It relies on complex syntax.

It requires extensive background knowledge.

It allows for universal understanding across different languages and cultures.

It is always lengthy and detailed.

It uses figurative language that makes problems more relatable.

It eliminates the need for lengthy explanations, providing clear formulas instead.

It allows for peer discussions about emotions involved in problem-solving.

It combines various languages into a single format.

It uses numbers and symbols instead of words.

It follows grammatical rules that allow communication of ideas.

It has more symbols than any spoken language.

It is only used for calculations.

It removes all unnecessary words and symbols.

It replaces long descriptions with standardized symbols and notations.

It limits the expression of mathematical ideas.

It avoids the use of variables in equations.

It allows complex ideas to be expressed efficiently and applied across different fields.

It simplifies all real-world problems into basic arithmetic.

It is only used by mathematicians and scientists.

It eliminates the need for natural language communication.

Computational fluency

Abstract reasoning

Pattern recognition

Visual perception

Approximation

Conciseness

Preciseness

Powerfulness

It often includes slang terms and idioms.

It relies on symbols, variables, and a set of rules for interpretation.

It is based on emotional appeal and subjective interpretation.

It allows for multiple interpretations of a single problem.

It simplifies complex problems into mere numbers.

It provides a standardized method for interpreting and communicating quantitative information.

It eliminates the need for logical reasoning.

It encourages reliance on trial and error rather than structured approaches.

The product of a number and four equals y minus one.

For every increase of one in x, y increases by four.

Four times x, plus one, is equal to y.

If you add four to x, you get y.

Twice the sum of a number and three equals sixteen.

The difference between twice a number and three equals sixteen.

The result of adding three to twice a number is sixteen.

Two times a number increased by three equals sixteen.

$$(x^2-4)$$

$$((x-4)^2)$$

$$(4-x^2)$$

$$(2x-4)$$

Three times the difference between twice a number and four is equal to negative twelve.

The total of three times the result of subtracting four from a number doubled plus twelve equals zero.

The product of three and the sum of a number minus four equals zero.

Three times the quantity of a number times two minus four plus twelve equals zero.

2 + (n + 6) = 20

2(n + 6) = 20

2n + 6 = 20

n + 2 + 6 = 20

Four more than three times a number is six more than the number.

Three times a number, decreased by four, is equal to six more than twice the number.

Four less than three times a number is the same as six less than twice the number.

A number multiplied by three and then reduced by four equals six.

3(x−7)=2(x+5)

3x−7=2(x+5)

3(x+7)=2(x−5)

3x−7=2x+5

The sum of twice a number and five is four times the sum of the number and three.

Twice a number decreased by five is equal to four times the sum of the number and three.

Five less than twice a number is four times the number increased by three.

The difference between twice a number and five is four times the sum of three and the number.

$$(x)(x^2)−3x=20$$

$$(x^2)(x^2)−3x=20$$

$$(x)(x)+3x=20$$

$$(x)(x^3)+3x=20$$

The sum of a number and five is twice ten.

Half of the sum of a number and five is ten.

Twice the sum of a number and five equals ten.

A number increased by five is equal to twenty.

$$3x+\frac{1}{x}=5$$

$$3+\frac{x}{3}=5$$

$$\frac{3}{x}+x=5$$

$$x+\frac{1}{3x}=5$$

3n - 4 = 8

n - 4 = 3 * 8

3n + 4 = 8

n + 3 = 8 - 4

$$2x+5=16$$

$$2(x+5)=16$$

$$x+2(5)=16$$

$$\frac{x}{2}+5=16$$

$$x^2+(x+2)^2=52$$

$$x^2+(x+1)^2=52$$

$$x^2+(x+2x)^2=52$$

$$x^2+(x+4)^2=52$$

Five times a number decreased by seven is less than thirteen.

Five times a number minus seven is greater than thirteen.

The product of five and a number is subtracted by seven, which is equal to thirteen.

Five times a number, when reduced by seven, results in thirteen.

x−2x=−x+5

x−2x=x+5

x−2x=−5−x

x+2x=−x+5

"Three more than the quotient of a number and five is twice the number decreased by seven."

"The quotient of a number and five, increased by three, is equal to seven more than twice the number."

"A number divided by five, then increased by three, equals seven less than twice the number."

"Three times a number divided by five equals seven more than twice the number."

$$3x−x^2=\frac{1}{x}+5x^3$$

$$3x−x^2=x−1+5x^3$$

$$x^2−3x=\frac{1}{x}+5x^3$$

$$3x−x^2=5x^3−\frac{1}{x}$$

Finite Set

Unit Set

Null Set

Infinite Set

It is smaller in size compared to any other set.

