Chapter IV
Addition and Subtraction Across the Tens Boundary
Why is it worth teaching addition and subtraction through action?
Addition and subtraction are the basic arithmetic operations with which learning mathematics begins. Understanding and mastering them is a foundation for further calculation, as well as for solving word problems, planning spending, measuring, and many everyday activities. Curiosity sparked in a child by frequently asking “how many” encourages action and discovering mathematical relationships.
In early-school education it is essential that a child not only knows the symbolic record of an operation, but understands what lies behind it: addition is joining, enlarging a set; subtraction is taking away, making smaller. Children learn most effectively through action, experience, and play — and not through abstract symbols.
Blocks support this stage of learning very well, because they make it possible to:
- handle concrete pieces while building models of operations,
- physically add and take away blocks in a construction,
- visually compare quantities and differences in height, length, and the number of pieces,
- introduce operations through movement, building, turning, and sliding, which activates different styles of learning.
Activities with blocks help children feel and understand mathematics with their whole body, and not only “in their head” or “on paper”. This chapter is a base for further learning and prepares children to think mathematically in a natural, dynamic, and engaging way.
Joining Towers
Learning goal
Introducing the idea of a sum.
Needed elements
Blocks
How it goes
The teacher builds a tower from 10 blue gears and asks one pupil to count the gears in the tower aloud. The teacher repeats to the children that this tower consists of 10 gears. The teacher writes the number on the board (under a drawing of a blue gear). Next the teacher builds a tower from 3 red gears and asks another pupil to count the red gears aloud. The teacher asks how many gears the second tower consists of. The teacher writes the number on the board (under a drawing of a red gear). The teacher asks another pupil to count all the gears in both towers and give the result using the phrase “What is the sum of all the gears?”. Then the teacher writes the result on the board and places the addition and equals signs between the values. The teacher also points out that the sum describes the total number of all pieces in this operation.
Variants
The teacher asks a pupil by how much the blue tower is taller than the red one.
The teacher moves one gear from the blue tower onto the red tower and asks the pupil how many gears there are altogether, how many gears are in the taller tower, and how many in the shorter one (by how many gears the two towers differ).
Colorful Addition
Learning goal
Practicing addition by building and joining colorful towers.
Needed elements
- Blocks
- Paper and a pencil for each group
How it goes
The teacher divides the children into teams of four; each team receives a worksheet with instructions and a place to write the result. All the sets are in a box under the board. The teacher tells the children that each of them has been “assigned” one color (one person in the group has purple, one yellow, one blue, one red); only the person in the group who has been assigned a given color may go for the gears they need. The teacher says:
● build a tower from 10 yellow gears
● build a tower from 4 purple gears and join it to the yellow tower
● count the gears in each tower and then add them together
● write the operation and at the end give the sum of all the gears
● take 2 yellow gears back to the box
● bring 4 red gears and build a tower from them
● count the gears in each tower, write the equation, and calculate the sum of all the gears
● write the result
● take 1 purple, 1 red, and 1 yellow gear back to the box
● bring 5 blue gears and build a tower from them — count the gears in each tower
● write the equation and count how many gears there are altogether, write the result
After carrying out all the instructions, take the gears back to the box.
Chain of Operations
Learning goal
Practice in addition and subtraction.
Needed elements
Blocks
How it goes
The teacher divides the children into pairs and places a box of gears under the board. The teacher tells the children that they will take turns going to the box for the gears they need. The children carry out the instructions written on the worksheet:
● bring 10 purple and 6 yellow gears and build one tower from them. Give the sum of all the gears in the tower. Write the operation.
● take back 4 purple gears and count how many gears are left; write the operation.
● take back 2 yellow and 1 purple gear; how many gears are left? Write the result.
● from the result of the first operation subtract the result of the second
● how many yellow gears did you take back altogether, and how many purple ones?
How Many Gears Are Missing?
