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SMP6: Attend to Precision #LL2LU

We want every learner in our care to be able to say

I can attend to precision.

CCSS.MATH.PRACTICE.MP6

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But what if I can’t attend to precision yet? What if I need help? How might we make a pathway for success?

 

Level 4:
I can distinguish between necessary and sufficient conditions 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.

 

How many times have you seen a misused equals sign? Or mathematical statements that are fragments?

A student was writing the equation of a tangent line to linearize a curve at the point (2,-4).

He had written

y+4=3(x-2)

And then he wrote:

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He absolutely knows what he means: y=-4+3(x-2).

But that’s not what he wrote.

 

Which student responses show attention to precision for the domain and range of y=(x-3)2+4? Are there others that you and your students would accept?

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How often do our students notice that we model attend to precision? How often to our students notice when we don’t model attend to precision?

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Attend to precision isn’t just about numerical precision. Attend to precision is also about the language that we use to communicate mathematically: the distance between a point and a line isn’t just “straight” – it’s the length of the segment that is perpendicular from the point to the line. (How many times have you told your Euclidean geometry students “all lines are straight”?)

But it’s also about learning to communicate mathematically together – and not just expecting students to read and record the correct vocabulary from a textbook.

[Cross posted on Experiments in Learning by Doing]

 
 

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Classifying Triangles

We look specifically at 45-45-90 triangles on the first day of our Right Triangles unit. I’ve already written specifically about what the 45-45-90 exploration looked like, but I wanted to note a conversation that we had before that exploration.

Jill and I had recently talked about introducing new learning by drawing on what students already know. I’ve always started 45-45-90 triangles by having students think about what they already know about these triangles (even though many have never called them 45-45-90 triangles before). After hearing about one of Jill’s classes, though, I started by asking students to make a column for triangles, right triangles, and equilateral triangles, noting what they know to always be true for each. This short exercise gave students the opportunity to attend to precision with their vocabulary.

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It occurred to me while we were talking that having students draw a Venn Diagram to organize triangles, right triangles, and equilateral triangles might be an interesting exercise. How would you draw a Venn Diagram to show the relationship between triangles, right triangles, and equilateral triangles?

In my seconds of anticipating student responses, I expected one visual but got something very different.

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What does it mean for an object to be in the intersection of two sets? Or the intersection of three sets? Or in the part of the set that doesn’t intersect with the other sets?

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Then we thought specifically about 45-45-90 triangles. What do you already know? Students practiced look for and make use of structure.

One student suggested that the legs are half the length of the hypotenuse. Instead of saying that wouldn’t work or not writing it on our list, I added it to the list and then later asked what would be the hypotenuse for a triangle with legs that are 5.

10.

I wrote 10 on the hypotenuse and waited.

But that’s not a triangle?

What?

5-5-10 doesn’t make a triangle.

Why not?

It would collapse (students have a visual image for a triangle collapsing from our previous work on the Triangle Inequality Theorem).

Does the Pythagorean Theorem work for 5-5-10?

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Students reflected the triangles about the legs and hypotenuse to compose the 45-45-90 triangle into squares and rectangles.

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And they constructed an altitude to the hypotenuse to decompose the 45-45-90 triangle into more 45-45-90 triangles.

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And then we focused on the relationship between the legs and the hypotenuse using the Math Nspired activity Special Right Triangles.

And so the journey continues … listening to and learning alongside my students.

 
 

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Special Right Triangles: 45-45-90

I gave my students our learning progression for SMP 8 a few weeks ago as we started a unit on Right Triangles and had a lesson specifically on 45-45-90 Special Right Triangles.

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The Geometry Nspired Activity Special Right Triangles contains an Action-Consequence document that focuses students attention on what changes and what stays the same. The big idea is this: students take some kind of action on an object (like grabbing and dragging a point or a graph). Then they pay attention to what happens. What changes? What stays the same? Through reflection and conversation, students make connections between multiple representations of the mathematics to make sense of the mathematics.

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Students start with what they know – the Pythagorean Theorem.

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Looking at the side lengths in a chart helps students notice and note what changes and what stays the same:

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The legs of the triangle are always the same length.

