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The Treehouse-Projekt - Team Architects
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1. The Story
- The Treehouse- Team A
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2. Well connected
- In little Slices - Team A
- More Accurate - Team A
- A Picture Tells A Thousand Words - Team A
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3. Nice views
- With Good Prospects - Team A
- Cubic Steps - Team A
- Clear View - Team A
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4. New Coat of Whitewash
- lick of paint - Team A
- Picture Perfect - Team A
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5. Scaffold
- Tooth and nail - Team A
- Nail it down - Team A
- Woodwork concrete - Team A
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6. We made it!
- Congratulation Team Architects!
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7. Research & Results
- Qualitative Analysis
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The Treehouse-Projekt - Team Architects
Susanne Digel, Jul 10, 2020

This book gives an experimental approach to the topic of functional relationships. Hands-on material and simulations are combined for a multi-facetted examination.
Table of Contents
- The Story
- The Treehouse- Team A
- Well connected
- In little Slices - Team A
- More Accurate - Team A
- A Picture Tells A Thousand Words - Team A
- Nice views
- With Good Prospects - Team A
- Cubic Steps - Team A
- Clear View - Team A
- New Coat of Whitewash
- lick of paint - Team A
- Picture Perfect - Team A
- Scaffold
- Tooth and nail - Team A
- Nail it down - Team A
- Woodwork concrete - Team A
- We made it!
- Congratulation Team Architects!
- Research & Results
- Qualitative Analysis
The Treehouse- Team A
Welcome students!
Max and Sarah are planning to build a treehouse. They make a plan beforehand and thus ask themselves what material they need.
But they face some problems … even after careful consideration some of their problems remain unsolved.
They need YOUR HELP!
Come with us and join the team of treehouse architects!

In little Slices - Team A
Diameter and Circumference
Sarah and Max have to tie ropes around the tree trunk and several branches. They know the thickness (= diameter) of the branches and trunk at the various points, but still have to find out how long the rope has to be.
Max borrows a tree slice from his grandfather. It is a decoration hanging over the fireplace in grandpa's living room.
It has a diameter of 26 cm and a circumference of 82 cm.

Baumscheibe
When looking at the slice, Max and Sarah reckon with your aid, it should be fairly easy to estimate diameter and circumference.
Take the following material out of the box:

MATERIAL: Six circular discs
Now estimate without measuring which disc (they are numbered) has a circumference of approx. 15 cm.
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Again without measuring, estimate the circumference of disc No. 4.
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With Good Prospects - Team A
Sarah found some cubic boxes (edge length: 40 cm) and imagines a staircaise out of boxes on the outside of the house to get to its roof - creating a lookout!!
How many cubes would be needed?
Sarah hasn't counted the boxes yet but there are definetly more than 10 but less than 20.
Would that be enough?

Cube staircase with 3 steps
Estimate: How many cubes are needed to build a staircase with 6 steps?
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How many steps would a staircase made with 30 cubes have?
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lick of paint - Team A
Max can contribute another find from the cellar - blue paint that his grandpa had filled into an old vessel. The strange shape of the vessel makes it quite difficult to estimate the amount of paint inside.

Vessel with paint
Sarah and Max would like to be able to estimate whether enough paint is left to paint the rest of the treehouse after they are finished with the front wall (including door).
And again they would like to know if enough is left after completing front wall and both side walls.
The front wall is of the same size as the back wall and as both side walls together.
Estimate:
What is the fill height in the vessel when one third, or two third respectively have been used?
Sketch these fill heights in the following simulation "Fill Height" by using the Pen-Tool
, that can be found in the pull down menue which appears when clicking on the arrow
.


Fill Height


Now Max and Sarah can estimate if the remaining paint will suffice.
But it might be more usefull to know this beforehand would be even better!
You can help with the simulation Vessel.
Vessel


