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Math Expression (A+B)^3 for School Teaching

by eashwarps, published

Math Expression (A+B)^3 for School Teaching by eashwarps Sep 5, 2017

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Summary

To verify (A+B)^3 as a mathematical expression based on physical measurement by modeling and 3D printing.

Takeaways post Session (3 Hours)

  1. Apply the knowledge of modeling primitive shapes using TINKERCAD
  2. Utilize physical measurement and modeling technique to other complex mathematical expressions and formulae
  3. Explain the concept of Pascal’s Triangle in an intuitive and novel approach
  4. Make use of technology to simplify the teaching approach for better and quicker understanding amongst students

Print Settings

Printer Brand:

MakerBot

Printer:

Rafts:

Yes

Supports:

No

Resolution:

0.2mm

Infill:

10%


Notes:

No. of Shells - 2

How I Designed This

Modeling

TinkerCAD software was used to design the cubes. The dimensions are provided in the images for reference. The side A was chosen to be 2cm and B was chosen to be 3cm. Rest of the calculations follow as per the equation

(A+B)^3 = A^3+B^3+3A^2B+3B^2A

A^3 and B^3 Model Dimensions

A^2B and B^2A Model Dimensions

Standards

Overview and Background

Project Goal

To verify (A+B)3 as a mathematical expression based on physical measurement by modeling and 3D printing.

Pre-requisites

The prior knowledge of the following concepts would aid for a quicker and faster hands-on session

• Pascal’s Triangle
• Expansion of (A+B)3 = A3+3A2B+3AB2+B3
• Measurement Units
• Google Chrome Web Browser and
• Google Search
• Basic Operating Knowledge of a Computer

Lesson Plan and Activity

Cubes to be placed on the Drawing Desk (Ones without Dimensions) - 5 Minutes

The physical models without any letters engraved on them will be kept on the table for display and the students to be asked to arrange them in the order they wish to.

Take the Students through the TinkerCAD Basic Tutorials

STEP 1 - Open the TinkerCAD homepage https://www.tinkercad.com/
STEP 2 - Click on Sign Up and fill the details
STEP 3 - Enter your email address and create a password (remember to use an easy one)
STEP 4 - Click on the logo on the top right side corner of your portal page and then click on ‘+New’
Step 5 – Click on the Learn Tab in the main dashboard or visit https://www.tinkercad.com/learn/

Learn the basic lessons as per the instructions showcased on the screen

Dimension Sheet for the Cubes to be handed to Students

A print out OR display the dimensions of the cubes to the students. You may use the images attached in the modeling section for the same.

Sample Instructions for Cube with side 2cm

STEP 1 – Create a new project in the TinkerCAD Dashboard.
STEP 2 – Click on Create  3D design and name your design on the top left corner. Keep the naming simple like Cube 1 so as to be able to re-open when you need it.
STEP 3 – Click on Edit Grid (bottom right corner) to determine whether the drawing work plane caters to the model that you would be creating. Adjust the width to 200mm, height to 200mm and unit to 1mm in the edit grid option.
STEP 4 – Drag and Drop a Box in the Basic Shapes available to your right. The Solid indicates that you want a solid object and by clicking on the Hole you can make it to be hollow.Step 5 – Since this is a box, the radius parameter is kept at a value of 0. Click on the small white box near the corner of the box and enter the dimensions for its length and width to be as 20mm respectively.
Step 5 – Since this is a box, the radius parameter is kept at a value of 0. Click on the small white box near the corner of the box and enter the dimensions for its length and width to be as 20mm respectively.
STEP 6 – Click on the small white box determining the height of the box and set the value to be 20mm.
STEP 7 – Click on Export (top right corner) and select everything in the design. Then click on the .STL format to download the file for 3D printing the same.

Click on the TinkerCAD logo on the top left hand corner of the page to go to your projects and create similar 3D designs for the remaining three models as per the drawing sheets placed on your desk.

Teach the Students Basics of 3D Printing

A process for making a physical object from a three-dimensional digital model, typically by laying down many successive thin layers of a material.

Digital Fabrication

  • it takes a model

       └a digital design
    
          └turns it into a real

HOW IS 3D PRINTING CARRIED OUT?

