Giving Math Students the Opportunity to Make Mistakes

After finishing an involved project or presentation, I usually breathe a sigh of relief and give myself a pat on the back. However, several years ago I would often hear not-quite-so-positive comments from my friend, a "negative Nancy" coworker. In fact, the first thing I'd be likely to hear from her after my presentation was that I had made a mistake or done something wrong. Then she would go on to tell me what a good job I'd done overall. But by then I was so focused on the mistake or oversight, I could no longer think about the things I had done well. My mood changed from upbeat and positive to completely discouraged.

A totally different process happens when I'm completing a crossword puzzle. I can often tell, through trial and error, that the word I've chosen will or will not fit the space I have. The more words I find, the easier it becomes (sometimes) to check my work and get the correct answers so I can eventually complete the puzzle. With the crossword puzzle, I can often learn from my mistakes and then continue. I don't have to focus on the mistakes or get bogged down with what I did wrong. Instead, I'm encouraged to keep trying in order to get the words right. My whole focus changes from being fixed on my mistakes to learning from them and moving on.

In the classroom, our students can learn and work independently when we allow mistakes to be made (they are going to happen, after all) and encourage the students to discover for themselves how to fix them and move on. We want our students to focus on what they do know rather than get discouraged about what they don't. This is especially true in math class, where some students come into the class intimidated and discouraged that they "can't do math." As teachers, that's what we're there to help them learn!

When we point out mistakes, our students ignore the majority of problems they may have solved correctly and focus their attention on the few problems they've gotten wrong. As a result, they may get discouraged, feel like giving up, stop trying, or conclude that math "just isn't my thing." Rather than set them up for success, we've pointed out their failures, and that's what they remember.

Noted educator and speaker, Fred Jones, suggests,

One of the main times in which a teacher interacts one-on-one with a student is when that student seeks help during Guided Practice. Beware! The way in which we give corrective feedback has a strong biological component that can do damage if we are unaware of it.

Imagine that the student’s work is part right and part wrong. As you scan the work, which part catches your eye? Your eye scans past the part that is right but it stops as soon as the pattern of correct performance is broken by an error. 

Do you want to spend valuable instructional time clarifying for the student something that you never want them to repeat – while making them feel bad? Instead, take a relaxing breath, keep the error to yourself, and simply give a prompt – a description of exactly what to do next. 

Instead of simply pointing out errors, it's more helpful to the student when we give him or her the tools to continue the problem-solving process. Building on that success, the student begins to relax. You've eliminated any reason for the student to be defensive. Instead, the student gains confidence and realizes "I can do it!" and "it's not that hard." Read more in Learning Math Without the Anxiety by Dr. Fred Jones.

David Warlick, in his blog post, What Does It Mean to Be Learned, noticed four potent components of the learning process while watching his own children play. Here's the first:

The experiences are responsive. Every decision and action is responded to. It is a particularly powerful form of assessment, because the response message is not, “You got that right!” or “You got that wrong!” The message becomes, “That worked!” Or “That didn’t work!” And, as is often the case, what doesn’t work can be as instructive as what does. Read more.

Wow! That instant feedback and letting students learn from their mistakes (and from what doesn't work) is exactly what Excel Math writer and founder, Dr. Janice Raymond, discovered worked so well as she began to test and refine Excel Math lessons with students around the country over 30 years ago. One of the unique features she included in these lessons was the CheckAnswer system.

The Checkanswer is a special part of Excel Math lessons because it gives students a chance to make mistakes and fix them on their own. It is used in throughout Excel Math for Grades 2-6. This process enables students to check their work, and verify for themselves that they understand the concepts in the day's Guided Practice or Homework.

With Excel Math, students are encouraged to solve the problems, show their work, check their answers, and then go back and fix any errors they've made. At the same time, students are given an immediate feedback loop so they don't have wait until the next day to find out which problems they've missed. Excel Math lessons help students stop what they're doing and fix the errors before moving on. In most math lessons, students continue making the same mistake over and over until they get back to class and discover they've been doing it all wrong. By then they may not even remember why they tried to solve the problem as they did. With Excel Math, they can correct their answer before moving on to the next problem, and they can do this at home without waiting for the teacher to correct them.

Here's how it works:

All three problems in block A are shown here. In red you can see the student's the work and the solutions. The results of problems 1, 2 and 3 are added in the bottom right corner of block A, and the sum of those numbers (2377) equals the CheckAnswer shown in the box at the top next to A.