Click on the button "+20 ml" in the simulation to fill 20 ml into the vessel.
Observe the fill height in the vessel.
Repeat filling the vessel with 20 ml until it is full.
Write down your first observations.
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How much colour fits into the vessel in total?
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Now click on "Gefäß leeren" to empty the vessel. Fill it up to one third with paint by clicking repeatedly. Then fill it up to two third. Compare the two fill heights in the simulation with your estimations from tast No. 34. How well did you do?
If you were successfull, reveal to Sarah and Max how you did it.
If your guess was not very accurate, try to find out why and describe what went wrong.
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Great! The vessel in the simulation is of the same shape as Grandpa's vessel of paint - but the latter fits double the amount of paint.
Now you can calculate for Sarah and Max whether the amount of paint will suffice.
100 ml paint is needed per squaremeter.
The side walls are rectangular, 1,50 m broad and 2 m high.
Make notes of your calculations.
(what you know: front and back wall have the same size. Both side walls together have the same size as well.)
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Tooth and nail - Team A
Before they start building, Sarah and Max need to keep track of the remaining material.
Nails
Max‘ Dad already bought a package of nail, but it doesn't say how many nails are inside. Only the total weight of the package is given. That makes the think... are there 100 inside, or rather 1000?
Further research is needed! Take the unlabeled package of nails out of the box.

MATERIAL: unlabeled package of nails
Take a close look at it! Now everyone make an estimation on how many nails it contains:
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MATERIAL: 5 packages of counted nails
Take out the counted packages of nails one by one and weigh them in your palm.
Compare the number of nails with their perceived weight.
Write down your observations.
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Which variables correlate in this experiment? Write down a sentence describing the correlation.
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Beam
Som uch for the nails, put the packages aside for the moment.
Max and Sarah are now planning the roof. Max' father suggests a half-timber (see picture below).
He read about it in a DIY guidebook.

Fachwerk
But how to plan and build such a construction?
The easiest kind of half-timber is like a "card house".
Sarah wants to visualise it through a model with matches.

half-timber model out of matches
Estimate without counting how many matches are needed for the half-timber model above.
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Now take the matchbox out of the box and build your own half-timber model WITH ONE STORAGE.
Expand your model to two storages and observe how many matches are needed additionally.
Write down your observations.
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Which variables correlate in this experiment? Write down a sentence describing the correlation.
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Correlation, Correlation, Correlation
Max and Sarah feel somewhat dizzy and are having a hard time keeping the different variables and correlations apart. They need your help.
Sort it out for them:
Look back in your notes what you wrote about the different correlations.
There were discs, cubes, a vessel of paint, nails and beams!
Try to build pairs according to similar correlations and write down what they have in common.
One material does not fit - describe what is different with this correlation.
Which pairs did you come up with?
Congratulation Team Architects!
WOW - that was a hard row to hoe!
Sarah and Max are impressed by your maths skills!
You turned out to be real treehous architects!
Congratulations!!!!


Once the treehouse is ready you will be invited to the house warming party!
Maybe you acquired a taste for it - go ahead and build your own treehouse!
Sarah, Max and the team from the university's student lab "Mathe ist mehr" want to thank you for your great support!



Qualitative Analysis
Summary of Study
Covariation is a major aspect of functional thinking, but not easily accessible for students. Even more it is somewhat underrepresented in mathematics teaching at school compared to the correspondence aspect. Since experiments have proven to be particularly beneficial for functional thinking, two experiment-based approaches have been developed and evaluated with regards to the learning of functional thinking. One approach uses a numerical setting, which emphasises the correspondence aspect. A second, qualitative approach (as presented in this GeoGebra-book) sets the focus on covariation. Previous quantitative studies have shown, that the qualitative setting leads to significantly higher learning gains in FT than the numerical setting (Digel & Roth, 2022).
In a qualitative study we examined the development of the covariation aspect during intervention in the numerical and the qualitative setting respectively in eight focus groups (high-/low-achieving), using the levels of covariational reasoning (Thompson & Carlson, 2017). In accordance with quantitative findings the groups in the qualitative setting show higher levels of covariational reasoning and more consistently dynamic argumentations.
Study Design
In this study the following questions guide the analysis:
(RQ 1) How does the aspect of covariation develop during training?
(RQ 2) Are there differences in the development between the qualitative and the numerical approach?
16 high- and 16 low-performing students in the 6th and 7th grades of two grammar schools were videotaped when participating in the intervention with the numerical and the qualitative learning environments at the mathematical school lab of the University in Landau.
Participation took place in July and August 2021.
They worked in groups of four three times 90 minutes on one morning. All had little or no previous experience with functions in mathematics lessons.
Two groups of four of the low-achievers and two groups of four of the high achievers worked on the numerical setting, the other two groups of low-achievers and high-achievers worked on the qualitative setting. Teachers classified the students as high-/low-performers.
Coded units in qualitative/numeric setting for high-/low-achiever
Timeline of high-achievers qualitative setting
Timeline of high-achievers numeric setting
Timeline of low-achievers qualitative setting
Timeline of low-achievers numeric setting
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