The ten pizzas and ten cakes arrived and were quickly devoured by the two hundred kids who came to the party. After several hours, they all showed up with food that they promptly ate, leaving Moira with the problem of a messy house, which she solved using the presents kids had brought for her birthday. After all the kids left, the remainder of her orders were delivered and dumped on her front lawn— cakes and pizzas. It also provides a context for collecting and using information to estimate how much pizza to order for a birthday party of two hundred kids.

The data collected provides a context to introduce the concept of average using mean, median, and mode. Focus on the problem of too much pizza. Following this discussion, let students know that you will help them collect and organize information about slices of. Have each student write his or her number of slices on a sticky in writing large enough to be seen across the room.

Guide students in posting their data from smallest to largest along one section of the board, like the following example. Be ready to explain your reasons for choosing that number. As you introduce those ways, link them to the ideas students have shared. Let students know that mathematicians sometimes use the median in a set of data to solve problems. Ask students how they could figure out which response is in the middle of the set of pizza slices data posted on the board. Label the middle piece of data with the word median.

Mathematicians sometimes use the median in a set of data to make some estimates about a group. Pose this question:. Mathematicians also use the mode to make estimates about a group. Pose the following question about the data:. Call this the mode , the most common number of slices. Ask students for their ideas on how the data could be reorganized to easily show the number of slices chosen most often.

Reorganize the data and label the mode. For example:. Let students know they are going to work as a class to figure out how many slices of pizza each person would get if they evenly distributed the number of pieces represented by your data. Ask each student to take Snap Cubes to represent his or her number of slices of pizza. So if someone chose zero pieces of pizza, he or she would have zero cubes; if someone chose three pieces of pizza, then he or she would have three cubes.

Ask everyone to stand and pair up with someone who has a different number of cubes. In pairs, have students make a stack of all cubes from the partnership. Then they should take the cubes and split them into two equal stacks of cubes or as near equal as possible. Continue the process of pairing up with partners having a different number of cubes, combining the cubes, and then splitting them into equal or near equal stacks. Continue the pairing and splitting until everyone has the same or almost the same number of cubes. In this example, students would have either two or three cubes after the process of pairing and sharing.

Let students know that they just went through a process of evenly distributing cubes in the group to represent evenly distributing slices of pizza.

You may want to model the process mathematicians would use to find the mean number of slices: adding up all the numbers of slices students reported in this case, 75 and dividing by the number of students in the group in this case, 27 , which would give 2. After taking a look at the data and these three statistical benchmarks in the data mean, median, and mode , ask the following questions:. Distribute the newsprint and markers. Pose the following problem for students to work on as a group of four:.

What is a reasonable number of pizzas Moira could have ordered to feed the kids at her birthday party? Also, let students know they should be prepared to share their strategies with the class in a whole-class discussion. While observing students at work, decide on a general order you will use to ask students to share their solutions, providing a scaffolded discussion for students that supports access for all students to the variety of ways the problem has been solved.

You may want to alert groups that you will ask to present early in the discussion.

Call students together for a class discussion of their solutions to the problem. The goal of the discussion is to reach a common understanding of the problem and its solution and to see how solutions compare and how approaches to solving the problem are the same and different. As each group presents, classmates should listen with the following questions in mind:. In the course of discussion, revisit the ideas of mean, median, and mode, and help students identify how they used those ideas in their problem solutions.

In this dice activity, the class works together to generate the numbers one through twelve in order. Students need to think flexibly and consider many possibilities in order to find the solutions to the computation challenges involved. Their response was positive. Our goal is to systematically get rid of each number I listed. We have to get rid of the numbers in order, beginning with one first and then moving to two, and all the way to twelve.

Then you need to think about how you can use these numbers to make the number one. You may use any operation—addition, subtraction, multiplication, or division—or a combination of operations. Here I go. Amy was surprised that students appeared to be struggling. Then she realized her directions might not have been clear enough.

You can use one, two, or all three of the numbers rolled to make the number you need. Adriana suggested the need for parentheses around 5 — 3. Technically parentheses are not required in this situation; however, their use is not incorrect and adds clarity. Amy decided to include them. She then pointed out that since she had rolled a 1, they could use just that 1, and she recorded that on the board too.

