## More Or Less Than One

*In this lesson, Marilyn Burns shows fifth-grade students a fraction and they decide if its more or less than one-half and then explain their reasoning. This focus on one-half helps establish it as an important and useful benchmark. The lesson also provides practice with mental computation of whole numbers as students compare numerators and denominators. More or Less Than One-Half will appear in Marilyns forthcoming book *Teaching Arithmetic: Lessons for Introducing Fractions, Grades 45*, to be published in fall 2001 by Math Solutions Publications.*

I used this activity when I had some time left at the end of a periodor as a warmup at the beginning of class. Id write a fraction on theboard and ask if it was more or less than one-half. Students who answered also had to explain their thinking. After each students response, Id ask, Does anyone have another way to explain that? In this way, students focused not only on answering and explaining their reasoning but also on trying to think of different ways to explain answers. Id continue discussing the fraction until all students who wanted had had a chance to explain.

Heres how the activity went with a class of fifth graders in the fall of the year. I wrote on the board:

Is this more or less than one-half? I asked.

Davy began by saying, There are three thirds in a whole, and two-thirds is more than halfway to the whole.

How do you know its more than halfway? I probed. Davy wasnt sure.

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## Question: How Is Each Number Below Different From Each Of The Others

*81 81 36 14*

List students reasons as they offer them, encouraging them to incorporate math vocabulary into their reasons, such as *divisor*, *factor*, and *divisible by*. Discuss the meanings of the math terms they use and the relationships among them. For example, suppose one student says, The number fourteen is the only number that doesnt have nine as a factor, and another student says, The number fourteen doesnt belong because its the only number thats not divisible by nine. Use these two statements to discuss the relationship between the terms *factor *and* divisible by*.

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. Finally, have the student who suggested the numbers describe any differences that the class didnt find.

## Identifying The Mathematical Goals

Students will:

- learn a method for finding the volume of a rectangular prism based on the number of layers in the prism
- connect the method of multiplying the dimensions to finding the number of cubes in the base layer and
- multiplying by the height h
- review the units used when determining volume of rectangular prisms

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## Learning Vocabulary With Tangrams

Vocabulary instruction is a large part of geometry instruction throughout the elementary grades. To learn geometric terms and their meanings, students need opportunities to interact with and use the language of geometry. In this lesson, Maryann Wickett used the experience of making tangrams as an opportunity to help a class of third graders expand their

## Line Segments On The Geoboard

*Geometry comes to life in this lesson, as Rusty Bresser has fourth graders use geoboards to explore making pairs of line segments that touch exactly nine pegs, record them on dot paper, and label them as parallel, intersecting, or perpendicular. Rusty is the author of the new book *Math and Literature, Grades 46, Second Edition* .*

Robin Gordons fourth graders had been studying geometry, and I visited the class to provide them with an additional experience. The students watched attentively as I switched on the overhead projector and showed them my clear plastic geoboard and two geobands.

Today were going to use geoboards in our math lesson, I told the students. Since you havent worked with geoboards yet this year, Im going to give you about five minutes to explore and see what sorts of shapes you can make.

I directed one student from each table to get one geoboard and two geobands for each student at the table. As soon as the students received their materials, they got to work.

After five minutes, I interrupted them and gave a new direction: Now Id like you to come to the rug area, but I want you to leave your geoboards and geobands at your table. Well use them later in the lesson.

They go on infinitely, said Brandon.

I drew a line on the whiteboard like this:

When mathematicians draw lines, they add something, I began.

Arrows! several students exclaimed.

Thats correct, I said, adding arrows on each side of the line Id drawn:

Figure 1. Henrys work

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## Math Is About Wondering And Asking Questions

Some people enjoy the mental exercise of working through problem after problem. Others want calculators or computers to do the work for them. Either way, it starts with a question or a wondering. Sometimes when practicing many problem sets to get a process down students forget why they are doing it. Use student generated questions and wonderings to build problems for solving. Talk about problem sets like piano scales or drills in sports.

## What Is Another Word For Math

The word *math *comes from a shortening of *mathematics*. This is often what *math* is formally called when its a school subject. In the U.K. and other places, its shortened to *maths*.

