Seriously Addictive Mathematics https://seriouslyaddictivemaths.com.sg Singapore Math Enrichment Program Tue, 20 Feb 2024 23:41:44 +0000 en-US hourly 1 https://wordpress.org/?v=6.5.2 https://seriouslyaddictivemaths.com.sg/wp-content/uploads/2023/02/faviconV2.png Seriously Addictive Mathematics https://seriouslyaddictivemaths.com.sg 32 32 Singapore Math Heuristics: Draw a Table, Make Suppositions and Use Before-After Concept https://seriouslyaddictivemaths.com.sg/math-heuristics-model/ Thu, 19 Jan 2023 00:18:33 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=306

Heuristics, in the context of problem-solving, are a set of strategies to help students solve mathematical problems.

Although problem-solving is by and large the process of working towards a goal to which a solution may not be immediately present, it is important that problem solvers (or students) are not only aware of what they are doing and why they are doing it, but also have the ability to self-regulate these processes.

This is where the role of metacognition in problem-solving comes in. Metacognition is the awareness of how your mind works and the ability to control your thinking process.

When students understand and recognise how they learn and are given opportunities to monitor and regulate their thinking during problem-solving, not only do they improve their metacognitive skills but they may also be more successful in solving the problem.

Singapore Math Heuristics

In part one of our Math Heuristics series, we gave an overview of the 12 heuristics in Singapore Primary Math syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to solve various math word problems using them.

In part two of the Math Heuristics series, we expanded on the heursitics – Act It Out, Draw A Diagram and Look For Patterns, and also demonstrated how to apply the Polya’s 4-step problem-solving process in sample word problems.

In part three of the Math Heuristics series, we focused on the heuristics: Make a Systematic List, Guess and Check, and Restate the Problem in Another Way.

In part four of the Math Heuristics series, we looked at three heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards.

In the final part of the series, we will explore the remaining heuristics: Draw a TableMake Suppositions and Use Before-After Concept.

_______________________________________________________________________________________________________

Heuristic: Draw a Table

Word Problem (Grade 4):

Mary cycles to the park every 6 days. John cycles to the same park every 8 days. Mary and John cycle to the park on 3 June, Tuesday. On which date and day will they cycle to the park together on the same day?

Solution:

1. Understand:
What to find: date and day Mary and John next cycle to the park on the same day.
What is known: Mary cycles to the park every 6 days. John cycles to the park every 8 days. Mary and John cycle to the park on 3 June, Tuesday.

2. Choose: Draw a table

3. Solve:
Let’s compare the days Mary and John cycle to the park.
Mary cycle to the park on days in multiples of 6.
John cycle to the park on days in multiples of 8.

Mary and John will cycle to the park on the same day every 24 days.
3 June + 24 days is 27 June
24 days = 3 weeks and 3 days
Tuesday + 3 days is FridayMary and John will next cycle to the park on 27 June, Friday.4. Check
Is 24 a multiple of 6? Yes
Is 24 a multiple of 8? Yes
Is 27 June Friday 24 days after 3 Jun Tuesday? Yes

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 4):

Janice wanted to distribute stickers equally among some children. If each child received 8 stickers, she would have 3 stickers left. If each child received 11 stickers, she would need another 9 stickers. How many children were there?
Answer: Janice had 35 stickers. There were 4 children.

_______________________________________________________________________________________________________

Heuristic: Make Suppositions

Word Problem (Grade 5):

Mary took a Science test. She answered all 30 questions and obtained 42 marks. For each correct answer, 3 marks were awarded. For each wrong answer, 1 mark was deducted. How many questions did she answer correctly?

Solution:

1. Understand:
What to find: Number of questions Mary answered correctly.
What is known: Mary answered 30 questions. She obtained 42 marks. 3 marks were awarded for each correct answer. 1 mark was deducted for each wrong answer.

2. Choose: Make suppositions

3. Solve:
Suppose that Mary answered all 30 questions correctly.
30 x 3 = 90
Mary obtained 92 supposed total marks.

90 – 42 = 48
The supposed total marks are 48 more than the actual total marks.

3 + 1 = 4
By replacing 1 correct answer with 1 wrong answer, 4 marks are deducted from the total marks.

48 ÷ 4 = 12
Mary answered 12 questions wrongly.

30 – 12 = 18
She answered 18 questions correctly.

4. Check:
What are the total marks awarded for 18 correct answers?
18 x 3 = 54
What are the total marks deducted for 12 wrong answers?
12 x 1 = 12
What are the total marks obtained by Mary?
54 – 12 = 42

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 4):

Farmer James has some ducks, horses and cows on his farm. He has 30 ducks and cows altogether. The total number of legs the ducks and cows have is 82. The total number of legs the horses have is 28. How many ducks and how many cows are there on the farm?

Answer: There are 19 ducks and 11 cows on the farm.

_______________________________________________________________________________________________________

Heuristic: Use Before-After Concept

Word Problem (Grade 5):

The ratio of Joy’s age to her uncle’s age is 1 : 4 now. In 21 years’ time, the ratio of Joy’s age to her uncle’s age will be 3 : 5. How old is Joy’s uncle now?

Solution:

1. Understand:
What to find: Joy’s uncle age now.
What is known: The ratio of age now is 1 : 4. The ratio of age 21 years later is 3 : 5.

2. Choose: Use before-after concept

3. Solve:

(9 – 2) units = 7 units = 21 years
1 unit = 21 ÷ 7 = 3 years
8 units = 3 x 8 = 24
Joy’s uncle is 24 years old now.4. Check:
How old is Joy now? 3 x 2 = 6 years old
Ratio of Joy’s age to her uncle’s age now = 6 : 24 = 1 : 4
How old is Joy 21 years later? 3 x 9 = 27 years old
How old is Joy’s uncle 21 years later? 3 x 15 = 45
Ratio of Joy’s age to her uncle’s age 21 years later = 27 : 45 = 3 : 5

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 5):

Ray and Sam each brought some money for shopping. The ratio of the amount of Ray’s money to the amount of Sam’s money was 3 : 4. After each of them bought a laptop for $1250, the ratio of the amount of Ray’s money to the amount of Sam’s money became 1 : 3. How much money did Sam bring for shopping?
Answer: Sam bought $2000 for shopping.

