Solution of Area on the Coordinate Plane (Questions published on Dec 8, 2025)

Area on the Coordinate Plane

Full Step-by-Step Solutions (Problems 1–15)

Working conventions / reminders

• For an equation of the form \(|x-h|+|y-k|=R\) the graph is a diamond (rotated square) with area \(2R^2\).
• For \(a|x-h| + b|y-k| = c\): horizontal half-width \(=c/a\), vertical half-height \(=c/b\). Diagonals: \(d_x=2c/a,\ d_y=2c/b\). Area \(=\dfrac{d_x d_y}{2}.\)
• Area of a circle: \(A=\pi r^2\). Distance from \((x_0,y_0)\) to line \(Ax+By+C=0\) is \(\dfrac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}\).

Solutions

1. Problem 1.
   \(y = -|x+4| + 7\) and \(y = |x+4| - 5\). Find the enclosed area.
   1. Set the two expressions equal to find intersection points: \[ -|x+4|+7 = |x+4|-5 \Longrightarrow 12 = 2|x+4| \Longrightarrow |x+4|=6. \] So \(x+4 = \pm 6 \Rightarrow x = -10,\; x = 2.\)
   2. At those \(x\)-values, compute \(y\): \(y = |x+4|-5 = 6-5 = 1\). So intersections are \((-10,1)\) and \((2,1)\).
   3. The top vertex of the top curve \(y=-|x+4|+7\) is at \(x=-4\), \(y=7\). The bottom vertex of the lower curve \(y=|x+4|-5\) is at \((-4,-5)\).
   4. Horizontal diagonal length \(=2-(-10)=12\). Vertical diagonal length \(=7-(-5)=12\).
   Area \(=\dfrac{12\cdot12}{2}=72.\) \(\boxed{72}\)
2. Problem 2.
   \(y = 12 - |x-6|\). Find the area of the region enclosed by the V.
   1. The vertex is at \((6,12)\). Find x-intercepts where \(y=0\): \[ 12 - |x-6| = 0 \Rightarrow |x-6| = 12 \Rightarrow x = -6,\; x=18. \]
   2. This forms an isosceles triangle with base length \(18-(-6)=24\) and height \(12\).
   Area \(=\tfrac12\cdot 24\cdot 12 = 144.\) \(\boxed{144}\)
3. Problem 3.
   \(|x-8| + |y-1| = 10.\)
   This is a diamond centered at \((8,1)\) with parameter \(R=10\). Formula: area \(=2R^2\).
   Area \(=2\cdot 10^2 = 200.\) \(\boxed{200}\)
4. Problem 4.
   \(3|x| + 2|y| = 18.\)
   1. When \(y=0\): \(3|x|=18\Rightarrow|x|=6\) so x-intercepts \((\pm6,0)\) ⇒ horizontal diagonal \(=12\).
   2. When \(x=0\): \(2|y|=18\Rightarrow|y|=9\) so y-intercepts \((0,\pm9)\) ⇒ vertical diagonal \(=18\).
   Area \(=\dfrac{12\cdot18}{2}=108.\) \(\boxed{108}\)
5. Problem 5.
   \(4|x-2| + |y+3| = 20.\) Find the absolute difference between the diagonals.
   1. Horizontal diagonal: set \(y=-3\) so \(|y+3|=0\Rightarrow 4|x-2|=20\Rightarrow |x-2|=5\). Solutions \(x=-3,7\). Horizontal diagonal \(=7-(-3)=10\).
   2. Vertical diagonal: set \(x=2\) so \(|x-2|=0\Rightarrow |y+3|=20\Rightarrow y+3=\pm20\Rightarrow y=17\) or \(-23\). Vertical diagonal \(=17-(-23)=40\).
   Absolute difference \(=|40-10|=30.\) \(\boxed{30}\)
6. Problem 6.
   \((x-4)^2 + (y-9)^2 = 25.\) Area?
   Radius \(r=\sqrt{25}=5\). Area of circle \(A=\pi r^2=\pi\cdot5^2=25\pi\).
   \(\boxed{25\pi}\)
7. Problem 7.
   \(|x-a| + |y-2| = b\) has area \(72\). Find \(b\).
   Area of such a diamond \(= 2b^2\). So \(2b^2 = 72 \Rightarrow b^2 = 36 \Rightarrow b = 6\).
   \(\boxed{6}\)
8. Problem 8.
   \(6|x+1| + 3|y-5| = k\) and area \(=54\). Find \(k\).
   Use formula: for \(a|x-h|+b|y-k|=c\), area \(= \dfrac{2c^2}{ab}\).
   Here \(a=6,\;b=3,\;c=k\). So area \(=\dfrac{2k^2}{6\cdot3}=\dfrac{2k^2}{18}=\dfrac{k^2}{9}.\)
   Set \(\dfrac{k^2}{9}=54 \Rightarrow k^2 = 486 = 81\cdot6 \Rightarrow k = \sqrt{486} = 9\sqrt{6}\).
   \(\boxed{9\sqrt{6}}\)
9. Problem 9.
   \(7|x| + 24|y| = 84.\) A circle is inscribed — find its area.
   Consider the first-quadrant boundary line: \(7x + 24y = 84\). The inscribed circle will be tangent to that line and centered at the origin \((0,0)\).
   Distance from origin to line \(7x+24y-84=0\) is \[ r = \frac{| -84 |}{\sqrt{7^2 + 24^2}} = \frac{84}{25}. \]

