Study Plan D

Every rule this week is the same observation written differently: an exponent is a count of how many times the base is multiplied. Write the factors out and each rule becomes obvious.

Multiplying — add the exponents. x³ · x² = (x·x·x)(x·x) = x⁵. You counted the x’s: 3 + 2.

Dividing — subtract the exponents. x⁵ ÷ x² = (x·x·x·x·x)/(x·x). Two cancel from the top → x³. That is 5 − 2.

Power of a power — multiply. (x³)² = x³ · x³ = x⁶. Two groups of three.

Power of a product — give it to everyone. (2x)³ = 2³x³ = 8x³.

Worked example. Simplify (3x²y)² · x. → 3²x⁴y² · x → 9x⁴⁺¹y² = 9x⁵y².

Oops! You add exponents when multiplying, which feels backwards the first ten times. And the rules only work with the same base: x³ · y² stays x³y², it does not become anything with exponent 5.

Oops! (2x)³ = 8x³, not 2x³. The exponent lands on everything inside the bracket, the 2 included.

These two look like arbitrary facts to memorise. They are not — both fall out of the division rule.

Zero exponent. x³ ÷ x³ is obviously 1 (anything over itself). By the subtract rule it is x³⁻³ = x⁰. Both are right, so x⁰ = 1. (Any base but 0.)

Negative exponent. x² ÷ x⁵ = (x·x)/(x·x·x·x·x) = 1/x³. By the subtract rule it is x²⁻⁵ = x⁻³. So x⁻³ = 1/x³.

Read a negative exponent as an instruction: “this belongs on the other side of the fraction bar.” It does not mean the answer is negative.

Worked example. 2⁻³ = 1/2³ = 1/8. Positive, just small.

Worked example. Simplify x⁵y⁻² → x⁵/y².

Oops! 3⁻² is not −9 and not −6. It is 1/9.

One number times a power of ten, where the first part is at least 1 and less than 10.

Big numbers → positive power. 47,000 = 4.7 × 10⁴ (the point moved 4 places left).

Small numbers → negative power. 0.0032 = 3.2 × 10⁻³ (the point moved 3 places right).

That is where Tuesday’s two topics meet: you need negative exponents to write small numbers at all.

Worked example. Write 0.00081 in scientific notation. Move the point to sit after the 8: 8.1. You moved 4 places right, so the power is −4 → 8.1 × 10⁻⁴.

Check yourself: a number smaller than 1 always has a negative power. If you wrote a positive one, you moved the wrong way.

Oops! 47 × 10³ is not scientific notation even though it equals the right amount — the first part must be between 1 and 10. It should be 4.7 × 10⁴.

The test title covers the whole of Chapter 4, which started before this week. Nothing in this week’s posted rows says what 4.1–4.4 contain — the list below comes from last week’s posted Pre-Algebra rows (14–18 September), so treat it as a reminder rather than as this week’s instructions:

• 4.1–4.2 prime factorisation and GCF — break into primes; GCF takes shared primes at the lower power.
• 4.3 simplifying fractions — divide top and bottom by the GCF.
• 4.4 least common multiple — every prime at the higher power.
• 4.5 and 4.6 exponents — this week, with the homework also naming 4.7.

The check that catches errors: GCF × LCM = the product of the two numbers.

If you have two evenings before the test, spend the first on 4.1–4.4 (the part you have not seen for a week) and the second on 4.5–4.7 (fresh, but more rules to confuse).

Monday hands you three stations and a test two days later. The order they are in is the order to use them:

1. Station 1 — the study guide. Do this honestly first. The questions you have to look up are your actual revision list.
2. Station 2 — Blooket. A game is good at showing you what you have forgotten, fast. Note the ones you got wrong; do not just enjoy the ones you got right.
3. Station 3 — help. Come here with a specific question from station 1 or 2. “I do not get it” is much harder to answer than “why is this one B?”

