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Use units to decide what a calculation means and how precise its answer may be, and read an expression's structure before rewriting or factoring it.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you learn to let the units do the thinking: kilometres divided by hours is a speed, and a result is only as precise as the ruler it came from. Then you learn to look at an algebraic expression before touching it — to see $(1 + r)^n$ as one growth factor and $x^2 - 49$ as one square minus another — so that rewriting and factoring become recognition rather than guesswork.
You already divide to find a rate: $150$ km in $3$ hours is $50$ km each hour. You already multiply to scale a recipe up. What is new is treating the units as part of the calculation rather than a label stuck on the end — because when the units come out right, the arithmetic almost always is right too.
Unit: what one of the thing is — a metre, a second, a dollar.
Rate: a quantity per one of something else, written as a fraction of units: km/h, $\$$/kg, mL/min.
Conversion factor: a fraction worth exactly $1$, such as $\frac{1000\ \text{m}}{1\ \text{km}}$, used to change the unit without changing the quantity.
Precision: how finely a measurement was taken. A ruler marked in millimetres gives a reading to the nearest millimetre and no finer.
Units multiply and divide exactly as numbers do. Metres times metres gives square metres, an area. Kilometres divided by hours gives kilometres per hour, a speed. So the unit you want tells you which operation to do: to get $\frac{\text{mL}}{\text{min}}$, millilitres must finish on top and minutes underneath. Converting works the same way — multiply by fractions worth $1$ until the unwanted units cancel. And a result is only as precise as the measurement it came from: a ruler marked in millimetres gives $174$ mm, not $174.000$ mm.
Another way: diagram
A chain of three fractions written side by side — $\frac{90\ \text{km}}{1\ \text{h}}$, $\frac{1000\ \text{m}}{1\ \text{km}}$, $\frac{1\ \text{h}}{3600\ \text{s}}$ — with 'km' struck out top and bottom, and 'h' struck out top and bottom, leaving m over s.
Another way: story
Treat the unit as a piece of algebra. You would cancel an $x$ that appears above and below a fraction without thinking twice; km and h cancel in exactly the same way, and what survives is the answer's unit.
"The units are just a label." They are part of the number. $\frac{\text{km}}{\text{h}}$ and $\frac{\text{m}}{\text{s}}$ are different quantities, and a calculation that ends in the wrong unit has gone wrong somewhere earlier.
"To convert, multiply — or was it divide?" Neither: write the conversion as a fraction worth $1$ and cancel. $\frac{90\ \text{km}}{1\ \text{h}} \times \frac{1000\ \text{m}}{1\ \text{km}} \times \frac{1\ \text{h}}{3600\ \text{s}}$ leaves m/s and there is nothing to remember.
"More decimal places is more accurate." Writing $174.000$ mm from a millimetre ruler claims a thousandth of a millimetre you did not measure. Report to the precision of the tool.
$72$ km is $72 \times 1000 = 72\,000$ m.
Multiply by $\frac{1000\ \text{m}}{1\ \text{km}}$, which is $1$.
One hour is $3600$ s, so the speed is $\frac{72\,000\ \text{m}}{3600\ \text{s}}$.
Multiply by $\frac{1\ \text{h}}{3600\ \text{s}}$.
$72\,000 \div 3600 = 20$, so $72$ km/h is $20$ m/s.
Per person: $300 \div 4 = 75$ g, and the unit is grams per person.
Divide by people to get a rate.
For $10$ people: $75 \times 10 = 750$ g.
Grams per person times people leaves grams.
$54$ km is $54\,000$ m, and one hour is $3600$ s.
$54\,000 \div 3600 = 15$, so the speed is $15$ m/s.
A car travels $400$ km in $5$ hours. Which unit does $400 \div 5$ carry?
A speed of $108$ km/h is how many metres per second?
answer
A recipe for $5$ people uses $265$ g of rice. How many grams of rice are needed for $10$ people?
Answer:
A ruler marked in whole millimetres measures a pencil as $188$ mm. Which statement is honest?
