# Standard 4: Model with Mathematics

## Classroom Observations

Teachers who are developing students’ capacity to "model with mathematics" move explicitly between real-world scenarios and mathematical representations of those scenarios. A middle childhood teacher might pose a scenario of candy boxes containing multiple flavors to help students identify proportions and ratios of flavors and ingredients. An early adolescence teacher might represent a comparison of different DVD rental plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted. A teacher of adolescents and young adults might pose a "kite factory" scenario, in which advanced students are asked to determine the conditions for always creating a particular shape of kite given the dimensions of the diagonals and the angle of intersection. Visit the video excerpts below to view multiple examples of teachers engaging students in mathematical modeling.

## The Standard

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

## Connections to Classroom Practices

### 1st Grade

*Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation... They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.*

Liz O'Neill works with her first grade students engaging them in composing and decomposing numbers within twenty. Her students play a game called "How Many are Hiding?" Pairs are given a bag with 10 cubes, a paper plate, and the "How Many Are Hiding Recording Sheet." One partner takes some of the cubes and "hides" them under the plate. The remaining are placed on the top. The second partner uses sentence frames to answer the questions "What number do you see?", "How many are hiding?", "How do you know __ are hiding"? In addition, the answers are recorded. Roles are then reversed. The partner game gives students practice in composing and decomposing numbers within ten. Students who have completed the original task are challenged with changing the total number of cubes to 15 or 20.

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

**(Parts 2-4)**

### 5th Grade

*Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation... They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense...*

Hillary Lewis-Wolfsen invites students to examine a problem about proportions and ratios with a strategy used by a student to organize the information in the problem. In this clip, she gives students “private think time” to work the problem again, refreshing their memories about the problem, asks them to use their “think sheets” to record their ideas, then has them turn to a partner and share and defend their thinking. In the discussion, the pairs share where they think the example student’s strategy reflected misunderstandings of the quantities.

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

### 7th/8th Grade

*Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as ...tables.... They can analyze those relationships mathematically to draw conclusions.*

Cecilio Dimas leads a lesson on making comparisons between three different financial plans, helping students use multiple representations of mathematical problems: verbal, tabular, graphical, and algebraic generalization. In this clip, Dimas connects to the prior day’s lesson, in which the class “started a conversation about the economic status of our world [and] about making responsible decisions when we’re spending our money.” His students share that they were to represent various DVD rental plans using verbal and tabular representations.

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

### 8th Grade

*Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.… They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense….*

Antoinette Villarin begins her lesson on graphing constant rates of change, reviewing the learning goals and mathematical practices, naming Standards for Mathematical Practice 1, 3, 6, and 7. She notes that it is important that her students understand how to build a mathematical argument, and she shares sentence frames and key vocabulary that the students will use as they build their arguments.

Antoinette presents a model of two bottles attached to each other so that fluid can flow between them, and she asks her students to make sense of the problem by describing what they see happening.

Students share that as the amount of fluid in the top container/prism, decreases, the amount in the bottom container/prism increases.

This clip also relates to standard 1 (make sense of problems and persevere in solving them), standard 3 (construct viable arguments & critique the reasoning of others), standard 6 (attend to precision), and standard 7 (look for and make use of structure).

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

*Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.*

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 5 (use appropriate tools strategically).

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

**lesson part 1**

*Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.*

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 5 (use appropriate tools strategically).

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

**lesson part 2**

### 9th/10th Grade

*Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation... They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense...*

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 introduces the task by posing a problem as a real-life investigation in which a kite manufacturer who “only manufactures quadrilateral kites”, and needs to know the properties of convex quadrilaterals that will always result in a given kite shape, saying “how to do the sticks is the issue.” The students work in groups to give prototype advice to this manufacturer, so that any time an order comes in, the manufacturer will always know “what kind of sticks to put in the kit and how they are to be put together.”