# data[].attributes.body
Quizziz

# data[].attributes.attachment_link.readable_text
Galileo Galilei once said: "Mathematics is the language with which God has written the universe." Too bad most schools are awful at helping us become fluent in Universe.

So, instead, here's a bunch of mathematical games which can help you learn the Universe's language, through play:

# data[].attributes.attachment_link.description
Galileo Galilei once said: "Mathematics is the language with which God has written the universe." Too bad most schools are awful at helping us become fluent in Universe. So, instead, here's a bunch of mathematical games which can help you learn the Universe's language, through play:

# data[].attributes.attachment_link.description
Discover Mathigon, the Mathematical Playground. Learning mathematics has never been so interactive and fun!

# data[].attributes.attachment_link.readable_text
An advanced and easy-to-use 2D/3D home design tool.
Join a community of 96 746 864 amateur designers or hire a professional designer.

Home Design Made Easy

Just 3 easy steps for stunning results

Layout & Design

Use the 2D mode to create floor plans and design layouts with furniture and other home items, or switch to 3D to explore and edit your design from any angle.

Furnish & Edit

Edit colors, patterns and materials to create unique furniture, walls, floors and more - even adjust item sizes to find the perfect fit.

Visualize & Share

Use the Renders feature to capture your design as a realistic image - this adds shadows, lighting and rich colors to make your work look like a photograph!

Create Anywhere, Anytime

We cover all platforms with full synchronization between them!

Created for Amateurs

Use Planner 5D for your interior house design needs without any professional skills

HD Vizualizations

Use the Renders feature to capture your design as a realistic image - this adds shadows, lighting and rich colors to make your work look like a photograph!

2D/3D Modes

Experiment with both 2D and 3D views as you design from various angles. Arrange, edit and apply custom surfaces and materials.

5000+ Item Catalog

Explore our project gallery and browse our content. We have something great in store for everyone in our user-generated library.

Easy User Interface

The intuitive and user-focused interface provides an easy design process without any tutorials or instructions.

Item Editing

Apply custom colors, patterns and materials to furniture, walls and floors to fit your interior design style.

Community

Be a part of a growing community. Upload and customize projects. Get inspired by designs created by other users.

See What Users Have Created

Easily capture professional 3D house design without any 3D-modeling skills.
Get Started For Free

# data[].attributes.attachment_link.title
Planner 5D: House Design Software | Home Design in 3D

# data[].attributes.attachment_link.description
An advanced and easy-to-use 2D/3D house design tool. Create your dream home design with powerful but easy software by Planner 5D.

# data[].attributes.body
http://www.mathpathontario.ca/career_pathways.php

# data[].attributes.attachment_link.readable_text
This demo allows you to enter a mathematical expression in terms of x and y. When you hit the calculate button, the demo will calculate the value of the expression over the x and y ranges provided and then plot the result as a surface. The graph can be zoomed in by scrolling with your mouse, and rotated by dragging around. Clicking on the graph will reveal the x, y and z values at that particular point.

The table below lists which functions can be entered in the expression box.

Expression

Description

sin(x)

The sine of x in radians

cos(x)

The cosine of x in radians

tan(x)

The tangent of x in radians

asin(x), acos(x), atan(x)

The inverse of the three trigonometric functions listed above

sqrt(x)

The square root of x (for positive x only)

log(x)

The natural logarithm of x

pow(x, y)

The power of x to the y

You can also apply certain constraints/inequalities to the graph. If, for example, you wanted to show the funtion \(x^2\) - \(y^2\) but only in areas where \(x\) is greater than \(y\), you would enter "if (x > y, x*x - y*y)" in the expression box. You can also enter a third option which will be plotted if the first condition is not met - for example "if (x > y, x*x - y*y, x)" will plot \(x^2\) - \(y^2\) in all areas where \( x \) is greater than \( y\), and \(x \) in all areas where x is not greater than y.

The resolution slider can be used to increase the number of data points displayed on the graph, which gives a smoother final result, but since this needs more computational power, you may notice a slight decrease in frame rate when interacting with the graph.

Every time you hit the "Calculate" button, the URL updates with your current settings meaning you can share a link directly to a graph of your choice without having to keep typing in the settings.

Please note, if your surface includes complex numbers, only the real part will be plotted.

Please enable JavaScript to view the comments powered by Disqus.

# data[].attributes.attachment_link.description
This demo allows you to enter a mathematical expression in terms of x and y. When you hit the calculate button, the demo will calculate the value of the expression over the x and y ranges provided and then plot the result as a surface.

# data[].attributes.attachment_link.description
Use logic to solve visual, interactive mathematical puzzles. Create your own puzzles to share with your friends or the SolveMe Community.

# data[].attributes.attachment_link.readable_text
The Problem of the Week is designed to provide students with an ongoing opportunity to solve mathematical problems. Each week, problems from various areas of mathematics will be posted here and e-mailed to teachers for use with their students from grades 3 and up.

Current Problems

Subscribe Now!

Information for Teachers

Problems are organized into themes. A problem may have more than one theme.

These problems may be used in a variety of different ways including:

posting the problems in a classroom,

discussing the problems with students, and

integrating the problems into the curriculum.

We know that teachers will use these problems in many creative ways!

Solutions to the problems will be sent out to teachers one week after the problems are e-mailed.

Problems & Solutions from 2023-2024

Problem and Solution Archive

The following table has links to booklets containing all the problems and solutions from particular years. The problems are organized into themes, grouping problems into various areas of the curriculum. A problem often appears in multiple themes.

# data[].attributes.attachment_link.authors[].name
Design - Graphics, University of Waterloo

# data[].attributes.attachment_link.description
The CEMC has become Canada's largest and most recognized outreach organization for promoting and creating activities and materials in mathematics and computer science.

# data[].attributes.attachment_link.readable_text
Cignition’s FogStone Isle lets kids explore, create and engage with math through real-world scenarios. Unlike most game-based learning programs, FogStone […]

Polyup is a free and open computational thinking playground where students can experiment with numbers and functions. Students will “mod”, […]

Refraction focuses on teaching fractions and discovering optimal learning pathways for math education. Refraction lets you bend, split, and redirect […]

This lets you play with math while exploring a 3-D virtual world, filled with number creatures and mathematical machines. It […]

Dragon Box teaches you algebra through visuals. You solve for x and have fun doing it. It really encourages algebraic […]

SET is a board game where any table becomes the board. It is a highly addictive, original game of visual […]

In this app you have to release the creature from jail by turning the gears. It helps you develop number […]

Mancala is a family of board games played around the world, sometimes called “sowing” games, or “count-and-capture” games, which describes […]

Muggins math board games and manipulatives are not only educational but make learning FUN! Built the old fashioned way with […]

# data[].attributes.attachment_link.description
Mancala is a family of board games played around the world, sometimes called "sowing" games, or "count-and-capture" games, which describes [...]

