7.2: Addition- Dots and Boxes

Addition as combining

For now, we’ll focus on the base-10 system. Here’s how we think about the number 273 in that system:

And here is the number 512:


We can add these in the natural way: just combine the piles of dots. Since they’re already in place-value columns, we can combine dots from the two numbers that are in the same place-value box.

We can count up the answer: there are 7 dots in the hundreds box, 8 dots in the tens box, and 5 dots in the ones box.

[egin{split} 273& +; 512 hline 785& end{split}]

And saying out the long way we have:

  • Two hundreds plus five hundreds gives 7 hundreds.
  • Seven tens plus one ten gives 8 tens.
  • Three ones plus two ones gives 5 ones.

This gives the answer: 785.

: 163+489

Let’s do another one. Consider 163+489.

[egin{split} 1; ; ; 6; ; ; 3& +; 4; ; ; 8; ; ; 9& hline 5 | 14 | 12& end{split}]

And this is absolutely correct:

  • One hundred plus four hundreds is 5 hundreds.
  • Six tens plus eight tens is 14 tens.
  • Three ones plus nine ones is 12 ones.

The answer is 5 | 14 | 12, which we might try to pronounce as “five hundred and fourteeny-tenty twelvety.” The trouble with this answer is that most of the rest of the world wouldn’t understand what we are talking about.

Since this is a base 10 system, we can do some explosions.

The answer is “six hundred fifty two.” Okay, the world can understand this one!

[egin{split} 1; ; ; 6; ; ; 3& +; 4; ; ; 8; ; ; 9& hline 5 | 14 | 12& = 652 end{split}]

Think / Pair / Share

Solve the following exercises by thinking about the dots and boxes. (You can draw the pictures, or just imagine them.) Then translate the answer into something the rest of the world can understand.

[egin{split} 148& +; 323& hline end{split} qquad qquad egin{split} 567& +; 271& hline end{split} qquad qquad egin{split} 310462872& +; 389107123& hline end{split}]

Problem 1

Use the dots and boxes technique to solve these problems. Do not covert to base 10! Try to work directly in the base given. It might help to actually draw the pictures.

[egin{split} 20413_{five}& +; 13244_{five}& hline end{split} qquad qquad egin{split} 4052_{nine}& +; 6288_{nine}& hline end{split} qquad qquad egin{split} 3323_{seven}& +; 3555_{seven}& hline end{split}]

The Standard Algorithm for Addition

Let’s go back to the example 163+489. Some teachers don’t like writing:

[egin{split} 1; ; ; 6; ; ; 3& +; 4; ; ; 8; ; ; 9& hline 5 | 14 | 12& = 652 end{split}]

They prefer to teach their students to start with the 3 and 9 at the end and sum those to get 12. This is of course correct — we got 12 as well.

But they don’t want students to write or think “twelvety,” so they have their students write something like this:

[egin{split} 1& 16&3 +; 48&9 hline &2 end{split}]

This can seem completely mysterious. What’s really going on? They are exploding ten dots, of course!

Now we carry on with the problem and add the tens. Students are taught to write:

[egin{split} 1& 1&63 +; 4&89 hline &52 end{split}]

But what this means is better shown in this next picture. Notice the “exploded” (or regrouped) dot at the very top, which is added to the tens box in the answer.

And now we finish the problem by combining the dots in the hundreds boxes:

[egin{split} 1& 1&63 +; 4&89 hline 6&52 end{split}]

In the standard algorithm, we work from right to left, doing the “explosions” as we go along. This means that we start adding at the ones place and work towards the left-most place value, “carrying” digits that come from the explosions. (This is really not carrying; a better term for it is regrouping. Ten ones become one ten. Ten tens become one hundred. And so on.)

In the dots and boxes method, we add in any direction or order we like and then we do the explosions at the end.

  • Why do we like the standard algorithm? Because it is efficient.
  • Why do we like the dots and boxes method? Because it is easy to understand.

My Party Games

Dots and boxes also called dots and lines or connect the dots is a very popular classic game for kids. Today I have made 3 different printables for this game with various difficulty levels. Two and even more than two kids can enjoy playing this fun game. This game is an excellent travel game and kids can also play it during summer break or even long evenings to keep busy.

