Two ideas that have been slippery — this time we draw them before we calculate.
This is a slow-down lesson. We are going back to two things that keep tripping us up: writing the formula for a shrinking geometric pattern, and solving an inequality when the number in front of the letter is negative. No new topic — just a clearer picture. Read the two explanations first, then try the six problems.
Part A · The formula for a shrinking pattern
Picture it: each step is half the last
Mia photocopies her horse sketch, and each copy comes out half the height of the one before. The first sketch is 16 cm tall. Look at how the height shrinks:
Each bar is half the bar above it
16
$a_1 = 16$ — copy 1 (shrunk 0 times)
8
$a_2 = 8$ — copy 2 (shrunk 1 time)
4
$a_3 = 4$ — copy 3 (shrunk 2 times)
2
$a_4 = 2$ — copy 4 (shrunk 3 times)
1
$a_5 = 1$ — copy 5 (shrunk 4 times)
Every arrow multiplies by $\tfrac{1}{2}$. That multiplier is the ratio $r$. The big question that keeps catching us is: how many times have we multiplied by $r$? Copy 5 has been shrunk only 4 times — one less than its number. That is why the exponent is $n-1$, never $n$.
Geometric formula: $a_n = a_1 \cdot r^{\,n-1}$ — start value $a_1$, times the ratio $r$, raised to (the term number minus 1).
Worked example — colour-coded so each part is separate
■ $a_1$ = start value■ $r$ = the ratio■ $n-1$ = shrinks so far
Sketch heights: 16, 8, 4, 2, … The start is $\textcolor{#2255cc}{16}$ and each step is $\times \textcolor{#d1495b}{\tfrac{1}{2}}$. Write the formula, then find the 5th copy.
Check against the picture: the 5th bar really is 1 cm. The formula and the drawing agree.
Part B · When you divide by a negative, flip the sign
Picture it: multiplying by −1 is a mirror at 0
An inequality like $-x > 3$ is asking: which numbers, once you flip their sign, land to the right of 3? Multiplying by $-1$ reflects every point across zero — like a mirror standing at 0. When the whole line flips left-to-right, the direction of the arrow must flip too.
Reflect across 0 — the arrow turns around
Top: $-x$ sits to the right of 3. Mirror it across 0 and $x$ sits to the left of −3. The > became <.
The rule: multiplying or dividing an inequality by a negative number reverses its direction ($> \leftrightarrow <$). Adding or subtracting never flips it — only a negative multiply/divide does.
Worked example — Äquivalenzumformung, watch the flip
At a taekwondo endurance drill, Mia's score starts at 40 and drops 6 points each time she stops. She passes if her score stays above 10. For how many stops $s$ does she still pass?
So she still passes with fewer than 5 stops (0, 1, 2, 3 or 4). The parentheses around $(-6)$ are there on purpose — they remind us the divisor is negative, which is exactly when the sign flips.
Practice — start easy, build up
Problem 1 Geometric
Mia draws a row of nested squares, each with half the side length of the one before. The first square has a side of 24 cm. Using the formula given, find the side of the 4th square.
Formula: $a_n = 24 \cdot \left(\tfrac{1}{2}\right)^{n-1}$ where $a_n$ = side length in cm
put $n = 4$ → the exponent is $n-1$ →
Problem 2 Inequality
On a hike down from a mountain hut in Austria, solve for the number $x$: $\;-3x < 12$. Divide by a negative — remember what happens to the sign.
$\div (-3)$ → flip →
Problem 3 Geometric
A poster at a martial-arts expo in Tokyo is reprinted each year at 0.8 of the previous year's height. The first edition is 50 cm tall. Write the formula $a_n = a_1 \cdot r^{n-1}$, then find the height of the 4th edition.
write formula → put $n = 4$ →
Problem 4 Inequality
During a sparring round, Mia's stamina bar starts at 30 and drops 4 points each minute $m$. She stays in the round while the bar is more than 6. Write the inequality $30 - 4m > 6$, solve it, and say how many whole minutes she lasts.
A training ball is dropped in the dojo from 243 cm. Each bounce reaches $\tfrac{2}{3}$ of the height of the bounce before (call the drop height $a_1 = 243$). Write the formula, then find the height of the 5th term in the pattern.
write formula → $n = 5$, exponent 4 →
Problem 6 InequalityOpen
At an art market in Lisbon, Mia has €60. Sketchbooks cost €7 each, and she must keep more than €20 for the bus home. Write your own inequality for the number of sketchbooks $b$ she can buy, solve it, and say the most she can buy.
write inequality → isolate the $-7b$ term → $\div(-7)$ flip →
Show answers
Problem 1
$a_4 = 24 \cdot \left(\tfrac{1}{2}\right)^{3} = 24 \cdot \tfrac{1}{8} = 3$ cm · exponent is $4-1 = 3$
Why this lesson exists: geometric explicit formula (fractional/decimal $r$) and negative-coefficient inequalities were both marked wrong across lessons 09–11 and did not clear in the targeted review. Ledger status: both struggling.
Part A watch-point: the exponent $n-1$, not $n$. If she writes $r^n$, point back to the bar picture — "copy 5 was shrunk only 4 times." Have her count shrinks on the drawing before writing the exponent.
Part B watch-point: the flip only happens on a negative multiply/divide, never on add/subtract. Problems 4 and 6 hide a positive-looking setup ($30 - 4m$, $60 - 7b$) that turns into a negative coefficient once isolated — that is where the flip must appear.
Clearing the flag: these skills only leave struggling by the mastery criterion (≥90% across ≥5 attempts, ≥2 lessons). One good day here is not enough — they reappear as review slices in the next lessons.
Coming up next → Lesson 13: Proportionality
Shrinking by a fixed ratio (Part A) and growing by a fixed rate are two sides of the same idea. Next we look at relationships that keep a constant ratio as they grow — the foundation for slope and graphs later on.