When children bring home a "difficult" math problem, parents often find it strange. But most thinking problems in elementary school actually belong to about ten familiar types. Recognizing the pattern is halfway there: the child knows what to draw, what to try, and avoids the familiar trap of that pattern.
Below are the 10 most common patterns, each with recognition signs, ways to think and an example with solution. We have checked all answers in the article by listing all the cases.
1. Rules for sequences of numbers and letters
Identifying signs: give a sequence of numbers, letters or pictures and ask for the next term or missing term.
How to think: find the difference between two consecutive numbers first. If the effect is uneven, see if the effect gradually increases. If you still don't see it, try splitting the sequence into two alternating sequences (odd positions and even positions).
For example: Find the next number: 1, 2, 4, 7, 11, ?
Solution: Middle difference The consecutive numbers are 1, 2, 3, 4, increasing by one unit each time. The next difference is 5, so the number to find is 11 + 5 = 16.
The sequence of letters also follows the same way of thinking. In the image below, the sequence E, X, F, W, G, ? are essentially two alternating sequences: odd positions E, F, G go forward according to the alphabet, even positions X, W go backward. The missing letter is in the even position, behind W when going backwards, so it's V.
2. Count pictures
Identification signs: "How many triangles / squares / rectangles are there in the picture?"
How to think: don't count haphazardly. Count by size: first the smallest shapes (simple shapes), then the compound shapes of 2 simple shapes, from 3 simple shapes... until the largest shape. Record the quantity of each type and then add them up.
For example: A large triangle has two straight segments drawn from the top to the bottom. How many triangles are there?
Solution: Two lines divide the large triangle into 3 small triangles. Triangle assembled from 2 adjacent parts: 2 shapes. Triangle assembled from all 3 parts: 1 picture. Total 3 + 2 + 1 = 6 triangles. Another way of looking at it: from the top there are 4 straight lines going down to the bottom, each way choose 2 segments for a triangle, and there are 6 ways to choose.
3. Calculate quickly by pairing
Identifying signs: long series of calculations, many numbers, with alternating plus and minus signs, or written "quick calculation", "reasonableness".
How to think: find pairs with a round sum (or difference): round ten, equal to 1, equals 0. Remember to keep the sign accompanying each number when combining.
For example: Quick calculation: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
Solution: Match the first number with the last number: 1 + 10, 2 + 9, 3 + 8, 4 + 7, 5 + 6. There are 5 pairs, each pair equals 11, so the total is 5 × 11 = 55.
Same with alternating minus signs: 2 − 3 + 4 − 5 + 6. The number 2 stands alone, combining (−3 + 4) = 1 and (−5 + 6) = 1, yielding 2 + 1 + 1 = 4.
4. Age problem
Identifying signs: talks about the age of two people at present, in the past or in the future; There are often the phrases "when born", "in a few years".
How to think: the difference in the ages of two people never changes, because every year everyone gets 1 year older. Draw two straight lines representing age, the difference remains the same.
For example: Mom gave birth to a bear when she was 25 years old. How old was the mother when the bear was 13 years old?
Solution: When the bear was born (0 years old) the mother was 25 years old, so she was always 25 years older than the bear. When the bear is 13 years old, the mother is 13 + 25 = 38 years old.
5. Line up, order
Identification signs: "ranked … from the beginning of the line, second … from the end of the line", "how many people are in the middle".
How to think: draw the line out with dots. When adding up the counting positions from both sides, the person mentioned is counted twice, so 1 must be subtracted.
For example: In a row, the pig is 14th from the beginning of the row and 40th from the end of the row. How many friends are there in the row?
Solution: In front of the pig there are 13 friends, behind the pig there are 39 friends, plus the pig himself: 13 + 39 + 1 = 53 friends. Shorthand: 14 + 40 − 1 = 53.
6. Chickens and dogs (temporary hypothesis method)
Identifying signs: there are two types of things (chickens and dogs, bicycles and cars, 3-legged and 4-legged chairs), know the total number of animals and the total number of legs, ask how many of each type.
How to think: assume are all types with fewer pins, calculated in the number of pins, than the true number. The missing legs are because some animals are actually multi-legged. Each time a chicken is changed to a dog, add 2 legs.
For example: In the yard there are 6 chickens and a dog, count all 18 legs. How many dogs are there?
Solution: Suppose all 6 are chickens: 6 × 2 = 12 legs. There are actually 18 legs, missing 18 − 12 = 6 legs. Each dog has two more legs than chickens, so the number of dogs is 6 : 2 = 3, and the number of chickens is 6 − 3 = 3. Check: 3 × 2 + 3 × 4 = 6 + 12 = 18 legs.
Try one more problem: 10 chickens and dogs have 28 legs. Assuming all chickens have 20 legs, 8 legs are missing, so there are 8 : 2 = 4 dogs and 6 chickens. Check: 6 × 2 + 4 × 4 = 12 + 16 = 28.
7. Planting a tree
Identifying signs: plant a tree, plant a flag, place a pole along a road or fence, know the length and distance between two trees.
How to think: count the number approximately distance first (length divided by distance), then count the number of trees. If planted at both ends, the number of trees is 1 more than the number of distances. If planted at only one end, the number of trees is equal to the number of distances.
For example: A road is 20 m long. People planted trees at both ends of the road, two adjacent trees 5 m apart. How many trees can be planted?
Solution: Number of distances: 20 : 5 = 4. Plant at both ends so the number of trees is 4 + 1 = 5 trees (at the marks 0 m, 5 m, 10 m, 15 m, 20 m). If only planted at one end, the number of trees is 4.
