Archive for the ‘geometry’ Category

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Smoke Jumpers and their Amazing Wildfire Maths . . . . . . . . . . . . . . METRIC UNITS

December 5, 2016

mathspig-smoke-jumpers 

With wildfires in Texas and now Australia is facing the fire season, it is time to think about fire fighter maths.

mathspig-metric-units-fire-math

METRIC UNITS

Background Story

On 5th August 1949 Wag Dodge was dropped by parachute with 14 other fire fighters into Mann Gulch, a steep-sided gully in a Montana pine forest. Fire fighters who parachute in to put out small blazes started by lightening are called Smoke Jumpers. As they worked their way down the sides of the gully the breeze was blowing away from them. But the wind soon shifted. This produced an updraft, which increases the speed of the fire front. The 15 Smoke Jumpers turned and started running for their lives uphill.

HOW FAST CAN YOU RUN?

METRIC UNITS

Time Trial:

Mark out a 10 m course. Make 3 time trials.

t1 =

t2 =

t3=

Average your time:

tav = (t1 + t2 + t3)/ 3 =

Your Speed S = 10/tav = ……… m/sec

mathspig-firefirghter-maths-1

HOW FAST IS A GRASS FIRE?

This will, of course, vary depending on the wind speed. A typical grass fire in Australia in a flat area can travel at 20kph (up to 30 kph) in a gentle breeze.

Fire Front Speed Grass Fire

Fire Front Speed = 20 kph = 20 x1000/(60 x 60)

                               = 20 x 0.27777777 = 20 x 0.28 m/sec

                               = 5.6 m/sec

mathspig-firefighter-maths-2

CAN YOU OUT RUN A FIRE?

Average Running Speed Boy 13–14 yo = 3.0 m/sec

Average Running Speed Girl 13–14 yo = 2.4 m/sec

We’ll assume, boy or girl, that you are really motivated and can run away from the fire at top speed of 3.0 m/sec. Now calculate the distance you can run and the fire front moves in 10 secs intervals up to 1 minute.

mathspig-fire-fighter-table-1

This is not looking good. See more Firefighters Need Maths here.

We can do very accurate calculations using simultaneous equations. Wildfire Algebra: Detailed Worksheet using simultaneous equations and solutions  here.

NOW YOU ARE RUNNING UP HILL. WHAT HAPPENS?

We’ll assume, due to being motivated by having a fire licking your heels, that you can run at your top speed up hill for a short time, at least. But here is the problem.

Heat rises and so there is a Chimney Effect pushing the fire uphill. The rule of thumb used by fire fighters is:

Each 10º increase in slope, the fire front speed doubles.

mathspig-cfa-diag

mathspig-fire-fighter-table-2

Now you can calculate the distance travelled by the fire front up a slope at a 30º angle.

Don’t forget you can use the WEB 2.0 Calculator here.

mathspig-fire-fighter-table-3

Even at your top running speed, which is unlikely up a slope, you can run 180 m in 1 minute. In that time the forefront has moved 2688 m or 2.7 km.

It depends how far away you are from the fire front, but it seems you cannot out run this fire front.

Again we can do very accurate calculations using simultaneous equations.

See Firefighters Need Maths here.

Wildfire Algebra: Worksheet and solutions here.

CAN YOU OUT RUN A WILD FIRE?

High winds can turn a bush or forrest fire into a WILD FIRE with wind speeds up to 110 kph and temperatures up to 2000 °C, which can and does melt glass and cars.

The fire front speed doubles with every 10º, so speeds for the fire front can reach 220 kph, 330kph and up to 550kph.

20o-angle-mathspig-2

What happened to the Smoke Jumpers?

When the fire front changed direction Wag Dodge and 14 other Smoke Jumpers found themselves running for their lives up a steep slope. What did Wag do next?

ANS: Here’s the amazing thing. Wag realised he could not out run the fire at that point. So he stopped. Took off his back pack. Took out some MATCHES and lit a fire in the grassy patch in front of him. Just before the firewall hit he threw himself face down on the burnt patch. He survived. The other 14 firefighters did not.

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Smoke Jumpers and their Amazing Wildfire Maths . . . . . . . . . . . . . . USA UNITS

December 5, 2016

 

mathspig-smoke-jumpers 

With wildfires in Texas and now Australia is facing the fire season, it is time to think about fire fighter maths.

mathspig-usa-units-fire-math

 

USA UNITS

Background Story

On 5th August 1949 Wag Dodge was dropped by parachute with 14 other fire fighters into Mann Gulch, a steep-sided gully in a Montana pine forest. Fire fighters who parachute in to put out small blazes started by lightening are called Smoke Jumpers. As they worked their way down the sides of the gully the breeze was blowing away from them. But the wind soon shifted. This produced an updraft, which increases the speed of the fire front. The 15 Smoke Jumpers turned and started running for their lives uphill.

