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Matthew wrote that there are weights between 2. Their new book helps students calculate with multidigit divisors and dividends using a method that makes sense to them! The lesson presented below teaches students a game that reinforces all of these goals. Be sure to write both your names on your recording sheet. What divisor would be a good choice? Five times twenty is one hundred. He has to cross off nineteen from the list of divisors.

We only get to use each number listed once in a game. I wrote 95 under the Start Number column. Skylar looked overwhelmed, so I called on Kenzie for advice. He reminded me to circle the remainder of 15, to put his initial beside the problem, and to cross off I did. Joanna suggested I use eighteen. Eight times four is thirty-two. So forty and thirty-two make seventy-two. Eighty minus eight is seventy-two. Skylar decided to use eleven as his divisor.

So seventy-two divided by eleven is six with a remainder of six. I get a score of six. After a few minutes, I noticed Sean and Lucas were involved in an intense discussion. What can be a remainder depends on the divisor. I noticed as I continued to circulate that Skylar and Sasha showed how they did the dividing and, after two rounds, changed their way of recording their work. Skylar and Sasha changed how they recorded the game but showed their thinking clearly. Lupe made an error when she divided 40 by At the end, you can get a lot of leftovers once the starting number gets below twenty.

We got to thirteen as the starting number, and fourteen was left as a divisor, so I took fourteen and got all thirteen because thirteen divided by fourteen is zero with a remainder of thirteen. Everett and Derek added up all their leftovers together and it came out to one hundred. When Kenzie and I added ours up, it was only eighty-eight.

I suggested to Beth and Kenzie that it was possible they had made an error somewhere and perhaps they needed to go back and check their work. They reported on the subtraction and division errors they had made and commented on what they had discovered. There would be a remainder of seven for that problem. Look where it says eighty divided by eighteen equals four remainder eight.

Four times eighteen is seventy-two, which is the next start number. There were no other comments. Over the next several days, children continued to play when they had free time. It was wonderful to see them so happily engaged while getting practice with division.

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. In this lesson, students continue their work on developing a method for determining the volume of any rectangular prism. They share their methods with each other and discuss similarities and differences between the methods. One goal of the lesson is to help students articulate how volume can be determined by finding the number of cubes in each layer of a prism.

If students know the number of cubes in one layer, they can multiply that amount by the number of layers, or height of the rectangular prism. The method of multiplying the dimensions is then connected to the idea of layers. Measurement and Data: Standard 5. MD Understand concepts of volume and relate volume to multiplication and addition.

Begin this lesson by reviewing the terms rectangular prism, volume, and cubic unit. Give the groups interlocking cubes to help them develop their methods. Withhold comments or corrections. If groups have identical methods, post the methods both times. With your partner, see if you can apply each of the methods displayed to a three-by-four-by-five-unit prism. If you get stuck, think about how you might edit the method so that it works. If the method does work, think about why it works and whether it will work for all rectangular prisms. Briefly discuss which methods may need a little revision or editing.

If a student makes a suggestion on how to revise a method that he or she did not write, check back with the students who wrote it. Do you think it makes sense to add it to what you wrote? Pick the methods you would like everyone to discuss based on the mathematics. Any time a student uses language associated with layering, ask at least two other students to repeat it. Ask students if they agree or disagree that the method would work for all rectangular prisms and why. Ask other students to explain why they agree or disagree that this method would work for all rectangular prisms.

How is it different? Summarize the key mathematical points. They all involve finding the volume by determining the number of cubes in one layer and then multiplying that by the number of layers. 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. 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.

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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.

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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. How are these numbers alike?

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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. Throughout the year, I continued helping the class learn ways to compare fractions. As always, I learned a great deal from the students, especially from their written work. Most revealing to me was the variety of strategies that students developed for comparing fractions.

Below, I describe some of what I learned from their writing and offer suggestions for how you can use writing with your students. In my early years teaching mathematics, I taught students to compare fractions the way I had learned as an elementary student—convert the fractions so they all have common denominators. However, in my more recent teaching of fractions, I do not teach one method. Instead, I prod students to think, reason, and make sense of comparing fractions, helping them learn a variety of strategies that they can apply appropriately in different situations.

