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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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MP6 – Mapping a Figure 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.

Our learning intention for the day was I can map a figure onto itself using transformations.

Performing a [sequence of] transformation[s] that will map rectangle ABCD onto itself is not the same thing as describing a [sequence of] transformation[s].

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We practiced both, but we focused on describing.

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I asked the student who listed several steps to share his work.

  1. rotate rectangle 180˚ about point A
  2. translate rectangle A’B’C’D’ right so that points A’ and B line up as points B’ and A. [What vector are you using?]
  3. Reflect rectangle A”B”C”D” onto rectangle ABCD to get it to reflect onto itself. [About what line are you reflecting?]

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What if we want to carry rectangle ABCD onto rectangle CDAB? How is this task different from just carrying rectangle ABCD onto itself?

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What about mapping a regular pentagon onto itself?

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Many students suggested using a single rotation, but they didn’t note the center of rotation. How could you find the center of rotation for a single rotation to map the pentagon onto itself?

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This student used the intersection of the perpendicular bisectors to find the center of rotation, but didn’t know what angle to use for the rotation. How would you find an angle of rotation that would work?

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What can you do other than a single rotation?

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This student reflected the pentagon about the perpendicular bisectors of one of the side of the pentagon.

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The descriptions students gave made it obvious that we needed more work on describing. The next day, we took some of the descriptions and critiqued them. Which students have attended to precision?

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It’s good work to distinguish precision from knowing what someone means as we learn to attend to precision. And so the journey continues …

 
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Posted by on September 21, 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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Who Will Work with Whom?

How do you and your students determine who will work with whom?

Elizabeth has been reflecting on teaming her speed demons with other speed demons and her katamari with other katamari. She is grouping and regrouping often, paying attention both to how students work and how students work together.

As part of SREB’s Mathematics Design Collaborative, we use the work from student pre-assessments to pair students homogenously on days when we are doing a formative assessment lesson from the Mathematics Assessment Project. Many of our teachers have worried about homogenous pairing. They wonder how two students who have little understanding of the material will learn anything if they are paired with each other. What we are finding, though, is twofold. Since we don’t have to spend as much time with pairs of students who have demonstrated understanding or some understanding, we can focus our time on the pairs of students who have little understanding. In addition, neither student can sit back and rely on the other student to do all of the work. Together, they end up doing something. The formative assessment lessons are written so that all students have entry to the content. Some items are more challenging than others, and we are slowly learning that every student doesn’t have to get to the same place in the collaborative activity. Students work for a certain amount of time and share what they have learned, even though they might not finish the entire activity.

Others (Alex Overwijk and Dylan Kane) cite Peter Liljedahl’s work on visible, random assignment of student teams.

It takes me a long time to get to know someone and feel comfortable sharing my ideas. For many years I let students choose their teams and work together for the entire year. More recently, though, my coworkers and I have used a card sort activity for teaming students on the first day of a unit. Teams work together throughout the unit unless we are enacting a Formative Assessment Lesson (FAL), in which case we team students homogenously based on their pre-assessment.

In geometry, we’ve made card sorts that introduce students to some of the terms and diagrams that we will study in the unit, often leading right in to the first lesson. It often takes a while for students to find their other team members since they don’t already know the content. Alternatively, we could use content/card matches from the previous unit to team them randomly and visibly on the first day.

For the first team sort, I emailed a preview to students the night before class so that they would have some idea of what to expect/what they might do with their card when they came to class.

Many students noted in their end of course feedback that we should keep the team sorts:

I think you should keep putting us into teams, as we can learn from others who think differently or similarly to us. I think you should also keep switching the classes some. I feel like this helped me a lot this year.

I would keep the different groups that are paired up. I feel that the groups helped me to see others point of view not just my own.

switching classes to see different teaching styles and having different groups throughout the year.

The changing of groups because it has helped me make friends and learn to work together with people who frustrate me.

All of our geometry team sorts are linked here.

I’ve heard others talk about teaming and re-teaming several times during a single lesson based on what students know and don’t know yet. I’m not there yet, but I am intrigued by the idea and would like to learn more both about the value of moving around so often and the logistics of what happens to students’ stuff.

And so the journey to figure out who will work with whom continues …

 

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MP5 – The Center of the Circle

How do you give students the opportunity to practice “I can use appropriate tools strategically”?

MP5

How would your students find the center of a circle?

Every year, I am amazed at the connections students make between properties of circles that we have explored and what the center of the circle has to do with those properties.

We started on paper.

Some students moved their thoughts to technology.

Whose work would you select for an individual and/or whole class discussion?

Could we use the tangents to a circle from a point to find the center of the circle?

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Could we use the intersection of the angle bisectors of an equilateral triangle inscribed in a circle to find the center of the circle?

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Could we use the perpendicular bisector of a chord of a circle to find the center of the circle?

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Could we use the intersection of the perpendicular bisectors of a pentagon circumscribed about a circle to find the center of the circle?

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Could we use the intersection of the perpendicular bisectors of several chords of a circle to find the center of the circle?

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Could we use a right triangle inscribed in a circle to find the center of the circle?

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And so the journey continues …

 
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Posted by on September 15, 2016 in Circles, Geometry

 

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Derivative Rules

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Learning Objective 2.1C: Calculate Derivatives

Essential Knowledge 2.1C2: Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

Essential Knowledge 2.1C3: Sums, differences, products, and quotients of functions can be differentiated using derivative rules.

Mathematical Practice for AP Calculus (MPAC) 1: Reasoning with definitions and theorems

Students can: develop conjectures based on exploration with technology.

 

How do you provide students the opportunity to develop conjectures?

After determining the derivative of a few quadratic functions using the definition, we use our TI-Nspire Computer Algebra System (CAS) software to explore derivatives.

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We use the power rule to make conjectures about the product rule. (I think that I saw this suggestion in a Mathematics Teacher magazine in the early 90s, but I can’t find the reference now.)

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We know what the derivative should be, because we know the derivative of x^5. How could we use f, f ‘, g, and g ‘ to get to what we know is the derivative from the power rule?

Once students made conjectures about the product rule, we formalized the rule.

I asked students to predict the derivative of f(x)=sin(3x). As expected, many thought that it would be f ’(x)=cos(3x). When we looked at the graph of the derivative of f(x), students realized that f ‘(x)=3cos(3x).

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We used CAS to explore the chain rule (power and composite) in more detail.

Students practiced “Notice and Note”. Several generalized the chain power rule before I asked.

Once students knew the chain rule, we used the chain rule to derive the quotient rule.

And so the journey providing opportunities for students to make sense of rules instead of just telling them rules continues …

 
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Posted by on September 14, 2016 in Calculus, Derivatives

 

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Seeing the Definition of Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

  1. relate the concept of a limit to all aspects of calculus
  2. Students can connect concepts to their visual representation with and without technology.

 

How do you introduce the definition of a derivative?

We start with the visual of a tangent line at a point and a secant line containing the point.

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We don’t need calculus to determine the slope of the secant line. We do need calculus to determine the slope of the tangent line.

How might we use the slope of the secant line to determine the slope of the tangent line?

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Posted by on September 9, 2016 in Calculus, Derivatives

 

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