Mathematical modeling, in which students use mathematics to explain or interpret physical, social, or scientific phenomena, is an essential component of the high school curriculum. The Common Core State Standards for Mathematics (CCSSM) classify modeling as a K-12 standard for mathematical practice and as a conceptual category for high school…

One method of proof is to provide a logical argument that demonstrates the existence of a mathematical object (e.g., a number) that can be used to prove or disprove a conjecture or statement. Some such proofs result in the actual identification of such an object, whereas others just demonstrate that such an object exists. These types of proofs are…

As teachers working with students in entry-level algebra classes, authors Amy Nebesniak and A. Aaron Burgoa realized that their instruction was a major factor in how their students viewed mathematics. They often presented students with abstract formulas that seemed to appear out of thin air. One instance occurred while they were teaching students…

Tossing a fair coin 1000 times can have an unexpected result. In the activities presented here, players keep track of the accumulated total for heads and tails after each toss, noting which player is in the lead or whether the players are tied. The winner is the player who was in the lead for the higher number of turns over the course of the game.…

Geometry students need challenges. They need to apply what they already know to new contexts. As a result, high school teacher Wayne Nirode is always looking for groups of related problems of theorems to challenge his geometry students. He came across one such group or problems when reading Jun's (2012) one-page abstract posted online for the…

Constructing viable arguments and reasoning abstractly is an essential part of the Common Core State Standards for Mathematics (CCSSI 2010). This article discusses the scenarios in which a mathematical task is impossible to accomplish, as well as how to approach impossible scenarios in the classroom. The concept of proof is introduced as the…

A mathematics classroom that reflects the vision of NCTM's "Principles and Standards for School Mathematics" will have the teacher posing problems, asking questions that build on students' thinking, and encouraging students to explore different solutions. In teaching about area, it is not sufficient to give students the…

For many students, making connections between mathematical ideas and the real world is one of the most intriguing and rewarding aspects of the study of mathematics. In the Common Core State Standards for Mathematics (CCSSI 2010), mathematical modeling is highlighted as a mathematical practice standard for all grades. To engage in mathematical…

Today, the Common Core State Standards for Mathematics (CCSSI 2010) expect students in as early as eighth grade to be knowledgeable about irrational numbers. Yet a common tendency in classrooms and on standardized tests is to avoid rational and irrational solutions to problems in favor of integer solutions, which are easier for students to…

At a time when the debate continues over whether homework is overused, optional, or essential or favors well-off students over those with little home support, teachers must understand ways in which effective homework strategies can help narrow the achievement gap. Vatterott (2009, p. 94) argues convincingly that the "old paradigm" of…

Retaining mathematical knowledge is difficult for many students, especially for those who learn facts and procedures without understanding the meanings underlying the symbols and operations. Repeated practice may be necessary for developing skills but is unlikely to make conceptual ideas stick. An idea is more likely to stick if students are…

Planning a lesson can be similar to planning a road trip--a metaphor the authors use to describe how they applied research and theory to their lesson planning process. A map and mode of transportation, the Common Core State Standards for Mathematics (CCSSM) and textbooks as resources, can lead to desired destinations, such as students engaging in…

The topic of continuity is typically not introduced until calculus and then reexamined in real analysis. Recognizing the connections between secondary school mathematics and the advanced mathematics studied at the college level allows teachers to better identify mathematical concepts in student ideas, motivate students by piquing their curiosity,…

Research shows that students often struggle with understanding empirical sampling distributions. Using hands-on and technology models and simulations of problems generated by real data help students begin to make connections between repeated sampling, sample size, distribution, variation, and center. A task to assist teachers in implementing…

The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…