tutorials.Rmd

---
title: "Math Lessons"
author: "R-Girls"
output: 
   learnr::tutorial:
    css: css/custom.css
runtime: shiny_prerendered

---

```{r setup, include=FALSE}
library(learnr)
library(tidyverse)
library(tibble)
library(fontawesome)
library(astsa)
library(grid)
knitr::opts_chunk$set(echo = FALSE)
```


## Generating Sequences Part 1


### Lesson objectives
- Generating sequences using R code

### Packages used in this lesson:

- `tidyverse`

### Success criteria
* Generate different sequences using R code

### Keywords
* sequence
* term = the numbers in the sequence
* nth term

### Worked Example 1

First we will tell R to generate a sequence.

This is how you generate a sequence of numbers from 1 to 100 where the terms increase by 1.
```{r chunk11, exercise=TRUE}
seq(from=1,to=100,by=1)
```

Use the above example R code in chunk1 to generate the following sequences.  Remember to edit the numbers in the code. 

#### Activity 1 Questions
Q1 Generate a sequence from 1 to 100 where the terms increase by 10

```{r chunk21, exercise=TRUE}
```

Q2 Generate a sequence from 5 to 95 where the terms increase by 5.
```{r chunk31,  exercise=TRUE}
```

Q3 Generate a sequence from 3 to 399 where the terms increase by 4.
```{r chunk41,  exercise=TRUE}
```

Q4 Generate a sequence from 1 to 9 where the terms increase by 3.
```{r chunk51,  exercise=TRUE}
```


### Worked Example 2
Now we will generate a sequence that starts from a number and goes up to the nth term, where the nth term is for example 10.

For example a sequence from 1 where the terms go up by 1 till the 10th term.
```{r chunk61,  exercise=TRUE}
seq(from=1, length.out=10, by=1)
```

Use the above example R code in chunk6 to generate the following sequences.  

Knit your document after each question to check your sequence.

#### Activity 2 Questions
Q1 Generate a sequence from 1 where the terms go up by 4 up to the 100th term

```{r chunk71,  exercise=TRUE}
```

Q2 Generate a sequence from 5 where the terms go up by 10 up to the 89th term
```{r chunk81,  exercise=TRUE}
```

Q3 Generate a sequence from 3 where the terms go up by 3 up to the 33rd term
```{r chunk91, exercise=TRUE}
```


### Worked Example 3
So the terms have increased by a particular number, but we can also decrease by a number and produce a backward sequence.

For example, we can sequence from 100 to 1 by -1

```{r chunk101, exercise=TRUE}
seq(from=100, to=1, by=-1)
```


Use the above example R code in chunk10 to answer the following questions.

Knit your document after each question to check your sequence is correct.

#### Activity 3 Questions

Q1 Generate a sequence that goes down from 50 to 0 where the terms go down by 5.
```{r chunk111, exercise=TRUE}
```

Q2 Generate a sequence from 10 to -10 where the terms go down by 1.
```{r chunk121, exercise=TRUE}
```

Q3 Generate a sequence that goes down from 20 to 10 where the terms go down by 2.
```{r chunk141, exercise=TRUE}
```


## Generating Sequences Part 2

### Lesson objectives
- Investigating sequences

### Packages used in this lesson:

- `tidyverse`

### Success criteria
* Find the relationship between the terms
* Does a number lie in a sequence
* Find a missing term in the sequence

### Keywords
* sequence
* term
* nth term


#### Worked Example 1
This is how you find the difference between the terms in a sequence.

The sequence is 3, 7, 11, 15.

Knit your document to see the sequence generated by the R code in chunk1.

```{r chunk1, exercise=TRUE}
x <- c(3,7,11,15)
diff(x)
```

The R code tells you the difference between each number in the sequence.

##### Activity 1 Questions

Now use the above example R code in chunk1 to find the difference between the terms in the following sequences

Knit the document after each question to see the answer.  Check your answer.

