How to use these guides
Let the learner attempt the example before revealing every step. Ask them to explain what each number in the diagram represents, then change one condition and discuss which parts of the method stay the same.
Visual problem-solving · Grades 3–6
See the structure before choosing a formula.
Each guide starts with a worked example. Use the diagram to sort out the quantities, follow the steps, and check the answer against the question.
01
Choose a problem pattern
The labels help you find a starting point; the real goal is to recognize the mathematical relationship when the story changes.
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A smaller quantity is one bar and the larger quantity is three equal bars, with a total of 48. 02
Twenty heads are first counted as two legs each; eight are then replaced by four-legged animals. 03
A parent aged 36 and child aged 8 both move six years forward to ages 42 and 14. 04
A 120-metre line is split into twelve 10-metre intervals, giving thirteen posts when both ends are used. 05
Two vehicles approach across 240 km at 50 km/h and 30 km/h. 06
Seven positions form a cycle; the 100th term lands on position 2 of the cycle. 07
Three shirt branches each split into two trouser branches, making six complete paths. 08
A two-by-three grid has four vertical and three horizontal lines; two of each determine a rectangle.
Worked visual guides
01Quantity relationships · Grades 3–5Bar-model ratio problems
Represent totals and multiplicative comparisons as equal-sized parts.
Open guide
Bar-model ratio problems
Represent totals and multiplicative comparisons as equal-sized parts.
How to recognize it
- A total and a difference or multiplicative comparison are both given.
- The unknowns can be represented by equal-sized parts.
Example
Two numbers total 48. The first is three times the second. Find both numbers.
Method
Unit-parts method
- Treat the second number as 1 part and the first as 3 parts.
- There are 4 equal parts altogether.
- Each part is 48 ÷ 4 = 12.
- The numbers are 12 and 36.
Answer36 and 12
Reusable ruleOne part = total ÷ total number of parts
Common pitfall“Three times as much” means 3 equal parts, not 3 more.
Check36 + 12 = 48 and 36 ÷ 12 = 3.
02Quantity relationships · Grades 4–6Two-type total problems
Assume every item is one type, then use the difference created by each replacement.
Open guide
Two-type total problems
Assume every item is one type, then use the difference created by each replacement.
How to recognize it
- There are two kinds of items with a known total count.
- Each kind contributes a different number of legs, wheels or points.
Example
A pen contains 20 animals, some chickens and some rabbits. They have 56 legs altogether. How many of each are there?
Method
Assume-and-adjust
- Assume all 20 are chickens: 20 × 2 = 40 legs.
- The actual total has 56 − 40 = 16 extra legs.
- Replacing a chicken with a rabbit adds 2 legs.
- There are 16 ÷ 2 = 8 rabbits and 12 chickens.
Answer12 chickens and 8 rabbits
Reusable ruleNumber replaced = total difference ÷ difference per replacement
Common pitfallThe total difference and per-item difference must measure the same attribute.
Check12 + 8 = 20 and 12 × 2 + 8 × 4 = 56.
03Quantity relationships · Grades 4–6Age problems
Everyone ages by the same amount of time, so age differences stay constant.
Open guide
Age problems
Everyone ages by the same amount of time, so age differences stay constant.
How to recognize it
- The wording includes “years from now” or “years ago”.
- Two ages are compared by a sum, difference or multiple.
Example
A parent is 36 and a child is 8. In how many years will the parent be three times the child’s age?
Method
Synchronized change
- Let x be the number of years; both ages increase by x.
- Write 36 + x = 3(8 + x).
- Simplify to 36 + x = 24 + 3x.
- Then 12 = 2x, so x = 6.
Answer6 years
Reusable ruleThe same elapsed time is added to every person
Common pitfallDo not add the elapsed years to only one person.
CheckAfter 6 years the ages are 42 and 14, and 42 ÷ 14 = 3.
04Applied problems · Grades 3–5Fence-post and spacing problems
Count intervals first, then decide whether endpoints are included or the path is closed.
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Fence-post and spacing problems
Count intervals first, then decide whether endpoints are included or the path is closed.
How to recognize it
- Objects are equally spaced along a path or boundary.
- The endpoints or a closed loop change the relationship between objects and gaps.
