Formative Assessment Lessons

This resource is intended to support teachers using the Classroom Challenges (formerly called Formative Assessment Lessons) from the Mathematics Assessment Project in their class. Classroom Challenges are intended to support teachers in understanding what students understand, and provide feedback to students on their own understanding.

This is a guide to using Formative Assessment Lessons (renamed to Classroom Challenges) in your classroom.

This lesson unit is intended to help you assess how well students are able to translate between words, symbols, tables, and area representations of algebraic expressions. It will help you to identify and support students who have difficulty in:

• Recognizing the order of algebraic operations.
• Recognizing equivalent expressions.
• Understanding the distributive laws of multiplication and division over addition (expansion of parentheses).

The domains directly supported in this challenge (A-SSE, A-APR) are directly tested (12.6%) and support skills necessary for both creating equations and reasoning with equations, inequalities and function rules.

This lesson is intended to support students in making connections between situations and graphs representing those situations. This lesson is intended to help you assess how well students are able to:

• Articulate verbally the relationships between variables arising in everyday contexts. 
• Translate between everyday situations and sketch graphs of relationships between variables.
• Interpret algebraic functions in terms of the contexts in which they arise.
• Reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.

The domains directly supported in this challenge (F-IF, F-LE) are directly tested and are crucial for students using function relationships in both equation-based and function-based test situations.

This lesson is intended to help you assess how well students are able to:

• Form and solve linear equations involving factoring and using the distributive property (of multiplication over addition).

In particular, this lesson aims to help you identify and assist students who have difficulties in:  

• Using variables to represent quantities in a real-world or mathematical problem.
• Solving word problems leading to equations of the form px + q = r and p(x + q) = r.

The domains directly supported in this challenge (7-EE), representing the key middle school content required for high school mathematics success, is directly tested in all expression work within both algebraic and function contexts.  These skills are crucial for students reasoning abstractly and quantitatively (both with numbers and variables) and for understanding and applying grade-level mathematical models.

This lesson is intended to help you assess how well students are able to identify linear and quadratic relationships in a realistic context: the number of tiles of different types that are needed for a range of square tabletops.  In particular, this lesson aims to identify and help students who have difficulties with:

• Choosing an appropriate, systematic way to collect and organize data.
• Examining the data and looking for patterns; finding invariance and covariance in the numbers of different types of tile.
• Generalizing using numerical, geometrical or algebraic structure.
• Describing and explaining findings clearly and effectively.

The domain directly supported in this challenge (F-BF) is directly tested and supports students ability to draw the connections between function representations (pattern, table, graph, rule) and interpretations.

This lesson is intended to help you assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation. In particular, the lesson will help you identify and help students who have the following difficulties:

• Understanding how the factored form of the function can identify a graph’s roots.
• Understanding how the completed square form of the function can identify a graph’s maximum or minimum point.
• Understanding how the standard form of the function can identify a graph’s intercept.

The domains directly supported in this challenge (A-SSE, F-IF) are directly tested and are crucial for students using algebraic structures in both equation-based and function-based test situations.

This lesson is intended to help you assess how well students are able to interpret exponential and linear functions and in particular to identify and help students who have the following difficulties:

• Translating between descriptive, algebraic and tabular data, and graphical representation of the functions.
• Recognizing how, and why, a quantity changes per unit interval.
• To achieve these goals students work on simple and compound interest problems.

The domains directly supported in this challenge (A-SSE, F-LE) are directly tested and build students practice in constructing and understanding algebraic modeling.

This lesson unit is intended to help you assess how well students are able to manipulate and calculate with polynomials.

• Switching between visual and algebraic representations of polynomial expressions.
• Performing arithmetic operations on algebraic representations of polynomials, factorizing and expanding appropriately when it helps to make the operations easier.

This lesson unit is intended to help you assess how well students are able to solve problems involving area and arc length of a sector of a circle using radians. It assumes familiarity with radians and should not be treated as an introduction to the topic. This lesson is intended to help you identify and assist students who have difficulties in:

• Computing perimeters, areas, and arc lengths of sectors using formulas.
• Finding the relationships between arc lengths, and areas of sectors after scaling.

This lesson unit is intended to help you assess how well students are able to identify and use geometrical knowledge to solve a problem. In particular, it aims to identify and help students who have difficulty in:

• Making a mathematical model of a geometrical situation.
• Drawing diagrams to help with solving a problem.
• Identifying similar triangles and using their properties to solve problems.
• Tracking and reviewing strategic decisions when problem-solving.

This lesson unit is intended to help you assess how well students can:

• Understand the concepts of length and area.
• Use the concept of area in proving why two areas are or are not equal.
• Construct their own examples and counterexamples to help justify or refute conjectures.

In particular, the lesson will help you identify and help students who have difficulty:

• Solving problems by determining the lengths of the sides in right triangles.
• Finding the measurements of shapes by decomposing complex shapes into simpler ones.

The lesson unit will also help students to recognize that there may be different approaches to geometrical problems and to understand the relative strengths and weaknesses of those approaches.

This lesson unit is intended to help you assess how well students are able to:

• Use the Pythagorean theorem to derive the equation of a circle.
• Translate between the geometric features of circles and their equations.

This lesson unit is intended to help you assess how well students are able to:

• Describe in words the transformation that maps an object to a transformed image.
• Given a geometric figure and a rotation, reflection or translation, draw the transformed figure (or the original figure if the image is given.)
• Describe transformations as algebraic functions that take points in the plane as inputs and give other points as outputs.

This lesson unit is intended to help you assess how well students are able to create and solve linear and non-linear equations. In particular, the lesson will help identify and help students who have the following difficulties:

• Solving equations where the unknown appears once or more than once.
• Solving equations in more than one way.

This lesson unit is intended to help you assess how well students understand the notion of positive correlation. In particular this unit aims to identify and help students who have difficulty in:

• Understanding correlation as the degree of fit between two variables.
• Making a mathematical model of a situation.
• Testing and improving the model.
• Communicating their reasoning clearly.
• Evaluating alternative models of the situation.

This lesson unit is intended to help you assess how well students are able to:

• Use the properties of exponents, including rational exponents and manipulate algebraic statements involving radicals.
• Discriminate between equations and identities.

In this lesson there is also the (optional) opportunity to consider the role of the imaginary number.

This lesson unit is intended to help you assess how well students are able to:

• Understand conditional probability.
• Represent events as a subset of a sample space using tables and tree diagrams.
• Communicate their reasoning clearly.

This lesson unit is intended to help you assess how well students understand conditional probability and, in particular, to help you identify and assist students who have the following difficulties:

• Representing events as a subset of a sample space using tables and tree diagrams.
• Understanding when conditional probabilities are equal for particular and general situations.