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Comprehension & Writing Strategies P5 and P6 Parent Workshop 11 March 2016, Friday

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P5 and P6 Parent Workshop Mathematics11 March 2016, Friday
Objectives
• Sharing of some strategies that children can apply to improve their understanding of texts in general.
• The strategies include the appropriate use of:
– chunking – a reading strategy
vital information/ instructions and understanding them
Chunking
- a reading strategy - break up a text into manageable sections so that
we better understand what we read
Let’s try Chunking!
Mary was dragging her feet home after a long, dreary day in school. Her hefty schoolbag weighed her down and she felt her shoulders droop all the way down to the ground. It was not only her bag. Her heart also felt the same. She dreaded to go home and break the news to her mother who was waiting for the good news anxiously.
Chunk 1 Mary was dragging her feet home after a long,dreary day in school.
How is Mary feeling? Why?
What does this phrase mean?
Chunk 2 Her hefty schoolbag weighed her down and she felt her shoulders droop all the way down to the ground.
What does this word mean? This phrase
tells me that the bag is heavy.
Why is her bag heavy?
Chunk 3
It was not only her bag. Her heart also felt the same.
What does the “same” tell me about the heart?
Chunk 4
She dreaded to go home and break the news to her mother who was waiting for the good news anxiously.
What is this news?
Since she is dreading to go home, it does not sound like a good news.
Chunking
-should be done regularly so that children are familiar with this technique
- can be used on any texts like newspaper articles, poems, etc
- size of chunking may vary - can break up one sentence into 2 parts or so
Breaking up a sentence into 2 or smaller parts
• Chunk 1 - She dreaded to go home and • Chunk 2 - break the news to her mother • Chunk 3 - who was waiting for the good news
anxiously.
Mind Movie - is a strategy where the readers visualise as they
read a text
Creating Mind Movies
- Identify words/phrases involving the 5 senses - have a mental image of what they are reading - they are also encouraged to draw should they
need visual help
Let’s try creating Mind Movies Samuel was woken up by the shrieking of his red alarm clock. He sat up in bed looking shocked, with his eyes wide open and his hair tousled! His white pyjama top was drenched in perspiration. He felt sticky and wet. His face was as red as a lobster. He was breathing heavily. Just then, the aroma of his favourite pancakes wafted into his room.
Effectiveness of mind movie
- enhances their understanding and retention of the text
- helps children to make connections by having a mental imagery of the characters and setting
Retelling
What is it? - include the main points of the story that they have
read - can be in written form - can be done orally
Let’s try
Predicting
-Looking at the title of the movie, jot down/tell us what you think it is about.
Let’s try
Recalling & Retelling -Write/give a summary of the scene that you
have watched. Be as detailed as possible. -This assesses the child’s understanding of the
scene they have watched. -Ensure that all main points have been
captured
Summary of the scene • Grandma asked Grandson what he wanted to eat
• Boy wanted to eat fried chicken and communicated that to her using actions and pictures
• Grandma packed some eggs to exchange for a chicken
• Then she was caught in the rain on the way back
• Boy fell asleep while waiting
• Grandma prepared steamed instead of fried chicken
• Boy got extremely upset
• But he got up in the middle of the night to eat the steamed chicken as he was hungry
Retelling with a Text
-Look at the title of the text/story, jot down/tell us what you think it is about.
Predicting
- Guess and jot down some of the words/phrases which you think will appear in the text.
Reading
-Read the story. -Is the story what you have predicted? -Highlight the words/phrases which you
have predicted and can be found in the text.
Recalling
- Without referring to the text, write/give a summary of it. Be as detailed as possible.
Retelling
Reviewing
-Now review what you have written. -Have you left out any vital information? -Use a pen of another colour and add in the
missing information.
Points to note -Retelling can be done in children’s own words -They don’t have to use the exact words to tell the story -This will reflect the extent of their understanding
Enhancing Comprehension
• Retelling • Chunking • Creating Mind Movie • These 3 strategies can be used together to help
your children better understand what they are reading.
Tips for Improving Writing –Writing about what they like/dislike
- Encourage your child to write more on topics they feel strongly about
- Strong feelings about a topic can inspire a child to write
- Come up with a list of things that the child truly likes or dislikes
- Then get the child to write a paragraph on that topic
- Focus on the content, not on the grammar
- Here, the focus is to instil the desire to write
Tips for Improving Writing –Writing about what they know/love
Things I do for fun
Play basketball
Things I Care About
Journal/Fun Write/Free Writing
- Keep a diary where they jot down eventful things in their life and invite them to share it with you.
- Travel Log & Itinerary - Movie/Book/Song Review - My Favourite Computer Game
References
• http://www.edutopia.org/blog/brain-movies- visualize-reading-comprehension-donna-wilson
11 March 2016, Friday
knowledge that are required in Math problem-solving.
2) Create awareness on the use of model method and
ratio method in problem-solving.
Whole Numbers
Money Decimals
& Volume
Knowledge Conceptual Knowledge
understanding of Mathematics
Metacognitive Knowledge
Conceptual Knowledge • A concept is a general idea or understanding of something.