   The distance from the center to each side is the radius of the inscribed circle. For the line \(7|x| + 24|y| = 84\), one side in the first quadrant is \(\displaystyle 7x + 24y = 84\). The distance from the origin to this line is: \[ r = \frac{84}{\sqrt{7^{2} + 24^{2}}} = \frac{84}{25} = \frac{84}{25}. \] Thus the area of the inscribed circle is: \[ \text{Area} = \pi r^{2} = \pi \left(\frac{84}{25}\right)^{2} = \frac{7056}{625}\pi. \]

   .
   \(\boxed{\dfrac{7056\pi}{625}}\)
10. Problem 10.
    \((x-3)^2 + (y+4)^2 = 32.\) A square is inscribed in the circle — find the area of the square.
    Radius \(r=\sqrt{32}=4\sqrt{2}\). Diameter \(=2r=8\sqrt{2}\) equals the square's diagonal.
    Area of square \(= \dfrac{\text{(diagonal)}^2}{2} = \dfrac{(8\sqrt{2})^2}{2} = 64.\)
    \(\boxed{64}\)
11. Problem 11.
    \(y = |x+10| - 3\) and \(y = -|x+10| + 9.\) Find enclosed area.
    Set equal: \(|x+10|-3 = -|x+10| + 9 \Rightarrow 2|x+10| = 12 \Rightarrow |x+10| = 6.\)
    So \(x+10 = \pm 6 \Rightarrow x = -16,\; -4\). At these points \(y = 6 - 3 = 3\). Intersection points \((-16,3)\) and \((-4,3)\).
    Vertex of top curve is \((-10,9)\). Vertex of bottom curve is \((-10,-3)\). Vertical diagonal \(=12\). Horizontal diagonal \(=12\).
    Area \(=\dfrac{12\cdot12}{2}=72.\) \(\boxed{72}\)
12. Problem 12.
    \(|x-2| + 3|y| = 15.\) Find the area.
    Horizontal half-width \(=15\) ⇒ diagonal \(=30\). Vertical half-height \(=5\) ⇒ diagonal \(=10\).
    Area \(=\dfrac{30\cdot10}{2} = 150.\) \(\boxed{150}\)
13. Problem 13.
    \(2|x-1| + 5|y+6| = p\). Diagonals differ by 6 — find \(p\).
    Horizontal diagonal \(d_x = p\).
    Vertical diagonal \(d_y = \frac{2p}{5}\).
    \(|d_x - d_y| = 6 \Rightarrow \left| p - \frac{2p}{5} \right| = 6 \Rightarrow \frac{3p}{5} = 6 \Rightarrow p = 10.\)
    \(\boxed{10}\)
14. Problem 14.
    \((x+12)^2 + (y-1)^2 = 18.\) Area of a regular hexagon inscribed in this circle?
    Radius \(r=\sqrt{18}=3\sqrt{2}\). For a regular hexagon, side length \(s=r\).
    Area \(= \dfrac{3\sqrt{3}}{2} r^2.\)
    \[ A = \dfrac{3\sqrt{3}}{2}\cdot 18 = 27\sqrt{3}. \]
    \(\boxed{27\sqrt{3}}\)
15. Problem 15.
    \(|x| + |y| = 14\). A rectangle with sides parallel to axes is inscribed. Maximize its area.
    Use first-quadrant point with \(x+y=14\).
    Area \(=4xy\).
    Max when \(x=y=7\).
    Maximum area \(=196.\) \(\boxed{196}\)

Answer summary (quick reference)

1. 72
2. 144
3. 200
4. 108
5. 30
6. \(25\pi\)
7. 6
8. \(9\sqrt{6}\)
9. \(\dfrac{7056\pi}{625}\)
10. 64
11. 72
12. 150
13. 10
14. 27\(\sqrt{3}\)
15. 196

Hope it helps.

Competitive exam || Area on the Coordinate Plane || Practice Set

Area on the Coordinate Plane

Practice Set — Absolute Values, Circles & Inscribed Shapes

Instructions

• You may use scratch paper and a calculator where appropriate.
• Show all steps — for absolute-value regions, identify vertices or diagonals; for circles give radius/diameter; for inscribed shapes justify your geometry.
• Answers requiring \(\pi\) may be left in terms of \(\pi\).

Problems (1–15)

