Category: JinuAcademy.com

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JinuAcademy.com Curriculum

目標: 60万、毎9年 유지、10번 재도전

For your 150-issue-per-grade format, I would organize each course into about 10 units × 15 issues:

https://chatgpt.com/share/6a81d41e-6750-83e8-8d21-ecd628f2846b

GRADE 5 – FOUNDATIONS OF MATHEMATICS

Unit 1: Place Value and Whole Numbers
Unit 2: Multiplication and Division
Unit 3: Fraction Concepts
Unit 4: Fraction Operations
Unit 5: Decimals
Unit 6: Numerical Expressions and Patterns
Unit 7: Measurement and Unit Conversion
Unit 8: Geometry and Coordinate Plane
Unit 9: Volume
Unit 10: Data Analysis and Problem Solving
GRADE 6 – MATHEMATICS & PROBLEM SOLVING

Unit 1: Ratios
Unit 2: Rates and Unit Rates
Unit 3: Fractions, Decimals, and Percents
Unit 4: Integers and Rational Numbers
Unit 5: Expressions
Unit 6: Equations and Inequalities
Unit 7: Coordinate Geometry
Unit 8: Area, Surface Area, and Volume
Unit 9: Statistics
Unit 10: Mathematical Modeling and Problem Solving
GRADE 7 – PRE-ALGEBRA 

Unit 1: Rational Number Operations
Unit 2: Ratios and Proportional Relationships
Unit 3: Percent Applications
Unit 4: Algebraic Expressions
Unit 5: Equations
Unit 6: Inequalities
Unit 7: Linear Relationships
Unit 8: Geometry and Scale
Unit 9: Probability
Unit 10: Statistics and Modeling
GRADE 8 – ALGEBRA I

Unit 1: Algebraic Foundations
Unit 2: Linear Equations
Unit 3: Linear Inequalities
Unit 4: Graphing Linear Functions
Unit 5: Systems of Equations
Unit 6: Exponents and Exponential Functions
Unit 7: Polynomials
Unit 8: Factoring
Unit 9: Quadratic Equations and Functions
Unit 10: Data Analysis and Mathematical Modeling
GRADE 9 – GEOMETRY

Unit 1: Foundations of Geometry
Unit 2: Logic and Proof
Unit 3: Transformations
Unit 4: Congruent Triangles
Unit 5: Similarity
Unit 6: Polygons and Quadrilaterals
Unit 7: Right Triangles and Trigonometry
Unit 8: Circles
Unit 9: Area, Surface Area, and Volume
Unit 10: Coordinate Geometry and Applications
GRADE 10 – ALGEBRA II

Unit 1: Functions and Their Graphs
Unit 2: Linear Systems
Unit 3: Quadratic Functions
Unit 4: Polynomial Functions
Unit 5: Rational Expressions and Functions
Unit 6: Radical Functions
Unit 7: Complex Numbers
Unit 8: Exponential and Logarithmic Functions
Unit 9: Sequences and Series
Unit 10: Probability, Statistics, and Modeling
GRADE 11 – PRECALCULUS

Unit 1: Advanced Function Analysis
Unit 2: Polynomial Functions
Unit 3: Rational Functions
Unit 4: Exponential and Logarithmic Functions
Unit 5: Trigonometric Functions
Unit 6: Trigonometric Identities and Equations
Unit 7: Analytic Trigonometry and Applications
Unit 8: Vectors, Matrices, and Parametric Equations
Unit 9: Conic Sections and Polar Coordinates
Unit 10: Sequences, Series, and Introduction to Limits
GRADE 12 – CALCULUS

Unit 1: Functions and Precalculus Review
Unit 2: Limits
Unit 3: Continuity
Unit 4: Derivatives
Unit 5: Derivative Rules
Unit 6: Applications of Derivatives
Unit 7: Optimization and Related Rates
Unit 8: Integration
Unit 9: Fundamental Theorem of Calculus
Unit 10: Differential Equations and Applications
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Regular Math Issue 001: Place Value

Lesson Focus

This issue practices Place Value in Regular Math.

Synopsis

Place value tells you the value of a digit based on where it appears in a number. In our base-ten number system, each position is ten times the value of the position immediately to its right. For example, in 4,582, the digit 4 represents 4,000, the digit 5 represents 500, the digit 8 represents 80, and the digit 2 represents 2. Understanding this structure helps you read large numbers accurately and recognize the value represented by each digit.

In this issue, you practice identifying ones, tens, hundreds, thousands, and larger place values. You also use place value to interpret numbers and distinguish between a digit itself and the value that digit represents. These skills form an important foundation for comparing numbers, rounding, performing arithmetic, and working with decimals later in mathematics.

Practice Problems

Problem 1. In the number 48,326, what is the value of the digit 8?