8. Deductive logic
Identifying signs: puzzles without complicated calculations: who is tallest, who sits where, how many people are in the house, who is telling the truth.
How to think: draw a diagram or table, write down each data one by one and gradually eliminate the possibilities ability. Read every word carefully, because the trap often lies in details that are easy to miss.
For example: In the house, the frog is a boy and has 3 younger sisters and 2 younger brothers. How many children do the frog's parents have?
Solution: In addition to 3 sisters and 2 brothers, there is also the frog himself. The number of children is 1 + 3 + 2 = 6 people. Many children answer 5 because they forgot to count the frogs.
Second example: An is higher than Binh, Chi is lower than Binh. Arrange on an axis from low to high: Chi, Binh, An. So An is the highest, Chi is the lowest.
9. Number problems
Identifying signs: "How many two (three) digit numbers...", "How many numbers can be made from cards...", or conditions about tens and units.
How to think: count the number of ways Choose each row one by one, remember the highest row cannot be 0. For small children, list in order (starting with 1, then 2...) so as not to miss out.
For example: How many two-digit numbers are there? whose units digit is greater than 4?
Solution: Tens has 9 choices (1 to 9, cannot be 0). Units greater than 4 have 5 ways to choose (5, 6, 7, 8, 9). The number of numbers is 9 × 5 = 45 numbers.
Easier lesson for first graders: How many different two-digit numbers can you form from three cards 1, 2, 3? List: 12, 13, 21, 23, 31, 32, all 6 numbers.
10. Combination: counting ways to choose, ways to arrange
Identifying signs: "how many ways to choose", "each of you shake hands once", "how many paths".
Way to think: systematically list or draw a tree diagram. Pay attention to distinguish when changing the order is a new way (lining up) and when it is not (shaking hands, choosing a group).
For example: Four people meet, each two of them shake hands with each other exactly once. How many handshakes are there?
Solution: Call four people A, B, C, D. List: AB, AC, AD, BC, BD, CD. There are 6 handshakes. Don't count THREE because it's still a handshake between A and B.
How to teach children to recognize cards?
- One or two forms each week. Deep learning of one form with many variables It might be better to go through the ten forms.
- Start with a small number. Chicken and dog card game with 2 animals and 6 legs let your child guess before using the temporary hypothesis method.
- Let your child name the form himself. After solving, ask "this problem". Is it similar to any problem you've done?".
- Always try the answer again. Substitute the answer into the question: Do 3 chickens and 3 dogs have exactly 18 legs?
- Mix the forms when you get used to them. The real question doesn't list the form, so it's time to Someday, let your child do a mixing test.
For children 5–6 years old, the above forms are still early; See suggestions in the article Math thinking for children 5–6 years old. If you plan to take the exam, the lesson What is TIMO? explains the 5 areas of the test, and practice questions have solutions by grade: grade 1, grade 2, grade 3. The full picture can be found in mental math handbook for children.
Practice these forms in the app Karotto
Transparency note: Karotto is an app developed by the team behind this website development. The photos in this article were taken from app.
Mathematics course "Number Garden" (504 lessons, 8 questions M1–M8, for children 5–8 years old) has rules, counting pictures, quick calculations, math problems with text and pictures for children to see and count. Before each new form there is a "Model Rabbit" step: Thỏ Bông solves a small numerical example with animation. When stuck, the child has 3 steps to suggest: light reminder, illustration, then answer with explanation. All topics have a Vietnamese voice.
Interspersed between stages are TIMO-style training stages in 5 areas (logic, arithmetic, number theory, geometry, combinatorics); At the end of the chapter there is a practice test with time, marking in arrays and reviewing incorrect sentences. The questions are automatically generated by the app, the real exam questions are not copied. The app has no advertising, learn a reasonable amount of lessons for free every day; Unlimited learning premium, buy once for lifetime use, no subscription required.
In short
Ten types of elementary math thinking have their own "key": find the difference for a series of numbers, count by size for counting shapes, the age difference remains constant, minus 1 when counting from both sides, assuming it's all chickens, count the distance before counting trees. Teach your child to understand that key, don't just teach the formula.
Frequently Asked Questions
How many types of mental math are there in elementary school?
There is no fixed number, each book is divided into one type. However, most elementary school thinking problems fall into about 10 familiar forms: sequence rules, counting shapes, quick calculation, age problems, lining up, chickens and dogs, growing trees, deductive logic, numbers and combinations of counting and choosing.
Which form of thinking math should first graders learn first?
Should start with the rules of number sequences and geometric sequences, Simple picture counting, lining up and family logic puzzles. These forms only require small addition and subtraction and are very intuitive. Chickens and dogs that grow trees are more suitable for children in grades 2–3.
Should children be taught to memorize the formula for each form?
Do not force children to memorize the formula before understanding it. Let your child draw a diagram, solve with small numbers, draw the rules themselves, and then name the formula. A child who understands why the queue must subtract 1 will be able to do it even if the question changes the way of asking.
How do you know which form you are weak in?
Let your child do a short question with all the forms, then write down the wrong sentence according to form. After 2–3 problems, any form that is repeated incorrectly is the form that needs more practice. Asking your child to explain how to do it also helps find out if he or she understands or is just guessing.
Practice mental math with Thỏ Bông
Rules, counting pictures, calculating quickly, math problems with text, with Rabbit as a model, 3-level suggestion and Vietnamese voice. Learn a reasonable amount of lessons every day for free.