MiddleSchoolPortal/Numbers at Work

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Numbers at Work! - Introduction

Students need to see numbers at work in the world they live in, if only to answer their persistent question: “Why do I have to learn this?” But it can be both difficult and time-consuming to find middle school-level problems that demonstrate how people actually use mathematical thinking in concrete settings. Our aim in this publication is to offer such problems and projects for your students.



In the section Real Numbers in Real Life, we offer scenarios that connect mathematics to questions that touch the world of the middle school student. Besides their real-world context, these activities challenge students to deepen their understanding of numbers — especially fractions, decimals, and percents. Another section, Number Skills, concentrates almost entirely on equivalent representations of these numbers as well as practice in computation and estimation. These activities appear in game formats, as most likely to engage students in the practice they need.

Some numbers have become part of our common language — the boiling and freezing points of water or pi, for example — and we introduce them in Famous Numbers. You can connect your students’ new numbers skills to other disciplines with the ideas in Projects. A section on careers offers insights into the use of mathematics in various jobs. The information and activities may open students to the professional possibilities available to them through their knowledge of mathematics.

In the final section, NCTM Standard: Number and Operations Standard, we discuss how the aim of this publication aligns with the Principles and Standards for School Mathematics.

Background Information for Teachers

Topics under the Number and Operations strand of the Principles and Standards of School Mathematics (NCTM) can be surprisingly complex. For those teachers who would like to explore number systems in depth, an online workshop heads this list of resources. The other sites offer a treasure house of math activities from which to choose those most suited for your classroom.

In addition, you may want to check out the book Navigating Through Number and Operations in Grades 6-8. Published by the National Council of Teachers of Mathematics, it contains well-developed activities in which students explore and consolidate their understanding of fractions, decimals, and percents, and are introduced to proportionality. An accompanying CD-ROM contains all handouts as well as applets for the activities. You will find ordering information as well as the Table of Contents and Preface online at

Learning Math: Number and Operations Created for K-8 teachers, this online workshop examines core topics — understanding numbers, representations, relationships, and number systems. Videos as well as immediate feedback on problems enliven this web-based course. Three sessions that relate particularly to the middle school curriculum are: Fractions and Decimals, Rational Numbers and Proportional Reasoning, Fractions, Percents, and Ratios.

New York Times Daily Lesson Plan: Mathematics Each of these interdisciplinary lesson ideas grows out of a New York Times article and connects mathematics to real-world issues and statistics. The mathematics activities include, for example, examining graphs, designing a playground, and using census data from various years to create an illustrated time line.

CIESE: K-12 Online Classroom Projects These interdisciplinary projects, designed by the Center for Innovation in Engineering and Science Education (CIESE), can enhance the science and mathematics curriculum through compelling use of the Internet. Teachers and students throughout the world can collaborate on projects and use real-time data available from the Internet. An excellent resource for middle school and high school classes!

Data Analysis: As Real World as It Gets Data analysis shows the power and influence of numbers when they are collected, analyzed, and displayed graphically. The digital resources offered here range from individual problems to full projects, all offering opportunities for students to work with real-world numbers.

Real Numbers in the Real World

Where do numbers actually come into everyday living? The following problems offer scenarios that aim to connect mathematics to questions that touch the world of the middle school student.

Figure This! Math Challenges for Families Here you have 80 mathematical challenges that immediately involve middle school students in the mathematics of their everyday world. Each challenge contains an initial problem with a solution hint, a complete explanation of the answer, and additional problems related to the same challenge. The next four resources listed below are a sampling of those challenges that deal most directly with number and operations.

Movie Money: Do Movies Make Money? Does showing movies make money for theater owners? With information about box office receipts, the percentage that theater owners must pay to movie distributors, and operating expenses, students answer the question by making profit-and-loss calculations.

How Far Can You Go on a Tank of Gas? In this activity, students are given the mileage per gallon of gas, as determined by the EPA, and the tank capacity of three cars and asked to determine which car will go the farthest on a single tank of gas.

Majority Vote: What Percentage Does It Take to Win a Vote? In this activity, students explore what percentage means when a class votes on an issue. The percentage of the boys and the percentage of the girls who supported the issue are given, but not the number of boys and girls in the class!