Q1  2,7,12,17,22
```{r chunk2, exercise=TRUE}
```

Q2  3,9,15,21,27
```{r chunk3, exercise=TRUE}
```

Q3 1,4,10,19,31
```{r chunk4, exercise=TRUE}
```


#### Worked Example 2
Now we will see if a number lies in a sequence.

First we will generate the first 20 terms of the sequence 3,7,11,15 and then we will see if the numbers 55 and 56 lie in the sequence. The computer will generate the sequence and if it finds the number it will report it below the sequence.  If it does not find it then it will say numeric (0).

```{r chunk5, exercise=TRUE}
x <- seq(from=3, length.out=20, by=4)
x
x[x==55]
x[x==56]
```

Use the above example R code in chunk5 to answer the following questions.

Remember to tell R how many terms you want it to generate by editing the length.out number. Remember to tell R what numbers to look for in the sequence by editing the x== numbers.


##### Activity 2 Questions
Q1 Generate the first 100 terms of the sequence 3,7,11,15 and see if the numbers 117 and 395 lie in the sequence. Do they?

```{r chunk6, exercise=TRUE}
```


#### Worked Example 3
Now we will find the missing number in a sequence.  Our sequence is 4, ?, 16, 22. Term 2 is missing.  So we generate the sequence based on what we know which is that the difference between 16 and 22 is 6.  The computer will put in the missing term.  

```{r chunk7, exercise=TRUE}
seq(from=4, to=22, by=6)
```
Answer: The missing term is 10.

Use the above example R code to answer the following questions.

Remember to knit to see your results.

##### Activity 3 Questions
Q1 What is the missing term in the sequence 3,12,?,30.

```{r chunk8, exercise=TRUE}
```


## Using Pythagoras' Theorem

### Lesson Objective

Use Pythagoras' Theorem to find the lengths of missing sides of a right angle triangle

### Packages used in this lesson:

- `tidyverse`

### Success criteria

Write the R code to work out the missing side

Calculate the missing side

### Keywords

* Pythagoras theorem
* Length of a side
* Hypotenuse
* square
* square root


#### Worked Example 1: Finding the hypotenuse

First knit the document so that you can see images of the triangles and the formulae you will be working with.

This is a worked example for you to follow.

The formula to find c is: $$c = \sqrt{a^2 + b^2}$$

Check how the formula is written in the output .

Run the R code in chunk1 by clicking on the little arrow to the right of the code chunk. This will draw you the triangle that you are working with below the code chunk.

```{r chunk21, echo=FALSE, exercise=TRUE}
x <- c(0, 0.5, 0)
y <- c(0, 0, 1.5)
plot(x,y,axes=F, ylab='', xlab='', xlim=c(-0.3,2), ylim=c(-0.3,2))
polygon(x, y, col = 'deeppink')
text(0.2,-0.1, "b=14cm")
text(-0.2,0.5, "a=20cm")
text(0.5,0.5, "c")
```

The length of side c is missing.

The R code in chunk2 below works out the missing side c. Click on the little arrow on the right of the chunk to run the code.  

The answer will be shown below the code chunk.

```{r chunk22, exercise=TRUE}
a <- 20
b <- 14
c <- sqrt(a^2+b^2)
c
```

The length of $c = r round(c,2)$ cm

The inline code above tells R to round c to 2 decimal places. Knit the document to check this.

#### Activity1: Your turn

Run R code chunk3. This will draw you the triangle with the length of side c missing.

```{r chunk23, echo=F, exercise=TRUE}
x <- c(0, 0.5, 0.5)
y <- c(0, 0, 1.5)
plot(x,y,axes=F, ylab='', xlab='', xlim=c(-0.3,2), ylim=c(-0.3,2))
polygon(x, y, col = 'deeppink')
text(0.2,-0.1, "b=4cm")
text(0.7,0.5, "a=15cm")
text(0.1,0.5, "c")
```

Write your own code in chunk4 to find length c in the triangle where a = 15 cm and b = 4 cm.

Use the code in chunk2 to help you.

```{r chunk24, exercise=TRUE}
```


#### Worked Example 2

Use Pythagoras' theorem to find one of the other sides.

So find the length of side a when b = 14 cm and c = 25 cm.

Knit the document to see the formula and triangle.