The type of *math* you learn early in schooladdition, subtraction, multiplication, and divisionis sometimes called *arithmetic*. It all starts with basic *counting*, but there are many branches of *math*, including *algebra*, *geometry*, and *trigonometry*, and *calculus*.

The word *calculus* is also sometimes used in a general way to mean *calculation*, and the word *math* can also be used in this way.

The word *math* is also used in other general ways. If you say the *math doesnt add up*, it means somethings not right with the *numbers* .

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## Directions For Playing The Game

## Developing Algebraic Thinking Using Manipulatives

Algebra Content Standards Create numeric patterns that involve whole-number operations. Apply procedures to find missing numbers in numeric patterns that involve whole-number operations. Illustrate situations that show change over time as increasing. Lesson Process Standards Generate descriptions and mathematical statements about relationships.

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## Why We Should Solve Riddles Every Day

To find the solution to the Riddles needs us to focus on each word, even the smallest that looks no so important to us. While finding out the solution to these riddles, Our one of the most important task is to focus on each detail of the given Riddles in order to find the correct and final answer to the riddle. This special inherent talent to focus on the smallest of the detail helps us in our real life, especially at our Job. We can improve our quality of Work just by solving these simple Riddles. Take a look at What Do You Call Two Math Friends? Riddle.

## I: Design A Task That Will Capture A Broad Range Of Responses

Open-ended tasks allow students to control some of the difficulty level themselves. In the example task below, students may limit their consideration to only a few shapes or by focusing exclusively on two-dimensional shapes. Similarly, students may choose to use drawings, charts, or diagrams to communicate their ideas, or they may rely more on prose.

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## + Fun Things To Do When Bored In Class

Okay, youve got a choice.

You can annoy your teacher and classmates because *youre *bored and *you *want to be the center of attention *all the time*. You can annoy the guys / girls in your class and get them mad at you.

Or, you can do something fun and entertaining thats not annoying.

These 21+ fun things to do when bored in class can be done pretty quietly without distracting people.

## Math Is Useful In Everyday Life

When will I ever use this? is a question that comes up often in math class. Brainstorm with your class ways you use math every day. Ive given some examples for cooking and shopping and creating. Other ideas to get you started: Use math to decide if you have enough of a particular item for everyone in class or to decide if you have enough time to play a game. Use math to figure out how many days of school before break or to add up a score in a game. Post your list in your classand add to it as new things come up!

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## Helping English Language Learners Make Sense Of Math Word Problems

Rusty Bresser, Kathy Melanese, and Christine Sphar Alison Williamss second-grade class has the full range of English language learners , from beginning to advanced, including a few native English speakers. Native languages spoken by the students include English, Vietnamese, Spanish, Somali, and Laotian. Alison knows that ELLs typically experience difficulty understanding and therefore solving math

## Introduction To Systems Of Equations

**MA, Stanford University**Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

A system of equations is two or more equations that contain the same variables. A solutions to a system of equations are the point where the lines intersect. There are four methods to solving **systems of equations**: graphing, substitution, elimination and matrices. Solving systems of equations is an important concept that shows up first in Algebra I, but is built upon in upper-level math.

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## Choose Other Guests To Flesh Out The Experience You Want

One call doesnt have to be the end! Would someone else have contrasting or differing views? Showing students both sides of an issue and letting them decide can be powerful and a great skill to prepare them for the real world.

Video calls are some of my favorite classroom experiences and can enrich practically any lesson! If you havent tried them, give it a shot.

**You probably already have the tech. You have curious students. Pick that low-hanging tech fruit!**

What questions do you still have about using video calls in the classroom? What would you add based on experiences youve had?

## The Game Of Tens And Ones

This game gives children practice with adding and subtracting ones and tens. Using a special die, two 099 charts, and two markers, children play in pairs. During the course of a game, they calculate between 20 and 30 addition and subtraction problems. The Game of Tens and Ones appears in Maryann Wickett and Marilyn Burnss

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## Consider A Virtual Field Trip As One Of Your First Calls

If you want to dip your toes in the video call waters, virtual field trips are great. If you book them through the Skype in the Classroom website, all of their field trips are totally free. Theyre great for beginners because, often, you connect on the call with a guide who takes over and talks with your students. You stand back and watch the magic happen!

