To the teacher:

This PowerPoint consists of the Structure and Logic

section of the AIMS. It covers:

• Conditional, Converse, Inverse, Contrapositive

• Logical Order of If-Then statements

• Use of Inductive and Deductive Reasoning

• Proofs

• Structural Similarities

• Algorithms and Algorithmic Thinking

AIMS Prep

Logic, Reasoning, Problem Solving,

Proof, and Algorithmic Thinking

Structure and Logic

1) Which of the statements below is the inverse of the

following conditional statement?

If I do my homework, then I can go to the

movies.

A. If I don’t do my homework, then I can not go to the

movies.

B. If I go to the movies, then I did my homework.

C. If I don’t go to the movies, then I did not do my

homework.

D. I can go to the movies if I do my homework.

1) Which of the statements below is the inverse of the

following conditional statement?

If I do my homework, then I can go to the

movies.

A. If I don’t do my homework, then I can not go to the

movies.

B. If I go to the movies, then I did my homework.

C. If I don’t go to the movies, then I did not do my

homework.

D. I can go to the movies if I do my homework.

5.2.9

2)

5.2.9

2)

5.2.9

3) Which statement has a false inverse?

A If x = 1, then x = 1.

B If x2 = 0 , then x = 0 .

C If x = −2 , then x2 = 4 .

D If x + 3 = 5 , then x = 2 .

5.2.9

3) Which statement has a false inverse?

A If x = 1, then x = 1.

B If x2 = 0 , then x = 0 .

C If x = −2 , then x2 = 4 .

D If x + 3 = 5 , then x = 2 .

5.2.9

4) Assume the statement shown below is true.

“If P, then Q.”

Based on this assumption, which of the following

must be true?

A If Q, then P.

B If P, then not Q.

C If not P, then not Q.

D If not Q, then not P.

5.2.9

4) Assume the statement shown below is true.

“If P, then Q.”

Based on this assumption, which of the following

must be true?

A If Q, then P.

B If P, then not Q.

C If not P, then not Q.

D If not Q, then not P.

5) The statements below are out of order.

W: If blitz, then kerd.

X: If mot, then det.

Y: If kerd, then mot.

Z: If toc, then blitz.

Which of the following puts the nonsensical if-then statements in logical order?

A. W Z X Y

B. Z W Y X

C. W Y X Z

D. Z X Y W

5) The statements below are out of order.

W: If blitz, then kerd.

X: If mot, then det.

Y: If kerd, then mot.

Z: If toc, then blitz.

Which of the following puts the nonsensical if-then statements in logical order?

A. W Z X Y

B. Z W Y X

C. W Y X Z

D. Z X Y W

6) Put the following statements in proper logical order.

I. If Dean graduates with a degree, then he will make a lot

of money.

II. If Dean studies hard, then his grades will be good.

III. If Dean makes a lot of money, then he will be able to

buy a new car.

IV. If Dean attends college, then he will graduate with a

degree.

V. If Dean has good grades, then he will be able to attend

college.

A. V, III, I, IV, II

B. II, V, I, IV, III

C. V, II, IV, I, III

D. II, V, IV, I, III

6) Put the following statements in proper logical order.

I. If Dean graduates with a degree, then he will make a lot

of money.

II. If Dean studies hard, then his grades will be good.

III. If Dean makes a lot of money, then he will be able to

buy a new car.

IV. If Dean attends college, then he will graduate with a

degree.

V. If Dean has good grades, then he will be able to attend

college.

A. V, III, I, IV, II

B. II, V, I, IV, III

C. V, II, IV, I, III

D. II, V, IV, I, III

7) The following number pattern was seen on the white

board: 2 4 8

Using inductive reasoning, what number will follow 8?

A. 16 (It’s a doubling sequence)

B. 14 (You are adding consecutive even numbers to each

term)

C. 10 (You are listing pairs of consecutive even

numbers, skipping the even number between the

pairs, and so on.)

