Ropes of Different Thickness and Equal Length - Knot

When we tie a knot in a rope we use up a bit of that rope. I took two ropes of different thicknesses and asked my students to guess too low, too high and best guess for how much rope would get used up if we tied a knot.

Groups then collected data to get how many centimeters per knot.

 As a class we took each group rates and averaged them to get a class rate per knot. We decided on the question shown on this whiteboard.

 

And some student work to solve this problem.

 

 

And some of my work to consolidate with the students how to use the equations to solve this problem.

 

 

 

 

 So we got 30 knots and a length of 4 meters when both ropes would have the same length.

This is what it looked like when we tied the 30 knots in each of the ropes.

 

And here is a close up of the lengths.

 

 

 Great discussion at this point about our model and how averaging the entire classes rates made it better. Students loved this.

Of course I could knot resist and we decided to look at figure 8 knots.

It took a little while for groups to figure out figure 8 knots. Data collection from various groups.

The class average.

  And some student work.

 I was knot prepared to do 60 knots.

 

 Knotted up for equal lengths!!!!! 

Volume = Length * Width * Height

This activity allows students to see the relationship between cubic polynomial equations in expanded form (volume form - y intercept form) and factored form  (length width height form - x intercept form)

Students were put in groups of 4 using visible random groupings.

Each group was given one of these equations.
y = x^3+4x^2+5x+2
y = x^3+5x^2+8x+4
y = x^3+5x^2+7x+3
y = x^3+6x^2+11x+6
y = x^3+7x^2+16x+12

Students were then told to get four different colors of cube-a-link blocks, enough of each color to generate each term from x = 1 to x = 4. Here is one groups work to figure out how many blocks they needed.

Groups were then told to make 4 piles of the blocks when x = 1, 2, 3 and 4. Once they had the blocks in piles they were to create the 4 stacks. Constant term on the bottom, then the linear term, and then the x^2 term and then the x^3 term. They looked like this.

The first one is y = x^3+4x^2+5x+2. The second one is y = x^3+5x^2+7x+3. The third one is y = x^3+6x^2+11x+6. Since the stacks are ordered from x = 1 to x = 4, looking at the second one, you see the tops of the black cubes is the graph of y = 3, the tops of the brown cubes is the graph of y = 7x + 3, the tops of the red cubes is y = 5x^2 + 7x + 3 and lastly the tops of the green cubes is y = x^3+5x^2+7x+3. Of course we are only seeing the graphs for the four points when x = 1 to x = 4.

Very cool!

 

Now if you take the x^3 cubes at each stage (x = 1 to x = 4) you can create a cube.

The x^2 cubes will allow you to create the appropriate number of squares.

The x cubes will allow you to create the appropriate number of lines.

And the constant cubes will always be what they are (ones). (points maybe?)

 

For example looking at this chart from before. (which is the middle stack picture above)

I will do it for x = 3. The 27 green cubes makes one cube 3 by 3 by 3. The 45 red cubes makes five 3 by 3 squares. The 21 brown cubes make 7 lines of cubes 3 long. Finally the 3 black cubes make 3 ones. Beautiful.

 

Now a few rules to creating the three dimensional rectangular prisms.

1. The Cube goes in the corner and the Ones can only touch the cube at a point. So basically they are in opposite corners.

2. The Squares must be attached on the faces of the cubes.

 

This is what they create. The first one is the example I have been using. The wooden blocks show it in general. So we end up with:

Volume =Length * Width * Height

x^3+5x^2+7x+3 = (x + 1) (x + 1) (x + 3)

I posed these photos to twitter and got a few responses.