Learning goal
Completing a tower to 10 gears.
Needed elements
Blocks
How it goes
The teacher asks a pupil to build a tower from 6 gears in one color of their choice, then asks another pupil to say how many gears are missing for the tower to have a height of 10 gears. The pupil adds the missing number of gears in another color of their choice and checks whether the sum is 10. The teacher writes the equation on the board.
The teacher asks a pupil to build a tower from 10 gears in any color, then asks them to take off as many gears as needed so that 6 gears remain in the tower. The teacher writes the equation on the board.
The teacher divides the pupils into pairs; each pair gets a set of blocks and a worksheet. Pupils take turns building towers according to the model on the worksheet and complete each tower to 10 gears. They record the number of missing gears on the worksheet.
Read, Build, Complete
Learning goal
Reading with understanding, completing to a given value.
Needed elements
- Blocks
- Worksheets
How it goes
The teacher asks a pupil to build a tower from 11 gears in one color of their choice, then asks another pupil to say how many gears are missing for the tower to have a height of 20 gears. The pupil adds the missing number of gears in another color of their choice and checks whether the sum is 20. The teacher writes the equation on the board.
The teacher asks a pupil to build a tower from 20 gears in any color, then asks them to take off as many gears as needed so that 11 gears remain in the tower. The teacher writes the equation on the board.
The teacher divides the pupils into pairs; each pair gets a set of blocks and a worksheet. Pupils take turns building towers according to the model on the worksheet and complete each tower to 20 gears. They record the number of missing gears on the worksheet.
Add and Build
Learning goal
Consolidating addition of one-digit numbers.
Needed elements
- Gears
- Cylinders
- Platforms
- Connectors
- Cards with digits
How it goes
The child draws two cards with digits (e.g. 4 and 3) and from the box chooses any pieces — first as many as on the first card, then as many as on the second. They count how many pieces they took out altogether and build a construction consisting of as many pieces as the sum (7).
The teacher asks a pupil to build a tower from 10 gears in any color, then asks them to take off as many gears as needed so that 6 gears remain in the tower. The teacher writes the equation on the board.
The teacher divides the pupils into pairs; each pair gets a set of blocks and a worksheet. Pupils take turns building towers according to the model on the worksheet and complete each tower to 10 gears. They record the number of missing gears on the worksheet.
Variants
When building with younger children it is worth using only gears, first choosing the right number in one color and then in another, and then building one tower.
We Subtract from a Construction
Learning goal
A practical understanding of subtraction.
Needed elements
Blocks
How it goes
The teacher asks each child to build a construction from 10 pieces. Then the teacher asks them to remove, e.g., three pieces from it. Next the teacher asks the pupils to count how many pieces remain.
Variants
The teacher creates a chain of operations (e.g. add another three pieces, how many blocks are there now? subtract 5, and so on).
Adding Machine
Learning goal
Illustrating addition as a process.
Needed elements
- Two platforms joined together for each pupil
- Gears
How it goes
The child arranges a “machine” on the platforms. First the teacher asks the children to add, e.g., 4 gears to their machine, and then to add two more in such a way that they all turn. After they are joined, the machine “adds them up” (the gears turn) and the child says the result. Then the children may add more gears. Gears can be stacked on one another (turning towers) — it is important that the children can always count all the gears that are turning.
Variants
Reversing the role — the machine subtracts.
Number Stick
Learning goal
Developing mathematical and spatial thinking in addition.
Needed elements
- Cross connectors
- Cylinders
How it goes
The child's task is to build a “stick” from as many pieces as the teacher gives. The first number is the number of cylinders, the second is the number of connectors, e.g. the teacher says: “Build a stick that consists of 6 cylinders and 3 connectors. What is the total sum of the blocks?”.
Variants
As an extension of the operation, gears can be added to the stick, placed on the connectors as “flowers” on branches.
Block Shop
Learning goal
Adding values and finding change.