The first example in block B asks students to select an operation symbol. Since these symbols don't have a numerical value, they cannot be added for the CheckAnswer. To get around this, we show four possible choices and an arbitrary value for each. The + sign is the correct choice, and its value is 25, so 25 is part of the CheckAnswer sum. We use this wherever answers are symbols, true/false, yes/no, etc.

(In case you are wondering, it is possible for a student to get multiple wrong answers which add up to the correct Checkanswer. But it's not very likely. And when we insert numerical value choices, as discussed above, we make sure that a wrong choice won't make a correct CheckAnswer.) 

Here are our original three blocks A, B, C with all the work and the answers provided. This is the view that the teacher would normally see in the Teacher Edition answer key.

The Associative Principle tells us that it doesn't matter in what order the students do the problems, or in what order they arrange the answers when adding to get the CheckAnswer.  If any answers are wrong, the result is like this — they get the wrong sum and have to go back to recheck their work. (The original work is shown in black below, with the corrections in red.)  Read more about the CheckAnswer system in our previous blog post, The CheckAnswer.

With the CheckAnswer system, students can tell where they got the right answer as soon as the problems in each set are completed. At this point, they can go back and check their work to find out where the mistake lies. During Guided Practice, the teacher can help students who are getting stuck by
giving them suggestions for solving the problem.

The error has already been pointed out through the CheckAnswer system so the teacher no longer needs to be the discourager. Instead, the teacher can focus on encouraging students, guiding them with problem-solving techniques and helping them be successful in math class. What a concept! Perhaps we can completely change students' attitudes about math so they walk into our classroom smiling and eager to begin the lesson.

Here's what one teacher from Idaho called to tell us:

Since returning to teaching, I chose Excel Math again because I had experience with it and knew how well it covered the curriculum, introduced new concepts, and provided spiral review.  

While teaching 3rd grade, 22 out of my 23 students received a score of "advanced" on the ISAT, and ALL students demonstrated significant growth on their MAP math scores. I had similar results with my 5th graders, with all students showing impressive growth. 

More important in my eyes, my students report LIKING math and liking learning through Excel Math. I love Excel Math...it has helped me become a stronger math teacher, and I KNOW it has increased my students' confidence in math (and their test scores!)."

— Anne Evans, Teacher, Meridian School District

Read more research on this topic in our previous blog posts, Research Agrees: Spaced Repetition Really Works! and Research Supports the Excel Math Spiraling Process.

Bulletin Board Ideas for the Math Classroom

A simple, low-cost and attractive way to decorate your math classroom is with bulletin boards. Place one of the the bulletin boards near the door so it will catch the children's interest as they enter. If you don't actually have a bulletin board, use part of a wall. Attach trim or a 3-inch wide strip of construction paper to form the border. The hallway outside your room is also a great place for a bulletin board and can be a handy spot to post announcements and upcoming class and all-school events. If you share the hallway bulletin board with other classes, brainstorm ideas with those teachers. Some simple bulletin board ideas include:

• Cover the bulletin board with a large sheet of poster paper, gift wrap, or cloth for a colorful background. Add a star trim border around the edges of the board to highlight your classroom tasks or math fact families. Change the trim to pumpkins in October, snowflakes in November, hearts in February, umbrellas in April, and flowers in May.

Class Favorites Bulletin Board
from Excel Math at excelmath.com

• Make a colorful border by cutting construction paper strips or using corrugated trim in a contrasting color to your bulletin board background. You can be as creative as you like by cutting the border strips in fancy or decorative shapes. Add stickers or glue star cutouts (or other seasonal shapes) around the border to make it pop.
• Adding a three-dimensional effect can give your bulletin boards added eye appeal. Make flowers out of egg carton cups. Put a layer of thick cardboard behind some of your bulletin board figures to make them stand out.
• Attach materials to the bulletin boards using staples, as they are much more difficult for younger children to remove and swallow. Be sure that no loose staples are left on the floor after you've completed the board. Use textured materials such as sandpaper for a sandy beach, cotton balls for clouds, and green felt or fabric for grassy hills.
• Make interactive bulletin boards: Mount a mirror on the board, covered with a piece of cloth. Place a picture of of your class beside the mirror. Guide a child to the board. Ask, Who is important in our classroom? Look and see. Create a header for the bulletin board with this question. Have the student lift the cloth to see himself. Let each student write a paragraph about his strengths and abilities. For the final sentence, have each student write, "Who am I?" Then have him fold up the bottom third of the page and print his name. Unfold the paper and fold it in half with the words inside. Let the child print a math problem on the outside, staple the paper to the board, and let the children read about each other and guess who is being described.
• Make a bulletin board titled, “We are learning and growing together.” Mount a large green stem with paper leaves so you have one for each child. Let each child draw a self-portrait of his or her face in the center of a paper plate. Have the child color the rim of the plate on both sides. Then let the child cut 1-inch slits along the edge of the plate at about 2-inch intervals and fold the edges in to frame the self-portrait. Glue or staple a  plate on top of each stem to form a garden of flowers. Add butterflies, ladybugs, etc. Print each child's name on a leaf of his or her self-portrait. Guide younger students to the board and help them identify their classmates by name. 

