Interaction Design: 4 Approaches to Balancing Whole Number Expressions

Mathematical Model:

A scale measures equality between two weights, represented by mathematical expressions. By adding opposites that sum to zero, expressions can be simplified. The balance responds to the relative weights in the virtual manipulative.

Problem Statement:

What is the best user experience that stays true to the mathematics, and is engaging to interact with?

1) MathPlayground (http://www.mathplayground.com/AlgebraEquations.html) offers a solution to the problem of Solving Whole Number Linear Equations (Fig. 1).

Fig. 1 Mathplayground's Algebra Equations Practice Mode

I like:

·      Realistic Functionality - balance responds to inequality

·      There is exactness to the equality of the symbols on each side

·      There is a symbolic representation of the equation
·      Tiles are in a convenient place to grab

I wish:           

·      You could scale the number of tiles you were adding/subtracting, and could have more than 20 tiles on a side

·      The game had multiple representations

o   Table

o   Graph

o   Abstract symbolic

·      That other representations could be chosen to work with and automatically updated when a move was made 
·      That the manipulatives were made into fun items that were less abstract
·      That tutorial mode was not the default user first experience, it is much more restrictive than the practice version of the manipulative. 

2) Utah State University's NLVM (http://nlvm.usu.edu/en/nav/category_g_3_t_2.html) has Algebra Balance Scales (and a Negative Integer version in Fig. 2)

Fig. 2 NLVM's Algebra Balance Scales - Negatives

I like:

·      Realistic functionality - balance responds to inequality

·      Exactness of symbols - equivalent on each side

·      Symbolic representation of the equation
·      Symbolic manipulation of the equation
·      Tiles are in a convenient place to grab, and they drop easily

I wish:           

·      Other forms of manipulation other than symbolic (The objects were still moveable after being placed on the scale)
·      You could scale the number of tiles you were adding/subtracting, and were able to have more than 10 of a given object on a side

·      The game had multiple representations

o   Table

o   Graph

o   Abstract symbolic

·      That other representations could be chosen to work with and automatically updated when a move was made 
·      That the manipulatives were made into fun items that were less abstract

3) DragonBox Algebra (http://www.dragonboxapp.com/) has a gamified approach to teaching the procedures for solving linear equations. (Fig 3)

Fig. 3 DragonBox's Algebra Equations Progression

I like:

·      The graphics are great and the functionality is intuitive
·      The symbols and operations are consistent
·      The way anti-objects are implemented, and the scaffolding pictures are slowly removed

·      The gamification elements make it an addicting experience
·      That a child will learn all of the procedures for solving linear equations without thinking they are doing math

I wish:

·      There was a relation to the real world, like the balancing of weight

·      There was a conceptual underpinning to things like dividing every term by a value
·      The game had multiple representations

o   Table

o   Graph

o   Abstract symbolic

·      That other representations could be chosen to work with and automatically updated when a move was made 

4)  Curtis Wang designed a Whole Number Linear Equation lesson that can be extended as a game concept. (Fig. 4)

Fig. 4 Curtis Wang's Solving Whole Number Linear Equation Lesson

I like:

·      The graphics and friendly design

·      The animations in the original version (https://app.box.com/s/204p4g5wnk3fr4n2pf5a)
·      That this could be more fun to play than the abstracted form

I wish:           

·      There was exactness to the symbols on each side

·      There wasn’t a change in the moment of inertia when placing objects

·      There were anti-objects like helium balloons to remove weight with

·      We could incorporate scaling (Whole number multiplication/Division)

Neighborhood Boundary Project

I tried to use some Buck Institute Project-Based Learning materials and methods for a few days with Algebra Project Cohort students.  If you are not familiar with their free DIY guide, I recommend it.
http://www.bie.org/diy/

Fig. 1

This came after we spent a good deal of time with students on the idea that when "solving" a linear inequality in one variable, we were putting an equals sign in place of the inequality symbol in order to identify the boundary points (Fig 1). After finding the boundary points we were then able to find the solution set by testing left/right of the boundary point on the number line.

The Neighborhood Boundary Project was an attempt to extend the idea for two variables.

Hook

The original hook involved overlaying multiple definitions of our neighborhood (zip codes, city council districts, community member-made street line drawing) and asking who was right?

Who decides the neighborhood boundary lines?

Fig. 2 Neighborhood Boundary deck. See the .key for the actual animation

Fig. 3 Eagle Rock product

Purpose

The idea was to teach a geometric interpretation of linear inequalities as boundary lines with a shaded region above/below the boundary line in the y-direction or left/right of the boundary line in the x-direction.

Students had to define their own system of linear inequalities such that the overlapping solution areas would generate the community they were assigned to.

Fig. 4 Highland Park product

Learning Outcomes

Students did a good job of explaining why they chose particular inequality symbols to get the correct shading. However, not all students were able to produce the symbolic representations of the boundary lines, which is the major skill on standardized tests. Therefore, in future iterations it would be important to provide an opportunity for students to practice and repeat this skill in a way that does not detract from the engagement of the project.

Extensions

We did not put together a composite graph due to time constraints, even though the students were all using the same coordinate system.

I would have loved to have students argue about the boundary lines that were defining adjacent regions. We alluded to gerrymandering of district lines, and this could have made that idea more relevant.

There need to be other ways to get students reflecting on their abstract symbolic inequalities.

Grab the files and adapt as you please. Let me know how it goes.
https://app.box.com/s/km6sgf2h1oybrc4fp1hw