what is our global goal? vision: every student empowered to do rigorous mathematics with confidence....

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What is our global goal?

Vision: Every student empowered to do rigorous mathematics with confidence.

What is our goal here?

Use GeoGebra to help us do rigorous mathematics

Examples in Workshop 2: Rule of Four

1. description 2. algebra 3. graph 4. table

Using GeoGebra in the Classroom

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Techniques & Tools in this Workshop

1. Spreadsheet techniques Heading row

Toggle formulas: Ctrl+Shift+D

2. Rescale the y-axis

3. Label the axes

4. Numbers and sliders


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5. Functions restricted to an interval (math)

6. GeoGebra Command Function[] Input Command

Select Command from Command List

7. Multiplication sign

8. Current value vs. Definition


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9. Spreadsheets - Undefined entries

10. Options -> Labeling

11. Lists with dynamic number of elements

12. Rounding errors

13. Input point

14. Check box for hiding/showing objects


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Problem 1: The Shirt Problem

Statement of Problem: My boss gives me 5 shirts to sell. I make a

$30 profit on each shirt I sell.

What do I need to do: Use the Rule of Four to completely describe

this problem.


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… some math solving …

Questions - Solve each using all four rules.

1. If I sell 2 shirts, what is my profit?

2. If my profit is $120, how many shirts mustI sell?

3. Can I make a profit of $100?


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Problem 1: The Shirt ProblemStatement of Problem: My boss gives me 5 shirts to sell. I make a $30

profit on each shirt I sell.

Rule 1: Description of variables with units, domain of x, rule for y

• Let x= number of shirts sold

• Then x can be: 0,1,2,3,4,5

• Let y= profit in $

• Then relationship of x and y is: y=30*x

No escaping it – this part is

“by hand”!

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Rules 2-4 with GeoGebra

Algebra – Graph - Table

Practice Learned Skills - DIY DIY: Open GeoGebra DIY: Move (0,0) to bottom left corner of DP.

Use or Shift + Click & Drag DIY: Open the Spreadsheet view with 2 columns

Command: View -> Spreadsheet or Ctrl+Shift+S

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Problem 1: The Shirt Problem• Let x= number of shirts sold• Then x can have values: 0,1,2,3,4,5• Let y= my profit in $• Then the relationship of x and y is: y=30*x

Build-in Flexibility DIY: Create the number sp=30 (or ShirtProfit=30).

Click in the Input Bar and then type: sp=30 and hit Enter. This is called “Direct Input”.

Notice: sp is a Free Object and a number.

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Problem 1: The Shirt Problem• This is what you should see.

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Problem 1: The Shirt Problem• Let x= number of shirts sold• Then x can have values: 0,1,2,3,4,5• Let y= my profit in $• Then the relationship of x and y is: y=30*x

• There are exactly 6 points that describe this problem.

• There is no line. There is no function.

• There is only a table of 6 values.

• Table=Spreadsheet!

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Problem 1: The Shirt Problem• Let x= number of shirts sold• Let y= my profit in $

We make Header Row

Click in A1 and type Shirts Sold and hit Enter.

OR type "x=Shirts Sold" and hit Enter.

OR type "x" and hit Enter. If you want to have the x or = you must tell GeoGebra

that it is text by using double quotes.

Click in B1 and type Profit and hit Enter OR type "y=Profit" and hit Enter. OR type "y" and hit Enter.

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Problem 1: The Shirt Problem• Let x= number of shirts sold• Then x can have values: 0,1,2,3,4,5

Let us fill in the x column first. What is the first value of x? DIY: Enter 0 in A2.

Click in A2, type: =0 and hit Enter.

Build in flexibility: We see that the step=1. DIY: Create number for step.

Click in Input Bar, type: step=1 and hit Enter.

0

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Problem 1: The Shirt Problem• Let x= number of shirts sold• Let y= my profit in $• step=1• sp=30 or ShirtProfit=30

Click in A3. What value do we want here? We want to be able to copy this cell entry down so type =A2+step and hit Enter.

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Critical points about spreadsheets– Addresses are always CAPITAL LETTERS

A1 NOT a1

– Spreadsheet objects are AUXILIARY OBJECTS

You can find them using Object Properties

– Ctrl+Shift+D toggles formulas

– To do ANYTHING with cell column• Click on column letter to select.

• Right-click on a highlighted CELL

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We want to copy the formula in A3 to A4:A7 relatively – we copy just like in Excel. Click in A3 – the cell will be highlighted. Find the little blue square and click & drag it down.

