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Tag Archives: look for and make use of structure

MP6 – Mapping a Parallelogram Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

We continued working on our learning intention: I can map a figure onto itself using transformations.

Perform and describe a [sequence of] transformation[s] that will map parallelogram ABCD onto itself.

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This task also requires students to practice I can look for and make use of structure. What auxiliary objects will be helpful in mapping the parallelogram onto itself?

The student who shared her work drew the diagonals of the parallelogram so that she could use the intersection of the diagonals as the center of rotation.

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Then she rotated the parallelogram 180˚ about that point.

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Could you use only reflections to carry a parallelogram onto itself?

You can. How can you describe the sequence of reflections to carry the parallelogram onto itself?

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How else could you carry a parallelogram onto itself?

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Posted by on September 22, 2016 in Geometry, Rigid Motions

 

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Squares on the Coordinate Grid

I’ve written before about Squares on the Coordinate Grid, an Illustrative Mathematics task using coordinate geometry.

CCSS-M G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

How do you provide opportunities for your students to practice I can look for and make use of structure?

SMP7 #LL2LU Gough-Wilson

How do you draw a square with an area of 2 on the coordinate grid?

It helped some students to start by thinking about what 2 square units looks like, which was easier to see in a non-special rectangle.

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What’s true about the side length of a square with an area of 2?

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How could we arrange 2 square units into a square?

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How do you know the figure is a square? Is it enough for all four sides to be square root of 2?

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CC made his thinking visible by reflecting on his learning after class:

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“Now drawing the square root of two exactly on paper is nearly impossible unless you know how to use right triangles.”

 
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Posted by on September 21, 2016 in Coordinate Geometry, Geometry

 

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Assessing the Centroid of a Triangle

The centroid of a triangle is often called the balancing point of the triangle. It is the point at which the medians of the triangle intersect.

Students used technology to explore the relationship between the vertices of a triangle in the coordinate plane and the vertices of the centroid.

If your students knew the relationship between the vertices of a triangle and the vertices of the centroid, how would you expect them to answer the following question? (I included this question on an end of unit assessment.)

The vertices of a triangle are (a,b–c), (b,c–a), and (c,a–b). Prove that its centroid lies on the x-axis.

A few of my student responses are below.

What learning opportunities could I have provided in class to better prepare my students for this question without just giving them a similar problem?

And so the journey to provide meaningful learning episodes that prepare students to answer questions they haven’t seen before continues …

 
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Posted by on August 22, 2016 in Angles & Triangles, Geometry

 

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MP7: The Diagonal of an Isosceles Trapezoid

 

SMP7 #LL2LU Gough-Wilson

I’ve written about the diagonals of an isosceles trapezoid before.

When we practice “I can look for and make use of structure”, we practice: “contemplate before you calculate”.

We practice: “look before you leap”.

We ask: “what you can you make visible that isn’t yet pictured?”

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We make mistakes; the first auxiliary line we draw isn’t always helpful.

Or sometimes we see more than is helpful to see all at one time.

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We persevere.

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Even with the same auxiliary lines, we don’t always see the same picture.

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We learn from each other.

And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

 
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Posted by on August 16, 2016 in Angles & Triangles, Geometry, Polygons

 

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SMP7 – The Triangle Sum Theorem

G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

How do you provide opportunities for your students to look for and make use of structure? I’m finding that deliberate practice in looking for and making use of structure is making the practice a habit for my students.

SMP7 #LL2LU Gough-Wilson

We ask: “what you can you make visible that isn’t yet pictured?”

We practice: “contemplate before you calculate”.

We practice: “look before you leap”.

We make mistakes; the first auxiliary line we draw isn’t always helpful.

We persevere.

We learn from each other.

 

Months ago, our goal was to prove the Triangle Sum Theorem.

We thought first about what we already knew … what we had already added to our deductive system.

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Then we practiced “I can look for and make use of structure”.

And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

 
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Posted by on July 1, 2016 in Angles & Triangles, Geometry

 

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Making a Better Question Worse

I recently read a post on betterQs from @srcav with an area question from Brilliant that I added to today’s opener.

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I knew something was up when I saw my students’ results.

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My mistyping was a good reminder of the importance of nouns.

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Without showing any results, I sent the corrected question as a Quick Poll.

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(I wonder whether the way the first question was asked prompted the misconception in the wrong answer, but I won’t find out until I have another group of students.)

We are learning to look for and make use of structure.

We are learning to contemplate, then calculate.

And we are learning how to ask better questions, as the journey continues …

 

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What’s My Rule?

We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?

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As students first started looking, I heard some of the following:

  • positive x axis
  • x is positive, y equals 0
  • they come together on (2,0)
  • (?,y*0)
  • when z is on top of w, z is on the positive side on the x axis

 

Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?

 

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As students looked for longer, I heard some of the following:

  • The length of the line segment from the origin to Z is the x coordinate of W.
  • w=((distance of z from origin),0)
  • The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

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And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?

 

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