# Factors

Factors and their multiples are so important to students’ work with fractions and number theory. These concepts come under the National Council of Teachers of Mathematics (NCTM) Number and Operations Standard for the middle grades. The resources here are all hands-on, if only virtually. Important to their success is the classroom talk that the resources generate.

The Factor Game
This is more than a game; it is a full lesson plan, complete with handouts, including the game board, and questions for discussion. Students practice finding factors and then analyze the winning strategies. It’s this analysis that leads them to talk about prime, composite, abundant, deficient, and perfect numbers. The plan is downloadable and printable. (See the online version below.)

The Factor Game (i-Math Investigations)
This is the same game but online and interactive. One advantage here is that a single student can play against the computer. What is missing, however, is the winning strategy analysis, so rich in discussion. If you decide to use this version, you can simply borrow the discussion questions from the pdf version above.

Coloring Multiples in Pascal’s Triangle
Good practice in finding multiples of small factors! Students roll a number cube, then color in the multiples of that number in Pascal’s triangle. As they click on all the multiples of the number rolled, they also practice multiplication and observe the surprising patterns that form in the triangle.

The activity starts with building a factor tree, but then moves to finding common factors and reviewing those old friends: GCF (greatest common factor) and LCM (least common multiple). The student must first find the prime factors for a pair of numbers. After two factor trees are built, the student drags the prime factors from each tree to a Venn diagram, showing the common factors of the two numbers in an overlapping area. Finally, the student enters the greatest common factor and the least common multiple, which can be checked immediately. This activity is interactive, visual, and good review!

# Action with Fractions!

Really understanding what fractions are, how they fit on the number line, and how to operate with them – add, subtract, multiply, divide – is central to learning decimals and percentages. According to the NCTM Principles and Standards, students in the middle grades should be expected to acquire a deeper understanding of fractions, decimals, and percents and an increased flexibility in using them to solve problems. Yet students may reach even the higher grades of middle school without a firm grasp of fractions. Perhaps a change in strategy could help? Let’s offer compelling visuals and hands-on manipulation of those fractured numbers!

COUNTDOWN: Number and Operations – Fractions
COUNTDOWN is an interactive television math program broadcast on cable television in Chicago. This web site contains the archives of those broadcasts. The 4-7 minute math movies consist of direct instruction and are reinforced with literature, manipulatives, activities and related computer instruction. Topics include logic, perimeter, area, probability, graphing, congruence, integers and much more. Movies are organized according to content standards established by the National Council of Teachers of Mathematics. Many of the movies have associated worksheets that can be printed.

Visual Fractions
A thorough tutorial on the topic—from identifying and comparing fractions to operations with them! Students work through interactive exercises and games. A complete step-by-step, illustrated explanation of each subtopic is included. Don’t miss the game of Finding Grampy, where students practice finding a mixed number in lowest terms as they look for Grampy on the number line.

Fraction Sorter
Using this online manipulative, students represent two to four fractions—such as 7/13, 2/7, 8/9, and 2/3—by dividing and shading areas of squares or circles. They then order the fractions from smallest to largest on a number line. The visual representation here is powerful.

National Library of Virtual Manipulatives
Under the middle school offerings, you will find the activity Adding Fractions. Students must do the usual exercise of finding equivalent fractions with common denominators, but here the fractions are represented visually as portions of a square. Once the computer checks that the fractions are correct, the students can drag the representations into a third box and enter the sum of the fractions. This is a learning experience! There are other activities on fractions as well, all worth checking out.

The Maths File Game Show
This BBC site offers a game that requires players to match fractions with equivalent decimals or percentages, Go to Saloon Snap. When a question is answered correctly, the player can move his or her piece across the board. The goal is to be the first to create a path across a 10-by-10 grid. It’s a real math exercise and it’s motivating!

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# Order of Operations Rules

The order of operations rules are important now in evaluating numerical expressions and later in the study of algebra. The featured resources offer explanations, examples of what goes wrong when the rules are misapplied, and problems to solve.