Two Dots

Join two brave dots as they traverse the arctic tundra, navigate fiery jungles, and plunge the ocean depths in this beautiful puzzle adventure.

• FREE to play for life
• CONNECT one dot to another, sink anchors, make a line, create bombs, fight fire, and much more in this fun free puzzle game
• ADVENTURE through 3475 fun and addicting levels
• ADORABLE fun, and thoughtful game design in this beautiful game about connecting
• ENJOY beautiful minimalistic flat design with relaxing game music and fun sound fx
• PLAY at your own pace to figure out the perfect move without having to worry about the old time clock
• CHALLENGE your friends to a Two Dots match on Facebook, beat their scores, and earn medals
• MAKE a line or a square, not a circle, to connect every dot of a single color




We love all of our friends and everyone who has played Dots and Two Dots. Why not connect? Add us on Facebook or your favorite social network and send us your suggestions and feedback, as we are always looking to add new levels and features!

Free, blank dots and boxes game printable sheets

Dots and boxes is a fun game that many people have played in the past which makes it really easy to play when you’re bored. The basic goal is to create boxes on a dot grid. If you don’t know how to play or you need more in-depth rules, check out this post. In this post, we have multiple dot grids for you to use for the game.

Click on each image to be redirected to the post where you can download them as a PDF, and see our tips and suggestions for playing.

Dot grids

A standard dot grid is most commonly used for dots and boxes. You can draw your own grid, but why go through that hassle? We have different sizes of grids for you, including a massive one which would take a while to finish! For younger players, you’ll want to use a smaller grid so the game doesn’t drag on for too long. We not only have regular square grids, but also hexagonal and triangle grids for you to challenge yourself while playing. If you want to use colored markers and use solid grid paper, check out this post.

Dots and Boxes Rules

The game’s rules are easy but you might need a refresher. Players go back and forth and draw lines between the dots with the goal of creating boxes. When a player completes a box, they get to color it in or write their initials inside. Then they are able to place another line. Here we have some diagrams showing how the game works and also some strategies that can be helpful while playing. The diagrams show the game starts and then how it might continue to progress. The final image shows ways that a player could get multiple boxes in one turn.

Are you interested in learning more about the strategies you can use in dots and boxes? Check out this post: it gives ideas about how to use every line to your advantage. It also gives some history to the game, as well as variations from different countries and cultures! There is also information on a board game that works in a similar manner.

Need other games to play? We have plenty for you! Check out these fun games to play when you’re bored. Are you more interested in puzzles? We have word searches for you to print out and solve.

Elizabeth Hampson is a freelance artist, graphic designer, and copywriter living in Edinburgh. Elizabeth received her Masters Degree at University of Edinburgh, She decided to stay in the Edinburgh, where she lives with her cat.

Download information

The following files are available for download from the Microsoft Download Center:

For all Windows operating systems except Windows RT 8.1:

For information about how to download Microsoft support files, see How to obtain Microsoft support files from online services.

Microsoft scanned this file for viruses, using the most current virus-detection software that was available on the date that the file was posted. The file is stored on security-enhanced servers that help prevent any unauthorized changes to it.

Issues that are fixed in this update

The following issues are fixed for .NET Framework 4.7.2.

Fixes an issue in which the .NET Framework API SqlConnection.ConnectionString property is used to set a null or empty connection string. In this situation, a Null Reference Exception (NRE) occurs when you use the API together with .NET Framework 4.7.2. [611802, System.Data.dll, Bug]

Fixes an issue in which you connect to Azure SQL DB and MultipleActiveResultSets=true is used in the connection string together with System.Data.SqlClient.SqlConnection. In this situation, the async query operations cause a bad TDS protocol request stream to be sent from the client. This causes the Async Query APIs to fail. [620109, System.Data.dll , Bug]

Windows Presentation Framework (WPF)

Adds an AppContext switch that opts out of some work that was done during AppDomain or process shutdown. This issue can reduce (but not eliminate) the possibility of a crash in applications that make unwarranted assumptions about the timing of the finalizer thread. [593963, WindowsBase.dll, Bug]

Fixes a crash in WPF that occurs when you multiple characters are replaced by a single character (in a different language than the original text) by using IMEPad. [605996, PresentationFramework.dll, Bug]

Combo box grouped items now report children correctly through UIAutomation. [605922, PresentationFramework.dll, Bug]

This version of .NET Framework runs side-by-side with .NET Framework 3.5 SP1 and earlier versions. However, it performs an in-place update for .NET Framework 4, 4.5, 4.5.1, 4.5.2, 4.6, 4.6.1, 4.6.2, 4.7, and 4.7.1.