The formula to find the length of side a is: $$a= \sqrt{c^2 - b^2}$$

The code in chunk5 will draw the triangle for you.

```{r chunk25, echo=F, exercise=TRUE}
x <- c(0, 0.8, 0)
y <- c(0, 0, 1.5)
plot(x,y,axes=F, ylab='', xlab='', xlim=c(-0.3,2), ylim=c(-0.3,2))
polygon(x, y, col = 'deeppink')
text(0.4,-0.1, "b=14cm")
text(-0.2,0.5, "a")
text(0.6,0.9, "c=25cm")
```

The code in Chunk6 calculates the length of side a when b and c are known.

Run the code by clicking on the little arrow on the right of the code.

```{r chunk26, exercise=TRUE}
b <- 14
c <- 25
a <- sqrt(c^2-b^2)
a
```

The length of a = r round(a,2) cm.

#### Activity2: Your turn

Use the R code from the above examples to help you find the missing sides in the table below

| a   | b   | c   |
|-----|-----|-----|
| 36  |     | 81  |
|     | 9   | 49  |
| 10  | 10  |     |


Find length b when a=36 and c=81

```{r chunk27, exercise=TRUE}
```

Answer b =

Find length a when b=9 and c=49

```{r chunk28, exercise=TRUE}
a
```



Find length c when a=10 and b=10

```{r chunk29, exercise=TRUE}
c 
```


### Extension

Can you round your answers from code chunks 7, 8 and 9 to 2 decimal places?  Use the space below and knit the document to see if you have been successful.

```{r chunk30, exercise=TRUE}
 
```

## Finding the Equation - Part 1

### Lesson objectives
- Plotting straight line graphs $y=mx+c$

### Packages used in this lesson:

- `tidyverse`

### Success criteria
- Plot a straight line graph $y = mx+c$
- Compare graphs

### Keywords
* slope
* gradient = m
* intercept = c
* x-axis
* y-axis


#### Worked Example 1
This is a worked example for you to follow.

We will show you how to plot the line graph for $y=2x + 5$.


```{r chunk16, exercise=TRUE }
x <- seq(from=-4, to=4, by=1) # sequence the x-axis from -4 to 4
y <- 2*x+5
mydata <- tibble (x,y)
ggplot(mydata) +
  aes(x,y) +
  geom_line (col='red')+
  geom_vline (xintercept = 0, col='black')+
  geom_hline (yintercept = 0, col='black')
```

Now close the image by clicking on the X to the right of the graph

#### Activity1:  
Write your own code to draw the following graphs.

Use code chunk1 from the example above to help you.

1. y = 2x (in code chunk2)
2. y = 3x + 10 (in code chunk3)
3. y = 5x + 2.5 (in code chunk4)


```{r chunk26, exercise=TRUE}
```

```{r chunk36, exercise=TRUE}
```

```{r chunk46, exercise=TRUE}
```

#### Worked Example 2
Drawing more than one line on a graph helps us to compare the lines and see what is the same and what is different.
Here is a worked example for four different lines.

1. y=3x+0 
2. y=3x+1
3. y=3x+2
4. y=3x+3


```{r chunk56, exercise=TRUE}
x <- seq(-4, 4) # sequence from -4 to 4
y1 <- 3*x+0
y2 <- 3*x+1
y3 <- 3*x+2
y4 <- 3*x+3
mydata <- tibble (x,y1,y2,y3,y4)
ggplot(mydata) +
  geom_line (aes(x=x, y=y1), col='red')+
  geom_line (aes(x=x, y=y2), col='blue')+
  geom_line (aes(x=x, y=y3), col='green')+
  geom_line (aes(x=x, y=y4), col='grey') +
  geom_vline (xintercept = 0, col='black')+
  geom_hline (yintercept = 0, col='black')
```
Question:  From the graph what looks the same and what looks different about these lines?

Answer:

#### Activity2:
Draw the following lines on a graph

1. y=1x+5
2. y=2x+5
3. y=3x+5
4. y=4x+5

Use code chunk5 from the example above to help you. Remember to update the R code with the new lines. 

