Comprehension & Writing Strategies

Comprehension & Writing Strategies

P5 and P6 Parent Workshop

11 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

– creating “mind movie”

– retelling

• Tips for improving writing

Tuning-in

• Rationale: Importance of skimming the text for

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

Title: The Way Home

Predicting

-Looking at the title of the movie, jot down/tell us what you think it is about.

Let’s try

Title: The Way Home

Watching the movie

- Is your prediction accurate? - What is different or similar?

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

-Share what you have written.

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

Take trips Play

computer games

Watch movies

Tips for Improving Writing –Writing about what they know/love

Things I Care About

My family

My pets

How others view me

My friends

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

• The Writing Teacher’s Strategy Guide by Steve Peha

http://www.edutopia.org/blog/brain-movies-visualize-reading-comprehension-donna-wilson














Understanding the

skills and knowledge

required in problem-solving (Mathematics)



Problem-solving in Mathematics

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.

Overview of Curriculum

P1 P2 P3 P4 P5 P6

Whole Numbers

Money Decimals

Fractions

Length Length, Mass & Volume Length, Mass

& Volume

Volume of

Solids

Volume of

Solids

Time

Area and Perimeter

Graphs Average Pie Charts

Geometry

Percentage

Ratio

Algebra

Speed

Connect within topics

Connect between topics

Connect across levels

Importance of the Types of

Knowledge Conceptual Knowledge

Procedural Knowledge

Factual Knowledge

Linguistic Knowledge

Connect types of knowledge to build

understanding of Mathematics

Metacognitive Knowledge

Conceptual Knowledge • A concept is a general idea or understanding of something.

• To develop conceptual understanding,

One Idea Many Ideas Connections

Concrete Representations Pictorial Abstract

(GOAL)

• Foundation for Mathematical understanding

• Make Mathematics meaningful

Percentage

• 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

40= 40%

Concrete -> Pictorial -> Abstract

43% = 100

43

Marbles Hundred Squares Symbols

Common Difficulty in Conceptual

Understanding • Which is the base (100%) in the question?

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

He has $833.

a) Jenny has 100%?

100% -> $1000

1% -> $10

80% -> $800

He has $800.

Infer that it is ‘MORE’

than the base (100%)

Point of

reference

for base

(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

3 : 5 ÷ 20 ÷ 20



Infer that it is ‘LESS’


125% 100%

100%

60%

÷ 25 ÷ 25

15 : 12

12 : 20 x 4 x 4 x 3

x 3

Ali : Calvin

15 : 20

3 : 4


• Knowing a set of sequential steps in order to solve a

problem (procedural strategies)

• Procedural and Conceptual Knowledge develop

interactively

Area of Triangle Conceptual Understanding Procedural Knowledge

• base must be one of the 3 sides of a

triangle

• height must be perpendicular to the

base

• able to quickly and accurately identify

the base and its corresponding height

• area of triangle equals to half the area

of its related rectangle

• able to recall and apply the formula

x base x height 2

1

Knowing the procedure well Automaticity

Procedural Fluency (GOAL)

5 cm

12 cm

10 cm

Which is the base and its corresponding height?

What is the area of triangle ABC as shown in the figure?

A

B C

3 cm

8 cm 4 cm

5 cm

(8 x 3) ÷ 2 = 12 cm 2

(4 x 3) ÷ 2 = 6 cm 2

12 + 6 = 18 cm 2

8 + 4 = 12 cm

(12 x 3) ÷ 2 = 18 cm 2

Conceptual Understanding & Procedural Fluency

Factual Knowledge • Information that pupils can retrieve from memory

(e.g. formulae, properties of geometrical shapes)

• Factual Fluency is the automaticity of retrieving the

required information

• Factual Fluency reduces cognitive load when pupils

acquire new concepts

• Requires regular practice and investment of time for

pupils to achieve fluency

• Knowing ≠ Factual Fluency (time factor)

Diagram Properties of Geometrical Shapes

Square

2 sets of parallel lines

4 sides are equal in length

4 right angles

Rhombus



diagonal angles are equal

2 angles between parallel lines adds to 180˚

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

M

Recall properties of square:

• all sides are equal length

• all angles are 90˚

Recall properties of equilateral triangle:



• sum of 3 angles in triangle = 180˚ QM = QP = QN = MN (infer)

MQN = 60˚ (infer equilateral )