This is the final part to S.A.M Math Heuristics series. Read part onepart twopart three and part four here.

Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world’s largest Singapore Math enrichment program for children aged four to 12. The award-winning S.A.M program is based on the global top-ranking Singapore Math curriculum with a focus on developing problem solving and thinking skills.

The curriculum is complemented with S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

]]>
Singapore Math Heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards https://seriouslyaddictivemaths.com.sg/math-heuristics-thinking/ Thu, 19 Jan 2023 00:17:11 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=309

Problem-solving in mathematics helps children develop reasoning and communication skills that are transferrable and important life skills.

Reasoning is required on three levels when children solve word problems. First, they use reasoning to recognise what information is provided or missing. Then, they use reasoning to figure out what information they need to find. Finally, they use reasoning to draw on relevant prior knowledge and choose the most suitable heuristic to work out the solution.

Communication is required for comprehension and expression during problem-solving. Children need to read and understand word problems and then write and express their solutions. When children are encouraged to explain their thinking verbally, visually and in written form, they gain a better understanding of math concepts and develop stronger communication skills.

Singapore Math Heuristics

In part one of our Math Heuristics series, we gave an overview of the 12 heuristics in Singapore Primary Math syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to solve various math word problems using them.

In part two of the Math Heuristics series, we expanded on the heursitics – Act It Out, Draw A Diagram and Look For Patterns, and also demonstrated how to apply the Polya’s 4-step problem-solving process in sample word problems.

In part three of the Math Heuristics series, we focused on the heuristics: Make a Systematic List, Guess and Check, and Restate the Problem in Another Way.

In part 4 of this series, we will zoom in on these 3 heuristics: Solve part of the problem, simplify the problem and work backwards.

_______________________________________________________________________________________________________

Heuristic: Solve part of the problem

Word Problem (Grade 3):

At a school library, each student could borrow up to 4 books. The bar graph below shows how many books students borrowed from the school library in one week. What was the total number of books borrowed from the library that week?

Solution:

1. Understand:
What to find: total number of books borrowed
What is known: 19 students borrowed 1 book. 27 students borrowed 2 books. 16 students borrowed 3 books. 11 students borrowed 4 books.

2. Choose: Solve part of the problem

3. Solve:

19 students borrowed 1 book = 19 × 1 = 19 books
27 students borrowed 2 books = 27 × 2 = 54 books
16 students borrowed 3 books = 16 × 3 = 48 books
11 students borrowed 4 books = 11 × 4 = 44 books

19 + 54 + 48 + 44 = 165 books

165 books were borrowed from the library that week.

4. Check:
Did I read the information from the bar graph correctly? Yes
Did I calculate the number of books for each bar correctly? Yes
Did I add the number of books for all bars? Yes

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 6):

The diagram below shows 4 shaded triangles in Triangle ABC. All the triangles in the diagram are equilateral triangles. If the area of Triangle ABC is 64 cm2, find the total area of the shaded triangles.

Answer: The total area of the shaded triangles is 28 cm2
See the solution in part one of our Singapore Math Heuristics series.

_______________________________________________________________________________________________________

Heuristic: Simplify the problem

Word Problem (Grade 3):

A bakery sold cupcakes at $4 each. For every 6 cupcakes bought, a discount of $2 was given. Ling bought 50 cupcakes. How much did Ling pay for the cupcakes?

Solution:

1. Understand:
What to find: How much Ling paid for the cupcakes.
What is known: Each cupcake cost $4. A discount of $2 was given for every 6 cupcakes bought. Ling bought 50 cupcakes.

2. Choose: Simplify the problem

3. Solve:

6 cupcakes form 1 set.
50 ÷ 6 = 8 remainder 2
Ling bought 8 sets of 6 cupcakes and 2 more cupcakes.

Each cupcake costs $4.
6 × $4 = $24
$24 – $2 = $22
Each set of 6 cupcakes cost $22.

Cost of 8 sets of 6 cupcakes = 8 × $22 = $176
Cost of 2 more cupcakes = 2 × $4 = $8
Total cost = $176 + $8 = $184

Ling paid $184 for the cupcakes.

4. Check:
How many cupcakes are there in 8 sets? 8 × 6 = 48
Were there 50 cupcakes? 48 + 2 = 50. Yes

Word Problem (Grade 4):

Square EFGH is made up of 4 rectangles. The perimeter of Square EFGH is 32 centimetres. Find the perimeter of each rectangle.

Solution:

1. Understand:
What to find: The perimeter of each rectangle.
What is known: Square EFGH is made up of 4 rectangles. The perimeter of EFGH is 32cm. EF is made up of the length of a rectangle. FG is made up of the breadths of 4 rectangles.

2. Choose: Simplify the problem

3. Solve:

A square has 4 equal sides.
32 ÷ 4 = 8
EF is 8cm. The length of a rectangle is 8cm.
FG is also 8cm.
8 ÷ 4 = 2
The breadth of a rectangle is 2cm.

8 + 2 + 8 + 2 = 20
The perimeter of each rectangle is 20 centimetres.

4. Check:
Is the perimeter of EFGH 32cm?
8 + 2 + 2 + 2 + 2 + 8 + 2 + 2 + 2 + 2 = 32. Yes.
Is EF the same length as FG?
EF = 8. FG = 4 × 2 = 8. Yes.

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 6):

The shaded figure below shows a semicircle and two quarter circles. Find the area of the shaded figure. (Take pi = 3.14)

Answer: The area of the shaded figure is 450 cm2.
See the solution in part one of our Singapore Math Heuristics series.

_______________________________________________________________________________________________________

Heuristic: Work backwards

Word Problem (Grade 3):

Mr Adam had some pens in his stationery store. He sold 318 of the pens at $1 each. He then put the remaining pens into packs of 5 and sold each pack for $3. He made $249 from selling all the packs. How many pens did Mr Adam have at first?