• To develop conceptual understanding,
Concrete Representations Pictorial Abstract
• The term percent means “parts per hundred”
• Relate to the concept of fractions with its denominator as one hundred and the concept of equivalent fractions
• Pave the way for the concept of ratio
eg = 7 % = 7 per hundred;
100
7
5
2 =
100
Jenny has $1000. She has 20% more money
than Ken. How much money does Ken have?
b) Ken has 100%? 120% -> $1000
1% -> $8.33
100% -> $833
than the base (100%)
(100%)
Ali has 25% more stickers than Ben. Ben has 40% fewer
stickers than Calvin. What is the ratio of Ali’s stickers to Calvin’s stickers?
Ali : Ben
125 : 100
5 : 4
Ben : Calvin
60 : 100
than the base (100%)
than the base (100%)
x 3
Ali : Calvin
15 : 20
3 : 4
Procedural Knowledge
• Knowing a set of sequential steps in order to solve a
problem (procedural strategies)
interactively
• base must be one of the 3 sides of a
triangle
base
• area of triangle equals to half the area
of its related rectangle
x base x height 2
1
Procedural Fluency (GOAL)
Which is the base and its corresponding height?
What is the area of triangle ABC as shown in the figure?
A
12 + 6 = 18 cm 2
8 + 4 = 12 cm
Conceptual Understanding & Procedural Fluency
(e.g. formulae, properties of geometrical shapes)
• Factual Fluency is the automaticity of retrieving the
required information
acquire new concepts
pupils to achieve fluency
Square
4 sides are equal in length
4 right angles
4 sides are equal in length
diagonal angles are equal
Equilateral Triangle
each angle is 60
Geometry
In the figure below, not drawn to scale, ABCD is a square
and QM = QP = QN. MN is parallel to AB and it is
perpendicular to PQ. Find MPN.
A B
C D
• all angles are 90
• all angles are 60
• sum of 3 angles in triangle = 180 QM = QP = QN = MN (infer)
MQN = 60 (infer equilateral )
NQP = 60 ÷ 2 = 30
MPN = 75 + 75 = 150
• 2 sides are equal length & angle
• sum of 3 angles in triangle = 180
Linguistic Knowledge
• Relating of symbols to Math Vocabulary to Math
concepts.
= 20 x 57 + 4 x 57
Algebra Math Vocabulary Meaning
eg (2p + 5)
Constant
the expression easier to use
Three boys collected drink cans for recycling.
John collected 2y drink cans which was half as many as
what Ken collected. Ken collected 8 drink cans more than
Larry. How many drink cans did they collect altogether?
Give your answer in terms of y in the simplest form.
2y + 2y = 4y (Ken)
Metacognitive Knowledge
• It is like the brain that controls the activation of the
other types of knowledge
Ans: ________________
• Subtraction of 2 decimals – Factual Knowledge
• Subtraction algorithm and renaming – Procedural Knowledge
13.2
12.23
Use all the digits 3, 0, 2, 9 to form the smallest multiple of 5.
Ans: ________________
__ __ __ __ TH H T O
2 390
A solid cuboid of height 11 cm has a square base of side 2 cm. What is its volume?
(1) 22 cm 3
(2) 44 cm 3
(3) 123 cm 3
(4) 242 cm 3
• Recall formula and Times Table Facts – Factual Knowledge
• Meanings for ‘cuboid’ and ‘square’ – Linguistic Knowledge
• Apply formula volume of solid – Procedural
11cm
The pie chart below shows how Joyce spent her pocket
money last month. What is the ratio of the amount of
money Joyce spent on toys to the amount of money
spent of books?
= (equivalent fractions)
3 : 4
• Find equivalent fractions – Factual Knowledge
• Meanings for ‘pie chart’ and ‘ratio’ – Linguistic Knowledge
• Subtraction of fractions from one whole – Procedural Knowledge
10
4
10
3
10
2
10
1
10
4
relational understanding in problem-solving
• Chinn, Steve. (2012). The Trouble with Maths New York: Routledge
• W. George, Cathcart, Yvonne, M. Pothier, James, H. Vance, Nadine, S. Bezuk (2006) Learning Mathematics in Elementary and Middle Schools New Jersey: Pearson Education
• PSLE Examination Questions (2015). Singapore: EPH
• PSLE Examination Questions (2006 – 2010). Hillview Publications
• The Straits Times, Monday, February 22, 2016. B10 Education
Model and Ratio Methods
11 March 2016, Friday
Understand
Plan
Do
Check
STEP STAGE QUESTIONS
1 UNDERSTAND • What information is given? • What is the problem asking us to do? • What are we trying to find out? • How can I make sense of the problem? • Can we restate the problem? • What can I infer from the given data? • Can I act the scenario out? • Do I have a concrete mental vision of the problem?
The 3I Strategy • Identify (Keywords/Topic) • Interpret (Re-state the Information) • Infer (Uncover hidden information)
Polya’s Four Steps to Problem Solving
STEP STAGE QUESTIONS
2 PLAN • What do we need to do to solve the problem? • Do we need more information? • Is there any hidden question? • What strategies are useful? • How should I represent the problem?