Interactive Addition Bulletin Board
from Excel Math at excelmath.com

• Make an interactive math fact bulletin board. Glue three clear plastic pockets to your board. Make sure each pocket can hold an index card. Place several cards with numbers on them in each pocket. For the last pocket, turn the number cards so they face the board. Add a plus, minus, multiplication, or division sign between the first two pockets and an equal sign before the last pocket. Let your students solve the equation during free time or as bell work when they enter the room. Have a child choose the correct answer (number card) from the last pocket and turn it facing forward. Change the numbers a few times each day and let the students solve the new problems.
• Head your bulletin board "It All Adds Up." Create several sets of pockets as described in the previous paragraph. Use only addition on this board. Place 6-8 problems on the board. Leave the answer pockets blank. Prepare several (correct) answer cards for each problem so you can give one to each of your students. Place the answer cards in a box or basket face down. As they enter the room, have each child take a number card from the box or basket. At your signal, the first 6-8 students you select may each take the card and place it in the correct answer pocket on the board. After each pocket is full, remove the cards. At a later time during the class, let the next group of students have a turn to add their cards to the answer pockets. Continue until most students have had a turn. Those who didn't get a turn can go first the next day. (Variation: Give each child a blank card on which to print the answer. Let the children take turns adding an answer to the pockets until they are filled. Then remove the answer cards and let the next group of children add their cards.)
• Copy the clock and hands page from below onto card stock and cut them out. (This clock is from the Manipulatives section of the Excel Math Teacher Edition.) Attach the hands to the clock using a paper fastener. Staple the clock to the bulletin board so the hands can be moved. Head the bulletin board "It's Time to …" Attach a clear pocket below the clock. Make several index cards with various times (such as 1:00 or one o'clock) written on each card (one time for each card). Place the index cards in the pocket so one card is showing. Have a child come to the front of the room and change the clock hands to match the time on the card. (Variation: Move the clock hands to a certain time and let each child key in the time on his iPad, computer, or student response system. Continue for 5-10 minutes at the beginning or end of class.)

Clock Pattern from Excel Math Teacher Edition Manipulatives
Click here for a PDF download.

For more math bulletin board ideas plus seasonal patterns, visit our September blog post, Seasonal Bulletin Board Ideas for the Math Classroom. Feel free to share your bulletin board ideas with us. Just add a comment below with your idea and blog address. If we use it in a future post, we'll credit you and include a link to your website or blog.

Find out how Excel Math can work for your students at excelmath.com. Excel Math is fully aligned to the Common Core and to state standards. Download correlations.

New to Excel Math? Learn how to get started.

Five-Minute Math Class Warm-Up Activities

Making every minute count in your math class means giving the students something to do the moment they enter the room. It also means teaching them the proper procedure so no time is wasted. Since the new school year is fast approaching, we wanted to give you some tips for getting off to a great start. How wonderful would it be if your students were on-task before the bell rings, already beginning the work of learning mathematics (or whichever subject you teach).

My husband is a middle school history teacher who has his students begin the class by filling out their planner, adding a proper heading to their page, and copying Geography Brief information into their notebooks from the board or computer projector. However you begin your class, discipline problems are kept to a minimum when your students know they must be on-task from the minute they enter the room. (And at the end of class, letting them know the bell does not dismiss them—the teacher does also helps them stay on task until the teacher signals it's time to leave.) Having control of the class, filling each moment with well-prepared activities, and setting up a structured way to begin the lesson that they can count on each day provides students with a sense of security and familiarity. 

Many of our students today come from chaotic, confusing and unreliable environments of parent job loss, disfunctional families, unstable home life, exposure to drugs and alcohol, helicopter or absent parents, foreclosed homes, deteriorating neighborhoods, questionable friends, and technology overload. Your classroom may be the most stable environment your students have. And you, the teacher or administrator, may be one of the few trusted adults in their lives. No wonder they crave structure, clear guidelines and a safe place to learn. Your classroom can be a safe haven in the midst of the storms of your students' lives. When you are a class manager, rather than a disciplinarian, the whole learning atmosphere for your students improves.