Solution:

1. Understand:
What to find: How many pens did Mr Adam have at first?
What is known: He sold 318 pens. He packed the remaining pens into packs of 5. He sold the packs at $3 each and made $249.

2. Choose: Work backwards

3. Solve:

Let’s label the changes C1, C2 and C3
C1: He sold 318 pens.
C2: He packed the remaining pens into packs of 5
C3: He sold the packs at $3 each.

We can organise the information like this:

C3 is he sold the packs at $3 each.
Before C3: 249 ÷ 3 = 83
He had 83 packs.C2 is he packed the remaining pens into packs of 5.
Before C2: 83 × 5 = 415
He had 415 pens remaining.C1 is he sold 318 pens.
Before C1: 415 + 318 = 733
Mr Adam had 733 pens at first.4. Check:
How many remaining pens did he have? 733 – 318 = 415
How many packs of 5 pens did he have? 415 ÷ 5 = 83
How much did he made from selling the packs? 83 × 3 = $249

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 3):

Darren had some stickers in his collection. He bought 20 more stickers and gave 33 stickers to his sister. He had 46 stickers left. How many stickers did Darren have in his collection at first?

Answer: Darren has 59 stickers in his collection at first.
See the solution in part one of our Singapore Math Heuristics series.

Look out for the final part of this series for the remaining Math Heuristics and word problems with step-by-step worked solutions.

This is the fourth part to S.A.M Math Heuristics series. Read part onepart two and part three here.

Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world’s largest Singapore Math enrichment program for children aged four to 12. The award-winning S.A.M program is based on the global top-ranking Singapore Math curriculum with a focus on developing problem solving and thinking skills.

The curriculum is complemented with S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

]]>
Singapore Math Heuristics: Make A Systematic List, Guess And Check, Restate The Problem In Another Way https://seriouslyaddictivemaths.com.sg/math-heuristics-approach/ Thu, 19 Jan 2023 00:16:27 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=312

The skills children pick up in math are indispensable; they can be applied to other academic subjects and to solve real-world problems in their daily lives and future work.

The Singapore Math curriculum focuses on problem solving. Through problem solving, children develop thinking skills such as creative thinking and critical thinking.

When children analyse math problems, they notice patterns, learn to generalise, form new ideas and activate their creative thinking. Children become critical thinkers when they are able to select the best strategy out of multiple methods to solve word problems.

Singapore Math Heuristics

In part one of our Math Heuristics series, we gave an overview of the 12 problem-solving methods or heuristics taught in the Singapore primary math education syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to use them to solve various math word problems.

In part two of the Math Heuristic series, we zoomed in on the heursitics – Act It Out, Draw A Diagram and Look For Patterns, and also showed how to apply the Polya’s 4-step problem-solving process in sample word problems.

In the third part of this series, we will focus on the next 3 heuristics – Make a systematic listguess and check, and restate the problem in another way.

Sample word problems are solved using these 3 heuristics and Poly’s 4-step process in the step-by-step worked solutions provided by the curriculum team at S.A.M.

_______________________________________________________________________________________________________

Heuristic: Make a systematic list

Word Problem (Grade 2):

Jimmy uses the number cards given below to form as many 3-digit odd numbers as he can. List all the numbers that Jimmy can form.

Solution:

1. Understand:
What to find: All the 3-digit odd numbers that Jimmy can form from the 4 number cards.
What is known: Odd numbers end with 5 or 7.

2. Choose: Make a systematic list

3. Solve:

Odd numbers that end with 5:
245
275
425
475
725
745

Odd numbers that end with 7:
247
257
427
457
527
547

Jimmy can form 12 3-digit odd numbers.

4. Check:

Did I form 3-digit numbers? Yes
Did I form odd numbers? Yes
Did I form all possible numbers? Yes

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 2):

A shop sells apples in bags of 3. It sells lemons in bags of 4. Paul buys some bags of apples and lemons. He buys the same number of each fruit. He buys more than 20 and fewer than 30 pieces of each fruit. How many apples does Paul buy?

Answer: Paul buys 24 apples.
See the solution in part one of our Singapore Math Heuristics series.

_______________________________________________________________________________________________________

Heuristic: Guess and check

Word Problem (Grade 3):

David sold a total of 15 $4 coupons and $5 coupons for a funfair. He received $65 for the sale of the coupons. How many $4 coupons and how many $5 coupons did he sell?

Solution:

1. Understand:
What to find: The number of $4 coupons and the number of $5 coupons David sold?
What is known: He sold 15 coupons. He received $65.

2. Choose: Guess and check

3. Solve:

$4 + $5 = $9. (7 × 9) = 63 is close to 65. I can start the first guess with 7 $4 coupons.

David sold 10 $4 coupons and 5 $5 coupons.

4. Check:
What is the total number of coupons? 10 + 5 = 15
What is the total value of coupons? $40 + $25 = $65

Word Problem (Grade 5):

In a quiz, 5 marks were awarded for each correct answer and 3 marks were deducted for each wrong answer. Darren answered 14 questions and scored 30 marks. How many questions did he answer correctly?

Solution:

1. Understand:
What to find: The number of questions Darren answered correctly.
What is known: Add 5 marks for each correct answer. Minus 3 marks for each wrong answer. He answered 14 questions. He scored 30 marks.

2. Choose: Guess and check

3. Solve:

I can start the first guess with the same number of correct answers and wrong answers.

4. Check:
What is the total number of questions? 9 + 5 = 14
What is the total marks scored? 45 – 15 = 30

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 5):

Vijay is presented with the equations below. Insert one pair of brackets in each equation to make it true.
4 × 11 + 18 ÷ 3 + 6 = 46

Answer: The equation is 4 × 11 + 18 ÷ (3 + 6) = 46.
See the solution in part one of our Singapore Math Heuristics series.

