{"id":309,"date":"2023-01-19T00:17:11","date_gmt":"2023-01-19T00:17:11","guid":{"rendered":"http:\/\/newsite.seriouslyaddictivemaths.com.sg\/?p=309"},"modified":"2025-01-22T06:34:09","modified_gmt":"2025-01-22T06:34:09","slug":"math-heuristics-thinking","status":"publish","type":"post","link":"https:\/\/seriouslyaddictivemaths.com.sg\/math-heuristics-thinking\/","title":{"rendered":"Singapore Math Heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards"},"content":{"rendered":"
Problem-solving in mathematics helps children develop reasoning and communication skills that are transferrable and important life skills.<\/p>\n
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.<\/p>\n
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.<\/p>\n<\/div>\n<\/div>\n
Singapore Math Heuristics<\/b><\/p>\n
In\u00a0part one of our Math Heuristics series<\/a>, 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.<\/p>\n In\u00a0part two of the Math Heuristics series<\/a>, we expanded on the heursitics \u2013 Act It Out, Draw A Diagram and Look For Patterns, and also demonstrated how to apply the Polya\u2019s 4-step problem-solving process in sample word problems.<\/p>\n In\u00a0part three of the Math Heuristics series<\/a>, we focused on the heuristics: Make a Systematic List, Guess and Check, and Restate the Problem in Another Way.<\/p>\n In part 4 of this series, we will zoom in on these 3 heuristics:\u00a0Solve part of the problem, simplify the problem<\/b>\u00a0and\u00a0work backwards.<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n _______________________________________________________________________________________________________<\/p>\n Heuristic: Solve part of the problem<\/strong><\/p>\n Word Problem (Grade 3)<\/b>:<\/p>\n 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?<\/p>\n<\/div>\n<\/div>\n Solution<\/b>:<\/span><\/p>\n 1. Understand: 2. Choose: Solve part of the problem<\/p>\n 3. Solve:<\/p>\n 19 students borrowed 1 book = 19 \u00d7 1 = 19 books 19 + 54 + 48 + 44 = 165 books<\/p>\n 165 books were borrowed from the library that week.<\/p>\n 4. Check: Try solving the following word problem using Polya\u2019s 4-step process.<\/p>\n Word Problem (Grade 6)<\/b>:<\/p>\n 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<\/sup>, find the total area of the shaded triangles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n _______________________________________________________________________________________________________<\/p>\n Heuristic: Simplify the problem<\/strong><\/p>\n Word Problem (Grade 3)<\/b>:<\/p>\n 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?<\/p>\n<\/div>\n<\/div>\n Solution<\/b>:<\/span><\/p>\n 1. Understand: 2. Choose: Simplify the problem<\/p>\n 3. Solve:<\/p>\n 6 cupcakes form 1 set. Each cupcake costs $4. Cost of 8 sets of 6 cupcakes = 8 \u00d7 $22 = $176 Ling paid $184 for the cupcakes.<\/p>\n 4. Check: Word Problem (Grade 4)<\/b>:<\/p>\n Square EFGH is made up of 4 rectangles. The perimeter of Square EFGH is 32 centimetres. Find the perimeter of each rectangle.<\/p>\n<\/div>\n<\/div>\n Solution<\/b>:<\/span><\/p>\n 1. Understand: 2. Choose: Simplify the problem<\/p>\n 3. Solve:<\/p>\n A square has 4 equal sides. 8 + 2 + 8 + 2 = 20 4. Check: Try solving the following word problem using Polya\u2019s 4-step process.<\/p>\n Word Problem (Grade 6):<\/b><\/p>\n The shaded figure below shows a semicircle and two quarter circles. Find the area of the shaded figure. (Take pi = 3.14)<\/p>\n<\/div>\n<\/div>\n _______________________________________________________________________________________________________<\/p>\n Heuristic: Work backwards<\/strong><\/p>\n Word Problem (Grade 3)<\/b>:<\/p>\n 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?