# data[].attributes.body
Math Frog

# data[].attributes.attachment_link.readable_text
Data Management and Probability
Geometry and Spatial Sense | Measurement
Number Sense and Numeration | Patterning and Algebra

Data Management and Probability Geometry and Spatial Sense
GeoGebra
This geometry tool is great for exploring slides, flips and turns.
GeoGebra Instructions
Measurement Number Sense and Numeration
MultipliACTION & Penguin Jump
Improve your multiplication skills through practice and by racing your penguin to the finish.
Print the Follow-up Exercises | Play MultipliACTION | Play Penguin Jump

Patterning and Algebra

For more great math games, check out our LINKS!

# data[].attributes.attachment_link.description
Data Management and Probability Geometry and Spatial Sense | Measurement Number Sense and Numeration | Patterning and Algebra Data Management and Probability Geometry and Spatial Sense GeoGebra This geometry tool is great for exploring slides, flips and turns.

# data[].attributes.attachment_link.title
ClassCalc - The calculator of the Future - ClassCalc

# data[].attributes.attachment_link.description
ClassCalc The lockdown calculator app for exploring math in the 21st century. No distraction, no cheating. Explore Calculator Free Trial Curious what our c

# data[].attributes.attachment_link.readable_text
You are here: Home → Worksheets
Here you can generate printable math worksheets for a multitude of topics: all the basic operations, clock, money, measuring, fractions, decimals, percent, proportions, ratios, factoring, equations, expressions, geometry, square roots, and more. We also offer pages that list worksheets by grade levels (grades 1, 2, 3, 4, 5, 6, and 7).

The worksheets are available as both PDF and html files. They are also very customizable: you can control the number of problems, font size, spacing, the range of numbers, and so on. The worksheets are generated randomly, so you get a different one each time.

All of the worksheets come with an answer key on the 2nd page of the file.

NEW! Square roots worksheets page has been updated with more options - including worksheets in both PDF and html formats.

Copying permission: you can freely print and copy unlimited copies of the worksheets for use in the classroom, home, tutoring center-anywhere you might be teaching. If you want to distribute the links or worksheets on a website or publication, please contact us.

Below you can see examples of the variety of the worksheets:

By grade levels

Elementary math

Special

Money (various currencies)

Fractions

Decimals

Number theory

Ratio, proportion, and percent

Geometry

Pre-algebra

Other math worksheet websites

DadsWorksheets.com - thousands of free math worksheets
This site has over 5,000 different math worksheets from kindergarten to pre-algebra and growing.

Math Maze
Generate a maze that practices any of the four operations. You can choose the difficulty level and size of maze.

10 Quickies Worksheets
Interesting math review worksheets... with 10 questions per page, but each page has a scenery that kids can color afterwards. A few free samples available.

# data[].attributes.attachment_link.description
You are here: Home → WorksheetsHere you can generate printable math worksheets for a multitude of topics: all the basic operations, clock, money, measuring, fractions, decimals, percent, proportions, ratios, factoring, equations, expressions, geometry, square roots, and more. We also offer pages that list worksheets by grade levels (grades 1, 2, 3, 4, 5, 6, and 7).

# data[].attributes.attachment_link.readable_text
Interactivate: Algebra Four

Algebra Four: Students play a generalized version of connect four, gaining the chance to place a piece on the board by solving an algebraic equation. Parameters: Level of difficulty of equations to solve and type of problem. Algebra Four is one of the Interactivate assessment games.

On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science. Additionally, the National Computational Science Institute (NCSI) provides nation-wide workshops portraying resources and instructional ideas to middle school, high school, and undergraduate instructors for use in the classroom. Resources and materials offered to these instructors are available free of charge from Shodor's website and are largely developed by Shodor student interns. Shodor's academic program efficiently guides participants from excitement to experience to expertise through computational explorations, research opportunities, and service.

We're sorry!

We have detected JavaScript as being disabled in your browser.

The links below provide instructions for enabling JavaScript dependent on your browser.

Mozilla Firefox Users
Internet Explorer Users
Safari Users

After enabling JavaScript, refresh the page.
You may also try using the help feature of your browser.

# data[].attributes.attachment_link.description
Interactivate: Algebra Four On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science.

# data[].attributes.attachment_link.readable_text
Fractions have always given students trouble, and we have had many questions about working with them. Even looking only at division of fractions, I have had to restrict my attention to a few sample answers. These show the reasons for the standard method, presented in a variety of ways, together with some alternative methods.

Note: In this post, I am going to replace the slash (/), which we had to use on the old site, with the obelus (÷) when referring to division (as opposed to fractions), to make it easier to follow what we say.

How do I divide fractions?

Here is a basic question, from 1999:

Dividing Fractions How do you divide fractions? I love math and I am good at it. This is the only thing slowing me down. I've tried and cannot find a way.

Doctor Rick answered Olivia's question, starting with pictures of fractions in order to make the ideas concrete:

Let's think about a few simple examples. Once you understand them, you will be able to put them together to make harder problems.

What is 2 divided by 1/5? Here are 2 blocks:

     +---------+---------+
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     +---------+---------+

Here is 1/5 of a block (the shaded part):

     +---------+
     |XXXXXXXXX|
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +---------+

To divide 2 by 1/5, we divide 2 blocks into pieces the size of the 1/5 block.

     +---------+---------+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +---------+---------+

There are 10 pieces this size in the 2 blocks. In other words,

     2 ÷ 1/5 = 10

That's the same as

     2 x 5 = 10

Do you see why? To make 1/5 size blocks, we had to make each whole block into 5 pieces. In other words, we multiplied the number of pieces by 5.

So, dividing by a fraction with 1 in the numerator amounts to multiplying by that numerator.

Next problem:

    2 ÷ 2/5 = ?

Here is 2/5 of a block (the shaded part):

     +---------+
     |XXXXXXXXX|
     +.........+
     |XXXXXXXXX|
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +---------+

Let's divide the 2 blocks into pieces this size.

     +---------+---------+
     |         |         |
     +         +         +
     |         |         |
     +.........+.........+
     |         |         |
     +         +         +
     |         |         |
     +.........+.........+
     |                   |
     +---------+---------+

The bottom block is a different shape, but it's the same size, because it's made of 2 1/5-size pieces. Since the pieces we want are twice as big, there are only 1/2 as many of them. The answer is

     2 ÷ 2/5 = 10/2 = 5

This is the same as

     2 x 5/2 = 5

Do you see how the two equations are related? You take the fraction, 2/5, and "turn it upside down" (in math words, we say "invert the fraction," or "take its reciprocal"). Then replace the division sign by a multiplication sign.

Here, in addition to multiplying by the denominator, 5, we had to divide by the numerator, 2. That amounts to multiplying by the reciprocal, 5/2. And that is the standard way to divide by a fraction.