_______________________________________________________________________________________________________

Heuristic: Restate the problem in another way

Word Problem (Grade 2):

Sally has some beds and sofas. All of them are equal in length. The total length is 14 metres. Each bed is 2 metres long. Sally has 1 fewer bed than sofas. What is the total length of the sofas?

Solution:

1. Understand:
What to find: Total length of the sofas.
What is known: Each bed is 2 metres long. Each sofa is 2 metres long. Total length of beds and sofas is 14 metres. Sally has 1 more sofa than bed.

2. Choose: Restate the problem in another way

3. Solve:

If we add 1 more bed, Sally will have the same number of beds and sofas.
New total length = 14 m + 2 m = 16 m
Sally has the same number of beds and sofas
Total length of sofas = 16 m ÷ 2 = 8 m

The total length of the sofas is 8 metres.

4. Check:
How many sofas are there? 8 m ÷ 2 m = 4
How many beds are there? 4 – 1 = 3
How many beds and sofas altogether? 4 + 3 = 7
What is the total length of beds and sofas? 7 × 2 m = 14m

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 2):

There are some identical pens and erasers. 2 pens and 3 erasers are 45 centimetres long altogether. 6 erasers and 2 pens are 60 centimetres long altogether. What is the length of 3 erasers?

Answer: The length of 3 erasers is 15 cm.
See the solution in part one of our Singapore Math Heuristics series.

These are just a few examples to show you how Singapore Math heuristics are used to solve basic and intermediate word problems in lower grade levels and complex word problems in upper grade levels.

Look out for parts four and five of this series for the other 6 remaining Singapore Math heuristics and word problems with step-by-step worked solutions.

This is the third part to S.A.M Math Heuristics series. Read part one and part two here.

Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world’s largest Singapore Math enrichment program for children aged four to 12. The award-winning S.A.M program is based on the global top-ranking Singapore Math curriculum with a focus on developing problem solving and thinking skills.

The curriculum is complemented with S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

]]>
Singapore Math Heuristics: Act It Out, Draw A Diagram, Look For Patterns https://seriouslyaddictivemaths.com.sg/math-heuristics-method/ Thu, 19 Jan 2023 00:15:43 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=315

In part one of our Singapore Math Heuristics series, we gave an overview of the 12 heuristics in Singapore Primary Math syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to solve various math word problems using them.

To recap, heuristics are methods or strategies students can use to solve complex math word problems. They are general guidelines of what students can do to tackle a word problem when the solution is not obvious.

Besides using heuristics to solve a word problem, Singapore Math also adopts Polya’s 4-step problem-solving process:

1. Understand the problem: What to find? What is known and unknown?
2. Devise a plan: Choose the most suitable heuristic
3. Carry out the plan: Solve the problem
4. Look back: Check the answer

To solve word problems efficiently, students must be familiar with both the problem solving methods (heuristics) and the problem solving process (Polya’s 4-step).

In this article, we will focus on 3 heuristics – Act it outDraw a diagram/model, and Look for pattern(s).

Sample word problems are solved using these 3 heuristics and Poly’s 4-step process in the step-by-step worked solutions provided by the curriculum team at S.A.M.

_______________________________________________________________________________________________________

Heuristic: Act it out

Word Problem (Grade 2):

Ben cuts a cake into 5 equal pieces. He wants to eat some of the pieces of cake so that the fraction of cake he eats is smaller than the fraction of the cake left. What is the greatest possible fraction of the cake that Ben can eat?

Solution:

1. Understand:
What to find: The greatest possible fraction of the cake that Ben can eat.
What is known: The cake is cut into 5 equal pieces. The fraction of cake he eats is smaller than the fraction of the cake left.

2. Choose: Act it out

3. Solve:

If Ben eats 1 piece, 4 pieces are left. 1/5 is smaller than 4/5
If Ben eats 2 pieces, 3 pieces are left. 2/5 is smaller than 3/5

If Ben eats 3 pieces, 2 pieces are left. 3/5 is greater than 2/5. This does not match the word problem.

The greatest possible fraction of the cake that Ben can eat is 2/5.

4. Check:
Did I give the answer as a fraction? Yes
Is the fraction he ate smaller than the fraction left? Yes
Is it the greatest possible fraction of cake he can eat? Yes

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 1):

Alan, Ben and Carol are in the school’s Art Club. Their teacher, Mr Tan, wants two of them to join a contest. How many ways can Mr Tan choose two pupils?

Answer: Mr Tan can choose two pupils in 3 ways.
See the solution in part one of our Singapore Math Heuristics series.

_______________________________________________________________________________________________________

Heuristic: Draw a diagram/model

Word Problem (Grade 2):

There were 158 children in a movie theatre. There were 267 more adults than children in the theatre. 236 of the adults were men. How many women were there in the theatre?

Solution:

1. Understand:
What to find: The number of women in the theatre.
What is known: There were 158 children. There were 267 more adults than children. There were 236 men.

2. Choose: Draw a diagram/model

3. Solve:

There were 158 children.

There were 267 more adults than children.
158 + 267 = 425
There were 425 adults in the theatre.
425 – 236 = 189
There were 189 women in the theatre.4. Check:
How many adults were there? 236 + 189 = 425
Were there 267 more adults than children? 425 – 158 = 267. Yes

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 3):

The smaller of two numbers is 1217. The greater number is 859 more than the smaller number.
(a) What is the greater number?
(b) What is the sum of the numbers?

Answer:
(a) The greater number is 1217 + 859 = 2076.
(b) The sum of the numbers is 2076 + 1217 = 3293.
See the solution in part one of our Singapore Math Heuristics series.

Word Problem (Grade 5):

Tammy has 3 boxes of apples. Box A is 550 grams heavier than Box B and 770 grams heavier than Box C. The average mass of the 3 boxes is 2150 grams. Find the average mass of Box B and Box C. Give your answer in kilograms and grams.

Solution:

1. Understand:
What to find: The average mass of Box B and Box C.
What is known: Box A is 550 grams heavier than Box B. Box A is 770 grams heavier than Box C. The average mass of Box A, Box B and Box C is 2150 grams.

