Word of Warning teachers: Be cognisance those students in the upper school who make have issues with their body issues.
Friday, 28 February 2020
Ratios, Using String and partners to develop ratios
Sometimes we are too quick to get out the tape measure, rototape, to get students involved in measurement. This great little activity just uses lengths of string, for the students to work out the ratios between different parts of the body.
Word of Warning teachers: Be cognisance those students in the upper school who make have issues with their body issues.
Enjoy: dont forget to feedback how you got on.
Word of Warning teachers: Be cognisance those students in the upper school who make have issues with their body issues.
Thursday, 27 February 2020
Going Loopy - Moebius Strip and Toplogy
For many years I explored with students and teachers the interesting Moebius Strip. It was great for predicting outcomes, physical involvement and of course the AHA or WOW moment! I enjoyed it and I am sure most of th students did as well, as "they did not see it as maths!"
The Moebius Strip is a subset within Topology another branch of Mathematics, that is often useful but really looked at or understood!
As a start you might like to show the "Mr Bean at the Seaside" and then see if the students can replicate his antics.
Secondly the handcuff problem is a great way to get students cooperating with each other, as they wont be able to do it on their own. WARNING it will bring out some levels of frustration
Third, the scissor problem could be a "Math's Table Activity" Have a couple of students set it up.
Finally an indepth exploration with strips of paper, twists and none twists, predictions and checking. Starting by reading the Paul Bunyan Story
Mr Bean at the Seaside https://www.youtube.com/watch?v=ZWCSQm86UB4
In “At the Seaside” Mr Bean managed to put his togs on without taking off his trousers first. Explore ways of putting your togs on without taking your jeans off first Be prepared to show, and/or explain how this can be done.
Handcuffs
Find a partner and two lengths of string, each about 1 metre long. Tie one length of string to your partner’s wrists.
Now tie the other piece of string to your wrist, loop it past your partner’s string and then tie it to your other wrist.
Without cutting, untying, breaking the string, disconnect yourself from you partner. Explain how you became “unjoined”.
Scissor Puzzle
Make a scissor puzzle.
Find a way to remove the string without cutting or unfastening.
Paul Bunyan versus the Conveyor Belt
By WILLIAM HAZLETT UPSON*
"One of Paul Bunyan's most brilliant successes came about not because of brilliant thinking, but because of Paul's caution and carefulness. This was the famous affair of the conveyor belt.
Paul and his mechanic, Ford Fordsen, had started to work a uranium mine in Colorado. The ore was brought out on an endless belt which ran half a mile going into the mine and another half mile coming out - giving it a total length of one mile. It was four feet wide. It ran on a series of rollers, and was driven by a pulley mounted on the transmission of Paul's big blue truck "Babe." The manufacturers of the belt had made it all in one piece, without any splice or lacing, and they had put a half-twist in the return part so that the wear would be the same on both sides.
After several months' operation, the mine gallery had become twice as long, but the amount of material coming out was less. Paul decided he needed a belt twice as long and half as wide. He told Ford Fordsen to take his chain saw and cut the belt in two lengthwise.
"That will give us two belts," said Ford Fordsen. "We'll have to cut them in two crosswise and splice them together. That means I'll have to go to town and buy the materials for two splices."
"No," said Paul. "This belt has a half-twist -- which makes it what is known in geometry as a Moebius strip."
"What difference does that make?" asked Ford Fordsen.
"A Moebius strip," said Paul Bunyan, "has only one side, and one edge, and if we cut it in two lengthwise, it will still be in one piece. We'll have one belt twice as long and half as wide."
"How can you cut something in two and have it still in one piece?" asked Ford Fordsen.
Paul was modest. He was never opinionated. "Let's try this thing
out," he said.
They went into Paul's office. Paul took a strip of gummed paper about two inches wide and a yard long. He laid it on his desk with the gummed side up. He lifted the two ends and brought them together in front of him with the gummed sides down. Then he turned one of the ends over, licked it, slid it under the other end, and stuck the two gummed sides together. He had made himself an endless paper belt with a half-twist in it just like the big belt on the conveyor.
"This," said Paul, "is a Moebius strip. It will perform just the way I said - I hope."