1. What is the area, in square units, enclosed by \[ y = -|x+4| + 7 \quad\text{and}\quad y = |x+4| - 5 ? \]
2. What is the area of the region enclosed by \[ y = 12 - |x-6|. \] (Interpret the V-shape and the x-intercepts.)
3. What is the area of the region described by \[ |x-8| + |y-1| = 10\; ? \]
4. Find the area of the region enclosed by \[ 3|x| + 2|y| = 18. \]
5. What is the absolute difference between the lengths of the diagonals of the region \[ 4|x-2| + |y+3| = 20\; ? \]
6. What is the area, in square units, enclosed by \[ (x-4)^2 + (y-9)^2 = 25 ? \] Express your answer in terms of \(\pi\).
7. The area enclosed by \[ |x-a| + |y-2| = b \] is \(72\) square units. Find \(b\).
8. If the region enclosed by \[ 6|x+1| + 3|y-5| = k \] has area \(54\), find the value of \(k\).
9. A circle is inscribed in the figure \[ 7|x| + 24|y| = 84. \] What is the area of the circle? Express your answer as a common fraction in terms of \(\pi\).
10. A square is inscribed in the circle \[ (x-3)^2 + (y+4)^2 = 32. \] What is the area of the square?
11. What is the area of the region enclosed by \[ y = |x+10| - 3 \quad\text{and}\quad y = -|x+10| + 9 ? \]
12. What is the area, in square units, of the region enclosed by \[ |x-2| + 3|y| = 15\; ? \]
13. A diamond described by \[ 2|x-1| + 5|y+6| = p \] has diagonals whose lengths differ by \(6\). Find \(p\).
14. The graph \[ (x+12)^2 + (y-1)^2 = 18 \] is a circle. What is the area of a regular hexagon inscribed in this circle? (Leave your answer in terms of \(\pi\).)
15. A rectangle is inscribed in the region \[ |x| + |y| = 14, \] with its sides parallel to the coordinate axes. What is the maximum possible area of such a rectangle? Explain your reasoning.

Hints & Helpful Formulas

• For \(|x-h| + |y-k| = R\) the graph is a diamond (a rotated square) whose area is \(2R^2\).
• For \(a|x-h| + b|y-k| = c\), the diamond's horizontal half-width is \(c/a\) and vertical half-height is \(c/b\). Full diagonals: \(d_x = 2c/a,\ d_y = 2c/b.\) Area \(= \dfrac{d_x d_y}{2}.\)
• Distance from origin to line \(Ax+By+C=0\) is \(\dfrac{|C|}{\sqrt{A^2+B^2}}\) when the line is written appropriately; use this to compute an inscribed circle's radius.
• Area of an inscribed square in a circle of radius \(r\): diagonal \(=2r\) so area \(= \dfrac{(2r)^2}{2} = 2r^2.\)
• Area of a regular hexagon inscribed in a circle of radius \(r\): \( \dfrac{3\sqrt{3}}{2}r^2\). (Useful for problem 14.)

Use the hints if you get stuck, but show the full steps in your written work.

Created for practice — absolute-value regions, circles, and inscribed figures. If you'd like, I can also generate a printable PDF, provide a worked answer key, or produce step-by-step solutions for each problem.

IB MYP 2 Mathematics || Summative Assessment || Topic: Expressions & Algebraic Sequences || Criteria A & B

IB MYP 2 Mathematics

Summative Assessment – Algebraic Expressions & Sequences

Instructions

• You have 60 minutes to complete this assessment.
• Show all your working clearly and explain your thinking using correct mathematical language, symbols, and steps.
• Use full sentences when describing patterns, rules, or calculations.
• This task assesses both Criterion A (Knowing and Understanding) and Criterion B (Expression).

Part A – Criterion A: Knowing and Understanding

1. Definitions and Understanding:
   In your own words, define the following and provide one numerical example for each:
   • Arithmetic Sequence
   • Geometric Sequence
   • Algebraic Expression
   • Term-to-Term Rule
2. Identifying Sequences:
   For each sequence below, state whether it is arithmetic, geometric, or neither, and explain your reasoning:
   • 2, 5, 8, 11, …
   • 3, 6, 12, 24, …
   • 1, 2, 4, 8, 16, 31, …
3. Finding the nth Term:
   For the sequences identified as arithmetic or geometric above, write an algebraic expression for the \(n^{th}\) term.

Part B – Criterion B: Expression

1. Simplifying Expressions:
   Simplify the following expressions step by step:
   • \( 4x + 7x - 5 \)
   • \( 3(a + 5) + 2a \)
   • \( 6y - 2(3y - 4) \)
2. Applying Sequences:
   Using the algebraic expressions you wrote for the nth term in Part A:
   • Find the 10th term of the arithmetic sequence: 2, 5, 8, 11, …
   • Find the 6th term of the geometric sequence: 3, 6, 12, 24, …
3. Explaining Reasoning:
   For at least one arithmetic and one geometric sequence, explain:
   • How you determined the sequence type
   • How you derived the nth term formula
   • How you used the formula to calculate a specific term
4. Create Your Own Sequence and Expression:
   Write an arithmetic or geometric sequence of at least 5 terms. Then:
   • Describe the rule in words
   • Write the nth term formula
   • Find the 7th term using your formula

Rubrics

Criterion A – Knowing and Understanding

Level	Description
0	The student does not reach a standard described by any of the descriptors below.
1–2	Limited understanding of mathematical concepts and sequences; explanations are mostly unclear.
3–4	Some understanding of concepts; explanations are partially correct and sometimes clear.
5–6	Good understanding of concepts; explanations are mostly correct and clear.
7–8	Thorough understanding of concepts; explanations are clear, precise, and detailed.

Criterion B – Expression

Level	Description
0	The student does not reach a standard described by any of the descriptors below.
1–2	Communicates mathematical ideas with limited clarity; little use of correct notation or reasoning.
3–4	Communicates ideas with some clarity; some correct use of notation and reasoning.
5–6	Mathematical ideas are clearly expressed; reasoning is mostly logical and notation is mostly correct.
7–8	Mathematical ideas are communicated effectively and coherently; reasoning is logical and detailed with consistent correct notation.

