Mr Mathematics Blog

Planes of Symmetry in 3D Shapes

March 12, 2024

This lesson for Key Stage 3 or foundation GCSE students delves into the symmetry of 3D shapes. Using isometric paper, it starts with simple shapes and progresses to more complex ones, helping students grasp symmetry concepts. The lesson includes a worksheet for further practice.

Success Criteria:

  • Visualize and identify symmetry planes in 3D shapes.
  • Accurately sketch symmetry planes using isometric paper.
  • Discuss shared properties of solids based on reflective symmetry.

Key Questions:

  1. “How many symmetry planes are in this shape?”
  2. “What does the number of symmetry planes say about a shape?”
  3. “How does isometric paper aid in sketching symmetry planes?”

Advice for New Teachers:

  • Begin with basic 3D shape properties to build a solid foundation.
  • Use diverse shapes to accommodate various learning speeds.
  • Encourage peer discussions for collaborative learning insights.
  • Utilize the worksheet for practice and explore digital 3D tools for varied learning styles.
  • Employ visual aids for better comprehension of 3D symmetries.

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GCSE Trigonometry Skills & SOH CAH TOA Techniques

March 8, 2024

GCSE trigonometry skills are essential for students to solve the three types of right-angled triangle problems presented, common in GCSE Mathematics and crucial for those aiming for grades 4 or 5. Here’s a summary of the key skills required for each problem, along with tailored advice for students and teachers, and probing questions to assess comprehension.

GCSE Trigonometry Skills & SOH CAH TOA Techniques

Skill: Understanding and applying the trigonometric ratios (Sine, Cosine, and Tangent) to find angles.

Advice: Remember that for any right-angled triangle, SOH-CAH-TOA represents the ratios of sides: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. Identify which sides are given and which ratio to use accordingly.

Key Questions:

  • Which trigonometric ratio would you use to find angle θ in this triangle?
  • How do you know if you should use the sine, cosine, or tangent function?

Finding Lengths in Right-Angled Triangles

Skill: Using trigonometric ratios to find unknown sides in right-angled triangles.

Advice: Students should focus on identifying the right ratio to use based on the sides given and the side they need to find. Drawing and labeling the triangle can help in visualising the problem.

Key Questions:

  • Given angle θ and side PR, which trigonometric function helps us find QR?”
  • “What is the relationship between side QR and side PR in this context?

Application of Pythagorean Theorem and Area Calculation

Skill: Demonstrating the Pythagorean theorem and calculating the area of irregular shapes by decomposing them into basic shapes.

Advice: Be sure to review the Pythagorean theorem (a² + b² = c²) where ‘c’ is the hypotenuse. For areas, students should break down complex shapes into simpler shapes like triangles or rectangles whose areas they can calculate.

Key Questions:

  • How can we show that triangle DCA is a right-angled triangle?
  • Can you decompose quadrilateral ABCD into shapes with areas you know how to calculate?

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Regions in the Complex Plane

March 7, 2024

When finding regions in the complex plane common misconceptions include confusing a number’s modulus with its components and misinterpreting loci inequalities, which can result in inaccurate shading. Recognizing geometric implications, like modulus inequalities forming circles, is key.

The tutorials below are designed to help students in identify complex regions within an Argand diagram, complete with summaries and teacher-crafted questions to solidify understanding.

Regions in the Complex Plane

Content Criteria:

  • Understanding of the geometric representation of complex numbers.
  • Interpretation and shading of regions based on inequalities involving complex numbers.
  • Familiarity with the modulus of a complex number as a distance in the Argand diagram.

Questions for Students:

  1. How would you represent the inequality ∣z − 3 − 4i ∣ = 4 on the Argand diagram, and what geometric shape does it correspond to?
  2. z−5∣>∣z−9∣ does not form a closed region. What does this inequality represent in terms of distance on the Argand diagram?

Locus and Combined Inequalities

Content Criteria:

  • Combining inequalities involving the argument of a complex number.
  • Understanding how to represent complex inequalities graphically on an Argand diagram.
  • Identifying the intersection of loci to shade the appropriate region.

Questions for Students:

  1. What does the inequality 0 < arg ⁡(z + 1 − i) < 3π/π​ represent on the Argand diagram?
  2. How would you describe the intersection of the loci described by the given inequalities? Why is the region bounded?

