Student Explorations in Mathematics

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  • Demystifying Division (March 2015)

    Do your students struggle with division? Many students have misunderstandings about division that relate to its definition. This exploration takes division from its definition and integer division through factoring and algebraic division using manipulatives and a variation on the standard division algorithm. It also links division, as well as factoring with integers, to algebra. This is an opportunity for students at all levels to strengthen their understanding of division and for upper-level students to explore polynomial division.

    Stuck in Traffic (January 2015)

    Why are we stuck in so much traffic?  Can't we just build more roads? Data collected by the Texas A&M Transportation Institute shows driving time in urban areas appears to be increasing each year. In this activity, we will graph and analyze traffic data and learn about transportation issues in many of our metropolitan areas.  How does your region compare to others? What options are available to ease traffic congestion in our urban areas?

    It's A Hit! The Mathematics in Baseball (September 2014)
    As the Major League Baseball season comes to a close this fall and fans eagerly anticipate the World Series, you and your students can explore the mathematics of baseball with It's a Hit.
    Black and White and Math All Over (May 2014)
    What's black and white with math all over? A newspaper, of course! How can you help your students apply fraction, decimal, percent, area, and ratio concepts while investigating a real-world task? Once again, the answer involves using a newspaper. A newspaper is not just for reading. Your local newspaper has many examples of real-world applications of mathematics, such as graphs, charts, sports statistics, and mathematics examples found in advertisements. In this activity, your students will flex their mathematical muscles and apply fraction, decimal, and ratio concepts to determine what percentage of your local newspaper is really news.
    Out of the Park! Exploring the Mean in Sports (March 2014)
    Capitalize on student interest in sports with this classroom-ready investigation of mean. Means are often used to compare athletes’ performances in sports. Avid student fans may know statistics for their favorite team or players, but do they know how these stats are determined? This activity explores the application of mean in baseball and basketball.
    Constructing Pythagoras (January 2014)
    What do President James Garfield and Pythagoras have in common? They were both lovers of mathematics, and in 1876, James Garfield constructed an original proof of the Pythagorean theorem. "Constructing Pythagoras," a hands-on investigation, challenges your students to derive the Pythagorean theorem (a 2 + b 2 = c 2 ) using the same proof as the USA's twentieth president, James Garfield.
    Demystifying Multiplication [November 2013]

    Multiplication is a mystery to many students. Their first experience with this basic arithmetic operation is as a table of facts to memorize. The purpose of this exploration is for students to understand the meaning behind multiplication, eliminating the mystery.

    This activity begins with a brief formative assessment of student understanding of multiplication of two-digit numbers. It supports students' understanding of multiplication and its application. Constructivist pedagogy is used to move learning from the concrete (using manipulatives) to the representational (using pictures and diagrams) to the abstract (using algorithms). Students build models of the operation 27 x 15 and its result in a variety of ways. The activity promotes student reasoning and sense making by analyzing various multiplication algorithms (area models, partial products, lattice multiplication, and the traditional method).

    Build It, and They Will Come! Experiences in 3-D [September 2013]
    This activity is designed to provide a series of experiences that develop student understanding of how two-dimensional drawings can be translated into three-dimensional models and vice versa. Beginning with rich experiences in building structures, students move from creating their own concrete models to building structures from a variety of visual representations, including isometric drawings and orthographic and mat views.
    Are We There Yet? A Journey through Our Solar System [May 2013]
    The May SEM activity, "Are We There Yet? A Journey through Our Solar System," helps students use proportional reasoning to build a football-field-size scale model of our solar system. This is a hands-on activity designed to help students experience the vast distances between celestial objects. The activity concludes with students developing a logarithmic scale to help represent the immense distances between planets and other celestial objects in our galaxy.
    Divide Like an Egyptian [March 2013]

    This activity is a reworking of the Student Math Notes ( SMN ) activities " Egyptian Fractions " from March 2009 and " Egyptian Mathematics " from May 1983. A 2010-2011 user survey identified the original activity as one of the most popular SMN activities in the current collection of over seventy activities available ( www.nctm.org/sem , click Back Issues ).

    Give students an opportunity to explore fractions as the Egyptians used them as they are introduced to the Egyptian notations, answer questions of division using that notation, and then make connections to our modern representations. Students also explore a variety of methods for comparing fractions without needing common denominators. In this updated version, students work on developing their own methods for rewriting fractions.

    NBA Salaries: A Statistical Dunk [January 2013]

    This activity is designed to help students understand which measure of central tendency-mean or median-is more appropriate when analyzing and reporting data. Students use estimation to gain an intuitive understanding of mean and then build on that knowledge through calculations. They discover some of the misconceptions about the mean and median of a data set and begin to understand the realities of how numbers can be used to inform readers in a persuasive manner.

    The activity also gives students experience with using technology for the statistical calculations as they investigate relationships within scatterplots and lines of best fit. The activity concludes with students making conjectures-with supporting evidence-about which measure is more appropriate when reporting the data.

    Modular Arithmetic (May 2012)
    This issue of Students Explorations in Mathematics introduces students to modular arithmetic and its applications. This activity will help develop students’ problem-solving strategies and provide opportunities to apply and extend ideas to unfamiliar contexts. Students begin with familiar problems involving an analog clock and soon realize they already have some experience with modular arithmetic. Using this knowledge, they develop mathematical notation and ideas to help solve problems involving basic number theory. Applications to serial number coding are also discussed. To facilitate using the activity in your classroom, download the companion Teacher version, including a Teacher Notes section with tips, instructional strategies, and a solution key.