HOW FAST CAN YOU RUN?

USA UNITS

Time Trial:

Mark out a 30ft course. Make 3 time trials.

t1 =

t2 =

t3=

Average your time:

tav = (t1 + t2 + t3)/ 3 =

Your Speed S = 30/tav   ft/sec

mathspig-firefirghter-maths-1

HOW FAST IS A GRASS FIRE?

This will, of course,  vary depending on the wind speed. A typical grass fire in Australia in a flat area can travel at 12mph (up to 20mph) in a gentle breeze.

Fire Front Speed Grass Fire

Fire Front Speed = 12 mph = 12 x 5280/(60 x 60)

                           = 17.6 ft/sec

                           = 18 ft/sec

mathspig-firefighter-maths-2

CAN YOU OUT RUN A FIRE?

Average Running Speed Boy 13–14 yo = 10 ft/sec

Average Running Speed Girl 13–14 yo = 8 ft/sec

We’ll assume, boy or girl, that you are really motivated and can run away from the fire at top speed of 10 ft/sec and -Wow! – this is easy math. Now calculate the distance you can run and the fire front moves in 10 secs intervals up to 1 minute.

mathspig-fire-fighter-table-4

This is not looking good. See more Firefighters Need Maths here.

We can do very accurate calculations using simultaneous equations. Wildfire Algebra: Detailed Worksheet using simultaneous equations and solutions  here.

NOW YOU ARE RUNNING UP HILL. WHAT HAPPENS?

mathspig-cfa-diag-usa-units

We’ll assume, due to being motivated by having a fire licking your heels, that you can run at your top speed up hill for a short time, at least. But here is the problem.

Heat rises and so there is a Chimney Effect pushing the fire uphill. The rule of thumb used by fire fighters is:

Each 10º increase in slope, the fire front speed doubles.

mathspig-fire-fighter-table-5

Now you can calculate the distance travelled up a slope at a 30º angle.

Don’t forget you can use the WEB 2.0 Calculator here

mathspig-fire-fighter-table-6

Even at your top running speed, which is unlikely up a slope, you can run 1080 ft in 1 minute. In that time the forefront has moved 8640 ft or 1.6 miles. It depends how far away you are from the fire front when you start running, but it seems likely that you cannot out run this fire front.

Again we can do very accurate calculations using simultaneous equations.

See Firefighters Need Maths here.

Wildfire Algebra Worksheet and solutions  here.

CAN YOU OUT RUN A WILD FIRE?

High winds can turn a bush or forrest fire into a WILD FIRE with wind speeds up to 70 mph and temperatures up to 2000 °C, which can and does melt glass and cars.

The fire front speed doubles with every 10º, so speeds for the fire front in a strong wind can reach 140 mph, 210 mph and up to 280 mph.

20o-angle-mathspig-2

What happened to the Smoke Jumpers?

When the fire front changed direction Wag Dodge and 14 other Smoke Jumpers found themselves running for their lives up a steep slope. What did Wag do next?

ANS: Here’s the amazing thing. Wag realised he could not out run the fire at that point. So he stopped. Took off his back pack. Took out some MATCHES and lit a fire in the grassy patch in front of him. Just before the firewall hit he threw himself face down on the burnt patch. He survived. The other 14 firefighters did not.

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FOOTBALL MATHS: THE LONGEST KICK

June 3, 2016

Mathspig Football Maths 1

Mathspig football Maths 1a

Mathspig Football Maths 2aMathspig Football Maths 3Mathspig Football Maths 4

………………………………………………

The following maths is suitable for Year 9+

but can be presented to lower grades just to show

maths is cool!

Mathspig Football Maths 5Mathspig Football Maths 6bMathspig Football Maths 7

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MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig …………………………………………………. with Kerry Cue

March 1, 2016

 

1. MATHSPIG CUBISTHellooooooooo My Sweet Little Picassos,

Mathspig has gone, like, totally Cubist this month. You too can turn any portrait of yourself into a Cubist Master piece (See Mathspig portrait above), here.

Mathspigs maths friends, Lyn and Erwin, who I met at the 13e Salon Culture & Jeux Mathematique, Paris, have sent me a reminder about their amazing Cryptocube construction kit (Mathspig is twirling one above) . This is not for the faint hearted. It’s a Big Maths challenge, but well worth the investment especially for schools. You can learn more about the Zometool Cryptocube construction kit here.