Next up is two. Amy wrote equations for their ideas next to the 2 and then crossed off the 2. Now that the students understood the mechanics of the activity, she gave them time to talk in groups about ways to make the rest of the numbers. Soon the board filled up with a variety of solutions for the next few numbers.

The students were more interested in finding multiple solutions for each number than moving on to the next number. To keep the game moving, Amy decided to limit responses to one equation for each number. In a few minutes the board looked like this:. The class got stuck on eleven, so Amy told the students it was time to roll again. Before rolling, she asked a question to assess their comfort with the various operations. What numbers would you like to get now? This quick question showed Amy that her students were comfortable breaking eleven apart in different ways but were not as comfortable thinking about operations other than addition.

Perhaps with more time and practice, they would open up their thinking. She rolled the dice and got two 5s and a 1. Then she turned to the class. Amy wrote the equation next to the 11 on the board. She pointed to the first 5 and the 1. Amy gave the students a minute to talk to a partner about how to use these rolls to make twelve. This time she got a 5, a 2, and a 1. She wrote these numbers on the board below the previous roll. The students took a minute to consider the possibilities.

Christine raised her hand. Amy wrote the equation on the board and had the class double-check to make sure everyone agreed. Amy crossed out the 12 on the board and congratulated the students for clearing the board. She asked how many rolls it took for them to accomplish their task. Since she had written each roll on the board, the students had an easy reference. Amy wondered aloud if they thought it would take more or less than three rolls when they played next time.

Leaving them with something to think about set the stage for future experiences with Clear the Board. This fifth-grade teacher knows that making connections among concepts and representations is a big idea in mathematics. She wants all of her students to be able to represent and connect number theory ideas.

The teacher will incorporate this idea into the unit, but first, she wants to informally pre-assess her students to find out how they might classify numbers. She asks students to get out their math journals and pencils and gather in the meeting area. She writes the numbers 4, 16, 36, 48, 64 , and 81 on the whiteboard and asks the students to copy these numbers in their journals.

She then writes:. I want everyone to have some time to think. Students are given time to work independently while the room is quiet. The teacher thinks about asking how many other students agree with Sheila, but as she wants them to eventually find more than one candidate, she decides not to have them commit to this thinking. Like Tara, Marybeth, and Dewayne, you might find different reasons for why a particular number does not belong. Like Melissa, if you change the rule, you might find a different number that does not belong. You will have about ten minutes.

You can work on your ideas the whole time or, if you finish early, find a partner who has finished and share. Some of you might find rules to eliminate each of the numbers, one at a time. Tara seems to recognize a visual difference in the numbers, while Marybeth uses more formal language to describe this attribute. Dewayne refers to the value of the numbers and Melissa is willing to verbalize another possibility. The teacher looks around and sees everyone has spread out a bit and begun to think and write.

After a couple of minutes, she notices that a few of the students have stopped after writing about the number four. Some do and others need to be reminded, but with a bit of coaching, all are able to identify eighty-one as different because it is an odd number. After almost ten minutes the teacher notices that all but two of the students are talking in pairs about their work.

She gives them a one-minute warning and when that time has passed,they huddle more closely and refocus as a group. The teacher asks them to draw a line under what they have written so far, and then to take notes about new ideas that arise in their group conversation.

As the students share their work, several ideas related to number theory terms and concepts are heard. Jason identifies the square numbers in order to distinguish forty-eight from the other numbers. Sometimes the teacher asks another student to restate what has been heard, or to define a term, or to come up with a new number that could be included in the list to fit the rule. Each time, the teacher makes sure there is time for students to take some notes and that the majority of them agree that the classification works. Two students use arithmetic to find a number that is different.

While students do not experience this activity as a pre-assessment task, it does give the teacher some important information that she can use to further plan the mini-unit on number theory. She has an indication of their understanding and comfort level with concepts and terms such as factors, multiples, divisible, primes, and square numbers.

From this information she can decide how to adapt her content for different students. The complexity of ideas can vary. Some students can reinforce ideas introduced through this activity, while others can investigate additional ideas such as triangle numbers, cubic numbers, and powers. Once content variations are determined, process is considered. Some students can draw dots to represent square and triangular numbers so that they have a visual image of them while others can connect to visual images of multiples and square numbers on a hundreds chart. Visual images of four on a hundreds chart and square numbers on a multiplication chart.