A. 8

B. 80

C. 800

D. 8,000

E. 80,000

Problem 2. Which digit is in the hundreds place in 7,294?

A. 7

B. 2

C. 9

D. 4

E. 0

Problem 3. What is the value of the digit 5 in 35,718?

A. 5

B. 50

C. 500

D. 5,000

E. 50,000

Problem 4. In 602,451, the digit 6 is in which place?

A. hundreds

B. thousands

C. ten-thousands

D. hundred-thousands

E. millions

Problem 5. Which number has a 3 in the thousands place?

A. 432

B. 3,210

C. 12,345

D. 23,100

E. 130

Problem 6. What is the value of the 9 in 91,408?

A. 9

B. 90

C. 900

D. 9,000

E. 90,000

Problem 7. In 5,064, which digit is in the tens place?

A. 5

B. 0

C. 6

D. 4

E. 1

Google Form

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Answer Key

  1. Problem 1: D
  2. Problem 2: C
  3. Problem 3: D
  4. Problem 4: D
  5. Problem 5: B
  6. Problem 6: E
  7. Problem 7: C
Posted on

Regular Math Issue 002: Reading and Writing Whole Numbers

Lesson Focus

This issue practices Reading and Writing Whole Numbers in Regular Math.

Synopsis

Whole numbers can be represented in several useful forms. Standard form writes a number using digits, word form expresses the number in words, and expanded form shows the value contributed by each nonzero digit. For example, 6,304 can be written as “six thousand three hundred four” or as 6,000 + 300 + 4. Moving between these forms strengthens your understanding of how our base-ten number system is organized.

In this issue, you practice reading whole numbers correctly and translating numbers between standard, word, and expanded forms. Pay close attention to the position of each digit, including zeros that serve as placeholders. By becoming comfortable with different representations of the same number, you build skills that will help with estimation, comparison, arithmetic, and more advanced numerical reasoning.

Practice Problems

Problem 1. How is 4,305 written in words?

A. four hundred thirty-five

B. four thousand three hundred five

C. four thousand thirty-five

D. forty-three thousand five

E. four thousand five hundred three

Problem 2. Which number is six thousand two hundred nineteen?

A. 6,219

B. 6,291

C. 62,019

D. 621

E. 6,029

Problem 3. Write 80,407 in expanded form.

A. 80,000 + 400 + 7

B. 8,000 + 400 + 7

C. 80,000 + 4,000 + 7

D. 80,000 + 40 + 7

E. 800,000 + 400 + 7

Problem 4. Which word form matches 12,050?

A. twelve thousand five

B. twelve thousand fifty

C. one thousand two hundred fifty

D. twelve hundred fifty

E. one hundred twenty thousand fifty

Problem 5. Which standard form matches thirty-five thousand eight?

A. 35,800

B. 35,080

C. 35,008

D. 3,508

E. 350,008

Problem 6. Which number is written correctly with commas?

A. 12345

B. 1,2345

C. 12,345

D. 123,45

E. 1234,5

Problem 7. What is 7,000 + 600 + 20 + 9 in standard form?

A. 7,629

B. 76,029

C. 762

D. 7,269

E. 7,602

Google Form

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Answer Key

  1. Problem 1: B
  2. Problem 2: A
  3. Problem 3: A
  4. Problem 4: B
  5. Problem 5: C
  6. Problem 6: C
  7. Problem 7: A
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Regular Math Issue 003: Comparing Whole Numbers

Lesson Focus

This issue practices Comparing Whole Numbers in Regular Math.

Synopsis

Comparing whole numbers means determining whether one number is greater than, less than, or equal to another. A reliable method is to compare the number of digits first and then, when the numbers have the same number of digits, compare corresponding digits from left to right. The first place where the digits differ determines which number is larger. The symbols >, <, and = are used to record these relationships.

In this issue, you practice comparing pairs of whole numbers and identifying their relative sizes. You use place-value reasoning rather than relying only on how a number looks. This skill is essential for ordering numbers, interpreting data, estimating quantities, and deciding whether numerical answers are reasonable.

Practice Problems

Problem 1. Which symbol makes this true: 4,582 ___ 4,528?

A. <

B. >

C. =

D. ≤

E. +

Problem 2. Which number is greatest?

A. 8,104

B. 8,014

C. 8,140

D. 8,041

E. 8,004

Problem 3. Which number is least?

A. 12,300

B. 12,030

C. 12,003

D. 12,330

E. 12,303

Problem 4. Compare 56,781 and 56,871.

A. 56,781 > 56,871

B. 56,781 < 56,871

C. 56,781 = 56,871

D. 56,781 + 56,871

E. Cannot compare

Problem 5. Which list is in order from least to greatest?

