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Primary 3/4 Mathematics Workshop For Parents. 14 April 2012. Workshop Outline. Introduction to Problem-Solving Model Method 3 Different types of Models 4 different Heuristics Format of assessment. Problem-solving Approach. Understand the Problem (Understand) Devise a Plan (Plan)

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Primary 3/4 Mathematics WorkshopFor Parents

14 April 2012

Endeavour Primary School

Mathematics Department 2012


Workshop Outline

  • Introduction to Problem-Solving

  • Model Method

  • 3 Different types of Models

  • 4 different Heuristics

  • Format of assessment


Problem-solving Approach

  • Understand the Problem (Understand)

  • Devise a Plan (Plan)

  • Carry out the Plan (Do)

  • Review and discuss the solution (Check)


Problem-solving Approach

  • Understand the Problem (Understand)

    • Read to understand.

    • If at first not clear, read again.

    • Still don’t get it? Read chunk by chunk.

    • Explain the question in another way.

    • Use visualisation tool – model, timeline, diagrams, table etc.


Problem-solving Approach

  • Devise a Plan (Plan)

    • Have I seen a similar or related question before?

    • Do I have a ready plan?

    • Do I have all the data? What data is missing?

    • Can I find the missing data?

    • Can I use a smaller number to try first?

    • Use a heuristics?


Problem-solving Approach

  • Carry out the Plan (Do)

    • Are all my steps accurate?

    • Are there traps I need to be alert of?

    • Have I used all the data given?

    • Do my steps make sense?


Problem-solving Approach

  • Review and discuss the solution (Check)

    • Does the answer make sense?

    • Did I answer the question?

    • Could this problem be solved in a simpler way?


Model Method

Draw a diagram


Why Model Drawing?

  • Visual representation of details

    • Majority of our children are visual learners

  • Helps children plan the solution steps for solving the problem

    • Useful in fractions, ratio & percentage


WhyModel Drawing?

  • Teaches mathematical language

  • Provides foundation for algebraic understanding

  • Empowers students to think systematically and master more challenging problems


Model Drawing does NOT

  • Work in every problem

  • Specify ONE RIGHT model

  • Specify ONE RIGHT operation


Concrete-Pictorial-Abstract Approach

Concrete – Manipulatives:

Base-Ten Blocks

Pictorial - Models:

100

30

?

Abstract – Symbols:

100 – 30 = 70


Concrete-Pictorial-Abstract Approach

4 + 2 = 6


Types of Models

  • Part-whole model

    a) Whole Numbers

    b) Fractions

    2. Comparison Model

    a) Comparing 2 items

    b) Comparing 3 items

    c) Other Comparison Models

  • Before-After Model

    a) Total unchanged

    b) Total changed


1. Part-whole Model

Find value of unknown part

Find value of whole


Part-whole Model: Whole Numbers

Calvin earns $2000 every month. He pays

$300 for food. He also spends $200 on his car, $500 on housing and saves the rest. How much does he save every month?

Calvin earns $2000 every month. He pays

$300 for food. He also spends $200 on his car, $500 on housing and saves the rest. How much does he save every month?


$2000

$300

$200

$500

?

food

savings

car

housing

Part-whole Model: Whole Numbers

Calvinearns $2000 every month. He pays

$300 for food.Healsospends $200 on his car,$500 on housingandsaves the rest.. How much does he saveevery month?


$2000

$300

$200

$500

?

food

car

housing

saving

Used

$300 + $200 + $500

= $1000

Savings

$2000 - $1000

= $1000

He saves $1000 every month.


How can we check if $1000 is a reasonable answer?

What is another way to solve this problem?


Part-whole Model: Whole Numbers

Qi Ying bought some sweets. She ate half of them and gave 5 sweets to Joy. She had 7 sweets left. How many sweets did Qi Ying buy?


Ate

5 (Joy)

7 (Left)

1 unit (half)

1 unit (half)

Part-whole Model: Whole Numbers

Qi Ying bought some sweets. She ate halfof them and gave 5 sweetsto Joy. She had7 sweets left.How many sweets did Qi Ying buy?

?


Ate

5 (Joy)

7 (Left)

1 unit

1 unit

Part-whole Model: Whole Numbers

?

1 unit

5 + 7

= 12

2 units

2 × 12

= 24

Qi Ying bought 24 sweets.


How can we check if ‘24 sweets’ is a reasonable answer?

What is another way of representing this problem?

÷ 2

?

5 + 7

× 2


Part-whole Model: Fractions


? girls

24 boys

Part-whole Model: Fractions

24

2 units

24 ÷ 2

1 unit

= 12

There are 12 girls.


How can we check if the answer is reasonable?


Part-whole Model: Fractions

¼ of the fish in an aquarium are goldfish. There are 4 more guppies than goldfish in the aquarium. The remaining 16 fish are carps. How many fish are there in the aquarium?


