Chapter V – Multiplication and Division.
Table of Contents:
- 1. Introduction to multiplication
- 2. Multiplication games on platforms
- 3. Tower transformation
- 4. Changing columns and rows
- 5. Colorful multiplication
- 6. Introduction to division
- 7. Division with remainder
- 8. What is the divisor?
- 9. Commutativity of multiplication and division
- 10. Division by building groups (division by 4)
- 11. Team division
- 12. Gear multiplication machine
- 13. Building according to an operation
- 14. Game “Who has more?”
- 15. Block equations
The ability to multiply and divide is one of the milestones in a child’s mathematical education. It allows not only for more efficient calculations, but also for solving word problems, analyzing data, logical thinking, and planning actions in everyday life.
For children at the early primary education stage, multiplication can be quite challenging. It already requires a certain level of abstract thinking, which is not easy for them to visualize, especially with higher numerical values.
Division is something that comes naturally to children. From an early age, they learn it by observing cutting, sharing, and sorting. It might seem that these skills would make performing division operations the easiest mathematical challenge to master. Meanwhile, in the world of mathematics, children encounter difficulties caused by translating these skills into abstract mathematical operations performed solely on numbers.
Students aged 7–9 learn best through activity, play, manipulation, and observation. Therefore, in this chapter we use Korbo blocks – a tool that perfectly fits these needs. Children can physically arrange groups, divide them, build repeating patterns, observe relationships, and draw conclusions.
Thanks to different types of blocks – gears, cylinders, platforms, and connectors – children not only understand the structure of mathematical operations, but also engage in the learning process on many levels: motor, visual, auditory, and verbal.
After completing work with this chapter, the student:
Understands multiplication as repeated addition, e.g. 3 × 2 means three times two.
Understands division as splitting a whole into equal parts or groups.
Builds models of multiplication and division using Korbo blocks.
Performs simple multiplication and division calculations mentally or with the help of concrete objects.
Records operations using mathematical symbols: ×, ÷, =.
Solves simple word problems that use multiplication or division.
Compares the results of operations, analyzing which are greater or smaller.
Uses inverse operations (multiplication–division) and recognizes the relationship between them.
Completes missing elements in an operation (e.g. _ × 3 = 12; 16 ÷ _ = 4).
Uses division with remainder and can represent it using concrete objects.
1. Introduction to multiplication
Educational objective: Understanding the concept of multiplication
Materials: Korbo blocks, worksheets with operations
Procedure:
The teacher builds three identical towers, each made of two gears. The teacher asks a student to say how many towers there are and then write the number on the board. The teacher asks another student to count how many gears are in one tower and write the number on the board. The teacher says that there are three identical towers, each tower has two gears, and asks the students how many gears there are in all the towers. The teacher writes the operation 3 × 2 = 6. The teacher explains to the students what the multiplication sign means.
The teacher gives another example (e.g. 4 × 3 = 12), emphasizing that the towers are the same.
The teacher divides the children into groups, each pair receives a Korbo set and worksheets with operations to solve. The children build towers and solve the tasks.
2. Multiplication games on platforms
Educational objective: Reinforcing multiplication through addition
Materials: Korbo blocks
Procedure:
The teacher builds an 8×8 mat using platforms. Then the teacher places three gears in the first row and asks a student how many gears are on the mat, writing the operation on the board (three gears in row one: 1 × 3 = 3). Next, the teacher removes the gears and places four gears in the first row and four gears in the second row, saying: we have four gears in the first row and we add another row with the same number of gears, which means two rows of four gears, 2 × 4 = 8. The teacher shows two more examples (e.g. 5 × 4 = 20 and 8 × 3 = 24).
The teacher divides the children into teams of four, each team receives worksheets. The children build an 8×8 mat and number the rows and columns. The children solve the tasks by placing the appropriate number of gears in rows and columns according to the written operation.
