DSE Maths | Binomial Theorem Teaching Guide (for tutors) | Locking down one term with the general term

2026-11-03

Tutor lesson-prep guides · Secondary Maths

Students can recite C(n,r) a^(n-r) b^r but expand everything when they see “constant term”. This lesson does one thing: write the general term, solve the index equation, then substitute back. Expansion is only for checking the first three terms.

Objectives — Students can: (1) write the general term C(n,r) a^(n-r) b^r and check the n = 4 row 1, 4, 6, 4, 1 against Pascal triangle; (2) solve 6 – 2r = 0 from (x + 2/x)^6 to get r = 3 and a constant term of 160; (3) find the coefficient of x^2 in (2x + 3)^4 as 216 and of x^3 in (1 – x)^10 as -120; (4) get the coefficient sum 2^8 = 256 by setting x = 1 and estimate 1.02^12 as about 1.266 from three terms.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Combinations	What is C(6,3)?	20
Index laws	What is x^6 divided by x^3?	x^3
Sign handling	What is (-1)^3?	-1
Solving	Solve 6 – 2r = 0	r = 3

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Warm up	Expand (x + 1)^3, then ask about (x + 1)^10	Write the first four terms and spot the pattern
8-20 min	General term	Move from Pascal triangle to C(n,r) and state the general term	Copy it and mark that r starts at 0
20-34 min	Constant term	Work (x + 2/x)^6, set the index to zero and solve	Solve the equation before substituting
34-44 min	Specified power	Find the x^2 coefficient in (2x + 3)^4, keeping numbers and x apart	Write C(4,2) x 2^2 x 3^2
44-52 min	Sums and approximation	Set x = 1 for 2^8, then estimate 1.02^12 with three terms	Compare the estimate with the true value
52-60 min	Exit check	Hand out four questions	Answer independently, redo errors at once

03 Teaching script (read it out)

Scene 1: The general term is the only formula to remember

The (r + 1)th term of (a + b)^n is C(n,r) a^(n-r) b^r with r starting at 0. For n = 4 the coefficients are 1, 4, 6, 4, 1, matching Pascal triangle. Tell students that “the 5th term” means r = 4 while “the coefficient of x^3” needs an index equation: both start from the general term, never from counting on fingers.

Ask students to state aloud which r the “5th term” uses and correct it on the spot.

Scene 2: Constant term: set the index to zero

For (x + 2/x)^6 the general term is C(6,r) x^(6-r) (2/x)^r = C(6,r) 2^r x^(6-2r). A constant term needs 6 – 2r = 0, so r = 3 and the term is C(6,3) x 2^3 = 20 x 8 = 160. Asking for the x^2 coefficient instead gives 6 – 2r = 2, r = 2, and C(6,2) x 2^2 = 60. One general term answers every “specific term” question, which is the spine of this lesson.

Have students write “6 – 2r” beside the question and solve before touching the calculator.

Scene 3: Coefficient sums and approximation: substitute x = 1 or x = 0.02

Setting x = 1 in (1 + x)^8 gives the coefficient sum 256, that is 2^8; for odd terms only, substitute x = -1 and subtract. For (1 + 0.02)^12 the first three terms give 1 + 12 x 0.02 + C(12,2) x 0.02^2 = 1.2664 against a true 1.2682, an error near 0.0018, so three decimal places need one more term. Students should justify dropping the rest: x is small, so higher powers are negligible.

Ask what happens if x = 0.5 and only three terms are kept, leading to the convergence condition.

04 Question types and worked steps

Type	Example	Steps	Answer
Constant term	(x + 2/x)^6	1) general term 2) set the index to zero 3) solve r 4) substitute	160
Coefficient of a power	x^2 in (2x + 3)^4	1) keep 2^2 and 3^2 apart 2) use C(4,2) 3) multiply all three	216
Negative term	x^3 in (1 – x)^10	1) b = -x 2) keep (-1)^3 3) answer carries a minus	-120
Coefficient sum	(1 + x)^8	1) set x = 1 2) get 2^8 3) use plus and minus 1 for odd or even	256

Sanity check (teach this): Pascal check: the middle coefficient of n = 6 is C(6,3) = 20, times 2^3 gives 160, matching the general term. The sum 256 exceeds the largest single coefficient 70, as it must. The estimate 1.2664 is slightly below the true 1.2682 because every remaining term is positive, so the direction is right.

05 Independent practice (3 questions)

Question	Answer	What to watch
Coefficient of x^3 in (x + 3)^5	C(5,2) x 9 = 90	Swapping the powers of 3 and x
Constant term of (2x – 1)^6	C(6,6) x (-1)^6 = 1	Handling the sign at r = 6
Coefficient sum of (1 + x)^7	2^7 = 128	Substituting x = 1, not x = 0

06 Common mistakes and what to say

Mistake	Student writes	Say this
Starting r at 1	“The 5th term means r = 5”	“r starts at 0, so the 5th term is r = 4. Write the r = 0 term first and count from there.”
Dropping the power of b in a constant term	“C(6,3) = 20, so that is the answer”	“b is 2/x, so r = 3 brings a factor 2^3. A bare combination number is incomplete.”
Multiplying the minus sign only once	“(1 – x)^10 gives +120 for x^3”	“(-x)^3 is negative, so every odd power carries a minus. Write (-1)^3 explicitly.”
Substituting x = 0 for the coefficient sum	“x = 0 gives 1, so the sum is 1”	“x = 0 leaves only the constant term. A coefficient sum needs x = 1.”

07 Homework (paper, section and question numbers)

Maths chapter index — Review permutations and combinations, then questions 1 and 4
Permutations and combinations — Question 2 needs C(n,r)
Indices and logarithms — Question 6 compares the binomial estimate with the exact value

08 Exit check (last 5 minutes)

Question	Answer	Decision
Constant term of (x + 2/x)^6	160	Wrong: index equation solved incorrectly
x^2 coefficient in (2x + 3)^4	216	Wrong: 2^2 or 3^2 omitted
x^3 coefficient in (1 – x)^10	-120	Wrong: sign flipped to positive
Coefficient sum of (1 + x)^8	256	Wrong: substituted the wrong x

09 Tutor FAQ

Can the binomial theorem be taught before combinations?

Yes. Build the pattern with Pascal triangle first, then introduce C(n,r) as “choosing which brackets supply b”.

How many terms should be written out?

Only the general term and the target term unless the question asks for an expansion; expanding ten terms is slow and error prone.

When is a fourth term needed in an estimate?

Judge by the required accuracy and the size of x: for x at most 0.05 three terms usually give three decimals, otherwise add one and say so.

Find a tutor — free Tutor Registration →

A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.