A. 3,102; 3,120; 3,210

B. 3,210; 3,120; 3,102

C. 3,120; 3,102; 3,210

D. 3,102; 3,210; 3,120

E. 3,210; 3,102; 3,120

Problem 6. Which number is between 6,430 and 6,450?

A. 6,405

B. 6,429

C. 6,441

D. 6,501

E. 6,399

Problem 7. Which statement is true?

A. 9,099 > 9,900

B. 7,070 < 7,007

C. 10,001 > 9,999

D. 5,555 < 5,055

E. 8,080 = 8,008

Google Form

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Answer Key

  1. Problem 1: B
  2. Problem 2: C
  3. Problem 3: C
  4. Problem 4: B
  5. Problem 5: A
  6. Problem 6: C
  7. Problem 7: C
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Pre-Algebra Issue 001: Reviewing Whole Numbers

Lesson Focus

This issue practices Reviewing Whole Numbers in Pre-Algebra.

Synopsis

Whole numbers are the nonnegative counting numbers 0, 1, 2, 3, and so on. Before beginning more advanced pre-algebra topics, it is important to be comfortable reading, comparing, rounding, and performing basic operations with whole numbers. These familiar skills become the numerical foundation for expressions, equations, variables, and problem solving.

In this issue, you review key ideas involving whole numbers and apply them in a variety of short problems. You may need to recognize number properties, identify even or odd numbers, use divisibility ideas, compare values, or round to a specified place. The goal is to strengthen number sense so that later algebraic work can focus on new concepts rather than basic numerical difficulties.

Practice Problems

Problem 1. What is 4,000 + 300 + 20 + 5?

A. 4,325

B. 4,235

C. 43,025

D. 4,305

E. 432

Problem 2. Which digit is in the thousands place in 28,614?

A. 2

B. 8

C. 6

D. 1

E. 4

Problem 3. What is 735 + 268?

A. 993

B. 1,003

C. 1,013

D. 903

E. 1,103

Problem 4. What is 5,000 – 1,875?

A. 3,125

B. 3,225

C. 4,125

D. 2,875

E. 3,075

Problem 5. Which number is even?

A. 341

B. 527

C. 608

D. 719

E. 935

Problem 6. Which number is divisible by 10?

A. 405

B. 410

C. 411

D. 419

E. 421

Problem 7. Round 6,482 to the nearest hundred.

A. 6,400

B. 6,500

C. 6,480

D. 6,000

E. 7,000

Google Form

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Answer Key

  1. Problem 1: A
  2. Problem 2: B
  3. Problem 3: B
  4. Problem 4: A
  5. Problem 5: C
  6. Problem 6: B
  7. Problem 7: B
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Pre-Algebra Issue 002: Place Value Review

Lesson Focus

This issue practices Place Value Review in Pre-Algebra.

Synopsis

Place value describes how the position of a digit determines its value. In a base-ten system, moving one place to the left multiplies a digit’s value by ten, while moving one place to the right divides its place value by ten. For example, the 7 in 72,409 represents 70,000 because it is located in the ten-thousands place. Expanded form makes this structure visible by separating a number into the values of its individual digits.

In this issue, you review how to identify place positions, determine the value represented by a digit, and convert between standard and expanded forms. You also consider the role of zero as a placeholder. A strong understanding of place value supports later work with decimals, scientific notation, estimation, and algebraic expressions involving powers of ten.

Practice Problems

Problem 1. What is the value of the 7 in 72,409?

A. 7

B. 70

C. 700

D. 7,000

E. 70,000

Problem 2. Which digit is in the ten-thousands place in 156,230?

A. 1

B. 5

C. 6

D. 2

E. 3

Problem 3. Write 9,000 + 80 + 4 in standard form.

A. 9,084

B. 9,804

C. 9,840

D. 984

E. 90,804

Problem 4. In 43,912, the digit 3 is in which place?

A. ones

B. tens

C. hundreds

D. thousands

E. ten-thousands

Problem 5. Which number has a 6 in the hundreds place?

A. 6,120

B. 4,681

C. 2,604

D. 9,061

E. 16,002

Problem 6. What is the value of 0 in 8,052?

A. 0

B. 5

C. 50

D. 500

E. Cannot be determined

Problem 7. Which expanded form equals 31,507?

A. 30,000 + 1,000 + 500 + 7

B. 30,000 + 1,000 + 50 + 7

C. 3,000 + 100 + 500 + 7

D. 31,000 + 50 + 7

E. 30,000 + 1,500 + 7

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Answer Key

  1. Problem 1: E
  2. Problem 2: B
  3. Problem 3: A
  4. Problem 4: D
  5. Problem 5: B
  6. Problem 6: A
  7. Problem 7: A
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Pre-Algebra Issue 003: Comparing & Ordering Numbers

Lesson Focus

This issue practices Comparing & Ordering Numbers in Pre-Algebra.