### Working with geometric series

The teacher can create some packets of logic problems, such as the one that follows, that require students to identify one number based on a series of clues involving number theory terminology:. When I put them in two equal stacks, there is one penny left over.

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When I put them in three equal stacks, there is one penny left over. When I put them in four equal stacks, there is one penny left over. Some students could write rap lyrics to help them remember the meaning of specific terms. Other students could play a two- or three-ring attribute game with number theory categories as labels; they would then place numbers written on small cards in those rings until they could identify the labels. A learning center on codes could help students explore how number theory is related to cryptology.

The teacher could think about pairs of students who will work well together during this unit and identify subsets of students that she wants to bring together for some focused instruction. Then the teacher must think about product —how her students can demonstrate their ability to use and apply their knowledge of number theory at the end of the unit. For example, students might write a number theory dictionary that includes representations, pretend they are interviewing for a secret agent job and explain why they should be hired based on their knowledge of number theory, create a dice game that involves prime numbers, make a collage with visual representations of number theory ideas, or create their own problem booklet.

In this lesson, excerpted from A Month-to-Month Guide: Fourth-Grade Math Math Solutions, , Lainie Schuster has her fourth graders start the school year with an investigation that offers them the opportunity to work in pairs to collect, represent, and analyze data. In this book, Martha, a dog who is able to talk as a result of eating a bowl of alphabet soup each day, finds herself in a pickle when the owner of the soup company reduces the number of letters in each can of alphabet soup. After listening to the story, students pair up to investigate the frequency with which each letter in the English alphabet is used.

They gather and organize their information, then summarize in writing what they notice about the letter frequencies. They create posters to show their findings and participate in whole-class discussions to share their thinking. To extend the activity, Lainie gives students pasta letters from which they form as many words as possible. I began the lesson with a read-aloud of Martha Blah Blah. Following the reading, we had a class discussion about letter distribution in words. Why might that be?

Do you think consonants are used more than vowels or vowels more than consonants? How could we collect the data? How could we chart the collected data? If Granny Flo, the owner of the soup company, was determined to eliminate the least-used letters from her soup, how could we help her make an informed decision at least for talking dogs? I directed the children to discuss these questions with their neighbor and come up with ideas for carrying out a letter-frequency investigation.

As the children talked, I circulated around the room, listening and handing out a record sheet containing writing prompts. See end of lesson for blackline master. When I felt everyone was ready, I called the class back together so students could share their ideas and discuss the writing assignment. In response, I asked the children if it was necessary to write down all the letters in the alphabet or if they could write down just the ones that were in their reading selection.

This led to a rich discussion about the importance of being able to quickly assess the holes in the collected data. Then I described the task. I explained that, working in pairs, they were going to determine how often each letter of the alphabet was used. I gave each pair a piece of newsprint on which to record all their work. I told them to post the data they collected in whatever format they were comfortable with and to leave room on their newsprint for writing summary statements and posting the results of another task, which I would give them the next day.

I also told them that they would have an opportunity to decorate their posters at the end of the investigation. Once we agreed on what they were being asked to do, the children set off to collect and represent their data. Many chose a paragraph from a book they were presently reading. Others picked one of the books in the classroom book display.

I had asked that the paragraphs be relatively short because it was the beginning of the school year, and I was more concerned about the manageability of the task than the length of the paragraph. Before all the students were finished collecting their data, I asked for their attention, and we discussed the written part of the task.

I asked the children to write two summary statements on their sheet of newsprint. I explained that a summary statement should explain what they noticed about the data they had collected. I had pairs work together to create summary statements but asked them to individually complete the writing prompts before discussing and comparing their opinions and answers. As the children went back to work, I circulated around the room to offer assistance.

## Aristotle and Mathematics

Talking about the mathematics is one thing—writing about it can be quite another. Sometimes it was necessary to remind students to refer to their collected data as they wrote. Writing Prompts for Martha Blah-Blah We would suggest that Granny Flo take out the following 7 letters: Without these 7 letters, it would be difficult for Martha to say the following words: Without these 7 letters, it would be easy for Martha to say : Why?

Since math time was almost over, I continued the lesson the next day. After students retrieved their newsprint, we began a class discussion of their findings and decisions. Most agreed that more consonants should be removed than vowels.