Needed elements
- Blocks as “goods”
- Cards with constructions of different point values (e.g. 3, 6, 10)
- A worksheet — block values
How it goes
An important rule: in one round each child may go to the box for one block or to the board for one card with a value.
Children collect blocks of different values, making a construction from them, so that they can then exchange it for a picture card and points. In each round:
Children go to the box together and each chooses one block of the value they need. Then they add them to their constructions. When everyone has added them to their constructions, the next round begins — children may go to the box again or exchange their construction for points.
Tower Operations
Learning goal
Comparing sums and differences in operations.
Needed elements
Gears.
How it goes
Children build towers corresponding to the sums of different operations (e.g. 4 + 2 vs. 5 + 1). Then they compare the height of the towers. The teacher asks: what is the difference in the number of blocks in the first and second tower.
Variants
Group work — each child arranges a tower from as many gears as a card shows. Then in pairs they join towers, to then check what the difference is in the number of blocks between the two groups (two towers).
Gear Equations
Learning goal
Completing the missing addends in an operation.
Needed elements
Cylinders.
How it goes
The teacher says: Everyone take three cylinders. And now I want you to have seven cylinders each. Check how many more cylinders you still need to take so that everyone has seven?
Variants
Subtraction operations (You have seven and I would like you to have three).
Inventing equations for one another in pairs.
Moving Number-Block Line
Learning goal
Spatial addition and subtraction on a number line.
Needed elements
● A long platform (10 platforms joined in a line)
● A cylinder — a token made from blocks
● A die
Two pairs may work at each platform (one in the top row, the other in the bottom row).
How it goes
Each point (peg) on the platform is the next space. In pairs, children take turns moving the token according to the number rolled and carrying out the operation, e.g.: they roll a 5, they go to the fifth space. Then they roll a 3. They say 5 + 3 and go three spaces farther. They count how many spaces they have already covered from the start (five and three is 8), and so on.
Variants
Instead of dice, pupils create stories related to addition to carry out on the line, e.g. (John walked four steps, stopped, and then took two more steps. How many steps did he take altogether?)
Block Operation Towers
Learning goal
Consolidating addition and subtraction operations.
Needed elements
- Gears
- Cards with gears (for assigning values to them)
How it goes
The teacher places cards with gears on the board. Under them, for each color of gear, the teacher assigns a matching value, e.g.:
● a yellow gear has the value three,
● a blue gear has the value four,
● a red gear has the value five,
● a green gear has the value six.
Then the teacher asks the pupils to build a tower that has the value, e.g., 10.
Variants
- build a tower from three blocks that has the value 10
- build a tower that has the value 12 and is built from four blocks
- build a tower that has the value 15 and has at least two yellow gears in it
- build a tower that has the value 12, one block is yellow and one is blue
- build a tower that has the value 16 and has three green gears in it
Block Operation Pyramids
Learning goal
Consolidating addition and subtraction operations.
Needed elements
- Cylinders
- Cards with cylinders (for assigning values to them)
How it goes
The teacher places cards with cylinders on the board. Under them, for each color of cylinder, the teacher assigns a matching value, e.g.:
● a yellow cylinder has the value three,
● a blue cylinder has the value four,
● a red cylinder has the value five,
● a green cylinder has the value six.
Then the teacher asks the pupils each to take, e.g.: two blue cylinders, two green, one red, and one yellow, and to build a given pyramid from them, e.g.:
● in the first row blue, red, blue,
● in the second row two green cylinders,
● in the third row a yellow cylinder.
Then the teacher asks questions and asks them to complete challenges:
● What is the sum in the first row?
● What is the sum in the first and second row?
● What is the sum of the whole pyramid?
● Change the first row so that its sum is 16.
● Change the second row so that its sum is eight (the first row can then also be rearranged).
● Arrange the pyramid so that the sum of the first two rows is, e.g., twenty-two.