IB MYP 2 Mathematics || Criterion B || Expression through Algebraic Sequences

IB MYP 2 Mathematics

Criterion B Task – Expression: Algebraic Sequences

Instructions

• You have 50 minutes to complete this task.
• Show all your working and explain your thinking using correct mathematical language, symbols, and steps.
• Use full sentences when describing patterns, rules, and formulas.

Task: Algebraic Sequences

1. Part A – Recognizing Patterns:
   The following sequences are given. Determine the rule for the \(n^{th}\) term in words and in algebraic form:
   • Sequence 1: 3, 7, 11, 15, …
   • Sequence 2: 2, 6, 18, 54, …
   • Sequence 3: 5, 9, 14, 20, …
2. Part B – Finding Terms:
   Using your rules from Part A, find the following:
   • The 10th term of Sequence 1
   • The 5th term of Sequence 2
   • The 8th term of Sequence 3
3. Part C – Explaining Your Reasoning:
   For each sequence, explain:
   • How you determined the type of sequence (arithmetic or geometric)
   • How you derived the algebraic expression for the \(n^{th}\) term
   • How you used the expression to calculate specific terms
4. Part D – Create Your Own Sequence:
   Write your own arithmetic or geometric sequence with at least 5 terms. Then:
   • Describe the rule in words
   • Write the algebraic formula for the \(n^{th}\) term
   • Find the 6th term using your formula

Criterion B Rubric – Expression

Level	Description
0	The student does not reach a standard described by any of the descriptors below.
1–2	Communicates mathematical ideas with limited clarity. Little use of correct notation or reasoning.
3–4	Communicates mathematical ideas with some clarity. Some correct use of notation and reasoning.
5–6	Mathematical ideas are clearly expressed. Procedures and reasoning are mostly correct and logical.
7–8	Mathematical ideas are communicated effectively and coherently. Procedures are correct and reasoning is logical and detailed, with consistent use of proper notation.

IB MYP 2 Mathematics Assessment || Criterion B || Algebra - Expression

IB MYP 2 Mathematics

Criterion B Task – Expression: Communicating Mathematical Ideas

Instructions

• You have 50 minutes to complete this task.
• Show all your working and explain your thinking using correct mathematical language, symbols, and steps.
• Use full sentences when explaining your reasoning, especially for word problems and formulas.

Task: Expression of Mathematical Ideas

1. Part A – Simplifying Expressions:
   Simplify the following expressions and show all steps:
   • \( 3x + 5x - 7 \)
   • \( 2(a + 3) + 4a \)
   • \( 5y - 3(2y - 4) \)
2. Part B – Translating Word Problems:
   Write an algebraic expression for each situation:
   • A rectangle has a length that is 3 meters longer than twice its width \(w\). Write an expression for the perimeter.
   • Sarah has \(n\) marbles. She buys 5 more packs of 4 marbles each. Write an expression for her total number of marbles.
3. Part C – Patterns and Formulas:
   The sequence of numbers is: 2, 5, 8, 11, …
   • Write an expression for the \(n^{th}\) term.
   • Find the 10th term using your expression.
4. Part D – Reasoning and Explanation:
   Show all steps in your calculations and explain your thinking clearly. Use proper mathematical notation.

Criterion B Rubric – Expression

Level	Description
0	The student does not reach a standard described by any of the descriptors below.
1–2	Communicates mathematical ideas with limited clarity. Little use of correct notation or reasoning.
3–4	Communicates mathematical ideas with some clarity. Some correct use of notation and reasoning.
5–6	Mathematical ideas are clearly expressed. Procedures and reasoning are mostly correct and logical.
7–8	Mathematical ideas are communicated effectively and coherently. Procedures are correct and reasoning is logical and detailed, with consistent use of proper notation.

IB MYP 2 Math Task || Topic: Number Operation Properties || Criterion C – Communicating

IB MYP 2 Mathematics

Criterion C Task – Communicating: Number Operation Properties

Instructions

• You have 50 minutes to complete this task.
• Show all your working and explain your thinking using correct mathematical language, symbols, and steps.
• Use full sentences when giving definitions and justifications.

Task: Understanding and Applying Number Properties

1. **Definitions:** In your own words, define the following properties and give one numerical example for each:
   • Commutative Property (Addition and Multiplication)
   • Associative Property (Addition and Multiplication)
   • Distributive Property
   • Identity Property (Addition and Multiplication)
   • Inverse Property (Addition and Multiplication)
2. **Identifying Properties:** For each expression below, state the property being used and explain why.
   • \( 6 + 9 = 9 + 6 \)
   • \( (4 \times 5) \times 2 = 4 \times (5 \times 2) \)
   • \( 3(4 + 7) = 3 \times 4 + 3 \times 7 \)
   • \( 12 + 0 = 12 \)
   • \( \frac{5}{6} \times \frac{6}{5} = 1 \)
3. **Explain the Error:** A student says that:
   
   \( 5 + (3 + 2) = (5 + 3) \times 2 \)
   
   Is this correct? Justify your answer using properties and show the correct evaluation of both sides.
4. **Create Your Own:** Write two of your own examples demonstrating:
   • One correct use of the distributive property
   • One incorrect use of any property (with explanation of the mistake)
   For each example, write a full explanation of what property is shown and why it is applied correctly or incorrectly.