Now we can tackle harder problems.The same method always works, even if the answer isn't a whole number as in the simple examples. For instance, 5 ÷ 3/8 = ? Invert the fraction 3/8; you get 8/3. Then multiply by this fraction: 5 x 8/3 = 40/3 = 13 1/3 We can even replace the whole number in the problem with a fraction, like this: 5/7 ÷ 3/8 = ? 5/7 x 8/3 = ? 5 8 40 19 --- x --- = ---- = 1 ---- 7 3 21 21 If you are comfortable with multiplying fractions, then you will have no trouble dividing fractions. Just remember:invert the fraction you are dividing by (the divisor), and multiply instead of dividing.

We have given many other explanations of this, both the reason for it and the process. But before I get back to those, there is another question:

Do I have to multiply by the reciprocal?

First, a question (also from 1999) asking "Why can't we?" triggers the answer, "Actually, we can!":

Dividing Fractions I know how to divide fractions. I want to know WHY you cannot divide them. Can you please help me?

What Alex meant, of course, was that you can't just divide them as they stand, but have to change it to a division. But is that really true? I played around with alternatives just to see what I could do, and replied:

There are several ways to divide fractions, some more direct than others. The method of multiplying the reciprocal is usually the easiest.

Suppose we want to divide 4/9 by 2/3. We COULD do it just bydividing the numerators and denominators: 4 2 4 ÷ 2 2 --- ÷ --- = ----- = --- 9 3 9 ÷ 3 3 You can see that it works. Why do we usually not do this? BecauseI chose that problem carefully so the divisions would work out. More typically you would have trouble: 3 5 3 ÷ 5 3/5 --- ÷ --- = ----- = --- 4 6 4 ÷ 6 2/3 You have not gained anything, since you still have one fraction divided by another. We have to fix this up; we can do this by multiplying numerator and denominator by the least common denominator (LCD) of the two fractions: 3 5 3/5 3/5 * 15 9 --- ÷ --- = --- = -------- = -- 4 6 2/3 2/3 * 15 10

There's a surprise! I usually tell students you don't need a common denominator to divide, only to add! But here, we ended up having to simplify a "complex fraction", and that is another place where common denominators are used. We can also use common denominators first, before dividing numerator and denominator:

Or, we could firstconvert both fractions to use the same denominator: 3 5 9 10 9 ÷ 10 9 --- ÷ --- = -- ÷ -- = ------- = -- 4 6 12 12 12 ÷ 12 10 Since I made the denominators the same, the "denominator" of the quotient works out to 1. But there is really no division in that method, is there? It is still multiplication.

Then I rewrote that division in fraction form, and found something interesting:

A variation on this method is:

      3       9
     ---     --
      4      12     9
    ----- = ---- = --
      5      10    10
     ---     --
      6      12

You justmultiply numerator and denominator by 12 in the last step to eliminate the fractions. Do you see what I did there? Let us write it out more carefully andyou will see where "multiplying by the reciprocal" is hidden: 3 3*6 --- --- 4 4*6 3*6 3*3 9 ----- = ----- = --- = --- = -- 5 5*4 5*4 5*2 10 --- --- 6 6*4 Here I have ignored LEAST common denominators and just used thesimplest common denominator, the product of the denominators 4 and 6. The product is simply the product of 3/4 and 6/5.

(These days, I sometimes call the product of the two denominators the OCD: Obvious Common Denominator. Sometimes you don't really need the least. In fact, schools today often downplay the LCD, because it tends to turn what could be an open playing field into a rule-studded minefield. Just find something useful to multiply by, without worrying whether it is the best choice; eventually you will learn how to save effort.)

The idea is that in converting both fractions to use this common denominator, and then canceling that denominator, I ultimately just multiplied the numerator of the dividend by the denominator of the divisor, and vice versa, 3*6 and 4*5.

So in my exploration of ways to divide without multiplying by the reciprocal, I ended up actually doing that.

To answer your question simply, we divide fractions by multiplying,not because we cannot divide, but because multiplication is easier than division, and because division by itself does not always produce whole numbers. All these methods just make you multiply the numerator of the dividend by the denominator of the divisor, and vice versa, with some division to simplify the fraction; the difference is only in how much you have to think about what you are doing.

So multiplying by the reciprocal is just the best way to make the process automatic. But sometime not being automatic means you are thinking more, which can be a good thing.

A more complicated example

Our next question, probably from a student in a "math for elementary teachers" course, called for a combination of the two ideas above: illustrating a division of fractions, and showing more than one way to think about it. This time, it was a complicated division:

(9/2)/(3/4): A Picture When dividing fractions,why do you have to invert and multiply? In my college class we were trying to find a division fraction question and everything we came up withturned into a multiplication fraction problem. Why is that?

The phrasing of the question interested me!

I'd like to know how your division questions "turned into" multiplications; it sounds as if you may have come close to seeing why we invert and multiply.Division problems naturally turn into multiplication problems when you see them the right way. Let's take a simple, practical division question: I want to cut a board that is 4 1/2 feet long (that is, 9/2 foot) into pieces that are 3/4 foot long. How many pieces will I get? Here's the board: +-----------+-----------+-----------+-----------+-----+ | | +-----------+-----------+-----------+-----------+-----+ Here's one piece: +--------+ | | +--------+ I could, of course, just cut until I use up the board: +--------+--------+--------+--------+--------+--------+ | | | | | | | +--------+--------+--------+--------+--------+--------+ But I'd like to think first and find the answer mathematically.

(In effect, one could say that what I just did was to repeatedly subtract 3/4 from 9/2; that is one way to think of division.)

Before getting to the invert-and-multiply rule (which is perhaps not the most natural thing to do when actually working with wood), I pointed out an alternative:

One way (believe it or not!) is to use acommon denominator. That means choosing a different unit to measure in. Since both 4 1/2 feet and 3/4 foot can be easily expressed in inches, I'll use that (which amounts to using 12 as a common denominator). So I want to divide 4 1/2 * 12 = 54 inches by 3/4 * 12 = 9 inches; the answer is 6. (In fact, that's how I drew my picture; each character is one inch.)

Once again, although we usually tell students not to use a common denominator for multiplication and division, it is a perfectly reasonable thing to do in many cases. We don't usually think of using a smaller unit as a common denominator, but what I did here was to rewrite 9/2 as 54/12, and 3/4 as 9/12, so that the problem became 54/12 ÷ 9/12; but I never had to think of it that way when I used inches.

But that's not the way we usually teach division of fractions. We want to divide in a way that uses the numerator and denominator of our fractions. So let's take those one at a time. Rather than work with the 3/4 foot boards, let's think about a 1/4 foot piece, paying attention to the denominator alone for now. How many 1/4 foot pieces will fit into our 4 1/2 foot board? Well, there will be 4 pieces in each foot, so we want to MULTIPLY 4 1/2 feet by 4, which is the DENOMINATOR of our divisor. We find that 18 of these smaller pieces will fit in our board.

But we don't want 1/4 foot pieces; we want 3/4 foot pieces. So we have to take those 1/4 foot pieces and glue each three of them back together! How many will we end up with? Well, for every three of the 18 pieces we have, we will end up with only one piece; so we will DIVIDE 18 by 3, the NUMERATOR of the divisor. That gives 6, which is our answer.