Paul took a pair of scissors, dug the point in the centre of the paper and cut the paper strip in two lengthwise. And when he had finished sure enough - he had one strip twice as long, half as wide, and with a double twist in it.
Ford Fordsen was convinced. He went out and started cutting the big belt in two. And, at this point, a man called Loud Mouth Johnson arrived to see how Paul's enterprise was coming along, and to offer any destructive criticism that might occur to him. Loud Mouth Johnson, being Public Blow-Hard Number One, found plenty to find fault with.
"If you cut that belt in two lengthwise, you will end up with two belts, each the same length as the original belt, but only half as wide."
"No," said Ford Fordsen, "this is a very special belt known as a Moebius strip. If I cut it in two lengthwise, I will end up with one belt twice as long and half as wide.'.
"Want to bet?" said Loud Mouth Johnson.
"Sure," said Ford Fordsen.
They bet a thousand dollars. And, of course, Ford Fordsen won. Loud Mouth Johnson was so astounded that he slunk off and stayed away for six months. When he finally came back he found Paul Bunyan just starting to cut the belt in two lengthwise for the second time.
"What's the idea?" asked Loud Mouth Johnson.
Paul Bunyan said, "The tunnel has progressed much farther and the material coming out is not as bulky as it was. So I am lengthening the belt again and making it narrower."
"Where is Ford Fordsen ?
Paul Bunyan said, "I have sent him to town to get some materials to splice the belt. When I get through cutting it in two lengthwise I will have two belts of the same length but only half the width of this one. So I will have to do some splicing."
Loud Mouth Johnson could hardly believe his ears. Here was a chance to get his thousand dollars back and show up Paul Bunyan as a boob besides. "Listen," said Loud Mouth Johnson, "when you get through you will have only one belt twice as long and half as wide."
"Want to bet?"
"Sure."
So they bet a thousand dollars and, of course, Loud Mouth Johnson lost again. It wasn't so much that Paul Bunyan was brilliant. It was just that he was methodical. He had tried it out with that strip of gummed paper, and he knew that the second time you slice a Moebius strip you get two pieces - linked together like an old fashioned watch chain.
*Reprinted from the Ford Time.v, by permission of the Ford Motor Company.
Investigating The Moebius Strip
Materials required: Sellotape, strips of paper (30cm x 3cm), scissors, coloured pens.
•Take a number of strips of paper and then in each strip make a loop by joining the ends. Try to make each loop different by twisting the strip a different number of times before joining the ends.
•Take a pair of scissors and cut each loop - lengthwise, predict what is likely to happen before you cut. Try by cutting down the middle, or cutting the strip in 1/3 or 2/3 sections length ways.
Make a table of your predictions and record your results.
Take one strip of paper, make a half twist before joining the ends to make a loop.
How many sides does this loop have? (Remember that a side is a flat surface -not an edge).
Find a way to prove the number of sides that your loop has?
More Loops For Investigation.
* Take two strips of paper and lay them on top of each other. Now connect the ends after making 1 twist.
* Predict: What do you have? A double Moebius strip? A single etc.?
* Now separate the two loops and compare with your results as in your table
* Repeat with three strips of paper.
Investigations Without Loops
* Make two loops of paper without twists.
Sellotape the loops at right angles to each other.
Predict - what do you expect to get if you cut around the middle of each strip?
Now check it out.
Investigate what happens when you use :
* 3 Loops at right angles
* 4 loops at right angles
* 2 loops joined at 60° to each other
* 3 loops joined at 60° to each other
* 4 loops joined at 60° to each other
* All of the above if each loop is a Moebius strip
* Explain how you get 5 squares from 4 joined loops.
For Excellence
You have been investigating ideas that are part of the area of mathematics called “Topology”.
Other areas of topology include: Map colouring
Inside and outside closed curves
The maths of knots
The Klein Bottle, Donuts
Choose an area of topology. Use the internet and other resources to make your own report on that aspect of topology.
The Moebius Strip is a subset within Topology another branch of Mathematics, that is often useful but really looked at or understood!
As a start you might like to show the "Mr Bean at the Seaside" and then see if the students can replicate his antics.