According to Harry Wong, educational speaker from Mountainview, California, "What you do the first day of school will determine your success for the rest of the school year. You would not expect a truck driver to haul an expensive load without first making sure he knew how to drive the truck. Neither can you expect students to succeed if they do not know the routines and procedures of your class." Read more from Harry Wong at http://teachers.net/gazette/wong.html.

Muxin Li, editor and writer for the Women of China magazine has recommended, "Decide on a short, five-minute warm-up activity to use at the beginning of each class. Beginning class with a consistent warm-up activity will prepare students for the lesson and can help them develop a valuable skill throughout the course. Using effective PowerPoint presentations can be a great teaching tool, but they shouldn’t replace instruction. " Read more from Muxin at Suite101.com:  http://suite101.com/article/writing-lesson-plans-for-teachers-a164597#ixzz2100A19qR

Here are some five-minute classroom warm-up activities to fill those first few minutes as students enter your room and get settled in for math class.

This is a well-thought-out 8th grade math class warm-up routine from the Teaching Channel: https://www.teachingchannel.org/videos/class-warm-up-routine

And here's a Basic Fact Practice section from Excel Math 2nd Grade Student Worksheet Lesson 106:

Basic Fact Practice from Excel Math Grade 2 Student Worksheet Lesson 106

You can use this free math worksheet with your students for bell work. Download and print a copy for each student. Give each student a blank sheet of paper along with the Basic Fact Practice sheet. Have the students print their names at the top of the page and then cover up all except the first line of Basic Fact Practice and begin solving the problems. They should do as many as they can in 5-10 minutes. When they finish the first line, they can move on to the next. When you give the signal that time's up, they stop where they are and continue solving the problems on the page the next day. 

You can go over the answers together and help students who may need extra practice. The idea is to provide your students with opportunities to practice basic math facts so they become easier as the year progresses. If students can't recall basic facts, it will be difficult for them to learn other concepts in later years. You can use this worksheet for timed exercises as well as just for individual practice. Click here to download your copy. The answers are given below.

Mental Math is a good way to help students practice calculating math problems in their heads. The teacher reads numbers aloud and the students practice adding or subtracting the numbers in their heads.  Visit our website for a free download of this resource. The Mental Math PDF file includes instructions and a table of numbers up through 100. After your students complete their bell work at the beginning of class, do 5-10 minutes of mental math with them before jumping into the lesson of the day. You can also do a few minutes of Mental Math at the end of class whenever you have a few minutes remaining before the bell.

Read more about Mental Math and how to use these sheets on our previous blog post: http://excelmathmike.blogspot.com/2010/03/mental-math.html

If your students have access to technology, they'll enjoy this timed math fact practice on the Excel Math website. (Thanks to our friends at the U.S. Department of Energy's Argonne National Laboratory for creating this timed math game and letting us use it.) You can choose which number should be at the high and low ends of the practice and whether to have your students practice addition, subtraction, multiplication, division (formatted as fractions), or random equations. Try out the Excel Math Timed Fact Practice and see how many problems you can solve in 60 seconds.

Learn more about how Excel Math can work for your students at excelmath.com. Excel Math is fully aligned to the Common Core and to state standards. Download correlations.

If you have additional math warm-ups you've used with your class, feel free to share them in the comment section immediately following this post. (Click on the word comment to begin.)

New to Excel Math? Learn how to get started.

Here are the answers to the Basic Fact Practice Sheet shown above:

You may also like these articles:

Fibonacci: 810 Years of Mathematical Magic

Roman Numeral Clock

Have you ever wondered where we got our decimal numbering system? The Roman Empire developed a system for numbering called the Roman numeral system, which we still see today in some copyright notices (1998 is MCMXCVIII) and on some watches and clocks.

The Romans didn't have a symbol for zero as we do with Arabic numerals (Arabic numerals are the ones we use today), and numeral placement within a number could sometimes indicate subtraction (if the smaller number was placed before the larger one) rather than addition (if the smaller number came after the larger one). So the Roman numeral I = 1 and V = 5. IV = 4 (5 - 1), and VI = 6 (5 + 1). L = 50, C = 100, D = 500, and M = 1000. You can try out a Roman numeral converter at novaroma.org.

