According to Gerardo Young in his book La Argentina Secreta , phone providers were paid millions of US dollars each year to set up interception systems. Nokia Siemens Network also reportedly was providing a Data Voice Call Recording and Acquisition Unit to Argentine phone providers and that the intelligence services had offices within the headquarters of phone providers to conduct interceptions.
Jaime Stiuso, an intelligence agent attached to the Argentine intelligence services, was exposed as the most powerful spy in Argentina for years, having managed the former Secretariat of Intelligence mostly outside of what was prescribed by law. It is alleged that during a year investigation Nisman had gathered phone recordings that revealed an impunity deal between the Iranian and Argentinian governments in exchange for economic benefits.
Nisman worked closely with Jaime Stiuso during his investigations, and it is alleged that the intelligence services were involved in his death. In March , a security and surveillance technology conference called Segurinfo was held in Buenos Aires, and in September Buenos Aires held another conference called Seguriexpo.
Among the government members who attended Segurinfo was Sergio Blanco head of the Secretary for Management Technology. Furthermore, Blue Coat was found to have an office in Argentina. A parliamentary inquiry in Germany had revealed that over the past decade the German government had sold surveillance technologies to Argentina.
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In July , after the leak of internal information from the Italian spyware vendor Hacking Team, it was established that the company negotiated with several intermediaries in Argentina that said to have ties to law enforcement agencies and State intelligence bodies, but it is not clear from the leaked emails that a transaction actually took place.
In early July , the Mayor of the Autonomous City of Buenos Aires, Horacio Larreta, together with the Security Minister, Patricia Bullrich, introduced the "Aerostatic Surveillance System" , with the aim of flying surveillance cameras attached to balloons in order to do a degree monitoring of a specific area. According to the Mayor's statement, the system will be used to monitor the entrace and exit routes into the City near the "Riachuelo", as well as "in mass events, when there is a big demonstration, a marathon, something that has to be covered in a large area of the City".
It can be raised in the City and areas bordering up to 30 meters, allowing a o viewing angle, identifying and following targets at distances greater than two kilometers, and recognition up to 4 kilometers".
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The Ministry of the Interior and Transport and the Ministry of Security largely oversee national security issues. The Argentine Supreme Court has held that the right to privacy and intimacy has its constitutional basis in Section 19 of the Constitution and that the right protects a wider sphere beyond domestic and family life. It has further stated that no one can interfere in the private life of a person or in areas of his or her activity that are not intended to be made public without his or her consent.
Only by statute can such interference be justified, provided that there is an overriding interest in safeguarding the freedom of others, the defense of society, morals or prosecution of crime. Recent cases involving Google and other search engines have ruled on the constitutional right to privacy. On 27 October , the Supreme Court declared that in most cases search engines are not liable for linking to content that could violate the constitutional right to privacy. The Court held that to establish the liability of search engines it must be shown that wrongdoing was committed with intention or negligence, and added that "search engines have no general obligation to monitor In Argentina, metadata is protected to the same degree and using the same legal basis as content.
In brief, the protection of privacy is manifested in the right that individuals other than those conversing may not illegally obtain information on the content of the telephone conversations or other aspects inherent in the communication process, such as those mentioned. Citing the former Telecommunications Act, which contained a similar general principle to the Argentina Digital Act, and Section of the Penal Procedural Code, which sets out the same procedure for the request of communications interception and of metadata, the Court held that "telephone communication registers for the purpose of criminal investigation While the law does not require telecommunication companies to collect and store metadata, they are allowed to do it and may have been doing so in the past.
However, Proyecto X was revealed to have been used to target left-wing activists and trade unionists. The project gathers as many personal details as possible on individuals. This includes general information such as addresses, bank details, properties, political affiliations and also private information such as sexual preferences, and whether the person smokes or not.
Reports have also revealed that policemen infiltrated protests and political meetings to gather information. Left wing activists, however, still feel targeted by the government. Speaking to Privacy International, human rights lawyer Nicolas Tauber says he has encountered several cases of people who had received voicemail messages containing tapped phone conversations they had had in the past. He also said that there has been multiple instances of human rights lawyers who had been victims of burglary and only the electronic devices had been taken.
