"Between-ness"
[b]This is a series of applets drawn from the website mathMINDhabits [[url]sites.google.com/site/mathmindhabits[/url]] that is designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn.[/b] 'Between-ness' and ordering are two sides of the same coin. From a pedagogic perspective, 'between-ness' offers an opportunity for us to pose problems to our students that do not have unique correct answers while simultaneously being well-defined and open-ended.
Table of Contents
- functions
- Parabolas between parabolas
- mean value theorem
- morphing polynomials
- morphing absolute value functions
- whole number operations
- 'between-ness' in addition/subtraction
- 'between-ness' in multiplication/division
- fractions
- Exploring the Ordering of Fractions
- Assessing Understanding of Fraction Ordering
- Between-ness with rational numbers
- shapes
- Square in a triangle
- Taking Sides
- Polygons in polygons
- BOOMERANG !
- triangle homotopy
- ellipse homotopy
- By all means!
- By all means! - averaging 2 numbers
- By all means! - averaging 3 numbers
Parabolas between parabolas
Drag the blue and green dots to form two non-intersecting parabolas - a blue f(x) and a green g(x).[br][br]What are possible expressions for f(x) and g(x)?[br]How can you tell from the expressions that f(x) and g(x) do not intersect?[br][br]Now check the 'build a "between" parabola' check box.[br]Drag the yellow dots to make a parabola p(x) such that f(x) < p(x) < g(x) for all x.[br][br]What is a possible expression for p(x)?[br]How can you tell from the expressions that p(x) is between f(x) and g(x) for all x?[br][br]How many possible functions, p(x), are there? How do you know?[br][br][color=#ff0000]What other questions [could,would] you ask?[/color]
'between-ness' in addition/subtraction
Exploring the Ordering of Fractions
Three fractions are represented as points in the [br]{Denominator, Numerator} plane. You can set the values[br]of these fractions by dragging the blue, yellow and green [br]dots in the left-hand panel.[br][br]Explore how the order of the three fractions on a [br]number line in the right-hand panel is related to the[br]positions of the blue, yellow and green dots in the[br]left-hand panel.[br][br][i][b][color=#ff0000]What questions could/would you pose to your students based on this applet ?[/color][/b][/i]
Square in a triangle
... adapted from Polya [br][br]The challenge is inscribe a square in a triangle - two vertices of the [br]square should lie on the base of the triangle.[br][br]You can vary the triangle by dragging the RED dots.[br]You can vary the inscribed rectangle by dragging the GREEN dot.[br][br]Can you think of a way to use the mean value theorem to show that[br]this can be done? Can it always be done?[br][br]Can you inscribe a cube in a tetrahedron - with one face of the cube[br]lying on base of the tetrahedron ?[br][br][i][b][color=#ff0000]What other questions [could,would] you ask of your students?[/color][/b][/i]
By all means! - averaging 2 numbers
Use the sliders to set the two numbers [br]whose arithmetic and geometric means [br]you wish to find.[br][br]Drag the dots on the coordinate axes to find [br]the geometric mean and the arithmetic mean.[br][br]How would you extend these ideas to the[br]averages of three numbers?[br][br]What other questions [could,would] you ask?