3 DO • Solve the problem using a suitable heuristic • Apply Math skills, concepts and strategies
4 CHECK • Have you computed accurately? • Read the question again • Estimate and check the solution • Is your solution reasonable? • Should we revise our plan?
Model Approach for Problem Solving
• Research has shown that the ability to solve word problems requires more than conceptual, procedural and factual knowledge.
• The ability to represent problems, e.g. the use of a diagram, is critical.
• Useful representations allow students to - Better visualise the problem - Make useful modifications - Link to suitable strategies, computations and procedures
Quoted from Bar Modeling, A Problem-Solving Tool Yeap Ban Har, PhD
Model Approach for Problem Solving Types of Models
• Part-Whole Model
Part-Whole Unitary Model
Concepts of Multiplication,
Division, Fractions, Ratio
• Comparison Model Find the Difference
• Before-After Model Incorporate comparison before and after a change
• Complex Models Involve a combination of the various types of models with advanced skills like
“shifting” and/or “cutting”
Ratio Method
• Model Method is a pictorial representation • It can progress to more abstract forms when solving
complex problems, e.g. percentage or when we need to “cut” into many equal parts.
• In such cases, ratio method may be a more efficient representation.
100%
• Ratio method is a more abstract representation
• It is efficient in solving problems involving fractions, percentage and ratio.
• To apply the strategy accurately, pupils must be proficient in these topics and proportion concept.
Percentage Decimal Ratio Fraction
100 =
4
25
Model Ratio
A
B
C
A : B 5 : 3 = 10 : 6 By multiplying by 2, we effectively cut every unit into two equal parts. We can cut the parts into any number of units required to solve the problem.
Model and Ratio Methods
1. Model Method (Whole Numbers)
Don had 130 fewer pencils than Howard at first. After Howard gave away 172 pencils and Don gave away 20 pencils, Don had thrice as many pencils as Howard. How many pencils did both of them have at first?
Identify key words Interpret the information
• at first/ After/ both of them have at first Before-After
• At first, Don had fewer Howard had 130 more than Don
• Howard gave away 172 Take away 172
• Don gave away 20 Take away 20
• At the end, thrice Don had 3 units and Howard had 1 unit
• To find out Total pencils of Don and Howard at first
1. Model Method (Whole Numbers)
Don had 130 fewer pencils than Howard at first. After Howard gave away 172 pencils and Don gave away 20 pencils, Don had thrice as many pencils as Howard. How many pencils did both of them have at first?
Infer hidden information
• At the end, Don had thrice as many as Howard
Don had more; Howard had fewer
1. Model Method (Whole Numbers)
Don had 130 fewer pencils than Howard at first. After Howard gave away 172 pencils and Don gave away 20 pencils, Don had thrice as many pencils as Howard. How many pencils did both of them have at first?
Howard
Don
After
Howard
Don
Before
172
?
2 units = 172 – 130 – 20 = 22 1 unit = 22 ÷ 2 = 11 4 units = 11 x 4 = 44 44 + 20 + 172 = 236 They have 236 pencils.
2. Model Method (Fractions)
Mr Goh had some muffins at first. After selling 52 of them in the
morning and 5
8 of the remainder in the afternoon, he was left
with 1
4 of the muffins. How many muffins did he sell altogether?
Identify key words Interpret the information
• at first/ After/ left Before-After
• Sold 52 in the morning Take away 52
• Sold 5
Left with 8 units after selling 52 in the morning
Sold 5 units in the afternoon
Left with 3 units
• To find out Total number of muffins sold in the morning and afternoon
2. Model Method (Fractions)
Mr Goh had some muffins at first. After selling 52 of them in the
morning and 5
8 of the remainder in the afternoon, he was left
with 1
4 of the muffins. How many muffins did he sell altogether?
Infer hidden information
• Left with 1
2. Model Method (Fractions)
Mr Goh had some muffins at first. After selling 52 of them in the
morning and 5
8 of the remainder in the afternoon, he was left
with 1
4 of the muffins. How many muffins did he sell altogether?
Before
? (9 units)
4 units = 52 1 unit = 52 ÷ 4 = 13 9 units = 13 x 9 = 117 He sold 117 muffins.
3. Model Method (Percentage)
Alvin has 50% more money than Betty. Betty has 30% less money than Candy. Given that Alvin has $63, how much do Betty and Candy have altogether?
Identify key words Interpret the information
• Alvin has 50% more than Betty
Base (Betty) 100% and Alvin is 150%
• Betty has 30% less than Candy
Base (Candy) 100% and Betty is 70%
• Alvin has $63 Start solving from “Alvin”
• To find out Total amount of money that Betty and Cathy have
3. Model Method (Percentage)
Alvin has 50% more money than Betty. Betty has 30% less money than Candy. Given that Alvin has $63, how much do Betty and Candy have altogether?
Infer hidden information
• Betty has 100% in the first comparison
• Betty has 70% of what Candy has in the second comparison
Equate the “100%” to the “70%”
3. Model Method (Percentage)
Alvin has 50% more money than Betty. Betty has 30% less money than Candy. Given that Alvin has $63, how much do Betty and Candy have altogether?
Betty
100%
?
