Mathematics

We help teachers, coaches, and administrators to define and work towards a common vision of what it means to know, learn, and teach mathematics. 

We provide teachers with the knowledge and tools to engage all students in rigorous mathematics instruction and implement the kinds of cognitively demanding tasks that are called for by the National Council of Teachers of Mathematics and 21st century standards.

We understand the power of shared experiences. We ask participants to implement common tasks, lessons, or units with their students and bring back artifacts of practice for discussion, reflection, and further planning.

PROFESSIONAL DEVELOPMENT

Our professional development courses help mathematics educators
  • Select cognitively demanding tasks that are consistent with 21stcentury standards 
  • Modify tasks to increase the cognitive demand
  • Engage students in cognitively demanding tasks in ways that maintain the cognitive demand 
  • Use targeted questions to assess and advance student learning so that all students have the opportunity to advance in their thinking
  • Engage students in discussions to make sense of mathematical concepts and make connections between and among mathematical representations in order to form mathematical generalizations 
  • Use quick writes to formatively assess student growth 
  • Study student work to identify what students know and can do and determine instructional next steps

Supporting Ongoing Achievement Responsively

SOAR tools and activities are designed to identify and support students' mathematical understanding that is in need of further inquiry and development

 

SOAR  Survey Tools

The survey tools are comprised of three parts: a set of survey questions, a monitoring checklist, and a tiered survey summary. The survey can be used as a stand-alone tool or as a follow-up to universal screeners that give only broad areas of need. Teachers gain valuable data on their students’ ability to make send of, represent, solve, and explain mathematical concepts.

SOAR Activities

The intervention activities are designed and sequenced to support students across tiers of intervention. These activities are aligned to the survey tool and are designed to support student learning in making sense, creating and using representations, using strategies, and providing explanations. Each activity has teacher facing materials that outline the mathematical foci, identify the essential mathematical understandings, as well as provide questions to target those understandings.

SOAR tools and activities are available for the following concepts: 

  • Adding and Subtracting Situation Problems (Grades 1-3)
  • Multiplication and Division of Situation Problems (Grades 3-5)
  • Fractions as Numbers and Operating with Fractions (Grades 3-5)
  • Ratios and Proportional Relationships (Grades 6-8)
  • Expressions and Equations (Grades 6-8) 

INSTRUCTIONAL MATERIALS

Our professional development courses range from short, targeted workshops to courses that are embedded in multi-year school, district, or state partnerships. Our courses are most effective when teachers work side-by-side with professionals who support classroom instruction, such as coaches, administrators, and central office educators. Additional courses are available for leaders to support mathematics teaching and learning.

The IFL's Sets of Related Lessons include eight high-level tasks designed around key mathematics concept, and accompanying lesson guides. Sets of Related Lessons can be strategically integrated into existing curricula.

A Set of Related Lessons has two types of tasks: developing understanding tasks and solidifying understanding tasks. Tasks are sequenced to provide students with opportunities to investigate concepts and skills in a coherent way. First, to make discoveries and then to form mathematical generalizations related to the concepts under study.

Sets of Related Lessons include:

  • an outline of the concepts students will study;
  • an articulation of strategies and representations for students to learn and/or use;
  • multiple and varied forms of formative assessment, including opportunities for students to write about their mathematical understanding; and
  • structures and routines to support students' engagement with tasks, including information about:
    • setting up the task so students have an entry point;
    • supporting students in exploring the mathematical ideas developed in the task both individually and in small groups; and
    • engaging students in teacher-orchestrated discussions to deepen and solidify their understanding of the key mathematical ideas.

A unique aspect of the Sets of Related Lessons is the identification of essential understandings (EUs)—underlying mathematical truths—within the lessons. EUs drive the sequence of the solution paths and the discussion questions outlined in the lesson guides.

The IFL also offers performance-based assessments (PBAs) to assess students' learning of concepts and practices in specific Sets of Related Lessons. Sets of Related Lessons with associated PBAs are marked with an asterisk (*).

