Words into expressions – write the relationship as an expression, in two steps
Each task has two steps for the same three sentences: first you count with a given number, then you write the same relationship as an expression with the letter x. After every answer you see the right method next to a typical mistake, and a picture with bars that shows why.
What next?
How do you play?
Every task has three kinds of objects, for example notebooks, pencils and pens. Three sentences describe how the numbers of these objects depend on each other. In step 1 you count with a concrete number: you know how many objects of the first kind there are, and you choose the right result from four answers. In step 2 you write the same relationship as an expression with the letter x. After every answer you see the right method next to the typical mistake behind one of the other answers, and a picture with bars: one bar is the number x (in step 1, the given number), and the small boxes are numbers added (+) or taken away (−). There are 5 tasks. At the end you will see where to find more practice. By default there is no time limit; the timed mode is optional and counts only the time while a question is waiting for your answer.
How do you write a relationship as an algebraic expression?
Write the sentences one by one, each as an operation on the number it talks about. If there are x notebooks and there are 3 times as many pencils as notebooks, there are 3x pencils. If there are 4 fewer pens than pencils, there are 3x − 4 pens. Notebooks, pencils and pens together are x + 3x + (3x − 4) = 7x − 4. Put an expression in brackets when it contains an addition or a subtraction, so that you do not lose a sign.
What is the difference between “3 times as many” and “3 more”?
“3 times as many as x” is a multiplication: 3x. “3 more than x” is an addition: x + 3. For x = 10 the first gives 30 and the second 13. In the same way, “4 fewer than” means taking away 4, not adding it. When a sentence says “than notebooks and pencils together”, first add both numbers: x + 3x = 4x. Only then subtract: 4 fewer than notebooks and pencils together is 4x − 4.
How do you check an expression with a number?
Choose a simple number, for example x = 10, and solve the task without the letter: 10 notebooks, 30 pencils, 26 pens, 66 together. Then put the same number into the expression: 7 · 10 − 4 = 66. If the result is the same, the expression is probably right. If it is different, the expression is certainly wrong. You do the same calculation in step 1 of the game and then compare it with the expression from step 2.
Why do we multiply the whole bracket?
If there are 2 fewer pencils than notebooks, there are x − 2 pencils. Notebooks and pencils together are x + (x − 2) = 2x − 2. When there are 3 times as many pens as notebooks and pencils together, we multiply the whole sum: 3 · (2x − 2) = 6x − 6. The number 2 is also 3 times as big, so you must not write 6x − 2. For x = 10: 10 notebooks, 8 pencils, 18 together, three times as many is 54, and 6 · 10 − 6 = 54.
What mistake is the most common?
Reading two relationships at once as if there were only one. When the sentence says “than notebooks and pencils together” and someone takes only the pencils into account, they get an expression with a different coefficient of x. The second common mistake is to change “times as many” into “so many more” (x + 3 instead of 3x), or to reverse the sign after “fewer”. Checking with a number usually reveals these mistakes.
How does this help in the eighth-grade exam?
In the Polish eighth-grade mathematics exam of 2025, task 6 – a closed task in which pupils had to choose the algebraic expression that describes the number of balls from a verbal description of how the numbers of balls of three kinds depend on each other – was solved correctly by 42% of the pupils. It was the hardest task checking skills from the area “using and interpreting representations”. Almost half of the pupils chose 9x − 3 or 6x − 3 as the correct expression, which, according to the report, comes from a wrong interpretation of the simultaneous dependence of the number of balls of one kind on the numbers of balls of the other two kinds.
Source: report of the Central Examination Board (CKE), 2025, p. 25 (in Polish). Curriculum: writing algebraic expressions (grades 7–8, III.3) – Dz.U. 2024 item 996.
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