D. Any of these numbers are valid using inductive

reasoning

7) The following number pattern was seen on the white

board: 2 4 8

Using inductive reasoning, what number will follow 8?

A. 16 (It’s a doubling sequence)

B. 14 (You are adding consecutive even numbers to each

term)

C. 10 (You are listing pairs of consecutive even

numbers, skipping the even number between the

pairs, and so on.)

D. Any of these numbers are valid using inductive

reasoning

8) Using inductive reasoning, what conjecture is

most likely derived from these statements?

My sister’s dog is a boxer.

My best friend’s dog is a boxer.

A. Dogs are animals.

B. All dogs are boxers.

C. A boxer is a type of dog.

D. My sister is my best friend.

8) Using inductive reasoning, what conjecture is

most likely derived from these statements?

My sister’s dog is a boxer.

My best friend’s dog is a boxer.

A. Dogs are animals.

B. All dogs are boxers.

C. A boxer is a type of dog.

D. My sister is my best friend.

5.2.8

9) The table shows a sequence of figures, the number of

squares in each figure, and the Perimeter of each figure.

Based on the pattern shown, which conjecture is valid?

A A figure with 4 squares has a perimeter of 10.

B A figure with 8 squares has a perimeter of 14.

C A figure with 20 squares has a perimeter of 14.

D A figure with 28 squares has a perimeter of 28.

5.2.8

9) The table shows a sequence of figures, the number of

squares in each figure, and the Perimeter of each figure.

Based on the pattern shown, which conjecture is valid?

A A figure with 4 squares has a perimeter of 10.

B A figure with 8 squares has a perimeter of 14.

C A figure with 20 squares has a perimeter of 14.

D A figure with 28 squares has a perimeter of 28.

10) Look at the figures below.

528

If the number of square units in the pattern of the figures

continues to increase arithmetically as shown, how many

square units will be in the 9th figure?

A. 9 B. 39 + 1 C. 9(3 + 1) D. 1 + (3 • 9)

10) Look at the figures below.

528

If the number of square units in the pattern of the figures

continues to increase arithmetically as shown, how many

square units will be in the 9th figure?

A. 9 B. 39 + 1 C. 9(3 + 1) D. 1 + (3 • 9)

11) Using deductive reasoning, what conclusion

can be made from the following statements?

All birds have feathers.

My pet is a bird.

A. Birds make good pets.

B. All pets are birds.

C. All pets have feathers.

D. My pet has feathers.

11) Using deductive reasoning, what conclusion

can be made from the following statements?

All birds have feathers.

My pet is a bird.

A. Birds make good pets.

B. All pets are birds.

C. All pets have feathers.

D. My pet has feathers.

Multiples

of 2

Multiples

of 3

12) Which of the following is the correct conditional statement for

the Venn Diagram?

A. If a number is a multiple of 2, then it is a multiple of 3.

B. If a number is a multiple of 3, then it is NOT a multiple of 2.

C. Some multiples of 2 are multiples of 3.

D. All multiples of 2 are multiples of 3.

Multiples

of 2

Multiples

of 3

12) Which of the following is the correct conditional statement for

the Venn Diagram?

A. If a number is a multiple of 2, then it is a multiple of 3.

B. If a number is a multiple of 3, then it is NOT a multiple of 2.

C. Some multiples of 2 are multiples of 3.

D. All multiples of 2 are multiples of 3.

5.2.5

13)

5.2.5

13)

5.2.8

14) Based on the diagram below, which of these

arguments are valid?

a. The triangles are congruent by side-side-side (SSS)

b. The triangles are congruent by side-angle-side (SAS)

c. The triangles are congruent by angle-side-angle (ASA)

d. The triangles are congruent by angle-angle-side (AAS)

5.2.8

14) Based on the diagram below, which of these

arguments are valid?

a. The triangles are congruent by side-side-side (SSS)

b. The triangles are congruent by side-angle-side (SAS)

c. The triangles are congruent by angle-side-angle (ASA)

d. The triangles are congruent by angle-angle-side (AAS)

5.2.1

15) If the sum of the measures of two angles is 90o, then the

angles are complimentary.

