In this investigation students explore the concept of ratios and their relationship with factors and multiples. Given an assortment of cogs (gears) with different numbers of teeth, solvers are asked to find which pairs allow for each tooth of one to fit into every gap of the other, and to form conjectures regarding which pairs would work generally. The problem is well-suited for group work. The Teachers' Notes page offers suggestions for implementation, key questions, student sheets, and links to articles, videos, and an interactive Cog Environment (cataloged separately).

This flash applet provides students with an activity to become more familiar with factors and multiples. The challenge is to arrange the four number cards (1, 2, 3 and 21) on a square of the grid to make as many different diagonal, vertical or horizontal lines as possible. The number card can be placed on a square of the grid if the square is the same number, a multiple of that number and or a factor of that number. Users have the ability to change the difficulty level. The Teachers' Notes page offers rationale, suggestions for implementation, key discussion questions, ideas for extension and support.

This problem helps learners improve their knowledge of factors, especially those in the usual multiplication tables, and encourages the problem solving strategy of trial and error. The goal of the game is to go around the track in as few moves as possible, keeping to the rules that a player can move any number of spaces which is a factor of the number the player is on, except 1. There is a "training" track to play on initially to see the rules in action and then a more complicated track for players to use. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support.

This problem offers opportunities for students to reinforce their understanding of factors and multiples and provides them the chance to justify their solutions. The goal is for the students to create number chains of four whole numbers that can range from 2 to 100 and each consecutive number is a multiple of the previous number. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support, and a link to a spreadsheet for students to experiment with placing numbers in specific boxes in the chain.

This puzzle, played with cards on a board (downloadable file), provides an interesting context in which students can apply their knowledge of number properties. Students attempt to arrange 25 numbers and 10 property headings into a 5 by 5 grid so that each number satisfies two conditions. Properties addressed include primes, square and triangular numbers, specific sets of multiples and factors, and parity. It can be worked individually or in small groups cooperatively. The Teachers' Notes page offers suggestions for implementation, discussion questions, ideas for extension and support, and links to an article, "Using Games in the Classroom" (catalogued separately).

This videos shows a few basics about finding the greatest common factor and least common multiple of 2 numbers.  Attribution:Rovnet, CC BY-SA 3.0 <http://creativecommons.org/licenses/by-sa/3.0/>, via Wikimedia Commons

This three-act math task utilizes videos and questioning to help students explore factors and multiples.

This problem reinforces a learner's understanding of factors and multiples. Students are presented with the hundred grid or parts of the grid where specific numbers have been shaded and must find out which factors have been chosen in order to produce the shading. A link to a spreadsheet which shades the squares according to the chosen factors, can be used by students to check their hypotheses. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, and ideas for extension and support.

Open Middle tasks provide opportunities for student to approach a mathematical task using different strategies and representations. They can be used as a warm-up/closing activity, as a formative assessment, or to facilitate discourse and discussion and get insite into student thinking and problem solving. These tasks provide a great opportunity for student to engage with the Standards for Mathematical Practice.

Open Middle tasks provide opportunities for student to approach a mathematical task using different strategies and representations. They can be used as a warm-up/closing activity, as a formative assessment, or to facilitate discourse and discussion and get insite into student thinking and problem solving. These tasks provide a great opportunity for student to engage with the Standards for Mathematical Practice.

Open Middle tasks provide opportunities for student to approach a mathematical task using different strategies and representations. They can be used as a warm-up/closing activity, as a formative assessment, or to facilitate discourse and discussion and get insite into student thinking and problem solving. These tasks provide a great opportunity for student to engage with the Standards for Mathematical Practice.

Open Middle tasks provide opportunities for student to approach a mathematical task using different strategies and representations. They can be used as a warm-up/closing activity, as a formative assessment, or to facilitate discourse and discussion and get insite into student thinking and problem solving. These tasks provide a great opportunity for student to engage with the Standards for Mathematical Practice.