecture that A is longer than B.
____
4. Jackie made the following conjecture.
The square of a number is always greater than the number.
Which choice, if either, is a counterexample to this conjecture?
____
1.
2.
0.52 = 0.25
(–5)2 = 25
a.
b.
c.
d.
Choice 1 and Choice 2
Choice 2 only
Neither Choice 1 nor Choice 2
Choice 1 only
5. Siddartha made the following conjecture.
When you divide two whole numbers, the quotient will be greater than the divisor and less
than the dividend.
Which choice, if either, is a counterexample to this conjecture?
1.
2.
a.
b.
c.
d.
____
Choice 2 only
Choice 1 and Choice 2
Choice 1 only
Neither Choice 1 nor Choice 2
6. Henry made the following conjecture:
The square of a number is always greater than the number.
Is the following equation a counterexample to this conjecture? Explain.
0.42 = 0.16
a.
b.
c.
d.
____
Yes, it is a counterexample, because 0.4 is less than 0.16.
No, it is not a counterexample, because 0.16 is less than 0.4.
No, it is not a counterexample, because 0.4 is less than 0.16.
Yes, it is a counterexample, because 0.16 is less than 0.4.
7. Loretta made the following conjecture:
When you add a multiple of 6 and a multiple of 9, the sum will be a multiple of 9.
Is the following equation a counterexample to this conjecture? Explain.
12 + 27 = 39
a. Yes, it is a counterexample, because 39 is not a multiple of 9.
b. No, it is not a counterexample, because 39 is a multiple of 3.
c. Yes, it is a counterexample, because 39 is a multiple of 9.
d. No, it is not a counterexample, because 39 is not a multiple of 9.
____
8. Bill made the following conjecture:
When you add a multiple of 6 and a multiple of 9, the sum will be a multiple of 6.
Is the following equation a counterexample to this conjecture? Explain.
12 + 27 = 39
a.
b.
c.
d.
____
Yes, it is a counterexample, because 39 is not a multiple of 6.
Yes, it is a counterexample, because 39 is a multiple of 3.
No, it is not a counterexample, because 39 is a multiple of 3.
No, it is not a counterexample, because 39 is not a multiple of 9.
9. Which of the following choices, if any, uses inductive reasoning to show
that the sum of three even integers is even?
a.
b.
c.
d.
2x + 2y + 2z = 2(x + y + z)
2 + 4 + 6 = 12 and 4 + 6 + 8 = 18
x + y + z = 2(x + y + z)
None of the above choices
____ 10. Which of the following choices, if any, uses inductive reasoning to show
that an odd number and an even number sum to an odd number?
a.
b.
c.
d.
3 + 6 = 9 and 4 + 5 = 9
2x + 2y + 1 = 2(x + y + 1)
(2x + 1) + 2y = 2(x + y) + 1
None of the above choices
____ 11. What type of error, if any, occurs in the following proof?
2
4(2)
4(2) + 3
8+3
11
a.
b.
c.
d.
=2+2
= 4(2 + 2)
= 4(2 + 2) + 3
= 16 + 3
= 19
a false assumption or generalization
an error in reasoning
an error in calculation
There is no error in the proof.
____ 12. Which type of reasoning does the following statement demonstrate?
All birds have feathers.
Robins are birds.
Therefore, robins have feathers.
a.
b.
c.
inductive reasoning
neither inductive nor deductive reasoning
deductive reasoning
____ 13. Determine the unknown term in this pattern.
1, 2, 4, ___, 16, 32, 64
a.
b.
c.
d.
6
12
8
10
____ 14. Determine the unknown term in this pattern.
1, 1, 2, 3, 5, ____, 13, 21
a.
b.
c.
d.
6
7
8
9
Short Answer
15. What conjecture could you make about the product of two even integers
and one odd integer?
16. Austin told his little sister, Celina, that horses, cats, and dogs are all mammals.
As a result, Celina made the following conjecture:
All mammals have four legs.
Use a counterexample to show Celina her conjecture is not valid.