Here's a picture of what we did:

  +-----------------------------------------------------+
  |                                                     | 4 1/2 ft
  +-----------------------------------------------------+
  times 4 fourths per foot
  +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 18
  +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
  divided by 3 pieces per fourth
  +--------+--------+--------+--------+--------+--------+
  |        |        |        |        |        |        | 6
  +--------+--------+--------+--------+--------+--------+

So we multiplied by the denominator, 4, and then divided by the numerator, 3, to get the answer.

And that, again, is what it means to multiply by the reciprocal, 4/3.

More practical examples

Here is one more question from 1999:

Explaining Division of Fractions My wife and I are helping some neighbor kids prepare for 6th grade. Since they will be entering an advanced program in science and math, those are the areas we're concentrating on. In a review of fractions, I stumbled when I could not come up with a practical explanation of how division of fractions works. Other than stating the rule about inverting the second fraction and multiplying to get the correct answer, I could not give a concrete example to help them understand the concept. For example, in the simple problem of 1/2 divided by 1/2, the correct answer is 1. However, I was unable to explain why division using whole numbers results in a smaller amount, but dividing by fractions produces an answer greater than what you started with. I hope I am clear. The kids left a few moments ago and my brain still feels like mush.

There are really several questions here: Why does dividing by a fraction yield a larger number, contrary to experience with whole numbers; where does invert-and-multiply come from; and how can we illustrate such divisions? Doctor Twe took it on, starting with an abstract view:

You're right, dividing fractions is confusing and seems counterintuitive. The logic, of course, is that division is the inverse function of multiplication. So if 1 * 1/2 = 1/2, then 1/2 ÷ 1/2 = 1. But that is distinctly unsatisfying. The "real world" doesn't provide much help, either. There simply aren't many examples in the real world of dividing by a fraction. When we visualize division, we picture splitting something into more than one part - not less than one part.

Note that his first attempt at an example used what is called a "partitive" model of division, splitting into a specified number of equal parts and asking how big each is. This can be difficult to picture with fractions. After giving a relatively unsuccessful example, he came back with some more. Not surprisingly, these are based on "quotative" models, where we split something into parts of a specified size, and ask how many there are (like my board cutting above):

I came up with a few more (and I hope better) examples. The first one is nice because it correlates directly with what we do with integers. Here they are:

Integer example:

I went to a dairy farm and bought a 10-gallon canister of milk. The canister won't fit in my refrigerator, so I want to pour it into several 2-gallon jugs.How many jugs do I need? Solution: 10 -- = 5 jugs needed. 2 Fraction example: I went to the store and bought 1/2 gallon of milk. The container won't fit in my refrigerator (I have a "mini-fridge"), so I want to pour it into several 1/8-gallon (one pint) containers.How many containers do I need? Solution: 1/2 --- = 4 containers needed. 1/8 In both cases, we divide the total quantity of milk by the capacity of the containers. This shows that dividing a fraction by a smaller fraction produces a value larger than one (you need more than one of the smaller containers).

This nicely answers the question about division increasing a number. Then, as I did in one answer above, he observed that this example motivates a different method of division:

This also demonstrates an alternative way to solve dividing fractions. The "real world" problem can be solved using integers byconverting the quantities to pints. 1/2 gallon = 4 pints, 1/8 gallon = 1 pint. Then 4/1 = 4 containers needed. The equivalent mathematical operation is called "eliminating the fraction," and is accomplished by multiplying both the dividend and divisor by a number that will eliminate the denominators. The most efficient value to use is the Least Common Multiple (LCM) of the denominators of the two fractions - in this example 8. 8 1/2 8*1/2 4 pints - * --- = ----- = ------- = 4 containers. 8 1/8 8*1/8 1 pint

But if you look carefully, you see that what he has done can also be described as multiplying 1/2 by 8, the reciprocal of the divisor.

A second example:
(A) How many 1/2-hour 'Simpsons' episodes can you watch in 1/2 hour?
(B) How many 1/4-hour 'Rugrats' episodes can you watch in 1/2 hour?

Solution:
(A)
     1/2     1   2
     ---  =  - * -  =  1 'Simpsons' episode.
     1/2     2   1

and

(B)
     1/2     1   4
     ---  =  - * -  =  2 'Rugrats' episodes.
     1/4     2   1

Here again, we can solve the problem byconverting to a smaller unit (minutes). 1/2 hour = 30 minutes, 1/4 hour = 15 minutes. So 30/30 = 1 'Simpsons' episode, and 30/15 = 2 'Rugrats' episodes. Eliminating the fraction can be accomplished by multiplying both the dividend and divisor by 30. (Note that 30 is _not_ the LCM, but it works because it is a common multiple of 2 and 4). (A) 60 1/2 60*1/2 30 min. -- * --- = ------ = ------- = 1 'Simpsons' episode. 60 1/2 60*1/2 30 min. (B) 60 1/2 60*1/2 30 min. -- * --- = ------ = ------- = 2 'Rugrats' episodes. 60 1/4 60*1/4 15 min.

A final thought:

We eliminate the fraction in the "real world" situations by converting to a smaller unit (pints instead of gallons, or minutes instead of hours). We can think of the mathematical method as converting to a smaller base unit. We are counting in 1/8ths (or /60ths, etc.) instead of 1's.

In closing, here are three more of the many explanations we have given for dividing fractions; all of these use a similar method that is more algebraic in flavor than the visual approaches I have focused on here:

Dividing Fractions [Doctor Rob, 1999]Dividing Fractions: Why Invert? [Doctor Greenie, 2001]Multiplying by the Reciprocal to Divide Two Fractions [Doctor Ian, 2008]

# data[].attributes.attachment_link.description
Fractions have always given students trouble, and we have had many questions about working with them. Even looking only at division of fractions, I have had to restrict my attention to a few sample answers. These show the reasons for the standard method, presented in a variety of ways, together with some alternative methods.

# data[].attributes.attachment_link.readable_text
Linear Equations

A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. In simple terms it is a mathematical sentence in which you can see only one letter (which might appear more than once) but there will be no powers (squared, cubed etc). Here is an example of a simple linear equation:

This equation can be "solved" to find which value is represented by the letter x.

The eQuation Generator above can make up unlimited equations for you to practise solving. You can change the options so that one of five different types of equation is displayed. It is not possible to predict how quickly you will develop confidence solving equations of a particular type but typically the examples will increase in difficulty very slightly each time you press the Next button. A Restart button is provided if the questions generated start to become a little too difficult. This button restarts the difficulty level but will present different equations.

Here are examples showing a good way to solve equations by thinking of the two sides of the equation as two sides of a balance. The equation will remain balanced only if you do the same thing (multiply, divide add or subtract) to both sides.

# data[].attributes.attachment_link.description
A never-ending supply of simple algebraic equations with answers for a teacher to project in front of a class practising algebra.