Equations and Inequalities Defining Loci

Content Criteria:

  • Understanding of the locus of a complex number given by an equation ∣zabi∣ = r.
  • Familiarity with loci given by an equality, such as ∣∣z∣ = ∣z − 4i∣, representing the perpendicular bisector of the line segment joining the origin and the point 4i in the Argand diagram.
  • Ability to combine these to find the region that satisfies both conditions.

Questions for Students:

  1. Draw the locus described by ∣z − 3 − 4i∣ = 5. What does this represent in terms of complex numbers?
  2. If ∣z∣ = ∣z − 4i∣, what geometric place does this describe? How does it divide the Argand diagram?

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GCSE Maths Equation Solving Strategies

March 5, 2024

For students aiming for grades 4 or 5, the ability to translate real-world problems into mathematical equations is essential. Teachers should emphasise the process of understanding the problem, clearly defining variables, and then constructing and solving equations.

Practice with a variety of contexts can build confidence and versatility. Additionally, teachers can use probing questions to ensure students are not only able to find the solution but also understand the underlying mathematical concepts.

GCSE Maths Equation Solving Strategies

These four examples show how to setup and solve linear equations from real-life problems. They build students’ confidence and skill, preparing them for similar exam questions.

Compound Shapes:

  • Key Skills: Decomposing a shape into simpler parts, calculating areas of rectangles, and understanding how to form and solve equations.
  • Advice: Practice breaking down complex figures and remember that the sum of the areas of the parts equals the area of the whole.
  • Check Questions: “How did you divide the shape?” “What equation represents the total area?”

Mean of Numbers:

  • Key Skills: Using the concept of the mean to set up an equation that balances two sets of numbers.
  • Advice: Ensure understanding of how the mean is calculated and how it relates to the sum of the numbers.
  • Check Questions: “What does the mean represent?” “How can we express the mean as an equation?”

Probability with Algebra:

  • Key Skills: Formulating algebraic expressions to represent quantities and solving for unknowns within probability scenarios.
  • Advice: Strengthen algebraic manipulation and understand probability principles, like the total number of outcomes.
  • Check Questions: “What do the expressions for each box represent?” “How do you calculate the total probability?”

Jug Filling Problem:

  • Key Skills: Setting up and solving linear equations from word problems involving proportional reasoning.
  • Advice: Focus on converting the written problem into mathematical statements and solve systematically.
  • Check Questions: “How do you represent the problem algebraically?” “What does each term in your equation represent?”

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Further Kinematics

January 11, 2024

Prerequisite Knowledge

Motion with Vectors

  • Basic understanding of vectors (addition, subtraction, magnitude, direction).
  • Fundamental kinematics concepts (speed, velocity, acceleration).

Variable Acceleration

  • Knowledge of constant acceleration formulas.
  • Familiarity with motion graphs (displacement-time, velocity-time).

Differentiation with Vectors

  • Proficiency in basic differentiation techniques.
  • Understanding of kinematic relationships using derivatives.

Integration with Vectors

  • Basic skills in integration (definite and indefinite integrals).
  • Application of integration in kinematics.

Success Criteria

Motion with Vectors:

Understand and apply the concept of displacement in vector form,

\vec{r} = \vec{r_0} + \vec{v}t

Successfully calculate the position vector at any given time using the initial position and velocity vectors.

Variable Acceleration:

Use kinematic equations in vector form for objects moving with constant acceleration, specifically

\vec{v} = \vec{u} + \vec{a}t   \; and \;  \vec{r} = \vec{u}t + \frac{1}{2}\vec{a}t^2

Understand the relationships between velocity, acceleration, and displacement vectors over time.

Differentiation with Vectors:

Differentiate position vectors with respect to time to find velocity vectors, and differentiate velocity vectors to find acceleration vectors.

Apply vector differentiation to kinematics, recognising that

\vec{r} = x\hat{i} + y\hat{j} 
\vec{v} = \frac{d\vec{r}}{dt}
\vec{a} = \frac{d\vec{v}}{dt}

Integration with Vectors:

Integrate acceleration vectors over time to find velocity vectors, and integrate velocity vectors to find displacement vectors.

