Standard 5: Use appropriate tools strategically
The Standard: Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
Classroom Observations:
Teachers who are developing students' capacity to "use appropriate tools strategically" make clear to students why the use of manipulatives, rulers, compasses, protractors, and other tools will aid their problem solving processes. A middle childhood teacher might have his students select different color tiles to show repetition in a patterning task. A teacher of adolescents and young adults might have established norms for accessing tools during the students' group "tinkering processes," allowing students to use paper strips, brass fasteners, and protractors to create and test quadrilateral "kite" models. Visit the video excerpts below to view multiple examples of these teachers
Connections to Classroom Practices
5th/6th Grade 
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models ... Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful...
Fran Dickinson leads a lesson on numeric patterning, helping students to investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, Dickinson tells his students that “the first step is to do a pictorial representation… I want you to play around with the tiles, and sketch out what you see happening in those first three patterns, but I want you to pay attention to colorcoding. You’re free to use those tiles like I said, or markers if you need them, I can make those available as well.” This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).

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9th/10th Grade 
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, ...[or] a ruler. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful...
Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, Humphreys circulates around groups of students as they use rulers, protractors, and sheets of paper to make different kinds of quadrilaterals and examine and “guarantee” their properties. A group of students debates whether or not one student’s assertion about the properties of a trapezoid hold in all circumstances, using the kite “sticks” to illustrate their points. 9th/10th grade first video
This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).

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Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, [or] concrete models... Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful...
In this clip, Humphreys closes day one of the Properties of Quadrilaterals lesson, and orients students to the second part of the investigation in which they will justify and prove their findings about the diagonals of the kites, saying “convince yourself, convince a friend, convince a skeptic” to describe for students the level of precision necessary to justify their conjectures. The students use definitions, postulates, and theorems to develop a proof about the diagonals of a quadrilateral and how they constrain the type of figure that is formed. Humphreys moves between groups, checking in on the progress students are making in developing their justifications. At the close of the period, she employs the resource manager to make sure all the manipulatives and materials are collected and stored.

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Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, [or] concrete models...
In this clip, Humphreys articulates the focus for day two of the group investigations of the properties of quadrilaterals: to prove that their conjectures are actually true. She shares examples of how students “kept track” of their own thinking, helping students in their metacognitive efforts at understanding how they’re thinking and how to document their understandings. The students mark a figure drawn from the given to help reason through the proof. This clip is also indicative of standard 6 (attend to precision).

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Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, [or] a protractor...
In this clip, groups are actively engaged in creating statements about the properties of their quadrilaterals that they can defend. The facilitator asks the resource manager to go get markers, paper, a ruler, and a compass for the group so that they can make their thinking visible. They continue to use the “kite sticks” from Tuesday’s lesson to experiment with different points of intersection.This clip is also indicative of standard 6 (attend to precision).

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9th  12th Grade 
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator...They are able to use technological tools to explore and deepen their understanding of concepts.
Carlos Cabana works with his high school English language learning students on algebraic reasoning and multiple representations around parabolas. In these clips, groups grapple with a new kind of problem for them. In the first clip, a group of students raises questions about mathematical tools and how to show and organize their work. In the second clip, a group of students uses graphing calculators to test their thinking around positive and negative numbers. Cabana observes that with their use of the calculator, they had to realize that, "Oh okay, we don't know this offhand, so we're going to figure it out this way."

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