Writing analogy: calligraphy copybooks, applications and books

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The analogy

Recently, I discovered a analogy about mathematical activities: what kind of writing task do you do?

Following 5 Practices for Orchestrating Productive Task-Based Discussions in Science  these categories rise up the demanding of knowledge. And I think that students really “write a book” if they do Project-based learning.

An example of this analogy

I give you an example of this analogy for practicing fractions as operator. I want students to calculate $$\frac{3}{4}$$ of $$16$$.

Calligraphy copybooks

• Activity: “Calculate $$\frac{3}{4}$$ of $$16$$
• Students possible response: “$$\frac{3}{4} \text{ of } 16 = \frac{3 \cdot 16}{4} = 12$$

Aplication

• Activity: “Two farmers want to divide a 4 x 4 plot but the first should have three times surface than the second.

Divide this plot to verify this requeriment"

• Students response:
• understand what is “three times”
• calculate somehow1 that one farmer has 12 squares and other 4 squares.
• draw Book

• Activity: “Two farmers want to divide a 4 x 4 plot but the first should have three times surface than the second. What’s the best way to do it? Consider costs like fencing, buying seeds, irrigation, etc. and crop benefits.”2
• Students responses: ?

Update: I change the book analogy from this:

Can you find three different ways to divide this plot verifying this requeriment?

What is the division which has the minimum cost? (each fencing side has a cost of \$10)?

Can you compare yours with your neighbours’?

Can you find out what is the minimum cost division among all possible divisions?"

to above.

 Cartier, J.L., Smith, M.S., Stein, M.K., and Ross, D.K. 5 practices for orchestrating productive task-based discussions in science. National Council of Teachers of Mathematics, 2013.

 Govern de les Illes Balears. Llibre d’estil. amadip.esment, 2006.

1. Try and failure, $$\frac{3}{4} \cdot 16$$, equations ($$3x+x = 16$$), etc.

2. Does the cost vary if we choose joined (arc-connex) regions than disjoint areas?