To help students learn to compare fractions, I used several types of lessons. I gave students real-world problems to solve, such as sharing cookies or comparing how much pizza different people ate, and had class discussions about different ways to solve the problems. I gave them experiences with manipulative materials—pattern blocks, color tiles, Cuisenaire rods, and others—and we explored and discussed how to represent fractional parts.

I taught fraction games that required them to compare fractions, and we shared strategies. At times I just gave them fractions, and we discussed different ways to compare them. We talked a good deal about fractions. By expressing their own ideas and hearing ideas from others, children expand their views of how to think mathematically. Also, talking and listening helps prepare them for writing, which I have them do individually several times a week in class and often for homework as well.

My students had many opportunities to explain in writing how they compared fractions. How would you decide which of the two fractions is larger? This is common when I read student work. But her method worked and was efficient. Laura converted the fractions so that they had common numerators. He had reasoned the same way that Laura had but expressed his thinking differently and with more detail. Jenny, however, reasoned differently. She compared both fractions to one whole.

Also, it was easy for me to understand because her approach mirrored the way I had thought about the problem. Jenny converted the fractions so that they had common numerators. As Jenny did, Donald compared both fractions to one whole and showed his understanding of equivalent fractions. But he also thought about common numerators. Donald needed to be reminded regularly to reread his papers before handing them in, and although this paper represented an improvement in his writing, more improvement was still needed.

Mariah relied on common denominators. I got this because I used common denominators. Mariah also showed how she had arrived at the converted fractions. She wrote:. They were doing what made sense to them as they tried to reason about the fractions. Sometimes I focus on their writing errors; other times I keep the focus just on the mathematics. Making this decision depends on the paper, the student, and the mathematics involved. For example, I asked Mariah to show the class how she arrived at fractions with common denominators. I knew when I hit fortieths that it would work for six-eighths.

Several students were eager to explain the methods they used. Raul was particularly animated. Then multiply 6 times 5 to get 30 and have that numerator for six-eighths and then multiply 4 times 8 and use 32 for the numerator for four-fifths. I had them compare their work with partners and then had volunteers demonstrate each method so that Mariah and Raul could judge if their methods had been correctly applied. I had others read their papers as well and began to compile a class list of strategies for comparing fractions.

I wrote the list on chart paper and kept it posted in the classroom for students to refer to. In my years of teaching, not all classes have come up with all strategies. This was the first class in which common numerators was such a prevalent strategy. Reproduce one of them, give copies to students in pairs, and have them see if they can figure out why it makes sense. Have them explain the method in their own words. Then give them practice applying it to other fractions. But, if possible, share the ideas from your own class.

In this beginning lesson, students first explore arithmetic sentences to decide whether they are true or false. For each, I had a student read it aloud, tell if it was true or false, and explain why. Few students knew how to read the third sentence. I then asked the students to write examples of arithmetic equations that were true and some that were false.

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A few minutes later I interrupted them. I drew two columns on the board, one for true mathematical sentences, and a second for false mathematical sentences. Then comes the equals sign. On the other side you do four plus five. I paused to give students time to think and then asked Rayna to come up to the board and write her equation. After a few moments, most students were clear that it was correct. I wrote her equation in the True column. After several other students shared their equations, even though more of the students wanted to do so, I moved on with the lesson.

As the students watched, I wrote the following on the board:. The class was quiet. Finally a few hands went up. I called on Jazmin. I did as Lizzie instructed. The box is called a variable, because you can vary what number you put into it or use to replace it. Most thought that it was.

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I called on Lizzie. The others showed thumbs up to agree that it was an open sentence. I called on Diego. Few hands were raised. I called on Keith. I did so and Keith came to the board and correctly wrote 15, I then called on Kenny to give another open sentence. Again, the students showed their agreement. I was pleased that Kenny had volunteered the use of a triangle. If no student had, however, I would have written an open sentence as Kenny did, talked about it with the students, and thenintroduced other symbols as well.

Since Kenny made his suggestion, I built on it at this time. I think you can use whatever you want. An x is OK, so is a box, so is a triangle. But mostly we see boxes, triangles, and letters used for variables in equations. By the end of class, the students all seemed comfortable with categorizing true, false, and open sentences and figuring out how to make open sentences true. Emma suggested a word problem. I wrote the number 40 in the first basket and the number 4 in each of the other two. The students nodded again. I gave a stab at this other interpretation of division—dividing forty-eight into groups of three, rather than into three groups.