Amby’s Math Resources: Order of Operations
Check out this online tutorial offering a pre-test, practice exercises, and a post-test. It is especially handy as a quick review for students working independently.

Explaining Order of Operations
This online article explains the how and why for the simplification of numerical expressions.

Expression Evaluation
Here is information about applying the order of operations rules, as well as an online quiz with a practical application.

Matho 2
Try to win a bingo game while practicing simplifying numerical expressions.

Number Card
Students must correctly combine four numbers on a number card to form a target number.

Operation Order
This online game is a good practice tool because the correct solution is given when a wrong answer is entered. The easiest level involves addition and subtraction while the hardest level requires the use of all operations.

Order of Operations
This site features a clear explanation of the rules for the order of operations with examples. It offers several problems for students to solve, including one practical application.

Please Excuse My Dear Aunt Sally
Here is an explanation of the popular mnemonic with problems and puzzles to solve.

# Let’s Play Math!

Math games open an arena for strategic planning and reasoning, one of the NCTM Process Standards. Many games also provide practice in basic operations or problem solving, and all motivate the active participation we hope for in the classroom.

The Maths File Game Show
A collection of 12 interactive games that deal with basic concepts of mathematics: data handling, numbers, algebra, and measurement. Students can use order of operations to ensure that letters reach the right mailbox, compete in a test of fractions and percentages, and even guide a spaceship across Cartesian coordinates.

Towers of Hanoi: Algebra (Grades 6-8)
An online version of the ancient Towers of Hanoi puzzle, featuring three spindles and a graduated stack of two to eight discs, as decided by the player, with the largest disc on the bottom. The challenge is to move all discs from the original spindle to a new spindle in the smallest number of moves possible while never placing a larger disc on a smaller disc. By observing the pattern of number of discs to number of moves, students can generalize the relation and answer the question “What if you had 100 discs?”

Traffic Jam Activity
Why the jam? There are seven stepping stones and six people. Three stand on the left-hand stones and three on the right-hand; all face center. Everyone must move so that the people on the right and the people on the left pass each other, eventually standing on the side opposite from where they started. But no two people may stand on the same stone at the same time! This problem requires reasoning, but its solution also reveals a pattern that leads to an algebraic expression. A lesson plan is provided.

Clever Games for Clever People
Here are 16 strategy games from the book On Numbers and Games by mathematician John Conway. Each develops those critical thinking skills so valued in math. As for materials, only paper and pencil or, occasionally, crayons or a checkerboard, are needed. Game rules and setup are clearly described and illustrated for each game.

# Scale and Powers of 10

Scale is a unifying concept in science. Whether one is considering the size and scope of the universe, an atom, or anything in-between, conceptual understanding of scale is a prerequisite to understanding. Other phenomena requiring knowledge of scale include geologic time, pH, and maps. Cognitively, most middle school students hold a concrete, incomplete conception of scale. Teachers can facilitate conceptual change to a more abstract conception with help from these resources.

Secret Worlds: The Universe Within
View the Milky Way at 10 million light years from the Earth. Then move through space toward the Earth in successive orders of magnitude until you reach leaf cells, the cell nucleus, chromatin, DNA, and finally the subatomic universe of electrons and protons. You can control the speed at which the images are flashed automatically or manually.

Powers of 10
This brief webcast approaches scale from the opposite direction of the tutorial above — moving from a picnic in a park to outer space and back to human cells — and indicates when the scale has changed by a magnitude of 10X.

Table of Images from Wordwizz.com
Bruce Bryson is the author of this award-winning site. It provides a table of contents in which teachers will find images of interest, such as the solar system, a bee’s eye, and the atomic nucleus. You may decide to have pairs of students investigate one item from the list and then share their findings in order of magnitude with the rest of the class.

Earthquake Scale and Magnitude
Not all scales are in powers of 10, as can be seen at this site.

Weather – Wind Chill
This interactive site explains how wind chill is estimated and allows students to calculate it. In doing so, students see a different manifestation of scale and magnitudes involving the interaction of variables, rather than a one-to-one corresponding change as in powers of 10.