Curves From Lines

This was prepared for the Family Maths Newsletter 2007. Copies of past issues are available for download from www.familymaths.org.nz  The Family Maths Trust is an Offspring from Family Math, Lawrence Hall of Science Berkeley California.  Organisations that encourage families to participate in fun math activities at home.

Cycloids

I was sent the following YouTube link http://showyou.com/v/y-pNe6fsaCVtI/crazy-circle-illusion?u=multimotion which shows a series of 'balls' moving around a circle.  The caption with this link was: "Eight Balls Moving in Straight Lines Interesting......it's all in the timing.  Eight Balls Moving in Straight Lines Who would think of this?"


A Cycloid created by tracing a point on a circle as it travels along a line, but they become more interesting when the point is traced as the wheel of circle is inside or outside of a circle.

The mathematics of cycloids is too difficult for most primary students, involves Trigonometry ratios but these students could have fun creating their own shapes, using a variation of the cycloid with the use of a drawing compass.  
Many moons ago I created a number of these off shoots and had them prepared on Overhead Projector Slides.
The ones I created consisted of drawing circles from equidistant points on a circle to a point inside or outside the circle and similarly with a point on a line with the points on the circle.
A great practice for measuring and using the compass and the students end up with interesting designs.

This from: http://www.cut-the-knot.org/pythagoras/cycloids.shtml

Halloween Problems

Some more problems from Charlotte Wilkinson www.thewilkieway.co.nz
Use these as starters and frameworks and then get your students to create and share their own Halloween Problems.  Remind them that they have to have worked out their own solutions before sharing!

Embrace Educational Change

This article from Charlotte Wilkinson www.thewilkieway.co.nz arrived by email today and I thought it was well worth sharing.  Her quotes remind me of a similar one, (I have lost) but referred to the likelihood of handwriting skills deteriorating with the advent of the Ball Point Pen as it replaced the dip or fountain pen.

CSI – Can you solve the case of the discovered bone?

Detectives have found a bone buried in a hole on the banks of the Waikato River. Forensic anthropologists have established that the bone is a radius bone, and that it is almost certainly human. The bone measures 28cm long. From some scraps of material found with the bone, detectives can conclude that the mystery person is almost certainly female. Another bone, a 44cm tibia, possibly belonging to the same person, has been found

• The detectives now need your help. 
• They need to find out the height of the mystery person along with some other measurements 
• Circumference – head (around forehead); 
• shoulders; 
• hips; & 
• knees (both knees together). 
• From these, the detectives need you to create a ‘skeleton’ model.

Can you solve it?

Teachers the students will need to collect data and find averages etc so that they can use the given information to work out the 'size' of the mystery person.   One way to may the 'skeleton' could be to have a length of adding machine tape(or card) for the backbone and then attached strips of card for the other details.  

I would love to hear the results of the investigations

Adapted from ‘The Case of the Mystery Bone’ by Doug Clarke (possibly out of print)

Another Quick and Easy Warm Up

As a visiting consultant one of the the things I always had in my pocket was a ten sided die (0-9), because invariably a teacher or class would want me to 'play' a maths game.  This one is so easy to set up but has the element of chance(probability) and place value.

Have the students to draw four boxes on a piece of paper.

Explain to them that you are going to roll the die 4 times and after each roll call out the number.  Their task is to (make the Largest 4 digit number, Smallest 4 digit number, Number closest to 5000 etc) The students are to decide which box to write the number in.  Once written it can not be moved.


Alternatives:

• Have 4 boxes with a discard box and roll the die 5 times, students are allowed to discard at any time one of the numbers

• Introduce a decimal point between one of the boxes, this will show you the teacher, that many students just do not understand very well our place value system

• For younger children just have two or three boxes

• For younger children have them record their numbers on cards and then use their cards as a lesson for 'Ordering" the numbers and statistics like Range, Mode.  (I had students display the cards in order on the classroom carpet with them sitting around so that they could all be involved)  Children love 'owning' their number

Len's Warm Up/Warm Down Maths Activity

This activity has Addition, Place Value and Probability all rolled in to one Great Warm Up.

Prepare a grid as below(this size is best for Upper Primary and Junior Secondary)  Younger children could just have 4 rows and then a total row. Distribute  OR have students prepare their own

Object: to be the student who has the highest total after all the dice rolls.

1. Teacher or student rolls a die (preferably a 0-9) the students use this number to fill in the square in ROW 1
2. The Die is rolled a second time and the students can now put the rolled number in either of the Ten's or the One's Place.
3. On the 3rd roll the number is placed in the vacant square in row 2.
4. Now it is Row 3 to be filled. After each roll of the die, the students decided whether to put the number in the Hundreds, or Tens or Ones.  (No storing of numbers OR changing once the number has been recorded.)
5. Continue in this manner for row 4 Th, H, T, O and then Rows 5-7
6. Hardest part is now to get the students to add each column to find their Grand Total!
7. The student who has the Largest Total wins, or scores points for their ‘school house’ etc.

 

Teachers: Worried about how to check the accuracy of the students answers?

• Have all students find the digital root of the TOTAL Row; This is when all the digits are added to get a single digit.  e.g. 14 666 789  becomes 1+4+6+6+6+7+8+9 OR 47 and then 4+7 is 11 and 1+1 is 2.
• Students with correct total should have the same digital root.
• Alternatively, ask one child to call out their answer and see how many have the same (if half the class or more)  ‘assume’ this is the correct one!