1718 WB1Z CH10 Lesson 1

• Wiskunde
• Middelbare school
• vwo
• Leerjaar 1

CH10 Equations

Refresh - §10.3: NO calculator
§10.4 and further: calculator

We will skip §10.1

Refresh

Recap:
- Making linear formulas from a graph
- Shortening formulas
- Plotting graphs

Graph --> Table --> Formula

How to make a formula from a linear graph

1) Make a table for the graph (horizontal axis = top row, vertical axis = bottom row), fill in values you can read off exactly (grid points)

Graph --> Table --> Formula

2) Read off the start value from the table
You can find the start value in the bottom row (vertical axis) when the value in the the top row (horizontal axis) is 0.

start  value = 4

Graph --> Table --> Formula

3) Calculate the gradient (how much does the graph rise or fall per horizontal step)
You can calculate the gradient by finding how much the bottom row (vertical axis) increases and dividing this by how much the top row (horizontal axis) increases.

gradient = 0.8

Graph --> Table --> Formula

4) Fill in the formula
All linear formulas have the form:
[Start Value] +/- [Gradient] x [Horizontal Variable] = [Vertical Variable]

start value = 4, gradient = 0.8,
horizontal axis = distance, vertical axis = height
graph is rising, so +

Formula: 4 + 0.8 x distance = height

Formulas

Sometimes formulas are written in a different order.  They all give the same result, because they follow the order of operations.

4 + 0.8 x distance = height
4 + distance x 0.8 = height
0.8 x distance + 4 = height
height = distance x 0.8 + 4
etc.

Writing formulas shorter

The formula 24 - 3 x time = height can be shortened.

When you write a formula shorter you do two things:
- replace the variable with a letter that represents that variable
- leave out the multiplication sign.

Writing formulas shorter

24 - 3 x time = height
- replace the variable with a letter that represents that variable.
24 - 3 x t = h
- leave out the multiplication sign.
24 - 3t = h

Mistakes to watch out for!

- replace the variable with a letter that represents that variable
12 + 3 x time = temperature
12 + 3 x t = t  is WRONG, because you use the letter t for two different variables. Possible ways to shorten the formula are:
12 + 3 x t = T
12 + 3 x t = c                (c for temperature in degrees Celcius)
12 + 3 x h = c               (h for time in hours)
etc.

Mistakes to watch out for!

- replace the variable with a letter that represents that variable
50 - 4 x time = water level
50 - 4 x t = w l  is WRONG, because you use two letters for one variable (water level).
Only use one letter per variable:
50 - 4 x t = w

Mistakes to watch out for!

- leave out the multiplication sign.
12 - a x 3 = t
12 - a3 = t is WRONG! When leaving out the multiplication sign in a x 3, you always write the number before the letter. a x 3 = 3 x a = 3a.

You shorten the formula like this:
12 - 3a = t

Shorten the formula
12 + 3 x time = height

Shorten the formula
price = 50 + time x 10

Shorten the formula
b = a x 15 -2

Shorten the formula
number of yellow beads = number of blue beads x 5

How to plot a graph from a table

Coordinate systems are drawn using a pencil and geotriangle.
Graphs are plotted (= drawn) using a pencil and sometimes a geotriangle.

How to plot a graph from a table

1. Draw a vertical axis using the information in the bottom row. Look at the highest and lowest value and decide which steps to use* and if you need to use a zigzag.

*An axis should be between 5 and 15 cm long. A good way to decide what steps to use is to look at the difference between the highest and lowest value and divide this by 10. Then round off to a whole number. Good steps are 1, 2, 5, 10, and multiples of 10.

Bottom row ranges from 5 to 220
(220 - 5) : 10 = 21.5, so use steps of 20 on the vertical axis.
A zigzag is not needed, because the lowest value is almost 0.

How to plot a graph from a table

2. Draw a horizontal axis using the information in the top row. Look at the highest and lowest value and decide which steps to use* and if you need to use a zigzag.

*An axis should be between 5 and 15 cm long. A good way to decide what steps to use is to look at the difference between the highest and lowest value and divide this by 10. Then round off to a whole number. Good steps are 1, 2, 5, 10, and multiples of 10.

Top row ranges from 0 to 60
60 : 10 = 6, so use steps of 5 on the horizontal axis.
A zigzag is not needed, because the lowest value is 0.

How to plot a graph from a table

3. Label the axes. This means you write down at each axis what the variable for that axis is and what unit it is measured in.

Top row says "time in seconds", so this is written at the horizontal axis.
Bottom row says "speed in miles per hour", so this is written at the vertical axis.

How to plot a graph from a table

4. Plot the points from the table and join them.
If a value is not on one of the lines of the grid paper, measure where you should plot the point.
If the graph is a straight line, use a geotriangle. If the graph is not a straight line, connect the points with a smooth line.

One grid square (1 cm) is 20 miles on the vertical axis, so 10 miles is
5 mm and 5 miles is 2.5 mm.

The graph is not a straight line, so draw a smooth line. You do not use a geotriangle.

How to plot a graph from a formula

- Make a table
- Use the formula to calculate some values
- Follow the steps for plotting a graph from a table

Homework Tuesday

Chapter 10, Refresh: Make and correct:
R1, R2, R3, R4, R5, R6, R7, R8