# data[].attributes.attachment_link.readable_text
In February 2014, I asked my subscribers for their favorite math puzzle. This collection is the result-a list of puzzles chosen by teachers and parents! Most only require the four basic operations or not even that, so they work well for elementary school children and on up. I have divide the puzzles into ones that are geometric in nature and ones that are math & logic puzzles. The challenging puzzles are on their own page. I also list several good puzzle websites and books at the bottom of this page.

Jump to:

Puzzles with shapes & sticks
Math puzzles
Puzzle websites

Puzzles with shapes and sticks

1. Stardoku

This is for you kindergarten folks-a coloring Sudoku game! The stars in the same row and column must have different colors.

Stardoku game

2. Turn the fish

Move three sticks and make the fish to swim in the opposite direction. You can see the solution and play aninteractive version of this puzzle at Puzzles.com (and many others too!).

3. Pigs need their pens

Farmer Black raises pigs and was raising them in modular pens, He currently has 6 pigs in the following arrangement:
+---+  +---+  +---+  +---+  +---+  +---+
|   |  |   |  |   |  |   |  |   |  |   |
+---+  +---+  +---+  +---+  +---+  +---+

He needed some walls for another project so he rearranged the pins like this:

+---+---+---+---+---+---+
|   |   |   |   |   |   |
+---+---+---+---+---+---+

He then found out he need 7 more wall pieces to finish his project. How did he rearrange the pens so that there was only 1 pig per pen?

See the solution.

4. Coin triangle

5. Dots in a square
This is one of my favorites! It is a simple puzzle, but illustrates beautifully the idea of "thinking outside the box."

Join all the dots using four straight lines and without lifting your pencil.

Click here for the answer, but think about it first!

6. Pentagon & star

How many triangles are there?

See a hint and the solution at Cool Math 4 Kids site.

7. Tower of Hanoi

Tower of Hanoi is an old favorite puzzle where you move disks from one tower to another. If you've never played, try thisonline version. You can either play it yourself, or let the computer solve it.

8. Trees in rows

A gardener planted 10 trees in five rows. Each row had four trees in it. How did he do this?

Click here for the answer, but don't peek too early-give it a good honest try first!

9. A matchstick puzzle

Move 3 matches to show 2 squares.

See the solution here

Math puzzles

1. Kenken
In Kenken, you need to place the counting numbers in the squares just like in Sudoku, but additionally, you are given boxes with clues with math operations. The numbers you place in each box have to make the "clue" number using the given operation.

So, Kenken is an excellent game to practice the four operations and logical thinking. Lower elementary children can use addition/subtraction and others can use multiplication/division or all four operations.

Play Kenken online here. You can choose the size of the game, operations, and the difficulty level.

2. Numbers in a triangle

Put the numbers from 1 to 9 in the circles so that the sum of the numbers on each side of a triangle is the same.

At CoolMath4Kids you can view a hint and the solution.

3. Kakuro
Kakuro puzzles are "cross-sum" puzzles-like mathematical crossword puzzles. Each "word" must add up to the number provided in the clue above it or to the left.

Download 5x5 Kakuro puzzles at KrazyDad. Other sizes exist also.

4. Same number - or twice as many

Here is a puzzle that seems simple, but is a good critical thinking challenge for lower elementary school kids. You can easily reword it for example to have two flocks of birds in two trees.

Farmer Brown and Farmer Green were ruminating one day on the fence between their farms. Farmer Brown says, "You know I was just thinking. If you gave me one of your cows, then we would have the same number of cows." Farmer Green replies, "If you gave me one of your cows, then I would have twice as many as you!"

How many cows does Farmer Brown have and how many cows does Farmer Green have?

Click here for the answer

5. More than two animals

This puzzle is fromGrace Church School's collection of Abacus problems 2013-2014 for grades 3-8. The site does not supply the answers, but students are encouraged to submit their answers and they will receive a reply.

6. Brick's weight

This is well-known problem that sounds so simple, yet it fools many people!

7. Average Miles per Hour

A car is going up a hill. The hill is one mile long. The driver goes up the hill at an average speed of 30 miles per hour. When the driver reaches the top of the hill, he starts down the other side. The downhill side is also one mile long. How fast must the driver go down the hill in order to average 60 miles per hour?

Solution

It is impossible to average 60 miles per hour. An average speed of 60 miles per hour requires two minutes to cover two miles. The first mile was covered at 30 miles per hour. This means that two minutes were used to travel that mile. Hence there are no minutes left to cover the second mile.

Source: Math Games and Number Tricks (Formerly published as Math Puzzles and Oddities) by Robert Yawin

Hide answer

8. Legs in the bus

1. There are 7 girls on a bus.
2. Each girl has 7 backpacks.
3. In each backpack, there are 7 big cats.
4. For every big cat there are 7 little cats.

How many legs are on the bus, not counting the driver?

Solution

There are a total of 10,990 legs.

1. There are 7 × 7 × 7 = 343 large cats. They have 343 × 4 = 1372 legs.

2. There are 7 × 7 × 7 × 7 = 2401 little cats. They have 2401 × 4 = 9604 legs.

3. There are 7 girls. They have 14 legs.

In total we get 1372 + 9604 + 14 = 10,990 legs.
Hide answer

9. How can these be equal?

Here's a short but puzzling puzzle that someone sent for the contest:

10. Coconut trader

11. Milkman's Containers

A milkman has an 8-liter container full of milk, and also two empty containers that measure 5 liters and 3 liters. He needs to deliver 4 liters of milk to a customer.

The milkman has no other spare containers and no way to mark any containers. He does not want to pour milk away. How will he measure the 4 liters of milk?

Solution

1. Fill the 5-liter container.

2. Pour out from the 5-liter container to the 3-liter one until it's full. The 5-liter container now has 2 liters left.

3. Pour all the milk from the 3-liter container back to the 8-liter container.

4. Pour the 2 liters from the 5-liter container to the 3-liter container. It has therefore just one liter of space left!

5. Fill the 5-liter container.

6. Pour out from the 5-liter container enough milk to fill the 3-liter container (which already has 2 liters). You will have 4 liters left in the 5-liter container.

Hide answer

12. Farmer and his ducks

13. Number puzzles with many operations Enter each of the numbers from 1 to 9 in the squares, each one only once. The order of operations doesn't apply! These puzzles are available in four different levels of difficulty.

Number puzzles

14. Jigsaw puzzle for fractions, decimals, and percents

You candownload a printable jigsaw puzzle to practice equivalent fractions, decimals, and percents, published here by permission from the creator Becky Colvin. You need to cut it out first.

15. Folding tissue paper

We fold a piece of paper, 1/10 mm thick, in a way as if we were to try and fit it in a little pocket: in half, in half again, in half once more, and so on.

Let's assume that in our minds (because in reality it's impossible - try it yourself!) we have folded our piece of paper 50 times. How thick will such a folded paper be? If you don't want to calculate, at least make an educated guess!

Solution.

It is 1/10 × 2 ⁵⁰ mm thick.
Converted to kilometers, we get that the thickness is (1/10) × 2 ⁵⁰ × (1/1000000) km = 112,589,991 km thick.