Example
Posts are placed every 10 m along a 120 m straight path, including both ends. How many posts are needed?
Method
Interval counting
- There are 120 ÷ 10 = 12 intervals.
- On a straight path with both endpoints included, posts = intervals + 1.
- Therefore 12 + 1 = 13 posts.
Answer13 posts
Reusable ruleBoth ends included: posts = intervals + 1; closed loop: posts = intervals
Common pitfallFirst identify whether the path is open or closed and whether endpoints are included.
CheckThirteen posts create twelve 10 m intervals, totaling 120 m.
05Applied problems · Grades 4–6Rate and motion problems
Use distance, rate and time; decide whether rates combine or differ.
Open guide
Rate and motion problems
Use distance, rate and time; decide whether rates combine or differ.
How to recognize it
- Objects move toward each other, in the same direction or in pursuit.
- Distance, rate and time information must be connected.
Example
Two vehicles start 240 km apart and travel toward each other at 50 km/h and 30 km/h. When do they meet?
Method
Relative rate
- Together they close 50 + 30 = 80 km each hour.
- Meeting time = distance ÷ combined rate.
- 240 ÷ 80 = 3 hours.
Answer3 hours
Reusable ruleToward each other: use the sum of rates; pursuit in one direction: use the difference
Common pitfallUse a rate difference for pursuit, not for two objects moving toward each other.
CheckIn 3 hours they travel 150 km and 90 km, totaling 240 km.
06Patterns and counting · Grades 3–6Repeating-pattern problems
Find the shortest repeating block and use a remainder to locate the requested term.
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Repeating-pattern problems
Find the shortest repeating block and use a remainder to locate the requested term.
How to recognize it
- Colours, numbers or shapes repeat in a fixed order.
- A far-away position is requested.
Example
The colours red, yellow, blue, green, purple, white and black repeat in that order. Which colour is in position 100?
Method
Cycle and remainder
- The shortest cycle has length 7.
- 100 ÷ 7 leaves remainder 2.
- Remainder 2 points to the second item in the cycle.
AnswerYellow
Reusable ruleDivide the position by the cycle length; remainder 0 means the final item
Common pitfallA remainder of 0 is not a “zeroth” item.
CheckPosition 98 ends a full cycle, so 99 is red and 100 is yellow.
07Patterns and counting · Grades 3–6Systematic counting
Separate a choice into stages and use a tree to avoid omissions and duplicates.
Open guide
Systematic counting
Separate a choice into stages and use a tree to avoid omissions and duplicates.
How to recognize it
- The question asks how many different choices or arrangements are possible.
- A complete outcome has two or more stages.
Example
There are 3 shirts and 2 pairs of trousers. How many outfits use one of each?
Method
Multiplication principle
- There are 3 choices for the shirt.
- Each shirt can be paired with either pair of trousers.
- There are 3 × 2 = 6 complete outfits.
Answer6 outfits
Reusable ruleUse multiplication when every stage is completed; use addition for mutually exclusive categories
Common pitfallCount complete start-to-finish paths, not the intermediate nodes.
CheckList A1, A2, B1, B2, C1 and C2.
08Geometry · Grades 4–6Counting rectangles
Count systematically from small shapes to composites, then verify by choosing boundary lines.
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Counting rectangles
Count systematically from small shapes to composites, then verify by choosing boundary lines.
How to recognize it
- A grid contains both smallest shapes and larger composite shapes.
- Counting by sight is likely to omit or repeat shapes.
Example
How many rectangles are in a 2-by-3 grid?
Method
Choose the boundaries
- The grid has 3 horizontal and 4 vertical lines.
- Choose 2 horizontal lines in 3 ways.
- Choose 2 vertical lines in 6 ways.
- Together they determine 3 × 6 = 18 rectangles.
Answer18 rectangles
Reusable ruleFor an m-by-n grid: C(m+1,2) × C(n+1,2)
Common pitfallRectangles include squares unless the question explicitly excludes them.
CheckClassifying by width gives 12 one-row and 6 two-row rectangles, totaling 18.
A note on scope
Use these pages as a reference for common methods, not as an official competition syllabus. Schools and competitions may use different names or difficulty levels, and a formula is only useful when the learner understands why it applies.