150% of money = $63 50% of money = $63 ÷ 3 = $21 100% of money = $21 x 2 = $42 (Betty) 70% of money = $42 10% of money = $42 ÷ 7 = $6 100% of money = $6 x 10 = $60 (Candy) $42 + $60 = $102 They have $102.
4. Ratio Method The figure below shows a square that is divided into 4 parts A, B, C and D. The line XY divides the square into 2 equal parts. The ratio of Area A to Area B is 2 : 5 and the ratio of Area B to Area D is 7 : 3. The area of the square is 490 cm2. What is the area of Area C?
Identify key words Interpret the information • Square Properties of a Square • XY divides the square into 2 equal parts One Half : A and B Other Half : C and D • A : B is 2 : 5 • B : D is 7 : 3 • Total Area is 490 cm2 • To find out Area of C
A
B
C
D
Y
X
4. Ratio Method The figure below shows a square that is divided into 4 parts A, B, C and D. The line XY divides the square into 2 equal parts. The ratio of Area A to Area B is 2 : 5 and the ratio of Area B to Area D is 7 : 3. The area of the square is 490 cm2. What is the area of Area C?
Infer hidden information • A : B is 2 : 5 and B : D is 7 : 3 Equate B “5 units” to the “7 units” using Equivalent Ratios • XY divides the square into 2 equal parts Total Area of A and B = Total Area of C and D
A
B
C
D
Y
X
4. Ratio Method The figure below shows a square that is divided into 4 parts A, B, C and D. The line XY divides the square into 2 equal parts. The ratio of Area A to Area B is 2 : 5 and the ratio of Area B to Area D is 7 : 3. The area of the square is 490 cm2. What is the area of Area C?
A
B
C
D
Y
X
A : B : C : D : Total 2 : 5 14 = 14 : 35 98 7 : 3 = 35 : 15 14 : 35 : : 15 : 98
Area (A + B + D) = 14 + 35 + 15 = 64 Area of C = 98 – 64 = 34 units 98 units = 490
1 unit = 490 ÷ 98 = 5 34 units = 5 x 34 = 170 The area of C is 170 cm2.
34
5. Ratio Method
The ratio of the number of boys to the number of girls in the Choir was 2 : 3. The ratio of the number of boys to the number of girls in the Band was 5 : 3. The Band had twice as many pupils as the Choir. Find the ratio of the number of girls in the Choir to the number of girls in the Band.
Identify key words Interpret the information
• (Choir) Ratio of Boys : Girls = 2 : 3
• (Band) Ratio of Boys : Girls = 5 : 3
• Band had twice as many
Band had 2 units and Choir had 1 unit
• To find out Girls (Choir) : Girls (Band)
5. Ratio Method
The ratio of the number of boys to the number of girls in the Choir was 2 : 3. The ratio of the number of boys to the number of girls in the Band was 5 : 3. The Band had twice as many pupils as the Choir. Find the ratio of the number of girls in the Choir to the number of girls in the Band.
Infer hidden information
• (Choir) Ratio of Boys : Girls = 2 : 3 Total pupils = 5 units
• (Band) Ratio of Boys : Girls = 5 : 3 Total pupils = 8 units
• Band had twice as many
Pupils (Band) = 20 units
Pupils (Choir) = 40 units
5. Ratio Method
The ratio of the number of boys to the number of girls in the Choir was 2 : 3. The ratio of the number of boys to the number of girls in the Band was 5 : 3. The Band had twice as many pupils as the Choir. Find the ratio of the number of girls in the Choir to the number of girls in the Band.
Choir Band B : G : T : B : G : T 2 : 3 : 5 = 8 : 12 : 20 5 : 3 : 8 = 25 : 15 : 40 8 : 12 : 20 : 25 : 15 : 40
Girls (Choir) : Girls (Band) = 12 : 15 = 4 : 5 The ratio is 4 : 5.
6. Ratio Method (Percentage)
In a school, 70% of the members in the Art Club and 60% of the members in the Computer Club are boys. Both the Art Club and Computer Club have the same number of girls. The Art Club has 15 more boys than the Computer Club. How many members are there in the Art Club?
Identify key words Interpret the information
• (Art Club) 70% are boys 30% are girls
• (Computer Club) 60% are boys 40% are girls
• (Art Club) Girls = (Computer Club) Girls
• Art Club has 15 more boys Computer Club has 15 fewer boys
• To find out Total number of boys and girls in Art Club
6. Ratio Method (Percentage)
In a school, 70% of the members in the Art Club and 60% of the members in the Computer Club are boys. Both the Art Club and Computer Club have the same number of girls. The Art Club has 15 more boys than the Computer Club. How many members are there in the Art Club?
Infer hidden information
6. Ratio Method (Percentage)
In a school, 70% of the members in the Art Club and 60% of the members in the Computer Club are boys. Both the Art Club and Computer Club have the same number of girls. The Art Club has 15 more boys than the Computer Club. How many members are there in the Art Club?
Art Club Computer Club
B : G : T : B : G : T 7 : 3 : 10 = 28 : 12 : 40 6 : 4 : 10 = 18 : 12 : 30 28 : 12 : 40 : 18 : 12 : 30
10 units = 15 boys 40 units = 15 x 4 = 60 There are 60 members.
