Applets as didactical tools for the learning of algebra

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1 Applets as didactical tools for the learning of algebra
Sonia Palha Freudenthal Institute The Netherlands

2 Question! Area 600 m2 and length is 10 m more than width
From: "Familie Boersma" To: Subject: Vraagje! Date: Thu, 10 Jul :06: 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 600m² de 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 = 600 Maar 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?

3 Overview for today Some things about the Welp-project
Algebra with applets Discussion

4 Applets small computer programs running over the internet
Dynamic, interactive, friendly interface Supplied with instructional materials Different uses in school practice: applying knowledge, concept construction, help tool, practice basic skills, assessment,…

5 Welp 2002-2004 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

6 Geometric algebra 2D Task
Go to the english version of the website . 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?

7 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:

8 WELP – student material, december 2002
Task The two rectangles below have areas: 12x and 20. 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? Someone says 'a rectangle with area y 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.

9 WELP – student material, december 2002
Solution: Task Build a rectangle with length x + 5 and height x What is the expression for the area? With the option ‘split’and’release’you can make four rectangles. What are the areas of these four rectangles? Complete (x+4)(x+5) = …..+…..+…... expression

10 WELP – student material, december 2002
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.

11 WELP – student material, december 2002
Task These four rectangles have together a total area of xy + x + y + 1. Join them together into just one big rectangle. What are the length and width of the big rectangle? Complete: xy + 2x + 2y + 4 = (......)(......) You can use the applet if needed.

12 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)

13 Geometric algebra problems 1
Start the applet Geometric problems 1 You will get the following picture: Task # Expand and simplify the expression by manipulating the rectangle in the figure. You can rotate, mirror, split, join, release, etc….  Click OK after finishing the task

14 Geometric algebra problems 2
Start the applet Geometric problems 2 You will get the following picture: 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 Click OK after finishing the task  

15 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

16 RE: Question! From: aad [mailto:A.Goddijn@fi.uu.nl]
Sent: vrijdag 11 juli :23 Subject: RE: Vraagje! Beste Tjeerd, Je vraagt hoe je X kunt uitrekenen als je weet dat X · (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 het plaatje.) 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 = 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 Klik 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!

17 RE: Question!

18 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 - The end -