Tower exchange
Educational objective: understanding the commutativity of factors in multiplication
Materials: Korbo blocks
Procedure:
The teacher builds three towers made of two gears each: the first tower with red gears, the second with blue gears, and the third with yellow gears. The teacher asks the children how many towers there are, how many gears are in each tower, and how to write the equation. A child writes the equation on the board (3 × 2 = 6). Then the teacher rebuilds the towers so that two identical towers of three gears each are created, with each gear in the tower being a different color. The teacher shows the towers to the children and asks them to check whether the number of gears in both towers has changed. The teacher explains that three towers were rebuilt into two towers, but using the same gears, so 2 × 3 = 6. The teacher gives two more simple examples explaining the commutativity of factors in multiplication.
The teacher divides the children into pairs; each pair receives a Korbo set and worksheets with operations to solve. The children build and rebuild towers and solve the tasks.
4. Swapping rows and columns
Educational objective: understanding the commutativity of factors in multiplication
Materials: Korbo blocks
Procedure:
The teacher builds an 8×8 mat and places five gears in the first, second, and third rows. The teacher asks a child to count the gears and write the operation (3 × 5 = 15). Then the teacher removes the gears and arranges them with five gears in the first, second, and third columns. The teacher asks a child to count the gears and write the operation (5 × 3 = 15).
The teacher divides the children into teams of four; each team receives two Korbo sets and worksheets. The children build an 8×8 mat and number the rows and columns. The children solve the tasks by placing the appropriate number of gears in rows and columns according to the written operation.
5. Colorful multiplication
Educational objective:
Materials: Korbo blocks
Procedure:
The teacher places images of four gears on the board, each in a different color. Each gear is assigned a specific numerical value, for example:
– red gear = 3
– blue gear = 5
– purple gear = 6
– yellow gear = 7
Next, the teacher asks a student to randomly select one gear, then the teacher draws another gear and hands it to the student in the other hand. The student writes on the board the values corresponding to both gears and the result of the multiplication.
The teacher divides the children into pairs; each pair receives several gears in different colors. The children take turns drawing gears, similarly to the teacher–student demonstration (one child takes one gear, the other chooses their gear and hands it to the first). The first student calculates the result and gives it to the second, who checks the correctness.
6. Introduction to division
Educational objective: Understanding the concept of division
Materials: Korbo blocks
Procedure:
The teacher builds a tower of eight gear wheels, shows it to the children, and writes the number 8 on the board. Next, the teacher divides the tower into two equal parts of four gear wheels each and asks the children into how many towers the large tower was divided and how many gear wheels each tower contains. The teacher writes the equation 8 ÷ 2 = 4 on the board and explains the “÷” symbol to the children.
Then the teacher rebuilds the tower of eight gear wheels, but this time divides it into four towers of two gear wheels each. The teacher asks the children to count the towers and say how many gear wheels each one consists of, and writes the equation 8 ÷ 4 = 2.
The teacher divides the children into pairs. The teacher gives the number of elements the children should take together from the container and the number of groups they should create (each group must contain the same number of blocks).
Variants:
The children sit back to back and simultaneously carry out the instructions given by the teacher, for example:
Build a tower of 6 gear wheels and divide it into 3 equal parts. How many gear wheels will be in each part?
Build a tower of 12 gear wheels and divide it into 4 equal parts. How many gear wheels will be in each part?
Build a tower of 4 cylinders and divide it into 2 equal parts. How many cylinders will be in each part?
Build a tower of 9 gear wheels and divide it into 3 equal parts. How many gear wheels will be in each part?
7. Division with Remainder
Educational objective: Explaining the concept of a remainder in division
Materials: Korbo blocks
Procedure:
The teacher points to a box containing 22 cylinders and asks three students to divide them among themselves. The students conclude that they are unable to divide the blocks equally. The teacher writes the operation 22 ÷ 3 and explains to the children what division with a remainder means. Next, the teacher asks one student to add enough cylinders to the set of 22 so that it can be divided equally among three people, and to write the corresponding equation.