The Roman numerals were not displaced until the 13th Century AD when Fibonacci published his Liber abaci, which means The Book of Calculations. This book showed the practicality of the new Arabic numbering system by applying it to bookkeeping, conversion of weights and measures, the calculation of interest, money-changing, and other everyday applications.

Leonardo da Pisa

Fibonacci, or more correctly Leonardo da Pisa (also called Leonardo Bigollo), was born in Pisa, Italy in 1175. He was the son of a merchant who also served as a customs officer in North Africa. Fibonacci travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence.

In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write Liber abaci in which he introduced the Latin-speaking world to the decimal number system. The first chapter of Part 1 begins:

"These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign which in Arabic is called zephirum, any number can be written, as will be demonstrated."

Fibonacci is perhaps best known for a simple series of numbers, introduced in Liber abaci in 1202 and later named the Fibonacci sequence in his honor.

The series begins with 0 or 1. After that, use the simple rule: Add the last two numbers to get the next number in the sequence. See the series in action at thinkquest.org. 

Fibonacci did not express his mathematics in the form of equations as we do today, but he did word them in such a way as to be easily translatable into modern equations. Here's an example from the Liber quadatorum [3, p. 4]:

"I thought about the origin of all square numbers and discovered that they arise out of the increasing sequence of odd numbers; for the unity is a square and from it is made the first square, namely 1; to unity is added 3, making the second square, namely 4, with root 2; if to the sum is added the third odd number, namely 5, the third square is created, namely 9, with root 3; and thus sums of square of consecutive odd numbers and a sequence of square always arise together";

this is readily translatable into the following set of equations [3, p4]

• 1 = 1²
• 1 + 3 = 2²
• 1 + 3 + 5 = 3²
• 1 + 3 + 5 + 7 = 4²
and suggests the general formula, which we would express as:
• 1 + 3 + ... + (2n-1) = n² 

The best known work of Fibonacci concerns the Fibonacci numbers, first examined in the Liber abbaci through the rabbit problem:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year (or n months) if it is supposed that every month each pair produces a new pair and the new pair from the second month on becomes productive.

1. At the end of the first month, they mate, but there is still one only 1 pair. 
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. 
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. 
4. At the end of the fourth month, the original female has produced yet another new pair and the female born two months ago produces her first pair, making 5 pairs. Read more at math.rutgers.edu.

The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

From this statement the famous Fibonacci numbers can be derived. 

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

where the first two numbers of the series are 1 and 1 and each numbers afterwards is defined as the sum of the two previous terms, Fn=Fn-2+Fn-1. (Though in Fibonacci's sequence the first number was 1 and the second number was two, the first one was assumed.) Read more at plus.maths.org.

Imagine that there are x_n pairs of rabbits after n months. The number of pairs in month n+1 will be x_n (in this problem, rabbits never die) plus the number of new pairs born. But new pairs are only born to pairs at least 1 month old, so there will be 

x_n -1 new pairs.

x_n+1 = x_n + x_n-1

And this is the rule for generating the Fibonacci numbers. If you prefer to have someone else do the calculations, this Fibonacci Calculator calculates thousands of Fibonacci numbers exactly and millions upon millions to the first few digits!

Fibonacci mathematics is a constantly expanding branch of number theory, with more and more people being drawn into the complex subtleties of Fibonacci's legacy. Here's a fun comic strip featuring the Fibonacci sequence from Bill Amend at FoxTrot.com.

The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). (The right panel instead applies the Perrin sequence). See more FoxTrot comic strips by Bill Amend at foxtrot.com.

Mathematicians celebrate Fibonacci Day on November 23 or 11/23, taking the date from the first four numbers in the Fibonacci sequence shown above.

Excel Math is a proven method that gives students a solid foundation of elementary math. Excel Math Student Lesson Sheets are much more than just math worksheets. Using strategically placed spaced repetition, Excel Math gives you a proven approach to teach math concepts for long-term retention, with powerful features and advantages, including our proprietary Spiraling Strategy. Much like the Fibonacci spiral (which we'll visit in November), concepts are reintroduced on a regular basis (with our unique method of spaced repetition) so they become a part of the student's longterm memory. Read the amazing reports from teachers, parents, and principals around the country.

Learn more about how Excel Math can work for your students at excelmath.com. Excel Math is fully aligned to the Common Core and to state standards. Download correlations.

New to Excel Math? Learn how to get started.

