№ lp_2_3_44698
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Year: 2022
Region / City: New South Wales
Subject: Mathematics
Document Type: Teaching Plan
Institution: NSW Education Standards Authority (NESA)
Author: Unknown
Target Audience: Students
Period of Validity: Not specified
Date of Approval: Not specified
Date of Changes: Not specified
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Going round in circles: lesson activity on deriving the equation of a circle using Pythagoras’ theorem
Year level: Stage 5 (Mathematics K–10)
Subject: Mathematics
Topic: Circles, Cartesian plane, Pythagoras’ theorem
Document type: Classroom lesson activity plan
Curriculum reference: MAO-WM-01; MA5-NLI-P-01
Curriculum document: Mathematics K–10 Syllabus
Educational authority: NSW Education Standards Authority (NESA)
Jurisdiction: New South Wales, Australia
Copyright year: 2022
Learning intention: To understand the connection between the graph of a circle and its equation
Success criteria: Find the distance between two points; derive the equation of a circle; determine the radius from an equation; graph a circle with centre at the origin
Teaching strategies mentioned: Visibly random groups, gallery walk, Two stars and a wish peer feedback, Think-Pair-Share, Pose-Pause-Pounce-Bounce questioning
Digital tools referenced: Desmos Graphing Calculator; GeoGebra applet “Circle Equation Anatomy and Exploration”
Associated materials: PowerPoint “Going around in circles”; Appendix A “Lost bushwalker” worksheet
Activity context: Scenario of a lost bushwalker navigating relative to a waterhole
Mathematical focus: Distance formula, right-angled triangles, Cartesian coordinates, equation
Note: ????
ON APPLICATION OF FIXED POINT THEOREM TO INSURANCE LOSS MODEL
Year: 2023
Region / city: Lagos, Nigeria
Topic: Insurance, Actuarial Science
Document type: Research Paper
Organization / institution: University of Lagos
Author: Abiola Bankole, Omotayo Johncally
Target audience: Researchers, Actuaries, Insurance Professionals
Period of validity: Not specified
Approval date: Not specified
Date of changes: Not specified
Entry Task: Consider the tank shown. The height is 3m, the width at the top is 2m and the length is 6m. We are pumping the water up to 4m above the ground out of a spout.Water has a density of 1000 kg/m3.If it starts full, how much work is done to pump it all out?If you don’t know how, that is fine, then instead do this…Label y = 0 and a typical yLabel a horizontal slice at y.For that slice…Find a formula for the width.Find a formula for the volume.Find a formula for the mass and weight.6.4 Work (continued) Recap: Constant force and given distance: Work = Force·Distance If force or distance changeWork=limn→∞i=1n(Force · Dist)=ab(Force)(Dist)PROBLEM TYPE 1 (“Leaky Bucket”):An single object is lifted, but gets lighter (or heavier) as it is lifted. FORCE = f(xi), DIST = ∆xPROBLEM TYPE 2 (“Stack of Books”)A bunch of objects are stacked up and each gets lifted a different distance. FORCE = (density)(horiz. slice length or vol) DIST = depends on labels (typically x or a-y)Old Exam Problems:Set up the integrals that give…1. A rocket loses mass as it burns fuel. The mass of the rocket when it is x meters off the ground is given by fx=409.8+209.8e-x2 kg.Set up the integral for the work done in the first 8 meters after launch.2. A 50 foot cable with density 4 lbs/ft is hanging over the side of a tall building. Find the total work done in lifting the cable halfway up.3. (Fall ‘17 exam) The figure shows a tank. The front face of the tank has the shape: f(x) = 4x2. Initially, there is fluid in the tank up to a height of 1 foot. The fluid weighs 15 lb/ft3. How much work is done to empty the tank by pumping all of the fluid to the top of the tank?3. (Fall ‘17 exam) The figure shows a tank. The front face of the tank has the shape: f(x) = 4x2. Initially, there is fluid in the tank up to a height of 1 foot. The fluid weighs 15 lb/ft3. How much work is done to empty the tank by pumping all of the fluid to the top of the tank?6.5 Average ValueThe average y-value