Sets of Related Lessons

 Elementary (K-5)

 

Grade, Focus & TitleDescription
Addition Situations: Solving for the Sum Grade K PURCHASE  This set of related lessons is designed to provide students with multiple opportunities to explore the concepts related to counting and cardinality. Students will
  • find the quantities of sets by counting;
  • explore number sequence by starting, stopping, and then continuing counting at various points along the number sequence;
  • investigate repeating patterns of numbers one through nine and counting across decades; and
  • grapple with the dual meaning of one by understanding one as a single item and one as a set of 10 items.
Approximately 8-15 days of instruction.
"Add to" Situations: Unknowns in All Positions* Grade 1 PURCHASE This lesson set provides a study of "adding to" situations that explore solving for unknowns in all positions (whole, change, start). Students will:
  • solve tasks with "add to" situations using a variety of strategies and part-part-whole maps;
  • determine if the "unknown" in a situation represents a part of the situation or the whole amount in a situation;
  • apply and discuss the commutative property of addition;
  • utilize subtraction to solve unknown-addend problems; and
  • use the part-part-whole map to respond to problems with situations.
Approximately 8-15 days of instruction.
The Relationship Between Addition and Subtraction* Grade 1 PURCHASE This lesson set provides an in-depth study of part-part-whole relationships as represented in "take from" and "take apart" situations and subtraction equations. Students will:
  • solve "take from" and "take apart" situations using a variety of strategies and part-part-whole maps;
  • determine if the "unknown" in a situation represents a part of the situation or the whole amount in a situation;
  • interpret tables and diagrams;
  • discuss subtraction as addition with an unknown-addend; and
  • write and discuss related addition and subtraction equations.
Approximately 8-15 days of instruction.
Put Together and Compare Situations* Grade 2 PURCHASE This lesson set provides a study of story situations that explore the part-part-whole relationship, and through it also, the relationship between subtraction and addition with an unknown addend. Students will:
  • make sense of and solve "put together" situations involving an unknown addend using a variety of strategies and representations;
  • determine if the unknown in "put together" situations represents a part or the whole amount in the situation;
  • make sense of and solve "compare" situations with an unknown difference using a variety of strategies and representations;
  • determine the unknown difference as being "how many more" or "how many fewer"; and
  • understand and utilize the relationship between subtraction and addition with an unknown addend.
Approximately 8-15 days of instruction  
Addition Strategies and Flexible Thinking* Grade 2 PURCHASE  This set of related lessons provides students with an opportunity to develop a deep understanding of addition with two-digit numbers while providing a bridge into addition with three-digit numbers. Students will:
  • solve "put together" situations using base ten blocks and/or a hundreds chart;
  • determine if the unknown in a situation represents a part or a whole;
  • understand that tables can be "read" and can represent part-part-whole relationships;
  • apply strategies to solve addition problems; and
  • use a variety of representations to respond to situations.
Approximately 8-15 days of instruction
A Focus on Compare Situations Grade 2 PURCHASE  This set of related lessons provides students with an opportunity to grapple with the relationship between subtraction and missing addend addition when using both operations to solve compare situations with an unknown difference. A variety of contexts provide opportunities for students to compare sets of items, heights, lengths, and distances traveled. Students will:
  • solve "compare" story situations with an unknown difference involving sets of items, lengths, heights, and distance traveled and discuss attributes of "compare" story situations;
  • discuss the relationship between subtraction and missing addend addition when determining a comparison situation;
  • use strategies to solve compare situations: adding on to the small set to equal the larger set and subtracting by removing aligned parts to determine the difference;
  • use strategies to solve subtraction equations: regrouping a ten into ones in order to subtract and making an easier number to subtract and adjusting;
  • select and use tools to make sense of situations (base ten blocks, diagrams, and number lines); and
  • interpret and make observations about data represented in bar graphs.
Measurement: Solving for Area of Figures Grade 3 PURCHASE  This set of related lessons provides students with a study of area measurement. Students will learn the meaning of area by tiling figures with square tiles of various sizes and solve for the unknowns in all positions (the area or the unknown length or width). Students will:
  • solve area problems using a variety of strategies;
  • determine that an area of a plane figure has two dimensions: length and width;
  • recognize that repeated addition and multiplication can be used to determine the area of figures; and
  • determine the unknown in a story situation and relate the unknowns to multiplication and division.
Approximately 8-15 days of instruction.
Understanding the Associative and Distributive Properties of Multiplication Second Edition * Grade 3 PURCHASE This lesson set provides a study of the associative and distributive properties of multiplication. Students will:
