PenPal Schools provides an excellent opportunity for students to develop and apply mathematics skills in an authentic, global context through project-based learning!
"I was impressed with how thoughtful my students were about their responses, particularly after they learned they had authentic audiences! The topics really hit home."
Wendi Pillars
North Carolina, USA
With the teacher dashboard, educators can assess their students' work across a variety of learning objectives to help them improve:
PenPal Schools learning objectives are aligned to a variety of academic standards, including Common Core State Standards, Texas Essential Knowledge and Skills, International Baccalaureate Organization Standards, and the National Council of Teachers of Mathematics (NCTM). Read on to see how you can use PenPal Schools learning objectives to assess and improve students' abilities relative to the standards that interest you.

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Expressions and Equations
PenPal Schools Learning Objective: Expressions & Equations
Student applies understanding of addition, subtraction, multiplication, and/or division to write and solve algebraic expressions
Anchor Standard: CCSS.Math.Content.7.EE.B.3: Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Grade Level Standards:
§111.26-28. Mathematics, Middle School
​(6)  Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships. The student is expected to:
(A)  identify independent and dependent quantities from tables and graphs;
(B)  write an equation that represents the relationship between independent and dependent quantities from a table; and
(C)  represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
(7)  Expressions, equations, and relationships. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:
(A)  generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization;
(B)  distinguish between expressions and equations verbally, numerically, and algebraically;
(C)  determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations; and
(D)  generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.
§111.48. Algebraic Reasoning
(7)  Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:
(A)  add, subtract, and multiply complex numbers;
(B)  add, subtract, and multiply polynomials;
(C)  determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two;
(D)  determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods;
(E)  determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping;
(F)  determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two;
(G)  rewrite radical expressions that contain variables to equivalent forms;
(H)  solve equations involving rational exponents; and
(I)  write the domain and range of a function in interval notation, inequalities, and set notation.
Criterion A: Knowing and Understanding
​Students select and apply mathematics to solve problems in both familiar and unfamiliar situations in a variety of contexts, demonstrating knowledge and understanding of the framework’s branches (number, algebra, geometry and trigonometry, statistics and probability)
Content Standards: Algebra
​Algebra is best learned as a set of concepts and techniques tied to the representation of quantitative relations and as a style of mathematical thinking for formalizing patterns, functions, and generalizations. Even young children can be encouraged to use algebraic reasoning as they study numbers and operations and as they investigate patterns and relations among sets of numbers.
Geometry
PenPal Schools Learning Objective: Geometry
Student solve real-world mathematical problems involving area, perimeter, surface area, volume, and other principals of geometry
Anchor Standard: CCSS.Math.Content.6.G.A.1: Student solve real-world mathematical problems involving area, perimeter, surface area, or volume
Grade Level Standards:
CCSS.Math.Content.5.G.A.1
CCSS.Math.Content.5.G.A.2
CCSS.Math.Content.5.G.B.3
CCSS.Math.Content.5.G.B.4
CCSS.Math.Content.6.G.A.1
CCSS.Math.Content.6.G.A.2
CCSS.Math.Content.6.G.A.3
CCSS.Math.Content.6.G.A.4
CCSS.Math.Content.7.G.B.4
CCSS.Math.Content.7.G.B.5
CCSS.Math.Content.7.G.B.6
CCSS.Math.Content.8.G.C.9
§111.26-28. Mathematics, Middle School
​(7)  Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to:
(A)  solve problems involving the volume of cylinders, cones, and spheres;
(B)  use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders;
(C)  use the Pythagorean Theorem and its converse to solve problems; and
(D)  determine the distance between two points on a coordinate plane using the Pythagorean Theorem.
§111.41. Geometry
(c)  Knowledge and skills.
