503: Circles

This is a straightforward question. I'd like to make all of you students into teachers for a minute ... Let's create a test question !

• Do we have to specify angle AOB?
• Is there a better way to say something without actually saying it?
• What other instructions and given information could we provide that would lead to the same answer? 
• What is the best question here?

498: Circles on a Lattice

On a square lattice, a circle can pass through 2, 3, or 4 points, as in the diagram below. The original question asks for a circle that passes through 5 points, but can you define a circle that passes through other numbers of points? and explain how the circle was created?

source:

On a square lattice, a circle can pass through exactly 2, 3, or 4 points. What about 5? #math #mathchat #MTBoS pic.twitter.com/CsMd6zzbLo

— Matt Enlow (@CmonMattTHINK) August 5, 2015

493: It Sure Would Be Easier

If we are told that we have a quadrilateral inscribed in a circle with diameter 10, do we KNOW if the angle at B and the angle at D are 90°? Because that sure would make it easier to find x.

484: Tempus Fugit

If the time is 10:10, what is the exact angle between the minute hand and hour hand?



How about 7:30?



How about 7:35?

481: Two Circles and One Square

What is the red area?

The two vertices of the square are the centers of two tangent and congruent circles. If the length of a side is 8√2, what is the area of the red part peeping out?

Here is the real question: Does it matter if the circles are congruent, as long as they're tangent and the centers are at the vertices of the square?

A Saturday #Geometry Delight #SATPrep #mathchat @intmath @JustinAion @Fouss @PDCorner @MathCurmudgeon @bowenkerins pic.twitter.com/V6zcjOfrqW

— David Marain (@dmarain) July 11, 2015

479: Dihedral Angle

What's the angle between the red lines? pic.twitter.com/IXK5GfvM0N

— Will Feavyour (@NextLevelMaths) April 3, 2015

478: Salinon Areas

source:

The Salinon was first introduced in the Book of Lemmas, a work attributed to Archimedes. http://t.co/crVrwtxeb8

— Cliff Pickover (@pickover) June 6, 2015

473: Overlapping Squares

I might have posted this puzzle before.

What do your students think? Can they generalize it?
What is the overlapped area?

source:

As the red square is rotated around point O, what is the area of the overlap with the yellow square? #geomchat

— David Wees (@davidwees)

472: How Many geometry Theorems Are There?

That's pretty much it: How many Geometry Theorems are needed to solve this extended problem?

source:

Old Exam Questions #2 : My solution in a few days... http://t.co/VDHpuoDY5a pic.twitter.com/OFpD7AT6G4

— solve my maths (@solvemymaths) May 24, 2015

468: Fields of Green

What is the area of the whole hexagonal shape?

Can your students generalize this result?


Source:

Farmer has 3 square fields & 4 triangular fields. What is the entire area of his estate? http://t.co/R8IyU2iyYF pic.twitter.com/3yEdcBrw3N

— Cliff Pickover (@pickover) May 19, 2015

450: One-third of a triangle

You've been asked to shade one-third of this triangle.

And how do you know?
Here's a response:

• Is this a valid way to get one-third of the triangle?
• Does this technique require that we stipulate a right triangle?
• Does this technique require that we stipulate an isosceles one?
• Do the lines have to be parallel to each other?
• Do the lines have to be parallel to a side for this to work? 
• Perpendicular to a side for this to work?

• How might we generalize this method (if it can be generalized)?

sources:

Try to shade 1/3 of this triangle. How do you know it is 1/3? #mathchat #elemmathchat pic.twitter.com/KaLtPglBfV

— David Wees (@davidwees) May 2, 2015

@davidwees this may be more than you want pic.twitter.com/Azlkv1kwAw

— David Sawtelle (@davesawtelle) May 2, 2015

446: ASN - Geometry

Always, Sometimes, Never?

444: Floorlamp

The floor lamp casts a shadow.

Straight line AB is drawn on a wide, spacious floor. A lamp is at a height of 8  m above point C, which is located on line AB.  Line CD is perpendicular to line AB.  A rectangular solid 5 m by 3 m by 4 m is on the floor as shown.  The box is 2  m from line AB and a meters from line CD. Assume the lamp is a point source and will cast a perfect shadow. Given that the rectangular solid casts a shadow with area 90  m², determine the distance a (in meters).

• Does the 2m distance from the wall matter to the shape of the shadow?
• How does the distance a change the area of the shadow?
• What shape is that shadow on the floor? 
• Are the edges of the shadow guaranteed to parallel with the edges of the box?
• How far away is that box from the light? 

It's another Five Triangles puzzle

428: Surface area of that cube again.

Let's use this Rubik's Cube again, shall we? The total surface area of the cube is 54 square units.

• Is it possible to remove a unit cube without changing the surface area?
• How about removing one that increases the area by 2 square units ....
• or maybe one that can be removed that increases area by 4 sq units?
  

427: More water in Boxes

A closed rectangular box whose dimensions are 8 feet by 5 feet by 3 feet has some water in it. When the box is resting on its smallest face, the water is 5 feet deep. How deep will the water be if the box is resting on its largest face?

source.

426: Pouring Water

Here are six sets of containers. The water in the top one is draining into the bottom one at a constant rate. The puzzle here is to determine, at five intervals during the pour, what the surface of the water looks like when viewed from above.

Here are the possible choices. There is one set for the surface of the upper one and one set for the surface of the lower one. Some of these cards have missing pieces. Students need to draw those.

Print them from the source below.


source.

422: Inscibed Semicircle, part three

Last question with this visual:

How could you draw the inscribed semicircle (area = π) so that the rectangle is of maximum size?

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, area of rect=? [4] #SATPrep #mathchat #Geometry
— David Marain (@dmarain) March 26, 2015

421: Inscribed Semicircle, part two

This problem was posed on Twitter the other day.

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, what's the area of the rectangle? [4]

My question yesterday was ... how could you draw the inscribed semicircle in a way that gives a rectangle of a very different (and larger) area?

My question today is, given the arrangement below, what points did I choose that had integer coordinates? I chose a larger semicircle - for convenience - how big was it?

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, area of rect=? [4] #SATPrep #mathchat #Geometry
— David Marain (@dmarain) March 26, 2015

420: Inscribed Semicircle

This problem was posed on Twitter the other day.

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, what's the area of the rectangle? [4]

My question is ... how could you draw the inscribed semicircle in a way that gives a rectangle of a very different (and larger) area?

A semicircle is inscribed in a rectangle. If the area of the semicircle is π, area of rect=? [4] #SATPrep #mathchat #Geometry
— David Marain (@dmarain) March 26, 2015

415: Parallels and Variables

Raw, pure math. Ummmmmm, tasty.

But can you make the question harder by rearranging the variables?

source:

Latest gems post - 5 maths teaching ideas http://t.co/tsIzPMVWzb #mathschat #mathstlp pic.twitter.com/Q4oW2J9sVO
— Jo Morgan (@mathsjem) March 9, 2015