Criterion C Rubric – Communicating

Level	Description
0	The student does not reach a standard described by any of the descriptors below.
1–2	Communicates mathematical ideas with limited clarity. Little use of appropriate terminology or representation.
3–4	Communicates mathematical ideas with some clarity. Some use of appropriate terminology and representation.
5–6	Communicates mathematical ideas clearly. Good use of appropriate terminology, representations, and logical steps.
7–8	Communicates mathematical ideas effectively and coherently, with consistent and accurate use of mathematical language and structure.

IB MYP 2 Math Task || Topic : Order of Operations || Criterion A (Knowing and Understanding) and Criterion B (Investigating Patterns)

IB MYP 2 Mathematics

Order of Operations – Criterion A & B Tasks

Criterion A: Knowing and Understanding

Topic: Order of Operations

Task: Answer the following questions to demonstrate your understanding of the order of operations rules.

1. Define the order of operations and explain why it is important in mathematics.
2. Simplify the expression:
   
   \( 7 + 2 \times 5 - (3^2 - 4) \)
3. Evaluate:
   
   \( (6 + 4) \div 2^2 + 3 \times 2 \)

Be sure to write your answer clearly and explain each step in your calculations.

Criterion B: Investigating Patterns

Topic: Order of Operations

Task: Investigate the pattern in the sequence of expressions below and answer the questions that follow.

1. Consider the sequence of expressions:
   
   \[ \begin{aligned} &E_1 = 1 + 2 \times 3 \\ &E_2 = (1 + 2) \times 3 \\ &E_3 = 1 + (2 \times 3) \\ &E_4 = (1 + 2 \times 3) \\ &E_5 = ((1 + 2) \times 3) \end{aligned} \] Calculate the value of each expression \( E_1 \) to \( E_5 \).
2. Describe how the placement of brackets affects the value of each expression.
3. Predict the value of the expression:
   
   \( E_6 = 1 + 2 \times 3 - 4 \)
   Then, write two different versions of this expression with brackets placed differently that would change the value, and calculate those values.
4. Reflect on how understanding the order of operations helps you make sense of these patterns.

IB MYP 2 Math Assessment || Topic : Order of Operations || Criterion C – Communicating & Criterion D: Applying Mathematics in Real-Life Contexts

IB MYP 2 Mathematics

Order of Operations – Criterion C & D Tasks

Criterion C: Communicating Mathematical Ideas

Topic: Order of Operations

Task: Simplify the following expressions step-by-step, showing all your working and explaining the order of operations you apply.

1. Simplify:
   
   \( 5 + 3 \times (8 - 4)^2 \div 2 \)
2. Simplify:
   
   \( (12 \div 3 + 7) \times (5 - 2)^2 - 4 \)
3. Simplify:
   
   \( 18 - 6 \times 2 + (3^3 \div 9) \)

Instructions:

• Write down each step clearly, explaining which operation you are doing and why, following the order of operations rules.
• Use correct mathematical notation and terminology (e.g., brackets, indices, multiplication, division, addition, subtraction).

Criterion D: Applying Mathematics in Real-Life Contexts

Topic: Order of Operations

Real-Life Scenario:

You are budgeting for a birthday party. You plan to buy:

• 4 boxes of cupcakes at $6.25 each
• 2 large pizzas, priced at $15 each, with a 25% discount on one pizza
• 5 liters of juice at $2.40 per liter
• Sales tax of 7%

1. Write a mathematical expression that calculates the total cost before tax, using appropriate operations and brackets.
2. Calculate the total cost including the 7% sales tax. Show all your working clearly and explain how you apply the order of operations.
3. If your budget for the party is $75, state whether you are within budget and justify your answer with calculations.
4. Reflect in 2–3 sentences on why it is important to use the correct order of operations in budgeting real-life expenses.

Assessment Rubrics

Criterion C: Communicating Mathematical Ideas

Level	Description
0	The student does not reach a standard described by any descriptors below.
1–2	Attempts to communicate mathematical ideas with limited clarity and accuracy.
3–4	Communicates mathematical ideas with some clarity and partial accuracy, using some appropriate terminology.
5–6	Communicates mathematical ideas clearly and accurately, using appropriate terminology and notation.
7–8	Communicates mathematical ideas precisely and coherently, with clear, accurate use of terminology and notation throughout.

Criterion D: Applying Mathematics in Real-Life Contexts

Level	Description
0	The student does not reach a standard described by any descriptors below.
1–2	Applies mathematical strategies with limited effectiveness to a real-life context.
3–4	Applies appropriate mathematical strategies to a real-life context with partial accuracy.
5–6	Effectively applies relevant mathematical strategies with mostly accurate results and some justification.
7–8	Consistently applies relevant, sophisticated mathematical strategies with accurate results and clear justification linked to the real-life context.

