unit 1.1 ma thema tics as a study of cr ea ting, discer ... · unit 1.1 ma thema tics as a study of...
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Unit 1.1
Mathematics as a study of
creating, discerning and
generalizing Patterns
4/21/2016 1 Dr Priya Mathew
St.Joseph's College of Education, Mysore
Understanding what is mathematics
• Mathematics as a study of patterns:
Example 1:
Stacey's example of working on numbers like 87:
• 87; 8+7=15; 87-15=72
• 86; 8+6=14; 86-14=72
• 79; 7+9=16; 79-16=63
• 78; 7+8=15; 78-15=63
• 65; 6+5=11; 65-11=54
• 69; 6+9=15; 69-15=54
• 23; 2+3=5; 23-5=18
• 28; 2+8=10; 28-10=18
4/21/2016 2 Dr Priya Mathew
St.Joseph's College of Education, Mysore
• Would this work for all two digit numbers?
• Would this work for all three digit numbers?
• How do you show that?
4/21/2016 3 Dr Priya Mathew
St.Joseph's College of Education, Mysore
Mathematics Pattern
• Mathematics is the study of patterns.
• Studying pattern is an opportunity to:
observe
hypothesise
experiment
discover
and create generalizations
• Study of patterns alternatively termed as a study of generalizations
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Dr Priya Mathew
St.Joseph's College of Education, Mysore
What are Patterns in Mathematics?
• In Mathematics, patterns are related to the type of any event or objects.
• If we take a set of elements or numbers in which all these elements or numbers are related to each other in a specific rule, then this rule or manner is called the pattern.
• Sometimes patterns are called as sequence.
• Patterns are finite or infinite in numbers.
• For example, in the sequence 2,4,6,8, ? ,
• each number is increasing by sequence 2. Here the pattern rule is start at 2. Add 2 to each time. So, last number will be 8+2 =10.
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Dr Priya Mathew
St.Joseph's College of Education, Mysore
Pattern Rule
• Pattern rule tells how to build a pattern. We
may be able to state a pattern rule in more
than one way.
• 1,3,5,7,9,11…
• Here one pattern rule is start at 1 & add 2
each ti e .
• Another pattern rule is The odd numbers
starting at 1
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Dr Priya Mathew
St.Joseph's College of Education, Mysore
Core of a pattern
• The core of a pattern is the smallest string of elements that repeats.
In this pattern the shape and colour changes
• The core of this pattern is
• The pattern rule is black triangle & white circle.
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Dr Priya Mathew
St.Joseph's College of Education, Mysore
Numerical patterns (Number Patterns)
• A sequence of numbers which follows a certain
pattern is called number patter.
• Like , , , , , ……… is a u ber patter whose first term is 1 and add 4 every time to get
the other terms.
Examples:
• Fibonacci Number Pattern
• Growing number pattern
• Repeating number pattern
• Geometric number pattern etc.
4/21/2016 9 Dr Priya Mathew
St.Joseph's College of Education, Mysore
Geometric pattern (Patterns of shapes)
• A geometric pattern is a kind of pattern
formed of geometric shapes and typically
repeating like a wallpaper.
• The geometric pattern is used in a variety of
ways in our daily life.
• For example, a dress pattern is a template,
and a wall paper pattern may be an infinitely
repeating design.
4/21/2016 12 Dr Priya Mathew
St.Joseph's College of Education, Mysore
Abstract Patterns
• Any of the five senses may directly observe
patterns.
• Conversely, abstract patterns in science,
mathematics, or language may be observable
only by analysis.
4/21/2016 14 Dr Priya Mathew
St.Joseph's College of Education, Mysore
Types of Patterns
• Repeating
• Growing
• Shrinking
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St.Joseph's College of Education, Mysore
1. Examples • 1, 1, 2, 2, 1, 1,…….
• It is of the form AA, BB, AA,….
• The core is 1,1, 2,2,
Repeating Patterns
• A repeating pattern is a cyclical repetition of
an identifiable core.
4/21/2016 16 Dr Priya Mathew
St.Joseph's College of Education, Mysore
2. Examples
• 120, 121, 123, 126, 130, 135, 141,…………..
• The pattern rule is start at 120 and Add 1. The amount you add increases by 1 each ti e .
• 870, 880, 889, 897, 904, 910,………
• 2, 3, 5, 8, 12, 17,……….
Growing Patterns:
• Growing pattern is a pattern in which the numbers increase, and the amount added changes each time in a predictable way.
• A growing pattern happens when something is added each time. It can be shapes, picture, objects, numbers, etc.
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Dr Priya Mathew
St.Joseph's College of Education, Mysore
3. Examples • , , , , , …………
• The pattern rule is start at 81. Subtract 9. The amount you subtract decrease by 1 each ti e .
• , , , , , ,……. Shrinking Patterns
• A shrinking pattern is a pattern in which the numbers decrease, and the amount subtracted changes each time in a particular way.
• A shrinking pattern happens when something is taken away each time.
4/21/2016 27 Dr Priya Mathew
St.Joseph's College of Education, Mysore
Patterns of Translation, Rotation and Reflection
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St.Joseph's College of Education, Mysore
Patterns in Daily Life
• What patterns are present?
• Are they from the natural or manmade world?
• What are the elements of pattern?
• What shapes do we see?
• How are colours used?
• Do the patterns have meaning or purpose?
• How many varieties are there?
• How do they differ?
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Dr Priya Mathew
St.Joseph's College of Education, Mysore