?

goldfish

Part-whole Model: Fractions

¼ of the fishin an aquarium are goldfish.There are 4 more guppies than goldfish in the aquarium. The remaining 16fish are carps. How many fishare there in the aquarium?

2 units

4 + 16

= 20

¼

¼

¼

¼

4

4 units

2 × 20

guppies

= 40

16 carps

There are 40 fish.


How can we check if the answer is reasonable?


2. Comparison Model

Find total sum given between difference and value of an item

Find value of an item given difference and sum


Comparison Model: 2 items

Sven collected 3426 stamps. He

collected 841 fewer stamps than Jerome.

How many stamps did they collect?


?

Comparison Model: 2 items

Svencollected3426 stamps. He

collected 841 fewer stampsthanJerome.

How many stampsdidtheycollect?

Who has more?

Sven

3426

Whose bar should be longer?

fewer

Jerome

841

?


?

Sven

3426

fewer

Jerome

841

?

Jerome

3426 + 841

= 4267

Total

3426 + 4267

= 7693

They collected 7693 stamps.


How can we check if ‘7693 stamps’ is a reasonable answer?

What is another way to solve this problem?


Comparison Model: 2 items


Comparison Model: 2 items

?

Smaller

Larger

¼


?

Smaller

Larger

¼

2 units

1 unit


How can we check if the answer is reasonable?


Comparison Model: 3 items

Kyle, Siti and Alice have a total of 290 stickers. Kyle has twice as many stickers as Siti. Alice has 50 stickers more than Siti. How many stickers does Alice have?


Comparison Model: 3 items

Kyle, Siti and Alice have a total of 290 stickers. Kyle has twiceas manystickers asSiti. Alicehas 50 stickersmorethanSiti. How many stickersdoes Alicehave?

Kyle

290

Siti

Alice

50

Note how ‘50’ is represented.


Kyle

290

Siti

Alice

50

4 units

290 – 50

= 240

1 unit

Alice

240 ÷ 4

60 + 50

= 60

= 110

Let Siti have x stickers.

Kyle 2x

Alice x + 50

4x + 50 = 290

4x = 240

x = 60

60 + 50 = 110

Alice has 110 stickers.


Comparison Model: 3 items

Kyle, Siti and Alice have a total of 270 stickers. Kyle has thrice as many stickers as Siti. Alice has half as many stickers as Siti. How many stickers does Siti have?


Comparison Model: 3 items

Kyle, Siti and Alicehaveatotal of 270stickers. Kyle hasthrice as manystickersas Siti. Alicehashalf as manystickersas Siti. How many stickersdoesSitihave?

Kyle

Siti

270

Alice


9 units

270

Kyle

Siti

Alice

270

270 ÷ 9

1 unit

= 30

2 units

30 x 2

= 60

Siti has 60 stickers.


How can we check if the answer is reasonable?


Other Comparison Models

2 files and 3 pens cost $18 altogether. A file costs 3 times as much as a pen. Find the cost of 1 file.


Other Comparison Models

2 files and 3 pens cost $18 altogether. A file costs 3 times as much as a pen. Find the cost of 1 file.

Files

Pens


Other Comparison Models

2 files and 3 pens cost $18 altogether. A file costs 3 times as much as a pen. Find the cost of 1 file.

Files

$18

?

Pens


2 files and 3 pens cost $18 altogether. A file costs 3 times as much as a pen. Find the cost of 1 file.

Files

$18

?

Pens

9 units

$18

= $2

1 unit

$18 ÷ 9

3 units

$2 x 3

= $6

1 file costs $6.


Other Comparison Models

2 crystal vases and 3 plates cost $161. The cost of 1 plate is half the cost of 1 vase.

What is the cost of 1 vase?


Other Comparison Models

2 crystal vases and 3 plates cost $161. The cost of 1 plate is half the cost of 1 vase.

What is the cost of 1 vase?

Vases

Plates


Other Comparison Models

2 crystal vases and 3 plates cost $161. The cost of 1 plate is half the cost of 1 vase.

What is the cost of 1 vase?

Vases

$161

?

Plates


2 crystal vases and 3 plates cost $161. The cost of 1 plate is half the cost of 1 vase. What is the cost of 1 vase?

Vases

$161

?

Plates

7 units

$161

1 unit

$161 ÷ 7

= $23

2 units

$23 x 2

= $46

1 vase costs $46.


3. Before and After Model

Total unchanged

Total changed


Before and After (total unchanged)

Alan

558

Ben


Alan and Ben had 558 cards altogether. Alan gave of his cards to Ben. After that, Ben had twice the number of cards as Alan.

How many cards did Ben have at first?

Alan

558

Ben

?