IB MYP 2 Math Task Sheet: Order of Operations || Criterion D: Applying Mathematics in Real-Life Contexts

IB MYP 2 Mathematics – Criterion D Task

Topic: Order of Operations

Criterion D: Applying Mathematics in Real-Life Contexts

Real-Life Context: Supermarket Shopping

You have a budget of $100 for your shopping. You plan to buy the following items:

• 3 packs of juice boxes, each costing $4.50
• 2 family pizzas, on sale as Buy 1 Get 1 50% off (original price: $11.00 each)
• 4 bags of apples, each weighing 1.5 kg and costing $2.60 per kg
• A 10% discount voucher applies only after calculating total before tax
• Sales tax is 8%

Task Instructions

1. Write a single mathematical expression using numbers, operations, and brackets to calculate your total bill before tax and discount.
2. Apply the order of operations correctly to calculate the total cost after applying the 10% discount and then the 8% sales tax.
3. Show all your steps clearly, explaining why you perform each operation in that order.
4. State whether your total cost is within your $100 budget.
5. Write a brief reflection (2–3 sentences) on how using the correct order of operations helps in budgeting or real-life purchases.

Sample Expression (for reference)

For example, the expression before discount and tax could look like this:
\(3 \times 4.50 + (11 + 11 \times 0.5) + (4 \times 1.5 \times 2.60)\)
Then apply discount and tax in correct order.

Extension (Optional Challenge)

Suppose you want to buy 2 movie tickets after shopping. Each ticket costs $12.75.

1. Recalculate your total cost including movie tickets.
2. Decide if your budget is sufficient.
3. If over budget, suggest how you could adjust your shopping to stay within $100.

Assessment Rubric – Criterion D

Level	Description
0	The student does not reach a standard described by any of the descriptors below.
1–2	The student applies mathematical strategies to some extent, with limited effectiveness in a real-life situation.
3–4	The student appropriately selects and applies mathematical strategies, providing a partially accurate solution to a real-life situation.
5–6	The student effectively applies relevant mathematical strategies and gives a mostly accurate solution with some justification.
7–8	The student consistently applies relevant, sophisticated mathematical strategies to reach a valid, accurate solution and provides a clear, well-justified explanation connected to the real-life context.

IB MYP 2 Math Task Sheet: Order of Operations || Criterion C – Communicating

MYP 2 Mathematics – Task Sheet

Topic: Order of Operations

Level: MYP Year 2 (Grade 7)

Assessment Focus: Criterion C – Communicating

Global Context

Scientific and Technical Innovation: How can we ensure accurate results in scientific and real-life calculations?

Learning Objectives

• Use and explain the correct order of operations (BIDMAS/PEMDAS).
• Communicate mathematical steps clearly and logically using appropriate notation.

Part A: Solve and Show Your Work

Solve each expression using the correct order of operations. Show ALL steps clearly.

1. \( 8 + 6 \div 2 \times (3 + 1) \)
2. \( (5 + 3)^2 - 4 \times 2 \)
3. \( 18 \div 3 + 2 \times (4 + 5) \)
4. \( [2 + 3 \times (6 - 4)]^2 \div 5 \)
5. \( 7 + \{8 - [3 + 2 \times (1 + 1)]\} \times 2 \)

Part B: Explain Your Reasoning

Choose one problem from Part A and explain your reasoning in full sentences. Example:

“First, I solved the brackets because they come first in the order of operations. Then I did the multiplication before the addition…”

Criterion C – Rubric (Communicating)

Level	Descriptor
0	The student does not reach a standard described by any of the descriptors below.
1–2	The student attempts to organize work using a limited logical structure. Attempts to use some appropriate mathematical language or representation.
3–4	The student organizes work using a logical structure. Uses appropriate mathematical language and representation.
5–6	The student organizes work using a coherent structure and appropriate mathematical language. Uses a variety of representations effectively.
7–8	The student organizes work using a complete, coherent structure. Uses accurate and appropriate mathematical language. Uses varied, appropriate representations and includes clear explanations.

Challenge (Optional)

Write your own multi-step expression that uses brackets, exponents, multiplication/division, and addition/subtraction. Solve it and explain why the order matters.

IBDP AAHL Mathematics Worksheet || Functions, Sequences, Algebra, Trigonometry, Differentiation, Vectors, Complex Numbers

DP Mathematics AA HL – Practice Set

Note: This worksheet excludes Integration, Statistics, and Probability.

🔢 Section A: Algebra & Functions

1. Solve for \( x \in \mathbb{R} \):
   
   \( \log_2(x^2 - 5x + 6) = 1 \)
2. Given \( f(x) = \dfrac{2x + 3}{x - 1} \):
   (a) Find the inverse function \( f^{-1}(x) \)
   (b) State the domain and range of \( f(x) \)
3. Let \( f(x) = ax^3 + bx^2 + cx + d \), with a local maximum at \( x = 1 \) and an inflection point at \( x = 2 \).
   Given \( f(1) = 4 \), find \( a, b, c, d \).

🔁 Section B: Sequences & Series

4. Given \( u_n = 2^n + 3n \), find the value of \( \sum_{n=1}^{6} u_n \).
5. The first three terms of a geometric sequence are \( x, 2x - 3, 4x - 6 \).
   Find the value of \( x \) and the common ratio \( r \).

📐 Section C: Trigonometry

6. Solve for \( \theta \in [0^\circ, 360^\circ] \):
   
   \( 2\sin\theta - \sqrt{3} = 0 \)
7. Prove the identity:
   
   \( \dfrac{1 + \tan^2x}{1 - \tan^2x} = \sec(2x) \)
8. The angle of elevation from the ground to a tower is \( 30^\circ \).
   After walking 50 m closer, the angle becomes \( 45^\circ \).
   Find the height of the tower.