2. Choose: Draw a diagram/model

3. Solve:

Box A is 550 grams heavier than Box B.

Box A is 770 grams heavier than Box C.
2150 x 3 = 6450
The total mass of the 3 boxes is 6450 grams.
770 – 550 = 220
Box B is 220 grams heavier than Box C.
3 units = 6450 – 770 – 220 = 5460
1 unit = 5460 ÷ 3 = 1820
The mass of Box C is 1820 grams.1820 + 220 = 2040
The mass of Box B is 2040 grams.2040 + 1820 = 3860
The total mass of Box B and Box C is 3860 grams.3860 ÷ 2 = 1930 grams
The average mass of Box B and Box C is 1 kilogram 930 grams.4. Check:
What is the mass of Box A? 1820 + 770 = 2590 grams.
What is the total mass of the 3 boxes? 1820 + 2040 + 2590 = 6450.
Is the average mass of the 3 boxes 2150 grams? 6450 ÷ 3 = 2150 grams. Yes
Did I give the answer in kilograms and grams? Yes

_______________________________________________________________________________________________________

Heuristic: Look for pattern(s)

Word Problem (Grade 4):

Carmen uses the letters in her name to form the pattern below.
C A R M E N C A R M E N C A . . .
What will be the 405th letter in the pattern?

Solution:

1. Understand:
What to find: The 405th letter in the pattern.
What is known: The pattern is formed by the letters CARMEN. The pattern starts repeating itself from the 7th letter.

2. Choose: Look for patterns

3. Solve:

CARMEN is the repeating block of letters.
There are 6 letters in each block.
405 ÷ 6 = 67 remainder 3
There are 67 such blocks.
The remainder of 3 means we have to count 3 more letters to get to the 405th letter.
C A R M E N
1 2 3
The 405th letter in the pattern is the letter R.

4. Check:
How many letters are there in 67 blocks of CARMEN? 67 x 6 = 402
Did I count to the 405th letter? 402 + 3 = 405. Yes

Try solving the following word problem using Polya’s 4-step process.

Word Problem (Grade 1):

Draw the shape that comes next.

Answer: The next shape is

These are just a few examples to show you how Singapore Math heuristics are used to solve basic and intermediate word problems in lower grade levels and complex word problems in upper grade levels.

Look out for parts three, four and five of this series for the other 9 Singapore Math heuristics and word problems with step-by-step worked solutions.

This is part two to S.A.M Singapore Math Heuristics series. Read part one here.

Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world’s largest Singapore Math enrichment program for children aged four to 12. The award-winning S.A.M program is based on the global top-ranking Singapore Math curriculum with a focus on developing problem solving and thinking skills.

The curriculum is complemented with S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

]]>
What are Singapore Math Heuristics? https://seriouslyaddictivemaths.com.sg/what-are-heuristics/ Thu, 19 Jan 2023 00:14:00 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=318

Ever tried to help your child with primary math homework and got stumped? Today’s math questions can be challenging – even for adults.

Math education is changing. While many parents spent time memorising procedures and formulas, today’s students are expected to not only understand and master the concepts, but also to have strong thinking skills and problem-solving skills to solve complex math questions.

As new concepts and strategies are being taught, and homework turns from arithmetic exercises to using multiple ways to solve a math word problem, you may feel unsure or have no idea how to help your child.

In this series, the curriculum team at Seriously Addictive Mathematics (S.A.M) shares expert tips on math heuristics and how to use them to solve math word problems.

What are Math Heuristics?

Heuristics – a word that baffles many primary school students and their parents.

To define it simply, math heuristics are strategies that students can use to solve complex word problems.

Word problems can be solved in several ways using different heuristics, while some word problems are solved using a combination of heuristics.

To solve word problems efficiently, students must be familiar with both the problem-solving methods (heuristics) and the problem-solving process.

How many Heuristics are there?

In Singapore Math, there are 12 heuristics in the primary math syllabus that can be grouped into four main categories:

  • To give a representation: Draw a diagram/model, draw a table, make a systematic list
  • To make a calculated guess: Look for pattern(s), guess and check, make suppositions
  • To go through the process: Act it out, work backwards, use before-after concept
  • To change the problem: Restate the problem in another way, simplify the problem, solve part of the problem
What is the Problem-Solving Process?

Another important point to note is that Singapore Math adopts Polya’s four-step problem-solving process:

1. Understand the problem: What to find? What is known and unknown?
2. Devise a plan: Choose the most suitable heuristic
3. Carry out the plan: Solve the problem
4. Look back: Check the answer

So, how can we use heuristics to solve math word problems? Here are examples of word problems with solutions provided by the curriculum team at S.A.M.

_______________________________________________________________________________________________________

Heuristic: Act it out

Word Problem (Grade 1):

Alan, Ben and Carol are in the school’s Art Club. Their teacher, Mr Tan, wants two of them to join a contest. How many ways can Mr Tan choose two pupils?

Solution:

Alan and Ben, Alan and Carol, Ben and Carol.

Mr Tan can choose two pupils in 3 ways.

_______________________________________________________________________________________________________

Heuristic: Draw a diagram/model

Word Problem (Grade 3):

The smaller of two numbers is 1217. The greater number is 859 more than the smaller number.
(a) What is the greater number?
(b) What is the sum of the numbers?

Solution:

The greater number is 1217 + 859 = 2076.

The sum of the numbers is 2076 + 1217 = 3293.

_______________________________________________________________________________________________________

Heuristic: Look for pattern(s)

Word Problem (Grade 1):

Draw the shape that comes next.

Solution:

Label all shapes, the pattern is in repeating blocks of A, B, C.

_______________________________________________________________________________________________________

Heuristic: Make a systematic list

Word Problem (Grade 2):

A shop sells apples in bags of 3. It sells lemons in bags of 4. Paul buys some bags of apples and lemons. He buys the same number of each fruit. He buys more than 20 and fewer than 30 pieces of each fruit. How many apples does Paul buy?

Solution:
Paul buys 24 apples.