Applications that rely on .NET Framework to initialize a COM component and that run with restricted permissions may fail to start or run correctly, and return "access denied," "class not registered," or "internal failure occurred for unknown reasons" errors.

For known issues with .NET Framework 4.7.2 see Known issues for .NET Framework 4.7.2.

Command-line switches for this update

For more information about the various command-line options that are supported by this update, see the “Command-Line options” section in .NET Framework Deployment Guide for Developers.

You may have to restart the computer after you install this update. We recommend that you exit all applications that are using .NET Framework before you install this update.

Supported operating systems

Microsoft .NET Framework 4.7.2 (Offline Installer) supports the following operating systems:


The film opens with an introduction that explains the interactive nature of the movie and introduces the Oogieloves, Goobie, Zoozie, and Toofie.The Oogieloves awake to prepare a surprise birthday party for their living pillow Schluufy, with the aid of magical window Windy Window, vacuum cleaner J. Edgar and fish Ruffy. However, J. Edgar accidentally releases the five magical balloons they bought for the party, so the Oogieloves set out to retrieve them.

The first balloon is found at the treehouse home of Dotty Rounder (Cloris Leachman) who is obsessed with circles and polka dots, and her granddaughter Jubilee (Kylie Dakota), who is obsessed with squares. The second is found at the milkshake cafe of Milky Marvin (Chazz Palminteri), who is holding a milkshake contest to win the second balloon in which the Oogieloves and their pet fish participate. The third balloon is found in possession of Rosalie Rosebud (Toni Braxton), a pop singer who denies her allergy to roses. The fourth balloon is by the truck of Bobby Wobbly (Cary Elwes), a cowboy with an unusual walk. The last balloon is found on top of a windmill, where the Oogieloves retrieve it with the help of Lola and Lero Sombrero (Jaime Pressly and Christopher Lloyd), who ride a giant flying sombrero.

Just before they reach home with all the balloons, the Oogieloves accidentally release them again but blow kisses to persuade them to return. They then hold the surprise party for Schluufy, who did not awake until just before their return.

  • Malerie Grady as the voice of Toofie
    • Alex Greene as Toofie (in-suit performer)
    • Carol Sweeney as Zoozie (in-suit performer)
    • Eric Dunman as Goobie (in-suit performer)

    The film was written by Scott Stabile and produced by Kenn Viselman, who was the producing partner of the Teletubbies in the Western Hemisphere as well as the American marketing executive for Thomas & Friends. Viselman claims that he and Teletubbies creator Anne Wood had multiple disputes with each other, because Wood refused to let Viselman pursue a film adaptation of the show. [6] The film is loosely based on the characters from the children's TV show My Bedbugs. [3] After seeing Madea Goes to Jail in a theater, where he saw how the audience members would shout out advice to the characters on screen, Viselman was partially inspired to create a children's film in the vein of Teletubbies with the interactive aspect, allowing the children to sing, dance, and respond to the characters on screen. [6] He felt that "The idea of interactivity isn't new, but the idea of interactivity in a theater is." [7] He also wanted to add something new to the Pixar-and-DreamWorks-dominated children's film market: "Why can't we have something that's all love, where we don't even have the color black? Pixar always has the triumph of good over evil. But why does there have to be evil in the first place?" [8]

    Being a fan of children's entertainment and having a past experience in it through an appearance on Blue's Clues, Braxton joined the project after learning that Viselman produced Thomas the Tank Engine, her son's favorite show. [9] She did no preparation for the role, reasoning that she "just went into the little kid that I am, that we all are, some times. I just said I'm going to have fun with her." [10] Braxton did actually have a severe cold when recording the singing voice for "Ode to Adelaide (The Scratchy Sneezy Cough Cough Song)," and the sneezes she performed were real. Due to her cold making her voice "deep" and "not-kid-friendly," it was pitch-shifted higher for the song. [9] The 86-year-old Leachman explained that she did the movements for her song through perseverance, and that the dancing influenced her to get back into exercising after fifteen years of not doing so. [11]

