# How We Develop the Standards for Mathematical Practice

## 1. Make sense of problems and persevere in solving them

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Our curriculum is comprehensive and aims to cover every type of context that a student might encounter when taking a course on the same subject elsewhere. We cover not only core concepts and skills, but also numerous applications where students are required to interpret a word problem, construct a mathematical representation, solve or otherwise manipulate the representation, and then interpret the result in the broader context of the problem.

Math Academy students are encouraged to consider straightforward problems first to gain insight into solving more complex ones. However, we scaffold our curriculum so that students begin by solving special cases and simpler forms of a problem. Every lesson is broken up into 2-4 segments or "knowledge points" of increasing difficulty, where the student solves progressively more advanced problem types on the topic that is the subject of the lesson. The more advanced problem types typically require not only more extensive manipulation, but also deeper understanding of the problem structure.

Every knowledge point starts with a fully worked out example for students to follow along with. As students attempt problems on their own, Math Academy provides fully worked out solutions to all problems so students have the opportunity to compare their solutions with a complete and correct solution, which may or may not be different than their own.

Our middle-school curriculum leverages dual-coding theory by including helpful visualizations and diagrams when possible and makes use of extensive concrete, pictorial models to help students develop a more general understanding of arithmetic objects, operations with these objects, and how to apply these operations to contextual settings.

For example:

• Students first learn to add, subtract, multiply, and divide fractions using models to gain valuable insights and intuition into how these processes work without models.
• In our 4th Grade Math course, students are expected to solve multi-step contextual problems involving the four basic operations and must apply area models appropriately to deal with multi-digit multiplication.
• In our high-school curriculum, beginning with Algebra I and Integrated Math I, students are asked to construct exponential growth expressions from first principles to represent a simple, concrete situation given in context. This experience equips students with the tools necessary to construct more general models of exponential growth (e.g., using functions), allowing them to generalize and solve more complex problems.
• When performing graph transformations, students are encouraged to study each transformation individually with familiar curves before combining these transformations. Conversely, students must be able to break a composite graph transformation into a series of simpler ones and state the order in which they must be applied.

Students often transform algebraic expressions or perform other manipulations to obtain the information that's necessary to solve problems.

For example:

• When working with more general exponential growth/decay models, students are expected to manipulate their model to determine some additional information that's of interest, e.g., finding an initial condition or the time taken for a certain event to occur.
• By completing the square on a quadratic function describing an object's height, students can determine its maximum height and the time it takes to reach it.
• Calculus students must approximate an improper integral using a graphing calculator by continually refining the limits of integration until a solution is found to an appropriate degree of accuracy. Students must be sure to increment appropriately in order to avoid technological errors.

## 2. Reason abstractly and quantitatively

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

An important part of the Math Academy curriculum is developing students to deal with situations where they must transition from a contextual setting to a purely mathematical one, perform some mathematical manipulations, and interpret the results in the original context. When performing mathematical manipulations, probing the contextual information for additional details needed to solve the problem may be necessary. Where relevant, students are expected to state answers with appropriate units.

For example:

• Middle school students are expected to solve a diverse range of real-world problems using manipulations with integers, fractions, decimals, percentages, units, unit rates, and proportions.
• At the Algebra I/Integrated Math I level, students are expected to construct linear models from real-world information, interpret quantities represented by the model’s coefficients within the original context, and solve problems using their models.
• Given contextual information about the rate of change of an evolving quantity, form a differential equation that models the system, find the general solution, and apply initial conditions to find a particular solution. Students must then reason quantitatively about the particular solution's properties within the broader context. In this setting, students are expected to understand the role of units in rate-of-change problems and make decisions about which units are appropriate.
• Use geometric series to model the growth of an investment account at various compounding rates and frequencies (including continuous compounding); use this information to calculate a fund's value at some point in the future and form a broader understanding of how compounding is instrumental in achieving retirement goals.

## 3. Construct viable arguments and critique the reasoning of others

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Students build proofs by induction and proofs by contradiction. CA 3.1 (for higher mathematics only).

Math Academy students must reason quantitatively about data and make decisions. Our curriculum contains numerous questions that require students to critically analyze statements to determine whether they are true or false.

For example, at the middle-school and prealgebra levels:

• Students must form correct logical arguments to draw conclusions about the relative sizes of two numbers. For example, use concrete models or other valid reasoning to determine why $\frac 1 8 < \frac 1 4$ or $0.01 > 0.00999$.
• Students must construct correct arguments to describe addition and multiplication equations as comparisons and vice-versa.
• Students must determine whether bivariate data (x,y) given in context forms a proportional relationship. They must form an appropriate model in the affirmative case and make predictions using their model.
• Students learn to use measures of skewness to determine which method of measuring centrality and spread (mean and mean absolute deviation vs. median and quartiles) is most appropriate in a given setting.