  • solve multiplication word problems with two and three factors;
  • interpret products of whole numbers; and
  • apply the properties of operation (associative or distributive) to multiply.
Approximately 8-15 days of instruction
Comparing Fractions with Common Denominators or Common Numerators* Grade 3 PURCHASE  This lesson set is designed to provide students opportunities to compare and reason about the comparison of fractions with either common denominators or common numerators. Students will:
  • solve real-world situations that require a comparison of fractional amounts;
  • reason about the meaning of the numerator and denominator to compare fractions;
  • work with fractional amounts between 0 and 1 and fractional amounts greater than or equal to 1; and
  • create and use regional and number line models to represent and compare fractions.
Approximately 8-15 days of instruction
Exploring Multiplicative Comparisons* Grade 4 PURCHASE  This lesson set provides a study of multiplicative comparisons, problems in which students have to solve for "how many times greater/less." Students will:
  • explore the meaning of the two factors in comparison multiplication problems;;
  • recognize the inverse relationship between multiplication and division, and determine that division can be used to solve comparison multiplication problems when either the group size or the scaling factor is provided; and
  • use a variety of representations to respond to a problem situation, and weigh the advantages and disadvantages of such representations.
Approximately 8-15 days of instruction
Fractions: Relating Repeated Addition to Multiplication* Grade 4 PURCHASE  This set of related lessons provides students with an introduction to addition and subtraction of fractions with like denominators in the same whole. Students will:
  • represent non-unit fractions;
  • understand non-unit fractions as a sum of unit fractions; and
  • prove equivalency of multiplicative expressions involving one fractional factor by connecting repeated addition to multiplication.
Approximately 8-15 days of instruction
Writing and Comparing Decimal Numbers* Grade 4 PURCHASE  This lesson set develops understanding of the relationship between fractions and decimals, decimal number notation and addition, as well as comparison of decimal numbers. Students will:
  • express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 and write fractions with denominators of 10 and 100 with decimal notation;
  • combine tenths and hundredths;
  • compare two decimals to hundredths and justify the comparisons using visual models; and
  • compare decimals and express their relationship using the symbols, >, <, or = and explain that comparisons are valid only when the two decimals refer to the same whole.
Approximately 8-15 days of instruction
Addition and Subtraction of Fractions with Unlike Denominators Grade 5 PURCHASE  This lesson set provides students an introduction to fraction addition and subtraction when the denominators of fractions differ; students are given problems in which they have to consider different names for fractions relative to a same-sized whole in order to create same-sized units to combine or separate quantities. Students will:
  • understand and explain the naming of units in relation to a value of one;
  • understand and explain the effects of partitioning a given fraction to form an equivalent value; and
  • use a variety of strategies and representations to respond to a problem situation involving addition and subtraction of fractions relative to a same-sized whole and weigh their advantages and disadvantages.
Approximately 8-15 days of instruction
Multiplication with Fractions: Finding Portions of Numbers* Grade 5 PURCHASE  This lesson set provides students with a study of multiplication when one or both factors are fractions. Students apply the meaning of factors to numbers less than 1 using visual representations. Students will:
  • solve contextual situations using a variety of strategies;
  • determine the fractional models that can be used to represent multiplication with fractions;
  • recognize that repeated addition can be used for some fractional multiplication problems but not for others; and
  • reason about why the product in fractional multiplication is less than one or both of the factors.
Approximately 8-15 days of instruction
Dividing Fractions: Understanding Division with Numbers Less than 1* Grade 5 PURCHASE  This lesson set provides an introduction to division with numbers less than one, both when the whole number amount is being divided by less than one and when the fraction is being partitioned by a whole number. Students will:
  • apply understanding of division with whole numbers to division involving numbers less than one;
  • understand the meaning of the dividend and the divisor and recognize what happens when a unit fraction is in either position (but not both); and
  • use a variety of representations to respond to a problem situation and weigh the advantages and disadvantages of such representations.
Approximately 8-15 days of instruction.
Decimal Operations: Multiplication and Division* Grade 5 PURCHASE  This lesson set provides an introduction to decimal multiplication and division problems in which students have to consider how many groups of a fixed size comprise a total. Students will:
  • apply understanding of division with whole numbers to division involving numbers less than one;
  • recognize the meaning of the dividend and the divisor and what happens when a unit fraction is in either position (but not both); and
  • use a variety of representations to respond to a problem situation and weigh the advantages and disadvantages of such representations.
Approximately 8-15 days of instruction.
 