15 more boys (10 units)
7. Ratio Method (Fractions)
4 of them were strawberry tarts
and the rest were lemon tarts. After selling 140 lemon tarts and 5
6 of the strawberry tarts, he had
1
tarts did the baker sell?
Identify key words Interpret the information Infer hidden information
• After/left Before-After

• Sold 140 lemon tarts
• At the end, 1

of the tarts were sold
• To find out Total number of strawberry and lemon tarts sold
7. Ratio Method (Fractions)
4 of them were strawberry tarts
and the rest were lemon tarts. After selling 140 lemon tarts and 5
6 of the strawberry tarts, he had
1
tarts did the baker sell?
SB : Lemon : Total Before 3 : 1 : 4 = 6 : 2 : 8 = 30 : 10 : 40 Sold 25 : 7 : 32 (140 tarts) After 5 : 3 : 8
7 units = 140 1 unit = 140 ÷ 7 = 20 32 units = 20 x 32 = 640 He sold 640 tarts.
8. Ratio Method (Fractions)
4 of it on a blouse and
2
5 of
the remainder on a bag. After that, her parents gave her $120. The total amount of money that she had at the end was
5
4 of the amount of
money she had at first. How much money did Gina have at first? Identify key words Interpret the information Infer hidden information • At first/After that/at the end Before-After
• Spent 1

• Spent 2
5 of the remainder on a bag
Left with 5 units after buying the blouse Spent 2 units on a bag Left with 3 units • $120 and 3 units left

of money at first
She had more money at the end than what she had at first • To find out Total amount of money at first
8. Ratio Method (Fractions)
4 of it on a blouse and
2
5 of
the remainder on a bag. After that, her parents gave her $120. The total amount of money that she had at the end was
5
4 of the amount of
money she had at first. How much money did Gina have at first?
Remainder Blouse : Bag : Left : Total Before 1 : 3 : 4 = 5 : 15 : 20 Spent 6 : 9 5 : 6 : 9 : 20 After (+ $120) 25
25 – 9 = 16 16 units = $120 4 units = $120 ÷ 4 = $30 20 units = $30 x 5 = $150 She had $150.
www.bit.ly/P5P6MAworkshop2016
11 March 2016, Friday
4.10 p.m. – 5.30 p.m.
Multiple Choice Questions (MCQ)
Short - Answer Questions (SAQ)
Long - Answer Questions (LAQ)
Objectives of workshop: 1) Create awareness of the essential skills and
knowledge that are required in Math problem-solving.
2) Create awareness on the use of model method and
ratio method in problem-solving.
Knowledge Conceptual Knowledge
understanding of Mathematics
Metacognitive Knowledge
Conceptual Knowledge • A concept is a general idea or understanding of something.
• To develop conceptual understanding,
Concrete Representations Pictorial Abstract
• The term percent means “parts per hundred”
• Relate to the concept of fractions with its denominator as one hundred and the concept of equivalent fractions
• Pave the way for the concept of ratio
eg = 7 % = 7 per hundred;
100
7
5
2 =
100
Jenny has $1000. She has 20% more money
than Ken. How much money does Ken have?
b) Ken has 100%? 120% -> $1000
1% -> $8.33
100% -> $833
than the base (100%)
(100%)
Ali has 25% more stickers than Ben. Ben has 40% fewer
stickers than Calvin. What is the ratio of Ali’s stickers to Calvin’s stickers?
Ali : Ben
125 : 100
5 : 4
Ben : Calvin
60 : 100
than the base (100%)
than the base (100%)
x 3
Ali : Calvin
15 : 20
3 : 4
Procedural Knowledge
• Knowing a set of sequential steps in order to solve a
problem (procedural strategies)
interactively
• base must be one of the 3 sides of a
triangle
base
• area of triangle equals to half the area
of its related rectangle
x base x height 2
1
Procedural Fluency (GOAL)
Which is the base and its corresponding height?
What is the area of triangle ABC as shown in the figure?
A
12 + 6 = 18 cm 2
8 + 4 = 12 cm
Conceptual Understanding & Procedural Fluency
(e.g. formulae, properties of geometrical shapes)
• Factual Fluency is the automaticity of retrieving the
required information
acquire new concepts
pupils to achieve fluency
Square
4 sides are equal in length
4 right angles
4 sides are equal in length
diagonal angles are equal
Equilateral Triangle
each angle is 60
Geometry
In the figure below, not drawn to scale, ABCD is a square
and QM = QP = QN. MN is parallel to AB and it is
perpendicular to PQ. Find MPN.
A B
C D
• all angles are 90
• all angles are 60
• sum of 3 angles in triangle = 180 QM = QP = QN = MN (infer)
MQN = 60 (infer equilateral )
NQP = 60 ÷ 2 = 30
MPN = 75 + 75 = 150
• 2 sides are equal length & angle
• sum of 3 angles in triangle = 180
Linguistic Knowledge
• Relating of symbols to Math Vocabulary to Math
concepts.