Synopsis

Comparing and ordering numbers requires you to understand the value of each digit and examine numbers systematically. For whole numbers, numbers with more digits are generally larger; when two numbers have the same number of digits, compare them from the greatest place value toward the right. Ordering extends this process by arranging several numbers from least to greatest or greatest to least.

In this issue, you practice using place value and comparison symbols to determine relationships among numbers. You also organize sets of numbers into the correct sequence. These skills strengthen number sense and prepare you for comparing integers, fractions, decimals, and algebraic quantities in later pre-algebra lessons.

Practice Problems

Problem 1. Which is greatest?

A. 4,308

B. 4,380

C. 4,083

D. 4,803

E. 4,038

Problem 2. Which is least?

A. 9,701

B. 9,170

C. 9,017

D. 9,710

E. 9,071

Problem 3. Put in order from least to greatest: 802, 820, 208.

A. 802, 820, 208

B. 208, 802, 820

C. 820, 802, 208

D. 208, 820, 802

E. 802, 208, 820

Problem 4. Which symbol makes this true: 15,208 ___ 15,280?

A. <

B. >

C. =

D. +

E. ×

Problem 5. Which number is between 3,100 and 3,200?

A. 3,020

B. 3,099

C. 3,150

D. 3,250

E. 3,300

Problem 6. Order from greatest to least: 6,010; 6,100; 6,001.

A. 6,001; 6,010; 6,100

B. 6,100; 6,010; 6,001

C. 6,010; 6,100; 6,001

D. 6,100; 6,001; 6,010

E. 6,010; 6,001; 6,100

Problem 7. Which statement is true?

A. 2,456 > 2,546

B. 8,008 < 8,080

C. 9,900 < 9,090

D. 1,111 = 1,101

E. 7,070 > 7,700

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Answer Key

  1. Problem 1: D
  2. Problem 2: C
  3. Problem 3: B
  4. Problem 4: A
  5. Problem 5: C
  6. Problem 6: B
  7. Problem 7: B
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Algebra I Issue 001: Evaluating Numerical Expressions

Lesson Focus

This issue practices Evaluating Numerical Expressions in Algebra I.

Synopsis

A numerical expression is a mathematical phrase made from numbers and operation symbols but does not contain an equals sign. Evaluating an expression means finding its numerical value by carrying out the indicated operations correctly. Expressions may include addition, subtraction, multiplication, division, exponents, and grouping symbols such as parentheses.

In this issue, you practice evaluating numerical expressions carefully and accurately. You identify the operations that must be performed and work through them in the proper sequence. Developing this skill is essential because algebraic expressions follow the same operational rules; later, numbers will often be replaced by variables whose values must be substituted before an expression is evaluated.

Practice Problems

Problem 1. Evaluate [latex]3 + 4 × 2[/latex].

A. [latex]14[/latex]

B. [latex]11[/latex]

C. [latex]10[/latex]

D. [latex]7[/latex]

E. [latex]24[/latex]

Problem 2. Evaluate [latex]18 ÷ 3 + 5[/latex].

A. [latex]3[/latex]

B. [latex]6[/latex]

C. [latex]11[/latex]

D. [latex]15[/latex]

E. [latex]30[/latex]

Problem 3. Evaluate [latex]2^3 + 4[/latex].

A. [latex]10[/latex]

B. [latex]12[/latex]

C. [latex]14[/latex]

D. [latex]16[/latex]

E. [latex]20[/latex]

Problem 4. Evaluate [latex](6 + 2) × 3[/latex].

A. [latex]14[/latex]

B. [latex]18[/latex]

C. [latex]21[/latex]

D. [latex]24[/latex]

E. [latex]30[/latex]

Problem 5. Evaluate [latex]5(4 – 1)[/latex].

A. [latex]9[/latex]

B. [latex]12[/latex]

C. [latex]15[/latex]

D. [latex]18[/latex]

E. [latex]20[/latex]

Problem 6. Evaluate [latex]7^2 – 9[/latex].

A. [latex]40[/latex]

B. [latex]42[/latex]

C. [latex]49[/latex]

D. [latex]56[/latex]

E. [latex]58[/latex]

Problem 7. Evaluate [latex]30 – 12 ÷ 3[/latex].

A. [latex]6[/latex]

B. [latex]18[/latex]

C. [latex]24[/latex]

D. [latex]26[/latex]

E. [latex]54[/latex]

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Answer Key

  1. Problem 1: B
  2. Problem 2: C
  3. Problem 3: B
  4. Problem 4: D
  5. Problem 5: C
  6. Problem 6: A
  7. Problem 7: D
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Algebra I Issue 002: Order of Operations (PEMDAS)

Lesson Focus

This issue practices Order of Operations (PEMDAS) in Algebra I.

