Topic: Area and Volume Math Lesson Plan, Thematic Unit, Activity, Worksheet, or Mathematics Teaching Idea

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Objective: Students will be able to describe the difference between area and volume and also be able to understand how various units of measure relate to one another.

Materials: Newspaper, scissors, masking tape, rulers and meter sticks, cardboard (and something to cut it with), markers to identify finished models.

Following an introduction to area and volume students will work in groups to build models of square centimeters, square inches, square feet, square meters, and then cubic centimeters, cubic inches, cubic feet, and cubic meters. This becomes a good cooperative team effort at problem solving. Students are provided with materials, but no initial instruction is given on how to build their models.

Welch, T. (1996). Area and volume. [On-line]. Available: gopher://bvsd.k12. co.us:70/00/Educational_Resources/Lesson_Plans/Big%20Sky/math/CECmath.30.

Objective: Students will learn how to use the "counting constant" function of the calculator, and using this function will explore patterns and relationships with numbers, including the concept of multiples and negative numbers. Students will demonstrate their mastery of the function with the calculator with the creation of "pattern puzzles" that they will share with other students.

Procedure: Students will need their own calculators, or an alternative would be to use a transparent calculator designed for use on an overhead projector. Introduce the idea of the "counting constant" and demonstrate how to make the calculator count. Students will discover what the calculator does after 0. This has never failed to generate curiosity and excitement. You can then explain further the concept of negative numbers, or simply allow

children to explore on their own, attempting then to explain the nature of these numbers, comparing them to other concepts of "negative."

Holsten, A. (1996). Calculator pattern puzzles. [On-line]. Available: gopher://bvsd.k12.co.us:70/00/Educational_Resources/Lesson_Plans/Big%20Sky/math/ CECmath.06.

Objective: Students will construct the tangram pieces from a square paper by following directions to fold and cut. They will also make observations on the pieces formed and compare how they are related to each other.

Materials: Square sheet of paper, plastic tangram sets, and tangram set for the overhead.

1. Students will fold and cut a square piece of paper by following these directions. Students should discuss and record observations in small groups after each step.

A. Fold the square sheet in half along a diagonal, unfold and cut along the crease. What observations can you make about the two pieces you have? How can you "prove" that your observations are correct?

B. Take one of the halves, fold it in half and cut along the crease. Make more observations and be able to support your statements.

C. Take the remaining half and lightly crease to find the midpoint of the longest side. Fold so that the vertex of the right angle touches that midpoint and cut along the crease. Continue with observations. Congruent and similar triangles may be discussed, as well as trapezoid.

D. Take the trapezoid, fold it in half and cut. What shapes are formed? Students may not realize that these shapes are trapezoids as well. What relationships do the pieces cut have? Can you determine the measure of any of the angles?

E. Fold the acute base angle of one of the trapezoids to the adjacent right base angle and cut on the crease. What shapes are formed? How are these pieces related to the other pieces?

F. Fold the right base angle of the other trapezoid to the opposite obtuse angle. Cut on the crease. You now should have the seven tangram pieces. Are there any more observations you can make?

Now, have the students mix up the pieces and try to put the pieces together to form the square which was the shape of the paper they originally started with. Students may be given plastic tangram pieces to do the remaining activities.

2. Have students order the pieces from smallest to largest and explain what criteria they used for their arrangement. Students should be able to verify their arrangement. Focus on the arrangement of pieces based on area. Use the small triangle as the basic unit of area. What are the areas of each of the pieces in triangular units?

3. Create squares using different numbers of tangram pieces and find the area of the squares in triangular units. For example, to form a square with one tangram piece, students should identify the square piece which is 2 triangular units in area. To form a square with two tangram pieces, students should use the two small triangles (2 triangular units in area) or the two large triangles (8 triangular units in area). Students should

continue to try to form squares with 3 pieces, 4 pieces, 5 pieces, 6 pieces and all 7 pieces. Are there multiple solutions for any? Are there no solutions for any? Do you notice any patterns?

Zenigami, F. (1996). Tangrams. [On-line]. Available: gopher://bvsd.k12. co.us:70/00/Educational_Resources/Lesson_Plans/Big%20Sky/math/CECmath.12.

Objective: Students will be able to explain that the value of a number increases when digits farthest to the left have greatest value.

Materials: Teacher-made number cards numbered 0-9, crayons, and reward for winners.

1. Predetermine the number of digits in the mystery number. Begin with three digit numbers and increase number of digits as student skill increases.

2. On scratch paper, have students draw lines so that there are the same number of lines as digits:

6. Select a student to arrange all the selected number cards from greatest to least.

Beal, K. (1996). I am the greatest. [On-line]. Available: gopher://bvsd.k12. co.us:70/00/Educational_Resources/Lesson_Plans/Big%20Sky/math/CECmath.18.

