Junior Maths | Directed Numbers and Algebra Prep Guide (for tutors) | Signs, number line and substitution in three steps

2026-11-05

Tutor lesson-prep guides · Secondary Maths

The first wall in S1 is the minus sign: students answer -10 for -3 + 7 and -13 for -5 minus -8. This lesson ties every sign rule to one arrow on the number line so students derive it instead of memorising it.

Objectives — Students can: (1) place and order directed numbers on a number line; (2) carry out the four operations with signs and state the rule used; (3) substitute into an expression such as 3x squared minus 2x plus 1, one line at a time.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Number line	What lies to the left of zero	Negative numbers
Meaning of subtraction	Is 7 minus 3 the same as 3 minus 7	No, the second is negative
Times tables	What is 6 times 7	42

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-6 min	Diagnostic opener	Write -3 + 7 and ask for one answer	Answer and justify
6-18 min	Direction on the line	From -3 move seven right to 4; subtracting a negative moves right	Draw the arrow
18-30 min	Mixed add and subtract	-5 minus -8 equals 3, one move two ways	Write both equations
30-44 min	Signs in multiplication	Same signs positive: -4 times -6 is 24; opposite signs negative: -36 over 4 is -9	Prove by repeated addition
44-56 min	Substitution	x = -2 in 3x squared – 2x + 1 gives 17	One term per line
56-60 min	Exit check	Three questions collected and marked	Work alone

03 Teaching script (read it out)

Scene 1: Addition and subtraction are direction, not size

Right is plus, left is minus. -3 + 7 moves seven places right to 4. -5 minus -8 removes a debt of eight, which also moves right, landing on 3.
Teach it as one arrow written two ways and let students name the rule themselves; a rule they state survives the exam, a rule you state does not.

Students do: use the number line to explain why -5 minus -8 and -5 plus 8 end in the same place.

Scene 2: Sign rules come from debt, not from a chant

-4 times -6 means taking away a debt of 4, six times: the debt falls by 24, so the answer is +24. Same signs positive and opposite signs negative both fall out of that picture.
Then anchor it in two measurements: temperature rising from -6 to 9 degrees is a change of 15, and altitude from -120 m to 80 m is a change of 200 m. Both are larger minus smaller, then a direction; the classic error is calling the temperature change 3.

Students say: why the change from -6 to 9 degrees is 15 rather than 3.

Scene 3: Substitution in three lines

With x = -2: write the expression, substitute term by term, then add. 3 times 4 gives 12, minus 2 times -2 gives +4, then +1, so 17.
The three killers are squaring a negative into a negative, turning minus-minus into minus-plus, and doing addition before multiplication. One line per step shows exactly where it broke.

Students do: write 12, +4 and +1 side by side, then total them in one sentence.

04 Question types and worked steps

Type	Example	Steps	Answer
Ordering	Place -4, 0 and 3 on a line	Find zero, then count outward	-4 less than 0 less than 3
Four operations	-36 over 4 and -4 times -6	Sign first, then size	-9 and 24
Substitution	x = -2 in 3x squared – 2x + 1	Expression, terms, then total	17
Applied change	Temperature and altitude differences	Larger minus smaller, then direction	15 degrees and 200 m

Sanity check (teach this): Sanity checks: -3 + 7 is 4, not -10; -5 minus -8 is 3, not -13; -4 times -6 is +24, because two negatives give a positive; substituting x = -2 gives 12 + 4 + 1 = 17; the mean of -6, 9, -2 and 5 is 1.5, still positive after cancellation.

05 Independent practice (3 questions)

Question	Answer	What to watch
Three on the line	Place -7, -1 and 2, then order them	Is -7 written as largest
Eight operations	Two each of add, subtract, multiply, divide	Signs decided first
Two substitutions	x = -2 and x = 3	Negative squared handled
One applied item	-120 m up to 80 m	Units written

06 Common mistakes and what to say

Mistake	Student writes	Say this
Negative squared as negative	3 times -2 squared written as -12	What is -2 times -2? Two negatives make what?
Minus-minus becomes minus-plus	-5 minus -8 answered as -13	On the line, does subtracting a negative move left or right? Draw it.
Direction lost in changes	-6 to 9 called 3 degrees	From -6 to 0 is how much? From 0 to 9? Add both legs.
Skipping lines	Answer 17 with no working	One line per step: square, multiply, add. Method marks live in those lines.

07 Homework (paper, section and question numbers)

Secondary Maths chapter index — Every chapter in the subject
All prep guides — Parallel chapters
Redo this chapter — Three number-line items and two substitutions

08 Exit check (last 5 minutes)

Question	Answer	Decision
-3 + 7	4	-10 means restart act one
-4 times -6	+24	-24 means redo act two
x = -2 in 3x squared – 2x + 1	17	5 or -1 means redo act three

09 Tutor FAQ

Should S1 students use a calculator?

Four operations and substitution by hand, with a calculator only to check. Slow hand work is what breaks algebra in S4.

Should the sign rule be memorised?

The chant is fine as a backup, but students must derive it once from the debt picture or three brackets will defeat them.

Why insist on every line?

Both school and public exams award method marks, and errors show up mid-line; students who skip lines do not know which sign failed.

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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.