How many groups would there be? I waited until more students raised their hands, and then I asked everyone to say the answer together, quietly. I continued drawing until I had sixteen groups of three circles. More experience with grouping contexts would be useful for this. Turn and talk to your neighbor and see if you can think of any.

After a moment, I called them to attention and had them report. They came up with a long list that included problems like the following:. Then the list evolved into problems that always had an answer of 1 R To make a shift from problems with this attribute, I asked them to think of problems that had the answer of 2 R1.

The list they came up with included these:. I continued until I had listed about a dozen or so problems. As I recorded, I noticed that the first numbers were all odd. I shared this with the class. But after some students made a few unsuccessful tries, Alexis came up with the answer of dividing by 4 I wrote her idea on the board:. At this point, a troop of parents came into the room. They were on a school tour, getting ready for their own children to go to kindergarten. I explained the problem we were working on and invited them to try to solve the problem with us, which produced a few looks of panic in their eyes.

I assured them that it would be a few years before their children brought home math work like this. Predictions for the Millennium New York: Puffin, is a collection of predictions about everyday and not-so-everyday events that will take place in the next three seconds, the next three minutes, the next three hours, days, weeks.

Here, fifth graders explore just one of the predictions made in the book and use estimation, multiplication, and division to make a prediction of their own. I held the book so the students could see the cover as they came and sat down in the meeting area. The cover shows a large circle, which on careful inspection you can see is the outline of a pocket watch. Inside the circle are illustrations and words radiating from a central picture of the Statue of Liberty. The students squinted to read the small print surrounding the Statue of Liberty.

When I opened the book, I purposely skipped the introduction, which gives directions for making your own predictions. Each two-page spread after the introduction covers a specific period of time, but always in increments of three: three seconds, three minutes, three hours, three days, three nights, three weeks, three months, three years, three decades, three centuries, three thousand years, and three million years.

On the first two-page spread, I read almost all the entries for what could happen in the next three seconds. We learned that in the next three seconds, Russians will mail more than four thousand letters or parcels and that Americans will buy fifty-six air-conditioning units. The author must be guessing. I turned back and read the introduction, which explains the role of counting throughout history, from caterers counting for imperial banquets in ancient China to the sophisticated counting we do today with computers.

The children agreed that the predictions must be estimates, and I moved on to read the next page, which is about what will happen in the next three minutes. There is more to this book than could possibly be read in one sitting, so I read just enough to give the students the flavor of each section. I then presented the problem I wanted the students to solve.

I suggested that they talk to someone sitting near them about their ideas. That would be easy! Then we discussed how to gather the information we needed to make the predictions. Ethan and Cecelia made a plan for timing. He made a tally mark every time she blinked until she motioned again for him to stop. Cecelia counted up her blinks and found that she had blinked twenty times.

To find their average number of blinks per minute they added twenty plus twenty-two and divided by two. Next they multiplied twenty-one by three to find out how many times, on average, they blinked in three minutes. They multiplied by three to get fifteen blinks in three minutes, then multiplied sixty by three. Then we multiplied that times five, because we blink five times every minute.

That gives us nine hundred blinks in three hours. Cameron and Mason clearly understood how to calculate the number of times they could blink in three minutes, three hours, and so forth. Ingrid, Douglas, and Ranna took turns timing one minute and counting each other blinking. They used twelve blinks per minute as their average.

Then we knew that there are sixty minutes in an hour, so we timesed sixty by three and got one hundred eighty. Because of this error, their other calculations were off. See Figure 2. Ingrid, Douglas, and Ranna made a computation error in one of the early steps and had to recalculate. 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.


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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. See Figure 1.

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. The Golden Ratio is a ratio of length to width and is approximately This ratio not only appears in art and architecture, but also can be observed in nature and in the human body.

Enter your data on a chart. How does your data compare to other members of your class? Things to consider How will you compare your data with other students in your class? See the PDF for a more complete version of this lesson. Guess My Number invites children to consider the structure of the number system while engaging in a logic game.