Objective: Students will use problem solving strategies such as guess and check and visualization to play the game. They will also use mental mathematics to decide on the placement of pattern blocks and look for patterns.

1. Two players are needed to play BLOCK IT. Each receives three each of the following pattern blocks: green triangle, blue rhombus, red trapezoid, yellow hexagon.

2. Players agree on assigned points for each color (e.g. green=1, blue=2, red=3, yellow=6).

3. The game begins with one yellow hexagon starting block placed on the playing surface. This piece does not belong to either player.

4. The first player must place one of her/his blocks such that one side of the block is completely touching on one side of the block(s) on the playing surface. The scoring for each play is the sum of the values of the block placed and those that it touches on a side. Play continues until both players use all of their pieces. For example, Player A selects a green triangle to play, therefore the green triangle (1 point) touches the yellow hexagon (6 points) so 7 points (1+6) are scored. Player B then places a red trapezoid (3 points) such that it touches one full side of the green triangle (1 point) and one full side of the yellow hexagon (6 points); Player B scores 10 points (3+1+6). Player A places a blue rhombus (2 points) that touches one full side of the green triangle (1 point) and one full side of the yellow hexagon (6 points) which scores another 9 points (2+1+6) giving Player A a total now of 16 points. Player B continues play in this manner.

6. The player with the most total points after all pieces have been used is the winner.

Zenigami, F. (1996). Block it. [On-line]. Available: gopher://bvsd.k12. co.us:70/00/Educational_Resources/Lesson_Plans/Big%20Sky/math/CECmath.33.

The students will estimate, measure, record, compare, and order objects and containers.

Have students bring in a variety of beverage containers, such as 1 liter milk containers or 2 liter drink bottles. Have them work together to estimate and record the volumes of the containers. They can read the labels to determine the actual volume of the containers and then arrange the containers from greatest volume to least volume. Provide students with opportunities to sort and order using different units of measure.

Ministry of Education. (1996). Grades 2-3 shape and space (measurement). [On-line]. Available: http://www.est.gov.bc.ca/.curriculum/www/irps/mathk7/gr23me00.htm.

Ask students who has the biggest mouth in the class. Then ask them the focus question of how can we tell for sure that that student has the biggest mouth? Introduce the unit of milliliters and tell students that they are going to measure the volume of their mouths in milliliters to find out who has the biggest mouth. Give students cups of water and have them fill their mouths with water by using a straw. When they have their mouths full, have them spit out the water into a beaker. Then teach students how to use a graduated cylinder to measure the volume of that water. Whoever has the largest volume of water after spitting it out has the biggest mouth in the class.

Adams, D. (1996). Big mouth. Student in T267, Teaching of Mathematics in the Elementary School.

Objective: Students will use manipulatives and counting strategies to derive addition and subtraction facts.

Materials: Sheets of number facts, addition and subtraction facts sheet, and a floor-size number line.

1. Given a series of number facts, ask student to explain ways to figure them out. For example, what strategies would help to solve these problems? Encourage student to "think out loud" to determine if child is using any strategies to make solving the fact easier.

2. Ask student to explain why adding or subtracting "one" or "two" can be figured out in one's head.

3. Have student "act out" number facts which are efficiently solved by counting on. Have student stand on a walk-on number line and demonstrate the solution to the number fact. For example, to find 9 + 4 the student steps to the nine and then advances four units more to land on 13. A desk top number line is best utilized if the student has a marker (teddy bear counter, toy dinosaur, car eraser, etc.) to physically model the fact. Notice that subtraction on a number line is a different model from subtraction that is illustrated by creating a set and removing part of it.

4. Given a worksheet, ask student to circle all facts already memorized and write the sums or differences. Have students find remaining facts using counters.

Public School of North Carolina. (1996). Strategies for instruction in mathematics. [On-line]. Available: http://www.dpi.state.nc.us/Curriculum/ Mathematics/Mth.LssnPlns/Mth.2.7.1.

Students will practice rounding numbers by trying to find numbers that will round to another specified number.

Pick a number and ask students to come up with the nearest numbers that round up and/or round down to that particular number. Ask "What is the greatest (or least) whole number that rounds to 50 (or 600)?" Use larger numbers as students catch on so that they can work with numbers that are more difficult. Students should notice a pattern. The closest whole number to round up will be just one number less than that number. The closest whole number to round down will be just one number higher than that number.

Harcourt Brace. (1994). Mathematics plus: teacher's edition, grade 3. New York: Harcourt Brace & Company.

Students will use problem solving skills and knowledge of fractions to help them solve riddles about fractions.