= 20 x 57 + 4 x 57
Algebra Math Vocabulary Meaning
eg (2p + 5)
Constant
the expression easier to use
Three boys collected drink cans for recycling.
John collected 2y drink cans which was half as many as
what Ken collected. Ken collected 8 drink cans more than
Larry. How many drink cans did they collect altogether?
Give your answer in terms of y in the simplest form.
2y + 2y = 4y (Ken)
Metacognitive Knowledge
• It is like the brain that controls the activation of the
other types of knowledge
Ans: ________________
• Subtraction of 2 decimals – Factual Knowledge
• Subtraction algorithm and renaming – Procedural Knowledge
13.2
12.23
Use all the digits 3, 0, 2, 9 to form the smallest multiple of 5.
Ans: ________________
__ __ __ __ TH H T O
2 390
A solid cuboid of height 11 cm has a square base of side 2 cm. What is its volume?
(1) 22 cm 3
(2) 44 cm 3
(3) 123 cm 3
(4) 242 cm 3
• Recall formula and Times Table Facts – Factual Knowledge
• Meanings for ‘cuboid’ and ‘square’ – Linguistic Knowledge
• Apply formula volume of solid – Procedural
11cm
The pie chart below shows how Joyce spent her pocket
money last month. What is the ratio of the amount of
money Joyce spent on toys to the amount of money
spent of books?
= (equivalent fractions)
3 : 4
• Find equivalent fractions – Factual Knowledge
• Meanings for ‘pie chart’ and ‘ratio’ – Linguistic Knowledge
• Subtraction of fractions from one whole – Procedural Knowledge
10
4
10
3
10
2
10
1
10
4
relational understanding in problem-solving
• Chinn, Steve. (2012). The Trouble with Maths New York: Routledge
• W. George, Cathcart, Yvonne, M. Pothier, James, H. Vance, Nadine, S. Bezuk (2006) Learning Mathematics in Elementary and Middle Schools New Jersey: Pearson Education
• PSLE Examination Questions (2015). Singapore: EPH
• PSLE Examination Questions (2006 – 2010). Hillview Publications
• The Straits Times, Monday, February 22, 2016. B10 Education
Model and Ratio Methods
Polya’s Four Steps to Problem Solving
Understand
Plan
Do
Check
STEP STAGE QUESTIONS
1 UNDERSTAND • What information is given? • What is the problem asking us to do? • What are we trying to find out? • How can I make sense of the problem? • Can we restate the problem? • What can I infer from the given data? • Can I act the scenario out? • Do I have a concrete mental vision of the problem?
The 3I Strategy • Identify (Keywords/Topic) • Interpret (Re-state the Information) • Infer (Uncover hidden information)
Polya’s Four Steps to Problem Solving
STEP STAGE QUESTIONS
2 PLAN • What do we need to do to solve the problem? • Do we need more information? • Is there any hidden question? • What strategies are useful? • How should I represent the problem?
3 DO • Solve the problem using a suitable heuristic • Apply Math skills, concepts and strategies
4 CHECK • Have you computed accurately? • Read the question again • Estimate and check the solution • Is your solution reasonable? • Should we revise our plan?
Model Approach for Problem Solving
• Research has shown that the ability to solve word problems requires more than conceptual, procedural and factual knowledge.
• The ability to represent problems, e.g. the use of a diagram, is critical.
• Useful representations allow students to - Better visualise the problem - Make useful modifications - Link to suitable strategies, computations and procedures
Quoted from Bar Modeling, A Problem-Solving Tool Yeap Ban Har, PhD
Model Approach for Problem Solving Types of Models
• Part-Whole Model
Part-Whole Unitary Model
Concepts of Multiplication,
Division, Fractions, Ratio
• Comparison Model Find the Difference
• Before-After Model Incorporate comparison before and after a change
• Complex Models Involve a combination of the various types of models with advanced skills like
“shifting” and/or “cutting”
Ratio Method
• Model Method is a pictorial representation • It can progress to more abstract forms when solving
complex problems, e.g. percentage or when we need to “cut” into many equal parts.
• In such cases, ratio method may be a more efficient representation.
100%
• Ratio method is a more abstract representation
• It is efficient in solving problems involving fractions, percentage and ratio.
• To apply the strategy accurately, pupils must be proficient in these topics and proportion concept.
Percentage Decimal Ratio Fraction
100 =
4
25
Model Ratio
A
B
C
A : B 5 : 3 = 10 : 6 By multiplying by 2, we effectively cut every unit into two equal parts. We can cut the parts into any number of units required to solve the problem.
Model and Ratio Methods
1. Model Method (Whole Numbers)
Don had 130 fewer pencils than Howard at first. After Howard gave away 172 pencils and Don gave away 20 pencils, Don had thrice as many pencils as Howard. How many pencils did both of them have at first?
Identify key words Interpret the information
• at first/ After/ both of them have at first Before-After
• At first, Don had fewer Howard had 130 more than Don
• Howard gave away 172 Take away 172
• Don gave away 20 Take away 20
• At the end, thrice Don had 3 units and Howard had 1 unit
• To find out Total pencils of Don and Howard at first
1. Model Method (Whole Numbers)
Don had 130 fewer pencils than Howard at first. After Howard gave away 172 pencils and Don gave away 20 pencils, Don had thrice as many pencils as Howard. How many pencils did both of them have at first?
