Primary 3/4 Mathematics Workshop For Parents 14 April 2012 Endeavour Primary School Mathematics Department 2012

Primary 3/4 Mathematics

WorkshopFor Parents

14 April 2012

Endeavour Primary SchoolMathematics Department 2012

Workshop Outline

• Introduction to Problem-Solving• Model Method• 3 Different types of Models• 4 different Heuristics• Format of assessment

Problem-solving Approach

1. Understand the Problem (Understand)

2. Devise a Plan (Plan)3. Carry out the Plan (Do)4. Review and discuss the solution

(Check)


1. 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.


2. 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?


3. 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?


4. 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

• Teaches mathematical language

• Provides foundation for algebraic understanding

• Empowers students to think systematically and master more challenging problems

Why Model Drawing?

Model Drawing does NOT

•Work in every problem

•Specify ONE RIGHT model

•Specify ONE RIGHT operation

Concrete-Pictorial-Abstract Approach

Concrete – Manipulative

s:

Base-Ten Blocks

Pictorial - Models:

100

30 ?

Abstract – Symbols:

100 – 30 = 70

4 + 2 = 6

Concrete-Pictorial-Abstract Approach

Types of Models1. Part-whole model

a) Whole Numbers b) Fractions

2. Comparison Modela) Comparing 2 itemsb) Comparing 3 itemsc) Other Comparison Models

3. 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?


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?

$300 $200

car

$500

food housing

?

savings

$2000

$300 $200

car

$500

food housing

?

$2000

saving

Used $300 + $200 + $500= $1000

He saves $1000 every month.

Savings $2000 - $1000= $1000

How can we check if $1000 is a reasonable

answer?What is another way to

solve this problem?


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?


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

1 unit (half)

5 (Joy) 7 (Left)

1 unit (half)

Part-whole Model: Whole Numbers?

Ate

1 unit

5 (Joy) 7 (Left)

1 unit

1 unit 5 + 7= 12

Qi Ying bought 24 sweets.

2 units 2 × 12= 24

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

answer?What is

another way of

representing this problem?

? 5 + 7× 2

÷ 2

Part-whole Model: Fractions


? girls24 boys

2 units 24

1 unit 24 ÷ 2

= 12

There are 12 girls.

How can we check if the answer is reasonable?


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?

¼


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?

¼

16 carps

¼

goldfishguppies

?

¼ 2 units 4 + 16

4 units 2 × 20 = 40

There are 40 fish.

= 20 4


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?


Sven collected 3426 stamps. He collected 841 fewer stamps than Jerome. How many stamps did they collect?

Who has more?

3426Sven

841

fewer

?

Jerome

Whose bar should be longer?

?

Jerome 3426 + 841= 4267

They collected 7693 stamps.

Total 3426 + 4267= 7693

3426Sven

841

fewer

?

Jerome?

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

answer?What is another way to

solve this problem?



Smaller ¼

?

Larger

2 units

1 unit

Smaller ¼

?

Larger



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?


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?

Siti

Kyle

Alice

290

50 Note how ‘50’ is represented.

Alice has 110 stickers.

4 units 290 – 50= 240

1 unit 240 ÷ 4= 60

Siti

Kyle

Alice

290

50

Let Siti have x stickers.Kyle 2xAlice x + 504x + 50 = 2904x = 240x = 6060 + 50 = 110

Alice 60 + 50= 110


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?


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?

Siti

Kyle

Alice

270

Siti

Kyle

Alice

270

1 unit

2 units

Siti has 60 stickers.

270 ÷ 9

30 x 2= 60

2709 units

= 30


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.



Pens

Files



Pens

Files$18?


Pens

Files$18

9 units

1 unit

1 file costs $6.

$18

$18 ÷ 9 = $2

3 units $2 x 3 = $6

?


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




Vases

Plates$161?

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$161

7 units

1 unit

1 vase costs $46.

$161

$161 ÷ 7 = $23

2 units $23 x 2 = $46

?

3. Before and After Model

• Total unchanged• Total changed

Before and After (total unchanged)

Alan

Ben558

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

Ben558

4

1

9 units

1 unit

Ben had 310 cards at first.

558

558 ÷ 9 = 62

5 units 62 x 5 = 310

?

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

Betty

Before and After (Total Changed)

Alice

Betty

1 unit

1 unit 1 unit $45

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

Betty

1 unit

1 unit 1 unit $45

?

1 unit

Each girl had $195 at first.

$120 - $45 = $75

$75 + $120 = $195


Alice

Betty

1 unit

1 unit 1 unit $45

?

1 unit

Each girl had $195 at first.

$120 - $45 = $75

$150 + $45 = $195

2 units $75 x 2 = $150

Before and After (Total Changed)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

Female


Male

Female

1 unit

1 unit 1 unit 461 unit 1 unit


Male

Female

1 unit

1 unit 1 unit 461 unit 1 unit

?

Male

Female

1 unit

1 unit 1 unit 46

3 units

There were 242 male passengers at first.

193 - 46 = 147

147 ÷ 3 = 49

1 unit 1 unit

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?

?

1 unit

49 + 193 = 242

Other Heuristics1. Work Backwards

2. Guess and Check

3. Make a Systematic List

4. Make a Table

Is model drawing the only method?No!

Work Backwards Find the missing number.

? 108 54 50- 4÷ 2x 3

+ 4x 2÷ 3

108 ÷ 3 = 36

The missing number is 36.

50 + 4 = 54

54 x 2 = 108

Work BackwardsA 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?

28÷2+ 7 - 8

?A

B C D

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?

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

28 + 8 = 36

72 36 28÷2+ 7 - 8

?

36 x 2 = 72

72 – 7 = 65

AB C D

+ 8x 2- 7

Work BackwardsJohn 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?

