Lesson outline

This task supplies Polypad’s number tiles to carry out a clear representation that helps students construct a much better understanding of factors, multiples and also prime numbers. The dynamic visual representations of number tiles developing rectangles give students a version for variable pairs of entirety numbers and for prime numbers. This representation likewise shows division with remainders: the number of columns reflects the divisor and the left over blocks in the last row reflects the remainder. The can also be offered for part enrichment job-related on elementary school number theory, such together modular arithmetic.

You are watching: How many rectangles can you build with a prime number of square tiles

Lesson objective: Understand exactly how to discover all the components of a number.

Lesson activity


Invite everyone in the course to pick a entirety number in between 1 and 100, compose it down, and then host it up for anyone to see. Climate ask them to look roughly the great – can they check out someone who determined a number which is a factor of their number? have the right to they check out someone who determined a number i beg your pardon is a multiple of their number? Take some time to discuss their findings, probably using the complying with prompts because that discussion:

Did anyone find it really simple to find multiples of your number? What number did you choose?Did anyone find it tough to find components of your number? What number did girlfriend choose?If who in the class chose 1: i m sorry number might everyone affix to?

This Polypad mirrors different division problems that can be represented using number tiles. Display students the blue examples, and ask them to write down the division problem, before revealing the green and red solution.

Main Activity

Create the number 32 utilizing three tens and two ones (left) and show students exactly how to pick all tiles and use the “merge tiles” button to produce a single block the 32 (right):


Next, show how dragging the black manage on the right changes the number of columns. Invite students to predict which arrangements that the 32 block will produce a complete rectangle, then test to view whether they are right. Class discussion should result in an commitment that there are six feasible rectangles that deserve to be made making use of the number 32: 1 by 32, 2 by 16, 4 by 8, 8 by 4, 16 through 2, and 32 by 1. Clarify that for this activity, we are considering 1 by 32 and also 32 by 1 together two separate rectangles also though they room rotations of every other.


Now provide students part time to check out the inquiry “How many rectangles?” through numbers of your choosing. Remind them that the prestige of recording their results. You might model just how to document all the rectangles because that a certain number making use of 32 together an example.

This Polypad shows an instance for the number 12. Students deserve to share their Polypad through yellowcomic.com’s teacher dashboard, or send friend a attach to your Polypad through your LMS.

To start off with, you should indicate that students select a number smaller sized than 30 – otherwise their widest rectangle may not right on the screen. The might likewise be crucial to talk about the importance of functioning systematically, by starting from an extreme rectangle and also moving one action at a time, come be sure not lacking any rectangles.

After students have actually tried a few examples, invite them come share any mathematical concerns that could have emerged to them. Here are a few suggested concerns that could be suggested, or the you could model if the course is unfamiliar with posing their very own questions:

Which numbers deserve to be made right into the greatest number of rectangles?Which numbers deserve to only it is in made into two rectangles?Are there any type of numbers which deserve to be made into an odd number of rectangles?Is over there a method to job-related out how numerous rectangles a number have the right to be made into, without having to do them all?

Once some fruitful questions have been posed, offer students some an ext time to explore the answers to your questions, probably working with a companion or in a tiny group. In the direction of the finish of the lesson carry the class earlier together to solve the answers come the inquiries – students can be invited to present their systems to a concern to the rest of the class, explaining exactly how they reached their conclusions.

Possible solutions

The numbers which deserve to be made right into the greatest number of rectangles space those through the greatest variety of factors. Both 24 and 30 have 8 factors, i beg your pardon is the most for numbers as much as 30. If students discover numbers as much as 100, lock may uncover several numbers through 12 factors. Consider complicated students come find every one of them: 60, 72, 84, 90, and also 96.

The numbers which have the right to only it is in made into two rectangles room the prime numbers. This deserve to be a valuable mental design for students that have challenge remembering the an interpretation of prime.

An odd variety of rectangles comes from a number with a repeated factor (that is a square number). Each of the rectangles have the right to be drawn two methods round, but the square just adds one brand-new rectangle to the total. Students could spot this indigenous their job-related with number under 30 through noticing that 1, 4, 9, 16 and 25 have 1, 3, 3, 5 and 3 determinants respectively. It is worth taking the moment to comment on why square numbers have actually an odd number of factors, together this help students do the connection in between different properties of numbers, and also will help them once they concerned work on prime factorisation.

Determining how plenty of factors a number has without in reality trying all feasible rectangle sizes could develop into an entire lesson ~ above its own – check out some suggestions below.

Support and also extension

For college student who are struggling, indicate that they job-related systematically by beginning with 1 and recording the variety of rectangles because that each number in turn. It can be helpful for students to work-related with a companion where one provides Polypad and also the other records the findings, swapping end after every activity.

The question “How can I uncover the number of factors the a number has?” is a rich question for further exploration. It might be provided to introduce factor trees to uncover the distinctive prime factorisation that a number, and yellowcomic.com’s element numbers course has some much more content around this. Students have the right to then consider how prime components raised to various powers contribute to each of the determinants of a number. For example, girlfriend could an obstacle students to find numbers that have a specific variety of factors (like 8), and also then look at your prime factorisation.

See more: How Tall Is Johnny Van Zant, Johnny Van Zant Net Worth, Age: Bio

Students might notice that part numbers have numerous factors while rather have very few. Questioning students come explore adding up the determinants of a number (other than the number itself) and see what patterns they notice. This introductory inquiry leads to an exploration of abundant, deficient, and also perfect numbers, and you have the right to read an ext about these in yellowcomic.com’s course about sequences.