 Middle (6-8)

 

Grade, Focus & Title Description
Integers: Equivalent Expressions*
Grade 6
PURCHASE  
This lesson set provides students with an introduction to the equivalence of expressions and to generating equivalent expressions. Students will:

  • determine when to apply the distributive property and the order of operations;
  • define and evaluate an expression for a variable; and
  • use a variety of strategies and representations to determine equivalence and non-equivalence of expressions and generate equivalent expressions.

Approximately 8-15 days of instruction.

Solving One-Variable Equations and Inequalities*
Grade 6
PURCHASE  
This lesson set provides an introduction to one-variable equations and inequalities. The set of lessons allows students to work on writing and solving one-variable equations then transition from equations to inequalities. Students will:

  • translate contexts into one-variable equations and inequalities;
  • determine the meaning of the variable and numbers in an equation as they relate to a context in order to solve problems with equations; and
  • use inverse operations to isolate a variable; or if the variable solution is known, use substitution and evaluation.

Approximately 8-15 days of instruction.

Ratios: A Focus on Part-Part and Part-Whole Relationships
Grade 6
PURCHASE
Learning to reason proportionally is one of the key mathematical studies of the middle school grades. In this lesson set for grade 6, students use a variety of representations to explore ratio thinking. Students will:

  • examine the meaning of ratio by exploring the composed units and iterations that underlie the multiplicative nature of ratio;
  • understand the distinction between fraction and ratio, and learn that while some ratios can be part-whole relationships, others are part-part relationships;
  • scale ratios up as well as down; and
  • use a variety of representations to solve real-world problems.

Approximately 8-15 days of instruction.

Proportional Relationships Second Edition*
Grade 7
PURCHASE
This lesson set provides a study of Proportional Relationships. Students will learn to move from the grade 6 study of ratios and equations of the form px = q to the study of proportional relationships. They combine their previous understanding of the topics of both ratio and equation to examine tests that establish or refute the presence of a proportional relationship between two variables. Students will:

  • develop an understanding of proportional relationships;
  • make and confirm conjectures about tests for proportionality;
  • recognize proportional relationships visually when they are presented in equation or graphic form;
  • recognize proportional relationships in tables and contexts; and
  • articulate how to test for proportional relationships.

Approximately 8-15 days of instruction.

Solving Linear Equations in One-Variable*
Grade 8
PURCHASE  
This lesson set provides students with the opportunity to write and solve linear equations in one variable, focusing particular attention on what it means for a value to be in the solution set of a linear equation in one variable. Students will:

  • determine whether or not a value is in the solution set of a linear equation in one variable;
  • use the properties of operations and the properties of equality to solve linear equations in one variable; and
  • classify linear equations as having no solution, a unique solution, or infinitely many solutions.

Approximately 8-15 days of instruction.

Understanding and Solving Systems of Linear Equations*
Grade 8
PURCHASE  
This lesson set provides the opportunity for students to extend their previous understanding of linear equations to include systems of linear equations. Students will:

  • analyze what it means to be a solution to a system of equations;
  • solve a system of equations graphically, using a table of values, and algebraically; and
  • develop an understanding of the conditions under which a system that consists of two linear equations has zero, one, or infinitely many solutions.

