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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.

SMP8

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?

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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.

8 Airport Parking

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 …

 
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Posted by on March 30, 2015 in Algebra 1

 

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SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 2)

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

I can reason abstractly and quantitatively.
(CCSS.MATH.PRACTICE.MP2)

SMP2

But…What if I think I can’t? What if I have no idea how to contextualize and decontextualize a situation? How might we offer a pathway for success?

We have studied this practice for a while, making sense of what it means for students to contextualize and decontextualize when solving a problem.

Students reason abstractly and quantitatively when solving problems with area and volume. Calculus students reason abstractly and quantitatively when solving related rates problems. In what other types of problem do the units help you not only reason about the given quantities but make sense of the computations involved?

What about these problems from The Official SAT Study Guide, The College Board and Educational Testing Service, 2009. How would your students solve them? How would you help students who are struggling with the problems solve them?

There are g gallons of paint available to paint a house. After n gallons have been used, then, in terms of g and n, what percent of the pain has not been used?

SMP2 SAT1

A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollar, on 2 cars that sold for $14,000 each?

SMP2 SAT2

In our previous post, SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 1), we offered a pathway to I can reason abstractly and quantitatively. What if we offer a second pathway for reasoning abstractly and quantitatively?

 

Level 4:

I can create multiple coherent representations of a task by detailing solution pathways, and I can show connections between representations. 

Level 3:

I can create a coherent representation of the task at hand by detailing a solution pathway that includes a beginning, middle, and end.  

Beginning:

I can identify and connect the units involved using an equation, graph, or table.

Middle:

I can attend to and document the meaning of quantities throughout the problem-solving process.

End:

I can contextualize a solution to make sense of the quantity and the relationship in the task and to offer a conclusion. 

Level 2:

I can periodically stop and check to see if numbers, variables, and units make sense while I am working mathematically to solve a task.

Level 1:

I can decontextualize a task to represent it symbolically as an expression, equation, table, or graph, and I can make sense of quantities and their relationships in problem situations.

 

What evidence of contextualizing and decontextualizing do you see in the work below?

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[Cross-posted on Experiments in Learning by Doing]

 

 
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Posted by on September 29, 2014 in Standards for Mathematical Practice

 

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SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 1)

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

I can reason abstractly and quantitatively.
(CCSS.MATH.PRACTICE.MP2)

SMP2

I wonder what happens along the learning journey and in schooling. Very young learners of mathematics can answer verbal story problems with ease and struggle to translate these stories into symbols. They use images and pictures to demonstrate understanding, and they answer the questions in complete sentences.

If I have 4 toy cars and you have 5 toy cars, how many cars do we have together?

If I have 17 quarters and give you 10 of them, how many quarters will I have left?

Somewhere, word problems become difficult, stressful, and challenging, but should they? Are we so concerned with the mechanics and the symbols that we’ve lost meaning and purpose? What if every unit/week/day started with a problem or story – math in context? If learners need a mini-lesson on a skill, could we offer it when they have a need-to-know?

Suppose we work on a couple of Standards of Mathematical Practice at the same time.  What if we offer our learners a task, Running Laps (4.NF) or Laptop Battery Charge 2 (S-ID, F-IF) from Illustrative Math, before teaching fractions or linear functions, respectively? What if we make two learning progressions visible? What if we work on making sense of problems and persevering in solving them as we work on reasoning abstractly and quantitatively. (Hat tip to Kato Nims (@katonims129) for this idea and its implementation for Running Laps.)

 

Level 4:

I can connect abstract and quantitative reasoning using graphs, tables, and equations, and I can explain their connectedness within the context of the task.

Level 3:

I can reason abstractly and quantitatively.

Level 2:

I can represent the problem situation mathematically, and I can attend to the meaning, including units, of the quantities, in addition to how to compute them.

Level 1:

I can define variables and constants in a problem situation and connect the appropriate units to each.

 

You could see how we might need to focus on making sense of the problem and persevering in solving it. Do we have faith in our learners to persevere? We know they are learning to reason abstractly and quantitatively. Are we willing to use learning progressions as formative assessment early and see if, when, where, and why our learners struggle?

Daily we are awed by the questions our learners pose when they have a learning progression to offer guidance through a learning pathway. How might we level up ourselves? What if we ask first?

Send the message: you can do it; we can help.

 

[Cross-posted on Experiments in Learning by Doing]

 
4 Comments

Posted by on September 28, 2014 in Standards for Mathematical Practice

 

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Visual: SMP-5 Use Appropriate Tools Strategically #LL2LU

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

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

SMP5 #LL2LU

Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display  confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

 

Suppose you are solving an equation.