Secondly the handcuff problem is a great way to get students cooperating with each other, as they wont be able to do it on their own. WARNING it will bring out some levels of frustration
Third, the scissor problem could be a "Math's Table Activity" Have a couple of students set it up.
Finally an indepth exploration with strips of paper, twists and none twists, predictions and checking. Starting by reading the Paul Bunyan Story
Mr Bean at the Seaside https://www.youtube.com/watch?v=ZWCSQm86UB4
In “At the Seaside” Mr Bean managed to put his togs on without taking off his trousers first. Explore ways of putting your togs on without taking your jeans off first Be prepared to show, and/or explain how this can be done.
Handcuffs
Find a partner and two lengths of string, each about 1 metre long. Tie one length of string to your partner’s wrists.
Now tie the other piece of string to your wrist, loop it past your partner’s string and then tie it to your other wrist.
Without cutting, untying, breaking the string, disconnect yourself from you partner. Explain how you became “unjoined”.
Scissor Puzzle
Make a scissor puzzle.
Find a way to remove the string without cutting or unfastening.
GOING LOOPY!
Paul Bunyan versus the Conveyor Belt
By WILLIAM HAZLETT UPSON*
"One of Paul Bunyan's most brilliant successes came about not because of brilliant thinking, but because of Paul's caution and carefulness. This was the famous affair of the conveyor belt.
Paul and his mechanic, Ford Fordsen, had started to work a uranium mine in Colorado. The ore was brought out on an endless belt which ran half a mile going into the mine and another half mile coming out - giving it a total length of one mile. It was four feet wide. It ran on a series of rollers, and was driven by a pulley mounted on the transmission of Paul's big blue truck "Babe." The manufacturers of the belt had made it all in one piece, without any splice or lacing, and they had put a half-twist in the return part so that the wear would be the same on both sides.
After several months' operation, the mine gallery had become twice as long, but the amount of material coming out was less. Paul decided he needed a belt twice as long and half as wide. He told Ford Fordsen to take his chain saw and cut the belt in two lengthwise.
"That will give us two belts," said Ford Fordsen. "We'll have to cut them in two crosswise and splice them together. That means I'll have to go to town and buy the materials for two splices."
"No," said Paul. "This belt has a half-twist -- which makes it what is known in geometry as a Moebius strip."
"What difference does that make?" asked Ford Fordsen.
"A Moebius strip," said Paul Bunyan, "has only one side, and one edge, and if we cut it in two lengthwise, it will still be in one piece. We'll have one belt twice as long and half as wide."
"How can you cut something in two and have it still in one piece?" asked Ford Fordsen.
Paul was modest. He was never opinionated. "Let's try this thing
out," he said.
They went into Paul's office. Paul took a strip of gummed paper about two inches wide and a yard long. He laid it on his desk with the gummed side up. He lifted the two ends and brought them together in front of him with the gummed sides down. Then he turned one of the ends over, licked it, slid it under the other end, and stuck the two gummed sides together. He had made himself an endless paper belt with a half-twist in it just like the big belt on the conveyor.
"This," said Paul, "is a Moebius strip. It will perform just the way I said - I hope."
Paul took a pair of scissors, dug the point in the centre of the paper and cut the paper strip in two lengthwise. And when he had finished sure enough - he had one strip twice as long, half as wide, and with a double twist in it.
Ford Fordsen was convinced. He went out and started cutting the big belt in two. And, at this point, a man called Loud Mouth Johnson arrived to see how Paul's enterprise was coming along, and to offer any destructive criticism that might occur to him. Loud Mouth Johnson, being Public Blow-Hard Number One, found plenty to find fault with.
"If you cut that belt in two lengthwise, you will end up with two belts, each the same length as the original belt, but only half as wide."
"No," said Ford Fordsen, "this is a very special belt known as a Moebius strip. If I cut it in two lengthwise, I will end up with one belt twice as long and half as wide.'.
"Want to bet?" said Loud Mouth Johnson.
"Sure," said Ford Fordsen.
They bet a thousand dollars. And, of course, Ford Fordsen won. Loud Mouth Johnson was so astounded that he slunk off and stayed away for six months. When he finally came back he found Paul Bunyan just starting to cut the belt in two lengthwise for the second time.