In the triangle ABC, m A = 25o, m B = 65o, m C = 90o.

Which valid conclusion follows directly from the previous

statements?

A C is a complimentary angle.

B B and C are complimentary angles.

C A and C are complimentary angles.

D A and B are complimentary angles.

5.2.1

15) If the sum of the measures of two angles is 90o, then the

angles are complimentary.

In the triangle ABC, m A = 25o, m B = 65o, m C = 90o.

Which valid conclusion follows directly from the previous

statements?

A C is a complimentary angle.

B B and C are complimentary angles.

C A and C are complimentary angles.

D A and B are complimentary angles.

5.2.7

16) Which pair of figures is

structurally similar?

A cone and cube

B cone and sphere

C cone and rectangular prism

D cone and triangular pyramid

5.2.7

16) Which pair of figures is

structurally similar?

A cone and cube

B cone and sphere

C cone and rectangular prism

D cone and triangular pyramid

5.1.1

17) Which expression below has been

simplified using the correct procedure?

5.1.1

17) Which expression below has been

simplified using the correct procedure?

5.1.1

18) Which procedure correctly simplifies

the expression below?

-(x + 3) – 2(4x – 3)

5.1.1

18) Which procedure correctly simplifies

the expression below?

-(x + 3) – 2(4x – 3)

5.1.3

19)

5.1.3

19)

5.1.2

20) Which of the following could be a correct procedure for solving the inequality shown below?

4x + 6 ≤ 6x + 15

A 4x + 6 ≤ 6x +15

–2x + 6 ≤ 15

–2x ≤ 9

x ≥ –4.5

C 4x + 6 ≤ 6x +15

–2x + 6 ≤ 15

–2x ≤ 9

x ≤ –4.5

B 4x + 6≤ 6x +15

–2x + 6 ≤ 15

–2x ≤ 21

x ≤ –2.5

D 4x + 6≤ 6x +15

–2x + 6 ≤ 15

–2x≤ 9

x ≥ –2.5

5.1.2

20) Which of the following could be a correct procedure for solving the inequality shown below?

4x + 6 ≤ 6x + 15

A 4x + 6 ≤ 6x +15

–2x + 6 ≤ 15

–2x ≤ 9

x ≥ –4.5

C 4x + 6 ≤ 6x +15

–2x + 6 ≤ 15

–2x ≤ 9

x ≤ –4.5

B 4x + 6≤ 6x +15

–2x + 6 ≤ 15

–2x ≤ 21

x ≤ –2.5

D 4x + 6≤ 6x +15

–2x + 6 ≤ 15

–2x≤ 9

x ≥ –2.5

5.1.1

Steps 1 and 2 describe an algorithm.

Step 1: Isolate the variable.

Step 2: Take the square root of both sides of the equation.

You now have your answer.

21) Which of these equations can be solved by the

algorithm above?

I. x2 – 2x – 3 = 0 III. x2 – 9 = 0

II. x + 5 = 0 IV. x3 + 2x + 6 = 0

A I B II C III D IV

5.1.1

Steps 1 and 2 describe an algorithm.

Step 1: Isolate the variable.

Step 2: Take the square root of both sides of the equation.

You now have your answer.

21) Which of these equations can be solved by the

algorithm above?

I. x2 – 2x – 3 = 0 III. x2 – 9 = 0

II. x + 5 = 0 IV. x3 + 2x + 6 = 0

A I B II C III D IV

5.1.1

22) Look at the expression.

–2a [–2a (–2a + 4b) + 3b (–a – 6)]

Which of the following correctly simplifies the expression?

