All practice pages

Math skills

The algebra the course runs on: solving for x, signs, fractions, proportions, rearranging the exam formulas, percentages and scientific notation. The Math Review, drills with every step shown, and rounds of ten.

Exam 1, Monday 5 October

Chapters 1 to 3. Almost every calculation on it needs one of these: an equation rearranged, a percentage, or scientific notation.

Three rules do all of it. Use the opposite function to move something across. Whatever you do to one side, do to the other. Get the unknown on top and by itself.

Solving for x

  1. The review document’s first example.

    2a=(27 − 3a)5

  2. Expand the right side by multiplying each term: 5 × 27 = 135 and 5 × 3a = 15a.

    2a=135 − 15a

  3. Group like terms. Add 15a to both sides so every a is on the left.

    2a=135 − 15a+15a+15a

  4. Isolate the unknown: divide both sides by its coefficient, the opposite of multiplying by 17.

    17a17=13517

  5. Solve.

    a=7.94

Signs travel with their numbers. −8 + 4 is −4, and a negative divided by a positive is negative. That minus is the easiest point on the page to lose.

Signs and the order of operations

The order

Brackets, then powers, then × and ÷ left to right, then + and − left to right. A fraction bar brackets its top and its bottom.

Two minus signs

Subtracting a negative is adding: 5 − (−3) = 8. Same signs multiply or divide to a positive, different signs to a negative.

A square and a minus

−3² = −9, but (−3)² = 9

A calculator reads −3^2 as −(3²). Type the bracket when the negative itself is squared.

Signs that mean something

ΔT is final minus initial, so cooling is negative. Heat or work into a system is positive, out of it negative. ΔH° is products minus reactants.

Fractions

  1. Adding or subtracting needs a common denominator. The Rydberg bracket for n = 2 to 3:

    12² − 13² = 936 − 436 = 536

  2. Multiplying goes straight across. Dividing by a fraction is multiplying by it flipped.

    23 ÷ 49 = 23 × 94 = 1812 = 32

  3. A fraction in front of the unknown comes off when both sides are multiplied by its flip, which is how the Fahrenheit formula turns into the Celsius one.

    95TC = TF − 32  →  TC = 59(TF − 32)

Proportions

  1. Two equal ratios, like units in like places. In N₂ + 3 H₂ → 2 NH₃, 3 mol of H₂ goes with 2 mol of NH₃. How much NH₃ comes from 4.50 mol of H₂?

    x mol NH₃4.50 mol H₂ = 2 mol NH₃3 mol H₂

  2. Cross-multiply, each top times the other side’s bottom, then divide.

    3x = 4.50 × 2  →  x = 3.00 mol NH₃

Rearranging a formula

Isolate P1, which is already on top

    Isolate T2, which starts on the bottom

      Rearrange with letters first and put the numbers in last. The letters show you what cancels, and one rearrangement works for every set of numbers.

      Squares and square roots

      1. The rms speed of a gas solved for T. A root undoes with a square, so square both sides first.

        urms = √3RTℳ  →  urms² = 3RTℳ

      2. Then clear the bottom and divide off what is left.

        T = urms²ℳ3R

      3. The other way round, a square undoes with a square root, as when the Bohr radius is solved for n. Typed, a root is sqrt( ).

        n = √rZa0

      Percentages

      1. A percentage is the part over the total, times 100.

        percentage=parttotal × 100%

      2. Thirty people in a room, seventeen of them male. The part is 17 males, the total is 30 people, and males are people, so the units cancel.

        17 males30 people × 100%=56.7%

      3. The hard part is naming the total. When a question gives two groups, the total is both of them added together, never the other group on its own.

      Weighted averages

      1. An average atomic mass weights each isotope by how common it is. Boron is 19.9% boron-10 at 10.013 amu and 80.1% boron-11 at 11.009 amu.

        (0.199)(10.013) + (0.801)(11.009) = 10.81 amu

      2. Backwards, call the unknown fraction x; the other isotope is then 1 − x. Multiply out the bracket, gather the x terms, divide.

        x(10.013) + (1 − x)(11.009) = 10.81  →  −0.996x = −0.199  →  x = 0.200

      Final temperature

      1. Heat lost by the hot object is heat gained by the cold one, so one q is minus the other. Each ΔT is the final temperature minus that object’s start.

        m₁c₁(Tf − T₁) = −m₂c₂(Tf − T₂)

      2. Multiply out both sides, gather the Tf terms on one side, and divide by what multiplies them.

        Tf = m₁c₁T₁ + m₂c₂T₂m₁c₁ + m₂c₂

      3. The answer always lands between the two starting temperatures, which is a free check.

      Scientific notation

      Which way the decimal moves

      Move it until one nonzero digit sits in front. Numbers under 1 get a negative exponent, numbers over 1 a positive one.

      0.00165 = 1.65 × 10−3

      On the calculator

      Use the EE or EXP key, which means “times ten to the”. The ^ key is not the same thing. The screen shows it as E.

      6.022E23 = 6.022 × 1023

      Multiply

      Multiply the digit parts, add the exponents, then fix the digit part.

      (3.4 × 106)(4.2 × 103) = 14.28 × 109 = 1.428 × 1010

      Divide

      Divide the digit parts, subtract the bottom exponent from the top.

      (6.4 × 106)/(8.9 × 102) = 0.719 × 104 = 7.2 × 103

      Add or subtract

      Rewrite both with the same power of ten first, then add the digit parts.

      4.215 × 10−2 + 0.032 × 10−2 = 4.247 × 10−2

      Typing it

      Quizzes on Brightspace want the form 2.3x10^-3. Keep every nonzero digit, and the zeros between them.

      6,926,300,000 = 6.9263 × 109

      Graphs

      What goes where

      The variable you change goes on the x axis; the one it affects goes on the y axis.

      Every graph needs

      A title, and each axis labeled with its unit. More than one line means each line is labeled.

      A line of best fit

      Plot points only, then add a linear trendline and show its equation and R² on the chart.