Comprehend and utilize integral calculus in the context of vectors to solve kinematics problems, with an understanding that

\vec{v} = \int \vec{a}\: dt \; and \; \vec{r} = \int \vec{v}\: dt 

Teaching Points

Motion with Vectors:

The displacement of a particle from its initial position can be represented by a position vector, which changes over time with constant velocity.

The position vector at any time t can be calculated using the initial position vector and the velocity vector , following the formula

\vec{r} = \vec{r_0} + \vec{v}t 

Variable Acceleration:

When an object is moving with constant acceleration, its velocity vector at time t can be found using the initial velocity vector and the acceleration vector, with the formula

\vec{v} = \vec{u} + \vec{a}t

The displacement vector at time t is determined by both the initial velocity and the acceleration, following

\vec{r} = \vec{u}t + \frac{1}{2}\vec{a}t^2

Differentiation with Vectors:

The velocity vector is the derivative of the position vector with respect to time, and the acceleration vector is the derivative of the velocity vector with respect to time.

This differentiation process can be represented in terms of the component functions, x(t) and y(t), of the position vector, such that ( \vec{v} = \frac{d\vec{r}}{dt} ) and ( \vec{a} = \frac{d\vec{v}}{dt} ).

\vec{v} = \frac{d\vec{r}}{dt} \;  and \;  \vec{a} = \frac{d\vec{v}}{dt} 

Integration with Vectors:

Integration is used to find the velocity vector from the acceleration vector and the displacement vector from the velocity vector over time.

The formulae

\vec{v} = \int \vec{a}\: dt \; and \; \vec{r} = \int \vec{v}\: dt

allow calculation of these vectors by integrating the respective component functions with respect to time.

Common Misconceptions

Motion with Vectors:

Misconception: The position vector is static and does not change with time.

Misconception: Displacement is the same as the distance traveled by the particle.

Variable Acceleration:

Misconception: Velocity and acceleration vectors do not affect each other.

Misconception: The formulae

\vec{v} = \vec{u} + \vec{a}t \; and \;  \vec{r} = \vec{u}t + \frac{1}{2}\vec{a}t^2

can be applied even when acceleration is not constant.

Differentiation with Vectors:

Misconception: The process of differentiation of vectors is unrelated to the differentiation of scalar functions.

Misconception: Velocity and acceleration vectors are independent of the position vector.

Integration with Vectors:

Misconception: Integration of the acceleration vector always yields the velocity vector without considering initial conditions.

Misconception: The definite integral of the velocity vector over a time interval gives the total distance traveled, not the displacement.

Resolving Forces and Friction

January 9, 2024

Prerequisite Knowledge

Vectors: Understanding vectors is crucial as forces are vector quantities, which have both magnitude and direction. Knowledge of vector addition, subtraction, and scalar multiplication is needed to resolve forces.

Trigonometry: Proficiency in trigonometry is essential. This includes understanding sine, cosine, and tangent functions, as well as how to use them to relate the sides and angles of right-angled triangles.

Newton’s Laws of Motion: An understanding of the three laws of motion is important for analysing problems in mechanics.

Mechanics Year 1: The study of statics and dynamics, which includes the concepts of equilibrium, motion on an inclined plane, and friction.

Mathematical Problem Solving: Ability to construct and solve mathematical models that represent physical situations.

Success Criteria

Understand Vector Resolution:

  • Demonstrate the ability to resolve a force into its components along two perpendicular directions, typically parallel and perpendicular to the direction of motion.
  • Apply trigonometric ratios to calculate the magnitude of the components when the angle of the force is known.

Trigonometric Application:

  • Correctly use the cosine function to find the component of a force in a given direction when the angle is given.
  • Utilize trigonometric identities and functions to relate forces in different directions.

Problem-Solving on Inclined Planes:

  • Set up free body diagrams for objects on inclined planes, showing all forces acting on the object.
  • Apply the concepts of parallel and perpendicular resolution of forces to problems involving inclined planes.
  • Solve for unknown forces, such as friction or the normal reaction force, using the equations of equilibrium.

Friction Calculations:

  • Understand the concept of maximum static friction and how it relates to the normal reaction force and the coefficient of friction.
  • Use the formula Fmax = μR to calculate the maximum or limiting value of friction between two surfaces.
  • Interpret the coefficient of friction μ and the normal reaction R in the context of given problems.