Write some riddles about fractions for students to solve. For example, "The sum of the digits in my numerator and denominator is 12. I am equivalent to ½. What fraction am I?" (4/8). The same question can be used over and over again with different numbers. Also, make up similar riddles that involve fractions in a similar way. The questions can be typed up on a worksheet for individuals or groups to work on, or they can be presented to the class to allow for them to think colletively.

Harcourt Brace. (1994). Mathematics plus: teacher's edition, grade 3. New York: Harcourt Brace & Company.

Students will use addition and subtraction to find doubles, triples, halves, and quarters of numbers.

Ask students to pretend they live in "Doubleland," and have them answer the questions that are asked with a double of the answer. For example, ask students "How many windows are there in our classroom?" If there are four windows, students would reply with eight. Use different objects in the classroom. After Doubleland has been conquered, place students in the context of Tripleland, Halfland, Quarterland, etc. For Halfland and Quarterland, students would reply with half the correct number and one fourth of the correct number, respectively. This is a good lesson to prepare students for multiplication and division too.

Harcourt Brace. (1994). Mathematics plus: teacher's edition, grade 3. New York: Harcourt Brace & Company.

Students will use problem solving skills and knowledge of money and coins to solve riddles using money.

The use of real coins and possibly dollar bills would be helpful in solving the riddles, but not absolutely necessary.

Create money riddles for students to solve such as "I have three coins that are worth fifty-five cents. What coins do I have?" (two quarters and one nickel). Similar riddles can be used with different numbers of coins and with different amounts. Once students are able to solve these riddles without any problems, give them similar riddles with larger, dollar amounts that require more thought. These riddles can also be presented to individuals or small groups of students on a worksheet.

Harcourt Brace. (1994). Mathematics plus: teacher's edition, grade 3. New York: Harcourt Brace & Company.

By moving geometric shapes on graph paper and seeing the way the appearance changes, but the shape itself does not, students will deepen their understanding of congruence and similarity.

Allow students to draw any polygon they desire, regular or irregular, but one that will fit in a 10 X 10 square on the graph paper. Demonstrate to students how to slide that polygon over so many spaces. Have students do this with their shape, then ask them if the second object is congruent to the first. After slides have been done, have them do flips of their polygon over a horizontal or vertical axis. Then ask them if these two shapes are congruent. Do the same with rotations. If students are doing okay with this, have them do more advanced movements such as a flip and a slide or a slide and a turn, etc. Each time ask students if the objects are similar or congruent and why. An extension on this activity would be to allow students to do the same on a geoboard. This would be better since it is more of a hands-on approach.

Scott and Foresman. (1991). Exploring mathematics: teacher's edition; grade 4. Glenview, Illinois: Scott, Foresman and Company.

Using various shapes, students will discover how many lines of symmetry they have and where these lines of symmetry are.

Numerous shapes on construction paper that students can cut out and fold to discover their lines of symmetry. Some good shapes to use are squares, rectangels, parallelograms, isosceles triangles, circles, octagons, and other polygons.

Provide students with the different shapes on construction paper. Introduce the topic of symmetry and its meaning. Ask students to fold the objects in different ways to determine where the lines of symmetry are and how many are on each shape. This process will need to be modeled for them with a couple of simple shapes first, to get them started. Tell them to look for patterns in what they find as well. Introduce the circle later in the lesson and ask students what they notice. For an extension, provide students with natural objects or pictures of them and look for the lines of symmetry in them.

Scott and Foresman. (1991). Exploring mathematics: teacher's edition; grade 4. Glenview, Illinois: Scott, Foresman and Company.

Students will learn that fractions with equal numerators but not equal denominators are not equal. They will also learn to discover which fractions are greater by using models.

Put two fractions on the board, such as 2/4 and 2/8. Ask students if these fractions are equal since their numerators are equal. Distribute cuisenaire rods and have students model why these fractions are not equal (the two parts of the wholes are different sizes). Do the same thing with other fractions. For most, however, the cuisenaire rods will not work and other models such as fraction bars will need to be used. Using fraction bars or some other model, ask students to represent 2/3 and 2/4 and ask them which is the larger fraction. Provide the students with other fractions and have them represent them with models to decide which ones are larger. Finally, give students larger fractions that are more difficult to represent with models and ask them how they would determine which is larger. Some direct instruction may be needed here.

Scott and Foresman. (1991). Exploring mathematics: teacher's edition; grade 4. Glenview, Illinois: Scott, Foresman and Company.

By using pictures of polygons with parts that are shaded, students will be able to write a mixed fraction. They will also be able to represent mixed fractions on these shapes.