✏️ Section D: Differentiation

9. Differentiate:
   
   \( f(x) = \dfrac{x^2 \sin x}{e^x} \)
10. Given \( f'(x) = 6x^2 - 18x \), find the stationary points and determine their nature.
11. Find the derivative of:
    
    \( f(x) = \ln\left(\sqrt{x^2 + 4}\right) \)

🧭 Section E: Vectors

12. Let \( \vec{a} = (3, -1, 2), \vec{b} = (1, 4, -2) \):
    
    (a) Find \( \vec{a} \cdot \vec{b} \)
    (b) Find the angle between vectors \( \vec{a} \) and \( \vec{b} \)
13. A line passes through point \( A(1, 2, -1) \) with direction vector \( \vec{d} = (2, -1, 3) \).
    Find the Cartesian equation of the line.

🔮 Section F: Complex Numbers

14. Solve for \( z \in \mathbb{C} \):
    
    \( z^2 + (3 - 2i)z + (4 + i) = 0 \)
15. Express \( \dfrac{3 + 4i}{1 - 2i} \) in the form \( a + bi \), where \( a, b \in \mathbb{R} \).

📌 Instructions

• Show all working clearly and neatly.
• Use GDC for verification only—full reasoning must be shown.
• Use correct mathematical notation at all times.

IB Mathematics AA HL – Derivatives: Product, Quotient & Chain Rule Practice Set

IB Mathematics AA HL – Derivatives Practice Set

Topic: Product Rule, Quotient Rule, Chain Rule

Level: Higher Level (AA HL)

Total Questions: 14

Section A: Short Questions (Fundamental Applications of Rules)

Apply the appropriate differentiation rule: product, quotient, or chain rule. Give your answers in simplified form.

1. Differentiate the function:
   
   \( f(x) = (3x^2 + 5)(\sin x) \)
2. Find \( \dfrac{dy}{dx} \) if
   
   \( y = x^2 e^x \)
3. Differentiate the function:
   
   \( f(x) = \ln x \cdot \tan x \)
4. Differentiate:
   
   \( f(x) = \dfrac{x^2 + 1}{\sqrt{x}} \)
5. Find the derivative of:
   
   \( y = \dfrac{\sin x}{x^2 + 1} \)
6. If \( f(x) = \dfrac{\ln x}{x^3} \), find \( f'(x) \)
7. Differentiate the following function:
   
   \( f(x) = \cos(3x^2 + 2) \)
8. Find \( \dfrac{dy}{dx} \) if
   
   \( y = \sqrt{1 + \tan x} \)
9. Differentiate:
   
   \( y = e^{(x^2 + 4x)} \)
10. Let \( f(x) = \ln(\sqrt{1 + e^x}) \). Find \( f'(x) \)

Section B: Extended Response Questions (IB-Style)

These problems require extended reasoning and clear, structured solutions. Use correct mathematical notation throughout.

11. Let \( f(x) = \sin(x^2 \cdot e^x) \)
    
    (a) Find \( f'(x) \).
    (b) Comment on the effect of the chain rule in this derivative.
12. The curve is defined implicitly by the equation:
    
    \( x^2 y + y^3 = \tan x \)
    
    Find \( \dfrac{dy}{dx} \) in terms of \( x \) and \( y \), using implicit differentiation.
13. Let \( f(x) = \dfrac{\ln(\sin x)}{x^2 + 1} \), where \( 0 < x < \pi \).
    
    (a) Find \( f'(x) \).
    (b) Hence, find the equation of the tangent to the curve at \( x = \dfrac{\pi}{4} \)
14. A balloon rises vertically such that its height in meters at time \( t \) seconds is given by:
    
    \( h(t) = 50 \ln(2t + 1) \)
    
    A camera is placed 100 meters horizontally from the point where the balloon was released. Let \( D(t) \) be the distance between the camera and the balloon.
    
    (a) Express \( D(t) \) in terms of \( t \).
    (b) Find \( \dfrac{dD}{dt} \) when \( t = 3 \), giving appropriate units.
    (c) Interpret your result in the context of the motion of the balloon.

IBDP Mathematics - Analysis and Approaches (HL) || Unit 1 || Topic 2 : Problem in Exponent and Logarithm

 

International Baccalaureate Diploma Program (IBDP) - Analysis and Approaches (AAHL) || Unit Circle and Trigonometric Ratios Beyond Acute Angles

Unit Circle and Trigonometric Ratios Beyond Acute Angles - Worksheet

Name: ___________________

Date: ___________________

Instructions:

• Answer the following questions.
• Show all your work, including any necessary diagrams.
• Ensure you clearly explain your steps where applicable.

Questions

1. Evaluate the following expressions using the unit circle:
   • a) \( \sin\left(\frac{5\pi}{4}\right) \)
   • b) \( \cos\left(\frac{7\pi}{3}\right) \)
2. Given \( \theta = \frac{5\pi}{6} \), find the exact values of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \) using the unit circle.
3. Find the exact value of \( \csc\left(\frac{3\pi}{2}\right) \).
4. The point \( P \) lies on the unit circle at an angle of \( \theta = \frac{4\pi}{3} \). What are the coordinates of \( P \)?
5. Prove the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \) using the unit circle.
6. For \( \theta = \frac{5\pi}{3} \), determine the value of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \). Then, verify that \( \sin(\theta) \) and \( \cos(\theta) \) satisfy the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
7. Given \( \tan(\theta) = -\frac{1}{\sqrt{3}} \), and \( \theta \) is in the fourth quadrant, find the exact values of \( \sin(\theta) \) and \( \cos(\theta) \).
8. A point \( P \) on the unit circle forms an angle of \( \theta = \frac{7\pi}{6} \). Determine the reference angle for \( \theta \), and find the exact values of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \).
9. A pendulum swings back and forth following the sine function \( y(t) = 5 \sin\left( \frac{\pi}{2} t \right) \), where \( t \) is in seconds. Find the period of the pendulum’s motion and the amplitude of the sine wave.
10. Using the unit circle, determine the exact value of \( \tan\left(\frac{11\pi}{6}\right) \) and explain your reasoning.