Teaching Points

Resolution of Forces:

  • The process of breaking down a force into two components acting at right angles to each other.

Components of Force:

  • Horizontal and vertical components of a force can be determined using trigonometry, specifically the cosine and sine functions.

Inclined Planes:

  • Forces on an inclined plane can be resolved into parallel and perpendicular components relative to the plane’s surface.

Friction:

  • Frictional force, which opposes motion, can reach a maximum value before an object begins to slide.
  • The formula for the maximum frictional force (F_max) is the product of the coefficient of friction (μ) and the normal reaction force (R).

Coefficient of Friction:

  • A dimensionless scalar value that represents the frictional resistance between two surfaces.

Normal Reaction Force:

  • The force exerted by a surface perpendicular to an object resting on it, equal and opposite to the object’s weight component perpendicular to the surface.

Common Misconceptions

Misconceptions about Force Components:

  • Believing that the largest component of a force must be in the direction of motion, when in fact it depends on the angle at which the force is applied.
  • Confusing the force’s component along the direction of motion with the net force acting on an object.

Misconceptions about Trigonometry in Forces:

  • Misapplying trigonometric functions, such as using the sine function where the cosine is appropriate, and vice versa.
  • Overlooking that the angle used in trigonometric functions is always relative to a reference direction, often leading to incorrect calculations of force components.

Misconceptions about Inclined Planes:

  • Assuming that the normal force is equal to the object’s weight regardless of the incline angle.
  • Misunderstanding the role of the incline angle in affecting the magnitude of the normal force and the force of friction.

Misconceptions about Friction:

  • Believing that the frictional force is always equal to the product of the coefficient of friction and the normal force, not recognizing that this is the maximum static friction.
  • Thinking that static and kinetic friction have the same value or that kinetic friction increases with relative motion speed.

Misconceptions about Coefficient of Friction:

  • Confusing the coefficient of static friction with the coefficient of kinetic friction, not realizing they are typically different values.
  • Assuming that a higher coefficient of friction always means more motion resistance, without considering the type of motion (starting vs. continuing).

Misconceptions about Equilibrium and Motion:

  • Misinterpreting an object at rest as having no forces acting on it, rather than having balanced forces.
  • Equating dynamic equilibrium with static equilibrium, not realizing that dynamic equilibrium also applies to constant velocity motion.

Resolving Forces and Friction

Grade 5 Maths Problems

January 7, 2024

In my experience students aiming for between grades 4 and 5 often struggle with the grade 5 maths problems in the latter half of the non-calculator paper.  Grade 5 non-calculator questions often require more method marks and greater knowledge of specific vocabulary.  For this reason, I created this lesson to remind students of key terms while developing their written methods.

Grade 5 Maths Problems

The eight questions cover the following topics:

  • Standard form
  • Turning points and roots of quadratic equations
  • Probability trees
  • Simultaneous equations
  • Composite area involving circles
  • Expanding quadratic expressions
  • Angles in polygons
  • Column vectors

Here is a sample of four questions from eight and with a brief description of how students got on with each problem.

Composite area involving circles

In this question, most students were able to work out the area of the composite shape in the form 90 – 4π cm2. About three quarters who wrote 90 – 4π cm2 as their area were able to correctly factorise it. Some students used the formula for the circumference of a circle but this was easily corrected.

Turning points and roots of quadratic equations

Most students either got full marks or left the answer blank.  This tells me that understanding the terms ‘turning point’ and ‘roots’ are often not understood.  Some students tried to solve the equation algebraically using the balance method but gave up after a few lines of working.

Probability Trees

Most students completed the tree diagram correctly.  However, when working out the probability of picking at least one blue counter a common mistake was to add the fractions rather than multiply them.  Of those who added the fractions about half did so incorrectly, without using a common denominator. 

Most students who realised to multiply the fractions worked out the correct probability using P(at least one blue) = 1 – P(not blue).

