A pplets as didactical tools for the learning of algebra

Applets as didactical tools for the learning of algebra

Sonia Palha

Freudenthal Institute

The Netherlands

From: "Familie Boersma" <[email protected]>To: <[email protected]>Subject: Vraagje!Date: Thu, 10 Jul 2003 19:06:18 +0200 Beste mensen,ik en mijn ouders hebben een vraag.ik heb een taak voor wiskunde (ik moet in de vakantie werken aan wiskunde).maar we kwamen ergens niet uit. stel je hebt een stuk grond met een oppevlakte van 600m²de lengte is 10 meter langer dan de breedte.dan krijg je de formule X · (X+10) = 600 of

X² + 10X = 600 Nu weet ik (uit mijn hoofd) dat X 20 moet zijn.Want 20 · 30 = 600Maar hoe kan je dit uitrekenen. Alvast Bedankt, Tjeerd Boersma

Question!

Area 600 m2 and length is 10 m more than width

So X(X+10) = 600

I know by heart that X=20 because 20X30 = 600

But.. How can you calculate this?

Overview for today

• Some things about the Welp-project

• Algebra with applets

• Discussion

• Dynamic, interactive, friendly interface

• Supplied with instructional materials

• Different uses in school practice:

– applying knowledge,

– concept construction,

– help tool,

– practice basic skills,

– assessment,…

Applets small computer programs running over the internet

• Implementation in grades 8 and 9 (13-14 years old)

• With an eye on Longterm Curriculum change of Algebra

• Using Area-model and GEOM2D-applet as a broad basis for

– Building/exploring the formula-area relation

– Variation of exercises

– Structure and dynamics of the formulas

Welp 2002-2004

Geometric algebra 2DGo to the english version of the website www.wisweb.nl .

Choose applets from the menu and look for the applet Geometric algebra 2D.

When you start this one you will see a screen like the one shown here.

You get a rectangle bij dragging the arrows to the screen and put them together as in the figure below.

Task

Calculate 4 times x with the applet. Which expression do you get?

Make a rectangle with area 7y. What are the lengths of the sides?

WELP – student material, december 2002

Task A rectangle with area 24 can have different shapes. Make some of them with the applet and investigate which one has the biggest perimeter and which one has the smallest.

Some possibilities:


Task

The two rectangles below have areas: 12x and 20.

a) Build these two rectangles with the applet and try to make it only one rectangle. What will be the length and height of the new rectangle?

b) Someone says 'a rectangle with area 4 + 16y has the same size as one with area 2(2+8y) and also the same size as one with area 4(1+4y)’

Do you agree? Explain your answer.

Task

a) Build a rectangle with length x + 5 and height x + 4. What is the expression for the area?

b) With the option ‘split’and’release’you can make four rectangles. What are the areas of these four rectangles?

c) Complete(x+4)(x+5) = …..+…..+…...

Solution:

expression



Task

Is it possible to design a rectangle for every expression below? Investigate this, using the applet

x2 + 6x + 5

x2 + 6x + 6

x2 + 6x + 7

x2 + 6x + 8

Explain your answers.


Task

These four rectangles have together a total area of xy + x + y + 1.

a) Join them together into just one big rectangle. What are the length and width of the big rectangle?

b) Complete: xy + 2x + 2y + 4 = (......)(......)You can use the applet if needed.

Area Algebra Start the applet Area Algebra.

You will get the following picture:

For every task, find the right expression that fits the dots. For square type ^ 2

Click OK after finishing the task

Possible activities Make the partial expressions

(simplify)

Make de rectangular expression (factor)

Challenging problems

Expressions with negative numbers

Expressions for other areas (empty)

Geometric algebra problems 1

Task #Expand and simplify the expression by manipulating the rectangle in the figure. You can rotate, mirror, split, join, release, etc….


Start the applet Geometric problems 1


Geometric algebra problems 2

Task #Find one rectangle with the pieces shown in the screen. What product is represented by your rectangle?

You can rotate, mirror, split, join, release the pieces


Start the applet Geometric problems 2


Some conclusions

• Students find applets attractive and motivating;• Flexible, simple to use; it can also be used at home• There is more attention for individual differences

between students• They help to create a learning environment• Allows a dynamical approach to algebra

From: aad [mailto:[email protected]]Sent: vrijdag 11 juli 2003 12:23Subject: RE: Vraagje! Beste Tjeerd, Je vraagt hoe je X kunt uitrekenen als je weet datX · (X+10) = X² + 10X = 45. Ik doet het met een plaatje voor, maar met kleinere getallen:X · (X+4) = X² + 4X = 600. Het plaatje is aangehangen aan deze mail. Je ziet links op het plaatje een rechthoek van X bij X + 4.Die bestaat uit een vierkant van X bij X en een rechthoek van X bij 4.Die rechthoek van X bij 4 wordt doorgeknipt. Nu zijn er twee van X bij 2. Door die twee langs het vierkant van X bij X te leggen,krijg je een groter vierkant, op een ontbrekend stukje na. (Middenin hetplaatje.)Dat missende stukje is een vierkantje van 2 bij 2 en is dus 4. Had je dat stukje wel, dan was de hele boel samen een vierkant van (X+2) bij(X+2)en samen was alles 45 + 4 = 49. (Rechts op het plaatje.)Dus X + 2 moet 7 zijn en X zelf 5. Je kunt het op dezelfde manier doen met jouw vraag! Het plaatje is gemaakt met een programmatje op internet.Ga naar www.wisweb.nlKlik op 'Applets' en zoek dan in de lijst op: "Geometrische Algebra 2D". Succes met je taak voor wiskunde en goede vakantie! Groet,Aad Goddijn

RE: Question!

RE: Question!

45 45 + 4 49

Technology-based tools can enhance student performance when they are integrated into the curriculum and used in accordance with knowledge about learning

White and Frederiksen 1998

But the mere existence of these tools in the classroom provides no guarantee that student learning will improve; they have to be part of a coherent education approach

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A pplets as didactical tools for the learning of algebra

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