Give students worksheets that have the shapes on them. Some of them should be shaded to represent fractions such as 2 ¾. Squares, triangles, hexagons, and other shapes can be divided into thirds, fourths, and halves for the students to write the fractions for. To reverse the process, provide students with mixed fractions and ask them to shade parts of the shapes to represent that fraction in pictures.

Scott and Foresman. (1991). Exploring mathematics: teacher's edition; grade 4. Glenview, Illinois: Scott, Foresman and Company.

Objective: Students will make simple bar graphs to represent different pattern blocks.

Materials: Pattern blocks, 18-by-24-inch paper, construction paper, crayons, and scissors.

1. Have children take a two-handed scoop of pattern blocks, sort them by shape, and place matching shapes in separate columns on 18-by-24-inch paper. It helps to draw a grid.

2. Ask students to trace and color the blocks, or paste construction paper shapes onto the grid in a graph form. Post one of the children's graphs and have children talk about what they notice. Over several days, repeat for all of the children's graphs.

3. In class have students write three sentences about their graph. Send home their graphs and ask parents to help them write three additional sentences.

Burns, M. (1996). Scoop and sort (k-2). [On-line]. Available: http://www.scholastic.com/Instructor/cover/activities.html.

Objective: Students will be able to build the yellow hexagon-shaped pattern block with other pattern blocks in different ways. They will record these different constructions using fractions.

1. Have students work in groups to find all the different ways to re-create the yellow hexagon using different assortments of blocks.

2. After they think they've found all the ways, have children record them using fractions, with the yellow hexagon assigned the value of 1. For example, if they build the hexagon with one red trapezoid and three green triangles, they'll write: 1/2 + 1/6 + 1/6 + 1/6 = 1. (Show students how to shorten that to 1/2 + 3/6 = 1).

Burns, M. (1996). Build the Yellow Hexagon (Grades 3-5). [On-line]. Available: http://www.scholastic.com/Instructor/cover/activities.html.

Objective: Students will be able to divide a circle into equal parts and shade parts of the circle to represent different fractions.

Materials: Circles on paper that are divided in different ways to represent different fractions. Some of these should already have different fractions shaded in them. Also, blank circles on paper will be needed.

Begin by distributing some sample circles with fractions already represented. Explain to students how these circles represent fractions and show them how some of the fractions should be named. Then ask students to name the rest of the fractions on their own. Show students how to divide the circles into roughly equal portions and shade parts of the circle to represent different fractions. Finally, present students with different fractions and have them draw them on the model.

Objective: Given answers with blanks for the numbers involved in the operation, students will be able to find possible numbers to make the solution correct.

Present students with arithmetic problems appropriate to their grade level that have the numbers missing that would be used to arrive at the answer. The following problems are examples.

Have students solve for the blank numbers. Request that they try to find as many possible solutions to each problem as they can. A variation on this activity to make it closed ended would be to only have one number missing and students have to solve to find that number. This is a good way to reverse students' thinking and to get them to use more advanced problem solving skills.

Burns, M. (1992). About teaching mathematics: a k-8 resource. New York: Math Solutions Publications.

Objective: Students will learn that objects that have different sizes (circumference, diameter, height, and length) may still have the same volumes.

Materials: Beans, conventional and non-conventional measuring devices, and 5-by-8-inch cards rolled into tubes, with some rolled the short way and some rolled the long way. Also, use something to close one end of the tube so it will hold materials.

Present students with the two different tubes and ask them to predict how much taller one tube is than the other and how much different the circumference and diameter are. Once students have made their predictions, have them measure the sizes using conventional or non-conventional means. Once students have gone through this process, ask them the following questions:

Finish by allowing students to pour beans into one tube and then empty the tube's contents into the other tube. Explain to students that just because different objects have different sizes, they may not have different volumes. Show them that the two containers were made from the same 5-by-8-inch cards.

Burns, M. (1992). About teaching mathematics: a k-8 resource. New York: Math Solutions Publications.

Objective: Students will be able to demonstrate that objects with the same perimeter do not necessarily have the same area.

Begin by having students trace their feet on the centimeter squared paper. Then have them find the area of their feet in square centimeters and record their measurements. Now, use string to put around the traced feet and cut it to represent the perimeter of their feet. Now look for other students that have the same or similar perimeters and see if the area is the same also. Most likely, there will be some students whose foot perimeter is the same, but their foot areas are different. Use these examples to show students that objects that have the same perimeters do not necessarily have the same area. To extend this to another situation, have students use the string that they cut to fit the perimeter of their feet, to make squares on the grid paper. Now measure the area of the square and compare it to the area of their feet. Also, ask students to try to make a shape using their string that does have the same area as their foot. Finally, have students make different shapes with the string that all have different areas.

Burns, M. (1992). About teaching mathematics: a k-8 resource. New York: Math Solutions Publications.

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