Run the code.  Knit the document.

```{r chunk66, exercise=TRUE}
```

Question: What is the same and what is different about the lines?

```{r chunk66, exercise=TRUE}
```


## Finding the Equation - Part 2

### Lesson objectives
Plotting and finding the equations of straight line graphs $y=mx+c$

### Packages used in this lesson:

- `tidyverse`

### Success criteria
* Plot a straight line graph $y = mx+c$
* Plot straight line graphs $y = mx+c$ with different intercepts
* Plot a straight line graph $y = mx+c$ with a negative gradient
* Find the equations of straight line graphs

### Keywords
* slope
* gradient
* intercept
* x-axis
* y-axis

#### Worked Example 1
This is a worked example for you to follow.

We will show you how to plot the line graph for y=2x + 5.

```{r chunk13, exercise=TRUE}
x <- seq(from=-4, to=4, by=1) # sequence the x-axis from -4 to 4
y <- 2*x+5
mydata <- tibble (x,y)
ggplot(mydata) +
  aes(x,y) +
  geom_point () +
  geom_line (col='red')+
  geom_vline (xintercept = 0, col='black')+
  geom_hline (yintercept = 0, col='black')
```


#### Worked Example 2
We will now see what happens when you change the intercept c to a negative value.

1. y=3x+5
2. y=3x-5

Run the code in chunk2 by clicking on the little arrow on the right of the code chunk.


```{r chunk55, exercise=TRUE}
x <- seq(-4, 4) # sequence from -4 to 4
y1 <- 3*x+5
y2 <- 3*x-5
mydata <- tibble (x,y1,y2,y)
ggplot(mydata) +
  geom_line (aes(x=x, y=y1), col='red')+
  geom_line (aes(x=x, y=y2), col='blue')+
  geom_vline (xintercept = 0, col='black')+
  geom_hline (yintercept = 0, col='black')
```
Question: What is the same and what is different about these lines?


Knit your document and check the output.


#### Activity 1

Draw these lines on a graph.  Use the R code from chunk2 to help you.  Remember to update the code with these new lines.

1. y=x+10
2. y=x-10

```{r chunk33, exercise=TRUE}
```
Question: What is the same and what is different about these lines?



#### Activity 2
Now we will investigate m (the gradient). Draw these lines on a graph. Use R code from chunk2 to help you. 

Run the code and knit the document.

1. y=-2x+5
2. y= 2x+5

```{r chunk43, exercise=TRUE}
```
Question: What is the same and what is different about these lines?



#### Activity 3 Answer the following questions

```{r chunk53, echo=FALSE, exercise=TRUE}
cat ("Q1 In the equation y=mx+c, what happens when you change c?")
cat ("Q2 What happens when you change m?")
```



#### Activity 4: Work out the equation from a line graph

Write down the equations of the following four lines on the graph below.

Knit the document to get a good view of the graph.

```{r chunk63, echo=FALSE, exercise=TRUE}
x <- seq(-10, 10)
y <- x
ggplot() +
  aes(x,y)+
  geom_blank()+
  geom_abline(slope=1, intercept=0, col='red')+
  geom_abline(slope=-1, intercept=2.5, col='cyan')+
  geom_abline(slope=2, intercept=10, col='blue')+
  geom_abline(slope=0, intercept=5, col='orange')+
  geom_vline (xintercept=0, col='black')+
  geom_hline (yintercept=0, col='black')
```


## Representing Data with boxplots


### Lesson objectives

- To understand how to visualise the shape of data using box plots\
- To produce box plots and the related five number summary\
- To interpret box plots\

### Packages used in this lesson:

- `tidyverse`
- `astsa`

### Success criteria

* Draw a box plot using R code
* Produce a five number summary using R code
* Read data from a box plot
* Compare two box plots


### Keywords

box plot, 
minimum, maximum, lower quartile, median, upper quartile, five number summary, inter-quartile range, range


### What is a box plot

A box plot is a way to show the shape or distribution of a set of data.
It shows five useful features of the data, known as the five number summary.