It is the only set that contains no numbers.

It has no elements, so it does not contradict subset conditions.

It is the largest set possible.

Universal sets

Disjoint sets

Equal sets

Joint sets

10

25

16

32

A set that contains no elements.

A set that contains exactly one element.

A set that contains an infinite number of elements.

A set that contains all possible elements in a given context.

A is a finite set.

A is a subset of the set of real numbers.

A and the set of odd numbers are joint sets.

A contains all numbers.

Finite sets contain real numbers, while infinite sets contain only integers.

Finite sets always have fewer elements than infinite sets.

Infinite sets are always larger than finite sets.

Finite sets have elements that can be counted, whereas infinite sets have uncountable elements.

{ }

{a, b, c, d, e}

{x | x is a natural number greater than 100}

{10, 20, 30, 40, 50}

Yes, because every whole number is also an integer.

No, because integers only include even numbers.

Yes, because integers and whole numbers are the same set.

No, because whole numbers include negative numbers, but integers do not.

The empty set is equal to the given set.

The empty set is always included as a subset of any set.

The empty set is never considered a subset.

The empty set is only a subset of the universal set.

A function where every input has a unique output.

A function where multiple inputs have the same output.

A relation where one input has multiple outputs.

A relation where every output has a unique input.

Check if any outputs are repeated.

Check if each input has exactly one output.

Check if the outputs follow an arithmetic sequence.

Check if the relation is linear.

Find the domain of f(x).

Evaluate g(x) first, then substitute the result into f(x).

Solve both functions independently before substituting.

Differentiate both functions before composition.

Identify whether the y-values are increasing or decreasing.

Check if any input value is repeated with a different output.

Determine if the function is quadratic or linear.

Look for a common difference in the outputs.

Solve for y and check if multiple values of y exist for some x.

Graph the equation and check if it is linear.

Substitute different values of y and check if they yield unique values of x.

Find the derivative and check its sign.

A set of ordered pairs where each input has only one output

A set of ordered pairs that can only form a straight line when graphed

Any set of ordered pairs, whether or not inputs repeat

A rule that assigns only positive outputs to inputs

A function must have repeating outputs.

A function cannot have repeating inputs with different outputs.

A relation must be a one-to-one correspondence.

A function is a set of numbers, but a relation is an equation.

A person and their phone numbers

A student and their grades in different subjects

A country and its capital city

A word and its possible meanings in a dictionary

The set of all possible input values.

The set of all possible output values.

The set of ordered pairs that define the function.

The set of all x-values in the function's graph.

A vertical line

A circle

A parabola opening upwards

A sideways parabola

A teacher assigns multiple students the same seat number.

A vending machine dispenses different snacks for the same button.

A city assigns each house a unique postal code.

A student is enrolled in multiple extracurricular activities.

The graph represents a function because each x-value has at most one y-value.

The graph does not represent a function because some x-values correspond to more than one y-value.

The graph is a function because it is a continuous curve.

The graph does not represent a function because it fails the Horizontal Line Test.

A person is assigned a unique passport number.

A country assigns a single dialing code for international calls.

A vending machine button dispenses different snacks at different times.

A student's ID corresponds to exactly one GPA.

It passes the horizontal line test.

It passes the vertical line test.

Every output has exactly one input.

Every x-value is positive.

Yes, because each student chooses one topic.

No, because a topic can be chosen a single time.

Yes, because every input has multiple outputs.

No, because a student can choose more than one topic.

f(x)=2x+1

f(x)=x+1

f(x)=3x-1

f(x)=2x-1

$$3x+2x+4$$

$$3x^2+10x-8$$

$$3x^2+10x-2$$

$$3x^2+12x-8$$

$$2(x^2-1)+2$$

$$2x^2-5$$

$$2x+32-1$$

$$2x^2+1$$

{(1,2), (2,3), (3,4), (1,5)}

{(1,4), (2,4), (3,4), (4,4)}

{(1,2), (2,3), (3,2), (4,3)}

{(1,2), (2,3), (3,4), (4,5)}

$$y=x^2$$

$$y=|x|$$

$$y=2x+3$$

$$y=-x^2+4$$

{1, -2, -3, -2, 1}

{-3, -2, 1}

{-3, -2, 1, 4}

{1, 4, 9}

$$x^2+2x+3$$

$$x^2+2x+2$$

$$2x^2+x+2$$

$$2x^2+x+3$$

2

10

16

24

x-4

x+2

x-2

x+4

$$\frac{x^2-1}{2x+3}$$

$$\frac{x^2+1}{2x+3}$$

$$\frac{2x+3​}{x^2-1}$$

$$\frac{x^2-2}{x-2}$$

-3

5

9

-15

$$\left(x^2+3\right)+3$$

$$\left(x+3\right)^2+2$$

$$x^2+5$$

$$x^2+6$$

All prime numbers greater than 2 are odd. 17 is a prime number. Therefore, it is odd.