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I concluded this investigation with a word search made from a cupful of alphabet pasta. I gave each pair of students a paper cup containing twenty-five pieces of uncooked alphabet pasta. I then gave the following directions:. They were especially curious about the frequency findings, the conclusions that were drawn, and the words that were made from the pasta letters. Day-by-Day Math: Activities for Grades 3—6, by Susan Ohanian, is an eclectic and quirky collection of events — and the mathematical investigations, problems, or activities that are suggested by them.

Each day of the year, from January 1 through December 31, lists historical events, each a lighthearted or serious moment. Here are some dates and investigations that your students can explore this coming year:. It flies at twice the speed — miles — and twice the altitude — 36, feet — of propeller-driven airplanes. At this altitude, planes are able to fly above unsettled weather. Find out how long it takes to make a typical commercial flight from New York to San Francisco. What is the hourly speed? Is the flight time from San Francisco to New York the same? Weather information is available from many online sources.

Keep a weather graph charting the temperature for a month. Then find the average temperature for the month. Check an almanac to find out whether this is above or below average. Elizabeth Blackwell, who had been turned down by 28 colleges before she found one that would let her study medicine, graduates from Geneva Medical College now Hobart and Williams Smith Colleges in Geneva, New York, at the head of her class and becomes the first woman doctor in the United States. Look at the list of doctors in the yellow pages of the phone book. How many are male and how many are female?

Can you determine whether female doctors are more apt to specialize in one field of medicine over another? The Coca-Cola Company announces it is replacing its year-old recipe with a new formula. Customers react so negatively that on July 10 the same year it reintroduces the old Coke under a new name, Coca-Cola Classic. Every minute, people around the world drink , Cokes. How many Cokes are consumed in one week? Kansas becomes the 34th state. The name Kansas comes from an Indian word meaning flat or spreading water. The state flower is the sunflower. The sunflower provides pioneer settlers in the Midwest with oil for their lamps and food for themselves and their stock.

Native Americans roast sunflower seeds and ground them into flour for bread or pound them to release an oil for cooking and for making body paint. Look at a live sunflower or a detailed picture of one. A sunflower has two distinct parallel rows of seeds spiraling clockwise and counterclockwise. The seeds are Fibonacci numbers, typically 34 going one way and 55 going the other way, although sometimes they are 55 and Find other natural examples of Fibonacci patterns. Good places to look include pinecones, pineapples, artichokes, and African daisies.

Gorman and 28 other Navajo volunteers turned their native language into a secret code that allowed Marine commanders to issue reports and orders and to coordinate complex operations. Although the highly respected Japanese code crackers broke U. I was a little less than halfway through reading K.

If each slice sells for one gold coin, and if I can sell 10 pizzas a day. I gave students some quiet think time before asking them to turn to their partners and share their thoughts. After partners had discussed their ideas, I called them back to attention. Then I finished reading the book. The students enjoyed the colorful illustrations by Giuliano Ferri and were happy when Chris Croc and Ben Bear solved their problem of being hungry by buying food from one another, passing the one gold coin back and forth until there were no pizzas or cakes left. I waited a few seconds and then called on Daniel.

Next, I asked the students to estimate which number they thought would be closest to the exact answer. With a show of hands, we discovered that 10 students thought 2, coins was closest, 10 students thought 30, was closest, 4 estimated 1,,, and no one thought coins could be possible. I then gave students some time to talk with their partners about the reasonableness of the estimates.

After a minute, I asked for their ideas. I gave the students a few seconds to think, then called on Jesus. Then I did sixty plus sixty is one hundred and twenty.

## Lectures on computational aspects of algebraic geometry

This time, everyone thought that the exact answer was closest to 30,, except for Henry, who stuck with 2, After Anton explained how he figured the answer using the standard algorithm, we checked his result with Amber. When Anton and Amber reported the answer—29,—I ended the lesson by asking the students which number was closer to the exact answer: 30, or , Three hundred fifty thousand is way too big!

A few hands sprung up. Raise your hand when you have a fraction in mind. I called on Josh. I called on other students and recorded the fractions as they offered them. I reminded him that everyone could change his mind at any time in math class, as long as he had a reason. Several other students raised their hands. Sam raised a hand. I continued by asking the class for fractions for the other two columns, each time having the student explain her reasoning for the fraction she identified. Then I repeated the activity using three-eighths and then one-fourth as starting fractions.