Minimum - the smallest value\ 
Maximum - the largest value\
Median - the middle or 50% value\
Lower quartile - the value half way between the minimum and the median or 25% value\
Upper quartile - the value half way between the median and the maximum or 75% value\

The difference between the upper and lower quartile is known as the inter-quartile range (IQR)

### Worked Example 1

Zaynab keeps a record of her journey times to school each morning. The times are to the nearest minute:

29,21,16,25,21,19,18,30,21,21,12,26,19,21,20,19,30,29,16,21,18,18,27,18,20

Let us draw a box plot and obtain the five number summary and the inter-quartile range (IQR).  We will do this in three steps.

The R code in chunk1 tells R to store all the times into a variable called y and then to use these times to draw a box plot of the data.

```{r chunk43, echo=FALSE, exercise=TRUE}
y <- c(29,21,16,25,21,19,18,30,21,21,12,26,19,21,20,19,30,29,16,21,18,18,27,18,20)
boxplot(y, xlab="Times (mins)", horizontal=T)
```

To help us read values from the box plot it is useful to have gridlines on the plot.\
Here is a simple trick to get gridlines on the plot.

The code in chunk2 tells R to use the numbers it store in variable y to draw a box plot and then to add grid lines.

Click on the little arrow on the right of the code chunk.  The box plot will be drawn below the chunk.

```{r chunk45, echo=FALSE, exercise=TRUE}
# we can calculate the five number summary like this
boxplot(y, xlab="Times (mins)", horizontal=T)
Grid()
boxplot(y, xlab="Times (mins)", horizontal=T, add=T)
```

We can also get R to calculate the five number summary for us.


```{r chunk46, echo=FALSE, exercise=TRUE}
# we can calculate the five number summary like this
summary(y) # five number summary but also provides the mean
IQR(y) # inter-quartile range
```


### Activity 1:

Produce a box plot, the five number summary and IQR of the following data

Aisha records the length of time spent by each member of her class on their last English homework assignment.

The times are recorded to the nearest minute.

85,124,55,140,120,61,95,105,118,180,55,78,130,112,70,126,60,90,115,60,142,100,105,65,100,75


Use the R code in chunk1 to help you draw the box plot.  Remember to replace the example data with the new dataset.

```{r chunk47, echo=FALSE, exercise=TRUE}
```

Now use the code in chunk2 to add grid lines to your box plot

```{r chunk48, echo=FALSE, exercise=TRUE}
```

Now calculate the five number summary using code in chunk3

```{r chunk49, echo=FALSE, exercise=TRUE}
```


### Worked Example 2

Below are the percentage exam marks out of 100 for Maths and English.

Maths (%):    98,79,51,54,62,61,56,87,70,60,93,51,52,54,68

English (%):  37,50,58,45,93,47,47,45,38,61,65,46,97,99,54

Produce two box plots side by side


```{r chunk59, echo=FALSE, exercise=TRUE}
maths <- c(98,79,51,54,62,61,56,87,70,60,93,51,52,54,68)
english <- c(37,50,58,45,93,47,47,45,38,61,65,46,97,99,54)
boxplot(maths, english, xlab="Marks%", names=c("Maths", "English"), horizontal=T)
Grid()
boxplot(maths, english, xlab="Marks%", names=c("Maths", "English"), horizontal=T, add=T)
summary(maths)
summary(english)
IQR(maths)
IQR(english)
```

Write down the five number summaries below

What are the inter-quartile ranges

### Activity 2:

Below are the heights of boys and girls of a similar age.

Boys (cm):    181,157,159,179,186,159,178,162,137,184,140,173,176

Girls (cm):   172,151,176,159,139,179,178,162,134,166,164,172,170

Produce two box plots side by  side with a grid and answer the following questions.

Remember to replace the example data with the new information (boys, girls, heights, data)

```{r chunk67, echo=FALSE, exercise=TRUE}
```

#### Questions

Are the statements below True or False - and explain why

1. The girls are taller on average

2. Half the girls are over 165 cm tall

3. The girls show less spread in height

4. The boys show less spread in height

5. The shortest person is a girl

6. The tallest person is a boy

7. Half the boys are over 172 cm tall

8. Half the girls are under 165cm tall