of y = f(x) from x = a to x = b is given byfave=1b-aabfxdxExample: The formula for the temperature of a particular object is T(t) = t2 degrees Fahrenheit where t is in hours. Find the average temp. from t = 1 to t = 4 hours.Mean Value Theorem for integrals:If f(x) is continuous on from x = a to x = b, then there is at least one value x = c at which f(c) = fave.Example:Using T(t) = t2 from t = 1 to t = 4 again.Find a time at which the temperature is exactly equal to the average value.Derivation:The average value of the n numbers: y1, y2, y3, …, yn is given by y1+y2+y3+⋯+ynn=y11n+⋯+yn1n.Goal: We want the average value of all the y-values of some function y = f(x) over an interval x = a to x = b.Derivation:Break into n equal subdivisions∆x=b-an, which means ∆xb-a=1nCompute y-value at each tick marky1=fx1, y2=fx2,…,yn=fxn Ave ≈fx1∆xb-a+⋯+fxn∆xb-aAverage ≈1b-ai=1nfxi∆xThus, we can defineAverage=1b-alimn→∞i=1nfxi∆xwhich means the exact average y-value of y = f(x) over x = a to x = b isfave=1b-aabfxdx
Year: Not specified
Region / City: Not specified
Theme: Work, integrals, physics problems
Document Type: Educational material
Organization / Institution: Not specified
Author: Not specified
Target Audience: Students studying calculus or physics
Period of Validity: Not specified
Date of Approval: Not specified
Date of Changes: Not specified
Work-Energy Theorem Laboratory Exercise
Year: 2026
Institution: PASCO Scientific
Document Type: Laboratory Manual
Author: Jon Hanks
Intended Audience: Physics students
Topic: Classical Mechanics, Work-Energy Theorem
Equipment: Motion sensor, Force sensor, Dynamics cart, Elastic cord, Masses, Track
Experiment Duration: Single lab session
Data Analysis: Force vs. position graphs, velocity vs. position graphs
Learning Objectives: Demonstrate the work-energy theorem, investigate the effect of mass on kinetic energy and velocity
Pre-lab Questions Included: Yes
Post-lab Analysis: Yes
Safety Considerations: Standard lab safety procedures
Extra Practice 9.19.1: Pythagorean Theorem and Triangle Exercises
Year: 2023
Subject: Mathematics
Type of document: Worksheet
Educational level: Middle/High School
Topics covered: Pythagorean triples, right triangles, triangle classification, application problems
Skills assessed: Algebra, Geometry, Measurement
Target audience: Students
Date created: 2023-09-19
Going round in circles: lesson activity on deriving the equation of a circle using Pythagoras’ theorem
Year level: Stage 5 (Mathematics K–10)
Subject: Mathematics
Topic: Circles, Cartesian plane, Pythagoras’ theorem
Document type: Classroom lesson activity plan
Curriculum reference: MAO-WM-01; MA5-NLI-P-01
Curriculum document: Mathematics K–10 Syllabus
Educational authority: NSW Education Standards Authority (NESA)
Jurisdiction: New South Wales, Australia
Copyright year: 2022
Learning intention: To understand the connection between the graph of a circle and its equation
Success criteria: Find the distance between two points; derive the equation of a circle; determine the radius from an equation; graph a circle with centre at the origin
Teaching strategies mentioned: Visibly random groups, gallery walk, Two stars and a wish peer feedback, Think-Pair-Share, Pose-Pause-Pounce-Bounce questioning
Digital tools referenced: Desmos Graphing Calculator; GeoGebra applet “Circle Equation Anatomy and Exploration”
Associated materials: PowerPoint “Going around in circles”; Appendix A “Lost bushwalker” worksheet
Activity context: Scenario of a lost bushwalker navigating relative to a waterhole
Mathematical focus: Distance formula, right-angled triangles, Cartesian coordinates, equation
Note: ????
Inequalities in Two Triangles and the Hinge Theorem
Year: 2026
Subject: Geometry, Triangle Inequalities
Document type: Educational Material
Author: Unknown
Intended audience: High school or introductory college students
Relevant theorems: Hinge Theorem (SAS Inequality Theorem), Converse of Hinge Theorem (SSS Inequality Theorem)
Exercises included: Yes
Mathematical concepts: Triangle congruence, angles, sides comparison
Topics: Pythagoras theorem