2pm+ 50 min + 1h 40 min

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?

He started washing his car at 11.30 am.

? 12.20pm 2pm- 1h 40 min- 50 min

12.20 pm 11.30 am

- 50 min

2 pm 1 pm 12.20 pm

- 40 min- 1 h

+ 50 min + 1h 40 min

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)

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? Conditions stated in the

question:1)Total vehicles: 252)Total wheels: 553)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 tricycle, how many bicycles are there at the park?

Guess No. of bicycles(more)

No. of bicycle wheels(no. x 2)

No. of tricycles(fewer)

No. of tricycle wheels(no. x 3)

Total number

of wheels

(55)

Check






Total number

of wheels

(55)

Check

1 15 15 x 2 = 30

25 ‒ 15 = 10

10 x 3 = 30

30 + 30 = 60






Total number

of wheels

(55)

Check

1 15 15 x 2 = 30

25 ‒ 15 = 10

10 x 3 = 30

30 + 30 = 60

2 17 17 x 2 = 34

25 ‒ 17= 8

8 x 3 = 24

34 + 24 = 58






Total number

of wheels

(55)

Check

1 15 15 x 2 = 30

25 ‒ 15 = 10

10 x 3 = 30

30 + 30 = 60

2 17 17 x 2 = 34

25 ‒ 17= 8

8 x 3 = 24

34 + 24 = 58

3 20 20 x 2 = 40

25 – 20 = 5

5 x 3 = 15

40 + 15 = 55

There are 20 bicycles 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

There are 20 bicycles at the park.

2 x 25 = 50

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)



Guess No. of 10¢

coins(more)

Value of 10¢

coins(no. x 10¢)

No. of 20¢

coins(fewer)

Value of 20¢ coins

(no. x 20¢)

Total value

($3.40)

Check


Guess No. of 10¢

coins(more)

Value of 10¢

coins(no. x 10¢)

No. of 20¢

coins(fewer)

Value of 20¢ coins

(no. x 20¢)

Total value

($3.40)

Check

1 20 $2 3 $0.60 $2.60 ×


Guess No. of 10¢

coins(more)

Value of 10¢

coins(no. x 10¢)

No. of 20¢

coins(fewer)

Value of 20¢ coins

(no. x 20¢)

Total value

($3.40)

Check

1 20 $2 3 $0.60 $2.60 ×

2 13 $1.30 10 $2 $3.30 ×


Guess No. of 10¢

coins(more)

Value of 10¢

coins(no. x 10¢)

No. of 20¢

coins(fewer)

Value of 20¢ coins

(no. x 20¢)

Total value

($3.40)

Check

1 20 $2 3 $0.60 $2.60 ×

2 13 $1.30 10 $2 $3.30 ×

3 12 $1.20 11 $2.20 $3.40

There are 11 20¢ 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



5 5 left 6 3 short


5 5 left5 5 + 5 = 10

10

15

20

25

30

35

40

45

50

6 3 short


5 5 left5 5 + 5 = 10

10 15

15 20

20 25

25 30

30 35

35 40

40 45

45 50

50 55

6 3 short


5 5 left5 5 + 5 = 10

10 15

15 20

20 25

25 30

30 35

35 40

40 45

45 50

50 55

6 3 short6 6 ‒ 3 = 3

12

18

24

30

36

42

48

54

60


5 5 left5 5 + 5 = 10

10 15

15 20

20 25

25 30

30 35

35 40

40 45

45 50

50 55

6 3 short6 6 ‒ 3 = 3

12 9

18 15

24 21

30 27

36 33

42 39

48 45

54 51

60 57


5 5 left5 5 + 5 = 10

10 15

15 20

20 25

25 30

30 35

35 40

40 45

45 50

50 55

6 3 short6 6 ‒ 3 = 3

12 9

18 15

24 21

30 27

36 33

42 39

48 45

54 51

60 57

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?

Receives Gives ResultBennyCindyDavidEvelyn


Receives Gives ResultBenny 19Cindy 19DavidEvelyn


Receives Gives ResultBenny 19Cindy 19 15David 15 Evelyn


Receives Gives ResultBenny 19Cindy 19 15David 15 + 3Evelyn 3


Receives Gives ResultBenny 19Cindy 19 15David 15 + 3 3Evelyn 3 3


Receives Gives ResultBenny 12 19Cindy 19 15David 15 + 3 3 + 12Evelyn 3 3


Receives Gives ResultBenny 12 19 Gives moreCindy 19 15 Receives moreDavid 15 + 3 = 18 3 + 12 = 15 Receives moreEvelyn 3 3 No change

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?


X 1 2 3 4 5 6

1

2

3

4

5

6


X 1 2 3 4 5 6

1

2

3

4

5 15

6


X 1 2 3 4 5 6

1 1 2 3 4 5 6

2 2 4 6 8 10 12

3 3 6 9 12 15 18

4 4 8 12 16 20 24

5 5 10 15 20 25 30

6 6 12 18 24 30 36


X 1 2 3 4 5 6

1 1 2 3 4 5 6

2 2 4 6 8 10 12

3 3 6 9 12 15 18

4 4 8 12 16 20 24

5 5 10 15 20 25 30

6 6 12 18 24 30 36

The score cannot be 7.

Format of Math Paper

P5/P6 Math Exam Paper Format

P2 – P4 P5 - P6

MCQ 20 Qns – 40 marks

15 Qns – 20 marks

SAQ 20 Qns – 40 marks

15 Qns – 20 marks

Word Problems

5 Qns – 20 marks

18 Qns – 60 marks

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

Documents

Primary 3/4 Mathematics Workshop For Parents 14 April 2012 Endeavour Primary School Mathematics Department 2012