Students try to guess a secret number from within a given range of possibilities. Through this activity, students learn the usefulness of number lines as tools for solving problems. I also marked 50 on the number line with an arrow, indicating all the numbers fifty and above were too large. Before I took any more guesses I decided to have a brief discussion about strategies for guessing. I had deliberately picked an easy number to start with so we could focus on the mechanics and thinking involved with the game.

So if you guess a number right in the middle, you can figure out which half the secret number is in and then you can just throw away the other half and not have to worry about it. I wrote the two prompts on the board to help them stay focused. Also, I was giving them a preview of the discussion to come. If you raise your hand to guess a number, you also have to be willing to explain why you think that number is a helpful guess.

I let the students talk to each other as I added the new information to the number line.


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Many students wanted to use the same strategy and pick the number that was halfway between 40 and We took a brief detour to establish that Christina guessed forty-two. I told her that the secret number was greater than forty-two and recorded this information on the board. Then Kenny guessed forty-nine. Some students expressed frustration with his guess since they already knew the number was less than forty-five. I stopped briefly to have a talk about maintaining a safe environment. It could have been any of one hundred different numbers, and it took you only eight guesses to get it.

That shows you used a lot of good mathematical thinking. Before I left the class, I called on a pair of students to lead the class in another round of Guess My Number. Guess My Number works equally well with fractions, decimals, or percents. Giving the students some visual tools is essential. Using a number line helps students compare numbers and order the numbers. A 1— chart is another tool that works nicely for Guess My Number.

Tell students that the secret number is somewhere on the 1— chart. Cross numbers off the chart as they are eliminated. Familiarity with a 1— chart gives upper-elementary students a distinctive edge when it comes to mental computation and understanding our number system. When students have a visual model of the chart in their heads, they can easily jump around using tens. They also have a useful geometric model the by square to get a feel for how numbers are related to one another.

Playing Guess My Number with a 1— chart gives students further exposure to the chart and pushes them to articulate some of the number relationships inherent in it. Bay-Williams and Sherri L.

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How much did he start with? The article explains that the problem was given to a class of eight-year-olds. Marilyn Burns presented the problem to a class of fifth graders. I was intrigued by the problem and before the lesson, as the article suggested, I took the time to solve it myself. I suggest that you do this, too.

Then, to begin the lesson, I gathered the students on the rug and showed them the journal. A few of the students had questions about the problem. After a few moments, Scott rushed up to me. Next, Mara and Natanya came up to me. Mara had a question. Soon I interrupted the students for a whole-class discussion.

Hassan shared first. So when he spent ten, he still had ten. And the same thing happened in the two other stores. Gissele giggled. In the first store, the storekeeper gave him twenty, and that was forty. He had thirty dollars left after spending ten. At the second store, the man gives him thirty dollars, so he has sixty dollars, and that gives him fifty left after spending ten.

And in the third store, fifty plus fifty is one hundred, so he spends ten and has ninety dollars left. Alexandra had a different idea. So it has to be some dollars and some cents. He explained to the class what he had told me before, that the man had to have five dollars going into the third store in order to have nothing left. Alvin raised a hand, excited. Five plus ten is fifteen, and half of that is seven dollars and fifty cents.

Some of the students now nodded in agreement; others were still confused or unsure. I decided that it was time for them to continue working on their own. Listen to what you should write. The information can help you later, and it can also help me understand how you were thinking. When you figure out the answer, explain in words how you finally got it. And finally, write about whether you think this problem was too easy, just right, or too hard for fifth graders.

The students were used to writing titles on their papers. No one else had a question and the students went back to their desks to work. The room was productively noisy, and the students stayed engrossed. I asked him to explain what he had written, and he did so clearly. As the students worked, I started a T-table on the board, labeling the first column Start and the second column End.

I recorded on the table the amounts that the students found as they worked, recording the end amounts as negative numbers when the man was in the hole and positive numbers when the man was ahead. After several entries, the table looked like this:. Several others did, too. I called the class back to the rug for a concluding conversation, and the interchange was lively.

The general consensus was that guessing and checking, working backward, acting it out, and looking for a pattern were the most used problem-solving strategies. Not all of the students had the time to write about whether the problem was too easy, just right, or too hard. But our class discussion revealed that most of the students thought that the problem was just right for fifth graders. Some students commented on their papers.