17. Bradley wanted to prove something about the sum of any six consecutive natural numbers. He
wrote this equation:
x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) = 6x + 15
What has Bradley proven?
18. Bob and Anne are playing darts. Bob has a score of 36.
To win, he must reduce his score to zero and
have his last counting dart be a double.
Give a strategy that Bob might use to win.
Problem
19. Alison discovered a number trick in a book she was reading:
Choose a number.
Add 3.
Multiply by 2.
Add 4.
Divide by 2.
Subtract 5.
Try the trick several times. Make a conjecture about the relation between the number picked and
the final result. Can you find a counterexample to your conjecture? What does this imply?
20. A set of 10 cards, each showing one of the digits from 0 to 9, is divided between five envelopes so that there
are two cards in each envelope. The sum of the cards inside each envelope is written on the envelope:
What pair of cards is definitely in an envelope marked 13? Explain.
21. In a magic square, the columns, rows, and diagonals all add up to the same total. Use the integers from –5 to
19 to complete this magic square. Use each number only once.
Math 11 Foundations Reasoning Exam
Answer Section
MULTIPLE CHOICE
1. ANS: A
PTS: 1
DIF: Grade 11
REF: Lesson 1.1
OBJ: 1.1: Make conjectures by observing patterns and identifying properties, and justify the reasoning.
TOP: conjectures and Inductive Reasoning
KEY: conjecture| inductive reasoning
2. ANS: A
PTS: 1
DIF: Grade 11
REF: Lesson 1.1
OBJ: 1.1: Make conjectures by observing patterns and identifying properties, and justify the reasoning.
TOP: conjectures and Inductive Reasoning
KEY: conjecture| inductive reasoning
3. ANS: B
PTS: 1
DIF: Grade 11
REF: Lesson 1.2
OBJ: 1.2 Explain why inductive reasoning may lead to a false conjecture.
TOP: Validity of conjectures
KEY: conjecture| validity of conjectures
4. ANS: D
PTS: 1
DIF: Grade 11
REF: Lesson 1.3
OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.
TOP: Disproving conjectures: Counterexamples
KEY: conjecture| disproving conjectures| counterexamples
5. ANS: B
PTS: 1
DIF: Grade 11
REF: Lesson 1.3
OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.
TOP: Disproving conjectures: Counterexamples
KEY: conjecture| disproving conjectures| counterexamples
6. ANS: D
PTS: 1
DIF: Grade 11
REF: Lesson 1.3
OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.
TOP: Disproving conjectures: Counterexamples
KEY: conjecture| disproving conjectures| counterexamples
7. ANS: A
PTS: 1
DIF: Grade 11
REF: Lesson 1.3
OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.
TOP: Disproving conjectures: Counterexamples
KEY: conjecture| disproving conjectures| counterexamples
8. ANS: A
PTS: 1
DIF: Grade 11
REF: Lesson 1.3
OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.
TOP: Disproving conjectures: Counterexamples
KEY: conjecture| disproving conjectures| counterexamples
9. ANS: B
PTS: 1
DIF: Grade 11
REF: Lesson 1.4
OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number
relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number
tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).
TOP: Proving conjectures| deductive reasoning
KEY: conjecture| proving conjectures| reasoning| deductive reasoning
10. ANS: A
PTS: 1
DIF: Grade 11
REF: Lesson 1.4
OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number
relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number
tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).
TOP: Proving conjectures| deductive reasoning
KEY: conjecture| proving conjectures| reasoning| deductive reasoning
11. ANS: A
PTS: 1
DIF: Grade 11
REF: Lesson 1.5
OBJ: 1.7 Determine if a given argument is valid, and justify the reasoning. | 1.8 Identify errors in a given
proof; e.g., a proof that ends with 2 = 1.
