← Back to Module 5
Mathematical Architects · Geometry · Foundations
Get Ready: Module 5 — Making Informed Decisions
Before we build the building, we pour the floor. This page rebuilds the everyday number skills
Module 5 leans on — so when probability shows up, it feels like something you already know.
Everyone starts somewhere. If fractions, percents, or tables felt shaky in the past, that
is completely okay — it just means we lay one brick at a time. Nothing here is timed and nothing here is graded.
Work through one skill, try a problem, peek at the answer, and move on when you feel ready. Skip the
ones you already own. When the readiness check at the bottom feels comfortable, you are ready for the module.
🧱
Skills to build first
Six everyday number skills. Each one is a brick in the floor.
Skill 01
Fractions, decimals & percents (and converting between them)
What it is: three different ways to write the same "part of a whole." \(\tfrac12\),
\(0.5\), and \(50\%\) are all the same amount. Why you need it: a probability is just a part
of a whole, and in this module you will report the same chance three ways — as a fraction, a decimal,
and a percent — so you need to slide between them comfortably.
Worked example — write \(\tfrac{3}{4}\) as a decimal and a percent
1Fraction → decimal: divide the top by the bottom. \(3 \div 4 = 0.75\).
2Decimal → percent: multiply by \(100\) (move the point two places right). \(0.75 \times 100 = 75\%\).
3Percent → fraction (to check): "percent" means "out of 100," so \(75\% = \tfrac{75}{100} = \tfrac34\). Back where we started.
So \(\tfrac34 = 0.75 = 75\%\). One amount, three outfits.
Try it
- Write \(\tfrac25\) as a decimal and as a percent.
Show answer
Divide: \(2 \div 5 = 0.4\). Multiply by 100: \(0.4 \times 100 = 40\%\).
Answer: \(\tfrac25 = 0.4 = 40\%\).
- Write \(20\%\) as a decimal and as a fraction in simplest form.
Show answer
Percent → decimal: \(20\% = 20 \div 100 = 0.20\). Percent → fraction: \(20\% = \tfrac{20}{100}\).
Simplify by dividing top and bottom by 20: \(\tfrac{20}{100} = \tfrac{1}{5}\).
Answer: \(20\% = 0.2 = \tfrac15\).
Skill 02
Ratios
What it is: a ratio compares two amounts — "3 reds to 2 blues," written \(3:2\).
Why you need it: probability is built from ratios. "Favorable outcomes to total outcomes"
is exactly a ratio, and you will turn ratios into fractions all module long.
Worked example — a jar has 3 red and 2 blue marbles
1Red to blue: there are 3 reds and 2 blues, so the ratio is \(3:2\).
2Find the whole: total marbles \(= 3 + 2 = 5\).
3Red to total (as a fraction): \(\tfrac{3}{5}\) of the marbles are red.
Red:blue is \(3:2\); red is \(\tfrac35\) of the whole jar.
Try it
- A class has 12 girls and 8 boys. Write the ratio of boys to girls, then simplify it.
Show answer
Boys to girls is \(8:12\). Divide both by 4: \(8:12 = 2:3\).
Answer: \(2:3\).
- In the same class (12 girls, 8 boys), what fraction of the class is boys?
Show answer
Total \(= 12 + 8 = 20\). Boys \(= 8\), so the fraction is \(\tfrac{8}{20}\).
Simplify by dividing by 4: \(\tfrac{8}{20} = \tfrac{2}{5}\).
Answer: \(\tfrac{2}{5}\) of the class is boys.
Skill 03
Part-to-whole reasoning
What it is: seeing a quantity as a part sitting inside a whole, and writing
it as \(\tfrac{\text{part}}{\text{whole}}\). Why you need it: every probability is a
part-to-whole fraction — the favorable outcomes (part) over all possible outcomes (the whole). Get this
habit and probability stops feeling new.
Worked example — 7 of 20 students walk to school
1Name the whole: the whole is all 20 students.
2Name the part: the part is the 7 who walk.
3Write part over whole: \(\tfrac{\text{part}}{\text{whole}} = \tfrac{7}{20}\), which is \(0.35 = 35\%\).
\(\tfrac{7}{20}\) of the students walk — that is 35% of the class.
Try it
- A bag has 9 sour candies out of 30 candies total. What fraction (in simplest form) is sour?
Show answer
Part over whole: \(\tfrac{9}{30}\). Divide top and bottom by 3: \(\tfrac{9}{30} = \tfrac{3}{10}\).
Answer: \(\tfrac{3}{10}\) (which is \(0.3 = 30\%\)).
- If \(\tfrac{3}{10}\) of the candies are sour, what fraction is not sour?
Show answer
The whole is \(1\) (or \(\tfrac{10}{10}\)). Subtract the part: \(\tfrac{10}{10} - \tfrac{3}{10} = \tfrac{7}{10}\).
Answer: \(\tfrac{7}{10}\) are not sour. (The part and its leftover always add to the whole.)
Skill 04
Reading tables
What it is: finding the right number by lining up a row with a column, and
using the totals along the edges. Why you need it: Module 5 lives in two-way tables.
If you can find a cell and read a total, you can compute every probability the table asks for.
Worked example — read this survey table
Students surveyed about liking pizza:
|
Likes pizza |
Doesn't |
Total |
| 6th grade |
8 |
2 |
10 |
| 7th grade |
12 |
3 |
15 |
| Total |
20 |
5 |
25 |
1Find a cell: "7th graders who like pizza" → go down the 7th grade row, across to the Likes pizza column. They meet at 12.
2Read a total: the bottom-right corner, 25, is the grand total — everyone surveyed.
3Check it makes sense: each row adds across (\(8+2=10\)) and each column adds down (\(8+12=20\)). The edges should agree.
Cells live where a row meets a column; the edges hold the totals.
Try it (use the table above)
- How many 6th graders do not like pizza?
Show answer
Go down the 6th grade row, across to the Doesn't column. They meet at 2.
Answer: 2 students.
- How many students total like pizza (both grades)?
Show answer
Read the bottom of the Likes pizza column — that is the column total, 20. (Check: \(8 + 12 = 20\).)
Answer: 20 students.
Skill 05
Basic counting
What it is: carefully counting how many possible outcomes there are — sometimes by
listing them, sometimes by multiplying. Why you need it: before you can find a probability you
must know the size of the whole. Counting the outcomes is finding the denominator.
Worked example — how many outcomes when you flip 2 coins?
1List the first coin: Heads (H) or Tails (T) — 2 choices.
2Pair each with the second coin: HH, HT, TH, TT. Writing them out, there are 4.
3Shortcut (multiply): 2 choices for coin one \(\times\) 2 choices for coin two \(= 2 \times 2 = 4\) outcomes. The list and the multiplication agree.
There are 4 equally-likely outcomes: HH, HT, TH, TT.
Try it
- You roll one 6-sided die. List all the outcomes — how many are there?
Show answer
The outcomes are 1, 2, 3, 4, 5, 6.
Answer: 6 outcomes.
- A lunch combo lets you pick 1 of 3 drinks and 1 of 2 sandwiches. How many different combos are possible?
Show answer
Multiply the choices: \(3 \text{ drinks} \times 2 \text{ sandwiches} = 6\).
Answer: 6 combos. (You could also list them to be sure.)