Access Ramp: How Steep Can a Ramp Be? An access ramp must be built over a three-step staircase, with steps 7 inches high and 10 inches wide. How far away should the ramp start? This involves students in talking about and figuring slope.

Village Math These problems are unique in that they relate to the everyday activities of Alaskan natives, such as stretching and sewing beaver skins, picking and transporting berries, net fishing, and tarpapering a cabin roof. The worksheets focus on standard and nonstandard measurement, money, percentages, and other topics that involve arithmetic.

Fermi Questions Fermi questions emphasize estimation, rely on numerical reasoning, and promote questioning skills. A Fermi question might be: How many piano tuners are there in New York City? The site offers a collection of questions and illustrates how such questions might be answered, while stressing the multiple approaches and multiple answers possible. Solutions are given for two classic Fermi questions to demonstrate the technique of questioning and estimating.

Quick Take on …Math in Spring and Summer Sports Working within baseball and track themes, students practice their skills in finding averages and compound interest, applying the distance-time-speed formula, and converting from fractions to decimals to percentages. These math activities really engage the middle school student!

Quick Take on… Give Us Our Daily Math This one-pager offers resources that tie calculation to everyday situations — from buying a soda to taking after-school jobs to understanding weather reports. Included here are quick introductions to a class project and to a newspaper resource that can provide lesson ideas for the entire year.

Number Skills

What skills are needed for mastery of number operations? Assuming that addition and subtraction are under control at the middle school level, I have emphasized here multiplication, factoring, and work with fractions, percentages, and estimation. Assuming also that you have a quantity of standard drill sheets on hand, I have selected exercises in a puzzle or game format, usually interactive. The hope is that these innovative formats will engage your students in needed practice.

Multiplication Matho Practice on multiplication facts! Students must select the correct response from choices set out in a grid, all the while competing against the clock. Five correct answers in a row spell Matho!

Math Cats Among the many interactive math activities on this site are opportunities to work on multiplication and fractions. For example, go to Broken Calculator, Egyptian Fractions, Interactive Multiplication Table and Multiply It.

The Maths File Game Show In this collection are four interactive number games that drill concepts of number and operation. Students can practice such skills as rounding off numbers to significant figures, matching fractions with equivalent decimals or percentages, and finding factors and multiples. Three levels of difficulty are available for each game.

The Factor Game Two students use a game board with the numbers 1 to 30. One student circles a number; the other circles all its factors. Play continues until no more numbers can be circled. Each student’s score is the sum of his or her circled numbers. A complete lesson plan guides the analysis of the game, resulting in students’ identifying the properties of prime, composite, abundant, deficient, and perfect numbers — and honing their winning strategies! MSP full record

Fraction Four A great way to practice fraction, decimal, and percentage skills! This two-player game allows players to choose which type of fraction-related problem they want to solve and at what level of difficulty.

Quick Take on … Action with Fractions Even students in the higher middle school grades may lack a firm grasp of fractions. This one-pager features online resources that offer visually compelling, hands-on interaction with these numbers.

Estimator Four In this online game, two players practice addition, multiplication, and percentage skills as they estimate answers to calculation problems. Players choose the level of difficulty, the time allowed for solving each problem, and which skill to practice. These calculations will challenge middle school students at all grade levels!

Famous Numbers

A few numbers have acquired individual fame and can immediately engage the curiosity of students. The following resources involve learners in inquiry activities as they delve into the origin and characteristics of the boiling point of water, the golden ratio, and pi.

Boil, Boil, Toil and Trouble: The International Boiling Point Project The boiling point of water is well known as 100° C or 212° F, but is that the actual number always and everywhere? In this inquiry, people from all over the world boil water at different elevations and post data to discover which factor in the experiment (room temperature, elevation, volume of water, or heating device) has the greatest influence on boiling point.

The Golden Ratio Expressed as a number, the golden ratio is an irrational: 1.6180339887 . . . . This well-illustrated site offers seven activities that introduce students in a friendly way to the golden ratio and guide them in recognizing it in art and architecture. A final activity has students construct a golden section and even a golden spiral.

Projects: Numbers on the Big Stage

If you are looking for an interdisciplinary project that involves numbers, one of these resources may work for your class.