Synopsis

When an expression contains more than one operation, the order of operations provides a consistent set of rules for deciding what to calculate first. A common memory aid is PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, followed by Addition and Subtraction from left to right. Multiplication does not automatically come before division, and addition does not automatically come before subtraction when they appear at the same level.

In this issue, you apply the order of operations to evaluate expressions step by step. You learn to respect grouping symbols, simplify powers, and then perform multiplication, division, addition, and subtraction in the correct order. Accurate use of these rules is fundamental throughout algebra because even a small change in operation order can produce a completely different result.

Practice Problems

Problem 1. Which operation should be done first in [latex]8 + 3 × 5[/latex]?

A. addition

B. multiplication

C. subtraction

D. division

E. comparison

Problem 2. Evaluate [latex]6 + 2^3[/latex].

A. [latex]14[/latex]

B. [latex]16[/latex]

C. [latex]64[/latex]

D. [latex]512[/latex]

E. [latex]10[/latex]

Problem 3. Evaluate [latex](10 – 4)^2[/latex].

A. [latex]12[/latex]

B. [latex]24[/latex]

C. [latex]30[/latex]

D. [latex]36[/latex]

E. [latex]96[/latex]

Problem 4. Evaluate [latex]20 ÷ 5 × 2[/latex].

A. [latex]2[/latex]

B. [latex]4[/latex]

C. [latex]8[/latex]

D. [latex]10[/latex]

E. [latex]20[/latex]

Problem 5. Evaluate [latex]3 + 18 ÷ (2 + 4)[/latex].

A. [latex]6[/latex]

B. [latex]9[/latex]

C. [latex]12[/latex]

D. [latex]21[/latex]

E. [latex]24[/latex]

Problem 6. Evaluate [latex]4 × 3^2[/latex].

A. [latex]24[/latex]

B. [latex]36[/latex]

C. [latex]64[/latex]

D. [latex]144[/latex]

E. [latex]48[/latex]

Problem 7. Evaluate [latex]7 + 5(6 – 2)[/latex].

A. [latex]27[/latex]

B. [latex]48[/latex]

C. [latex]75[/latex]

D. [latex]19[/latex]

E. [latex]35[/latex]

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Answer Key

  1. Problem 1: B
  2. Problem 2: A
  3. Problem 3: D
  4. Problem 4: C
  5. Problem 5: A
  6. Problem 6: B
  7. Problem 7: A
Posted on

Algebra I Issue 003: Properties of Real Numbers

Lesson Focus

This issue practices Properties of Real Numbers in Algebra I.

Synopsis

The real number system follows several properties that allow expressions to be rewritten without changing their value. The commutative properties allow the order of addition or multiplication to change, while the associative properties allow numbers to be regrouped. The distributive property connects multiplication with addition or subtraction. Identity and inverse properties describe special relationships involving 0, 1, opposites, and reciprocals.

In this issue, you practice recognizing and applying these properties to numerical and algebraic expressions. Rather than memorizing names alone, focus on what each property permits you to do. These rules provide the logical foundation for simplifying expressions, combining like terms, factoring, solving equations, and explaining why algebraic transformations are valid.

Practice Problems

Problem 1. Which shows the commutative property of addition?

A. [latex]a + b = b + a[/latex]

B. [latex]a(b + c) = ab + ac[/latex]

C. [latex]a + 0 = a[/latex]

D. [latex]a × 1 = a[/latex]

E. [latex]a + (b + c) = (a + b) + c[/latex]

Problem 2. Which shows the distributive property?

A. [latex]a + b = b + a[/latex]

B. [latex]a(b + c) = ab + ac[/latex]

C. [latex]a + 0 = a[/latex]

D. [latex]a × 1 = a[/latex]

E. [latex]ab = ba[/latex]

Problem 3. What is the additive identity?

A. 0

B. 1

C. -1

D. a

E. 10

Problem 4. What is the multiplicative identity?

A. 0

B. 1

C. -1

D. a

E. 10

Problem 5. Simplify using the distributive property: [latex]3(x + 4)[/latex].

A. [latex]3x + 4[/latex]

B. [latex]x + 12[/latex]

C. [latex]3x + 12[/latex]

D. [latex]7x[/latex]

E. [latex]12x[/latex]

Problem 6. Which property is shown by [latex](2 + 3) + 4 = 2 + (3 + 4)[/latex]?

A. commutative

B. associative

C. distributive

D. identity

E. inverse

Problem 7. Which is always true for real numbers [latex]a[/latex] and [latex]b[/latex]?