A square has four equal sides. This shape is a square, so it has four equal sides.

The past five math quizzes have been difficult. The next math quiz will also be difficult.

If a number is divisible by 4, then it is also divisible by 2. 16 is divisible by 4, so it is divisible by 2.

Moving from specific observations to general conclusions

Moving from general principles to specific cases

Relying only on intuition to make decisions

Using emotions to make judgments

Inductive reasoning

Deductive reasoning

Both are equally certain

Neither is certain

If all birds have feathers and a sparrow is a bird, then a sparrow has feathers.

Every swan I have seen is white, so all swans must be white.

If it is raining, then the ground is wet. It is raining, so the ground is wet.

All squares have four sides; a rectangle has four sides, so a rectangle is a square.

Inductive reasoning guarantees truth, while deductive reasoning does not.

Deductive reasoning uses specific cases to form general rules, while inductive reasoning moves from general rules to specific cases.

Inductive reasoning moves from specific cases to general rules, while deductive reasoning moves from general rules to specific cases.

They are the same process with different names.

If all humans are mortal and Socrates is human, then Socrates is mortal.

All chemistry students must take a lab course; John is a chemistry student, so he must take a lab course.

Every test in this class has been difficult, so the next test will probably be difficult.

If it is below freezing, water will turn to ice. It is below freezing, so water will turn to ice.

Deductive reasoning always leads to a true conclusion.

Inductive reasoning always leads to a true conclusion.

Inductive reasoning can suggest a likely truth, but deductive reasoning guarantees truth if the premises are correct.

Both inductive and deductive reasoning are based purely on guesswork.

It relies on past experiences to make generalizations.

It moves from a general statement to a specific conclusion.

It does not require logical consistency.

It relies purely on probability and chance.

All mammals have lungs; a dog is a mammal, so it has lungs.

I have seen many dogs bark, so all dogs must bark.

The sun has risen in the east every day, so it will rise in the east tomorrow.

If I eat too much sugar, I might get cavities.

The conclusion must be true.

The conclusion is likely but not certain.

The conclusion may still be false.

The conclusion depends on inductive reasoning.

Misapplying a mathematical formula or making calculation errors

Skipping the "Understand the Problem" step entirely

Ignoring the need to look back and verify the answer

All of the above

Checking the final answer for accuracy

Identifying the key information and restating the problem in your own words

Plugging in numbers without reading the question carefully

Skipping to a solution because the problem seems familiar

It helps break the problem into smaller, manageable parts.

It allows students to skip difficult problems.

It ensures that every solution is correct the first time.

It prevents students from thinking creatively.

Finding an entirely new way to solve the problem

Reviewing the solution to check for errors and alternative methods

Guessing if the answer is correct without verifying

Moving on to the next problem without reflecting

22

23

24

25

8 CM

10 CM

12 CM

16 CM

4

5

6

7

30°, 45°, 105°

20°, 30°, 130°

40°, 60°, 80°

36°, 54°, 90°

28 and 7

24 and 6

36 and 9

40 and 10

20

22

24

26

Mathematics is an art and a process of thinking.

Mathematics is a study of humanity.

Mathematics deals with the logic of shape, quantity, and arrangement.

Mathematics is immensely useful, practical, and powerful. 

spirals on a sunflower

a mountain range

number of petals on a daisy

pinecone spiral

Pattern

Rhythm

Sequence

Symmetry

Pattern of Flow

Pattern of Movement

Pattern of Rhythm

Pattern of Visuals

19

23

27

31

56

192

768

1023

13

46

72

97

Arithmetic Sequence

Geometric Sequence

Harmonic Sequence

Fibonacci Sequence

They decrease.

They increase.

They remain constant

They alternate between increasing and decreasing.

A sequence of random numbers

A consistent, systematic arrangement of numbers or objects

A series of unrelated mathematical operations

An irregular arrangement of shapes

Add the two previous terms

Multiply the two previous terms

Subtract the two previous terms

Divide the two previous terms

8

13

21

34

4.96

2.0161

2.1016

4.69

236

336

-146

-164

Leonardo Da Vinci

Leonardo DiCaprio

Leonardo Pisa

Leonardo Pisano