(1)  Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
(A)  apply mathematics to problems arising in everyday life, society, and the workplace;
(B)  use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
(C)  select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
(D)  communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;
(E)  create and use representations to organize, record, and communicate mathematical ideas;
(F)  analyze mathematical relationships to connect and communicate mathematical ideas; and
(G)  display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Content Standards: Geometry
​Students analyze characteristics of geometric shapes and make mathematical arguments about the geometric relationship, as well as to use visualization, spatial reasoning, and geometric modeling to solve problems.
Mathematics Modeling
PenPal Schools Learning Objective: Math Modeling
Student use mathematical models such as drawings, manipulative objects, and/or mathematical equations to represent and solve problems
Anchor Standard: Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.
Grade Level Standards:
High School: Modeling
§111.26-28. Mathematics, Middle School
​Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
(A)  apply mathematics to problems arising in everyday life, society, and the workplace;
(B)  use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
(C)  select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
(D)  communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;
(E)  create and use representations to organize, record, and communicate mathematical ideas;
(F)  analyze mathematical relationships to connect and communicate mathematical ideas; and
(G)  display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
§111.43. Mathematical Models with Applications
(b)  Knowledge and Skills:
(1)  Reading/Vocabulary Development. Students understand new vocabulary and use it when reading and writing. Students are expected to:
(A)  determine the meaning of grade-level technical academic English words in multiple content areas (e.g., science, mathematics, social studies, the arts) derived from Latin, Greek, or other linguistic roots and affixes;
(B)  analyze textual context (within a sentence and in larger sections of text) to distinguish between the denotative and connotative meanings of words;
(C)  produce analogies that describe a function of an object or its description;
(D)  describe the origins and meanings of foreign words or phrases used frequently in written English (e.g., caveat emptor, carte blanche, tete a tete, pas de deux, bon appetit, quid pro quo); and
(E)  use a dictionary, a glossary, or a thesaurus (printed or electronic) to determine or confirm the meanings of words and phrases, including their connotations and denotations, and their etymology.​
Criterion D: Applying mathematics in real-life contexts
​Students transfer theoretical mathematical knowledge into real-world situations and apply appropriate problem-solving strategies, draw valid conclusions and reflect upon their results.
Process Standards: Representations
​Mathematical ideas can be represented in a variety of ways: pictures, concrete materials, tables, graphs, number and letter symbols, spreadsheet displays, and so on. The ways in which mathematical ideas are represented is fundamental to how people understand and use those ideas. Many of the representations we now take for granted are the result of a process of cultural refinement that took place over many years. When students gain access to mathematical representations and the ideas they express and when they can create representations to capture mathematical concepts or relationships, they acquire a set of tools that significantly expand their capacity to model and interpret physical, social, and mathematical phenomena.
Number Sense
PenPal Schools Learning Objective: Number Sense
Student demonstrates fluency with single and multi-digit numbers
Anchor Standard: CCSS.Math.Content.4.NBT.B.4: Use place value understanding and properties of operations to perform multi-digit arithmetic.
​Grade Level Standards:
§111.26-28. Mathematics, Middle School
(2)  Number and operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms. The student is expected to:
(A)  classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers;
(B)  identify a number, its opposite, and its absolute value;
(C)  locate, compare, and order integers and rational numbers using a number line;
(D)  order a set of rational numbers arising from mathematical and real-world contexts; and
(E)  extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.
Content Standards: Number and Operations
​Understanding numbers, developing meanings of operations, and computing fluently. Young children focus on whole numbers with which they count, compare quantities, and develop an understanding of the structure of the base-ten number system. In higher grades, fractions and integers become more prominent. An understanding of numbers allows computational procedures to be learned and recalled with ease.
Problem Solving
PenPal Schools Learning Objective: Problem Solving
Students make sense of word problems and persevere in solving them
Criterion B: Investigating Patterns
Students work through investigations to become risk-takers, inquirers and critical thinkers.
Process Standards: Problem Solving
​Students require frequent opportunities to formulate, grapple with, and solve complex problems that involve a significant amount of effort. They are to be encouraged to reflect on their thinking during the problem-solving process so that they can apply and adapt the strategies they develop to other problems and in other contexts.