_______________________________________________________________________________________________________

Heuristic: Guess and check

Word Problem (Grade 5):

Vijay is presented with the equations below. Insert one pair of brackets in each equation to make it true.
4 × 11 + 18 ÷ 3 + 6 = 46

Solution:
The equation is 4 x 11 + 18 ÷ (3 + 6) = 46.

_______________________________________________________________________________________________________

Heuristic: Restate the problem in another way

Word Problem (Grade 2):

There are some identical pens and erasers. 2 pens and 3 erasers are 45 centimetres long altogether. 6 erasers and 2 pens are 60 centimetres long altogether. What is the length of 3 erasers?

Solution:

If we subtract the total length of 2 pens and 6 erasers from that of 2 pens and 3 erasers, we get the length of 3 erasers.

60 cm – 45 cm = 15 cm

The length of 3 erasers is 15 cm.

_______________________________________________________________________________________________________

Heuristic: Solve part of the problem

Word Problem (Grade 6):

The diagram below shows 4 shaded triangles in Triangle ABC. All the triangles in the diagram are equilateral triangles. If the area of Triangle ABC is 64 cm2, find the total area of the shaded triangles.

Solution:
Triangle ABC is made up of 16 small equilateral triangles.
7 out of 16 small equilateral triangles are shaded.
7/16 × 64 = 28The total area of the shaded triangles is 28 cm2.

_______________________________________________________________________________________________________

Heuristic: Simplify the problem

Word Problem (Grade 6):

The shaded figure below shows a semicircle and two quarter circles. Find the area of the shaded figure. (Take pi = 3.14)

Solution:

30 × 15 = 450

The area of the shaded figure is 450 cm2.

_______________________________________________________________________________________________________

Heuristic: Work backwards

Word Problem (Grade 3):

Darren had some stickers in his collection. He bought 20 more stickers and gave 33 stickers to his sister. He had 46 stickers left. How many stickers did Darren have in his collection at first?

Solution:

Label the changes as C1 and C2.

Before C2: 46 + 33 = 79
At first: 79 – 20 = 59Darren has 59 stickers in his collection at first.

_______________________________________________________________________________________________________

Heuristic: Draw a table

Word Problem (Grade 4):

Janice wanted to distribute stickers equally among some children. If each child received 8 stickers, she would have 3 stickers left. If each child received 11 stickers, she would need another 9 stickers. How many children were there?

Solution:
Janice had 35 stickers. There were 4 children.

_______________________________________________________________________________________________________

Heuristic: Make suppositions

Word Problem (Grade 4):

Farmer James has some ducks, horses and cows on his farm. He has 30 ducks and cows altogether. The total number of legs the ducks and cows have is 82. The total number of legs the horses have is 28. How many ducks and how many cows are there on the farm?

Solution:

Suppose that James has 30 ducks.
30 × 2 = 60
30 ducks have 60 legs altogether.
82 – 60 = 22
The total number of legs is 22 less than the actual total number.
A cow has 2 more legs than a duck.
22 ÷ 2 = 11
James has 11 cows.
30 – 11 = 19

There are 19 ducks and 11 cows on the farm.

_______________________________________________________________________________________________________

Heuristic: Use before-after concept

Word Problem (Grade 5):

Ray and Sam each brought some money for shopping. The ratio of the amount of Ray’s money to the amount of Sam’s money was 3 : 4. After each of them bought a laptop for $1250, the ratio of the amount of Ray’s money to the amount of Sam’s money became 1 : 3. How much money did Sam bring for shopping?

Solution:
(8 – 3) = 5 units = $1250
8 units = $1250 ÷ 5 × 8 = $250 × 8 = $2000Sam brought $2000 for shopping.

This is the first part to S.A.M Heuristics series. Look out for part two where we will share more expert tips on math heuristics.

Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world’s largest Singapore Math enrichment program for children aged four to 12. The award-winning S.A.M program is based on the global top-ranking Singapore Math curriculum with a focus on developing problem solving and thinking skills.

The curriculum is complemented with S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

Find out how to raise an intelligent child. Click here to download our free e-book!
]]>
Mathematics – A Friend For Life https://seriouslyaddictivemaths.com.sg/mathematics-a-friend-for-life/ Thu, 19 Jan 2023 00:13:40 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=331

“Math is useless. I don’t like Math. Why do I have to learn Math?” As a parent, you are likely to have heard these expressions from your child before. Here are some essential facts to help you explain to your child the importance of Math in everyday life and why an edge in Math today can go a long way in the future.

It’s a workout for your brain

Like any other part of our body, our brain also needs a workout. In fact, research has shown that children who know Math are able to use the brain regions associated with cognitive tasks more reliably. So, practising Math may help children improve their visual attention and decision-making skills. From everyday tasks like searching for a misplaced item in their rooms to solving puzzles and playing video games – studying Math can help them get better at a variety of activities!

It helps manage finances

From your child’s big dream of buying a supercar to his more short-term desire of buying a remote-control car, show your child that Math is necessary to plan and organise finances for purchases both big and small. Work out a simple financial plan with your child based on their allowance, expenses, and savings to buy that new toy or gadget that they have their eyes on. This simple task can help them understand the application of Math concepts and skills learnt in the classroom.

It helps with travel

Travel enriches our lives and opens our minds to new places and people. Get your child involved in planning your next family holiday, such as calculating the cost of hotel accommodation, planning the itinerary to make sure they have time for the activities they wish to enjoy, and factoring in currency exchange. Assisting with different aspects of the trip can help your child learn a lot about the importance of Math in making travel plans work out.

It helps with cooking

A basic yet necessary skill, cooking also requires knowledge of Math. From measuring ingredients to planning the quantity of food needed to feed your family or your guests, Math calculations are an important culinary skill. Planning family meals within the house, by factoring in the number of family members and the quantity of food needed, is a good example of how Math is used in everyday life. Even for their birthday parties, help your child understand how Math skills are needed to plan yummy treats for their friends or bake their delicious birthday cake.