I continued the lesson until only ten minutes remained in the period. Then I stopped to give the homework assignment. To avoid confusion when they were at home, I duplicated the directions for the homework and distributed them to the class:. Above the columns draw boxes for the numerator and denominator of the starting fraction. To find the starting fraction, roll a die twice.

Use the smaller number for the numerator and the larger number for the denominator. If both numbers you roll are the same, roll again so that the numerator and denominator of your starting fraction are different. Write at least five fractions in each column. The numerator and denominator in each fraction you write must be greater than the numerator and denominator in the starting fraction. Choose one fraction from each column and explain how you know it belongs there.

I emphasized the fifth rule. The next day, I had students report about what they had learned from the assignment. See Figures 1—3. No one reported having this problem. I ruled columns and started the activity. We continued for about fifteen minutes. To prepare for the lesson, I duplicated for each pair of students the shapes for the activity and a sheet of inch-squared paper.

I used yellow paper for the shapes so that there would be contrast when they pasted them on white paper. Also, I made an overhead transparency of each of the sheets I distributed. To begin, I projected an overhead transparency of the inch-squared paper, which was a 9-by-7 grid. From their study of multiplication, the students knew to multiply 7 by 9 to get the area of 63 square inches. I next placed a 5-byinch index card on the grid, positioning it carefully so its sides were on lines and it covered 40 of the squares completely.

I removed the 5-byinch index card, folded it in half the short way, and cut on the fold. I placed one half on the overhead grid. I then trimmed the 4-byinch card so it was a 4-byinch square. Next I cut the square in half on the diagonal, making two triangles. I placed one of them on the grid.

## Mathematical Techniques in Astronomy

Some students knew immediately that its area had to be 8 square inches. Tracing the triangle on the squared paper and then removing it helped the students see that it was, indeed, 8 square inches. Then two halves make a whole and two more halves make a whole. I then projected a transparency of the shapes the students were going to explore.

I explained what they were to do, also writing the directions on the board:. Students worked in pairs and I circulated, giving help as needed. I suggested to some students that they place a shape on the inch-squared paper and count the squares it covered. If the class had been a regular-length period, I would have collected their work and returned it to them to complete the next day. Then I called the students together to discuss what they had learned. Nicholas explained what he learned. That was cool.

Andrew and Hiroshi arranged their shapes from least area to greatest, as instructed. Roberto and Laura shared the writing of their discoveries. Danielle started reading One Riddle, One Answer aloud. Danielle stopped reading at the part where Aziza proposes that she write a number riddle. The princess tells her father that she would prefer to marry the person clever enough to answer her riddle.

The sultan agrees. You can use scratch paper to jot down your thoughts. Then, whisper your ideas about the answer to a partner. Danielle first gave students time to think and take notes. Then, as the students shared their ideas with their partners, she circulated, listening to their guesses. Some thought the answer was one; a few thought that it was zero; many were completely stumped.

She stopped at this point and addressed the class. Danielle continued to read. When Ahmed, the farmer in the story, guessed that the answer to the riddle was the number one, Danielle again checked with the students to see if it worked with all the clues, and it did. Danielle was taken aback. And it works for any number you multiply it by. Amanda nodded and continued with her argument. And when you count, zero comes before one on the number line! Getting students to argue passionately about their ideas in math class is often difficult to achieve. When the short debate was over, Danielle acknowledged that there could be more than one correct answer to the riddle.

Then she finished reading the story, including a section at the back of the book where the author explains how the number one works for each clue. First you have to think of a number to write about. Then you have to write some clues. I want you to practice by brainstorming some possible clues for the numberten. Danielle gave the students about five minutes to work in groups.

She then called the class together and wrote some of the clues the students came up with on the board:. Then Danielle gave feedback on the clues. When the class was finished brainstorming clues for the number ten, Danielle gave the homework assignment of writing one riddle sentence for the number one-half. The next day, Danielle asked for volunteers to share their riddles. Some students revealed some very sophisticated thinking about one-half, while others exposed misconceptions. Soon, the class began to hum with conversations as students shared their ideas with one another.