As the legs increase, the hypotenuse increases.

The hypotenuse is always the longest side.

 

Students begin to identify and describe patterns and regularities:

All of the hypotenuses have √2.

The ratio of the hypotenuse to the leg is √2.

 

Students practice look for and express regularity in repeated reasoning as they generalize what is true:

To get from the leg to the hypotenuse, multiply by √2.

To get from the hypotenuse to the leg, divide by √2.

hypotenuse = leg * √2

Teachers and students have to be careful with look for and express regularity in repeated reasoning. Are we providing students an opportunity to work with diagrams and measurements that make us attend to precision as we express the regularity in repeated reasoning that we notice?

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In a Math Practice journal, Kaci writes about “look for regularity in repeated reasoning”. We figured out that half of a square is a 45-45-90 triangle, and students were trying to determine the other two sides of the triangle given one side length of the triangle. She says “To find the length of the hypotenuse, you take the length of a side and multiply by √2. The √2 will always be in the hypotenuse even though it may not be seen like √2. In her examples, the triangle to the left has √2 shown in the hypotenuse, but the triangle to the right has √2 in the answer even though it isn’t shown, since 3√2√2 is not in lowest form. She says, “I looked for regularity in repeated reasoning and found an interesting answer.”

What opportunities can we provide our students this week to look for and express regularity in repeated reasoning and find out something interesting?

 
2 Comments

Posted by on April 7, 2015 in Uncategorized

 

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Visual: SMP-8 Look for and Express Regularity in Repeated Reasoning #LL2LU

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.
(CCSS.MATH.PRACTICE.MP8)

SMP8

But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

[Cross posted on Experiments in Learning by Doing]


Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.

 

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SMP8: Look for and Express Regularity in Repeated Reasoning #LL2LU

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.

CCSS.MATH.PRACTICE.MP8

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But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4

I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3

I can look for and express regularity in repeated reasoning.

Level 2

I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1

I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

 

What do you notice? What changes? What stays the same?

We use a CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning.

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What do we need to factor for the result to be (x-4)(x+4)?

What do we need to factor for the result to be (x-9)(x+9)?

What will the result be if we factor x²-121?

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What will the result be if we factor x²-a²?

We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1? (And then connect the result to the graph of y=x²+1.)

 

What happens if we factor over the set of real numbers?

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Or over the set of complex numbers?

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What about expanding the square of a binomial?

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What changes? What stays the same? What will the result be if we expand (x+5)²?

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Or (x+a)²?

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Or (x-a)²?

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What about expanding the cube of a binomial?

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Or expanding (x+1)^n, or (x+y)^n?

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What if we are looking at powers of i?

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We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Experiments in Learning by Doing]

 

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Which One Doesn’t Belong?

You’ve seen “which one is different” before.

(I first remember seeing this particular question from John Bament at a T3 session in 2014, although he might have gotten it from somewhere else. He sent it to the participants as a Quick Poll and showed us our quite varied results.)

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You’ve seen “Odd One Out” before.

These two images come from the Mathematics Assessment Project formative assessment lesson on Comparing Investments.

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I observed this lesson in a classroom a few weeks ago. It didn’t bother students that more than one answer can be correct, and they naturally explained why they chose what they did without the teacher even having to prompt them with “How did you get that?” or “Why?”

My coworker and I introduced Christopher Danielson’s Which One Doesn’t Belong to our beginning K-2 teachers recently. They began to think immediately about how they could do something similar with language as well as math. (And they were thrilled to learn something in PD that they could immediately take back to their classrooms.)

When I recently learned about Mary’s Which One Doesn’t Belong site, I decided to spend some time on it during our recent Math PLC meeting.

We started with a page from Christopher’s shape book. Our assistant principal (former history teacher) was thrilled to be able to immediately participate in our discussion. (How many of our students feel the same when we offer them low-floor, high-ceiling tasks?)

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We did a number WODB (one teacher fist-pumped another assistant principal when they figured out that 9 didn’t belong since the sum of its digits isn’t 7). Thanks, Pam!