Bonus Question (Optional)

1. Show that \( \cos(\theta + \pi) = -\cos(\theta) \) using the unit circle, and use this identity to find \( \cos\left( \frac{5\pi}{3} \right) \).

End of Worksheet

IBDP AAHL/AASL - Understanding Unit circle in Trigonometry

 

Understanding the Unit Circle in Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. One of the fundamental concepts in trigonometry is the unit circle, a powerful tool that provides a visual way to understand trigonometric functions. Whether you're new to trigonometry or looking to deepen your understanding, the unit circle is essential for working with sine, cosine, and tangent functions.

In this blog, we’ll explore what the unit circle is, how it’s constructed, and how it connects to trigonometric functions.

What is the Unit Circle?

The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) of a coordinate plane. In the Cartesian coordinate system, the equation for a circle is:

x² + y² = r²

For a unit circle, r = 1, so the equation becomes:

x² + y² = 1

This circle plays a crucial role in defining the six main trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) and helps us understand their values for any given angle.

The Basics of the Unit Circle

Imagine a point on the unit circle that makes an angle θ with the positive x-axis. The position of this point is determined by two coordinates: x and y.

• The x-coordinate of the point corresponds to the cosine of the angle θ.
• The y-coordinate corresponds to the sine of the angle θ.

In other words, for any angle θ, the coordinates of the point on the unit circle are:

(cos(θ), sin(θ))

This relationship is key to understanding the values of sine and cosine for any angle. The unit circle provides us with a way to easily visualize these values without relying on memorizing individual values for each angle.

Understanding Angles on the Unit Circle

Angles on the unit circle are typically measured in radians, though degrees can also be used. Here’s how radians and degrees relate to each other:

• 360° = 2π radians
• 180° = π radians
• 90° = π/2 radians

On the unit circle, we typically work with positive angles in the counterclockwise direction and negative angles in the clockwise direction. The most common angles you’ll encounter on the unit circle include 0°, 90°, 180°, and 270°, as well as their corresponding radian measures.

Key Points on the Unit Circle

Let’s now examine some key angles and their coordinates on the unit circle, as well as their sine and cosine values:

• At 0° (or 0 radians):
  • Coordinates: (1, 0)
  • cos(0) = 1, sin(0) = 0
• At 90° (or π/2 radians):
  • Coordinates: (0, 1)
  • cos(π/2) = 0, sin(π/2) = 1
• At 180° (or π radians):
  • Coordinates: (-1, 0)
  • cos(π) = -1, sin(π) = 0
• At 270° (or 3π/2 radians):
  • Coordinates: (0, -1)
  • cos(3π/2) = 0, sin(3π/2) = -1

These are just a few of the important points on the unit circle, but the circle contains infinitely many points corresponding to every possible angle.

Trigonometric Functions and the Unit Circle

Once we understand how the coordinates of the unit circle relate to sine and cosine, we can move on to the other trigonometric functions, which are also based on the unit circle.

• Tangent: The tangent of an angle θ is the ratio of the sine to the cosine:

  tan(θ) = sin(θ) / cos(θ)

• Cosecant: The cosecant function is the reciprocal of the sine:

  csc(θ) = 1 / sin(θ)

• Secant: The secant function is the reciprocal of the cosine:

  sec(θ) = 1 / cos(θ)

• Cotangent: The cotangent function is the reciprocal of the tangent:

  cot(θ) = 1 / tan(θ)

These additional functions are essential when solving trigonometric equations or understanding complex wave patterns in fields such as physics and engineering.

Why is the Unit Circle So Important?

The unit circle provides a way to visualize and understand trigonometric functions for any angle, not just the angles commonly found in right triangles. Here’s why it’s crucial:

• Simplifies Trigonometric Calculations: Instead of memorizing trigonometric values for specific angles, the unit circle allows you to compute these values easily by finding the corresponding point on the circle.
• Helps with Graphing: The unit circle helps us understand the behavior of trigonometric functions, especially when graphing them. The periodic nature of these functions becomes clear when you see the repeated patterns on the circle.
• Solving Complex Problems: In calculus, physics, and engineering, the unit circle is used to solve complex problems, such as those involving periodic motion, waves, or rotations.

Conclusion

The unit circle is an essential concept in trigonometry, providing a straightforward and visual way to understand the relationships between angles and trigonometric functions. By mastering the unit circle, you’ll have a deeper understanding of sine, cosine, and other trigonometric functions, enabling you to solve a wide range of problems in mathematics and its applications.

So next time you encounter a trigonometric function, remember the unit circle—it’s the key to unlocking the mysteries of angles and their relationships!