He said his own office had been targeted and while the other lawyers, who do not work on human rights issues, had not had anything stolen, his own USB sticks had disappeared. The Data Protection Act seeks to establish "the comprehensive protection of personal data in files, registers, data banks or other technical means for data processing, whether public or private …, to ensure the right to … privacy of individuals, as well as the access to the information that is held about them, in accordance with the provisions of Section 43 … of the Constitution" Section 1. The Act also states that "the processing of personal data is unlawful when the data subject has not given his or her express consent, which must be done in writing, or through any other similar means" Section 5.
Section Section 23 of the Act sets out the treatment of personal data when it is stored for law enforcement and intelligence purposes, as stated in the previous section. According to the Intelligence Act , judicial authorization to intercept communications shall be granted for a period that must not exceed sixty days. When it is necessary to complete the investigation, the period can be extended by the judge for a maximum of another sixty days. Once these time limits have passed, the judge shall order the initiation of a judicial case or otherwise order the destruction or deletion of the information and recordings obtained through the interception.
Section 23 of the Data Protection Act sets out the treatment of personal data when it is stored for law enforcement and intelligence purposes:. The treatment of personal data with national defense or public security purposes by the armed forces, security forces, police or intelligence agencies, without the consent of the parties concerned, is limited to those cases and categories of data as are necessary for the strict compliance with the duties legally assigned to such bodies for national defense, public security or the punishment of crimes.
In those cases, files must be specific and formed for the said purpose, and they shall be classified by categories, depending on their degree of reliability. Personal data registered with police purposes shall be erased when deemed unnecessary for the inquiries which gave rise to their storage.
In February , the Data Protection authority launched a consultation process for the drafting of a new bill on Data Protection, in which civil society was invited to make comments and contributions. The draft, which will eventually by reviewed by the Congress, is meant to reflect the societal and technological evolution that have impacted data protection regulations.
Among the changes suggested the current Data Protection Authority would be replaced by a new body that will have functional autonomy under the Ministry of Justice. This agency was established in September by the Decree of Necessity and Urgency There are a number of concerns surrounding this new agency. Firstly, the use of a decree of necessity and urgency DNU to carry out the modifications constituted an affectation to the principle of separation of powers of the State, since the Executive Power performs a function proper to the Congress.
The Law of Access to Public Information establishes that the Agency itself - through its Director- must give its own structure, without subjecting it to any approval. The decree went against the objective of having an independent control body for possible interference by the Executive Power. Law on access to public information , came into effect in September Chapter II, Section 8, establishes the exceptions under which a request can be denied, citing as the first case: "A Information expressly classified as reserved or confidential or secret, for reasons of defense or foreign policy.
Our research has not yet shown any case law related to data breaches in Argentina. Please send any tips or information to: research privacyinternational. In late , following the October elections, a blogger identified a code that was then used by a programmer to set up a site that enabled images to be retrieved from the the electoral registry. After this reached public attention through media reporting, the photographs were taken down. The decision was contested by experts on data protection law and members of opposition parties. They alleged that the data transfer does not align with the finality principle, because the data were collected for an efficient operation of the social security system, not for communication or public relations activities.
A lawsuit was filed by an Argentine citizen who argues that the Resolution is contrary to the Constitution, because it affects her right to privacy. The case is still pending. The information was uploaded to the code repository platform GitHub and shared through social media website Reddit, but when the news was published the links were already taken down.
The leak contained In the legislative elections of 18th August , the National Electoral Chamber CNE ran a pilot program of biometric voter identification. The process was voluntary, with the biometric data of , people collected. For example, one of the reasons given by the CNE to specify the operation was the problem of cross-border electoral migration.
However, as ADC argues , this aspect has not been proven to be an obstacle to the proper development of the electoral process. In fact, the test had as one of its purposes the construction of evidence in this regard. If the questions that justified the implementation of the pilot test were not answered, ADC pointed out the need to use biometric data of , people. In April , the Minister of Security announced that the Ministry would start working jointly with the Ministry of Communications to create a national registry of SIM cards in order to remove stolen phones from the market as well as to render them useless with the help of telephone companies.
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. 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.
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.
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. 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. 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.
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.
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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.
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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. Finally, ask students to write their own definitions of prime and relatively prime.