A –2a [4a2 – 8ab – 3ab – 18b]

–2a [4a2 – 5ab – 18b]

–8a3 + 10a2b + 36b

B –2a [4a2 – 8ab – 3ab – 18b]

–2a [4a2 – 5ab – 18b]

–8a2 + 10ab + 36ab

C –2a [4a2 – 8ab – 3ab – 18b]

–2a [4a2 – 11ab – 18b]

–8a3 + 22a2b + 36ab

D –2a [4a2 + 8ab + 3ab + 18b]

–2a [4a2 + 11ab + 18b]

–8a3 – 22a2b – 36ab

5.1.1

22) Look at the expression.

–2a [–2a (–2a + 4b) + 3b (–a – 6)]

Which of the following correctly simplifies the expression?

A –2a [4a2 – 8ab – 3ab – 18b]

–2a [4a2 – 5ab – 18b]

–8a3 + 10a2b + 36b

B –2a [4a2 – 8ab – 3ab – 18b]

–2a [4a2 – 5ab – 18b]

–8a2 + 10ab + 36ab

C –2a [4a2 – 8ab – 3ab – 18b]

–2a [4a2 – 11ab – 18b]

–8a3 + 22a2b + 36ab

D –2a [4a2 + 8ab + 3ab + 18b]

–2a [4a2 + 11ab + 18b]

–8a3 – 22a2b – 36ab

5.1.2

23) Which of the following quadratic equations is

solved correctly?

5.1.2

23) Which of the following quadratic equations is

solved correctly?

5.1.4

Four students each used a different method to find

the area of the rectangle below.

24) Which of the students used an INCORRECT method to find

the area?

A.Student 1 counted the

number of unit squares.

B. Student 2 used l • w.

C. Student 3 multiplied the

number of rows times the

number of columns.

D. Student 4 used 2l • 2w.

5.1.4

Four students each used a different method to find

the area of the rectangle below.

24) Which of the students used an INCORRECT method to find

the area?

A.Student 1 counted the

number of unit squares.

B. Student 2 used l • w.

C. Student 3 multiplied the

number of rows times the

number of columns.

D. Student 4 used 2l • 2w.

25) Simplify ( x + y )2

a) x2 + y2

b) x2 + xy + y2

c) x2 + 2xy + y2

d) 2x + 2y

25) Simplify ( x + y )2

a) x2 + y2

b) x2 + xy + y2

c) x2 + 2xy + y2

d) 2x + 2y

26) Simplify x( x + y + 1 )

a) x2 + xy

b) 2x + xy

c) x2 + y + 1

d) x2 + xy + x

26) Simplify x( x + y + 1 )

a) x2 + xy

b) 2x + xy

c) x2 + y + 1

d) x2 + xy + x

27) Determine whether the given

procedure for solving the equation is

valid. If not, solve using the appropriate

method. Show your work.

5(x + 2) (x + 2)

5(x + 2) (x + 2)

(x + 2) (x + 2)

5 1

5(x + 2) = (x + 2)

5(x + 2) = (x + 2)

(x + 2) (x + 2)

5 = 1

This procedure is invalid because when we divided each

side by (x + 2) we divided by a variable. Every time you

divide by a variable you lose a solution to the

equation. Since the equation is linear ( a line ) it only

had one solution to begin with. We lost the one and only

solution when we divided by a variable so the "solution"

5 = 1 is not really a solution and our procedure is invalid.

27) Determine whether the given procedure for solving the equation is valid. If not, solve using the appropriate method. Show your work

Actual answer:

x = -2

28) Determine whether the given

procedure for solving the equation is

valid. If not, solve using the

appropriate method. Show your work.

2(x + 3) – (x – 5) = 4

2x + 3 – x – 5 = 4

x – 2 = 4

x = 6

28) Determine whether the given procedure for solving the equation

is valid. If not, solve using the appropriate method. Show your work.

2(x + 3) – (x – 5) = 4

2x + 3 – x – 5 = 4

x – 2 = 4

x = 6

To test the method plug in 6 for x in the original

equation. We get 17 = 4. Obviously a mistake

was made and the method is not valid.

Error: Incorrect distributing of both

the 2 and the -1.

Actual Answer:

x = -7