```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = FALSE)
library(htmltools)
```
\usepackage{amsmath}
\usepackage{longdiv}
# Introduction: Fractions, Decimals, Percentages.
## Objectives
- To learn.
- To have fun.
- To find real-life applications.
Deal we the why and the how.
```{r, echo=FALSE, out.width='100%', eval=FALSE}
# \'cake.png\',
# \'burger.png\',
# \'icecream.png\',
# \'donut.png\',
# \'fries.png\',
# \'soda.png\'
#
# HTML content for the animation
html <- HTML('
Split Pizza Animation
')
# Save the HTML content to a file
htmltools::save_html(html, "animation.html")
# Include the saved HTML file in the RMarkdown slide
knitr::include_url("animation.html")
```
# 1. Fractions.
## Natural Numbers
Lets recall:
- **Natural Numbers**: Whole numbers starting from $1$ (e.g., $1$, $2$, $3$, $4$).
We use them in every day life to **count** all sort of things...
- Can you name examples?
## When Do We Use Fractions? - Sharing
- **Sharing**: When we need to divide something equally among people.
- Example: Splitting a pizza among friends:
## When Do We Use Fractions? - Spliting
- **Splitting**: When we need to break something into smaller parts.
- Example: Cutting an apple into *quarters*:
## When Do We Use Fractions? - Buy/Sell
- **Buying and Selling**: When we need to measure quantities that are not whole numbers.
- Example: Buying **half** kilogram of gummy bear candy.
## Fractions: 1.1 Formal Definition
- **Fractions**: A way to represent parts of a whole.
- Separated by a diagonal line (more common):
- Thirds $1/3$.
- Halves $1/2$.
- Quarters $1/4$.
- Separated by an horizontal line (more formal).
$$\frac{1}{2}, \frac{2}{3}, \frac{1}{8}, \frac{3}{5}$$
## Fractions: 1.2 Formal Definition
- *Numerator*: The top number represents the number of parts you have or take.
- *Denominator*: The bottom number represents the total number of equal parts the whole is divided into.
- *Example 1*: You buy a medium pizza ($8$ slices), and you take $3$ slices.
How will you represent your share using fraction notation?
## Fractions Examples
$$\frac{3}{8}$$
- *Example 2*: You have a chocolate bar with 12 pieces and you would like to *share it evenly* among 3 friends.
## Fractions Examples
- *Denominator*: The bottom number represents the total number of equal parts the whole is divided into.
$12$
- *Numerator*: The top number represents the number of parts you have or take.
We have $3$ friends, and we want the $12$ split evenly, so...
$$12 \div 3= 3\sqrt{12}=4$$
So in fraction notation:
$$\frac{4}{12}$$
Is this correct? Is it well expressed (fraction notation)?
## Fractions Examples
Picture your chocolate bar...
```{r, results='asis', warning=FALSE}
# Set up plot dimensions
plot(1, type="n", xlim=c(0, 4), ylim=c(0, 3), xlab="", ylab="", xaxt='n', yaxt='n', bty='n')
# Draw horizontal lines
for(i in 0:3) {
lines(c(0, 4), c(i, i))
}
# Draw vertical lines
for(i in 0:4) {
lines(c(i, i), c(0, 3))
}
```
## Fractions Examples
So... the correct answer is the simplified fraction:
$$\frac{4}{12}=\frac{1}{3}$$
## Fractions Activity (Game)
Open the QR:
## Fractions Activity (Game)
[Fractions Activity (Game)](https://phet.colorado.edu/sims/html/build-a-fraction/latest/build-a-fraction_en.html)
# 2. Decimals.
## Decimal Numbers
Lets recall:
- **Decimal Numbers**: Numbers that use a dot (called a decimal point) to show parts of a whole.
(e.g., $1.2$, $2.3$, $3.6$, $4.9$).
The number to the right of the a decimal point is less that a unit:
- $.1$ is less that $1$
- $.9$ is less than $2$
- $2.00...$ is the same as $2$ (trailing zeros.)
We use them in every day life to **measure** all sort of things...
## Counting vs. Measuring
Typically:
- *Counting* to find out whole items there are.
- *Measuring* involves determining the size, amount, or degree of something using a standard unit (like meters, centimeters, inches, etc.)
- We use *decimal* numbers to represent things that we measure.
## When Do We Use Decimals? length
- How tall are you?
$$1.73 \, \text{m}$$
- How far is Mérida from Cancún?
$$309.2 \, \text{km}$$
## When Do We Use Decimals? Temperature
- The freezing point of human blood is actually around:
$$-1.66 \, \text{°C}$$
- The temperature in the summer of Mérida is around:
$$37.5 \, \text{°C}$$
## When Do We Use Decimals? Money
- What is the price of Minecraft: Java & Bedrock Edition Microsoft PC?
$$\$569.99\, \text{MXN}$$
- How much does a Kinder Sorpresa?
$$\$18.50\, \text{MXN}$$
## Decimals: 2.1 Formal Definition
- Decimals is another way to write **rational numbers** used for things that we rather measure.