9 units

558

1 unit

558 ÷ 9

= 62

5 units

62 x 5

= 310

Ben had 310 cards at first.


Before and After (Total Changed)

Alice

Betty

Alice and Betty had the same amount of money each. After Alice spent $120 and Betty spent $45, Betty had twice as much money as Alice.

How much money did each girl have at first?


Before and After (Total Changed)

Alice and Betty had the same amount of money each. After Alice spent $120 and Betty spent $45, Betty had twice as much money as Alice.

How much money did each girl have at first?

Alice

1 unit

$120

$45

1 unit

1 unit

Betty


Alice and Betty had the same amount of money each.

After Alice spent $120 and Betty spent $45, Betty had twice as much money as Alice.

How much money did each girl have at first?

Alice

1 unit

$120

?

$45

1 unit

1 unit

Betty

1 unit

$120 - $45

= $75

$75 + $120

= $195

Each girl had $195 at first.


Alice and Betty had the same amount of money each.

After Alice spent $120 and Betty spent $45, Betty had twice as much money as Alice.

How much money did each girl have at first?

Alice

1 unit

$120

$45

1 unit

1 unit

Betty

?

1 unit

$120 - $45

= $75

2 units

$75 x 2

= $150

$150 + $45

= $195

Each girl had $195 at first.


Before and After (Total Changed)

Male

Female

There was an equal number of male and female passengers in a train at first. After 193 male passengers and 46 female passengers alighted, there were 4 times as many female passengers as male passengers left in the train. How many male passengers were in the train at first?


Before and After (Total Changed)

Male

1 unit

193

1 unit

1 unit

1 unit

1 unit

46

Female

There was an equal number of male and female passengers in a train at first. After 193 male passengers and 46 female passengers alighted, there were 4 times as many female passengers as male passengers left in the train. How many male passengers were in the train at first?


Before and After (Total Changed)

Male

1 unit

193

?

1 unit

1 unit

1 unit

1 unit

46

Female

There was an equal number of male and female passengers in a train at first. After 193 male passengers and 46 female passengers alighted, there were 4 times as many female passengers as male passengers left in the train. How many male passengers were in the train at first?


There was an equal number of male and female passengers in a train at first. After 193 male passengers and 46 female passengers alighted, there were 4 times as many female passengers as male passengers left in the train.

How many male passengers were in the train at first?

Male

1 unit

193

?

1 unit

1 unit

1 unit

1 unit

46

Female

3 units

193 - 46

= 147

1 unit

147 ÷ 3

= 49

49 + 193

= 242

There were 242 male passengers at first.


Other Heuristics

Is model drawing the only method?

No!

  • Work Backwards

    2. Guess and Check

  • Make a Systematic List

  • Make a Table


Work Backwards

- 4

x 3

÷ 2

50

?

54

108

+ 4

x 2

÷ 3

50 + 4 = 54

54 x 2 = 108

108 ÷ 3 = 36

The missing number is 36.

Find the missing number.


Work Backwards

C

B

D

A

+ 7

÷2

- 8

28

?

A train carrying some passengers left Station A.

At Station B, 7 passengers boarded.

At Station C, half of the passengers alighted.

At Station D, 8 passengers alighted.

As the train left Station D, there were 28 passengers on the train.

How many passengers were on the train when it left Station A?


A train carrying some passengers left Station A.

At Station B, 7 passengers boarded.

At Station C, half of the passengers alighted.

At Station D, 8 passengers alighted.

As the train left Station D, there were 28 passengers on the train.

How many passengers were on the train when it left Station A?

C

B

D

A

+ 7

÷2

- 8

28

?

36

72

- 7

+ 8

x 2

28 + 8 = 36

36 x 2 = 72

72 – 7 = 65

65 passengers were on the train when it left Station A.


Work Backwards

+ 1h 40 min

+ 50 min

2pm

John took 50 minutes to wash his car and another 1 h 40 min to polish it. He finished washing and polishing his car at 2 pm. At what time did he start washing his car?


John took 50 minutes to wash his car and another 1 h 40 min to polish it. He finished washing and polishing his car at 2 pm. At what time did he start washing his car?

+ 1h 40 min

+ 50 min

?

12.20pm

2pm

- 1h 40 min

- 50 min

- 40 min

- 1 h

2 pm 1 pm 12.20 pm

- 50 min

12.20 pm 11.30 am

He started washing his car at 11.30 am.


Guess and Check (1)

At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycles, how many bicycles are there at the park?


Guess and Check (1)

At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycles, how many bicycles are there at the park?


Guess and Check (1)

  • Conditions stated in the question:

  • Total vehicles: 25

  • Total wheels: 55

  • More bicycles than tricycles.

At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycles, how many bicycles are there at the park?


At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park?


At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park?


At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park?


At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park?