Infer hidden information
• At the end, Don had thrice as many as Howard
Don had more; Howard had fewer
1. Model Method (Whole Numbers)
Don had 130 fewer pencils than Howard at first. After Howard gave away 172 pencils and Don gave away 20 pencils, Don had thrice as many pencils as Howard. How many pencils did both of them have at first?
Howard
Don
After
Howard
Don
Before
172
?
2 units = 172 – 130 – 20 = 22 1 unit = 22 ÷ 2 = 11 4 units = 11 x 4 = 44 44 + 20 + 172 = 236 They have 236 pencils.
2. Model Method (Fractions)
Mr Goh had some muffins at first. After selling 52 of them in the
morning and 5
8 of the remainder in the afternoon, he was left
with 1
4 of the muffins. How many muffins did he sell altogether?
Identify key words Interpret the information
• at first/ After/ left Before-After
• Sold 52 in the morning Take away 52
• Sold 5
Left with 8 units after selling 52 in the morning
Sold 5 units in the afternoon
Left with 3 units
• To find out Total number of muffins sold in the morning and afternoon
2. Model Method (Fractions)
Mr Goh had some muffins at first. After selling 52 of them in the
morning and 5
8 of the remainder in the afternoon, he was left
with 1
4 of the muffins. How many muffins did he sell altogether?
Infer hidden information
• Left with 1
2. Model Method (Fractions)
Mr Goh had some muffins at first. After selling 52 of them in the
morning and 5
8 of the remainder in the afternoon, he was left
with 1
4 of the muffins. How many muffins did he sell altogether?
Before
? (9 units)
4 units = 52 1 unit = 52 ÷ 4 = 13 9 units = 13 x 9 = 117 He sold 117 muffins.
3. Model Method (Percentage)
Alvin has 50% more money than Betty. Betty has 30% less money than Candy. Given that Alvin has $63, how much do Betty and Candy have altogether?
Identify key words Interpret the information
• Alvin has 50% more than Betty
Base (Betty) 100% and Alvin is 150%
• Betty has 30% less than Candy
Base (Candy) 100% and Betty is 70%
• Alvin has $63 Start solving from “Alvin”
• To find out Total amount of money that Betty and Cathy have
3. Model Method (Percentage)
Alvin has 50% more money than Betty. Betty has 30% less money than Candy. Given that Alvin has $63, how much do Betty and Candy have altogether?
Infer hidden information
• Betty has 100% in the first comparison
• Betty has 70% of what Candy has in the second comparison
Equate the “100%” to the “70%”
3. Model Method (Percentage)
Alvin has 50% more money than Betty. Betty has 30% less money than Candy. Given that Alvin has $63, how much do Betty and Candy have altogether?
Betty
100%
?
150% of money = $63 50% of money = $63 ÷ 3 = $21 100% of money = $21 x 2 = $42 (Betty) 70% of money = $42 10% of money = $42 ÷ 7 = $6 100% of money = $6 x 10 = $60 (Candy) $42 + $60 = $102 They have $102.
4. Ratio Method The figure below shows a square that is divided into 4 parts A, B, C and D. The line XY divides the square into 2 equal parts. The ratio of Area A to Area B is 2 : 5 and the ratio of Area B to Area D is 7 : 3. The area of the square is 490 cm2. What is the area of Area C?
Identify key words Interpret the information • Square Properties of a Square • XY divides the square into 2 equal parts One Half : A and B Other Half : C and D • A : B is 2 : 5 • B : D is 7 : 3 • Total Area is 490 cm2 • To find out Area of C
A
B
C
D
Y
X
4. Ratio Method The figure below shows a square that is divided into 4 parts A, B, C and D. The line XY divides the square into 2 equal parts. The ratio of Area A to Area B is 2 : 5 and the ratio of Area B to Area D is 7 : 3. The area of the square is 490 cm2. What is the area of Area C?
Infer hidden information • A : B is 2 : 5 and B : D is 7 : 3 Equate B “5 units” to the “7 units” using Equivalent Ratios • XY divides the square into 2 equal parts Total Area of A and B = Total Area of C and D
A
B
C
D
Y
X
4. Ratio Method The figure below shows a square that is divided into 4 parts A, B, C and D. The line XY divides the square into 2 equal parts. The ratio of Area A to Area B is 2 : 5 and the ratio of Area B to Area D is 7 : 3. The area of the square is 490 cm2. What is the area of Area C?
A
B
C
D
Y
X
A : B : C : D : Total 2 : 5 14 = 14 : 35 98 7 : 3 = 35 : 15 14 : 35 : : 15 : 98
Area (A + B + D) = 14 + 35 + 15 = 64 Area of C = 98 – 64 = 34 units 98 units = 490
1 unit = 490 ÷ 98 = 5 34 units = 5 x 34 = 170 The area of C is 170 cm2.
34
5. Ratio Method
The ratio of the number of boys to the number of girls in the Choir was 2 : 3. The ratio of the number of boys to the number of girls in the Band was 5 : 3. The Band had twice as many pupils as the Choir. Find the ratio of the number of girls in the Choir to the number of girls in the Band.
