Teaching objectives
Word problems are the bridge between mathematics and real life, but many secondary-school students don't know where to start. This lab trains exactly that skill: reading a problem, choosing a variable, translating each sentence into an algebraic expression, and building the key equality that leads to the solution.
What you'll learn
Students practise the complete solving cycle: from everyday language to the equation, and from the equation to the answer. Across 13 rooms, difficulty grows gradually and the mechanics vary to keep engagement high.
- Identifying the unknown quantity and representing it with a letter (x, z, …).
- Translating verbal phrases into expressions: "triple", "3 years younger", "5 times smaller", "3 years ago he was twice as old".
- Matching linguistic expressions with their algebraic form (room 3, drag-and-match activity).
- Setting up the equality by expressing the same quantity in two different ways (rooms 4 and 5, family age problems).
- Solving visually using the balance scale metaphor: removing and distributing boxes while keeping both sides equal.
- Mastering the sign-change shortcut when crossing the equals sign (room 12), as a quick version of the balance operations.
- Solving a problem with two time conditions (4 years ago / in 2 years) that reduces to a single-unknown system (room 13).
Key mathematical ideas
- An equation is a balance scale: any operation applied to both sides preserves the equilibrium.
- "Moving a term to the other side" is equivalent to adding or subtracting the same thing from both members; the sign changes because the operation is reversed.
- Expressing the same quantity in two different ways generates the key equality: ax = bx + c (e.g., 5x = x + 36).
- Time expressions shift the variable: "3 years ago he was twice as old" is written (x − 3)·2, not 2x − 3.
- A problem with two conditions can be reduced to a single unknown when the variables are related to each other.
Room-by-room contents
Room 1 · The unknown quantity
Introduction to the concept of a variable: the quantity we do not know receives a letter (usually x). The student learns that everything else in the statement can be expressed in terms of that letter.
Student tasks
- Read the statement and decide which quantity is the unknown.
- Identify how the other quantities are described in terms of x.
Room 2 · Three years less
First translation exercise: the younger brother's age is expressed as z − 3, where z is Juan's age. The student practises writing expressions with subtraction.
Student tasks
- Write the algebraic expression that represents the brother's age (z − 3).
Room 3 · Match the expressions
Matching activity: given Olga's age as x, the student links verbal phrases ("four times Olga's age", "4 years older than Olga", "4 years younger than Olga") with their algebraic expressions (4x, x + 4, x − 4).
Student tasks
- Match each text phrase with the correct algebraic expression.
Room 4 · The key equality
Pedro and his father's problem is introduced: Pedro is 5 times younger than his father, and the father is 36 years older. The student learns to express the father's age in two different ways to build the equation 5x = x + 36.
Student tasks
- Express the father's age in two different ways (5x and x + 36).
- Equate both expressions to obtain the equation.
Room 5 · Solve with the balance (5x = x + 36)
Visualisation of the equation 5x = x + 36 as a balance with boxes on each side. The student solves it by removing and distributing boxes while always keeping the balance.
Student tasks
- Manipulate the balance by removing equal boxes from both sides.
- Find the value of x (Pedro's age) and verify the solution.
Room 6 · Another family member
Problem about Carolina and her nephew Hugo: Carolina is three times Hugo's age and is also 24 years older. The student translates the statement into the equation 3x = x + 24.
Student tasks
- Set up the equation from the verbal statement.
- Identify the two expressions for Carolina's age.
Room 7 · Solve 3x = x + 24
Using the same balance metaphor, the student solves 3x = x + 24 to find Hugo's age.
Student tasks
- Solve the equation step by step with the balance.
- Check that the solution satisfies the original statement.
Room 8 · Three generations
A grandmother, a mother, and a daughter's ages add up to 70; the mother is twice the daughter's age and the grandmother is four times the daughter's age. The student expresses the three ages in terms of x and solves x + 2x + 4x = 70.
Student tasks
- Express all three ages in terms of the daughter's age (x).
- Set up and solve the resulting equation.
Room 9 · Shared pizza
Problem with division and remainder: 18 pizza slices for 5 friends with 3 left over. The student sets up 5x + 3 = 18 and finds how many slices each person ate.
Student tasks
- Translate the statement into an equation with sharing and remainder.
- Solve the equation and verify the result.
Room 10 · Cards coming and going
Statement with two successive operations: give away 17 cards, then buy as many as were left, and now have 92. The student builds the corresponding equation and solves it.
Student tasks
- Track the changes in the number of cards with x as the starting point.
- Set up and solve the equation to find the initial quantity.
Room 11 · Three years ago
An age problem with a time reference: three years ago the father was twice the student's age. The student is now 12 years old. Introduces the idea of going back in time (x − 3) to set up the equation.
Student tasks
- Express the ages from 3 years ago in terms of the current ones.
- Set up and solve the equation to find the father's current age.
Room 12 · The shortcut: move to the other side
Presentation of the transposition shortcut: when a term crosses the = sign, it changes sign. Equivalent to adding/subtracting the same amount on both sides of the balance.
Student tasks
- Observe the equivalence between the balance method and transposition.
- Apply the shortcut to move all the x terms to the left and numbers to the right.
Room 13 · Two moments, one unknown
A two-condition age problem: four years ago the mother was 5 times older; in 2 years she will be 3 times older. The student sets up a system of two equations with a single unknown and finds the current ages.
Student tasks
- Set up two equations from the two conditions in the statement.
- Solve the system to find the current ages of both.
Rooms to project
The most striking ones to show and discuss in class.