A. [latex]a – b = b – a[/latex]

B. [latex]a ÷ b = b ÷ a[/latex]

C. [latex]a + b = b + a[/latex]

D. [latex]a + b = ab[/latex]

E. [latex]a^2 = a[/latex]

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Answer Key

  1. Problem 1: A
  2. Problem 2: B
  3. Problem 3: A
  4. Problem 4: B
  5. Problem 5: C
  6. Problem 6: B
  7. Problem 7: C
Posted on

Geometry Issue 001: What Is Geometry? Logic & Reasoning

Lesson Focus

This issue practices What Is Geometry? Logic & Reasoning in Geometry.

Synopsis

Geometry is the study of shapes, sizes, positions, distances, and spatial relationships. It begins with basic objects such as points, lines, planes, segments, and angles, but geometric problem solving also depends heavily on logical reasoning. You often start with known facts or definitions and use them to reach a justified conclusion.

In this issue, you are introduced to geometry as a system of mathematical reasoning. You practice distinguishing observations from conclusions and thinking about how statements can be supported by definitions, patterns, or established facts. Building clear reasoning habits at the beginning of geometry prepares you for conjectures, conditional statements, deductive arguments, and formal proofs later in the course.

Practice Problems

Problem 1. Geometry often uses logic to move from facts to conclusions. What is this process called?

A. estimation

B. deductive reasoning

C. rounding

D. measurement

E. random guessing

Problem 2. A statement accepted as true without proof is called a ____.

A. variable

B. postulate

C. counterexample

D. decimal

E. quotient

Problem 3. Which is an example of a geometric object?

A. triangle

B. verb

C. paragraph

D. continent

E. ingredient

Problem 4. A conclusion based on observed evidence but not yet proved is called a ____.

A. conjecture

B. product

C. sum

D. denominator

E. axis

Problem 5. What can disprove a conjecture?

A. one counterexample

B. ten examples that support it

C. a longer sentence

D. a ruler

E. a diagram only

Problem 6. Which sentence is a statement?

A. Close the door.

B. Is it raining?

C. [latex]7 + 5 = 12[/latex]

D. What time is it?

E. Please draw a line.

Problem 7. Why is reasoning important in geometry?

A. It replaces all diagrams.

B. It helps justify conclusions.

C. It avoids all calculations.

D. It makes numbers unnecessary.

E. It changes definitions.

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Answer Key

  1. Problem 1: B
  2. Problem 2: B
  3. Problem 3: A
  4. Problem 4: A
  5. Problem 5: A
  6. Problem 6: C
  7. Problem 7: B
Posted on

Geometry Issue 002: Recognizing Patterns

Lesson Focus

This issue practices Recognizing Patterns in Geometry.

Synopsis

Recognizing patterns is an important part of mathematical reasoning. A pattern may appear in a sequence of numbers, a collection of shapes, or a repeated geometric relationship. By examining what changes and what stays the same, you can identify a possible rule and use it to predict what comes next.

In this issue, you practice observing examples carefully, describing patterns, and extending them logically. Pattern recognition often leads to inductive reasoning, in which several specific observations suggest a broader conclusion. Although a pattern can provide strong evidence for a conjecture, later geometry lessons will show why additional reasoning or proof may be needed to establish that a statement is always true.

Practice Problems

Problem 1. What is the next number in the pattern 3, 7, 11, 15, …?

A. 17

B. 18

C. 19

D. 20

E. 21

Problem 2. What is the next square number after 1, 4, 9, 16?

A. 20

B. 21

C. 24

D. 25

E. 36

Problem 3. In a pattern, what should you look for?

A. only what changes

B. only what stays the same

C. what changes and what stays the same

D. only the largest number

E. only the first term

Problem 4. What is the next number in 2, 4, 8, 16, …?

A. 18

B. 20

C. 24

D. 32

E. 64

Problem 5. Which pattern adds 5 each time?

A. 1, 2, 4, 8

B. 3, 8, 13, 18

C. 10, 8, 6, 4

D. 2, 6, 12, 20

E. 5, 10, 20, 40

Problem 6. What is the next number in 50, 45, 40, 35, …?

A. 25

B. 28

C. 30

D. 32

E. 40

Problem 7. A pattern can help form a ____.

A. conjecture

B. fraction

C. ruler

D. circle

E. coefficient

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Answer Key

  1. Problem 1: C
  2. Problem 2: D
  3. Problem 3: C
  4. Problem 4: D
  5. Problem 5: B
  6. Problem 6: C
  7. Problem 7: A
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Geometry Issue 003: Making Conjectures

Lesson Focus

This issue practices Making Conjectures in Geometry.

Synopsis

A conjecture is a mathematical statement that you believe is true based on observations, examples, or patterns. Mathematicians often use inductive reasoning to form conjectures by studying several specific cases and identifying a consistent relationship. A good conjecture clearly states the general rule suggested by the evidence.

In this issue, you practice moving from observed examples to reasonable general statements. You also learn an important limitation: many supporting examples do not prove that a conjecture is always true. A single counterexample can show that a conjecture is false. This distinction between forming a conjecture and proving a statement is central to geometric reasoning and prepares you for deductive proof.