"What's the idea?" asked Loud Mouth Johnson.
Paul Bunyan said, "The tunnel has progressed much farther and the material coming out is not as bulky as it was. So I am lengthening the belt again and making it narrower."
"Where is Ford Fordsen ?
Paul Bunyan said, "I have sent him to town to get some materials to splice the belt. When I get through cutting it in two lengthwise I will have two belts of the same length but only half the width of this one. So I will have to do some splicing."
Loud Mouth Johnson could hardly believe his ears. Here was a chance to get his thousand dollars back and show up Paul Bunyan as a boob besides. "Listen," said Loud Mouth Johnson, "when you get through you will have only one belt twice as long and half as wide."
"Want to bet?"
"Sure."
So they bet a thousand dollars and, of course, Loud Mouth Johnson lost again. It wasn't so much that Paul Bunyan was brilliant. It was just that he was methodical. He had tried it out with that strip of gummed paper, and he knew that the second time you slice a Moebius strip you get two pieces - linked together like an old fashioned watch chain.
*Reprinted from the Ford Time.v, by permission of the Ford Motor Company.
Investigating The Moebius Strip
Materials required: Sellotape, strips of paper (30cm x 3cm), scissors, coloured pens.
•Take a number of strips of paper and then in each strip make a loop by joining the ends. Try to make each loop different by twisting the strip a different number of times before joining the ends.
•Take a pair of scissors and cut each loop - lengthwise, predict what is likely to happen before you cut. Try by cutting down the middle, or cutting the strip in 1/3 or 2/3 sections length ways.
Make a table of your predictions and record your results.
Take one strip of paper, make a half twist before joining the ends to make a loop.
How many sides does this loop have? (Remember that a side is a flat surface -not an edge).
Find a way to prove the number of sides that your loop has?
* Take two strips of paper and lay them on top of each other. Now connect the ends after making 1 twist.
* Predict: What do you have? A double Moebius strip? A single etc.?
* Now separate the two loops and compare with your results as in your table
* Repeat with three strips of paper.
Investigations Without Loops
* Make two loops of paper without twists.
Sellotape the loops at right angles to each other.
Predict - what do you expect to get if you cut around the middle of each strip?
Now check it out.
Investigate what happens when you use :
* 3 Loops at right angles
* 4 loops at right angles
* 2 loops joined at 60° to each other
* 3 loops joined at 60° to each other
* 4 loops joined at 60° to each other
* All of the above if each loop is a Moebius strip
* Explain how you get 5 squares from 4 joined loops.
For Excellence
You have been investigating ideas that are part of the area of mathematics called “Topology”.
Other areas of topology include: Map colouring
Inside and outside closed curves
The maths of knots
The Klein Bottle, Donuts
Choose an area of topology. Use the internet and other resources to make your own report on that aspect of topology.
Sunday, 23 February 2020
International Day of Maths - a starter activity below
Just a few days to get your students humming about Maths!
International Day of Maths
The website of the IDM is www.idm314.org
Also visit everywhere.idm314.org
Plan now for lots of school wide fun Maths activities. Click on the links for free activities and ideas
14 March (3/14) is also Pi Day 3.14…..
Write 34 on a piece of paper and put it in an envelope and seal it. (Dont show the kids what you have written on the paper.)
Now give it to a student and ask them to hold it and only open it when they are asked.
On the Board have a 4 x 4 grid with the numbers 1- 16.
Now have them cross out the numbers in the same ROW and COLUMN
Now have them cross out the numbers in the same ROW and COLUMN
Ask the Students to add the 4 numbers. Then with a MAGICIAN's FLOURISH ask the student with the envelope to open it and read out the number!! WOW it is the same!!
Let me know how you get on!!!
Tuesday, 11 February 2020
I just came across this article again. Admittedly it was written in 2010 but it is still relevant today! And it is worth sharing(changing math to maths etc) with families.
The best maths education is when there is a connection between, student, parent and school.
We used this diagram for years with Family Maths.