An infusion pump must deliver $301$ mL over $2$ hours. The pump is set in **millilitres per minute**. Which calculation gives the setting?
You already expand $3(x + 2)$ to $3x + 6$ and collect $5x - 2x$ into $3x$. You already know that $7^2 = 49$. This lesson asks you to look before you calculate: to see $x^2 - 49$ as one square minus another square, and $(1 + r)^n$ as one number rather than three symbols.
Term: a piece of an expression separated by $+$ or $-$. In $4x^2 - 7x + 2$ the terms are $4x^2$, $-7x$ and $2$ — the sign travels with the term.
Factor: something being multiplied. In $3x(2x + 3)$ the factors are $3$, $x$ and $(2x + 3)$.
Coefficient: the number multiplying a letter. The coefficient of $x$ in $-7x$ is $-7$.
Difference of squares: an expression of the form $a^2 - b^2$, which always factors as $(a - b)(a + b)$.
An expression can be read at more than one level. In $P(1 + r)^n$ you can stare at $P$, $r$ and $n$ separately, or you can see two factors: the starting amount $P$ and one growth factor $(1 + r)^n$ — and the second reading is the one that tells you what the formula does. The same habit factors things: $x^2 - 49$ is $x^2 - 7^2$, a difference of squares, so it is $(x - 7)(x + 7)$; and $x^4 - y^4$ is $(x^2)^2 - (y^2)^2$, the same pattern with squares in the slots. Rewriting works the other way — expand every bracket, then collect like terms — and the only trap is the sign in front of a bracket, which multiplies everything inside.
Another way: diagram
$x^4 - 16$ drawn as two boxes side by side, the first labelled $(x^2)^2$ and the second $(4)^2$, with an arrow to $(x^2 - 4)(x^2 + 4)$ and a second arrow from $x^2 - 4$ to $(x - 2)(x + 2)$.
Another way: story
Reading an expression is like reading a sentence: you can spell out the letters, or you can see the words. $(1 + r)^n$ is a word. Once you read it as one thing, the formula says 'start with $P$ and multiply by the growth factor'.
"$-2(x - 5)$ is $-2x - 10$." The minus multiplies both terms: $-2 \times (-5) = +10$, so it is $-2x + 10$.
"$x^2 + 49$ factors too." A sum of squares does not factor over the real numbers. Only the difference does.
"$x^4$ has nothing to do with squares." $x^4 = (x^2)^2$. Almost every 'clever' factorisation is an old pattern with bigger pieces in the slots.
Expand the first bracket: $3(x + 2) = 3x + 6$.
Every term inside is multiplied.
Expand the second, sign included: $-2(x - 5) = -2x + 10$.
$-2 \times (-5) = +10$.
Collect: $3x - 2x = x$ and $6 + 10 = 16$, so the answer is $x + 16$.
$49 = 7^2$, so the expression is $x^2 - 7^2$.
Look for two squares.
$a^2 - b^2 = (a - b)(a + b)$ with $a = x$ and $b = 7$: $(x - 7)(x + 7)$.
$36 = 6^2$, so this is $x^2 - 6^2$.
A difference of squares: $x^2 - 36 = (x - 6)(x + 6)$.
A deposit grows as $400(1 + r)^{6}$. How should you read $(1 + r)^{6}$?
Factor $x^2 - 25$. Fill in the two blanks.
$x^2 - 25 = (x - $ a $)(x + $ b $)$
Rewrite $2(x + 6) - 7(x - 3)$ as a single linear expression.
Answer:
How can you see that $x^4 - 256$ factors?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A car travels $280$ km in $4$ hours. Which unit does $280 \div 4$ carry?
A deposit grows as $300(1 + r)^{2}$. How should you read $(1 + r)^{2}$?
You can use units to choose and check a calculation, report an answer to a sensible precision, and read the structure of an expression. Explain why $90$ km/h is $25$ m/s, and why $x^4 - 16$ factors.
8. Your turn: $54$ km/h in metres per second, step 2
20. Your turn: factor $x^2 - 36$, step 2