<\/p>\n<\/div>\n<\/div>\n Solution<\/b>:<\/span><\/p>\n 1. Understand: 2. Choose: Work backwards<\/p>\n 3. Solve:<\/p>\n Let\u2019s label the changes C1, C2 and C3 We can organise the information like this:<\/p>\n<\/div>\n<\/div>\n<\/div>\n Try solving the following word problem using Polya\u2019s 4-step process.<\/p>\n Word Problem (Grade 3)<\/b>:<\/p>\n 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?<\/p>\n<\/div>\n<\/div>\n Look out for the final part of this series for the remaining Math Heuristics and word problems with step-by-step worked solutions.<\/p>\n This is the fourth part to S.A.M Math Heuristics series. Read\u00a0part one<\/a>,\u00a0part two<\/a>\u00a0and\u00a0part three<\/a>\u00a0here.<\/p>\n Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world\u2019s 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.<\/p>\n The curriculum is complemented with S.A.M\u2019s 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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":" 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 […]<\/p>\n","protected":false},"author":1,"featured_media":310,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[28],"tags":[],"class_list":["post-309","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-articles"],"yoast_head":"\r\n
\nWhat to find: total number of books borrowed
\nWhat is known: 19 students borrowed 1 book. 27 students borrowed 2 books. 16 students borrowed 3 books. 11 students borrowed 4 books.<\/p>\n
\n27 students borrowed 2 books = 27 \u00d7 2 = 54 books
\n16 students borrowed 3 books = 16 \u00d7 3 = 48 books
\n11 students borrowed 4 books = 11 \u00d7 4 = 44 books<\/p>\n
\nDid I read the information from the bar graph correctly? Yes
\nDid I calculate the number of books for each bar correctly? Yes
\nDid I add the number of books for all bars? Yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\nSee the solution in\u00a0part one of our Singapore Math Heuristics series<\/a>.<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\nWhat to find: How much Ling paid for the cupcakes.
\nWhat is known: Each cupcake cost $4. A discount of $2 was given for every 6 cupcakes bought. Ling bought 50 cupcakes.<\/p>\n
\n50 \u00f7 6 = 8 remainder 2
\nLing bought 8 sets of 6 cupcakes and 2 more cupcakes.<\/p>\n
\n6 \u00d7 $4 = $24
\n$24 \u2013 $2 = $22
\nEach set of 6 cupcakes cost $22.<\/p>\n
\nCost of 2 more cupcakes = 2 \u00d7 $4 = $8
\nTotal cost = $176 + $8 = $184<\/p>\n
\nHow many cupcakes are there in 8 sets? 8 \u00d7 6 = 48
\nWere there 50 cupcakes? 48 + 2 = 50. Yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\nWhat to find: The perimeter of each rectangle.
\nWhat 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.<\/p>\n
\n32 \u00f7 4 = 8
\nEF is 8cm. The length of a rectangle is 8cm.
\nFG is also 8cm.
\n8 \u00f7 4 = 2
\nThe breadth of a rectangle is 2cm.<\/p>\n
\nThe perimeter of each rectangle is 20 centimetres.<\/p>\n
\nIs the perimeter of EFGH 32cm?
\n8 + 2 + 2 + 2 + 2 + 8 + 2 + 2 + 2 + 2 = 32. Yes.
\nIs EF the same length as FG?
\nEF = 8. FG = 4 \u00d7 2 = 8. Yes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\nSee the solution in\u00a0part one of our Singapore Math Heuristics series<\/a>.<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\nWhat to find: How many pens did Mr Adam have at first?
\nWhat 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.<\/p>\n
\nC1: He sold 318 pens.
\nC2: He packed the remaining pens into packs of 5
\nC3: He sold the packs at $3 each.<\/p>\n
\nHe had 83 packs.C2 is he packed the remaining pens into packs of 5.<\/div>\n<\/div>\n<\/div>\n
\nHe had 415 pens remaining.C1 is he sold 318 pens.<\/div>\n<\/div>\n<\/div>\n
\nMr Adam had 733 pens at first.4. Check:
\nHow many remaining pens did he have? 733 \u2013 318 = 415
\nHow many packs of 5 pens did he have? 415 \u00f7 5 = 83
\nHow much did he made from selling the packs? 83 \u00d7 3 = $249<\/div>\n<\/div>\n<\/div>\n
\nSee the solution in\u00a0part one of our Singapore Math Heuristics series<\/a>.<\/div>\n<\/div>\n<\/div>\n<\/div>\n