You + Me = Us

Developing Thinking Skills and Personal Capabilities

Developing Knowledge, Understanding and Skills

• Challenging the routine method
• Generating possible solutions
• Valuing other people’s ideas to stimulate own thinking
• Valuing the unexpected or surprising
  (Being Creative)

• Using a range of methods for representing information
  (Managing Information)

• Developing active deep understanding
• Coping with challenges and making decisions
• Using appropriate vocabulary to enhance explanations
• Making predictions and examining evidence
  (Thinking, Problem-Solving and Decision Making)

• knowledge and understanding of:
  • Number,
  • Algebra,
  • Shape, Space and Measures,
• the creative use of emerging technologies to enhance mathematical understanding;

• creative thinking in their approach to solving mathematical problems,
• increasing competence in pencil and paper methods,
• increasing confidence in the use of mathematical language and notation.

How have different cultures used mathematics and number?

Learning Intentions
Pupils are learning…

Possible Learning, Teaching and Assessment Activities

… about number systems based on counting in 20s rather than 10?
Problem-Solving

… to use a range of methods for representing information.
Managing Information

Yoruba numbers Yoruba people live in West Africa. Their number system is structured differently to ours.

For example, 35 = 2 x 20 – 5, 65 = 3 x 20 + 5. 11 = 20 – 10 + 1.

Start with a multiple of 20. Either add on or subtract 1,2,3,4 or 5.
It may be necessary to subtract 10 first.
Get students to find all the Yoruba numbers up to 200.

Why are numbers after 200 more difficult to find? As an extension activity, pupils could research why the Yoruba number system is based on 20. They could also find out if other cultures developed a number system based on 20 or around a number other than 10.

Mathematical Skills:

Revise the concept of multiples.
Revise order of operations (BODMAS)
Practise multiplication and addition)

… about different methods of multiplication. Thinking, Problem-Solving and Decision Making

… to challenge the routine method.
Being Creative

Russian multiplication . This involves doubling and halving the 2 numbers involved.

Mathematical Skills:

Reinforce their experience of odd and even numbers.
Practise both mental and pencil & paper methods for doubling and halving numbers.

… to develop active deep understanding.

… to cope with challenges and make decisions.
Thinking, Problem-Solving and Decision Making

 … to use appropriate vocabulary to enhance explanations.

 … to challenge the routine method.
Being Creative

Egyptian multiplication An alternative method of multiplying two 2-digit numbers based on powers of 2. Write down one of the numbers to be multiplied in the first column and the digit 1 in the second column. Double each number. The column beginning with 1 will form powers of 2. Continue doubling until the power of 2 is greater than the second number in the multiplication. Choose the powers of 2 that form a sum equal to the second number in the multiplication. Highlight these rows. Add numbers in the highlighted first column to get the answer. Check answer with a calculator. E.g. 24 x 17:

answer = 24 + 384 = 408

Mathematical Skills:

Practise using powers of 2.

… to work confidently with fractions.

… to do conversions with equivalent fractions.

…about trial and error methods.

… to generate possible solutions, try out alternative approaches and evaluate outcomes.
Being Creative

… to experiment with different actions.

 … to value other people’s ideas to stimulate own thinking. Working with Others

 … to see opportunities in mistakes and failures;

 … how to learn from and build on others’ ideas.

Egyptian Fractions

With the exception of 2/3, the Egyptians expressed all fractions as a sum of unit fractions. For example, ¾ = ½ + ¼, 2/7 = ¼ + 1/28.

Get students to write out other fractions in this way. They may need a hand to get started by listing the following: ¾, 2/5, 3/5, 4/5, 5/6. 2/7, 3/7, 4/7, 5/7, 6/7, 3/8, 5/8, 7/8 etc.

Discuss possible reasons why the Egyptians adopted this complicated system for representing fractions. This links to how they ran their economy.

… to appreciate that other cultures have developed a rich mathematical tradition.

… to use appropriate technology for the calculations.

…to develop active, deep understanding, cope with challenge and, make decision. Thinking, Problem-Solving and Decision Making

… to examine evidence and distinguish fact from opinion.

… to challenge the routine method.

Chinese primes

Revise the concept of prime numbers. Using ‘Sieve of Erastothenes’, find all the prime numbers below 200. Use the following Chinese method to generate prime numbers and compare with the list of prime numbers you made.

For any number, n: Calculate 2ⁿ; subtract 2; if the answer is divisible by n then n is prime.