At the high school level:

• Students are expected to construct and make predictions using a linear regression model and distinguish between cases of strong, weak, and no correlation. They must show awareness of the difference between interpolation and extrapolation and why caution should be exercised in the latter case.
• Students are expected to distinguish between linear and nonlinear regression models and select a model that best fits a given data set. They must also calculate the parameters of an exponential regression model using data from semi-log scatter plots.
• Students must demonstrate knowledge of the difference between correlation vs causation in various settings (reverse causation, outside causations, coincidence, and multiple causations).

Students are also expedited to break situations that arise into cases when appropriate. For example:

• Determining values of parameters that make a matrix or linear transformation invertible.
• Students are expected to distinguish between cases of unique, multiple, and no solutions when solving equations and systems of equations.
• Uses cases when solving discriminant problems, such as determining the number of roots of a quadratic function.

At the higher level, our Methods of Proof course requires students to construct all types of proofs including not only direct proof but also proof by induction, contrapositive, contradiction, cases, and counterexample. Students learn the high-level structure of a proof, practice generating proofs, and deeply analyze the nuanced details of complicated proofs.

In particular:

• Proof by induction: Students will achieve mastery in proving statements related to sums of finite series, divisibility, inequality, matrix identities, and first-order recurrence relations. They will also develop an understanding of strong induction by proving statements relating to second-order recurrence relations.
• Proof by contradiction. Students will master proving statements related to parity, divisibility, irrationality, properties of rational and irrational numbers, and solutions to linear congruences.

## 4. Model with mathematics

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 additional 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.

The Math Academy curriculum contains many topics where students are expected to construct, manipulate, and interpret models.

For example:

• At the middle school level, students are given multiple opportunities to apply their knowledge of fractions, decimals, and percentages to real-world situations. This typically involves correctly constructing an equation that describes a given situation and calculating a quantity of interest. Students are also expected to model and find solutions to contextual multi-part problems.
• Modeling exponential growth and decay.
• Modeling with trigonometry, including cases with non-right triangles, and constructing models for sinusoidal phenomena using trigonometric functions.
• Modeling economic quantities (e.g., revenue, cost, and profit) and physical situations using quadratics and other polynomials.
• Modeling speed, distance, time, work, and mixture problems using linear equations.
• Modeling with direct and inverse variation.
• Modeling situations using one-variable and two-variable inequalities and, in the case of two-variable inequalities, interpreting a feasible region within a given context.
• Creating and working with probabilities, statistical models, and random variables.
• Modeling situations with differential equations, emphasizing exponential growth, decay, and logistic growth models.

Furthermore, before students are given lessons where they are asked to apply mathematics to solve problems, they are first given lessons where they practice all the underlying concepts and skills that are exercised in the application. In particular, we have topics that cover the following non-exhaustive list of categories:

• translating between pictorial and symbolic representations of quantity
• translating between verbal and mathematical expressions
• describing relationships depicted in graphs, and identifying graphs with relationships described verbally
• interpreting the meaning of different terms in mathematical expressions and functions
• identifying and carrying out manipulations that will transform a given algebraic expression into a form where a desired feature can be determined
• changing the viewing window on a graphing calculator to obtain information that is located outside the default view

## 5. Use appropriate tools strategically

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.

By engaging students in regular retrieval practice (quizzes and multi-part problems), students are continually assessed on their ability to characterize general features of problems (category induction learning) and apply appropriate techniques to solve problems (discrimination learning). This includes identifying which tools are appropriate to solve the problem at hand.

The vast majority of our topics require students to work problems out using pencil and paper, and we communicate to students and parents the importance of writing one’s work down in an organized manner. Students are encouraged to draw diagrams and tables as part of the solution process where appropriate, and these resources are explained when necessary. At the middle school level, students must demonstrate that they can correctly read lengths from a ruler in varied settings and solve angle problems that require the use of a protractor.

Students are expected to analyze and interpret tabular data at multiple points throughout the curriculum. For example:

• Determining whether bivariate data presented in tabular form follows a linear or exponential trend, comparing linear and exponential data sets, and making predictions (e.g. determining when an exponential growth function first exceeds a linear function).
• Interpreting two-way frequency tables, calculating experimental probabilities from this data, and using their findings to make predictions.