An aid to every job

If your child thinks Math is only for school, it’s time to correct their misconception. Your child may be surprised that almost every job requires some understanding and application of Math skills. From fashion designers to event organisers, from nurses to pilots, every profession uses Math skills on a regular basis.

Problem-Solving

Through every situation in life – personal and professional – problem-solving skills are crucial to getting ahead and dealing with tough times. Math problems help us develop problem-solving skills – a lifelong lesson that helps us tackle problems in an efficient and step-by-step manner.

These are just some of the skills that Math lends to our lives. One must also remember that while languages may vary across countries, numbers remain universal. From analytical and logical abilities to numerical and quantitative skills, it is important for children to learn Math from an early age to strengthen their understanding of the concepts and enjoy learning the subject. For a unique and comprehensive Math enrichment program, check out Seriously Addictive Mathematics (S.A.M) to help your child build a strong foundation and enjoy learning Math at an early age.

]]>
There’s More To Learning Than IQ https://seriouslyaddictivemaths.com.sg/theres-more-to-learning-than-iq/ Thu, 19 Jan 2023 00:12:53 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=334

In the face of a highly competitive academic environment, it is easy for children and parents alike to feel the pressure of performance cloud what could otherwise be a pleasurable learning experience. When a child sits down with a subject, their focus from the word ‘go’ is to excel at it, but the method needed forgetting there often includes faulty practices such as rote-memorisation.

Without developing a positive attitude before approaching it, even the brightest of children tend to develop a fear of the subject. This fear is one of the biggest impediments to any child reaching their fullest potential.

A child’s schooling days are the most important in their life, and it is important for them to learn earlier in life that learning is an enjoyable and fruitful lifelong journey. So then, how exactly does a so-called ’positive mindset’ really improve a child performance, and how can a parent help them achieve it?

Attitude is everything

Let us take one of the most commonly feared subjects – Math. We often see students who admit to being afraid of numbers, without even realising that all it really takes to excel at the subject is conceptual clarity and frequent practice.

While different students will undoubtedly learn at different paces, a student learning at a slightly slower pace need not fear the subject or believe they cannot excel at it the way their peers can. In order to avoid this, it is important for parents and teachers to remind the child that they need not compete with other students during the learning process itself.

For students showing a lack of interest and attention while learning, it is important to incorporate innovative and light-hearted teaching methodologies that can keep them engaged in the subject. When a student begins to look forward to going for a class, their enthusiasm will automatically translate to an increased interest and better quality performance in the subject.

Backed by research

It is a common misnomer that only quantifiable measures of intelligence (i.e., IQ) can indicate how well a child is able to perform at a subject. But IQ is not the only factor that influences a child’s performance.

A scientific study conducted at the Stanford University School of Medicine has recently found an actual brain pathway that links a positive attitude toward Math to higher achievement in the subject. It was found to be connected to the better functioning of the hippocampus, which is an important memory centre in the brain, especially as a child solves arithmetic problems in Math.

The research administered standard questionnaires to 240 children aged 7 to 10, and found that even with factors such as IQ accounted for, students with a positive attitude toward math were linked to better performance in the subject. It suggested that those students who develop a strong interest and self-belief in their potential in the subject perform certifiably better at it.

The study, published on 24 January 2018in Psychological Science, also gives an explanation as to why a passionate and inspiring teacher can influence the attitude of a child towards a subject, and subsequently enhance their performance in it.

What you can do

It’s time to look beyond conventional methods of education and inculcate a passion for learning in your child. In subjects like Math, it is important to ensure that the student enjoys learning concepts, and willingly partakes of practice sessions outside of their schooling hours.

One of the ways is to ensure that your child is getting ample play time and is engaged in extracurricular activities that he/she enjoys. This prevents them from getting bored of learning. If it is enrichment you are looking for, centres such as S.A.M (Seriously Addictive Mathematics) help inculcate a positive spirit in the classroom. With innovative teaching methodologies and hands-on activities, they can help your child enjoy learning rather than resent it. When your child volunteers to go for a lesson in Math, you made the right choice.

]]>
How To Raise An Intelligent Child https://seriouslyaddictivemaths.com.sg/how-to-raise-an-intelligent-child/ Thu, 19 Jan 2023 00:11:38 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=337

How To Raise An Intelligent Child

What does it take to raise an intelligent child in today’s society? Learn how early math exposure can benefit your child’s overall development and success in life by downloading the S.A.M e-book: How To Raise An Intelligent Child.

Table of contents

  • Why early math matters
    In a study on 35,000 preschoolers, early math skills was found to be the strongest predictive power for later success. Learning math also supports the development of literacy. Math is not just the foundation for careers in science or engineering; it is the pillar of a well-rounded foundation for every child.
  • Can IQ be improved by doing maths
    IQ was initially considered to be genetic and fixed. However, researchers found that fluid intelligence, which is the ability to solve abstract problems without depending on previous knowledge, skills or experience, can be improved with specific and targeted training of working memory.
  • How to improve your child’s IQ
    Developing a child’s IQ is not about getting them to do tons of IQ questions or assessment books. Neither is it about improving their memory. It is the everyday activities of what parents do and say that matters. Here are 5 things you can do to improve your child’s intelligence.
  • 4 things you can do at home to enhance your child’s IQ
    Research shows it is possible to create a highly conducive environment for cognitive and emotional growth. Here are more ideas on how you can create a stimulating home to develop your child’s fluid intelligence, sharpen their creativity and capacity to learn.
  • Besides IQ: qualities that every child needs
    Studies have shown that IQ predicts future success in life, such as academic achievement, income and health. Besides IQ, research also reveals other factors involved in determining a child’s success later in life. Here are two big qualities that will complement a child’s IQ.

Download the S.A.M e-book: How To Raise An Intelligent Child.

]]>
Easy Ways To Improve Your Kid’s IQ https://seriouslyaddictivemaths.com.sg/easy-ways-to-improve-your-kids-iq/ Thu, 19 Jan 2023 00:10:56 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=340

As your child begins to show signs of learning, through interactions they experience on a daily basis, you might be mindful as a parent to give them the best learning environment possible. But just what should you do to stimulate your child’s mind? Here’s four simple methods you can try to help develop your child’s learning for a better IQ.