- **Division**: To convert a fraction to a decimal, divide the numerator by the denominator.
- Example 1: Convert $\frac{1}{2}$ to a decimal.
$$\frac{1}{2} = 1 \div 2 = 0.5$$
## Decimals: 2.1 Formal Definition
## Decimals Examples
How do we do it?
1. Add a zero to the right of the dividend (numerator).
$$2\sqrt{1} \rightarrow 2\sqrt{10}$$
## Decimals Examples
2. Divide: Find the largest integer (quotient) that, when multiplied by the divisor (denominator), is less than or equal to the current dividend.
$$2\sqrt{10}$$
- What about $2 \times 3 = 6$?
- What about $2 \times 4 = 8$?
- What about $2 \times 5 = 10$?
- What about $2 \times 6 = 12$?
2.1 Add a decimal point to the right of quotient:
$$.05$$
## Decimals Examples
3. Multiply the divisor by this integer and write the result below the current dividend.
$$2 \times 5 = 10$$
4. Subtract: Subtract this result from the current dividend to find the remainder.
$$10-10=0$$
**If the remainder is $0$ STOP**
## Decimals Examples
**If the remainder is not $0$ Carry on**
5. Bring Down: Bring down the next digit (or add a zero if there are no more digits) to the right of the remainder.
6. Iterate: Repeat steps 2-5 until the remainder is zero or you have enough decimal places.
## Decimals Examples
- Context: You want to bake a cupcake, according to the recipe, you need $3/8 \, \text{kg}$ of flour for 12 pzs.
- You buy $1 \, \text{kg}$, and you need a scale to measure the flour.
How much flour is $3/8 \, \text{kg}$ (three-eighths) in grams (three decimal units of a kg)?
## Decimals Examples
## Decimals Examples
- Context: You are going on a trip, and you want to leave enough water for your dogs.
- Looking at the container, you see that approximately $2/3 \, \text{lts}$ are gone...
- You are going for 7 days and you need at least 1 lt of water per day.
- If the container is of 20 lts, do you have enough water?
## Decimals Examples
## Decimals: 2.2 Mixed Number
- A mixed number is a combination of a whole number and a fraction.
- It’s like saying you have $x$'s number of whole units and certain remainder...
- The quotient $6$
- The divisor $3$
- The remainder $2$
$$6 \frac{2}{3}$$
# 3. Percentages.
## Percentages.
Lets recall:
- **Definition:** A percentage is a way of expressing a number as a fraction of 100 (base).
- **Symbol:** %
- **Example:** 10% means $10$ out of $100$.
- $100 \%$ represents the whole.
## When Do We Use Percetages? Discounts
- Picture your favorite video game has 20% off.
- If the price is 600 MXN.
How much is the discount, and how much will you pay?
## When Do We Use Percetages? Discounts
- To calculate the discount:
1) Estimate the decimal:
$$20/100=.2$$
2) Estimate the discount
$$.2*600=120.0$$
3) How much will you pay?
$$600 - 120 = 520$$
## Percentages Examples
- Picture, you want to save money for trip at the end of the year.
- You are serious and you are committed to save 15 % of your weekly allowance.
- If your weekly allowance (pocket money) is 1000 MXN, and you started saving since the begging of the year..
How much will you have at the end of the year?
## Percentages Examples
- To calculate the weekly savings:
1) Estimate the decimal:
$$15/100=.15$$
2) Estimate the weekly savings
$$.15 \times 1000=150.00$$
3) How many weeks in a year?
$$365/7=56$$ (approx)
## Percentages Examples
4) What are your savings at the end of the year?
$$150*56=8400$$
## Percentages Activity (Game)
Open the QR:
## Percentages Activity (Game)
[Percentages Activity (Game)](https://www.mathplayground.com/bingo-find-a-percent-of-a-number.html)
## Wrap Up!
- Fractions to represent parts of a whole (proportions).
- Decimals to measure units (time, temperature, length)
- Percentages to represent rational numbers in an friendly way (base of 100%).
## Questions?
- Feel free to ask any questions you have about fractions.
- Let's make sure we all understand before moving on.
## Gaby knows fractions very well.
[Dj Gaby](https://www.youtube.com/shorts/EnaSwtjN6uk)
## How does Gaby manages to mix that?
She knows that she always needs to fit an equal number of beats in a bar.
For instance, the most common signature is $1/4$, meaning 4 beats in one bar.
## Signature & Beats
[Signature, Beats](https://www.youtube.com/shorts/XMfy63r4igI)
## The END
Thank you!