Support for Parents and Families: Helping your Math Students
by NCTM President J. Michael Shaughnessy NCTM Summing Up, December 2010
Recently I’ve received a flurry of media requests from reporters about what to tell parents whose children ask for help with their mathematics schoolwork. Family members might ask, “What do I tell my child when I don’t remember ever seeing this type of mathematics when I was a student?” Or, “I was never good at math, so what can I do to help my child?” When I’m asked these questions, three things come to mind.
(1) Remember, mathematics is important, and we can all do it.
(2) Work together as a team with your child—don’t show how to do it.
(3) Investigate the NCTM resources that can provide assistance when helping your children with their math work.
First and foremost, I implore family members not to say, “I was never good at mathematics, either.” That response only widens the spread of our national mathematical cultural disease—that it’s acceptable not to be good at math. There is no such thing as a math gene. That is both a myth and an excuse. Doing math just takes perseverance and a positive attitude, but everyone can enjoy success with mathematics.
It is also important that as parents, family members, and adult mentors, we thank our students for asking for our help with their math work. This provides us with a golden opportunity to point out how important mathematics is in our lives—that it is essential in building structural and technological tools; empowers new discoveries in the physical, biological, and social sciences; and is also extraordinarily beautiful, as seen in the visual, spatial, and musical arts.
Second, I encourage family members to ask their students to share what they know about the particular math problem at hand. One good strategy is to ask, “Tell me what the problem is, and help me understand how you are thinking about it. What do you think we need to do?” This can buy a panicky parent a bit of time to get his or her thoughts together, and it also puts parents and children on a level playing field and quickly gets them working together as a team. This is a strategy that math tutors can also use effectively, to get a student talking about a problem and sharing what he or she already knows. Often, in the process of talking it through, a student will figure out what is needed, or the next step to take. Students always know more than they think they know—and often more than they will admit to knowing. If you do know the answer, or how to proceed, then ask more questions and try to help the students figure it out for themselves. Never give away how to do the problem. Doing that disenfranchises students. If instead we help them figure out how to proceed, then we empower them, and their mathematical confidence grows. Remember, it’s about them, not about us, when they ask for help.
Finally, I heartily recommend a visit to the Family Resources page at NCTM’s Web site. Featured resources include A Family’s Guide: Fostering Your Child’s Success in School Mathematics; Involving Families in Mathematics Education; and Figure This: Math Challenges for Families.
Resources at the site also point to mathematics in children’s literature. The Family’s Guide includes questions that parents can ask their children and their teachers about the math that they are studying, as well as information about NCTM’s Content and Process Standards. In addition, the site offers a collection of one-page responses to frequently asked questions. FigureThis is a set of 80 mathematics problems and challenges for families to investigate and enjoy. Originally created for middle school students, many of these problems are appropriate for a wide range of students, from grades 5–10, and beyond.
The Six DO’s for Families and Their Math Students
1. Be positive
2. Link mathematics with daily life
3. Make mathematics fun
4. Learn about mathematics-related careers
5. Have high expectations for your students
6. Support homework—don’t do it!
The best maths education is when there is a connection between, student, parent and school.
We used this diagram for years with Family Maths.
Support for Parents and Families: Helping your Math Students
by NCTM President J. Michael Shaughnessy NCTM Summing Up, December 2010
Recently I’ve received a flurry of media requests from reporters about what to tell parents whose children ask for help with their mathematics schoolwork. Family members might ask, “What do I tell my child when I don’t remember ever seeing this type of mathematics when I was a student?” Or, “I was never good at math, so what can I do to help my child?” When I’m asked these questions, three things come to mind.
(1) Remember, mathematics is important, and we can all do it.
(2) Work together as a team with your child—don’t show how to do it.
(3) Investigate the NCTM resources that can provide assistance when helping your children with their math work.
First and foremost, I implore family members not to say, “I was never good at mathematics, either.” That response only widens the spread of our national mathematical cultural disease—that it’s acceptable not to be good at math. There is no such thing as a math gene. That is both a myth and an excuse. Doing math just takes perseverance and a positive attitude, but everyone can enjoy success with mathematics.
It is also important that as parents, family members, and adult mentors, we thank our students for asking for our help with their math work. This provides us with a golden opportunity to point out how important mathematics is in our lives—that it is essential in building structural and technological tools; empowers new discoveries in the physical, biological, and social sciences; and is also extraordinarily beautiful, as seen in the visual, spatial, and musical arts.