Have them share their ideas, first in pairs and then with the whole class. This question aims to help students generalize about the relationship between the sign of the sum and the numbers in an integer addition problem. Students may need to make a list of integer addition problems whose sums are negative and look for commonalities among them in order to answer this question. In this lesson, fourth and fifth graders gain experience multiplying by ten and multiples of ten as they make choices about the numbers to use to reach the target amount of three hundred.
In this game you will be multiplying by ten, twenty, thirty, forty, or fifty. The goal of the game is to be the player closest to three hundred. So the player with three hundred ten wins. Remember, you want to get closest to three hundred, and you must take all six turns. I called on Ben because I knew he had a good grasp of multiplying. Ben did the same on his. I went first so I could model out loud my thinking process as well as how to record. I rolled a 1.
If I multiply one by ten that will only give me ten. That seems like a lot. Maybe I should multiply by thirty; one times thirty equals thirty. Thirty is closer, but I still have two hundred seventy to go. Do you agree that one times fifty equals fifty? Ben nodded. I recorded my turn on my side of the chart. Once Ben had recorded my turn on his chart, I handed him the die, indicating it was his turn. Ben rolled a 2. This time I rolled a 4. That gives me eighty for this turn. Add the two hundred to the fifty from your first turn and that would be two hundred fifty. You could almost win on your second turn.
Several students put their hands up to respond. I called on Cindy. This is Mrs.
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If she got two hundred fifty by the end of her second turn, then she could only get fifty more to get three hundred! I decided to move on rather than continue to discuss this point. I handed the die to Ben. Ben rolled a 1. Now I have fifty. He gave me the die. What would work better? Hands immediately went up. I called on Allie. Subtract that from three hundred and you still have three turns to get one hundred ten more points. That equals two hundred fifty. Two hundred fifty and fifty is three hundred!
That only equals fifty, so my total is one hundred. After our next two turns, I had and Ben had There are six sides on a die. One is on only one side of the die so it has one out of six chances of being rolled. He could get forty by rolling a one and multiplying by forty, or getting a two and multiplying by twenty, or getting a four and multiplying by ten. Ben looked delighted. Giggling with delight and anticipation of getting exactly , Ben rolled.
He got a 3. The class cheered and Ben did a little victory dance. I waited for a few moments for the students to settle down and then showed them what else to record when they played. I wrote on the board under my chart:. The students played the game with great enthusiasm and involvement as partners participated in every turn. In this two-person game, students take turns identifying factors of successive numbers, continuing until one of them can no longer contribute a new number. To play the game you need a partner. One of the partners begins by picking a number greater than one and less than Can anyone tell me a number that goes evenly into 36?
Another way to think about it is by skip counting. Which numbers can you skip count by and get to 36? By introducing several ways to think about factors, I hoped to explain the game more quickly. Several students nodded or vocalized their assent. I pushed for more of a commitment. Those are the two main rules of this game. Can you think of any other factors of two? Chrissy had confused factors and multiples.
I was glad she had made the multiplication connection, but I needed to prompt her a bit to get her back on track. Like 36 is a multiple of six because six times six is The class consensus was no. I raised my eyebrows in feigned surprise as I looked at the numbers on the overhead. I wonder if that always happens in this game. I hoped that in subsequent games students would pay more attention to patterns in general as they played. Looking for patterns is a powerful way to build number sense, particularly when students have opportunities to think about the patterns and their relationships to numbers and operations.
I referred to the string of numbers on the overhead, which now looked like this:. I also wanted the students to see that math involves taking time to think. Talk at your tables for a minute or two and see what you can come up with. Four and two are used already. You want to get your partner stuck so she or he is unable to add a number to the string. The important part of the game is the mathematical thinking that you do. I played one more game with the whole class. The factor concept had been reinforced, the term multiple had been introduced in context, and the students knew how to identify prime numbers.
The students were ready to play with their partners. In addition to having practice with multiplication facts, students who play One Time Only search for winning strategies by thinking about relationships among numbers and factors. In so doing, they build their number sense. In previous lessons, students built rectangular prisms using cubic units and determined the volume of the prisms by counting cubes. Students started to devise methods for finding the volume of any rectangular prism without counting.
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