Common Core Math Resources

Common Core is a phrase hitting the web, the nation, and local school districts in a big way. If your state is not moving to Common Core, you may be interested in the debate surrounding its implementation and impact below. Then jump to our links and math correlations at the bottom of this post. Our Texas friends can visit our Texas Connection for more information on the TEKS and tools to prepare students for STAAR testing.

We've been sifting through lots of resources, blog posts, videos and articles on moving to the Common Core and we thought we'd share some of our favorites. Here's an interactive map showing which states have adopted the CCS, with links to each state's board of education for more information on how they're implementing the Common Core. NCTM has prepared an overview PowerPoint presentation and other presentations for Pre-K–Grade 3, grades 4–5, grades 6–8, and high school to inform math teachers about these new standards and to support them in implementation of the Common Core.

Excel Math is fully correlated to Common Core (CCS) and State Standards. One of the great benefits of Excel Math is that it is continually enhanced. We print Excel Math throughout the year so you will always have the latest lesson sheets, unlike textbooks, which are static until you buy a new book.

However, Excel Math gives your students even more. With Excel Math, students learn higher-order thinking skills beyond what is required of CCS or state standards. Our correlations allow you to focus on the standards or go even further, with additional concepts we provide. View and download Excel Math CCS Correlations.

If you want to see the Common Core standards or have questions about the Common Core, these first websites are the place to begin. Those of  you already familiar with the standards may be interested in learning more about the debate, discovering how to implement the Common Core, and finding websites with CCS resources. If your school is moving to the Common Core, take a look at the links we've pulled together to help you out.

CCS website information from the source:
Common Core State Standards Initiative
FAQ about the Common Core
Math Common Core Standards
Key Points in Mathematics Standards

Explaining the Common Core:
Common Core: Six Fundamental Shifts « Critical Thinking « Critical Thinking Works
Making the connection: Common Core and National Educational Technology standards

TheHuntInstitute – YouTube
New Guide on Development of Assessments for the Common Core State Standards
Understanding the Common Core Standards from edutopia.org
You’ve Got To Read This from achievethecore.org

Bastille Day Math July 14

Bastille Day celebrates the storming of La Bastille prison by French revolutionaries and mutinous troops who dismantled the Bastille and set free the seven prisoners inside in 1789. The Bastille was a medieval fortress that was designed and built around the city of Paris for protection. This royal fortress had come to symbolize the tyranny of the Bourbon monarchs. The dramatic action on July 14, 1789 signaled the beginning of the French Revolution, a decade of political turmoil and terror in which King Louis XVI was overthrown and tens of thousands of people, including the king and his wife Marie Antoinette, were executed. 

The Bastille was originally constructed in 1370 as a bastide, or "fortification," to protect the walled city of Paris from English attack. It was later made into an independent stronghold, and its name—bastide—was corrupted to Bastille. This French national holiday is officially called the Fête Nationale, and commemorates the Fête de la Fédération, a huge feast and official event begun in 1790 to celebrate the establishment of the short-lived constitutional monarchy in France.

The Bastille was first used as a state prison in the 17th century, and its cells were reserved for upper-class felons, political troublemakers and spies. Most of its prisoners were imprisoned without a trial under direct orders of the king. Read more at history.com.

Standing 100 feet tall and surrounded by a moat more than 80 feet wide, the Bastille was an imposing structure. It had eight towers, each about 78 feet tall, that were linked by walls of equal height and surrounded two courtyards and the armoury. The outer stone walls, 15 feet (4.5m) thick at the base, were pierced with narrow slits by which the cells were lighted. In early times, the Bastille had entrances on three sides, but after 1580 only one remained, with a drawbridge over the moat on the side toward the river, leading to outer courts and a second drawbridge. Learn more and see illustrations at discoverfrance.net.

The capture of the Bastille provided the French revolutionary cause with an unstoppable momentum. Joined by four-fifths of the French army, the revolutionaries seized control of Paris and then the French countryside, forcing King Louis XVI to accept a constitutional government. In 1792, the monarchy was abolished and King Louis XVI and his wife Marie Antoinette, were sent to the guillotine for treason.

Arc de Triomphe
on the Champs Elysées

By order of the new revolutionary government, the Bastille was torn down. On February 6, 1790, the last stone of the hated prison-fortress was presented to the National Assembly. Bastille Day was created on July 14, 1880. Today, Bastille Day is celebrated as a national holiday in France. Parisians celebrate this national holiday with a grand military parade up the Champs Elysées, colorful arts festivals, and elaborate parties marking the holiday. Read more at infoplease.com.