Appropriate usage of calculators (including graphing calculators) is sometimes expected. At the calculus level (AB/BC), students are provided with detailed instructions on how to operate graphing calculators with a high degree of proficiency. Students are then expected to use this knowledge to find solutions to problems that are, practically speaking, intractable using pen and paper alone.

## 6. Attend to precision

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Our curriculum explicitly assesses students on the meaning of symbols and units. Answers must be numerically or symbolically correct to receive credit – including the negative sign, as well as the form of the answer and the number of decimal places when appropriate. We also incentivize careful work by configuring our gamified XP system so that students receive outsized rewards for perfect performance (and penalties for excessive sloppiness).

The foundations of precision, estimation, and units are introduced in our middle-school curriculum. On completing the middle-school curriculum, students are expected to be proficient in:

• Rounding with integers and decimals.
• Estimating addition, subtraction, multiplication, and division with integers, fractions, and decimals.
• Solving problems with units (customary and metric), including converting between units.

At the Algebra I/Integrated Math I level, students perform an extensive study of units and degrees of accuracy, including the following:

• The lower and upper bounds of a measurement.
• The greatest possible error of a measurement.
• Finding upper and lower bounds of areas and volumes.

In addition, a more extensive algebraic treatment of units and unit conversions is covered at this level.

Students are expected to apply this knowledge throughout the curriculum whenever they encounter numerical problem-solving situations. Answers to numerical problems must always be given with an appropriate degree of accuracy. Students are expected to state appropriate units whenever appropriate.

## 7. Look for and make use of structure

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see $7 \times 8$ equals the well-remembered $7 \times 5 + 7 \times 3$, in preparation for learning about the distributive property. In the expression $x^2 + 9x + 14$, older students can see the $14$ as $2 \times 7$ and the $9$ as $2 + 7$. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see $5 - 3(x - y)^2$ as $5$ minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Students are often expected to look for and make use of structure. Our curriculum was designed with the intention that every single pattern or structural insight should be made explicit to students. In addition to pointing out these patterns and structural insights to students, we also craft problems that require the student to leverage the patterns and structural insights in problem-solving contexts.

For example:

• Students must understand the reasoning behind why the vertex form of a parabola gives the coordinates of the vertex.
• At the middle school level, students are expected to be fluent in converting between expanded and standard forms, regrouping, and understanding how these manipulations are utilized when applying standard algorithms.
• Recognize various forms of structure in quadratic expressions (e.g., perfect square, difference of squares, or equations that can be factored) and use this structure to solve quadratic equations using a method appropriate to the structure.

When teaching new concepts, we often leverage patterns so that the new concept seems like a natural extension of existing knowledge. For instance, to introduce students to the concept of raising a number to the zeroth power, we start by showing the following pattern:

\begin{align*} 2^4 &= 16 \\ 2^3 &= 8 \\ 2^2 &= 4 \\ 2^1 &= 2 \\ 2^0 &= ? \end{align*}

Each time we decrease the exponent, the result gets cut in half, so we expect that $2^0$ is half of $2^1$.

## 8. Look for and express regularity in repeated reasoning

​​Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing $25$ by $11$ that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through $(1, 2)$ with slope $3$, middle school students might abstract the equation $(y - 2)/(x - 1) = 3$. Noticing the regularity in the way terms cancel when expanding $(x - 1)(x + 1)$, $(x - 1)(x^2 + x + 1)$, and $(x - 1)(x^3 + x^2 + x + 1)$ might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Students are expected to identify situations where calculations are repeated and a clear pattern can be found and then use this information to find a more efficient method or a generalization.

Some examples:

• Students deduce the equation of a vertical line as $x = k$ (for some real number $k$) by plotting several points on the line and recognizing that the first coordinate of each point is $k$. They then deduce that only partial information about one point on the line needs to be known to determine the equation of the line. Analogously for horizontal lines.
• At the Precalculus/Integrated Math III level, students identify infinite limits of rational functions by iteratively evaluating the function near an asymptote using a calculator, approaching the asymptote from both the left and right. They then deduce that the limit can be found by evaluating the function at a maximum of two points very close to the asymptote.

Similar deductions can be made, for example, when considering the infinite limits of a polynomial, where only the leading term is important.

• Observe, by repeated reasoning, that most terms in the partial sums of a telescoping series always cancel, and use this observation to calculate the sum of the series.
• By differentiating a power series term by term, we can sometimes deduce that the nth term of the differentiated series follows a particular pattern and use this to write down a general formula for the nth term.