NQP = 60˚ ÷ 2 = 30˚

NPQ = (180˚- 30˚) ÷ 2 = 75˚

MPN = 75˚ + 75˚ = 150˚

P

N

Q

Recall properties of isosceles triangle:

• 2 sides are equal length & angle

• sum of 3 angles in triangle = 180˚


• Understanding the Math Vocabulary

eg product, multiples, factors

• Understanding the use of symbols eg. +, -, x, ÷

• Relating of symbols to Math Vocabulary to Math

concepts.

eg. Product of 24 and 57 = 24 x 57

= 20 x 57 + 4 x 57

Algebra Math Vocabulary Meaning

Algebraic expressions

Consists of numbers, variables and operations

eg (2p + 5)

Variable

An unknown number represented by a letter

Constant

A fixed value

Simplify

Grouping together all the variables and all the

constants using operations +, -, x, ÷ , so as to make

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)

4y – 8 (Larry)

2y + 4y + 4y – 8

= 10y – 8

Ken

John

Larry

8


• It is like the brain that controls the activation of the

other types of knowledge

Find the value of 13.2 – 0.97

Ans: ________________

• Place Value System – Conceptual Knowledge

• Subtraction of 2 decimals – Factual Knowledge

• Subtraction algorithm and renaming – Procedural Knowledge

13.2

- 0.97 No concept of Place Value

12.23

Use all the digits 3, 0, 2, 9 to form the smallest multiple of 5.

Ans: ________________


• Multiples of 5 – Factual Knowledge

• Meaning of ‘multiple’ – Linguistic Knowledge

• Place Value Chart – Procedural Knowledge

__ __ __ __ 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

• Properties of square – Conceptual Knowledge

• Recall formula and Times Table Facts – Factual Knowledge

• Meanings for ‘cuboid’ and ‘square’ – Linguistic Knowledge

• Apply formula volume of solid – Procedural

11cm

2 cm

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?

5

1

10

1

10

3

Books

Toys

Food

Pens

Pie chart overlap with fractions

Infer whole => 10 equal parts

= (equivalent fractions)

1 - - - =

$ spent on toys : $ spent on books

3 : 4

5

1

10

2

• Fractional parts and whole – Conceptual Knowledge

• 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

Integration of knowledge is required for

relational understanding in problem-solving

Procedural

Knowledge

Conceptual

Knowledge

Factual

Knowledge

Linguistic

Knowledge

Metacognitive

Knowledge

References

• 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)



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

Concepts of Addition and Subtraction

Part-Whole Unitary Model

Concepts of Multiplication,

Division, Fractions, Ratio

Whole

Part Part

Part

Whole


• 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”

Difference

B

A

Y X

Y X

Before

After

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%

72% 28%

42 units

18 units

B

A

260

Ratio Method

• 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

16% 0.16 16 : 100 = 4 : 25 16

100 =

4

25

Ratio Method How it linked to Model Method

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.


For Effective Problem-Solving

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



Infer hidden information

• At first Don had fewer; Howard had more

• At the end, Don had thrice as many as Howard

Don had more; Howard had fewer



Howard

Don

After

Howard

Don

Before

172

20 130

?

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?


• at first/ After/ left Before-After

• Sold 52 in the morning Take away 52

• Sold 5

8 of the remainder in the afternoon

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



morning and 5


with 1



• Left with 1

4 of the muffins Left with 3 units

• At first 4 x 3 units = 12 units



morning and 5


with 1


Before

52 (morning) afternoon

12 units After

3 units left

? (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?


• 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




• Betty has 100% in the first comparison

• Betty has 70% of what Candy has in the second comparison

Equate the “100%” to the “70%”



Betty

100%

Alvin 150%

Betty

70%

Candy

100%

$63

?

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


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


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.


• (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



• (Choir) Ratio of Boys : Girls = 2 : 3 Total pupils = 5 units

• (Band) Ratio of Boys : Girls = 5 : 3 Total pupils = 8 units


Pupils (Band) = 20 units

Pupils (Choir) = 40 units

5. Ratio Method


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?


• (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




(Art Club) Girls = (Computer Club) Girls

Equate “30% (girls)” and “40% (girls)”



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)

A baker made some tarts to sell. 3

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

5 of the tarts left. How many

tarts did the baker sell?