There are 20 bicycles at the park.


At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park?

Method 1: Guess and Check

Method 2: Supposition

Suppose all the vehicles are bicycles, the number of wheels

But there are 55 wheels altogether.

55 ‒ 50 = 5 extra wheels

Each tricycle has 1 wheel more than a bicycle, 5 ÷ 1 = 5

There are 5 tricycles.

25 ‒ 5 = 20

2 x 25 = 50

There are 20 bicycles at the park.


Guess and Check (2)

Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there?


Guess and Check (2)

Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there?


Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there?


Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there?


Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there?


Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there?

There are 1120¢ coins .


Make a Systematic List

Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?


Make a Systematic List

Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?


Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?


Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?


Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?


Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?


Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?


Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?

Mr John has 45 stickers.


Make a Table

Benny, Cindy, David and Evelyn give picture cards to one another.

Benny gives Cindy 19 cards.

Cindy gives David 15 cards.

Evelyn gives David 3 cards but David returns them to Evelyn.

David gives Benny 12 cards.

Who has fewer picture cards in the end than before?


Benny, Cindy, David and Evelyn give picture cards to one another.

Benny gives Cindy 19 cards.

Cindy gives David 15 cards.

Evelyn gives David 3 cards but David returns them to Evelyn.

David gives Benny 12 cards. Who has fewer picture cards in the end than before?


Benny, Cindy, David and Evelyn give picture cards to one another.

Benny gives Cindy 19 cards.

Cindy gives David 15 cards.

Evelyn gives David 3 cards but David returns them to Evelyn.

David gives Benny 12 cards. Who has fewer picture cards in the end than before?


Benny, Cindy, David and Evelyn give picture cards to one another.

Benny gives Cindy 19 cards.

Cindy gives David 15 cards.

Evelyn gives David 3 cards but David returns them to Evelyn.

David gives Benny 12 cards. Who has fewer picture cards in the end than before?


Benny, Cindy, David and Evelyn give picture cards to one another.

Benny gives Cindy 19 cards.

Cindy gives David 15 cards.

Evelyn gives David 3 cards but David returns them to Evelyn.

David gives Benny 12 cards. Who has fewer picture cards in the end than before?


Benny, Cindy, David and Evelyn give picture cards to one another.

Benny gives Cindy 19 cards.

Cindy gives David 15 cards.

Evelyn gives David 3 cards but David returns them to Evelyn.

David gives Benny 12 cards. Who has fewer picture cards in the end than before?


Benny, Cindy, David and Evelyn give picture cards to one another.

Benny gives Cindy 19 cards.

Cindy gives David 15 cards.

Evelyn gives David 3 cards but David returns them to Evelyn.

David gives Benny 12 cards. Who has fewer picture cards in the end than before?


Benny, Cindy, David and Evelyn give picture cards to one another.

Benny gives Cindy 19 cards.

Cindy gives David 15 cards.

Evelyn gives David 3 cards but David returns them to Evelyn.

David gives Benny 12 cards. Who has fewer picture cards in the end than before?

Benny has fewer picture cards than before.


Make a Table (2)

In a game, two dice are thrown and the two numbers shown are multiplied to give a score.

What whole number less than 10 cannot be a score of this game?


Make a Table (2)

In a game, two dice are thrown and the two numbers shown are multiplied to give a score.

What whole numbers less than 10 cannot be a score of this game?


In a game, two dice are thrown and the two numbers shown are multiplied to give a score.

What whole number less than 10 cannot be a score of this game?


In a game, two dice are thrown and the two numbers shown are multiplied to give a score.

What whole number less than 10 cannot be a score of this game?


In a game, two dice are thrown and the two numbers shown are multiplied to give a score.

What whole number less than 10 cannot be a score of this game?


In a game, two dice are thrown and the two numbers shown are multiplied to give a score.

What whole number less than 10 cannot be a score of this game?

The score cannot be 7.


Format of Math Paper


P5/P6 Math Exam Paper Format


P5/P6 Math Exam Paper Format

  • Paper 1 - MCQ and SAQ

  • Paper 2 - a combination of 2, 3, 4 and 5 marks word problems

  • Paper 1 to be completed in 50 minutes without calculator

  • Paper 2 to be completed in 100 minutes with calculator


Challenges due to Paper format

  • Paper 1 to be completed within 50 minutes (30 questions – less than 2 minutes per question)

  • Paper 2 – focuses on thinking skills as well as heuristics

  • Culture shock in P5 for pupils


Changes to P3 and P4 Format

  • 2012 – P3 and P4 SA2 Papers Section C total marks changed from 20 to 30.

  • 2013 – P4 SA1 and SA2 Papers Section C total marks changed from 30 to 40.

  • Heuristics and thinking skills come into play more.

  • Concept and syllabus becomes basic skills.


Thank You


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