Those who do not know this problem usually assume that the stack will not be thicker than up to a couple of meters. However, the thickness of a piece of paper folded in half 50 times will be surprisingly big, more than 112 million kilometers! (The distance from the Earth to the Sun is about 150,000,000 km).

But... it is not physically possible to do this. The length of paper you would need for this folding would be many light years long...!!! It is given by the equation

L = π/6 · t · (2 n + 4)(2 n − 1)

where t is the thickness of the material and n the number of folds. ( Formula by Britney Gallivan)

Hide answer

Even more puzzle websites & books

The list above already has many links to websites with puzzles, but here are some more.

Challenging puzzles sent to my puzzle contest.

The Moscow Puzzles: 359 Mathematical Recreations
Our whole family enjoys this book! It is divided into 14 chapters, each with different kinds of puzzles, such as matchstick puzzles, dominoes and dice, divisibility, cross sums and magic squares, "Measure Seven Times Before You Cut" (dissection puzzles), "With Algebra and without It" (tricky word problems), and so on. The puzzles and brainteasers range from simple fun riddles to difficult problems, yet none requiring advanced math. They are almost always illustrated and often presented as amusing stories. The book was originally published in Russia in 1956. Martin Gardner has edited the book to make it as easy as possible for an English-reading public to understand, while carefully retaining nearly all the warmth and humor of the original. Price: About $5 used, about $9 new.

Bedtime Math
Fun and lively daily math problems centered around real-life topics and offered in three levels: Wee Ones, Little Kids, and Big Kids. You can also get Bedtime Math problems in a free app.

Sally's Hexagon Number Puzzles
These are "number pyramids" where you get each number by adding the numbers just below it. Puzzles increase in difficulty so that grade 4 puzzles use negative numbers and grade 5 puzzles use decimals.
1st Grade - 2nd Grade - 3rd Grade
4th Grade - 5th Grade

Brainteaser puzzles from ThinkFun
Twelve printable (PDF) puzzles for the whole family. Most are visual kind of puzzles where you rearrange pennies and sticks into another shape.

MathMania
A children's magazine that is packed full of puzzles, mazes and teasers specially created to appeal to kids ages 7 and up.

Matchstick Puzzles
A large collection of puzzles where you need to rearrange matches.

Chess Snake Puzzles
You draw a snake (or a path) on a 5x5 grid from the lower left square to the upper right square. The grid has some chess pieces, and each piece attacks the same number of segments of the snake.

MathSphere Maths Puzzles
About two dozen printable puzzles (PDF) for upper elementary children.

Math Riddles
Fun riddles based on math concepts.

# data[].attributes.attachment_link.description
In February 2014, I asked my subscribers for their favorite math puzzle. This collection is the result-a list of puzzles chosen by teachers and parents! Most only require the four basic operations or not even that, so they work well for elementary school children and on up.

# data[].attributes.attachment_link.title
Breakout EDU - Educational Games for the Classroom

# data[].attributes.attachment_link.description
Breakout EDU gamifies learning to create an engaging and empowering experience for students of all grade levels.

# data[].attributes.attachment_link.readable_text
YOHAKU JUNIOR BOOKS 1, 2 AND 3 ARE NOW ON SALE!

A yohaku is a new type of number puzzle that will test your number sense and problem solving skills. Each yohaku is either an additive or a multiplicative puzzle (as indicated by the symbol in the bottom right of the grid). Your task is to fill in the empty cells such that they give the sum or product shown in each row and column.

Great for use in classrooms as a lesson starter, or for the morning commute, or for something to accompany an afternoon cup of tea. Some yohakus will take a few minutes to solve, others will have you puzzling for longer (and you're welcome).

Each yohaku is lovingly created by hand!​

# data[].attributes.attachment_link.description
YOHAKU JUNIOR BOOKS 1, 2 AND 3 ARE NOW ON SALE! A yohaku is a new type of number puzzle that will test your number sense and problem solving skills. Each yohaku is either an additive or a multiplicative puzzle (as indicated by the symbol in the bottom right of the grid).

# data[].attributes.attachment_link.description
Math explained in easy language, plus puzzles, games, worksheets and an illustrated dictionary. For K-12 kids, teachers and parents.

# data[].attributes.attachment_link.title
Online Latex Equation Editor - Sciweavers

# data[].attributes.attachment_link.description
Tex2Img is a free online Latex equation editor that converts Latex equations to high resolution images to embed in documents and presentations. Render Latex equations into plain text ASCII to insert as comments in source-code, e-mail, or forum. You can also use Tex2Img to learn Latex by trial and error.

# data[].attributes.attachment_link.title
Why Do We Need a 23 Million Digit Prime Number?

# data[].attributes.attachment_link.description
Watch Why Do We Need a 23 Million Digit Prime Number?, a Tech video from Seeker.

# data[].attributes.attachment_link.description
The earliest known reference to the sieve ( Ancient Greek: κόσκινον Ἐρατοσθένους, kóskinon Eratosthénous) is in Nicomachus of Gerasa's Introduction to Arithmetic , which describes it and attributes it to Eratosthenes of Cyrene, a Greek mathematician.

# data[].attributes.attachment_link.description
Play Spider Match Integers at Math Playground! Combine the correct numbers to make the given sum.

# data[].attributes.attachment_link.readable_text
Mental Math - combine positive and negative numbers
Problem Solving - match math expressions with their correct values

Common Core Connection for Grades 6 and 7

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers.

# data[].attributes.attachment_link.description
Play Orbit Integers at Math Playground! Combine integers to power up your space racer.

# data[].attributes.attachment_link.readable_text
Number Properties - identify prime factors

Common Core Connection for Grades 4, 5, 6

Determine whether a given whole number in the range 1-100 is prime or composite.
Find common factors and multiples.
Find the greatest common factor of two whole numbers less than or equal to 100.
Find the least common multiple of two whole numbers less than or equal to 12.

# data[].attributes.attachment_link.title
Factor Trees and Prime Factorization | Math Playground

# data[].attributes.attachment_link.description
Play Factor Trees at MathPlayground.com! Use prime factorization to find the GCF and LCM of number pairs.

# data[].attributes.attachment_link.description
Code to Learn offers free professional learning to Canadians on coding with young people. MicroWorlds coding software is available at no cost to engage youth across Canada. We work with educators and other community members to co-design curriculum activities & Coding Challenges-including the popular micro:bits!

# data[].attributes.attachment_link.title
The Pythagorean Theorem: Interactive Puzzle

# data[].attributes.attachment_link.description
An interactive demonstration of the Pythagorean theorem.

# data[].attributes.attachment_link.description
Welcome to math as you’ve never seen it before. With our wide range of exciting educational games, math homework and review just got a whole lot more fun.

# data[].attributes.body
Short Ontario Teacher videos on Math

# data[].attributes.attachment_link.description
Welcome to the official YouTube channel for Homework Help. Homework Help is funded by the Ontario government and administered by TVO's Independent Learning Centre.