Identify key words Interpret the information Infer hidden information

• After/left Before-After

•3


𝟏

𝟒 of them were lemon tarts

• Sold 140 lemon tarts

• Sold 5

6 of strawberry tarts

𝟏

𝟔 of strawberry tarts left

• At the end, 1

5 of the tarts left

𝟒

𝟓 of the tarts were sold

• To find out Total number of strawberry and lemon tarts sold






1



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.


At first, Gina had some money. She spent 1

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

4 of the money on a blouse

𝟑

𝟒 of the money left

• 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

𝟓

𝟒 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




2

5 of


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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Understanding the

skills and knowledge

required in problem-solving

(Mathematics)



4.10 p.m. – 5.30 p.m.

Problem-solving in Mathematics

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.

Connect within topics

Connect between topics

Connect across levels

Importance of the Types of

Knowledge Conceptual Knowledge


Factual Knowledge


Connect types of knowledge to build

understanding of Mathematics


Conceptual Knowledge • A concept is a general idea or understanding of something.

• To develop conceptual understanding,

One Idea Many Ideas Connections

Concrete Representations Pictorial Abstract

(GOAL)

• Foundation for Mathematical understanding

• Make Mathematics meaningful

Percentage

• 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

40= 40%

Concrete -> Pictorial -> Abstract

43% = 100

43

Marbles Hundred Squares Symbols

Common Difficulty in Conceptual

Understanding • Which is the base (100%) in the question?

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

He has $833.

a) Jenny has 100%?

100% -> $1000

1% -> $10

80% -> $800

He has $800.



Point of

reference

for base

(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

3 : 5 ÷ 20 ÷ 20



Infer that it is ‘LESS’


125% 100%

100%

60%

÷ 25 ÷ 25

15 : 12

12 : 20 x 4 x 4 x 3

x 3

Ali : Calvin

15 : 20

3 : 4


• Knowing a set of sequential steps in order to solve a

problem (procedural strategies)

• Procedural and Conceptual Knowledge develop

interactively

Area of Triangle Conceptual Understanding Procedural Knowledge

• base must be one of the 3 sides of a

triangle

• height must be perpendicular to the

base

• able to quickly and accurately identify

the base and its corresponding height

• area of triangle equals to half the area

of its related rectangle

• able to recall and apply the formula

x base x height 2

1

Knowing the procedure well Automaticity

Procedural Fluency (GOAL)

5 cm

12 cm

10 cm

Which is the base and its corresponding height?

What is the area of triangle ABC as shown in the figure?

A

B C

3 cm

8 cm 4 cm

5 cm

(8 x 3) ÷ 2 = 12 cm 2

(4 x 3) ÷ 2 = 6 cm 2

12 + 6 = 18 cm 2

8 + 4 = 12 cm

(12 x 3) ÷ 2 = 18 cm 2

Conceptual Understanding & Procedural Fluency

Factual Knowledge • Information that pupils can retrieve from memory

(e.g. formulae, properties of geometrical shapes)

• Factual Fluency is the automaticity of retrieving the

required information

• Factual Fluency reduces cognitive load when pupils

acquire new concepts

• Requires regular practice and investment of time for

pupils to achieve fluency

• Knowing ≠ Factual Fluency (time factor)

Diagram Properties of Geometrical Shapes

Square



4 right angles

Rhombus



diagonal angles are equal

2 angles between parallel lines adds to 180˚

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

M

Recall properties of square:



Recall properties of equilateral triangle:



• sum of 3 angles in triangle = 180˚ QM = QP = QN = MN (infer)

MQN = 60˚ (infer equilateral )

NQP = 60˚ ÷ 2 = 30˚

NPQ = (180˚- 30˚) ÷ 2 = 75˚

MPN = 75˚ + 75˚ = 150˚

P

N

Q

Recall properties of isosceles triangle:

• 2 sides are equal length & angle

• sum of 3 angles in triangle = 180˚


• Understanding the Math Vocabulary

eg product, multiples, factors

• Understanding the use of symbols eg. +, -, x, ÷

• Relating of symbols to Math Vocabulary to Math

concepts.

eg. Product of 24 and 57 = 24 x 57

= 20 x 57 + 4 x 57

Algebra Math Vocabulary Meaning

Algebraic expressions

Consists of numbers, variables and operations

eg (2p + 5)

Variable

An unknown number represented by a letter

Constant

A fixed value

Simplify

Grouping together all the variables and all the

constants using operations +, -, x, ÷ , so as to make

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)

4y – 8 (Larry)

2y + 4y + 4y – 8

= 10y – 8

Ken

John

Larry

8


• It is like the brain that controls the activation of the

other types of knowledge

Find the value of 13.2 – 0.97

Ans: ________________


• Subtraction of 2 decimals – Factual Knowledge

• Subtraction algorithm and renaming – Procedural Knowledge

13.2

- 0.97 No concept of Place Value

12.23

Use all the digits 3, 0, 2, 9 to form the smallest multiple of 5.