As Danielle circulated, she encouraged students to think of subtle rather than obvious clues. After about thirty minutes, she pulled the class back together. How are these numbers alike? Students may offer that the numbers are all less than Accept this, but push students to think about the factors of the numbers. Following are several possible responses: All have a factor of one. All have exactly two factors one and itself. Each can be represented by two rectangular arrays. All of them are prime. This question has one right answer the least common multiple for the numbers 4 and 6 is 12 ,but students may arrive at the answer in different ways.

But because 4 and 6 both share 2 as a factor, the least common multiple is less than the product of this pair of numbers. If numbers do not have a common factor, however, then the least common multiple is their product. To help students think about these ideas, consider presenting additional questions for them to ponder:. Can you find pairs of numbers for which the least common multiple is equal to the product of the pair? Can you find pairs of numbers for which the least common multiple is less than the product of the pair?

This question provides students with a real-world context—telling time—for thinking about a situation that involves numerical reasoning. The problem also provides a problem context for thinking about multiples. You might also ask students if they think it makes sense to have an amount other than 60 minutes in each hour, perhaps minutes, for example. What effect would such a decision have on how time is displayed on watches and clocks? Discuss the meanings of the math terms they use and the relationships among them.

This question can be asked for any set of four numbers. As an extension, ask students to choose four numbers for others to consider. Then have them list all the ways the numbers differ from one another. Use their sets of numbers for subsequent class discussions. Adding consecutive odd numbers produces the sums of 4, 9, 16, 25, 36, 49, 64, 81, , and so on. Of these, only some are reasonable predictions for Mr.

All of these sums, however, are square numbers. Using different-colored square tiles or by coloring on squared paper, represent square numbers as squares to help students see that they can be represented as the sum of odd numbers. Start with one tile or square colored in. Then, in a different color, add three squares around it to create a 2-by-2 square, then five squares to create a 3-by-3 square, and so on. Discuss the terms prime and relatively prime and the distinction between them. Then have students work on answering the question.

Also, the sides opposite one another will always measure the same length. To use the formulas for perimeter and area, you will need to measure the rectangle's length l and its width w. The square is even easier than the rectangle because it is a rectangle with four equal sides. That means you only need to know the length of one side s in order to find its perimeter and area. The trapezoid is a quadrangle that can look like a challenge, but it's actually quite easy. For this shape, only two sides are parallel to one another, though all four sides can be of different lengths.

This means that you will need to know the length of each side a, b 1 , b 2 , c to find a trapezoid's perimeter. To find the area of a trapezoid, you will also need the height h. This is the distance between the two parallel sides. A six-sided polygon with equal sides is a regular hexagon. The length of each side is equal to the radius r. While it may seem like a complicated shape, calculating the perimeter is a simple matter of multiplying the radius by the six sides. Figuring out the area of a hexagon is a little more difficult and you will have to memorize this formula:.

A regular octagon is similar to a hexagon , though this polygon has eight equal sides. To find the perimeter and area of this shape, you will need the length of one side a. Share Flipboard Email. Deb Russell is a school principal who has taught mathematics at all levels. She is also a freelance writer with more than 14 years of experience. To find the volume of a sphere, you only need the radius and the height. To use this formula, you must know:. Semiminor Axis a : The shortest distance between the center point and the edge. Semimajor Axis b : The longest distance between the center point and the edge.

To find out the perimeter of any parallelogram, use this simple formula:. Continue Reading. ThoughtCo uses cookies to provide you with a great user experience. These compactifications were introduced in full generality by Looijenga, inspired by the work of Heckman and Looijenga on the moduli space of rational elliptic surfaces.

The geometric examples discussed in the lecture of Heckman were all related to groups generated by order four complex reflections also called tetraflections and go back to the work of Deligne—Mostow and of Kondo. Allcock presented a surprising conjecture about a ball quotient and the monster group. In this lecture the complex reflection groups were related to order three complex reflections triflections in Conway's terminology , and in a sense the lecture by Heckman also served as a kind of introduction for Allcock's lecture.

The conjecture by Allcock is truly amazing, and would in fact reveal a natural geometric surrounding via generalized hypergeometric periods for most of the sporadic groups. Holzapfel and his collaborators discussed towers of ball quotients and morphsims among these.