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Then we moved to Rachel Fruin’s geometry Which One Doesn’t Belong. Our history teacher-turned assistant principal was still able to participate. She didn’t have the same vocabulary that the rest of the math teachers in our department had when stating why one doesn’t belong, but she learned some math vocabulary and we learned to see the images through different eyes during our shared experience.

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We ended our PLC with Hunter Patton’s Graphs & Equations 7.

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I recently heard that one measure of the success of professional development is whether the teacher’s practice changes as a result of what was learned. (Another part to this would of course be how long the teacher’s practice changes … one lesson? A few lessons? Or permanent change in lessons?) So I was thrilled to notice that the teacher with whom I share a room gave her precalculus students a WODB to try at the end of their opener later that day.

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They were studying rational functions. Which one doesn’t belong?

Before I knew it, students were in different corners of the room based on their initial responses.

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They shared thoughts with each other before sharing with the whole class.

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I tried the geometry WODB with my geometry students yesterday. I asked them to send me their response so that I could decide whether moving to one of the four corners of the room would be worthwhile. I asked bottom left to gather, bottom right to gather, and then top left & top right to gather. Why doesn’t your choice belong?

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Now work on your mathematical flexibility. Instead of being satisfied with one way to answer, find multiple responses.

Find a reason that each one doesn’t belong, and let me know when you do by selecting that choice on the new Quick Poll (now multiple response).

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Now sorted by individual responses so I can see which students need support:

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I’ve offered problem solving points for students who create their own WODB, and I look forward to seeing the results. Thank you, Mary, for creating a place for us to share and learn together … for creating a site that our teachers were able to immediately incorporate into their own learning and their students’ learning.

 

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Piecewise Functions

Piecewise Functions

We started a unit on piecewise functions in Algebra 1 with the following leveled learning progression:

Level 4: I can sketch a graph of a piecewise-defined function given a verbal description of the relationship between two quantities.

Level 3: I can interpret key features of a piecewise-defined function in terms of its context.

Level 2: I can determine the domain and range of a function given a context.

Level 1: Using any representation of a function, I can evaluate a function at a given value of x, and I can determine the value of x for a given value of f(x).

We started with an opener to ensure that students were successful with Levels 1 and 2 so that we could reach our target (Level 3) during the lesson.

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Questions 1-4 gave us evidence that most students could evaluate a function at a given value of x and determine the value of x for a given value of f(x) using any representation of a function.

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Questions 5-6 gave us evidence that our students needed more support determining the domain and range of a function given a context.

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Without showing the results from the opener that are pictured above, we talked all together about the context, reading the graph, but not explicitly discussing the domain and range. When we sent the question as a Quick Poll, we saw evidence that more students could determine the domain and range of a function given a context.

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We probably could have spent more time on Level 2 in class. But we didn’t. Instead, we had to provide additional support for Level 2 outside of class, through homework practice, zero block, and after school help.

To open our discussion of piecewise functions, we showed this picture from the front of the Jackson airport parking garage.

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What do you notice? What do you wonder?

Students wrote down a few observations individually, then shared their thoughts with a partner. We selected some for our whole class discussion. In particular, it was helpful that one student specifically said, “pay depends on time”.

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How much would you pay for a certain number of hours of parking?

Similar to an idea from the Internet Plans Makeover, we asked students to choose a number between 0 and 24. If you park that many hours, how much will you pay?

We asked students to check work with a partner before submitting. The result wasn’t quite as disastrous as when we tried the Internet Plans Makeover.

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Showing the grid helps some.

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And having a whole class discussion about the stipulations of the sign helped even more.

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Did anyone choose a non-whole number?

What would happen if you parked for 1.5 hours?

Or 2 hours and 20 minutes?

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We also used the Math Nspired lesson Dog Days or Dog Years with good success. What we are still trying to decide is which comes first … the structure from the Dog Days or Dog Years lesson about creating piecewise functions? Or the less structured conceptual introduction from the cost of parking at the airport? I’m not sure it’s wrong (or even better) to start with either one. But we still wonder, as the journey continues …

 
2 Comments

Posted by on March 30, 2015 in Algebra 1

 

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