E.g. n = 3: 2³ = 8; 8 – 2 = 6; 6/3 = 2 à so 3 is a prime number

n = 9; 2⁹ = 512; 512 – 2 = 510; 510 is not divisible by 9 so 9 is not a prime number.

Does the list of primes generated by this Chinese method agree with their first list? Discuss how prime numbers are different. Research what the biggest known prime number. Give reasons why people have a fascination with primes. Find out who uses them.

Mathematical Skills:
Understand concept of prime numbers.
Accurately work with powers of 2.

What new Mathematics did the Indian mathematician Kaprekar discover in the 20th century?

Learning Intentions Pupils are learning…

Possible Learning, Teaching and Assessment Activities

… to spot patterns.

… to verbally describe these patterns.

… to use technology such as calculators and spreadsheets to aid progress.

… to value the unexpected or surprising.
Being Creative

…to make prediction and examine evidence. Thinking, Problem-Solving, Decision Making

Kaprekar was an Indian Mathematician. In 1949 he discovered the Kaprekar constant, the number 6174 .

Introduce the activity by demonstrating with 2 digit numbers. Avoiding trivial examples, (11, 33 etc), choose 2 digits. Subtract the smallest number they form from the largest number. The difference will always be a multiple of 9.

If you continue the activity, using the digits of the difference, you will be left with 9.

For example:

Either individually or in groups, pupils investigate 3 digit numbers in the same way. Let them figure out and then compare their answers. 3 digit numbers always end up as 495. This would be an opportunity to use a spreadsheet for the calculations. Ask pupils to describe any patterns that they see.

Extend the task to investigate 4 digit numbers. These always end up as 6174.

Mathematical Skills:
Consolidate number skills.
Practise working with place values.

… to explore a branch of mathematics developed in another culture.

… to use spreadsheet for calculations. Thinking, Problem Solving, Decision Making

… to generate possible solutions.

When a Kaprekar number is squared, the digits of the answer add up to the Kaprekar number.

1² = 1 » 1 + 0 = 1

9² = 81 » 8 + 1 = 9

So 1 and 9 are Kaprekar numbers. Are there any more?

Calculators or spreadsheets will be necessary for this task.

Mathematical Skills:
Understand and use the term ‘square’.

How are shapes and patterns important to other cultures?

Learning Intentions
Pupils are learning…

Possible Learning, Teaching and Assessment Activities

... to appreciate how this pattern reflects Islamic culture.

... to link Maths with Art and Design.

… to experiment with different designs.
Being Creative

… to use mathematical terms such as congruent and intersect.

… to value other people’s ideas to stimulate own thinking.
Being Creative

… to confidently use a compass.

… to explore patterns that can be made with circles.

Instructions for Islamic tiles can be found at the following link:
http://www.counton.org/explorer/
morphing/islamic-tilings

The following example of Islamic pattern is based on circles:

1. Draw a horizontal line across the page.
2. Using a compass, draw a circle in the middle of the line.
3. Placing the compass at the points of intersection between the line and the circle, draw 2 more circles.
4. Draw 4 more circles placing the compass at the points of intersection.
5. Continue outwards, drawing more circles in this manner.
6. Connect the centres of each circle with straight lines to get characteristic Islamic pattern

This activity could also be carried out using LOGO. As well as drawing the circles individually, pupils could put together a programme to draw the pattern.

… to describe transformations through this context using mathematical language.

… to explore symmetry.

… to experiment with different designs, actions and outcomes.
Being Creative

… to sequence, order, classify, make comparisons.
Thinking, Problem-Solving, Decision Making

This could be carried out in collaboration with the Art and Design or Home Economic Department.

Give pupils a selection of quilt designs. Ask them to describe the patterns and relationships between the shapes under the following headings:

• Lines of symmetry
• Rotational symmetry
• Congruency
• Similarity

Examples of Amish quilt designs can be found on the internet.

http://quilting.about.com/gi/dynamic/
offsite.htm?site=

http://206.204.3.133/dir%5Fnii/nii%
5Fesprit.html

An interactive task manipulating patterns and shapes on Amish Quilts can be found at:

http://m759.freeservers.com/puzzle.html

How have different cultures influenced the mathematics and numbers that we use today?

Learning Intentions
Pupils are learning…

Possible Learning, Teaching and Assessment Activities

 … to use the Internet effectively to research how Maths developed in other cultures and communicate that information.
Managing Information

Investigate how maths developed in countries such as China and India independently of Europe and other Western countries.

This would be an opportunity for group work and using ICT.