TOP: invalid proofs| deductive reasoning
KEY:
12. ANS:
OBJ:
TOP:
13. ANS:
OBJ:
TOP:
14. ANS:
OBJ:
TOP:
valid proofs| invalid proofs| deductive reasoning
C
PTS: 1
DIF: Grade 11
REF: Lesson 1.6
1.9 Solve a contextual problem involving inductive or deductive reasoning.
reasoning to solve problems
KEY: reasoning| inductive reasoning| deductive reasoning
C
PTS: 1
DIF: Grade 11
REF: Lesson 1.6
1.9 Solve a contextual problem involving inductive or deductive reasoning.
reasoning to solve problems
KEY: reasoning| inductive reasoning| deductive reasoning
D
PTS: 1
DIF: Grade 11
REF: Lesson 1.6
1.9 Solve a contextual problem involving inductive or deductive reasoning.
reasoning to solve problems
KEY: reasoning| inductive reasoning| deductive reasoning
SHORT ANSWER
15. ANS:
For example, the product will be an even integer.
PTS: 1
DIF: Grade 11
REF: Lesson 1.1
OBJ: 1.1: Make conjectures by observing patterns and identifying properties, and justify the reasoning.
TOP: conjectures and Inductive Reasoning
KEY: conjecture| inductive reasoning
16. ANS:
For example, dolphins are mammals, and they don’t have any legs.
PTS: 1
DIF: Grade 11
REF: Lesson 1.3
OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.
TOP: Disproving conjectures: Counterexamples
KEY: conjecture| disproving conjectures| counterexamples
17. ANS:
For example, that the sum of six consecutive natural numbers is divisible by 3.
PTS: 1
DIF: Grade 11
REF: Lesson 1.4
OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number
relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number
tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).
TOP: Proving conjectures| deductive reasoning
KEY: conjecture| proving conjectures| reasoning| deductive reasoning
18. ANS:
For example, score double 18 on first throw and not throw any more darts.
PTS: 1
DIF: Grade 11
REF: Lesson 1.7
OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and
correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle
or a game, and describe a strategy for solving the puzzle or winning the game.
TOP: analyzing puzzles and games
KEY: reasoning| solving puzzles
PROBLEM
19. ANS:
For example, I chose to start with 8.
Choose a number:
Add 3:
Multiply by 2:
Add 4:
Divide by 2:
Subtract 5:
8
8 + 3 = 11
2(11) = 22
22 + 4 = 26
= 13
13 – 5 = 8
The final result is the same as the number I chose, so I made the conjecture that the final result will always be
the same as the starting number. I tried this several times and could not find a counterexample, which implies
that my conjecture is valid.
PTS: 1
DIF: Grade 11
REF: Lesson 1.3
OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture.
TOP: Disproving conjectures: Counterexamples
KEY: conjecture| disproving conjectures| counterexamples
20. ANS:
Make a list of possible pairs in each envelope:
4: (0, 4), (1, 3)
5: (0, 5), (1, 4), (2, 3)
10: (1, 9), (2, 8), (3, 7), (4, 6)
13: (4, 9), (5, 8), (6, 7)
Suppose the two envelopes marked 13 contain the pairs
(4, 9) and (5, 8). Since the 4, 8, and 9 cards are used, the only possible
pair in the envelope marked 10 is (3, 7). But that leaves no possible pairs
for the other envelopes because 3, 4, and 5 are all used. That means
(6, 7) must be in one of the envelopes marked 13.
If it is, there are two possible solutions:
(0, 4), (2, 3), (1, 9), (5, 8), (6, 7) and (1, 3), (0, 5), (2, 8), (4, 9), (6, 7).
PTS: 1
DIF: Grade 11
REF: Lesson 1.6
OBJ: 1.9 Solve a contextual problem involving inductive or deductive reasoning.
TOP: reasoning to solve problems
KEY: reasoning| inductive reasoning| deductive reasoning
21. ANS:
PTS: 1
DIF: Grade 11
REF: Lesson 1.7
OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and
correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle
or a game, and describe a strategy for solving the puzzle or winning the game.
TOP: analyzing puzzles and games
KEY: reasoning| solving puzzles