#1 Read to them

Part of increasing someone’s IQ is through expanding their imagination, and what better way to do this than by the power of books? Reading to your kid helps them create a world of possibilities in their mind, and this strengthens their brain muscle for better development of concepts.

Besides the simple effectiveness of verbal storytelling, enhance their learning experience with picture books. Through pictorial guidance along the way, younger ones will be better able to match words to images in their mind, which improves their IQ from an early age. Simple stories with math elements weaved into them also help to expand their abilities for both language and calculus.

Learning centres such as S.A.M (Seriously Addictive Mathematics) also provide various programmes for different aged groups with a focus on enriching knowledge through mathematics.

#2 Include mathematical topics in your interactions

Through your everyday conversations with your little one, they’re already picking up on learning elements. Take advantage of this fact and develop their brain further with mathematical information thrown in casually.

Saying sentences like “I will let you watch two cartoons at 6PM”, or “Do you want one apple or two?” firms up their understanding of different concepts. Soon enough, they will react faster to your questions, and even grow to requesting for what they want in specific quantities.

#3 Expand their horizons

Creativity plays an integral part in improving your child’s IQ, which you can still practice through your own imagination. Turn everyday objects into fun scenarios for your kid, such as making a spoon pretend to be an aeroplane while feeding them, stationery transformed into cartoon characters, or paper planes as fighter jets in the sky. Having various ways to tell stories will certainly make children more receptive to their surroundings and learn better!

#4 Let them practice what they watch

Whether it’s on traditional television or on YouTube, letting kids watch educational shows that encompass different educational topics in moderation is an easy way for them to learn. After all, didn’t most of us grow up on Sesame Street and Barney and Friends? The ages of 3 to 6 years old are vital for kids’ learning, as that’s when their brain development is at the most.

But learning from the TV isn’t sufficient either. Learners should put what they watch into practice, and a small class where they get to hone their new abilities such as counting and telling time will do them a world of favours in improving their IQ.

]]>
Besides IQ: Important Qualities Every Child Needs https://seriouslyaddictivemaths.com.sg/besides-iq-important-qualities-every-child-needs/ Thu, 19 Jan 2023 00:09:57 +0000 http://newsite.seriouslyaddictivemaths.com.sg/?p=343
IQ predicts future success in life, and there are studies have shown that IQ results can be a strong predictor of his or her future life outcomes, such as academic achievement, income and health.

Besides IQ, research also reveals other factors involved in determining a child’s success later in life. Here are two big qualities that will complement a child’s IQ.

1. Love for learning

Having a positive attitude towards learning plays a huge part in future success. Children begin exploring their bodies and their capabilities at a very early age. They may not know that this is “learning” but it is exactly what they are doing when they see, hear, taste and touch anything they come in contact with. Here’s what you can do to nurture their love for learning:

• Foster their interest

If your child loves cars and trucks, read stories to them on cars and trucks. Take them to a car show. When you teach them how to count or spell, weave their favourite cars or trucks in. A University of Chicago study found that the common trait among exceptionally high-achieving athletes and artists was having parents who recognised their interest and provided support during their early years.

• Make it relevant to everyday life

Learning is not just teaching our children to reach certain academic standards. It is a process, a series of “Oh I see!” moments. For example, if you want your child to learn about maths, show them how math ties to real life. Help them understand that two $10 Lego figurines cost $20 (two sets of ten rather than just memorising the multiplication table (10 x 2).

• Never shoot down their ideas

Create a safe environment to allow children to develop their ideas, express their feelings, take risks, and make choices. As parents, we wear many hats in this process. As an observer, we listen and watch. As a supporter, we accept and encourage. As a facilitator, we assist and inspire. As a role model, we demonstrate and surprise.

 

2. Motivation

A study revealed effective studying techniques and those who were self-driven that had the most improved math skill. So how can parents help children build internal motivation? According to Edward Deci and Richard Ryan at the University of Rochester, addressing these needs can help to build children’s intrinsic motivation.

• Competence

Being able to master a skill is extremely motivating. When children are bad at doing something, they feel embarrassed and tend to avoid these activities. Isn’t this the same for adults too?

The solution? Focus on effort and strategy. Praise children for their effort. Remind them and point out their progress of how they struggled initially and then succeed. You could say, “You used to have trouble with these kinds of problems, but now you really get them!” A simple sentence like this can empower children to feel more capable.

Next, equip them with study strategies to address skills gaps. For instance, if they always get a certain math sum wrong, find out what is the cause of the mistake instead of quickly prescribing them to more math practice. It could be a misunderstanding of a certain concept that they might need to relearn.

If your child tends to make careless mistakes or jump in without reading the instructions, show them how to circle or underline key instructions.

• Autonomy

Who likes to feel controlled? When children feel that they are being restrained, their motivation is thrown out the window. We cannot offer children total freedom, but we can minimise resistance by offering choices. Don’t overwhelm them with too many alternatives. Keep your options to just two or three.

Sometimes, children don’t understand the reason why they need to do certain things. Providing a rationale that makes sense and relates to their life might help.

For instance, if your child laments the purpose behind learning “stuff” that he’s never going to use later in his life, you can reply, “It may not seem relevant or important to you now, but it gives you a chance to practice skills that will be useful in your life in the future. For example, getting your work done on time, building your memory skills so you can learn to remember more important things.”

• Connection

A powerful source of motivation is relationships. Children are motivated to do things that bring them closer to parents or peers. When they are low in motivation, there could be underlying negative emotions. It could be feelings of being judged unfairly by an adult, or having feelings being left out in a social setting.

Having a role model can help children with their motivation too. It can be parents, or a beloved teacher. It could also be being part of a team or a peer group that inspires them to try hard.

It takes time and growth to muster intrinsic motivation, especially for mundane and tasks that are less fun. Always exercise patience, reasonable expectations and shower children with encouragement.

Who knows, your child might just surprise you one day.

]]>