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Reducible Differential Equations

January 5, 2024

Prerequisite Knowledge

  • Basic understanding of differential equations, including the concept of a derivative as a rate of change.
  • Proficiency in basic calculus, including differentiation and integration of functions.
  • Familiarity with the methods for solving first-order differential equations, like separation of variables and the use of integrating factors.
  • Knowledge of how to find the general solution of a differential equation.
  • An understanding of the particular solution of a differential equation given initial or boundary conditions.
  • Algebraic skills to manipulate and solve equations, including completing the square and factorization.
  • Competence in handling exponential and logarithmic functions, as they frequently appear in solutions to differential equations.
  • Experience with the concept of linear independence of functions, particularly when dealing with the complementary function and particular integral.

Success Criteria

First Order Reducible Differential Equations:

  • Ability to identify a first-order differential equation that can be reduced to a separable form.
  • Proficiency in using an integrating factor to simplify a first-order linear differential equation.
  • Skill in applying the appropriate substitution to transform the equation into a solvable form.

Second Order Reducible Differential Equations:

  • Able to recognise a second-order differential equation that can be reduced to a first-order equation.
  • Competence in applying a given substitution to transform a second-order differential equation into a standard form.

Modelling with Reducible Differential Equations:

  • Be able to formulating real-life problems into reducible second-order differential equations.
  • Proficiency in solving these equations to model and predict the behavior of various systems.

Teaching Points

First Order Reducible Differential Equations:

  • Teaching the method of using substitutions to transform non-separable first-order equations into separable ones.
  • Illustrating the process of finding an integrating factor for non-homogeneous first-order linear differential equations.

Second Order Reducible Differential Equations:

  • Demonstrating how to apply specific substitutions to reduce complex second-order differential equations to a simpler form that can be solved using known techniques.

Modelling with Reducible Differential Equations:

  • Not explicitly mentioned in the image, but typically, this would involve:
    • Showing how to translate real-life situations into second-order differential equations.
    • Teaching how to use substitutions to simplify these real-life models into solvable equations.

Common Misconceptions

First Order Reducible Differential Equations:

  • Misconception that all first-order differential equations can be made separable with a simple substitution.
  • Confusion between linear and nonlinear first-order differential equations, and when to use an integrating factor.
  • Misunderstanding the role and calculation of an integrating factor, often forgetting to multiply through by this factor.

Second Order Reducible Differential Equations:

  • Difficulty in recognising when a second-order differential equation can be reduced to first-order.
  • Overlooking the necessity for a substitution or incorrectly identifying the type of substitution needed.
  • Assuming solutions to second-order equations will always involve two arbitrary constants, not considering the nature of the equation after reduction.

Modelling with Reducible Differential Equations:

  • Misinterpreting real-life scenarios and setting up incorrect differential equations.
  • Over-simplifying the model by not considering all relevant factors or over-complicating it by including unnecessary elements.
  • Struggling to link the physical situation with the appropriate mathematical model, particularly in selecting the correct form and order of the differential equation to use.

Modelling Projectile Motion


Prerequisite Knowledge

From Year 1 Mechanics

  • Understanding of basic kinematic equations for constant acceleration.
  • Concepts of velocity and acceleration vectors.
  • Ability to resolve a force into its components.
  • Knowledge of Newton’s laws of motion.
  • Familiarity with the concepts of work, energy, and power.
  • Basic understanding of the principles of moments and equilibrium.
  • Ability to perform vector addition and subtraction.
  • Knowledge of trigonometric functions to resolve vectors.

Success Criteria

Understanding Projectile Motion:

  • Grasp the concept of projectile motion as a form of two-dimensional motion.
  • Recognize that projectile motion can be analyzed by separating it into horizontal and vertical components.

Equations of Motion:

  • Be able to use the formula for the horizontal motion, understanding that the horizontal velocity is constant since a = 0.
  • Know that the vertical motion of a projectile is subject to constant acceleration due to gravity a = g.

Calculating Components of Velocity:

  • Determining Time of Flight:
  • Use the formula for time of flight and understand its derivation.

Finding Maximum Height:

  • Calculate the time to reach the greatest height and recognise this as the point where the vertical component of the velocity is zero.

Calculating Range:

  • Apply the formula for range on a horizontal plane and understand its implications for the angle of projection.

Understanding the Trajectory Equation:

  • Analyze the trajectory equation and be able to plot it or derive it from the motion equations.

Problem Solving:

  • Solve problems that involve finding the range, maximum height, time of flight, and final velocity of a projectile.