Math in Daily Life These activities connect numbers to everyday decision-making. A cooking exercise introduces ratios and proportion; an essay on population introduces exponential growth and bar graphs; a home decorating exercise explains how to calculate area; a banking and credit-card scenario introduces simple and compound interest.

Down the Drain: How Much Water Do You Use? Students first collect data from their household members and their classmates and then determine the average amount of water used by one person in a day. The power of numbers becomes even more evident as students compare their average to the average amount of water used per person per day in other parts of the world. Through the Internet, they can collect and share information with other students from around the country and the world. A teacher's guide is included as well as guidelines on how students can publish reports, photos, or other work directly to the project web site.

The Stowaway Adventure This multidisciplinary set of lessons has students use data from the Internet to determine the actual location of a real ship at sea, calculate its speed and destination, then predict through map and math skills when it will arrive. You will need access to the Internet but not necessarily in the classroom; all the needed details for finding ships at sea are provided, plus lesson plans and handouts.

The Global Sun Temperature Project In this interdisciplinary project, students record the temperature and the number of minutes of sunlight per day over one week. They then compare these results with those collected by classes around the world. With this information in hand, students answer the core question: How does proximity to the equator affect average daily temperature and hours of sunlight? Included for the teacher are lesson plans, enrichment activities, and information on joining the online project.

Annenberg/CPB: Cast Your Vote This site takes students into a fictional political campaign and its attendant focus on polls. The main topic is statistics; in this case, analyzing numbers collected through surveys of potential voters. Throughout the discussion, the site challenges students to consider a central question: What do the numbers tell us? To help in answering that question, concepts such as random sampling and margin of error are explained at a level open to older middle school students.

Careers in Numbers

How do numbers play out in the workplace? These online resources offer insights and can help you make the connection more relevant and engaging for your students.

Math at the Movies Through conversation, illustrations, and activities, students learn how numbers figure in making movie stunts, building camera lenses, and creating computer graphics.

Mathematics of Cartography: Mathematics Topics Here are answers to such fundamental questions as “What is a map?” and “What mathematics do you use with maps?” Problems are included, plus a full section on careers in surveying and mapmaking.

Ohio Math Works This site for the middle school level shows how mathematics figures in journalism, meteorology, the snack food industry, the amusement park industry, and the fashion world.

Quick Take on…Exploring Careers in Engineering Here you will find a set of links to excellent, student-friendly information on what it takes to be an engineer, the different pathways within the profession, and personal stories of young engineers.

Apply Lessons: Applications of Mathematics 9 and 10 These problems and activities connect mathematics to 17 careers in such varied fields as firefighting, veterinary science, event planning, and sports broadcasting. For each career featured, a professional describes how mathematics is used in real-life scenarios.

The West Point Bridge Design Contest A challenge developed by the U.S. Military Academy at West Point for upper middle and high school students! The contest aims to provide a realistic, engaging introduction to engineering. All the logistics for registering, the contest rules, and free software are offered at the site.

The FunWorks Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

SMARTR: Virtual Learning Experiences for Students

Visit our student site SMARTR to find related virtual learning experiences for your students! The SMARTR learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities. Visit the virtual learning experience on Integers.

NCTM Standards

Historically, number has been a cornerstone of the entire mathematics curriculum. — Principles and Standards for School Mathematics, p. 32.

The Principles and Standards for School Mathematics (NCTM, 2000) places Number and Operations as the first among its five content standards for prekindergarten through grade 12. At the middle grades level, the emphasis is on rational numbers — understanding, representing, and calculating them using a range of strategies. The goal is “developing fluency,” which requires “a balance between conceptual understanding and computational fluency” (p. 35).

Numbers at Work! highlights activities that help create balance by involving students in problem scenarios that demand flexible thinking as well as computational skills in working with fractions, decimals, and percents. When students work in contexts that come from the everyday world, it is expected that they relate more genuinely and creatively to numbers and to the operations on them. As stated in the Standards, middle grades students “acquire an appreciation for, and develop understanding of, mathematical ideas if they have frequent encounters with interesting, challenging problems” (p. 211).

You can find the 14 specific expectations for middle school students identified in the Principles at

Author and Copyright

Terese Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.

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Copyright October 2007 - The Ohio State University. Page last updated October 30, 2010. This material is based upon work supported by the National Science Foundation under Grant No. 0424671 and since September 1, 2009 Grant No. 0840824. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.