# data[].attributes.body
Diagnostic Assessments for Personalized remediation

# data[].attributes.attachment_link.description
If the swf fails to load, check the address bar for a swfURL. If it is missing, the calling location may have been untrusted. If the swfURL starts with "http://", check that you are connected to the Internet.

# data[].attributes.attachment_link.title
Home | Spark Curiosity to Fuel Sense Making

# data[].attributes.attachment_link.description
Home. Spark Curiosity With Tasks That Are Contextual, Visual and Concrete

# data[].attributes.attachment_link.title
MFM1P Optimum Values Rectangle Area and Perimeter

# data[].attributes.attachment_link.description
2016 Aug 6. Here is video of this task structure implemented with elementary students. 2013 May 14. Here's a brief series on how to teach with three-act math tasks. It includes video. 2013 Ap...

# data[].attributes.attachment_link.title
Talkers & Drawers - Linear Relations * Activity Builder by Desmos

# data[].attributes.attachment_link.description
Students will generate a need to use less informal language and more formal language when describing linear relations. One student will attempt to "describe" (without using their hands) a graph for their partner to sketch (who can't talk). Then the two will swap roles.

# data[].attributes.attachment_link.title
Interview for MFM2P - Quadratic Unit Test

# data[].attributes.attachment_link.description
Unit: Quadratic Expressions Overall Expectation: * identify characteristics of quadratic relations

# data[].attributes.headline
The Robertson Program Intermediate Math Lessons

# data[].attributes.subject
The Robertson Program Intermediate Math Lessons

# data[].attributes.attachment_link.provider_name
The Robertson Program for Inquiry-based Teaching in Mathematics and Science

# data[].attributes.attachment_link.title
Junior/Intermediate Math - The Robertson Program for Inquiry-based Teaching in Mathematics and Science

# data[].attributes.attachment_link.description
All Number Sense Geometry and Spatial Sense Data Management Measurement Patterning and Algebra 21 Students aim to beat the dealer by getting as close to 21 as possible without going over.​ 99 In this collaborative game, students try to keep the cumulative total under 99 until the deck runs out.

# data[].attributes.attachment_link.description
What's New on mathies.ca? Relational Rods Plus has been updated to include the latest annotation improvements: rulers, number lines, hops, grids and polygons. All annotation objects can now be copied and rotated. Annotation is now scaled as rods are resized. Save/load functionality now includes annotation. A French version of the support wiki is now available.

# data[].attributes.headline
MFM2P Quadratics Unit Test Review Feedback

# data[].attributes.subject
MFM2P Quadratics Unit Test Review Feedback

# data[].attributes.headline
Logger Pro - Track the Motion of Objects

# data[].attributes.subject
Logger Pro - Track the Motion of Objects

# data[].attributes.attachment_link.description
Logger Pro is award-winning, data-collection and analysis software for Windows and Mac computers.

# data[].attributes.attachment_link.title
MEL4E Unit Plan Personal Finance Kaitlyn Fitzpatrick

# data[].attributes.headline
MEL4E Personal Finance Unit Test Review Survey

# data[].attributes.subject
MEL4E Personal Finance Unit Test Review Survey

# data[].attributes.attachment_link.title
Unit Test Review Feedback - MEL4E Personal Finance

# data[].attributes.attachment_link.description
A lot of time has gone by since the beginning of the unit. Some of the math we have done may be fresh in your mind and clear as day. Other math may still be confusing or in need of more review.

# data[].attributes.attachment_link.title
Student Unit Progress Report - MEL4E Personal Finance

# data[].attributes.attachment_link.description
Dear Parents and Students, This Unit progress report is a way for you to see how things are going in the math classroom. There are criteria for success in the math classroom, and below is a checklist of Learning Skills that are being demonstrated so far.

# data[].attributes.body
Lessons by handwriting

# data[].attributes.attachment_link.keywords[].name
math mathematics algebra "algebra 2" calculus precalculus help "how to" homework

# data[].attributes.attachment_link.description
Algebra and calculus help from an MIT graduate. More videos with Nancy coming in 2017!

# data[].attributes.body
Calculus and Vectors + advanced math theory animated

# data[].attributes.attachment_link.description
3Blue1Brown, by Grant Sanderson, is some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations and for difficult problems to be made simple with changes in perspective. For more information, other projects, FAQs, and inquiries…

# data[].attributes.headline
MEL4E Applications of Measurement Interview

# data[].attributes.subject
MEL4E Applications of Measurement Interview

# data[].attributes.headline
MEL4E Personal Finance Types of Accommodations

# data[].attributes.subject
MEL4E Personal Finance Types of Accommodations

# data[].attributes.attachment_link.title
Student Unit Progress Report - MEL4E Personal Finance

# data[].attributes.attachment_link.description
Dear Parents and Students, This Unit progress report is a way for you to see how things are going in the math classroom. There are criteria for success in the math classroom, and below is a checklist of Learning Skills that are being demonstrated so far.

# data[].attributes.headline
MEL4E Personal Finance Can You Afford It?

# data[].attributes.subject
MEL4E Personal Finance Can You Afford It?

# data[].attributes.attachment_link.description
Use this calculator to compare your income with your current or planned expenses and debt payments and see what kind of mortgage you can comfortably fit into your budget.

# data[].attributes.attachment_link.title
Exceptionalities in Math - Kaitlyn Fitzpatrick

# data[].attributes.attachment_link.readable_text
Whether you're calling attention to some feature of a graph, or asking students to reflect on something they've just done, note components can play a key role in an activity. Add text, math, or a combination.

# data[].attributes.attachment_link.description
Whether you're calling attention to some feature of a graph, or asking students to reflect on something they've just done, note components can play a key role in an activity. Add text, math, or a combination.

# data[].attributes.attachment_link.description
Math calculators & answers: elementary math, algebra, calculus, geometry, number theory, discrete & applied math, logic, functions, plotting & graphics, advanced mathematics, definitions, famous problems, continued fractions.

# data[].attributes.attachment_link.description
For more information, please visit us at: www.gatfl.org

# data[].attributes.attachment_link.description
Workplace Math: Is it for me? Why Choose the Workplace Mathematics Pathway? My destination is either: 1. Workplace, 2. Apprenticeship 3. College in the field of arts - What are the Benefits of Workplace Math? - develop skills that are practical to every day life and the workplace - MEL3E and MEL4E course topics include: 1.

# data[].attributes.attachment_link.description
By converting our sims to HTML5, we make them seamlessly available across platforms and devices. Whether you have laptops, iPads, chromebooks, or BYOD, your favorite PhET sims are always right at your fingertips.Become part of our mission today, and transform the learning experiences of students eve…

# data[].attributes.attachment_link.title
Jo Boaler on the Good and Bad of Mathematics Education

# data[].attributes.attachment_link.title
Math Catcher | Mathematics through Aboriginal Storytelling

# data[].attributes.attachment_link.description
It is summer time and Small Number and his friend Big Circle spend every day playing and exploring the surroundings of their village. A few days ago, they tripped over an old canoe buried in the sand; yesterday they found the biggest cedar tree that they had ever seen; and today they discovered a beach with an ancient totem pole in the middle of it.