The evening before the test: cover the answers on the study guide and do it again. Re-reading a completed study guide feels like studying and mostly is not — you recognise your own handwriting, not the content.

Friday is the history of cell theory, which means names and what each one added. The story is short and in order:

• Robert Hooke (1665) — looked at a thin slice of cork through a microscope, saw little empty boxes, and called them “cells” because they reminded him of monks’ rooms. He was looking at dead cell walls.
• Antonie van Leeuwenhoek — better lenses, and the first to see living single-celled things: animalcules moving in pond water (1670s), and later in scrapings from his own teeth (1683).
• Matthias Schleiden (1838) — all plants are made of cells.
• Theodor Schwann (1839) — all animals are made of cells. Put with Schleiden, that is parts 1 and 2 of the theory.
• Rudolf Virchow (1855) — cells come only from other cells. This is the statement that set up the end of spontaneous generation; Pasteur’s experiments a few years later were what actually disproved it.

The three statements of cell theory: (1) all living things are made of cells; (2) the cell is the basic unit of life; (3) all cells come from existing cells.

Oops! Hooke did not discover cells as living things — he named empty boxes in dead cork. Leeuwenhoek was the one who saw life. Keeping those two straight is usually worth a mark.

The posted target names exactly three things, so revise exactly three things.

1. Principles of Bible study. The habits of reading well: look at what the passage actually says before deciding what it means; read a verse in its surrounding paragraph rather than alone; notice who it was written to; ask what it shows about God before asking what it asks of you.

2. Lectio Divina. Latin for “divine reading” — slow re-reading of a short passage, four times, doing something different each time:

1. Read — what does it say?
2. Reflect — which word or phrase stands out?
3. Respond — say something back about it.
4. Rest — be quiet with it.

3. Spiritual disciplines. Practices that make room for God rather than earn anything: reading, prayer, worship, silence, fasting, service, giving, gathering with others. The distinction most likely to be asked: a discipline is a means, not a scorecard.

Oops! Lectio Divina has four steps and they are all verbs starting with R in English — Read, Reflect, Respond, Rest. If you can only remember three, it is usually Rest that goes missing, and it is the one that makes it different from ordinary study.

This is the only piece of work this week with a due date in next week, so it is easy to lose. It asks you to actually try Lectio Divina rather than describe it.

Pick a short passage — five or six verses, not a chapter. Short is the point; the method needs something you can read four times without getting tired of it. A psalm works well.

Then run the four steps and note what happened at each. Writing “nothing stood out” honestly is a real answer.

Wednesday asks you to identify the elements of a plan and then build one. A plan that lasts has three:

• When — a time already anchored to something you always do, not “whenever I can”.
• What — a specific book, not “the Bible”. Mark or a psalm beats Leviticus for a first plan.
• How much — small enough that you can still finish it on a bad day.

Oops! The obstacle is almost never “no time”. It is having no decided time. “Whenever I can” is the plan that quietly fails.

“Macro” means large — these are the nutrients the body needs in large amounts, as opposed to vitamins and minerals (micronutrients). There are three, and this week does one on each posted Wellness day — carbs Monday, protein Tuesday, fats Thursday.

• Carbohydrates (Monday) — the body’s first-choice fuel. Broken down to glucose, which is what your brain and muscles actually run on. Bread, rice, pasta, potatoes, oats, fruit. 4 calories per gram.
• Protein (Tuesday) — the building material. Broken into amino acids and used to build and repair muscle, skin, enzymes, everything. Meat, eggs, fish, beans, lentils, dairy, nuts. 4 calories per gram.
• Fats (Thursday) — stored energy, insulation, and the carrier that lets your body absorb vitamins A, D, E and K. Also what your cell membranes and brain are largely built from. Oils, butter, avocado, nuts, cheese, oily fish. 9 calories per gram — more than double the other two.

The one number worth remembering: fat has 9 calories per gram against 4 for the other two. That is not a reason to avoid fat — it is why fat is such an efficient way for a body to store energy.