Approximately 8-15 days of instruction

 

Understanding Congruence and Similarity*
Grade 8
PURCHASE
This lesson set provides the opportunity for students to explore congruence and similarity as a result of transformations in the plane. Students will:

  • examine the results of reflecting, translating, rotating, and dilating polygons in the plane;
  • recognize which transformations produce congruent figures and which do not;
  • recognize which transformations produce similar figures and which do not; and
  • determine a center of dilation.

Approximately 8-15 days of instruction.

 

 Algebra I, Algebra II, and Geometry 

Grade, Focus & TitleDescription
Creating and Interpreting Functions* Algebra I PURCHASE   This set of related lessons provides the opportunity for students to create and integrate functions, rate of change, and average rate of change of the function. Students will
  • create and interpret functions;
  • calculate and interpret the rate of change and average rate of change of functions;
  • examine the behavior of functions over given intervals; and
  • interpret key features of the graphs of functions.
Approximately 8-15 days of instruction
Developing and Understanding Quadratics* Algebra 1 PURCHASE  This set of related lessons examines the graphs of quadratic functions and differentiates characteristics of quadratic functions from those of linear functions. Students will
  • create and interpret graphs for quadratic equations;
  • learn strategies for solving quadratic equations; and
  • understand how the root of a function is connected to its graphical representation.
Approximately 8-15 days of instruction
Solving Problems Using Linear and Exponential Models Algebra I PURCHASE   This set of related lessons provides students the opportunity to extend their previous understanding of functions to include exponential functions. Students will
  • examine the way that exponential functions grow and compare this to the growth of linear functions;
  • build linear and exponential function rules from contexts and tables of values for both increasing and decreasing functions;
  • graph linear and exponential functions and connect features of these graphs to the parameters in the function rules; and
  • determine whether situations can be modeled by linear or exponential functions and identify distinguishing features of these situations.
Approximately 8-15 days of instruction.
Building Polynomial Functions Algebra II PURCHASE   This set of related lessons provides students the opportunity to use arithmetic to combine polynomial functions and explore the relationship between the original and new functions. Students will
  • deepen their understanding of how multiple representations are used to represent the same relationship between two variable quantities;
  • learn how to combine polynomial functions arithmetically using multiple representations; and
  • understand the relationship between factors and zeros of a polynomial function.
Approximately 8-15 days of instruction.
Investigating Rational Exponents Algebra II PURCHASE   This set of related lessons provides students the opportunity to investigate the characteristics of expressions and equations with rational exponents. Students will
  • deepen their understanding of the properties of rational exponents;
  • write rational expressions in equivalent forms;
  • sketch graphs of equations with rational exponents in the first quadrant; and
  • compare the relative size of numbers written using rational exponents.
Approximately 8-15 days of instruction.
Reasoning with Rational Equations Algebra II Coming Soon COMING SOON!
Investigating Coordinate Geometry and Its Use in Solving Mathematical Problems Geometry PURCHASE This set of related lessons provides students the opportunity to explore coordinate geometry as a mechanism for proving simple geometric theorems and solving problems. Students will
  • generalize midpoint and distance formulas using coordinate geometry;
  • use coordinate geometry to explore patterns and make conjectures about midpoint, distance, perpendicular bisectors, and midsegment; and
  • use coordinate geometry to prove or disprove conjectures.
Approximately 8-15 days of instruction
Investigating Congruence in Terms of Rigid Motion Geometry PURCHASE  This set of related lessons provides students the opportunity to formalize their understanding of the rigid motions: translation, rotation, and reflection. Students will
  • formalize their definition of rigid motion in terms of the movement of figures in the coordinate plane;
  • describe and predict the transformation of shapes undergoing a sequence of rigid motions;
  • use the definition of congruence in terms of rigid motion to prove the criteria for triangle congruence; and
  • apply the criteria for triangle congruence in solving problems and proving theorems.
Approximately 8-15 days of instruction
Trigonometric Ratios: Solving the Mystery of Similar Right Triangles Geometry Coming Soon COMING SOON!