## It All Adds Up Engaging 8

When I rst saw a copy of It All Adds Up!, by Australian teacher Penny Skinner, I began reading it eagerly. I was searching for ways to teach arithmetic with the same excitement I had for the other areas of the math curriculum. In the introduction, Penny explains that her book explores teaching strategies for

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## Math Homework That Counts

**Homework: If 3 is 5% of a number, what is 30% of that number?**

This assignment, at rst glance, doesnt seem very difcult. But when I asked my sixth graders to solve this problem for homework, I also asked them to take notes about how they solved it. When we shared our approaches, I was convinced that the rote teaching of the three methods of solving percent problems is far less satisfactory than developing students wider sense of number.

Deema said, I found out that one percent of the number is three divided by ve, since three is ve percent. The answer is three fths, which I know is point six. Since I want thirty percent, I just multiply point six by thirty, and I get eighteen. The answer is eighteen. I wrote what she said in shorthand on the board:

*Find 1% and then x 5 by 30.*

Joan said, I knew that there were six ves in thirty, so there must be six threes in the number. That number is eighteen. I wrote:

*5 x 6 = 30 so 3 x 6 = the number were looking for.*

Simone said, I made two ratios. The rst was to nd the whole number. I wrote three over *n* equals ve over one hundred. Then I cross-multiplied, and I knew that ve n is the same as three hundred, so the whole number is sixty. Then I wrote *n* over sixty is the same as thirty over one hundred to see what thirty percent of the whole number would be. I got eighteen also.

Wow, that makes so much sense! said John, amazed by Charlottes simple explanation.

So, John, I asked, how did you solve this problem?

## Iv: Review Students Responses And Plan Next Steps

Discover what each student chose to include perhaps it is what he knows best, or what she believes is most important, or what he finds most interesting. Also note what concepts students did not provide evidence for, or for which the evidence is

incomplete or inaccurate. Share your findings with other teachers. Look at the similarities and differences across grade levels. Following are observations and plans that teachers made upon reviewing responses to the sample task.

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## Combinatorics And Pascals Triangle

Lets calculate some values of *n***C***r*. We start with 0**C**0. Then we find 1**C**0 and 1**C**1. Next, 2**C**0, 2**C**1 and 2**C**2. Then 3**C**0, 3**C**1, 3**C**2 and 3**C**3. We can write down all these results in a table:

5C4 = 5 |
5C5 = 1 |

This is exactly Pascals triangle which we explored in the article on sequences. It can be created more easily by observing that any cell is the sum of the two cells above. Hidden in Pascals triangle there are countless patterns and number sequences.

Now we also know that the *r*th number in the *n*th row is also given by *n***C***r* . If we apply what we know about creating Pascals triangle to our combinations, we get

This is known as **Pascals Identity**. You can derive it using the definition of *n***C***r* in terms of factorials, or you can think about it the following way:

We want to choose r+ 1objects from a set of n+ 1objects. This is exactly the same as marking one object of the n+ 1, to be called X, and either choosing X plus r others , or not choosing X andr+ 1others .

Many problems in combinatorics have a simple solution if you think about it the correct way, and a very complicated solution if you just try to use algebra

1 of type 5 |

In total there are *r* stars and there are *n* 1 bars . This makes *r* + *n* 1 places in total. Any ordering of *r* stars and *n* 1 bars corresponds to precisely one valid selection of fruit.

Now we can apply our combinatorial tools: there are *r* + *n* 1 places and we want to choose *n* 1 of them to be bars . That there are exactly **C** possibilities to do that!

## Start Targeting A List Of Potential Guests

I list six places to find virtual guests and field trips in the section below this one But before we do that, know this first

This is the part that gives teachers anxiety the most. Part of it is interacting with people they dont know. The other part is fear of the unknown and fear of failure.

Now that we know what the common fears are, we can start to confront them.

- Interacting with people you dont know: People are generally helpful especially when kids are involved.
- Fear of the unknown: You might not have done this before, but teachers all over the world do it all the time and rave about the results.
- Fear of failure: If you cant find anyone that will help, youre back where you were when you started no worse off than before. The potential consequences are nearly non-existent, but the potential benefits are off the charts.

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