Practice Problems

Problem 1. A conjecture is best described as a ____.

A. proved theorem

B. careless mistake

C. reasonable guess based on evidence

D. calculator answer

E. definition

Problem 2. What can show a conjecture is false?

A. counterexample

B. another supporting example

C. a title

D. a longer proof

E. a variable name

Problem 3. You test several rectangles and find equal diagonals. What may you make?

A. a conjecture

B. a quotient

C. a percent

D. an equation only

E. a table of contents

Problem 4. Which is a counterexample to ‘All even numbers are multiples of 4’?

A. 4

B. 8

C. 12

D. 14

E. 20

Problem 5. Which statement is a conjecture?

A. The next term may be 21.

B. The answer key is printed.

C. A triangle has three sides by definition.

D. The page number is 2.

E. The word geometry has 8 letters.

Problem 6. Why do examples not prove a conjecture true for all cases?

A. Examples are never useful.

B. They may not cover every possible case.

C. Examples are always false.

D. They are not numbers.

E. They must be decimals.

Problem 7. What is the role of evidence in making a conjecture?

A. It supports a reasonable prediction.

B. It guarantees a theorem.

C. It removes all need for proof.

D. It changes the problem.

E. It replaces logic.

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Answer Key

  1. Problem 1: C
  2. Problem 2: A
  3. Problem 3: A
  4. Problem 4: D
  5. Problem 5: A
  6. Problem 6: B
  7. Problem 7: A
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Algebra II Issue 001: Review of Algebraic Expressions

Lesson Focus

This issue practices Review of Algebraic Expressions in Algebra II.

Synopsis

Algebraic expressions combine numbers, variables, operation symbols, and sometimes exponents or grouping symbols. To work effectively in Algebra II, you need to recognize the parts of an expression and simplify it using established algebraic rules. This may involve evaluating expressions after substituting values, combining like terms, applying the distributive property, and following the order of operations.

In this issue, you review the core expression skills developed in earlier algebra courses. The emphasis is on performing valid transformations while preserving the value of the original expression. A strong command of algebraic expressions is necessary for later work with equations, inequalities, polynomials, rational expressions, exponential functions, logarithms, and other Algebra II topics.

Practice Problems

Problem 1. Simplify [latex]3x + 5x[/latex].

A. [latex]8[/latex]

B. [latex]8x[/latex]

C. [latex]15x[/latex]

D. [latex]2x[/latex]

E. [latex]x^8[/latex]

Problem 2. Simplify [latex]4a – a[/latex].

A. [latex]3a[/latex]

B. 4

C. [latex]5a[/latex]

D. [latex]a^4[/latex]

E. 0

Problem 3. Which terms are like terms?

A. [latex]3x[/latex] and [latex]4y[/latex]

B. [latex]2a[/latex] and [latex]5a[/latex]

C. [latex]x[/latex] and [latex]x^2[/latex]

D. 7 and [latex]7b[/latex]

E. [latex]m[/latex] and [latex]n[/latex]

Problem 4. Simplify [latex]2(x + 6)[/latex].

A. [latex]2x + 6[/latex]

B. [latex]x + 12[/latex]

C. [latex]2x + 12[/latex]

D. [latex]12x[/latex]

E. [latex]8x[/latex]

Problem 5. Evaluate [latex]3x + 2[/latex] when [latex]x = 4[/latex].

A. 9

B. 10

C. 12

D. 14

E. 20

Problem 6. Simplify [latex]5y + 2 – 3y[/latex].

A. [latex]2y + 2[/latex]

B. [latex]8y + 2[/latex]

C. [latex]2y – 2[/latex]

D. [latex]10y[/latex]

E. [latex]5y[/latex]

Problem 7. Which expression is equivalent to [latex]x + x + x[/latex]?

A. [latex]x^3[/latex]

B. [latex]3x[/latex]

C. [latex]x + 3[/latex]

D. [latex]3 + x[/latex]

E. [latex]x/3[/latex]

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Answer Key

  1. Problem 1: B
  2. Problem 2: A
  3. Problem 3: B
  4. Problem 4: C
  5. Problem 5: D
  6. Problem 6: A
  7. Problem 7: B
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Algebra II Issue 002: Properties of Real Numbers

Lesson Focus

This issue practices Properties of Real Numbers in Algebra II.

Synopsis

The properties of real numbers explain why many common algebraic manipulations are valid. Commutative and associative properties govern the order and grouping of addition and multiplication. The distributive property allows multiplication to be applied across a sum or difference. Identity and inverse properties help simplify expressions involving 0, 1, opposites, and reciprocals.

