18 Association for Middle Level Education www.amle.org

By Erin Muschla

The Common Core State Standards represent a significant shift in the curriculum for many schools. Although the standards for each grade level are different, the Common Core addresses standards for mathematical practice that transcend all grade levels.

The mathematical practice standards concentrate on understanding and persevering in problem solving, making and supporting conjectures, examining the reasoning of others, modeling mathematics, using tools appropriately with precision, and finding patterns through reasoning.

In addition to the mathematical practice standards, the Common Core highlights the knowledge students should acquire at each grade level. For example:Grade 6: The domains are ratios and proportional relationships, the number system, expressions and equations, geometry, and statistics and probability. Grade 7: The domains are the same as in grade six, but are extended. Students must master ratios and proportional relationships by the end of grade seven because these skills no longer are a domain in grade eight. Grade 8: The domains include the number system, expressions and equations, functions, geometry, and statistics and probability. Ratios and proportional relationships in sixth and seventh grade are replaced

Erin Muschla teaches middle school math in Monroe Township, New Jersey, and is coauthor of Teaching the Common Core Math Standards with Hands-On Activities. E-mail: erin.muschla@monroe.k12.nj.us

with functions in eighth grade. The other domains of grade seven are extended or mastered in grade eight.

Problem-Based Learning ActivitiesProviding hands-on activities and problem-based learning are the most effective ways for teachers to address the grade-level standards and mathematical practice standards. Hands-on activities and problem-based learning engage students and require them to think critically, experiment, and revise conjectures. Students work with manipulatives or conduct investigations as they solve problems.

Grade 6 Number SystemFor example, one of the domains in grade six Common Core State Standards for math is the number system. The first standard listed under the number system focuses on dividing fractions. Fractions often are challenging to even the best math students. However, visual models help students understand not only the algorithm for dividing fractions, but why the algorithm works.

To help your students master this standard, send them to the National Library of Virtual Manipulatives website at www.nlvm.usu.edu with the mission to find the activity, Number Line Bars–Fractions. This activity focuses on division, but also can be used with any operation with fractions.

Hands on�with theCommonCoreMath Standards

19Middle Ground October 2012

Grade 8 Geometry… PlusIn grade eight, students build upon the knowledge of geometry they gained in grade seven. In one standard, students are asked to use coordinates to describe the effects of transformations, specifically dilations, translations, rotations, and reflections.

To address this standard, students can use graph paper or math visualization software such as Geometer’s Sketchpad (www.dynamicgeometry.com), which allows students to dilate, translate, rotate, and reflect two-dimensional figures on the coordinate plane. Have students create a right triangle on the coordinate plane and write the original coordinates. Then ask them to dilate, translate, rotate, and

reflect this image, recording the coordinates after each step. Be sure your students are transforming the pre-image, not the image they just created.

When students complete this activity, have them create another shape and perform the same transformations. Discuss any patterns your students found.

Math InvestigationsThese activities are examples of the kinds of activities teachers can use to help students meet the math standards of the Common Core. Additional hands-on and problem-based activities can be found at NCTM’s Illuminations at http://illuminations.nctm.org and The Teaching Channel at www.teachingchannel.org.

By providing your students with hands-on activities and allowing them to investigate mathematics, your students will become proficient in the Common Core math standards. MG

When students click on the activity, a model and question appear. To guide your students through the activity, ask questions such as: • What do you think this model represents? • What is the common denominator shown on the model? • Why is a common denominator shown on this model? • How can you find the answer from the model?

When students understand the model, discuss how to construct new fraction bars. (The directions are listed on the website.) Allow students to work with a partner to make new fraction bars and create problems. As they make the fraction bars, have them write down the problems they are creating as well as the answers to their problems. After about 15 minutes, have pairs of students exchange their work. Each pair will use the website to check the problems and solutions.

You may extend this activity by writing a few division problems on the board and having students draw models that match the problems. This task will be more difficult without the website and will truly assess if your students have mastered this skill. As students develop problems and draw models, they will gain a deeper understanding of what it means to divide fractions.

Grade 7 GeometryOne of the domains in grade seven is geometry. The first standard within the geometry domain focuses on scale drawings. Many times when students work with scale, they use maps or buildings in the activity; however, they often have little sense of the distance between two cities on a map or the height of a building. Focus instead on smaller objects, such as items in the classroom. Begin by asking students if they have ever heard of the term “scale.” Conduct a mini-lesson on what scale is and why it is used.

Ask students to work in groups of three or four to create a scale drawing of the classroom. You may allow them to select an appropriate scale or provide one for them, such as 1 inch equals 2 feet.

Provide students with yardsticks or meter sticks (depending upon the scale they select) and graph paper and give them some guidance with regard to how detailed you want their scale drawing to be. For example, should they include all the furniture (the teacher’s desk, students’ desks, bookshelves)?

Group members present their completed drawings to the class, discussing the scale they selected, how they can be certain their scale drawing is accurate, and how they can obtain the real measurements from the scale. Students should discuss why some scale drawings might be different from others. Did they measure accurately? Did they select an inappropriate scale? Encouraging students to engage in conversation uncovers underlying misconceptions and encourages critical thinking.

Hands-on activities allow students to investigate mathematics and develop a deeper understanding of math concepts.