# data[].attributes.attachment_link.description
By the creators of TVOkids, TVO mPower supports children in building math skills and math confidence through fun and practical online games. Ad free.

# data[].attributes.attachment_link.description
This site is designed for Ontario K-12 students, educators and parents/guardians.

# data[].attributes.attachment_link.readable_text
This online virtual oscilloscope allows you to visualise live sound input and get to grips with how to adjust the display. If you find this useful, our online spectrum analyser may also be of interest to you.

An oscilloscope is a useful tool for anyone working with electrical signals because it provides a visual representation of the signal's shape, or waveform. This allows you to measure properties of the wave, such as amplitude or frequency.

The initial signal above is a 200Hz sine wave, which has an amplitude of 5 volts. The frequency of this wave can be adjusted by using the "Input Wave Frequency" slider. (You can also choose to display a square wave.)

If you are browsing using the latest version of Google Chrome, the input dropdown box allows you to select "live input". This will take data from any microphone connected to your computer and display the live audio data. (Different microphones send different voltages to the computer, so for consistency we have normalised the input so the raw input signal will always be limited to somewhere between -5 and +5 volts.)

Since waveforms come in a wide variety of shapes, amplitudes and frequencies, oscilloscopes need to have a number of controls to adjust the display of the waveform so it can comfortably fit inside the viewport.

Freeze live input
This tickbox freezes the input allowing you to effectively take a snapshot of what is displayed on the oscilloscope at a given instant in time. This is especially useful because you can still adjust the time base and volts per division setting. Try whistling and freezing the input. Adjust the timebase to a convenient scale allows you to calculate the frequency of your whistle by counting the period of one complete waveform.

Oscilloscope gain
This is a number that the incoming signal is multiplied by. A gain of 1 will have no effect, a gain of less than 1 will make the signal smaller and a gain of more than 1 will make it larger.

seconds / div
This control allows you to adjust the length of time that each square of the grid represents. When the oscilloscope is first loaded, this setting is set at 1ms, and shows one complete waveform over 4 squares. This means that the period of the wave is 4ms, or 0.004s, giving a frequency of (1/0.004) = 250Hz. If you change the timebase to 500µs (half of what it started at), you should see the waveform now takes 8 squares to complete one full oscillation. The period (and hence the frequency) remain constant because 8 times 500µs still equals 0.004s.

volts / div
This setting is very similar to the timebase setting described above, but instead of stretching the wave along the x-axis, it involves stretching it along the y-axis. The sine wave has an amplitude of 5V, meaning when volts/div is set to 5, the waveform just reaches the top of the first square. If you were to change the setting to 10 volts/div, the waveform now only reaches up half of a square.

Horizontal and Vertical Offsets
These two sliders allow you to adjust the position of the oscilloscope's trace on the grid. They are particularly useful for lining up parts of the waveform with the gridlines (this can make it easier for you to count the squares when determining wavelength, for example).

If you would like to embed the oscilloscope on your own website, please copy and paste the following html onto your web page.

Please enable JavaScript to view the comments powered by Disqus.

# data[].attributes.attachment_link.description
This online virtual oscilloscope allows you to visualise live sound input and get to grips with how to adjust the display. If you find this useful, our online spectrum analyser may also be of interest to you.

# data[].attributes.headline
DESMOS GRAPHING CALCULATOR - Natalie Hope

# data[].attributes.subject
DESMOS GRAPHING CALCULATOR - Natalie Hope

# data[].attributes.body
With graphing calculators quickly becoming obsolete, students are expected to be know and understand how to use one of a number of available free online graphing tools. The most commonly used tool in our Math Department is Desmodo, which is available both as a web based version, and an app for both iOS and Android. Practice (and proficiency)  with this tool allows students to develop several different mathematical processes that can be used throughout the course.

# data[].attributes.attachment_link.description
Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

# data[].attributes.body
https://mslwheeler.wordpress.com/2014/11/09/visibly-random-groups-vertical-non-permanent-surfaces/

# data[].attributes.headline
Communicating through Google Classroom - Matt Woody

# data[].attributes.subject
Communicating through Google Classroom - Matt Woody

# data[].attributes.body
Math journaling is an excellent way to get informal feedback from students on how the lessons are going and which areas they are struggling with. If your school lacks access to computers phones can be used or it can be done in a physical paper binder. With computers it is easiest to do through google classroom on either a google document or through a google form that must be submitted a few times a week!

# data[].attributes.headline
INQUIRY-BASED LEARNING (IBL) - Clarissa Sooklal

# data[].attributes.subject
INQUIRY-BASED LEARNING (IBL) - Clarissa Sooklal

# data[].attributes.body
http://www.mathgiraffe.com/blog/structuring-guided-inquiry-in-secondary-math

# data[].attributes.attachment_link.title
Study Skills - Learning From Mistakes (Jo Boaler)

# data[].attributes.attachment_link.title
Practical Differentiation With Fractions Mosaics | Scholastic

# data[].attributes.attachment_link.description
Fractions mosaics provide a hands-on, differentiated context to explore equivalence, common denominators, and different "wholes." Here's the entire lesson with directions, differentiated task cards, and video.

# data[].attributes.attachment_link.title
GeoGebra - the world's favorite, free math tools used by over 100 million students and teachers

# data[].attributes.attachment_link.description
Free digital tools for class activities, graphing, geometry, collaborative whiteboard and more

# data[].attributes.attachment_link.description
Discover thousands of math skills covering junior kindergarten to grade 12, from counting to calculus, with infinite questions that adapt to each student’s level.

# data[].attributes.attachment_link.readable_text
Free, fun, and effective
math software for the classroom.

Fully Curriculum-Aligned K-10 Assessments

Differentiate your math class with consistent quality assessments that meet the pedagogical standards required by school boards.

District-Stamped Classroom Fun

Confidently engage your students with the excitement of popular game-based learning, fortified by the learning science that school boards trust.

Learn more on how we engage students

Goodbye Empty Dashboards

Rely on classroom dashboards that tell the full story with unmatched assignment completion rates and a vibrant view of your students' progress.

Transform how you teach with Classroom Connected Professional Learning. ClassOS Pro is designed around your unique classroom challenges.

Explore ClassOS Pro

Uniting Classroom Insights with Home Tutoring

Introduce parents to a tutoring solution that aligns with your teaching, ensuring students receive consistent, affordable, and quality support.

Learn more about KH After School

Privacy Matters

At Knowledgehook, we make sure to maintain each and every student's privacy in all products. See how we protect students, teachers, and parents

# data[].attributes.attachment_link.title
Knowledgehook - Change how children feel about math

# data[].attributes.attachment_link.description
Experience engaging math lessons with Knowledgehook. Perfect for classrooms, at-home learning, and districts. Free for teachers. Start today!