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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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The Area of a Trapezoid: Differentiating Success Criteria … Not Learning Intentions

I am enjoying our slow book chat on Dylan Wiliam’s Embedding Formative Assessment. (You can download the first chapter here, if you are interested.)

Chapter 3 is called Strategy 1: Clarifying Sharing, and Understanding Learning Intentions

How do we support students who need scaffolding while at the same time pushing students who need a bigger challenge?

I struggle with differentiation. But as we focus more on mathematical flexibility, I am learning to understand what Wiliam means by differentiating success criteria instead of learning intentions.

Consider this learning progression on mathematical flexibility from Jill Gough.

Flexibility #LL2LU GoughWhat if we pair that with a content learning progression on the area of trapezoids?

4: I can prove the formula for the area of a trapezoid more than one way.

3: I can prove the formula for the area of a trapezoid.

2: I can calculate the area of a trapezoid by composing it into a rectangle and/or decomposing it into triangles and other figures.

1: I can calculate the area of a trapezoid using the formula.

Our practice standard for this lesson is “I can look for and make use of structure”.

SMP7 #LL2LU Gough-Wilson

Wiliam says that there are 13 conceptually different ways to find the area of a trapezoid. Some of them are more challenging algebraically than others. Some of them are more challenging geometrically than others.

How many ways can you prove the formula for the area of a trapezoid?

How might we use this exercise to differentiate success criteria for our learners?

I got to try this with 6th-12th grade teachers in a recent Mississippi Department of Education geometry institute. In our Geometric Measure and Dimension session we moved from areas of special quadrilaterals in the coordinate plane to proving the area formulas for a kite and a rhombus. Then we proved the area formula for a trapezoid. We had some teachers for whom it was a challenge to generalize the height of the trapezoid as h and the bases as b1 and b2 instead of using numbers to represent the lengths.

(1&2)

The first instinct for many teachers was to either compose the trapezoid into a rectangle with dimensions b2 × h and subtract the areas of the two extra right triangles

Or to decompose the trapezoid into a rectangle with dimensions b1 × h and add the areas of the two right triangles.

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The algebra can be challenging, especially when deciding how to represent the lengths of the bases of the triangles. Will you call one of them x and the other b2x – b1? Or will you recognize that together, the bases have a sum of b2 – b1?

(3)

One of the least instinctive methods in the 200+ teachers in my sessions was to decompose the trapezoid into two triangles using a diagonal. It is also one of the most accessible methods algebraically. A few times I asked a teacher who was stuck what would happen if you drew one diagonal. Then I walked away. I almost always came back later to a successful proof.

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How might we use this exercise to differentiate success criteria for our learners?

(4)

Once they were successful with decomposing into two triangles, they were ready to consider decomposing into three triangles. A few teachers breezed through the algebra and were ready for another challenge. (We noted the freedom to connect the endpoints of b1 to a point on b2 that partitions b2 into any ratio, 1:1 or 1:2 or 1:x.)

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(5)

Some decomposed the trapezoid into a parallelogram and a triangle.

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(6)

Some used rigid motions to make sense of the area of the trapezoid, rotating the trapezoid 180˚ about the midpoint of one of its legs, creating a parallelogram with base b1 + b2 and height h. For others, rigid motions was a challenge. They asked for scissors so that they could cut out trapezoids and physically translate and rotate them.

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(7)

Others decomposed the trapezoid into two trapezoids using the median, and then rearranging the top trapezoid into pieces to form a parallelogram with base b1 + b2 and height ½h.

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(8)

Or a rectangle with the same dimensions.

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(9)

A few used the median to create the “average rectangle” with area equal to the trapezoid.

Or the “average parallelogram” with area equal to the trapezoid.

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(10)

One decomposed the trapezoid by constructing a segment from one endpoint of b1 to the midpoint of the other leg, and then rearranging the triangle formed to make the trapezoid into a triangle with base b1 + b2 and height h.

Another did the same from one endpoint of b2.