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Toggle to see formulas – Ctrl+Shift+D.

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Problem 1: The Shirt Problem• Let x= number of shirts sold• Let y= my profit in $• step=1• sp=30 or ShirtProfit=30

Now we work on the y-column. Click in B2 - we must use our rule. Type =A2*ShirtProfit and hit Enter.

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DIY: Copy the formula B2 to B7 Click in B2 to highlight cell. Click and drag little blue square down to B7.

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Now we make a list of points (x,y) from our data.

GeoGebra will also create the points in the Drawing pad so we get 2 for 1.

We want to show you how smart GeoGebra is.

We will select columns A and B of the spreadsheet.

– To do ANYTHING with cell column• Click on column letter to select.• Right-click on a highlighted CELL

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Problem 1: The Shirt Problem Click on A and drag to B to select both columns. Right-click on a highlighted CELL – NOT on A or B Select "Create list of points" from the menu.

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GeoGebra will "complain" that the data in first row is not numbers, but no worries ...

Just click on OK. This technique is REALLY useful with long lists. Plus – good to remember the trick about

right-clicking on cell (not on column head).

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We see the list list1 at left, but only one point P1 in DP.

The y-values are too big!

Where are the rest of the

points?

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x –axis is a good scale, but we need 150 on the y-axis.

Press Shift and click on the y-axis close to arrow and drag down. This rescales just the y-axis.

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Problem 1: The Shirt Problem This is what you should see.

We want values on the points and not "names".

DIY: Do you remember how to do this?

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Problem 1: The Shirt Problem Right-click on a point and choose "Object Properties". Move this window to side so you can see the action. Find category Points and click on it to select all points. Click on the Basic tab. Click on the down

arrow next to Name on Show label.

Select Value. Click on Close.

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Problem 1: The Shirt Problem Label the Axes and add Units

The x-axis is "Shirts Sold". There is no unit. The y-axis is "Profit". The unit is $.

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Problem 1: The Shirt Problem Right-click in empty space in Drawing pad and

select Drawing Pad from menu.

Click in Label field and type: Shirts Sold

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Click on yAxis tab

Click in Unit field and type: $ Click in Label field and type:

Profit Click on Close

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Problem 1: The Shirt Problem Worksheet is done – save it: shirt_problem.ggb!

Answering the questions

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Answer to Question 1:


Question 1 asks: x=2, y=?

Rule 1: y=30*x

y=30*2

y=60

Answer: My profit is $60.

Rules 2-4: Check this on our GeoGebra worksheet.

No escaping it Rule 1 is “by hand”.

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Answer: $60 since when x=2, y=60.

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2. If my profit is $120, how many shirts must I sell?

Question 2 asks: y=120, x=?

Rule 1: y=30*x

120=30*x

x=4

Answer: I must sell 4 shirts.



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Problem 1: The Shirt Problem2. If my profit is $120, how many shirts do I sell?

Answer: 4 shirts since when y=120, x=4.

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Answer to Question 3: 3. Can I make a profit of $100? Question 3 asks: Is y=100 for some x?

Rule 1: y=30*x 100=30*x x=3.3Answer: No, since x cannot be 3.3.

(x can only be 0, 1, 2, 3, 4 or 5.)



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3. Can I make a profit of $100?

Answer: No. There is no y=100.

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Problem 1: The Shirt ProblemWhy did we define ShirtProfit? Build-in Flexibility

Suppose we want a new problem where ShirtProfit=25 …

We created the number ShirtProfit=30. A slider is just a visible number. Right-click on ShirtProfit

and select Show Object.

It becomes a slider.

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DIY: Change the slider properties to be min=10, max=30 and increment=1

Right-click on slider; choose "Object Properties".

Click on Slider tab and type in new values. Close.

Click and drag slider ShirtProfit to 25. Everything changes properly!

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Problem 1: The Shirt Problem Everything changes properly!

list at left, points in DP, and spreadsheet values

Get your students to make up and answer new questions with this new problem!

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GeoGebra Views

Algebra – Graphing - SpreadsheetThese three aspects of definition are described in

classical philosophy of science under the following headings:

• Intensional Definition – formulas, procedures, programs, rules

• Extensional Definition – graph of all ordered pairs (x, f(x))

• Enumerative Definition – finite sample of cases under concept


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Problem 2: The Speed Problem

Statement of Problem: I drive 5 hours at 30mph.

What do I need to do: Use the Rule of Four to completely describe

this problem.