Meanwhile, here are links to 10 Amazing Ways to See a Cube:

1. Tube Cube

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

2. Folded Paper Cube

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

3. Anamorphic Cube

>>>>>>>>>>>>>>>>>>>>>>>>

4. Floating Cube

>>>>>>>>>>>>>>>>>>>>>>>>

5. Street Art Cube

>>>>>>>>>>>>>>>>>>>>>>>>

6. EDIBLE CUBE

>>>>>>>>>>>>>>>>>>>>>>>>

7. GIANT CUBE

>>>>>>>>>>>>>>>>>>>>>>>>

8. Spaghetti Cube

>>>>>>>>>>>>>>>>>>>>>>>>

9. Fashion Cube

>>>>>>>>>>>>>>>>>>>>>>>>

10. CUBIC BUBBLE

…………………………………………………………….

Toodles

…………………………………………………….

Mathspig

 

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1. Tube Cube

February 17, 2016

10 Amazing ways to see a cube

The Tube Cube is made from straws and hat elastic (Steps 1 – 9 below). The effect is quite amazing. The TUBE CUBE can then be used to make a CUBIC BUBBLE here.

Don’t show your Middle School students these instructions. Just give them access to some straws, hat elastic, rulers and scissors and ask them to make and then photograph their cube. That’s the challenge Mathspiggies. But the end result (See  Step 9) is awesome.

Mathspig Cube 1.1

Mathspig Cube 1.2

Mathspig Cube 1.3

Mathspig Cube 1.4

Mathspig Cube 1.5

Mathspig Cube 1.6

Mathspig Cube 1.7

Mathspig Cube 1.8

Step 8: The TUBE CUBE can be flatened into a hexagon.

Mathspig Cube 1.9

Step 9: The TUBE CUBE can be turned into an art work. This pic was taken in daylight.  The cube was positioned at an angle on a black sheet of paper with one corner set in Blu Tack. WOW!

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2. Folded Paper Cube

February 15, 2016

10 Amazing ways to see a cube

You can find a number of ways to fold an origami cube on the web. Jeremey Shafer will show you how to fold a seamless cube (below) here.

It’s a bit tricky. Wikihow has very clear instructions on how to fold a simple paper cube here. 

Screen shot 2016-02-15 at 3.39.33 PM

But, Mathspig prefers the paper cube designed by Phillip Stromberg of the Netherlands.His cube calendar (below) comes inside one of these paper cubes. This was a very spooky calendar as Mathspig could see her life disappearing in front of her eyes for one whole year. ARrrrgh!

Stromberg Cube Calendar

Here is the way to fold a Phillip Stromberg cube:

Mathspig Cube 2.1

Step 1: Draw up a grid on cardboard 7 x 6 square.

Mathspig used 8 cm squares on paper. Cardboard would make a stronger cube.

Mathspig Cube 2.2

Step 2: Count of squares and draw this pattern.

Mathspig Cube 2.3

Step 3: Cut out the cube template.

Mathspig Cube 2.4

Step 4: Use scissors to score all folding edges.

Mathspig Cube 2.5

Step 5: Score the perpendicular bisectors of the isoceles triangles. Ha Ha! I’ve always wanted to say that!!!!

Mathspig Cube 2.6

Step 6: Fold the cube sides up, tucking the extended flaps over the triangles.

Mathspig Cube 2.7

Step 6: Fold down the cube lid!!!

OK! It may take some practice. But mathspig likes her cube.

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3. Anamorphic Cube

February 11, 2016

10 Amazing ways to see a cube

This is quite a challenge. The idea is to draw an anamorphic cube so that the image, once projected onto a curved surface looks like a cube. You will find the template or graph for this exercise on Mathspig here.

It took me several goes to get it … sort of … right.

Mathspig Cube 3.1

Step 1: Make silver foil/cardboard cylinder to fit the dotted circle on the grid below.

Mathspig Cube 3.2

Step 2: Draw in the corners of the cube on the square grid and match these corners on the curved grid.

Mathspig Cube 3.3

Step 3: Draw in the three vertical sides of the cube and match these lines on the curved grid.

Mathspig Cube 3.4

Step 4: Draw in the top and bottom horizontal sides of the cube on the square grid and match these on the curved grid.

Getting tricky now.

Mathspig Cube 3.5

Step 5: Draw in the blue, orange and green sides of the cube on the square grid and match these on the curved grid.

Mathspig Cube 3.6

Step 6: Place silver cylinder on dotted circle.

Mathspig Cube 3.7

Step 7: Can you see the cube?

Mathspig Cube 3.8

Step 8: Here’s a closer look.

An even better Anamorphic Cube.

Anamorphic Illusion Art of a Rubik’s Cube

By John Snow

Amazing Anamorphic Illusions by Brusspup