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(11)

I asked those who finished quickly what would happen if they extended the legs of the trapezoid to form a triangle. It took a lot of algebra for them to prove the area of a trapezoid using similar triangle relationships but once they started, they wouldn’t stop.

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I think that these would be considered 11 conceptually different methods for proving the area of a trapezoid. I can’t remember that anyone found 2 others, and I’m sure there’s a site out there somewhere that I can find two more ways. But I’m not going to succumb to Google yet. I’m going to continue working on my mathematical flexibility, and I’m going to keep practicing look for and make use of structure, as the journey continues …

 

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The Area of a Rhombus and a Kite

I recently participated in the Mississippi Department of Education Geometry Institute. In our session on Geometric Measure & Dimension, we moved from areas of special quadrilaterals in the coordinate plane to proving the area formulas for a kite and a rhombus.

We had practiced look for and make use of structure on two kites, c and f. Participants had shared their thinking on kite c.

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Could they transfer what they had done in the coordinate plane with known segments lengths to a rhombus and a kite with diagonals d1 and d2?

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Just like our students would, some teachers struggled with the idea of generalizing the formula. Several used rulers to measure the lengths of the diagonals or made up numerical lengths for the diagonals and calculated the area.

Can you tell how these participants generalized their work?

I gave the kite as an assessment item for my students last year, and I asked them to practice “I can look for and make use of structure” along with “I can show my work”.

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Whose work can you understand without asking for clarification?

What opportunities do you give your students to practice “I can look for and make use of structure” along with “I can show my work”?

 

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Calculating Area – Looking for and Making Use of Structure

Several months ago when I was preparing a session on Geometric Measure & Dimension for the Mississippi Department of Education Geometry Institute, I noticed a blog post by Kate letting us know about some new IM tasks. Areas of Special Quadrilaterals (and one triangle) caught my attention, and so I took a look. I had no idea at the time how perfect this task was for starting the session.

Jill Gough and I often talk about pairing a content standard with a practice standard.

For this activity, the content standard was 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; apply these techniques in the context of solving real-world and mathematical problems.

And the practice standard was look for and make use of structure.

SMP7 #LL2LU Gough-Wilson

How would you find the area of each figure, composing into rectangles or decomposing into triangles and other shapes?

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Many teachers used color to make their thinking visible. I learned quickly, however, not to assume that they had found the area a certain way just because of a certain auxiliary line. I walked around, asking about their thinking, selecting and sequencing for our whole group discussion.

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We started with figure a.

Some decomposed into a rectangle and two triangles.

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Some decomposed into unit squares.

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Some composed into a rectangle to think about the areas of the two small triangles.

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Others decomposed into a rectangle and two triangles but then rearranged the two triangles into a rectangle.

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Others translated one triangle horizontally to rearrange into one rectangle.

One lady had not taught geometry for a very long time. She said she thought she knew a formula that would work for calculating the area, but she wasn’t confident about her work. Her formula? A=½h(b1+b2). Most participants knew that formula as the area of a trapezoid. Does it work for the parallelogram? They tried it. It worked. Maybe that adds to our reasons for considering the inclusive definition of trapezoid?

We looked next at figure d.

Some composed into a rectangle and subtracted the area of the right triangle.

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Some decomposed into a triangle and rectangle.

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One decomposed into a triangle and parallelogram.

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Some decomposed into unit squares.

Some decomposed and rearranged into a rectangle.

Then we looked at figure c.

So many ways!

Composing into a rectangle. From there, some subtracted the area of each right triangle. Some halved the area of the rectangle.

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Decomposing into triangles. Some into 4 triangles with both diagonals. Some into 2 triangles with the vertical diagonal. Some into 2 congruent triangles with the horizontal diagonal.

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Some decomposed and rearranged.

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Everyone practiced look for and make use of structure in ways they hadn’t thought to before. Everyone worked on their mathematical flexibility to find more than one way to determine the areas of the figures. Everyone learned at least one new way to look at the figure from the others in the room.

What opportunities do you provide your learners to look for and make use of structure and then share what they’ve made visible that wasn’t pictured before?

 

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