In this issue, you review these properties at an Algebra II level and use them to analyze or rewrite expressions efficiently. Understanding the underlying property is more powerful than memorizing a procedure because it helps you choose valid steps in unfamiliar problems. These ideas support equation solving, factoring, polynomial operations, function manipulation, and many other advanced algebraic techniques.

Practice Problems

Problem 1. Which shows the commutative property of addition?

A. [latex]a + b = b + a[/latex]

B. [latex]a(b + c) = ab + ac[/latex]

C. [latex]a + 0 = a[/latex]

D. [latex]a × 1 = a[/latex]

E. [latex]a + (b + c) = (a + b) + c[/latex]

Problem 2. Which shows the distributive property?

A. [latex]a + b = b + a[/latex]

B. [latex]a(b + c) = ab + ac[/latex]

C. [latex]a + 0 = a[/latex]

D. [latex]a × 1 = a[/latex]

E. [latex]ab = ba[/latex]

Problem 3. What is the additive identity?

A. 0

B. 1

C. -1

D. a

E. 10

Problem 4. What is the multiplicative identity?

A. 0

B. 1

C. -1

D. a

E. 10

Problem 5. Simplify using the distributive property: [latex]3(x + 4)[/latex].

A. [latex]3x + 4[/latex]

B. [latex]x + 12[/latex]

C. [latex]3x + 12[/latex]

D. [latex]7x[/latex]

E. [latex]12x[/latex]

Problem 6. Which property is shown by [latex](2 + 3) + 4 = 2 + (3 + 4)[/latex]?

A. commutative

B. associative

C. distributive

D. identity

E. inverse

Problem 7. Which is always true for real numbers [latex]a[/latex] and [latex]b[/latex]?

A. [latex]a – b = b – a[/latex]

B. [latex]a ÷ b = b ÷ a[/latex]

C. [latex]a + b = b + a[/latex]

D. [latex]a + b = ab[/latex]

E. [latex]a^2 = a[/latex]

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Answer Key

  1. Problem 1: A
  2. Problem 2: B
  3. Problem 3: A
  4. Problem 4: B
  5. Problem 5: C
  6. Problem 6: B
  7. Problem 7: C
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Algebra II Issue 003: Solving Linear Equations Review

Lesson Focus

This issue practices Solving Linear Equations Review in Algebra II.

Synopsis

A linear equation is an equation in which the variable has a first-degree power, and solving it means finding the value that makes the equation true. The central principle is to preserve equality by performing equivalent operations on both sides. Depending on the equation, you may simplify expressions, use the distributive property, combine like terms, and apply inverse operations to isolate the variable.

In this issue, you review solving one-step and multi-step linear equations. You work carefully through each transformation and can check a solution by substituting it back into the original equation. These equation-solving habits are essential in Algebra II, where linear techniques are frequently used inside more complex problems involving formulas, systems, functions, inequalities, and nonlinear equations.

Practice Problems

Problem 1. Solve [latex]x + 7 = 15[/latex].

A. [latex]x = 6[/latex]

B. [latex]x = 7[/latex]

C. [latex]x = 8[/latex]

D. [latex]x = 9[/latex]

E. [latex]x = 22[/latex]

Problem 2. Solve [latex]3x = 21[/latex].

A. [latex]x = 6[/latex]

B. [latex]x = 7[/latex]

C. [latex]x = 8[/latex]

D. [latex]x = 18[/latex]

E. [latex]x = 24[/latex]

Problem 3. Solve [latex]x – 5 = 12[/latex].

A. [latex]x = 7[/latex]

B. [latex]x = 12[/latex]

C. [latex]x = 17[/latex]

D. [latex]x = 60[/latex]

E. [latex]x = -7[/latex]

Problem 4. Solve [latex]x/4 = 9[/latex].

A. [latex]x = 13[/latex]

B. [latex]x = 36[/latex]

C. [latex]x = 5[/latex]

D. [latex]x = 2.25[/latex]

E. [latex]x = 45[/latex]

Problem 5. Solve [latex]2x + 3 = 11[/latex].

A. [latex]x = 3[/latex]

B. [latex]x = 4[/latex]

C. [latex]x = 5[/latex]

D. [latex]x = 7[/latex]

E. [latex]x = 8[/latex]

Problem 6. Solve [latex]5x – 10 = 20[/latex].

A. [latex]x = 2[/latex]

B. [latex]x = 4[/latex]

C. [latex]x = 6[/latex]

D. [latex]x = 10[/latex]

E. [latex]x = 30[/latex]

Problem 7. Which step solves [latex]x – 9 = 14[/latex]?

A. subtract 9

B. add 9

C. multiply by 9

D. divide by 9

E. square both sides

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Answer Key

  1. Problem 1: C
  2. Problem 2: B
  3. Problem 3: C
  4. Problem 4: B
  5. Problem 5: B
  6. Problem 6: C
  7. Problem 7: B
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