Two ideas about "less than zero" that keep slipping. This time we picture them on a number line before we touch a formula.
This is a slow-down lesson — no new topic. We're going back to the two places negatives trip us up: a sequence that goes down by a fixed amount, and an inequality where the number in front of the letter is negative. The rule for each is easy to memorise and easy to forget. So instead we're going to see why each rule is true. Look at the two pictures first, then try the six problems.
Part A · When the pattern goes downhill
Picture it: every step takes the same amount away
Mia's stable starts the month with 50 bales of hay, and the horses eat 4 bales every week. Instead of a formula, let's just plot it: week number along the bottom, bales up the side.
Bales of hay, week by week — a straight line heading downhill
Every step drops by the same amount, so the points fall in a straight line. That fixed drop is the common difference: $d = -4$. A downhill line = a negative $d$. (Remember this picture — it comes straight back as negative slope next lesson.)
The number that keeps catching us is the sign of $d$. Going up adds; going down subtracts. So here $d$ isn't $4$, it's $\textcolor{#d1495b}{-4}$. The explicit formula is the same one you already know — you just feed it a negative $d$:
Arithmetic formula: $a_n = a_1 + (n-1)\cdot d$. When the pattern goes downhill, $d$ is negative, and $(n-1)\cdot d$ subtracts as $n$ grows.
Worked example — colour-coded, one part at a time
■ $a_1$ = start value■ $d$ = the step (here negative)■ $n-1$ = steps taken
Bales: 50, 46, 42, … The start is $\textcolor{#2255cc}{50}$ and each week is $\textcolor{#d1495b}{-4}$. Write the formula, then find week 6.
Check against the picture: week 6 really is at 30. If you'd used $d = +4$ by accident, the line would climb uphill to 70 — which the drawing clearly says is wrong. The picture catches the sign slip.
Part B · Why dividing by a negative flips the sign
Picture it: negatives reverse the order on the line
Here's the real reason — not a rule to memorise, a fact you can see. On the number line, "less than" just means "further to the left." Look what happens to two friendly numbers when you make them both negative:
Flip both numbers to negative — their order reverses
Top: $2$ is left of $5$, so $2 < 5$. Make both negative and they jump to the other side of 0 — now $-5$ is left of $-2$, so $-5 < -2$. The smaller number became the bigger one. Negating reverses the order — that's the whole reason the sign must flip.
The rule (now you can see it): multiplying or dividing an inequality by a negative reverses its direction ($> \leftrightarrow <$, $\geq \leftrightarrow \leq$). Adding or subtracting never flips it — only a negative multiply/divide does.
Worked example — Äquivalenzumformung, watch the flip
Solve $\;-2x \leq 6$. Divide both sides by $-2$ — and because $-2$ is negative, the $\leq$ turns into $\geq$.
Test it: $x = 0$ gives $-2(0) = 0 \leq 6$ ✓ (and $0 \geq -3$ ✓). $x = -4$ gives $-2(-4) = 8 \leq 6$ ✗ — correctly left out, because $-4$ is not $\geq -3$. The flip keeps the true answers and throws out the false ones.
Practice — start easy, build up
Tip: for every one, sketch the picture first — a downhill line for Part A, a number line for Part B — then write the maths.
Problem 1 Sequence
A water trough starts at 60 litres and drops the same amount each hour. Using the formula given, find how much is left at hour 5.
Formula: $a_n = 60 + (n-1)\cdot(-5)$ where $a_n$ = litres at hour $n$
put $n = 5$ → the step is $-5$ →
Problem 2 Inequality
Solve for $x$, and picture it on a number line: $\;-4x < 20$. Dividing by a negative — remember what happens to the sign.
$\div(-4)$ → flip →
Problem 3 Sequence
On a trek in the Alps the temperature on day 1 is 12 °C and falls 3 °C each day. Write the formula $a_n = a_1 + (n-1)d$ with the correct sign for $d$, then find the temperature on day 6. (Don't be surprised if the answer goes below zero.)
what is $d$? → write formula → put $n = 6$ →
Problem 4 Inequality
Mia's phone starts a horse-video shoot at 80% battery and drops 6% per clip $c$. She needs to keep more than 20%. Write $80 - 6c > 20$, solve it, and say how many whole clips she can film.
$-80$ → isolate $-6c$ → $\div(-6)$ flip →
Problem 5 Sequence
An arithmetic sequence of belt-test scores has $a_1 = 100$ and $a_5 = 72$. The scores drop by the same amount each step. (a) Find the common difference $d$ (it's negative). (b) Then find $a_{10}$.
Hint: from $a_1$ to $a_5$ is 4 equal steps.
$d = (72-100)\div 4$ → then $a_{10} = 100 + 9d$ →
Problem 6 InequalityOpen
At an art market in Lisbon, Mia has €50. Small canvases cost €8 each, and she must keep more than €10 for lunch. Write your own inequality for the number of canvases $c$, solve it (a negative coefficient will appear), and say the most she can buy.
write inequality → isolate the $-8c$ term → $\div(-8)$ flip →
Why this lesson exists: the two skills still marked struggling in the ledger are both negatives — arith-seq-negative-d (negative common difference, missed in 10 & 11) and inequality-negative-coeff (sign-flip, her misconception #3, missed in 07, 11, and again in the 12a remediation). This lesson isolates just those two, visual-first.
Part A watch-point: the sign of $d$, not the exponent. If she writes $d = 4$, point at the downhill line — "the line falls, so the step must be negative." Have her read week 6 off the graph (30) before trusting the formula.
Part B watch-point: the flip is a fact about the number line, not a rule to recite. If she forgets, ask her to place the two answers on a line and check which is really smaller. Problems 4 and 6 look positive ($80-6c$, $50-8c$) but become a negative coefficient once isolated — that's the moment the flip is needed.
The bridge to next lesson: Part A's downhill line is negative slope. When Lesson 17 introduces slope, refer back to this hay graph — a falling line has a negative rate of change.
Clearing the flag: both skills only leave struggling by the mastery criterion (≥90% across ≥5 attempts spanning ≥2 lessons). A good day here isn't enough — they should reappear as review slices inside Lesson 17 onward.
Coming up next → Lesson 17: Slope
You just drew a line falling by −4 every week. Slope is exactly that idea — how steeply a line rises or falls per step — and a line heading downhill has a negative slope. The picture from Part A is the whole next lesson, waiting for you.