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Classroom Resources || Professional Resources || Illinois Schools || Education Data

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Sample Units Designs

Designing Units Around Math and Science Topics

University of Illinois at Chicago College of Education

ED312, Spring 1995; Professor Maria Varelas

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Lesson by Lora Duet, Michele Mosch, Patrice Pyla

Multiplication and Division

Lesson for:



K-4



Project description



Standards for Mathematics K-4



In grades K-4, the mathematics curriuclulm should develop whole number computation so that students can first model, explain, and develop reasonable proficiency with basic facts and algorithms. Second, the curriculum should use a variety of mental computation and estimation techniques. Third, it should use calculators in appropriate computational situations. Lastly, the mathematics curriculum should select and use computation techniques appropriate to specific problems and determine whether the results are reasonable.



The Big Ideas and Activities



The first big idea that we cam across in our research is the use of calculators in conjunction with the teaching of multiplication and division. Our technological age forces us to rethink the use of only paper and pencil computation. Paper and pencil computation cannot continue to dominate the curriuclum or there will be less time for our children to learn other, more important mathematics they need now and in the future.



With this in mind, we brought up the use of calculators in our classroom. In the discussion we covered both sides of the argument. On the pro side of the argument is the advancement for the future and the fact that a calculator is considered a manipuative. This is because it allows children a hands-on experience with seeing numbers on a screen and lets children punch in the numbers. This can be useful in teaching divison, because they learn what the divisor and dividend is by punching in the dividend first. Calculators also allow children much more time to spend on more involved problems with multiple steps. On the other hand, children might begin to rely on the calculator for computations and never learn simple multiplication and division algorithms.



A second big idea in multiplication and divison is the idea of grouping. Many things come in groups and some products are art and office supplies and food items such as eggs and soda pop are usually sold in the same amounts. Since multiplication is the process of repeated addition, grouping should be the stepping stone between addition/subtraction and multiplication/division.

A good way to start of a unit in multiplication would be to ask the students, "What comes in groups of two? Three? Four? etc." Some common replies might be ears and eyes for two, juice boxes and wheels on a tricycle for three, and legs on a dog, and quarters in a dollar for four. Then with the students' information, have them illustrate a multiplication situation with both words and numbers. Referring to the lists, each student can choose items and have them illustrate a few groups of that item. This idea leads into multiplication as repeated addition. Using four quarters in a dollar, a question which could be posed is "We said that there are four quarters in a dollars; how many quarters would there be if we had three dollars?"



For example:



    DOLLARS        QUARTERS IN A DOLLAR          TOTAL QUARTERS
       0              4            (0X4)             = 0
       1              4            (1X4)             = 4
       2              4            (2X4)             = 8
       3              4            (3X4)             =12





Another big idea in multiplication is the use of manipulatives such as chopsticks in the classroom so that stduents get the hands on understanding of the material and concept. One manipulative we used in our presentation are arrays. An array is a rectangular arrangement of objects in rows and columns. The arrays are a different way of thinking about multiplication versus groups of things. Cube-O-Grams can be used to make arrays. These arrays allow for additional visual comprehension and help to develop efficient computation strategies. One activity we did with the arrays was to build a multiplication table. This activity is extremely important for the understanding of how the table was built originally, and how to use it. This way they are using their hands and minds to come up with their own table to get their own solutions.

To build a multiplication table in a classroom, the materials needed for the students are Cube-O-Grams and graph paper. The instructor should use the overhead to show the arrays when possible. Start by asking students to arrange the cubes into squares and rectangles (boxes). This is important so they become familiar with the cubes. Then assign numbers to groups of children so they can work together for solutions. Have them create all possible boxes for a certain number. Then have them place the boxes onto their graph paper one at a time. Tell them that they are to start in the upper left hand corner, and they are to place each box on the graph paper horizontally and vertically and write the numberof cubes used in the lower right corner. When they come to the number four, make sure to ask them for all the ways to make boxes of four. There are two ways: four across or a block of two by two. As the numbers get larger talk about huge numbers like 12 x 13 which won't fit into the table.

One important thing to remember when using arrays, is to go slow with the students and to give them enough time to become familiar with how the arrays work. Starting with small arrays will help them to go on to the larger numbers. For instance, having students try to place the larger arrays on the chart will let them see for themselves that 1 x 20 will not fit. One alternative to the Cube-O-Grams is paper arrays. They are inexpensive and can be a helpful tool in learning multiplication. The children can cut them out and count the number of cubes in the rectangles. They could write 8 by 6 for example is 48 on the back. Or, they could use them like flash cards and write 8x6 across the front, and 48 on the back. If they don't know the answer they can count the rows of 6, count the columns of 8, or just count the 48 cubes.

One lst big idea we found was the relationship of multiplication and division. After all, division is the undoing of multiplication. We found that it is recommended that children be taught along with multiplication because they go hand-in-hand. For example, 24/8=3 and 3x8=24. And while teaching both processes, children need to start associating the word problems with the appropriate notation. This can be achieved by having students make pictures, sentences, and algorithms for each problem they do. For example, going back to the dollars and quarters, students might know the multiplication of 3x4 or how many quarters are in three dollars. They know the number of groups and the group size, and they are looking for the totals. If they know the gro up size and the total, they look for number of groups; or if they know the total and the number of groups, then they look for the group size by dividing. These are two different types of division. One type of division is called sharing. It involves dividing a collection of objects into a known number of groups of an unknown size. For example sharing 12 cookies among three friends. The second kind of division is grouping or splitting a collection of objects into groups of a know size. Having to find how many 15-cent pencils can be purchsed with 2 dollars is one example of grouping.