Are you practicing use appropriate tools strategically if you use the numerical solve command on your graphing calculator?

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Or what about using your calculator to substitute values of x until you find a value that makes a true statement?

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Are you practicing use appropriate tools strategically if you use a computer algebra system to explain your steps?

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Or what if you use the graphing capability of your handheld?

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Consider each of the following learning goals:

I can explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution, and I can construct a viable argument to justify a solution method.  CCSS-M A-REI.A.1.

I can solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. CCSS-M A-REI.B.3.

I can explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. CCSS-M A-REI.D.11.

Does use appropriate tools strategically depend on the learner? Or the learning goal? Or the teacher? Or the availability of tools?

 

[Cross posted on Experiments in Learning by Doing]

 
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Posted by on September 15, 2014 in Standards for Mathematical Practice

 

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SMP5: Use Appropriate Tools Strategically #LL2LU

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

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

SMP5

But…What if I think I can’t? What if I have no idea what are appropriate tools in the context of what we are learning, much less how to use them strategically? How might we offer a pathway for success?

Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

We still might need some conversation about what it means to use appropriate tools strategically. Is it not enough to use appropriate tools? Would it help to find a common definition of strategically to use as we learn? And, is use appropriate tools strategically a personal choice or a predefined one?

Strategic

How might we expand our toolkit and experiment with enough tools to display confidence when explaining why the selected tools are appropriate and effective for the solution pathway used?  What if we practice with enough tools that we make strategic – highly important and essential to the solution pathway – choices?

What if apply we 5 Practices for Orchestrating Productive Mathematics Discussions to learn with and from the learners in our community?

  • Anticipate what learners will do and why strategies chosen will be useful in solving a task
  • Monitor work and discuss a variety of approaches to the task
  • Select students to highlight effective strategies and describe a why behind the choice
  • Sequence presentations to maximize potential to increase learning
  • Connect strategies and ideas in a way that helps improve understanding

What if we extend the idea of interacting with numbers flexibly to interacting with appropriate tools flexibly?  How many ways and with how many tools can we learn and visualize the following essential learning?

I can understand solving equations as a process of reasoning and explain the reasoning.  CCSS.MATH.CONTENT.HSA.REI.A.1

What tools might be used to learn and master the above standard?

  • How might learners use algebra tiles strategically?
  • When might paper and pencil be a good or best choice?
  • What if a learner used graphing as the tool?
  • What might we learn from using a table?
  • When is a computer algebra system (CAS) the go-to strategic choice?

Then, what are the conditions which make the use of each one of these tools appropriate and strategic?

[Cross posted on Experiments in Learning by Doing]

________________________

“The American Heritage Dictionary Entry: Strategically.” American Heritage Dictionary Entry: Strategically. N.p., n.d. Web. 08 Sept. 2014.

 
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Posted by on September 14, 2014 in Standards for Mathematical Practice

 

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Visual: SMP-3 Construct Viable Arguments and Critique the Reasoning of Others #LL2LU

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

I can construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP3

SMP3

But…what if I can’t? What if I’m afraid that I will hurt someone’s feelings or ask a “stupid” question? How might we facilitate learning and grow our culture where critique is sought and embraced?

From Step 1: The Art of Questioning in The Falconer: What We Wish We Had Learned in School.

By learning to insert feedback loops into our thought, questioning, and decision-making process, we increase the chance of staying on our desired path. Or, if the path needs to be modified, our midcourse corrections become less dramatic and disruptive. (Lichtman, 49 pag.)

This paragraph connects to a Mr. Sun quote from Step 0: Preparation.

But there are many more subtle barriers to communication as well, and if we cannot, or do not choose to overcome these barriers, we will encounter life decisions and try to solve problems and do a lot of falconing all by ourselves with little, if any, success. Even in the briefest of communications, people develop and share common models that allow them to communicate effectively.  If you don’t share the model, you can’t communicate. If you can’t communicate, you can’t teach, learn, lead, or follow.  (Lichtman, 32 pag.)

How might we offer a pathway for success? What if we provide practice in the art of questioning and the action of seeking feedback? What if we facilitate safe harbors to share thinking, reasoning, and perspective?

 

Level 4:

I can build on the viable arguments of others and take their critique and feedback to improve my understanding of the solutions to a task.

Level 3:

I can construct viable arguments and critique the reasoning of others.

Level 2:

I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

Level 1:

I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

 

SMP3 #LL2LU

[Cross-posted on Experiments in Learning by Doing]

________________________

Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

 

 
2 Comments

Posted by on September 8, 2014 in Standards for Mathematical Practice

 

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