On this page
Algebra often starts badly because it looks like a new language. In the new NCERT book it starts with something simple: if Shabnam is 3 years older than Aftab, her age is always "Aftab's age + 3". This post explains letter-numbers, shows how to write, use and simplify expressions with worked examples, and lists the slips Class 7 students make most.
What is a letter-number?
Chapter 4 of NCERT's Class 7 Ganita Prakash, Expressions using Letter-Numbers, uses a letter to stand for a number that can change. Call Aftab's age a. Then Shabnam's age is a + 3. If a is 23, Shabnam is 23 + 3 = 26. The book calls letters used this way letter-numbers, and an expression containing them, such as a + 3, an algebraic expression. The contents page of Part 1 places this chapter right after decimals and says it builds on the chapter on arithmetic expressions.
Two habits matter from the start:
A letter stands for a number, not an object. In c × 35, c is the number of coconuts, not "coconut".
4 × n is written 4n, number first. So 5m + 3 with m = 2 means 5 × 2 + 3 = 13.
How do you turn a situation into an expression? Worked examples
Example 1 (from the chapter). A coconut costs ₹35 and 1 kg of jaggery costs ₹60. Write the total cost for c coconuts and j kg of jaggery.
Cost of coconuts: c × 35, or 35c.
Cost of jaggery: j × 60, or 60j.
Total: 35c + 60j.
For 7 coconuts and 4 kg: 35 × 7 + 60 × 4 = 245 + 240 = ₹485.
Example 2 (from the chapter's exercises). A flour mill takes 10 seconds to start, then 8 seconds to grind each kg. How long for y kg, starting with the machine off?
The start-up happens once; the 8 seconds happen once per kg. So the time is 10 + 8y seconds. For 5 kg: 10 + 8 × 5 = 50 seconds. The option (10 + 8) × y would charge the start-up for every kg, which is why it is wrong.
A good check for any expression: put in a small number you can work out by hand and see if the answer makes sense.
How do you simplify an expression?
Simplifying means writing the same expression with fewer terms. The chapter's shop example sells pencils at c rupees on three days: 5 on day 1, 3 on day 2, 10 on day 3.
Pencil money: 5c + 3c + 10c = (5 + 3 + 10)c = 18c.
Eraser money, at d rupees each, for 4, 6 and 1 erasers: 4d + 6d + d = 11d.
Total: 18c + 11d.
You cannot simplify 18c + 11d further, because c and d are different numbers. If c = ₹50, the pencil money alone is 18 × 50 = ₹900.
The book also checks that a simplified form is really equal. For a rectangle, l + b + l + b = 2l + 2b. With l = 3 and b = 4, both give 14.
Why do students make mistakes with letters?
Researchers Mollie MacGregor and Kaye Stacey studied students aged 11 to 15 in Students' understanding of algebraic notation: 11-15 (1997). They traced misreadings to a few sources, including intuitive assumptions about the new notation, analogies with familiar symbol systems, interference from other new maths, and misleading teaching materials. In other words, wrong answers usually come from a sensible-looking rule used in the wrong place.
The chapter's "Mind the Mistake, Mend the Mistake" box gives exactly these. Here are the errors and their fixes:
If d = 6, someone writes 3d = 36. They read 3d like the digits of a number. Correct: 3 × 6 = 18.
If r = 8, someone writes 2r + 1 = 29. Same slip, 28 + 1. Correct: 2 × 8 + 1 = 17.
If a = −4, someone writes 10 − a = 6. Subtracting a negative adds. Correct: 10 − (−4) = 14.
If m = −6, someone writes 3(m + 1) = 19. Correct: 3 × (−5) = −15.
If t = 4 and b = 3, someone writes 2t + b = 24 by multiplying everything. Correct: 8 + 3 = 11.
If h = 5 and n = 6, someone writes h − (3 − n) = 4. Correct: 3 − 6 = −3, and 5 − (−3) = 8.
A 20-minute practice routine
Say it, then write it. For three situations (bus fare, steps climbed, pocket money saved), first describe the rule in words, then write the expression.
Substitute small numbers. Check each expression with a value you can do in your head.
Mend three mistakes. Pick three items from the list above, cover the corrections, and explain each slip in one sentence.
Study one worked example, then try a twin. In Sweller and Cooper's experiments (1985), students who studied worked algebra examples solved similar problems faster and with fewer errors. Our post on worked examples versus problem solving explains when to use each.
When you are stuck on how to start an expression, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. Try your own first line before you look. For the order of operations behind all of this, see arithmetic expressions in Class 7.
Next step: write an expression for the number of wheels on b bicycles and t tricycles, then check it with b = 2 and t = 1.