    Several investors, including a Michigan real estate investor named Michael Chirco, spent much of its marketing money on billboards, television ads, and bidding with mommyblogs to cover the film. [8]

    The film was released on August 29, 2012. [4] On its opening weekend, the film became the biggest box office bomb of all time for films released in at least 2,000 theaters. [12] To alleviate the problem, the amount of screens showing the film were lessened, and it would be shown in early mornings ("kind of like The Rocky Horror Picture Show, but in the morning instead of midnight," explained Viselman). [13]

    The film was issued on Amazon Prime on February 5, 2013, [14] with the DVD issued exclusively at Walmart the same day by Lionsgate Home Entertainment. [15] On July 16, the DVD began selling at other retailers. [16]

    Box office Edit

    In its debut weekend, Oogieloves proved to be one of the least successful films ever released in at least 2,000 theaters. [12] Its production budget was $20 million, in addition to another $40 million in marketing costs. [17] On August 29, 2012, the film opened at #17 at the box office to $102,564 in 2,160 theaters, with a per-theater average of $47. [2] Box Office Mojo said the film needed "at least $5 million to avoid being dubbed a legendary flop, and it's not going to come anywhere close". [18] It grossed only $445,000 in its opening weekend, surpassing Delgo for the lowest opening weekend of a film in 2,000 or more theaters. [19] Delgo also played in the same number of theaters as Oogieloves. The film has the second worst opening weekend per-theater average for a widely released film at $206. [19] "To put that in perspective, if each location played Oogieloves five times a day on one screen at an average ticket price of $7, that would translate to fewer than two people per showing", according to Box Office Mojo. [19] Over the life of its exhibition in theaters, the film grossed a grand total of $1,065,907 as measured by total box office gross. Only Delgo had a worse theatrical gross by total gross however, The Oogieloves played for 23 days while Delgo played for only 7. [2]

    Following the film's opening weekend, Viselman attributed the poor box office to the film's marketing, suggesting it would've been more successful if its television spots aired earlier. [8] He was, however, appreciative of the press covering the film's horrendous opening weekend, suggesting it would help in marketing its home media releases and possible sequels to the film. [13]

    Critical reception Edit

    The film has a 30% "Rotten" approval rating on Rotten Tomatoes, based on 22 reviews, [20] and a score of 32% on Metacritic, based on 11 reviews. [21] Rotten Tomatoes' critical consensus says "Although it may serve as a passing diversion for very young viewers, The Oogieloves fails to offer much more than several brightly colored examples of the worst stereotypes of modern children's entertainment.". Loren King of the Boston Globe considered The Oogieloves to be a "dumbed-down mash-up of the least creative parts of Teletubbies, Barney & Friends and Pee-wee's Playhouse" which preschoolers would enjoy due to its interactivity, but would be a waste of time for parents "in a world where Sesame Street is on TV every day [and/or] even in a world where Sesame Street didn’t exist." King also questioned whether the intent of the film was to set the stage for future merchandising of its characters. [22] Mark Olsen of the Los Angeles Times criticized The Oogieloves for being a "prefab construction meant to appear like a beloved set of characters", and for being "so ineffectual and disengaging that it may be better to call it just plain dumb." [23] A. O. Scott of the New York Times reviewed the film from the perspective of the seven-year-old daughter of a family friend. She stated that she "thought it was for babies" and observed, among other things, that none of the children in the audience were paying attention to the film, and that the toddler whose family she saw it with fell asleep partway through the movie. [24]

    Accolades Edit

    The film was nominated for Worst Picture and Worst Screen Ensemble (as the entire cast of the film) at the 33rd Golden Raspberry Awards, but lost in both categories to The Twilight Saga: Breaking Dawn – Part 2. [5]

    Many styles of dashed lines are possible via the lineDashStyle option, which takes an array of numbers. The first number indicates the length of a dash, and the second indicates the gap after it. If there is a third number, that's the length of the next dash, and a fourth number, if present, is the length of the next gap.