Oops! “Carbs are bad” and “fat makes you fat” are both too simple to be true. All three macronutrients are required; the useful questions are how much, and which kind.

Two homeworks this week are literally “bring a food”. Bring something that clearly shows the nutrient, so it works in the activity:

A carb — posted Monday as “bring a carb to class tomorrow”: a slice of bread, a bagel, rice, pasta, oats, a banana or a potato. Clear examples beat clever ones.

A protein — posted Tuesday as “bring a Protein to class tomorrow”: a boiled egg, a tin of tuna, a handful of nuts, beans, cheese or a yoghurt.

Heads up — check with your teacher. The carb instruction is straightforward: bring it to Tuesday’s lesson. The protein one is not, because there is no Wellness row posted for Wednesday — the next posted Wellness class after Tuesday is Thursday. So “tomorrow” either means a lesson that was not posted, or it means the next class. This page does not guess which. Safest move: have the protein with you on Wednesday anyway.

Oops! Most real foods are a mixture. Cheese has protein and a lot of fat; a biscuit has carbs and fat. If you bring something mixed, know which nutrient you are claiming it for and why — that is a good answer, not a wrong one.

The third driving question comes back on every posted Wellness row: what does God say about fuelling my body?

The verse behind it is 1 Corinthians 6:19–20 — “do you not know that your body is a temple of the Holy Spirit…”. The argument is about value, not rules: a temple is cared for because of what it houses.

Which is why the unit is framed as fuelling rather than restricting. The question “what does my body need to do what I am here to do?” is a different question from “what am I allowed to eat?”

The objective names three kinds of drawing. They are a sequence, not three separate skills:

• 1-dimensional — lines and measurements. Length only.
• 2-dimensional — flat views: length and width. This is where the grid does the work, because you can count squares instead of measuring, and every proportion stays honest.
• 3-dimensional — depth added. On a grid the usual trick is to draw the front face, then draw the same shape offset diagonally behind it, then join the matching corners.

Why the grid matters for Tinkercad: Tinkercad works in real units. If your paper sketch has counted squares on it, you already know your measurements and can build the model without guessing.

The posted row says it will be printed “if the design meets specifications for printing” — that condition is part of the assignment. Four things decide it:

• Closed, watertight shapes. Every surface has to join up into one sealed object so the printer knows what is inside and what is outside. (A print can still be hollow inside — that is normal.)
• Overhangs. Plastic cannot be laid on air. Anything leaning out more than about 45° needs support under it — so a design that stands on its own base prints much better.
• Thin parts snap. Fine spikes and thin lettering fail. Thicken anything delicate.
• A flat bottom on the build plate stops the print peeling up as it cools.

Thursday is the presentation. Five beats, roughly 30 seconds each:

1. The brief — what “identity” meant, and what constraints you had.
2. The design — what you made, in one sentence a stranger would understand.
3. The reasoning — why this represents identity. This is the beat that carries the marks on an identity challenge.
4. The change — what did not work and what you did about it.
5. Next iteration — what you would change with one more day.

Oops! Hiding the failure is the usual mistake. “The first version was too thin to print, so I thickened the base” is more impressive than “it worked first time”, and far more believable.

The bell work asks for three things per assignment, and the middle one is where the marks are:

1. Which assignment — just the name and the grade.
2. What went wrong — and this is the part to be specific about. “I did badly” is not a cause. “I did not know the reading had to be finished before class”, “I ran out of time on the last two questions”, “I misread what the prompt asked for” — those are causes, and each one has a different fix.
3. What step or support would help — and it should follow from step 2. Ran out of time → start earlier or check the clock at the halfway point. Misread the prompt → underline the task words before writing. Did not know it was due → check the portal on a fixed day each week.

Oops! A reflection that says “I will try harder” describes an intention, not a change. Anything you could not explain to someone else as a concrete action is probably too vague to work.