Teaching Points

Horizontal Motion:
Recognize horizontal motion as having constant velocity due to zero acceleration.

Vertical Motion:
Understand vertical motion as having constant acceleration due to gravity.

Initial Velocity Components:
Calculate the horizontal component of initial velocity.
Calculate the vertical component of initial velocity.

Maximum Height of Projectile:
Identify the greatest height reached when the vertical component of velocity becomes zero.

Common Misconceptions

  • Constant Velocity and Acceleration:
    • Confusing horizontal constant velocity with zero acceleration, and not understanding that constant horizontal velocity implies that the horizontal acceleration is indeed zero.
    • Believing that the vertical acceleration changes as the projectile moves upwards or downwards, rather than understanding that it remains constant due to gravity.
  • Components of Initial Velocity:
    • Thinking that the horizontal component of the velocity affects the vertical component, or vice versa, rather than treating them as independent.
    • Misconceiving that the angles in the trigonometric functions for velocity components should change over time.
  • Maximum Height and Time:
    • Assuming that the time to reach the maximum height is half of the total time of flight, which is only true for projectiles launched from and landing on the same level.
    • Misunderstanding that the projectile’s maximum height occurs at the midpoint of the horizontal range.
  • Range and Optimal Angle:
    • Believing that increasing the launch angle always increases the range, not recognizing that there is an optimal angle for maximum range (45 degrees when disregarding air resistance).
    • Confusing the range equation with the horizontal displacement at any point during the trajectory.
  • Equation of Trajectory:
    • Misinterpreting the trajectory equation as linear, not realizing it describes a parabolic path.
    • Overlooking the fact that the trajectory equation accounts for the initial launch angle and the effect of gravity on the path of the projectile.
  • Influence of Gravity:
    • Assuming gravity only acts during the descent of the projectile, not understanding that it acts throughout the entire flight.
  • Initial Speed and Angle:
    • Thinking that a greater initial speed always results in a greater range, not taking into account the influence of the launch angle.
    • Assuming that the vertical component of velocity is always equal to the initial velocity multiplied by the sine of the launch angle, regardless of any changes during the motion.

Taylor’s Series

January 1, 2024

Prerequisite Knowledge

Taylor’s Series Approximation of Functions:

  • Fundamental understanding of Maclaurin series.
  • Concept of function derivatives at a point.
  • Basic polynomial functions and their behaviors.

Evaluating Limits of Accuracy:

  • Rules for evaluating limits, especially L’Hôpital’s rule.
  • Understanding of continuity and differentiability of functions.
  • Familiarity with common limit forms and indeterminate forms.

Series Solution of Differential Equations:

  • Knowledge of differential equations and their solutions.
  • Differentiation techniques, including the product rule and chain rule.
  • Concept of initial conditions in differential equations.

Success Criteria

Taylor’s Series Approximation of Functions:

  • Correctly apply the Taylor series expansion to a given function.
  • Determine the convergence interval for the series.

Evaluating Limits of Accuracy:

  • Accurately evaluate the limits of functions using Taylor series.
  • Determine the error or accuracy of approximations.

Series Solution of Differential Equations:

  • Find the series solution of a given differential equation using Taylor’s series.
  • Apply initial conditions correctly to find specific solutions.

Teaching Points

Taylor’s Series Approximation of Functions:

  • Taylor’s series can approximate functions as an infinite sum of terms.
  • Expansions are centered around a specific point.

Evaluating Limits of Accuracy:

  • Limits help evaluate the behavior of functions as they approach a specific value.
  • The accuracy of Taylor’s approximation depends on the degree of the polynomial.

Series Solution of Differential Equations:

  • Differential equations can be solved using series methods.
  • Taylor’s series provide a powerful tool for approximating solutions near a point.

Common Misconceptions

Taylor’s Series Approximation of Functions:

  • Believing Taylor series can only be centered at x=0 (Maclaurin series).
  • Assuming the series always converges to the function for all x.

Evaluating Limits of Accuracy:

  • Misunderstanding that limits can provide exact values for the function at a point.
  • Overestimating the accuracy of a Taylor series approximation without considering the remainder term.

Series Solution of Differential Equations:

  • Confusing the series solution with other methods like separation of variables.
  • Thinking that a series solution always exists and is easy to find.