January 2017

• NCTM is excited to offer a featured resource in your grade band this month to help you make the most of your NCTM membership. As we launch our new Classroom Resource Collaboration Center, we'll keep members informed through Summing Up and social media. Check out the #NCTM_CRCC hashtag on Twitter to follow along as we share and discuss these and other NCTM classroom resources.

Primary

A common context for subtraction in word problems is “take away.” But subtraction helps us answer all sorts of interesting questions, such as those that ask about “How many more?” The NCTM Illuminations lesson Comparing Columns on a Bar Graph engages primary mathematicians in collecting data about their classmates, displaying the data visually, and then using subtraction in the context of comparison to answer questions they are curious about. Both virtual and physical manipulatives are in play to help students make sense of both data and subtraction when used as comparison.

Does your curriculum allow your young mathematicians to build growing patterns and predict what will happen in the 25th or 100th step? Kimberly A. Markworth’s article, Growing Patterns: Seeing beyond Counting, from Teaching Children Mathematics supports you in understanding the value of these types of problems, offers great question prompts to help students become more proficient at working with visual patterns and thinking algebraically, and presents some sample patterns for you to introduce to students to supplement your existing curriculum.

NCTM also has great resources for students to work with virtual pattern blocks or for you to generate visual pattern handouts for your students. Check out the virtual pattern blocks from Illuminations where students can build the patterns you describe and also create their own.

In middle school, what is the role of manipulatives? How can whole-body movement help students problem solve in new ways? How does technology support students’ problem solving, and when should the technology  be introduced? Suzanne Alejandre of the Math Forum at NCTM grapples with all of these questions in Developing Algebraic Thinking (PDF). This article describes a lesson plan about algebraic reasoning that used multiple formats to engage all students, including English language learners and students struggling with math. Alejandre describes why hand-scale manipulation, followed by technology, followed by paper-and-pencil work, and culminating with whole-body experiences worked best for her learners. Try the Traffic Jamgame yourself or with your students. Another idea is to reflect on an activity you use and think about what order your students might physically and/or virtually interact with the task, and then reflect and generalize about the experience.

Note: The Java applet accompanying Alejandre’s  lesson may not run on your browser. For a newer Java applet, try the version at the National Library of Virtual Manipulatives.

Algebra 1

Studying systems of equations gives students a unique opportunity to connect algebra to coordinate geometry, if they have the right contexts to think geometrically about intersecting lines. The Over the Hill (PDF) task from NCTM's Reasoning and Sense Making Task Library gives students a rich context to use both geometric reasoning and algebraic reasoning to determine where to place a cellular tower to reach customers on both sides of a hill. Since the task can also be approached through the use of similar triangles, students are given opportunities to connect slope, proportion, and similarity.

Geometry

In the Tapping into Trapezoids article from Mathematics Teacher, author Jeffrey J. Wanko shows how trapezoids (perhaps the least beloved of the special quadrilaterals in the high school curriculum) can provide rich contexts for exploring such topics as conjecture and proof, similarity and tiling, the Pythagorean theorem, and area. No matter which of these topics you are exploring, “Tapping into Trapezoids” can offer students opportunities to grapple, explore, problem solve, conjecture, and justify.

Algebra II/Pre-Calculus

Now that it’s cold outside, Newton's Law of Cooling is a great topic to explore with students who are learning about exponential functions, logarithms, and nonconstant rates of change. The Math Forum’s Trigonometry/Calculus problem Cooling Colas (PDF) gives students an opportunity to explore what it means for an object to cool at a rate that is proportional to the difference in its temperature and the air temperature. Be sure to read the teacher packet (PDF) to see how students solve this problem using exponential models, calculus, tables, and patterns.

Probability and Statistics

One way to describe statistics is that it is the study of variability. But how well do our students understand the different kinds of variability that statistics attempts to quantify and account for? Daniel L. Canada’s The Known Mix: A Taste of Variation from Mathematics Teacher  provides an activity in which students  notice, describe, and wonder about sampling variability. Interesting conceptions and misconceptions about why and how samples vary from the population will emerge as students explore the familiar context of drawing colored chips from a jar. The article includes students’ informal or "rough draft" thinking to give readers a sense of the kinds of thinking that their own students might engage in and what it might sound like.

Be sure to also check out these additional resources and tools for your classroom.

Math Forum - Problems of the Week

Get your weekly dose of math problems and puzzles from the Math Forum. You will also find more math resources and tools, as well as a Math Forum: Problems of the Week Blog, furthering discussion.

Brain Teasers

Want quick ideas for great back-to-school icebreaker classroom activities? We've got you covered. Challenge your new students and mathematics enthusiasts alike with these staff-picked puzzles. In need of more? Browse the entire Illuminations library and discover what's in store in this amazing resource.