GoalSolve word problems by writing an equation: choose the unknown, set up the equation, solve, check.
New words
mathematical model · matematik modelchoosing the unknown · noma’lumni belgilashcondition of the problem · masala shartichecking · tekshirish
Explanation
Solving a problem with an equation has two stages: first we set up an equation from the condition (create the mathematical model of the problem), then we solve it. We denote the required quantity by x and express the other quantities through x; for example, “a number 4 greater than x” is x + 4. Then we write the equality given in the condition (a sum, a total, two quantities being equal) as an equation. In motion problems we recall that distance = speed · time and that speeds add when objects move towards each other. After getting the answer we check it against the condition: the number must be positive, natural or otherwise sensible. One problem can be solved by different equations.
Worked examples
Of three consecutive natural numbers, the first and the third add up to 64. If the first is x: x + (x + 2) = 64, 2x + 2 = 64, x = 31. The numbers are 31, 32, 33.
Two walkers set off at the same time towards each other from places 21 km apart, at 4 and 3 km/h. If they meet after t hours: (4 + 3)t = 21, so t = 3 hours.
Class activity
“Problem theatre”: pupils act out a problem (for example, two cyclists). The audience chooses the unknown and writes the equation on the board; then the actors check the answer.
Practice
1
Three times a number plus 5 exceeds twice the number by 9. Find the number.
4
2
A rectangle has perimeter 46 cm and its length is 7 cm more than its width. How wide is it (cm)?
8
3
A brother is 3 times as old as his sister, and in 4 years he will be twice as old as she is. How old is the sister now?
4
4
Why should we check the answer against the problem after solving the equation?
A root of the equation may not fit the problem (for example, a negative or fractional number), and checking reveals it.