Opportunity to assess Using ICT and Communication.

Explore how European digits developed from Arabian numbers.

Convert Egyptian number plates into a European format.

Explore Egyptian hieroglyphics:

http://atschool.eduweb.co.uk /ufa10/hierogly.htm

… about contexts in which Roman numerals are used in modern life.

 … to consider the effectiveness of different methods.
Thinking, Problem -Solving, Decision Making.

Teacher puts up a list of years with the equivalent written in Roman numerals. Pupils guess which numbers the different Roman symbols represent.

Having learned about Roman numerals, think about where they are used. Discuss advantages and limitations (how easy would it be to multiply with them; use them for phone numbers etc.?)

On a given evening record year of production of TV programs watched. Convert the Roman numeral them into the standard format for writing the year.

Mathematical Skills:
Write and understand Roman numerals. 

Are there other ways of structuring numbers?

Learning Intentions Pupils are learning…

Possible Learning, Teaching and Assessment Activities

… to understand a language better by analysing its number structure.

… about numbers in other languages.

… about exploring further and questioning.
Being Creative 

Explore how numbers are structured in different languages.

For example, in French, 80 =4 x 20; 75 = 50 + 15; 93 = 4 x 20 + 13

How can Mathematical skills and concepts be applied to issues faced in developing countries?

Learning Intentions
Pupils are learning…

Possible Learning, Teaching and Assessment Activities

…to apply mathematical skills to real life contexts.

… to break a task into subtasks and plan next steps.

…to select the most appropriate method for a task.

… to develop active, deep understanding and cope with challenges and make decisions.
Thinking Skills Problem-Solving, Decision Making

… to experiment with questions and ideas.
Being Creative   

Problem solving in contexts of water and food supply, in contexts of trading, locating water supplies, etc., using case studies and simulations.

A number of resources are published for this:

• Summing up the World Mathematics Activities with a Global Perspective published by DEED
• Human Rights in the Curriculum Mathematics published by Amnesty

Can you discover the tactics behind mathematical games from other cultures?

Learning Intentions
Pupils are learning…

Possible Learning, Teaching and Assessment Activities

… to develop logical thinking and mathematical skills using games from different cultures.

… to generate possible solutions, try out alternative approaches and evaluate outcomes.
Being Creative

Below are some maths/logic games from around the world.

Mancala game from Africa http://imagiware.com/mancala

Cayuga , a game based on probability, played by indigenous Americans; a similar game based on expected values known as Dish is played throughout Africa http://www.nativetech.org/ games/dicegame

Mu Torere – Maori – played on eight pointed star design http://home.aut.ac.nz /staff/jcrawfor/java/
muTorere/ MuTorere.htm

http://www.nrich.maths.org.uk
/public/viewer.php? obj_id=1255&part=index
&refpage=monthindex.php

http://www.barcelona2004.org
/eng/eventos/juegos/ bancos/mu.htm

Mathematical games on a cross cultural theme: http://atschool.eduweb.co.uk/ufa10/games.htm

What did you learn in this unit about maths, other cultures and yourself?

Learning Intentions Pupils are learning…

Possible Learning, Teaching and Assessment Activities

 …to reflect on their own learning.

 …to assess their strengths and weaknesses.

 … to identify aspects of work that might be improved.
Self Management

3-2-1 activity

Pupils identify and record:

• 3 new things they have learned in this unit
• 2 areas of Maths they have improved in
• 1 area in Maths to focus on for improvement

Links with Key Elements: 

Links with Learning for Life and Work:

Cultural Understanding

Mutual Understanding through collaborative working.

Ethical Awareness

Education for Sustainable Development

Moral Character

Local and Global Citizenship
Key Concept - Diversity and Inclusion

Development of Learning Outcomes

• Demonstrate mental mathematical capability with simple problems.
• Decide on the appropriate method and equipment to solve problems ­– mental, written, calculator, mathematical instruments or a combination of these.
• Demonstrate financial capability in a range of relevant everyday contexts.
• Research and manage information effectively to investigate and solve mathematical problems, using ICT where appropriate.
• Show deeper mathematical understanding by thinking critically and flexibly, solving problems and making informed decisions, using ICT where appropriate.
• Demonstrate creativity and initiative when developing ideas and following them through;
• work effectively with others.
• Demonstrate self-management by working systematically, persisting with tasks, evaluating and improving own performance.
• Communicate effectively in oral, visual, written, mathematical and ICT formats, showing clear awareness of audience and purpose.