Identify key words Interpret the information
• (Choir) Ratio of Boys : Girls = 2 : 3
• (Band) Ratio of Boys : Girls = 5 : 3
• Band had twice as many
Band had 2 units and Choir had 1 unit
• To find out Girls (Choir) : Girls (Band)
5. Ratio Method
The ratio of the number of boys to the number of girls in the Choir was 2 : 3. The ratio of the number of boys to the number of girls in the Band was 5 : 3. The Band had twice as many pupils as the Choir. Find the ratio of the number of girls in the Choir to the number of girls in the Band.
Infer hidden information
• (Choir) Ratio of Boys : Girls = 2 : 3 Total pupils = 5 units
• (Band) Ratio of Boys : Girls = 5 : 3 Total pupils = 8 units
• Band had twice as many
Pupils (Band) = 20 units
Pupils (Choir) = 40 units
5. Ratio Method
The ratio of the number of boys to the number of girls in the Choir was 2 : 3. The ratio of the number of boys to the number of girls in the Band was 5 : 3. The Band had twice as many pupils as the Choir. Find the ratio of the number of girls in the Choir to the number of girls in the Band.
Choir Band B : G : T : B : G : T 2 : 3 : 5 = 8 : 12 : 20 5 : 3 : 8 = 25 : 15 : 40 8 : 12 : 20 : 25 : 15 : 40
Girls (Choir) : Girls (Band) = 12 : 15 = 4 : 5 The ratio is 4 : 5.
6. Ratio Method (Percentage)
In a school, 70% of the members in the Art Club and 60% of the members in the Computer Club are boys. Both the Art Club and Computer Club have the same number of girls. The Art Club has 15 more boys than the Computer Club. How many members are there in the Art Club?
Identify key words Interpret the information
• (Art Club) 70% are boys 30% are girls
• (Computer Club) 60% are boys 40% are girls
• (Art Club) Girls = (Computer Club) Girls
• Art Club has 15 more boys Computer Club has 15 fewer boys
• To find out Total number of boys and girls in Art Club
6. Ratio Method (Percentage)
In a school, 70% of the members in the Art Club and 60% of the members in the Computer Club are boys. Both the Art Club and Computer Club have the same number of girls. The Art Club has 15 more boys than the Computer Club. How many members are there in the Art Club?
Infer hidden information
6. Ratio Method (Percentage)
In a school, 70% of the members in the Art Club and 60% of the members in the Computer Club are boys. Both the Art Club and Computer Club have the same number of girls. The Art Club has 15 more boys than the Computer Club. How many members are there in the Art Club?
Art Club Computer Club
B : G : T : B : G : T 7 : 3 : 10 = 28 : 12 : 40 6 : 4 : 10 = 18 : 12 : 30 28 : 12 : 40 : 18 : 12 : 30
10 units = 15 boys 40 units = 15 x 4 = 60 There are 60 members.
15 more boys (10 units)
7. Ratio Method (Fractions)
4 of them were strawberry tarts
and the rest were lemon tarts. After selling 140 lemon tarts and 5
6 of the strawberry tarts, he had
1
tarts did the baker sell?
Identify key words Interpret the information Infer hidden information
• After/left Before-After

• Sold 140 lemon tarts
• At the end, 1

of the tarts were sold
• To find out Total number of strawberry and lemon tarts sold
7. Ratio Method (Fractions)
4 of them were strawberry tarts
and the rest were lemon tarts. After selling 140 lemon tarts and 5
6 of the strawberry tarts, he had
1
tarts did the baker sell?
SB : Lemon : Total Before 3 : 1 : 4 = 6 : 2 : 8 = 30 : 10 : 40 Sold 25 : 7 : 32 (140 tarts) After 5 : 3 : 8
7 units = 140 1 unit = 140 ÷ 7 = 20 32 units = 20 x 32 = 640 He sold 640 tarts.
8. Ratio Method (Fractions)
4 of it on a blouse and
2
5 of
the remainder on a bag. After that, her parents gave her $120. The total amount of money that she had at the end was
5
4 of the amount of
money she had at first. How much money did Gina have at first? Identify key words Interpret the information Infer hidden information • At first/After that/at the end Before-After
• Spent 1

• Spent 2
5 of the remainder on a bag
Left with 5 units after buying the blouse Spent 2 units on a bag Left with 3 units • $120 and 3 units left

of money at first
She had more money at the end than what she had at first • To find out Total amount of money at first
8. Ratio Method (Fractions)
4 of it on a blouse and
2
5 of
the remainder on a bag. After that, her parents gave her $120. The total amount of money that she had at the end was
5
4 of the amount of
money she had at first. How much money did Gina have at first?
Remainder Blouse : Bag : Left : Total Before 1 : 3 : 4 = 5 : 15 : 20 Spent 6 : 9 5 : 6 : 9 : 20 After (+ $120) 25
25 – 9 = 16 16 units = $120 4 units = $120 ÷ 4 = $30 20 units = $30 x 5 = $150 She had $150.
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