The teacher divides the children into pairs. Each pair receives a task card with division problems. Using the cylinders, the children solve the equations and write down the results for divisions with and without a remainder.
8. What Is the Divisor?
Educational objective: Practicing division
Materials: Korbo cylinders
Procedure:
Children work in pairs and receive worksheets with equations in which the divisor is replaced by an image of a colored gear wheel. Using the cylinders as aids, the children assign the correct value of the divisor to the color of the gear wheel shown in the image. The teacher explains that the divisor indicates into how many groups the total number of blocks should be divided.
9. Commutativity of Multiplication and Division
Educational objective: Understanding the concept of the relationship between multiplication and division
Materials: Korbo blocks
Procedure:
The teacher prepares a tower made of twelve gear wheels, places it on a platform, and writes the equation 12 ÷ 4 = 3 on the board, asking a student to give the result. The teacher then divides the tower into four equal parts and asks a student how many towers they can see and how many gear wheels each tower contains. Next, the teacher writes the equation 4 × 3 on the board and asks how many gear wheels there are in total, writing 12 as the result.
In the same way, the teacher presents further examples, such as 15 ÷ 5 = 3 and 5 × 3 = 15, and explains to the students that the result of a division can be checked by multiplication.
10. Dividing by Building Groups (Division by 4)
Objective: Division as creating equal groups.
Required elements: Gear wheels, a platform.
Procedure:
The student receives 12 gear wheels and is asked to distribute them evenly onto 4 pegs on the platform.
Conclusion: 12 ÷ 4 = 3.
Variations:
Changing the number of groups.
“Uneven” division, e.g. 13 ÷ 4, and discussing the remainder.
11. Team Division
Objective: Division through assigning elements to groups.
Required elements: 12 cylinders, cards with the number of children in a group.
Procedure:
The student has 12 “team members” (cylinders). They divide them into, for example, 3 teams with 4 members each. Then into 4 teams with 3 members each.
Variations:
– Students check how many team members there are in different team sports (e.g. football, volleyball, basketball, futsal, rugby, etc.), choose the same number of blocks as there are team members, and check whether these teams can be divided into several equal groups.
12. Gear Multiplication Machine
Objective: Strengthening automatic recall of multiplication facts.
Required elements: Gear wheels, a platform as a machine (5 platforms arranged side by side).
Procedure:
Work in pairs or groups. The group chooses a number (e.g. 5) and builds a “machine” from 5 gear wheels rotating in a row. Then they place 3 additional gear wheels on each of them: 5 × 3 = 15. They count the total number of blocks.
Variations:
– Team game: which group can “produce” the result of the operation indicated by the teacher the fastest.
13. Building According to an Operation
Objective: Creating mathematical models.
Required elements: All Korbo elements.
Procedure:
A student draws a card with an operation (e.g. 6 ÷ 2). The student builds a construction using 6 elements and then divides it evenly into 2 parts.
Variations:
– Adding a verbal description, for example: “The construction has 6 elements, each group has …”.
– Creating operations based on one’s own constructions.
14. Game: “Who Has More?”
Objective: Comparing the results of multiplication and division operations.
Required elements: Sets of blocks, cards with operations.
Procedure:
Children work in pairs, draw operation cards, perform the operations using the blocks, and then compare the results to determine who has more.
15. Block Equations
Objective: Completing missing values in an operation.
Required elements: Gear wheels, cards with equations, e.g. _ × 4 = 16.
Procedure:
The student tries to find the missing value by building, for example by arranging gear wheels into groups of 4 until the total reaches 16.
Variations:
Completing division equations, e.g. 24 ÷ _ = 6.
Team game – guess what is missing. One group shows the operation on a card and what they have built. Another group’s task is to discover what is missing (e.g. an entire tower or the number of gear wheels in the towers), for example: the group has the operation 5 × 4 = 20, but the construction shows five towers of three blocks each.