One way to integrate literature into mathematics and achieve advancement in association is to read a book like One Hundred Hungry Ants . This book is about one hundred ants who are on their way to invade a picnic. They are int two rows of 50 to begin with. One ant says to get their faster, they should split into 4 rows of 25. Then the story goes to the little ant suggesting 5 rows of 20, and lastly 10 rows of 10. In the end they are too late for the picnic.

Start off by reading this book for enjoyment. Then go back and have them write out the number sentences for both multiplication and division. Another activity could be to change the number of ants so there are different numbers to work with. The possibilities are endless. Also, this book can assist with the children who have limited Englishproficiency because they too would be involved in reading and mathematics, especially if the students are working in groups. Working in groups is helpful in all aspects of mathematics when dealing with students who have a limited English proficiency.

What we know About the Topic and Foreseeable Problems

The topic of learning multiplication and division is complicated for children. Children tend to feel more comfortable with using multiplication than division when they are taughtas separate concepts of math. They need to work with manipulatives to get the hands on experience before they can go on to memorizing and algorithms. Young students tend to relate to topics which they experience; therefore, they can learn multiplication and division when it's related to their lives.

Students do have some problems when learning multiplication and division. They might not know when to use the appropriate procedure when faced with a word problem. In teaching, we need to create problem situations for our students using both multiplication and division to help them relate and make sense of these operations. Conceptual connections are needed.

Another problem might be that a student refuses to go past the repeated addition portion of multiplication. This can be for a number of reasons. They might be more comfortable with addition, or theyjust might not want to memorize the multiplication because they get the correct solution their way. A solution might be to have them use calculators or see how much more time they are taking. Aso, extrinsic rewards might motivate children to work toward the new go al of learning.

One last problem might be a difficulty in understanding remainders. It's suggested that children explain what they do with the remainder instead of just putting an R on top of the dividend. This might help children understand that remainders occur in every day situations. For example, "If there are 5 children and 16 cookies, how many cookies would each child get?" The children might split the extra cookie between the five of them or give it to someone else, give it to someone in the group, or even throw it away. The important thing is that they are doing something with a remainder and showing that they understand.

In having a chance to teach about multiplication and division, we learned that there are a few successful ways to do it. It's beneficial to use a variety of methods, explain them completely, and go back if a concept seems too difficult. We should always demonstrate and model for our students. Lastly, once again try to instill real life situations into the lessons.

Assessment

In assessing multiplication an ddivision, we as teachers should begin by only assigning work which is meaningful. Worksheets are fine for games or extra credit, but for basic comprehension and assessment we should assign problems which focus on real life situations. We should ask them to make a picture, and when assessing, check if the picture makes sense. We should ask for math sentences to explain in words what their answer is and how they got that answer. For example, My answer is _________ ______________, and I think this because _______________________. Continuously ask questions and ask them to create their own problems. Have them work in groups to solve some of the problems. Allow for calculator use when using large numbers.

Another idea in assessing improvement and comprehension is journal writing. Have them write in a journal once a week. Beofre beginning to teach multiplication, have them write what they think it is. After the unit is ended, ask them to write again what they think multiplication and division are, and what they think they have learned. This is a good way to assess conceptual understanding without putting them under the pressure of an examination.

Examinations are a tool used for assessing comprehension of concepts. In giving exams, use word problems with both processes. Use similar problems to those discussed in class. For division exams, give a problem four different ways. First ask them to divide 21 balloons by eight people, 21 dollars by eight people, and lastly 21 divided by 8 on a calculator. This way you are testing them on the two types of division and on remainders and what to do with them. Again, have instructions which ask them how they came to their answers and why the chose the process they did.

Resources

STORY BOOKS

Giganti, Paul. Each Orange Had 8 slices. Greenwillow Books; New York, 1992.

This book has examples of thinkgs that come in groups, how to count them, and posing questions about them. It is a good source for beginners in multiplication.

Hutchins, Pat. The Doorbell Rang. Greenwillow Books; New York. 1986.

This is a story about little kids having to share their cookies with each new visitor that rang their doorbell. It is a creative way to introduce division and integrate literature.

Mathews, Louise. Bunches and Bunches of Bunnies. Dodd, Mead and Company; New York. 1978.

This is a book about beginning multiplication. Each page you add a bunny and the new total is seen through the illustrations.

Pinczes, Elinor. One Hundred Hungry Ants. Houghton Mifflin Company; Boston. 1993.

This book can be used for implementing both multiplication and division. Activity ideas are stated under big ideas.

MATH CURRICULUM

Akers, Berle-Carman, Tierney. Things that Come in Groups: Multiplication and Division. Grade 3. Dale Seymour Publications; California. 1995. Pgs. 21-24.

Burns, Marilyn. Math by All Means: Multiplication. Grade 3. Math Solutions Publications; New York. 1991. Pgs. 16-33;42-44;86-90;106-109.

Burns, Marilyn. About Teaching Mathematics. K-8. Math Solutions Publications; New York. 1991. Pgs. 194-211.

Burns and Tank. A Collection of Math Lessons. Grades 1-3. Math Solutions Publications; New York. 1988. Pgs. 27-30.

Economopoulos, Russell, and Tierney. Arrays and Shares: Multiplication and Division. Dale Seymour Pubications; California. 1995. Pgs. 24-53.

Economopoulos, Russell, and Tierney. Packages and Groups: Multiplication and Division. Dale Seymour Publications; New York. 1995. Pgs. 8-13;16-21;42-59.

Wilde and Whitin. Read Any Good Math Lately?. K-6. Heinemann; New Hampshire. 1992. Pgs. 81-99.

National Council of Teachers of Mathematics. NCTM INC.; Virginia. 1992. Pgs. 44-45.

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