Mastering Numbers: Strategies and Concepts
Overview
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Part whole relationship
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- role of the denominator and numerator
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Partitioning into equal sized parts is an important concept
which lies at the heart of fractions, decimals and
percentages.
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Ordering or comparing fractions
What is 2/3 + 3/5?
Jane got 10 out of 15 for her test and
Mark got 15 out of 20. Anita said they
both did equally well because they
both got 5 wrong. Is Anita correct?
Marian won a prize in the local lotto.
She put ½ of her winnings into a savings account.
She gave 1/3 of the remainder to her sister
She spent the rest on buying a car.
Her sister received €2000 from Mary. How much did Marian win
? How much did she spend on the car?
(
Use fraction strips if necessary to model the problem.)
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Compare
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Compare
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Compare
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Have they got the concepts?
Diagnostic Test
Activity on Partitioning
Where there are difficulties ...
Identify what the unit or “
whole”
is
Parts
which make up the unit must be
equal in area
Develop
meaning for fractions
using concrete models –
fraction strips, circles, uni fix cubes
Where there are difficulties......
Good
ordering strategies
in order to make estimates and
make sense of answers
Strong mental representations of
equivalence of
fractions
W
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Using a Picture – Fraction Wall
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Using ordering strategies – estimate the answer
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What to do? What type of fractions can you add and how
do you add them ?
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How do you create equivalent fractions?
Equivalent fractions
Activities on generating equivalent fractions
Example:
Research indicates that trouble spots in
Algebra come from an incomplete
understanding of fraction concepts!
Addition and Subtraction
First estimate using ordering strategies
When finding common denominators use strategies for
generating equivalent fractions
Multiplication
(rational numbers)
What does 4 x 3 mean?
4 x 2/3 = 4 groups of 2/3 each
=
4X2/3 =2/3+2/3+2/3+2/3+2/3 = 8/3
Why is it not 8/12?
Multiplication –
number line model
Aoife earns €12 per hour. What would she earn in 2, 3, 4, 3/4
hours?
Notice “of “ becoming multiplication
3/4 x12 = 12 x ¾ =9
Multiplication – Area Model
Cara had 2/5 of her birthday cake left from her party. She ate ¾ of the leftover cake. How
much of the original cake did she eat?
2/5 cake
Divide into
quarters
¾ of 2/5
Area of 3x2 out
of area of 4x5
http://www.learner.org/courses/learningmath/number/session9/part_a/try.html
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Division by a fraction – making sense of “invert
and multiply”
Cara has 4 pizzas for her party. She decides that a serving
will be 3/5 of a pizza. How many servings from 4 pizzas?
Answer
Making sense of “invert and multiply”
How many servings will one pizza give ?
How many servings from 4 pizzas
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A fraction as a part of a whole
A fraction helps us measure continuous quantities
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Unit fractions – equal sized portions or fair shares
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Equivalent fractions
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Common denominators (common subdivisions)
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Mixed numbers
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Improper fractions
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Adding and subtracting fractions
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Multiplying a fraction by a whole number, a whole number
by a fraction and a fraction by a fraction
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Dividing a whole number by a fraction
Project Maths
projectmaths.ie
Develop computation strategies applicable to any number, including generalizations of numerical relationships and operations. Explore arithmetic operations, equality concepts, problem-solving strategies, mathematical proof, percentages, and comparisons of fractions, decimals, and percentages. Enhance understanding through oral language, pictures, manipulatives, models, symbols, and real-life contexts.
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Presentation Transcript
Percentages Decimals Multiply divide fractions Add Subtract Equivalence Ordering fractions Partitioning Fractions Diagnostic Test Overview
Strand/Topic Title Learning Outcomes Students will be able to Strand 3 : 3.1 Number systems Students will devise strategies for computation that can be applied to any number. Implicit in such computational methods are generalisations about numerical relationships with the operations being used. Students will articulate the generalisation that underlies their strategy, firstly in common language and then in symbolic language. generalise observations of arithmetic operations investigate models to help think about the operations of addition, subtraction, multiplication and division of rational numbers consolidate the idea that equality is a relationship in which two mathematical expressions have the same value analyse solution strategies to problems begin to look at the idea of mathematical proof calculate percentages use the equivalence of fractions, decimals and percentages to compare proportions
Oral language Pictures Manipulative Models Written symbols Real life contexts
1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 unit or 1 whole fraction wall What is 2/3 + 3/5?
Fraction concepts Ordering strategies Fraction Equivalence Fraction Operations Sense making of the algorithms
Jane got 10 out of 15 for her test and Mark got 15 out of 20. Anita said they both did equally well because they both got 5 wrong. Is Anita correct?
Marian won a prize in the local lotto. She put of her winnings into a savings account. She gave 1/3 of the remainder to her sister She spent the rest on buying a car. Her sister received 2000 from Mary. How much did Marian win ? How much did she spend on the car? (Use fraction strips if necessary to model the problem.)
Ordering Strategies -not relying on common denominators 3 5 and 7 7 1. Compare Same denominator, different numerator 2 3 2 5 2. Compare Same numerator, different denominator and 5 9 3 7 7 8 8 9 3. Compare Using 1/2 , 0, and 1 as benchmarks 4. Using equivalent fractions and , and
Have they got the concepts? Diagnostic Test Activity on Partitioning
2 3 3 5 5 8 + Why not? . Using a Picture Fraction Wall Using ordering strategies estimate the answer What to do? What type of fractions can you add and how do you add them ? How do you create equivalent fractions?
Equivalent fractions Activities on generating equivalent fractions
Research indicates that trouble spots in Algebra come from an incomplete understanding of fraction concepts! Example: + + 5 5 x y x y = Help!!
Addition and Subtraction First estimate using ordering strategies When finding common denominators use strategies for generating equivalent fractions
Multiplication (rational numbers) What does 4 x 3 mean? 4 x 2/3 = 4 groups of 2/3 each = 4X2/3 =2/3+2/3+2/3+2/3+2/3 = 8/3 Why is it not 8/12?
Multiplication number line model Aoife earns 12 per hour. What would she earn in 2, 3, 4, 3/4 hours? Notice of becoming multiplication 3/4 x12 = 12 x =9
Multiplication Area Model Cara had 2/5 of her birthday cake left from her party. She ate of the leftover cake. How much of the original cake did she eat? Divide into quarters of 2/5 2/5 cake 3 4 5 2 6 20 3 = = 10 Multiplication making smaller! Area of 3x2 out of area of 4x5 http://www.learner.org/courses/learningmath/number/session9/part_a/try.html
Division by a fraction making sense of invert and multiply Cara has 4 pizzas for her party. She decides that a serving will be 3/5 of a pizza. How many servings from 4 pizzas? Answer 2 63
Making sense of invert and multiply How many servings will one pizza give ? How many servings from 4 pizzas? 3 5 5 3 20 3 2 3 = = = 4 4 6