Ans: ________________


• Multiples of 5 – Factual Knowledge

• Meaning of ‘multiple’ – Linguistic Knowledge

• Place Value Chart – Procedural Knowledge

__ __ __ __ 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

• Properties of square – Conceptual Knowledge

• Recall formula and Times Table Facts – Factual Knowledge

• Meanings for ‘cuboid’ and ‘square’ – Linguistic Knowledge

• Apply formula volume of solid – Procedural

11cm

2 cm

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?

5

1

10

1

10

3

Books

Toys

Food

Pens

Pie chart overlap with fractions

Infer whole => 10 equal parts

= (equivalent fractions)

1 - - - =

$ spent on toys : $ spent on books

3 : 4

5

1

10

2

• Fractional parts and whole – Conceptual Knowledge

• 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

Integration of knowledge is required for

relational understanding in problem-solving

Procedural

Knowledge

Conceptual

Knowledge

Factual

Knowledge

Linguistic

Knowledge

Metacognitive

Knowledge

References

• 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


P5 and P6 Parent Workshop 2016


Understand

Plan

Do

Check



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)



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


• Part-Whole Model

Concepts of Addition and Subtraction

Part-Whole Unitary Model

Concepts of Multiplication,

Division, Fractions, Ratio

Whole

Part Part

Part

Whole


• 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”

Difference

B

A

Y X

Y X

Before

After

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%

72% 28%

42 units

18 units

B

A

260

Ratio Method

• 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

16% 0.16 16 : 100 = 4 : 25 16

100 =

4

25

Ratio Method How it linked to Model Method

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.


For Effective Problem-Solving




• 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




• At first Don had fewer; Howard had more

• At the end, Don had thrice as many as Howard

Don had more; Howard had fewer



Howard

Don

After

Howard

Don

Before

172

20 130

?

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.



morning and 5


with 1



• at first/ After/ left Before-After

• Sold 52 in the morning Take away 52

• Sold 5

8 of the remainder in the afternoon

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



morning and 5


with 1



• Left with 1

4 of the muffins Left with 3 units

• At first 4 x 3 units = 12 units



morning and 5


with 1


Before

52 (morning) afternoon

12 units After

3 units left

? (9 units)

4 units = 52 1 unit = 52 ÷ 4 = 13 9 units = 13 x 9 = 117 He sold 117 muffins.




• 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




• Betty has 100% in the first comparison

• Betty has 70% of what Candy has in the second comparison

Equate the “100%” to the “70%”



Betty

100%

Alvin 150%

Betty

70%

Candy

100%

$63

?

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.


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


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


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



• (Choir) Ratio of Boys : Girls = 2 : 3

• (Band) Ratio of Boys : Girls = 5 : 3


Band had 2 units and Choir had 1 unit

• To find out Girls (Choir) : Girls (Band)

5. Ratio Method



• (Choir) Ratio of Boys : Girls = 2 : 3 Total pupils = 5 units

• (Band) Ratio of Boys : Girls = 5 : 3 Total pupils = 8 units


Pupils (Band) = 20 units

Pupils (Choir) = 40 units

5. Ratio Method


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.




• (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




(Art Club) Girls = (Computer Club) Girls

Equate “30% (girls)” and “40% (girls)”



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)






1



Identify key words Interpret the information Infer hidden information

• After/left Before-After

•3


𝟏

𝟒 of them were lemon tarts

• Sold 140 lemon tarts

• Sold 5

6 of strawberry tarts

𝟏

𝟔 of strawberry tarts left

• At the end, 1

5 of the tarts left

𝟒

𝟓 of the tarts were sold

• To find out Total number of strawberry and lemon tarts sold






1



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.




2

5 of


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

4 of the money on a blouse

𝟑

𝟒 of the money left

• 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

𝟓

𝟒 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




2

5 of


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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