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… some math solving …


1. How far have I driven after 2 hours and 30 minutes?

2. How long does it take me to drive 100 miles?

3. I drive 30 minutes and then I look at my odometer. I drive another 75 miles? In total how long and how far have I driven??


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Problem 2: The Speed ProblemStatement of Problem: I drive 5 hours at 30mph.

Rule 1: Description of variables with units, domain of x, rule for y

• Let x= time in hours

• Then x can be: [0,5]

• Let y= distance in miles

• Then relationship of x and y is: y=30*x

No escaping it – this part is

“by hand”!

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Problem 2: The Speed Problem• Let x= number of hours • Then x can have values: [0,5]• Let y= distance in miles• Then the relationship of x and y is: y=30*x

Rules 2-4 with GeoGebra

Practice Learned Skills - DIY DIY: Open GeoGebra DIY: Input speed=30 . DIY: Label the x and y axes. Put in units if you want.

Time in hours and Distance in miles.

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DIY: Move (0,0) to bottom left corner of DP. The domain is [0,5]. What is the range? DIY: Use shift and pull the y-axis down to 150 plus. DIY: Open the Spreadsheet with 2 wide columns. DIY: Move DP and Zoom DP and change the size of

the GeoGebra window until you get it to look nice.

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x is defined on an interval – infinitely many valuesThe values (x,y) form a line segment.There is a function, but it is restricted to [0,5].

NEW STEP: Use GeoGebra command FUNCTION[]

Define our function only on [0,5].

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Problem 2: The Speed Problem• A function restricted to an interval

Function[function of x, xmin, xmax]

What are the parts of this command?Notice the brackets! Math commands have ()This command has 3 arguments.Command arguments are separated by commas.Here, the 1st argument is a function of x Here, the 2nd and 3rd arguments are the left and

right endpoints of the interval.

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Problem 2: The Speed Problem• Then x can have values: [0,5]• Then the relationship of x and y is: y=30*x

We could just use: Function[30* x, 0, 5]

Build-in Flexibility We will use speed for 30. DIY: Input xmin=0 and input xmax=5. Input command. Type: fun

The command appears: Use the arrow keys to move between the brackets

and complete function[speed* x, xmin, xmax].

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Problem 2: The Speed Problem• This is what you should see.

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Problem 2: The Speed Problem• Function[speed* x, xmin, xmax]

We get a line segment from (0,0) to (5,150).

x is always the argument of the function here!

Instead of typing fu, we can

use the Command List.

Notice f(x) = 30 x Where is * for multiplication? Where is speed?

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Problem 2: The Speed Problem1. Multiplication sign in GeoGebra

GeoGebra uses a BLANK SPACE for multiplication

We can write * but GeoGebra will replace it!

2. In the Algebra view is the current "value" of f(x) To see the definition of f(x), double-click on this

current value f(x)=30 x.

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• x is defined on an interval – infinitely many values

• There is a function f(x)=30 x

• Here, we can only make a partial table of values.

• NEW STEP: With GeoGebra we can change the number of values in the table dynamically!

• Result: Students understand: Table -> Function

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We make a slider n for the number of values in our table. (You may name this NumberValues.)

We choose 0 as our minimum (no table) and 50 as our maximum (lots of points).

What tool do we need?

Our min=0, our max=50.

What is our increment?

DIY: Make this slider.

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Click on Click in the

Drawing Pad. Fill in data. Click on Apply. Click on Move tool. Click & drag slider to 15.

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Now what is our step? It varies …

step=(xmax-xmin)/NumberValues DIY: Create the number step

We make Header Row

• Let x= number of hours • Let y= distance in miles DIY: A1 and B1 are our heading cells. Create text. In A1, input Time OR "x=Time" OR "x" and hit Enter.

In B1, input Distance OR … and hit Enter.

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Let us fill in the x column first. DIY: Fill in value in A2 and formula in A3. Click in A2 and type =0 and hit Enter.

Click in A3 and type =A2+step and hit Enter.

Our maximum number of table values is 50. DIY: Copy A3 down 50 cells to A52.

Do not worry – we will find a way to only "look" at the correct number of cells (momentarily 15).

But first, let us complete our "old" steps.

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Now for column B DIY: Click in B2. Type in the formula using A2 and

the function f(x) .

In B2 type: =f(A2) and hit Enter. DIY: Copy B2 down 51 cells to B52.

Notice that after B17 – this column has only "?"

It is because our function is restricted to x-values in the interval [0,5] and so undefined for x>5.