Travon wrote: This was hard because there is no very helpful pattern that I know of. Kaisha had a different point of view. She wrote: I think it was pretty easy. In this lesson, the marbles and beans, along with the book Great Estimations by Bruce Goldstone, are used to provide students with an opportunity to explore and apply strategies for estimation. Also, collections of grid paper with different-sized grids, measuring cups of various sizes, a few unifix cubes, and balance scales and masses provide engaging estimation tools for students.

Before class, place one cup of kidney beans in each quart-size sandwich bag, filling enough bags for one bag per pair of students. Place the bags of beans in a large paper bag along with a pint-size and quart-size jar of marbles. I think there are more than twenty-three marbles because I could get about ten marbles in a handful and I know there are more than two handfuls in the jar. Two tens equal twenty so there must be more than twenty-three.

As I read the book, listen carefully for strategies you could use to estimate. Yainid shared the first strategy. Continue reading and recording estimation strategies as students notice them. Following is the list of strategies students may suggest:. Ask students to choose strategies from the list to estimate the number of marbles in the jar.

Record their reasoning on the board as appropriate. Gaby thought making rows of ten would be helpful. I asked her to make a row of ten. I think if we took out another ten marbles the jar would be about one-fourth empty. I used twenty because that would be how many marbles were taken out of the jar for it to be one-fourth empty. I multiplied by four because there are four-fourths in the whole jar.

That would be a good estimate of how many marbles were in the jar when it was full. I asked Kaitlin to try her idea for us by taking out ten more marbles to make a total of twenty marbles removed from the jar. The jar looked like it was more than three-fourths full so Kaitlin took out six more marbles for a total of twenty-six marbles. We agreed that it now looked about three-quarters full or one-fourth empty. Twenty-six plus twenty-six equals fifty-two so there are fifty-two marbles in one-half. Fifty-two and fifty-two equal one hundred four marbles.

I know that four twenty-fives is one hundred. Then I added four more because I changed twenty-six to twenty-five. I have to put back the one. Four times one equals four. One hundred plus four equals one hundred four.

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Then I counted, 20, 40, 60, I put the eighty in my memory. Then I counted 6, 12, 18, I took the eighty out of my memory and added eighty and twenty-four. In my class, our initial estimates were , 23, , , 1,,, , , and Our class estimate was Finally, tell the students how many marbles you had counted when you filled the jar.

To verify your count and to reinforce the value of using a strategy to make an estimation, count the marbles by placing them into groups of ten. I had counted marbles in the jar. My class counted 11 groups of ten with two extra to figure this out. The class strategy for estimating marbles got them very close to the actual total of marbles. Next, pull the quart-sized jar of marbles from the bag and hold it up. How can you use what you know about the number of marbles in a pint to help you estimate the number of marbles in a quart? So if we know how many in a pint we can just double that to get a good estimate for how many in a quart.

Hold up the estimation tools: grid paper, measuring cups, unifix cubes, and balances and masses. Ask: How might these tools help you carry out the strategies listed on the board? In my class, the students shared that the grid paper could be used with the box-and-count strategy, the balance and masses could help with the weighing strategy, and the measuring cups could help with thinking about how many tens or hundreds fit in a measuring cup or even a unifix cube if the items were small like popcorn or lentils. Katie had a new idea. She suggested that we could use the small geoboard rubber bands as a way to help with the clumping strategy.

The class agreed this was a good idea and geoboard rubber bands were added to the estimation tools. Give partners time to work together on recording their estimates and notes about what tool they used to make their estimate. When students finish, ask them to record their information on a class chart:. Gather students for a discussion. Focus the discussion on the similarity of the estimates. Each bag had approximately the same number of kidney beans. In most cases, the estimates were similar even though partner pairs used different estimation tools.

This usually delights the students. The students in my class concluded that differences in the estimations came from varying sizes of beans, how precisely the beans filled an area of measurement, how carefully people measured and counted, and how the number of beans in the bags varied. Subscribe Now Subscribe Now. Final Say. Long reads. Lib Dems. US Politics. Theresa May. Jeremy Corbyn. Robert Fisk. Mark Steel. Janet Street-Porter. John Rentoul. Chuka Ummuna. Shappi Khorsandi. Gina Miller. Our view. Sign the petition. Spread the word. Steve Coogan.

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