Around the United States, you can celebrate Bastille Day on July 14 this year with a 5K run in Chicago, a Bastille Day festival in Seattle and Milwaukee (and numerous other cities), a celebration at Eastern State Penitentiary in Pennsylvania, brunch or dinner with a French flair in San Diego, a concert in Hamilton (New Jersey), and a street party in Boston, to name just a few.

Tour de France 2010

You can also watch bikers riding through France and Belgium on the Tour de France now through July 22. See the latest yellow shirts and watch the riders compete at http://www.letour.fr/indexTDF_us.html. 

Here's a photo my friend Mike took in 2010 when he visited France and followed the riders. The 99th Tour de France this year consists of 1 prologue and 20 stages and will cover a total distance of 3,497 kilometers. The race will finish in Paris on the Champs Elysées.

In Excel Math, we challenge students to solve word problems and then create their own after they've mastered the necessary concepts. Here's a Student Lesson Sheet with a Create A Problem exercise called "Tour de Vacation" in which students create a graph, calculate distance, and write their own word problem. Click here to download a PDF file you can use with your students:

Excel Math Grade 4 Student Lesson Sheet Create A Problem 18
Click here to download a PDF file.

Learn more about how Excel Math can work for your students at excelmath.com. (Excel Math is fully aligned to the Common Core and to state standards. Download correlations.) Here are the suggested answers to Create A Problem 18:

Excel Math Grade 4 Student Lesson Sheet Create A Problem 18 Answers

Summer Math: Bridging the Achievement Gap

According to The HechingerEd blog, "When kids return to school in the fall, on average they’ve slipped by about a month from where they were in the spring." This quote is from Catherine Augustine, a senior policy researcher at the RAND Corporation, a nonprofit research group, and co-author of a report released last year on summer learning programs. She also noted that both low-income and high-income children lose ground in math over the summer during the elementary school years. Read more at http://hechingered.org/content/how-summer-increases-the-achievement-gap_5072/.

Gary Huggins, chief executive officer of the National Summer Learning Association, has said, "Most youth lose about two months of grade-level equivalency in math computational skills over the summer."

"The effects of this 'summer slide' are cumulative and lead to a widening achievement gap, placement in less rigorous high school courses, higher high school dropout rates, and lower college attendance. In fact, summer learning loss accounts for two-thirds of the achievement gap in reading by the ninth grade," according to a Johns Hopkins University study.

Here's a video of our young friend Tyler, showing how some California students spend their summers. During the school year, he enjoys completing Excel Math lessons. But when summer comes, it's off to the beach with family and friends to catch a wave:

Surfing with Tyler [1:09 min]

Just as it takes practice to be able to stay on the surfboard, it takes regular practice for elementary students to keep from losing ground in math over the summer. That's where Excel Math Summer School Edition can help. Some teachers are recommending that parents use the Excel Math Summer School Edition with their students to help close the achievement gap over the summer. Here's what one teacher in Illinois called to tell us:

“We are thrilled with Excel Math! We even have parents ordering the summer school edition to keep the math concepts fresh in their students' minds over the summer break.” — Teacher at St. Patrick Catholic School in Washington, Illinois

Excel Math Summer Edition is used by teachers, camp directors and tutors to provide remedial instruction for students who need extra help. It's also used with more advanced students to help them stay focused during the summer and to challenge them with additional brainteasers (Stretches) and word problems (Create A Problem exercises) that teach higher-order thinking skills and combine math with literacy. Learn more about Excel Math Summer School Edition for Pre-Kindergarten through Pre-Seventh grades. Here's an example of an Excel Math Create A Problem exercise for grade 3:

Create a Problem exercise from Excel Math Grade 3 Student Sheet

The Excel Math Summer School Editions are graded to prepare students for entering that grade. So Kindergarten Excel Math Summer School is for children entering Kindergarten (to prepare them for Kindergarten), and Seventh Grade Summer School is for students entering seventh grade. Each Student Book has a companion Teacher Edition sold separately containing the answer keys, teaching suggestions, stretches (brain teasers) and grade charts. Thanks to Anne in Idaho for her wonderful report about Excel Math Lessons. Read more from Anne and others at excelmath.com/about/successes.html:

"While teaching 3rd grade, 22 out of my 23 students received a score of "advanced" on the ISAT, and ALL students demonstrated significant growth on their MAP math scores. I had similar results with my 5th graders, with all students showing impressive growth. Furthermore, and more important in my eyes, my students report LIKING math and liking learning through Excel Math. 

I love Excel Math...it has helped me become a stronger math teacher, and I KNOW it has increased my students' confidence in math (and their test scores!)." 