    When the chart is drawn, these lengths are repeated, so [4, 4] means a succession of 4-length dashes and 4-length gaps. [5, 1, 3] means a 5-length dash, a 1-length gap, a 3-length dash, a 5-length gap, and so on. Some examples:

    Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License, and code samples are licensed under the Apache 2.0 License. For details, see the Google Developers Site Policies. Java is a registered trademark of Oracle and/or its affiliates.

    To take advantage of substitution when rendering a graph, simply use the grViz() function with Graphviz DOT code:

    The notation @@ within GraphViz DOT will indicate where the substitution is to take place and corresponding R expressions will provide values to be substituted. The grViz() function will autodetect whether the working Graphviz graph specification contains @@ s between the curly braces of the [graph|digraph] <. >DOT construction.

    If you specify a subsitution with @@ , you must ensure there is a valid expression for that substitution. The expressions are placed as footnotes and their evaluations must result in an R vector object (i.e., not a data frame, list, or matrix). Because there is the possibility to have multiple substitutions, numbering is required. Thus, the @@ notation is immediately followed by a number and that number should correspond to the number of the footnoted R expression.

    The substitution construction is:

    Importantly, the footnote expressions should reside below the closing curly brace of the graph or digraph expression. It should always take the form of:

    In the simple example of specifying a single node, the following substitution syntax would be used:

    In the above example, the [1]: footnote expression evaluates as 'a' , and, that is what's substituted in place of the @@1 (where, in turn, it will be taken as the node ID).

    Substitutions can also be used to insert values from vector indices into the graph specification. Simply use this format:

    Here's an example of how specific values from vectors can be inserted into the graph:

    After evaluation of the footnote expressions and substitution, the graph specification becomes this:

    Also, some books and software will include the overall median ( Q2 ) when computing Q1 and Q3 for data sets with an odd number of elements. The Texas Instruments calculators do not include Q2 in this case, so you may encounter a book answer that doesn't match the calculator answer. And different software packages use all different sorts of formulas. Be careful to use the formula from your book when doing your homework!

    Additionally, the box-and-whisker plot may include a cross or an "X" marking the mean value of the data, in addition to the line inside the box that marks the median. The difference between the "X" and the median line can then be used as a measure of "skew".

    Please don't ask me to explain "skew".

    Draw the box-and-whisker plot for the following data set:

    My first step is to find the median. Since there are eight data points, the median will be the average of the two middle values:

    This splits the list into two halves:

    Since the halves of the data set each contain an even number of values, the sub-medians (or first and third quartiles) will be the averages of the middle two values of each half.

    The minimum value is 77 and the maximum value is 99 , so I have:

    Then my plot looks like this:

    As you can see, you only need the five values listed above (min, Q1, Q2, Q3 , and max) in order to draw your box-and-whisker plot. This set of five values has been given the name "the five-number summary".

    If you are asked to produce the five-number summary, they're wanting the same values as I found in the example above, but without doing the associated box-and-whisker plot.

    Give the five-number summary of the following data set:

    The five-number summary consists of the numbers I need for the box-and-whisker plot: the minimum value, Q1 (being the bottom of the box), Q2 (being the median of the entire set), Q3 (being the top of the box), and the maximum value (which is also Q4 ). So I need to order the set, find the median and the sub-medians (or hinges), note the largest and smallest values, and then list the required values.

    I'll start with ordering the list:

    . so the minimum is 53 and the maximum is 98 .

    To find the median, I'll take the average of the middle two values of the ordered list:

    The median splits the data into two halves. The lower half of the list is:

    With an even number of values, the first quartile's value will be the average of the middle two values of this half of the list:

    I'll do the same thing with the upper half of the list, which is:

    This gives me all the values for the five-number summary. Then my answer (with labelling, just to be clear) is:

    Part of the point of a box-and-whisker plot is to show how spread out your values are. But what if one or another of your values is way out of line? For this, we need to consider "outliers".

    You can use the Mathway widget below to practice finding the five-number summary (or skip the widget and continue with the lesson). Try the entered exercise, or type in your own exercise. Then click the button and scroll down to "Find the Five Number Summary" to compare your answer to Mathway's.

    (Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)

    Watch the video: Grundlagen Vektoren Analytische Geometrie. Gehe auf u0026 werde #EinserSchüler (December 2021).