Curated OER
Derivatives of Exponential Functions
Young scholars take derivatives of exponential functions. In this taking derivatives of exponential functions lesson, students prove the derivative of an exponential function is the exponential function. Young scholars find derivatives...
Curated OER
Pythagorean Theorem Proof
Tenth graders investigate the Pythagorean Theorem. Then they type up a formal paragraph proof of a proof of their choice.
EngageNY
Characteristics of Parallel Lines
Systems of parallel lines have no solution. Pupils work examples to discover that lines with the same slope and different y-intercepts are parallel. The 27th segment of 33 uses this discovery to develop a proof, and the class determines...
EngageNY
Properties of Parallelograms
Everyone knows that opposite sides of a parallelogram are congruent, but can you prove it? Challenge pupils to use triangle congruence to prove properties of quadrilaterals. Learners complete formal two-column proofs before moving on to...
EngageNY
Ptolemy's Theorem
Everyone's heard of Pythagoras, but who's Ptolemy? Learners test Ptolemy's Theorem using a specific cyclic quadrilateral and a ruler in the 22nd installment of a 23-part module. They then work through a proof of the theorem.
EngageNY
There is Only One Line Passing Through a Given Point with a Given Slope
Prove that an equation in slope-intercept form names only one line. At the beginning, the teacher leads the class through a proof that there is only one line passing through a given point with a given slope using contradiction. The 19th...
EngageNY
The Volume Formula of a Sphere
What is the relationship between a hemisphere, a cone, and a cylinder? Using Cavalieri's Principle, the class determines that the sum of the volume of a hemisphere and a cone with the same radius and height equals the volume of a...
Old Dominion University
Introduction to Calculus
This heady calculus text covers the subjects of differential and integral calculus with rigorous detail, culminating in a chapter of physics and engineering applications. A particular emphasis on classic proof meshes with modern graphs,...
West Contra Costa Unified School District
Law of Sines
Laws are meant to be broken, right? Learners derive the Law of Sines by dropping a perpendicular from one vertex to its opposite side. Using the Law of Sines, mathematicians solve for various parts of triangles.
Curated OER
Highs and Lows
Solve problems using integration and derivatives. By using calculus, learners will analyze graphs to find the extrema and change in behavior. They then will identify the end behavior using the derivatives. Activities and handouts are...
EngageNY
Unknown Angle Proofs—Writing Proofs
What do Sherlock Holmes and geometry have in common? Why, it is a matter of deductive reasoning as the class learns how to justify each step of a problem. Pupils then present a known fact to ensure that their decision is correct.
EngageNY
Unknown Angle Proofs—Proofs of Known Facts
Lead the class in a Greek history lesson with a geometric twist. Pupils relate a short video about geometric properties to modern-day methods of solving for unknown angles. They discuss parallel line theorems and complete practice...
EngageNY
Proofs of Laws of Exponents
Apply pupil understanding of exponent properties to prove the relationships. In the sixth lesson of the series, individuals are expected to prove relationships using mathematical statements and reasoning.
EngageNY
Analytic Proofs of Theorems Previously Proved by Synthetic Means
Prove theorems through an analysis. Learners find the midpoint of each side of a triangle, draw the medians, and find the centroid. They then examine the location of the centroid on each median discovering there is a 1:2 relationship....
Virginia Department of Education
Circles in the Coordinate Plane
Make the connection between the distance formula and the equation of a circle. The teacher presents a lesson on how to use the distance formula to derive the equation of the circle. Pupils transform circles on the coordinate plane and...
Curated OER
Getting It Right!
Pupils construct right triangles, measuring the sides and recording the measurements in spreadsheets. They analyze their measurements for patterns and research the Internet for information on Pythagoras.
Curated OER
Some Elementary Derivative Formulas and Rules
In this derivative formula worksheet, students solve problems concerning the derivative of constant functions and linear functions. They explore the derivative of the cosine and sums and differences functions. This six-page worksheet...
Curated OER
Derivatives of Exponential Functions
Learners listen as the teacher develops the derivative formula for if y=e^x, They find dy/dx by showing two proofs. Students determine the derivative of y=e^tan3x, and if y=e^2x they determine the hundredth derivative. Learners listen as...
Alabama Learning Exchange
Triangle Area: No Height? Use the Sine
No height? No problem! Learners use their knowledge and a little help from GeoGebra to develop the Law of Sines formula. The Law of Sines helps to determine the height of triangles to calculate the area.
EngageNY
Circles, Chords, Diameters, and Their Relationships
A diameter is the longest chord possible, but that's not the only relationship between chords and diameters! Young geometry pupils construct perpendicular bisectors of chords to develop a conjecture about the relationships between chords...
College Board
Differentiability of Piecewise Defined Functions
Differentiable or not, that is the question. An informational website gives two theorems on the differentiability of piecewise functions at their endpoints. It goes on to apply the theorems to present two solution methods to a past AP®...
EngageNY
Proving the Area of a Disk
Using a similar process from the first lesson in the series of finding area approximations, a measurement resource develops the proof of the area of a circle. The problem set contains a derivation of the proof of the circumference formula.
EngageNY
Congruence Criteria for Triangles—AAS and HL
How can you prove it? Guide classes through an exploration of two possible triangle congruence criteria: AAS and HL. Learners connect this criteria to those previous learned and also explore criteria that does not work. The lesson...
EngageNY
Law of Sines
Prove the Law of Sines two ways. The ninth segment in a series of 16 introduces the Law of Sines to help the class find lengths of sides in oblique triangles. Pupils develop a proof of the Law of Sines by drawing an altitude and a second...