— Anne Evans, Teacher in Idaho

Excel Math Student Lesson Sheets are much more than just math worksheets. Using strategically placed spaced repetition, Excel Math gives you a proven approach to teach math concepts for long-term retention, with powerful features and advantages, including our unique Spiraling Strategy. Take a look at Excel Math online at excelmath.com. (Summer School is available as an Individual Student Set padded book for Kindergarten through grade 7. A companion Summer Teacher Edition is also available for each grade level.)

Top 10 Reasons Your Students Will Love Excel Math

TOP 10 REASONS 

Your Students Will Thank You for Using Excel Math

1. Excel Math lessons go deeper. 
Our lessons present concepts in a more complete and complex manner so students learn higher-order thinking skills. See a lesson sample.

2. Your lessons will have increased rigor. 
With the Excel Math True Spiraling of concepts, lessons move smoothly from one to the next, gradually building on existing layers. With this approach, students see that math concepts are related and progressive.

3. Students have a built-in feedback loop. 
Our CheckAnswer system lets students correct their own work. Feedback is immediate. Students demonstrate understanding and mastery of math concepts as they build proficiency.

4. There's no busy work. 
Student Lesson Sheets eliminate the need for students to copy problems out of a book, but they give space for students to show their work. Once students have mastered a concept, you can choose to let them skip those problems in their Guided Practice.

5. Your lessons will combine math with literacy. 
Create A Problem exercises on the back of test pages along with word problems and stretches combine math with literacy and help students develop higher-order thinking skills.

6. You’ll interact more with your students. 
Guided practice after the daily lesson lets teachers help students individually while other students work independently. Activities and manipulatives in the Teacher’s Edition provide hands-on learning, challenge based learning, and problem based learning opportunities.

7. You can use technology in your math classroom.
Projectable Lessons provide slides for 155 lessons on CD so you and your students can focus together on the concepts for the day using a white board, overhead document projector, interactive white board, or simply a wall or screen.

8. You’ll have built-in tools for assessment. 
Homework is included on 4 out of 5 lessons and you get weekly quizzes, quarterly tests, and end-of-year tests. Your Teacher’s Edition shows when tested concepts were initially taught so you can see at a glance where your students need more help and where they’ve achieved mastery.

9. Your test scores will go up. 
Schools across the country report consistently improved test scores when they use Excel Math. Our unique Spiraling Process (see illustration below) and regular assessment ensures concepts are mastered and retained for the long term. (See Anne Evan's testimony below.)

10. You’ll save money! 
Excel Math Student Classroom Sets cost just $11.00 per student (Individual Padded Student Sets are only $19.00 each). Excel Math is a proven approach to teaching elementary math! Learn more.

Excel Math's Unique Spiraling Strategy
Click here to see the full chart.

Here's what teachers are saying about Excel Math:

"I have used Excel Math for four of the five years that I have taught. Two of these years were in the Clark County School District about ten years ago, where I taught a 4th/5th grade combination class with over 30 students each year. Excel Math allowed me to effectively teach math at two grade levels...without it I don't know how I would have hit all standards and provided the review and practice the students required to succeed. I recently had the opportunity to visit with twin sisters who were in my class during this time, and their mother shared with me that "those long math sheets" really prepared her girls for math throughout the rest of their schooling. Both girls are in college now, are strong math students, and both reported that they started liking math when they were in my class!

Since returning to teaching, I have taught 3rd and 5th grade in the Meridian School District in Idaho. I chose Excel Math again because I had experience with it and knew how well it covered the curriculum, introduced new concepts, and provided spiral review. While teaching 3rd grade, 22 out of my 23 students received a score of "advanced" on the ISAT, and ALL students demonstrated significant growth on their MAP math scores. I had similar results with my 5th graders, with all students showing impressive growth. Furthermore, and more important in my eyes, my students report LIKING math and liking learning through Excel Math.

Excel Math allows the teacher to put their own flavor and flare into each lesson or concept. I love Excel Math...it has helped me become a stronger math teacher, and I KNOW it has increased my students' confidence in math (and their test scores!)."

— Anne Evans, Teacher in Idaho

Excel Math is extremely teacher friendly and is fully correlated to Common Core (CCS) and State Standards. Read more from teachers here. We love teachers and we love serving you! Any time you call, we’re happy to hear from you. We’re willing to do whatever we can to make your experience with Excel Math a pleasure. Visit our Excel Math website at excelmath.com for lots of math resources.