Week 6

Ratios and Ratio Language

  • Comprehend the word “ratio” (in written and spoken language) and the notation (in written language) to refer to an association between quantities.
  • Describe (orally and in writing) associations between quantities using the language “For every of these, there are of those” and “The ratio of these to those is a:b (or a to b).”
  • Draw and label discrete diagrams to represent situations involving ratios.

Student interpretations of a 1:2 ratio:

Area of Irregular Shapes

CCSS.MATH.CONTENT.6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes

Week 3 – Parallelograms

Finding Area of Parallelograms by Decomposing and Rearranging

Learning Goals
  • Calculate the area of a region by decomposing it and rearranging the pieces, and explain (orally and in writing) the solution method.
  • Recognize and explain (orally) that if two figures can be placed one on top one other so that they match up exactly, they must have the same area.
  • Show that area is additive by composing polygons with a given area.

Week 2 – Understanding Area

Students have started to reason about what it means for two shapes to have the same area. We will continue to reason about area, and to learn to talk about mathematics, do mathematics, and use tools to help us.

Pair and Share Activity: Students decide and discuss which shape is covering more of the plane –  trapezoids, rhombuses, or triangles. They discover that counting the shapes does not give the area, and that shapes are composed of other shapes.

Student Task: Create a tiling pattern that is composed of at least wo different shapes, and the same amount of the plane is covered by each type of shape.

24 Task: Draw as many interesting polygons that you can with an area of 24 square units. Observing students working on this task was very informative (Several asked, “What is a polygon?”).  Initially students drew rectangles, but then a triangle appeared, and eventually some lovely and seriously complicated polygons. I asked students to explain one of their drawings in such a way that others would be able to recreate the shape by following the steps. This idea of explaining and verbalizing your thought process is new to many students, and will be a strong focus in this class.

Mathematical Practice Standard #3: Construct viable arguments and critique the reasoning of others

Mathematically proficient students … justify their conclusions, communicate them to others, and respond to the arguments of others. They … distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

… Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. … Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. —CCSS

Children love to talk, but explanation is very difficult for children, often  into middle school. To “construct a viable argument,” let alone understand another’s argument well enough to formulate and articulate a constructive “critique,” depends heavily on a shared context. and a safe environment where their reasoning is valued.

Children learn language, including mathematical language, by producing it as well as by hearing it used. When students are given a suitably challenging task and allowed to work on it together, their natural drive to communicate helps develop the academic language they will need in order to “construct viable arguments and critique the reasoning of others.

Learning to know why you are right, to explain your reasoning, and to disagree with others politely and constructively are skills that develop through practice.  My hope is that, given an interesting task, students can show their method and narrate their demonstration. The key is not the concreteness, but the ability to situate their words in context—to show as well as tell.

Finding Area by Decomposing and Rearranging