# Standard 5: Use Appropriate Tools Strategically

## 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.

## 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.

## Connections to Classroom Practices

### 3rd 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, recognizing both the insight to be gained and their [the tools’] limitations.*

Mia Buljan’s 3rd-grade students elect to use manipulatives to help them with a multiplication/division equal-grouping-scenario problem following a structure similar to “Sam's dad bought 24 hot dogs for Sam and his 3 friends. How many hot dogs can they each have?”

**See this video in the context of an entire lesson.**

### 4th Grade

*Mathematically proficient students consider the available tools when solving a mathematical problem.*

On the first day of the learning segment, Michelle Makinson engages her learners in a math talk focused on unit fractions, combining into wholes, “parts of,” and the idea of equivalence, using manipulatives to create and explain a visual representation of a contextualized representation / word problem. Her students then share with their partners, explaining their approach. This clip also relates to standard 1 (make sense of problems & persevere in solving them), standard 3 (construct viable arguments & critique the reasoning of others) and standard 6 (attend to precision).

**See this video in the context of an entire lesson.**

### 5th Grade

*Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper…[or] a calculator….*

Erika Isomura works with her 5th graders to develop their understanding of how decimals work. She begins her lesson having her students work in pairs to do a “calculate and check” mental math decimal task, using calculators to check, then evaluating their work with symbols and writing (e.g. “We disagree because,” “I have a question because”).

She reminds her students to do the mental math, then first discuss what they got “in their brains,” and then check with their calculators. As the pairs work, Erika circulates around the classroom and probes their thinking (e.g. “Did you agree? Did that make sense?”). This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others).

**See this video in the context of an entire lesson.**

### 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 color-coding. 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 & persevere in solving them).

**See this video in the context of an entire lesson.**

### 8th 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.*

Antoinette Villarin transitions her students to a matching activity, having her students use a recording sheet to make sense of different quantities and representations, find models, match graphs, and explain their justifications for the match.

She distributes a stack of twelve graphs that represent the flow of liquid between containers, and challenges students to match the two graphs that show movement for a given pair of containers. She notes that for some cards, information has intentionally been left out, and recommends that students begin with the cards that have complete information.

She reminds her students that they are trying to build a mathematical argument, so they must explain to their partner why a given pair of cards is a match, and the partner in turn needs to agree or disagree with each explanation. Antoinette explains how to use the cards, reminding students to refer back to the sentence frames for conversation if they don’t know what to say.

This clip also relates to standard 2 (reason abstractly & quantitatively).

**See this video in the context of an entire lesson.**

*Mathematically proficient students consider the available tools when solving a mathematical problem.*

Antoinette Villarin’s students continue their partner work, matching graphical representations of rates of change with each other.

In this clip, two different student pairs engage in mathematical discourse, comparing graphs to each other to discern which graph represents the top prism out of which the fluid is draining, and which represents the corresponding receptacle prism. Students offer ideas and defend their thinking to each other.

This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others).

**See this video in the context of an entire lesson.**

*Mathematically proficient students consider the available tools when solving a mathematical problem.*

Patty introduces her lesson by charging students to identify the “big ideas” they should be thinking about when they work with right triangles. Students pair-share their ideas, and Patty notes when they are making reference to available tools and supports, such as anchor charts, around the room. In her commentary, Patty notes that this lesson is intended to develop students’ capacity to engage in modeling mathematical situations. Students identify the Pythagorean Theorem, and Patty prompts them to attend to precision and communicate precisely. In a whole-group sharing, she engages all students to add on to, critique, extend, and clarify each other’s thinking. Students deepen their capacity to make sense of the problem or situation. Patty presents student work from a previous assessment and asks students to critique the person’s strategies and precision, giving advice to each exemplar learner about how to improve their approach. This clip also relates to standard 1 (make sense of problems & persevere in solving them), standard 3 (construct viable arguments & critique the reasoning of others), and standard 4 (model with mathematics).

**See this video in the context of an entire lesson.**

**lesson part 1**

*Mathematically proficient students consider the available tools when solving a mathematical problem.*

Patty’s students give advice to their peers about perseverance and strategies they can use to help themselves in an upcoming MARS performance assessment task. She asks “What do you do when you’re stuck? What strategies should you try?” Students think, write, then share their strategies with each other. They identify strategies using anchor charts and calculators, drawing pictures, consulting a peer, taking their time, double-checking their work and ensuring that it makes sense. This clip also relates to standard 1 (make sense of problems & persevere in solving them), standard 3 (construct viable arguments & critique the reasoning of others), and standard 4 (model with mathematics).

**See this video in the context of an entire lesson.**

**lesson part 2**

### 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 & persevere in solving them).

**See this video in the context of an entire lesson.**

*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.

**See this video in the context of an entire lesson.**

*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 meta-cognitive 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).

**See this video in the context of an entire lesson.**

*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).

**See this video in the context of an entire lesson.**

### 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."

**See this video in the context of an entire lesson.**

**(Parts 4 & 6)**