• The starting point for any 3D printing process is a 3D digital model, which can be created using a variety of 3D software programs (like 3D CAD) or scanned with a 3D scanner.
• The model is then ‘sliced’ into layers, thereby converting the design into a file readable by the 3D printer.
• The material processed by the 3D printer is then layered according to the design and the process.

Firing a print of the Models that was created in TinkerCAD in MakerBot Print Software

STEP 1 – Adding a 3D model to the build plate
To open an object to your local computer, open the Project Panel, click Add Models, and navigate to the location of the saved file. Select the file and click Open to import it into MakerBot Print. You can also click File>Insert File to import the object into MakerBot Print. Once a model has been added, it is available to be used on your build plate.
STEP 2 – Model Info
Allows you to change the units that your 3D model is measured in. You can cycle between mm, cm, m, in, and ft.
STEP 3 – Arrange Objects on Build Plate
If there are multiple objects on the build plate, select the Arrange Build Plate to move the objects to the most optimal placement on the build plate. Use the right-click menu to copy, paste, hide, or delete your model from the build plate.
STEP 4 – Print Settings
• Select Extruder Type as Smart Extruder+. This is based on the type of extruder you are using.
• Select Layer Height to 0.2mm (Range is 0.1mm to 0.4mm). This is to set the thickness of your printed object’s individual layers. A lower layer height will result in a smoother surface. An object with a higher layer height will print faster.
• Select the number of Shells to 2 (Range is 1-10). This indicates the number of outlines the extruder prints on each layer before printing the infill. More shells make an object stronger and heavier and increase print times.
• Adjust the Infill percentage to 10% (Range is 1-95%) set the density of your printed object’s internal support structure. A higher percentage will result in a heavier, stronger object.
• Select the Supports checkbox to have your model print with support structures. MakerBot Print will automatically generate supports for any overhanging sections of your object. Supports will be easily removable once you remove your finished object from the build plate.
• Select the Rafts checkbox to have your 3D object printed on a raft. The raft acts as a base for your object and any support structures, and ensures that everything adheres well to the build plate. The raft will be easily removable once you remove your finished object from the build plate.
STEP 5 – Slice Previews
Slices the 3D model into layers so that you can view the path the Smart Extruder+ tool will make while printing the model. Play preview shows you how your 3D model will be printed, before any printing is even done.
STEP 6 – Print or Export for Print
The print option is available at the bottom right corner. It is activated only when MakerBot Print in a computer is connected to a 3D printer via USB, Wifi or Ethernet. The fourth option would be to export the print setting to a file in a USB and plug it in to the printer.

The other models need to be loaded and sliced in the MakerBot print software for printing purpose following similar steps as shared above. Remember in order to print them in different colors, you will have to load them separately. Also the A^2B and B^2A models need to printed in quantities of 3 each. The A^3 and B^3 models need one quantity each.

Theoretical Explanation of the Math Expression with 3D Models

Pascals Trinagle
(a+b)^0 = 1
(a+b)^1 = a^1+b^1 = a+b
(a+b)^2 = a^2+2ab+b^2
(a+b)^3 = a^3+3a^2b+3ab^2+b^3
(a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4

Pascals Triangle Expansion

Stepwise Assembly of Cubes

Extra Knowledge - Golden Ratio 1

Materials Needed

Material Needed

For a group of 30 Students, you may have to split them into a batch of 5 students each. So for each batch you would need one set of cubes without names engraved on them and one more set of cubes with name engraved in them. During the first 75% of the teaching pedagogy you would need only the cubes without the names engraved. After the printing of a cube is initiated, you may collect the cubes (without names) back and then hand over the cubes with names and teach them the mathematical expression and ask them to assemble.

(A+B)^3 Pedagogy 1

(A+B)^3 Pedagogy 2

Skills Learned

  • Logical Reasoning
  • Systematic Planning
  • Interpretation
  • Execution
  • Aesthetic Sense
  • Visualization
  • Observation
  • Problem Solving
  • ICT Literacy
  • Math Skills

Duration of Lesson

3 hours

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