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Problem 2: The Speed Problem This is what you should see:

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Problem 2: The Speed Problem Click and drag slider NumberValues to 5 and look at

column B. Drag this slider to 50 and look at column B. Drag it back to 15.

If you want, turn off labeling before making list of points. Options -> Labeling -> No new objects

Make a list of points DIY: Make a list of points using columns A and B.

Click on A and drag to B. Right-click on any cell and select "Create list of points".

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Problem 2: The Speed ProblemNotice that the list contains only those points that

are defined. Drag the slider to 50. The list automatically changes and the points are

automatically added!

Small problem with rounding Drag the slider to 19. Notice there is no end point at x=5. This is

because of the rounding error with 5/19. Define P=(xmax,f(xmax))

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Problem 2: The Speed Problem This is what you should see now:

We will add a couple more things when answering the questions.

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1. How far have I driven after 2hours and 30 minutes?

Question 1 asks: x=2.5 hours, y=?

Rule 1: y=30*x

y=30*2.5

y=75

Answer: I have driven 75 mi.



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Problem 2: The Speed Problem1. How far have I driven after 2hours and 30 minutes?

Notice that x=2.5 is NOT in the table with n=15. Drag our slider left or right until we see 2.5 in the table.

Drag n=10. Now there is a table entry with x=2.5

Hence a point in the DP. Suppose we want to label that point.

How do we find that point and label it? or Suppose we want to find that point algebraically

and not from the points on the table. Can we turn off the points?

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Problem 2: The Speed Problem Suppose we want to label that point.

How do we find that point and label it?

We have n=10 and a table entry with x=2.5 Look at the table entry for x=2.5 – it is A7. Use Move tool,

run your mouse over the points untilyou see the one that says A7.

Problem: "Show label" will show name and not value.

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Problem 2: The Speed Problem DIY: Go to Object Properties and on all the points

change the label type to value.Right-click on any object and choose Object Props.Find the Points category in the left pane.At right, on the Basic tab, first select "Show Label".Then change Name to Value.Then deselect "Show Label".Click on Close.

Now right-click on the point and select "Show Label".

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Problem 2: The Speed Problem NEW STEP: Suppose we want to find that point

algebraically and not from the points on the table.

Can we turn off the points?

Using Check Boxes to show/hide objects• Check Box Tool: (slider menu)

• It is a BOOLEAN object value: true=1 or false=0 (word or number)

• Visible:

Checked = true, Unchecked = false.

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Create Check Box to turn on/off points

Activate Check Box tool

Click in the Drawing Pad (underneath slider) Type in caption (can have spaces)

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Problem 2: The Speed Problem Click on Down arrow to Select objects… and Scroll down to bottom.

So for now, just click on Apply.

• Only one object can be selected at a time and

• Only the "good" points are listed.

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Look in Drawing Pad– you will see Show Points Look in Algebra View– you will see a=true

Momentarily – nothing is "connected" to a. Try selecting and deselecting the Check Box

(use - undo with ) a = true/false

DIY: Make all points "good" by sliding n=50. (use - undo with )

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Let us connect all of the points to a. DIY: Make all points "good" by sliding n=50.

(again: use ) Right-click on any point and choose Object Props. In left pane, select category Point. In right pane, click on Advanced tab. Click in the field: Condition to Show Object. Type: a Click on Close.

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Let us connect all of the points to a.

Use to check and uncheck box.

If a=true (checkbox selected), points will show.

If a=false (checkbox empty), points will not show.

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We are done – save your file: speed_problem.ggb

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Problem 2: The Speed Problem The idea with the variable number of table points

is not just to learn cool things with GeoGebra.Result: Students understand: Table -> Function

• That is, it becomes clearer that If the number of function values is infinite, – then we CHOOSE the number of table values,– the more table values, the more information and– with more and more table values, the more

connected the points are and the more the points look like the function.

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Problem 2: The Speed Problem1. How far have I driven after 2hours and 30

minutes?

Drag n=10. Now there is a table entry with x=2.5 DIY: If you want, label that point or

Find the point on the graph algebraically: DIY: Turn off the points. DIY: Input x=2.5. You will get a vertical line. DIY: Use to find the intersection point of line

and function. DIY: Label this point with its value.

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Answer: I have driven 75 mi.

what is our global goal? vision: every student empowered to do rigorous mathematics with confidence....

Documents

type x

shirt problemlet x

relationship of x

number of shirts soldthen

classroom problem

type y

ylet x

domain of x