Second, I encourage family members to ask their students to share what they know about the particular math problem at hand. One good strategy is to ask, “Tell me what the problem is, and help me understand how you are thinking about it. What do you think we need to do?” This can buy a panicky parent a bit of time to get his or her thoughts together, and it also puts parents and children on a level playing field and quickly gets them working together as a team. This is a strategy that math tutors can also use effectively, to get a student talking about a problem and sharing what he or she already knows. Often, in the process of talking it through, a student will figure out what is needed, or the next step to take. Students always know more than they think they know—and often more than they will admit to knowing. If you do know the answer, or how to proceed, then ask more questions and try to help the students figure it out for themselves. Never give away how to do the problem. Doing that disenfranchises students. If instead we help them figure out how to proceed, then we empower them, and their mathematical confidence grows. Remember, it’s about them, not about us, when they ask for help.
Finally, I heartily recommend a visit to the Family Resources page at NCTM’s Web site. Featured resources include A Family’s Guide: Fostering Your Child’s Success in School Mathematics; Involving Families in Mathematics Education; and Figure This: Math Challenges for Families.
Resources at the site also point to mathematics in children’s literature. The Family’s Guide includes questions that parents can ask their children and their teachers about the math that they are studying, as well as information about NCTM’s Content and Process Standards. In addition, the site offers a collection of one-page responses to frequently asked questions. FigureThis is a set of 80 mathematics problems and challenges for families to investigate and enjoy. Originally created for middle school students, many of these problems are appropriate for a wide range of students, from grades 5–10, and beyond.
The Six DO’s for Families and Their Math Students
1. Be positive
2. Link mathematics with daily life
3. Make mathematics fun
4. Learn about mathematics-related careers
5. Have high expectations for your students
6. Support homework—don’t do it!
Monday, 10 February 2020
Maths Mind Reading
(Dont forget to put your magic Clock and hat on)
1. Have your child or student to choose a number from 1 to 31, but keep it secret, or write it on paper and keep hidden
2. Now ask them to point to the cards that their number appears on, still keeping their number secret.
3. With a flourish of your magic Wand and "Abracadabra" Add, in your head, the first numbers on each card and give the answer to the child.
If they chose 23 you will find 23 on Cards 1, 2, 3, 5.
The first numbers on each card are, 1 + 2 + 4 + 16 which add to 23!!!
4. Repeat with another child or student!!!!
Please DO NOT tell them what you are doing. The real joy for the children is to investigate and discover what you are doing. They could also make their own cards.
For Older Children/Students: Use these cards
Sunday, 9 February 2020
Palindromes
02 02 2020
0202 02 02
A palindrome reads the same forward or backwards. The second of February 2020 as written above was a Palindrome.
We usually look for palindromes in words: MUM, DAD, LEVEL etc but we can also explore them in numbers.
99 is a Palindrome as are 1, 2, 3, 4, 5, 6, 7, 8, 9, but we have a problem when we reach 10. In reverse this is 01.
But wait: if we take 10 and then add its reverse 01 (10 + 01 = 11) we get 11 a Palindrome.
We call 10 a 1 step palindrome as it takes just one step to turn it into a palindrome
37 is not a palindrome so lets see what happens when we reverse and add:
37 + 73 = 110
110 + 011 = 121
WOW a two step palindrome
Most of the numbers from 1 to 100 can be turned into palindromes by reversing adding and continuing until a palindrome is reached. Each time we reverse and add it is called a step.
Find all the 1 step palindromes, 2 step palindromes, 3 step palindromes and colour accordingly on the 10 chart below.
Can you find any numbers that don't seem to be able to be converted into a palindrome?
Find palindromes for some larger numbers, what is the largest number of steps you can find for one number?
Write a sentence or phrase that is a palindrome
Read ROBERT TREBOR by Marilyn Burns in the book: Good Times: Every Kid's Book of Things to Do ISBN-10: 0553011626 or ISBN-13: 978-0553011623
Teachers check out this link N-Rich Maths a maths website from Cambridge University England. Great ideas for all ages
Subscribe to:
Posts (Atom)