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Written directly to the student and using simple materials, this is a good guide for the classroom or to use independently.

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Introducing sophisticated mathematical ideas like fractals and infinity, these hands-on activity books present concepts to children using interactive and comprehensible methods. With intriguing projects that cover a wide range of math content and skills, these are ideal resources for elementary school mathematics enrichment programs, regular classroom instruction, and

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Introducing sophisticated mathematical ideas like fractals and infinity, these hands-on activity books present concepts to children using interactive and comprehensible methods. With intriguing projects that cover a wide range of math content and skills, these are ideal resources for elementary school mathematics enrichment programs, regular classroom instruction, and home-school programs. Reproducible activity sheets lead students through a process of engaged inquiry with plenty of helpful tips along the way. A list of useful terms specific to each activity encourages teachers and parents to introduce students to the vocabulary of math. Projects in this first of the two *Big Ideas* books include “Straw Structures,” where children get hands-on experience with measurement and 3-D visualization; “Kaleidoscopes,” in which students use geometry to build a mathematical toy; and “Crawling Around the Möbius Strip,” where kids build a physical example of infinity.

Written directly to the student and using simple materials, this is a good guide for the classroom or to use independently.

— Sally Niezgoda

- ISBN-13:
- 9781613741368
- Publisher:
- Chicago Review Press, Incorporated
- Publication date:
- 08/01/2007
- Sold by:
- Barnes & Noble
- Format:
- NOOK Book
- Pages:
- 160
- File size:
- 6 MB

All rights reserved.

ISBN: 978-1-61374-136-8

CHAPTER 1

**Activity 1**

PATTERN SHAPES

**The BIG Idea**

Geometric patterns are the very essence of discovering new mathematical relationships.

**Content Areas in This Activity**

• Geometric patterning

• Pattern rules

**Process Skills Used in This Activity**

• Communication (optional)

• Creativity

• Aesthetics of mathematics

**Prerequisite Knowledge and Skills**

None

**Age Appropriateness**

This simple activity is appropriate for all ages.

**The Mathematical Idea**

Patterns are a central idea in mathematics. Almost anything that has a pattern contains some mathematics, and most mathematical ideas contain some rule or pattern. Recognizing increasingly subtle patterns is an important mathematical skill. This activity will introduce children to the concept of patterns, with an emphasis on enjoying their visual appeal. Showing examples of aesthetically pleasing patterns will enhance the activity and encourage children to be creative. Many quilts, for example, show remarkable patterns.

The central notion of a pattern is that it is predictable, once we see what is repeated. The pattern may change as we progress, but in a predictable way. Children should be able to identify what is repeated in their patterns, and what would come next. Patterns can be linear, nonlinear, or rotational. (See the box on page **2** for definitions of these terms.)

Wherever there is a pattern, there will be some math. Even if they can't yet name the patterns mathematically, children can still enjoy inventing them. Have fun creating!

**Pattern Shapes: Making It Work**

**Objectives**

• Children will create geometric patterns.

• Children will connect the skills of pattern recognition, pattern creation, and spatial reasoning.

• The activity will encourage creativity and an aesthetic sense of mathematics.

**Materials**

[check] a few 8 ½"x 11" pieces of light cardboard or construction paper in several colors for each child

[check] scissors for each child

[check] glue stick for each child

[check] 11" x 17" piece of background paper for each child

[check] photographs of quilts and other geometric designs

[check] colored pens or pastels (unless using multicolored cardboard or paper)

[check] photocopy of the Pattern Shapes Activity Sheet (on page **5**) for each child

**Preparation**

• You could speed up the process by creating shapes ahead of time. Copy and cut out the shapes on the activity sheet (on page **5**). You can enlarge the shapes as desired. Trace them onto light cardboard or construction paper, cut them out, and color them (or use different colors of construction paper).

Precut shapes in a variety of colors may help inspire children to be creative. You could even put together a few patterns as examples for the children.

• If you don't cut out the shapes ahead of time, make enough copies of the activity sheet for each child to have one.

• Enhance child creativity and comprehension by gathering photographs of quilts and other geometric designs.

**Procedure**

1. Show children pictures of geometric patterns, such as quilts, to stimulate discussion and understanding of what a pattern is. Ask children what they think a pattern is and what it is not. Guide the discussion to include the idea that a pattern involves repetition.

2. Hand out the activity sheet (page **5**), the background paper, glue sticks, cardboard or construction paper (and heavy wrapping paper or textured paper, if desired), and colored pens or pastels (unless using multicolored paper).

If you precut the shapes, just hand out the shapes themselves. Give each child several of each shape so that everyone has plenty of shapes to choose from and can repeat shapes to form a pattern as desired.

3. Tell the children that they are to create any pattern they wish with the shapes. Encourage them to make a pattern, not just a pretty picture or set of randomshapes. They do not need to use all of the shapes. For example, a young child could simply make a checkerboard pattern with two colors of squares.

4. Choose questions to prompt children appropriately, depending on the situation. For example:

• "How about a star?"

• "In what other ways could you arrange the pieces?"

• "Can you make this pattern repeat?"

• "What would it look like if you made another ring of shapes?"

• "Can you make your own quilt pattern?"

• After they design patterns they like, children glue them onto their big piece of background paper.

**Suggestions**

• Doing this activity in a group will encourage the children to share ideas.

• Encourage children to be creative. Remember, math can be visual, creative, and just plain fun!

**Assessment**

Children are successful with this activity if they have created a clear pattern of their own.

CHAPTER 2**Activity 2**

SQUARES and ODD NUMBERS

The BIG Idea

There is often a geometric illustration of a complicated idea that makes the idea easier to see.

**Content Areas in This Activity**

• Square numbers (optional)

• Addition, single digit (optional)

• Multiplication, single digit (optional)

• Areas of squares

• Geometric patterning

• Numeric patterning

• Pattern rules

**Process Skills Used in This Activity**

• Reasoning

• Hypothesizing

• Concept of proof (optional)

• Communication (optional)

• Creativity

• Aesthetics of mathematics

**Prerequisite Knowledge and Skills**

• Activity 1

• Understanding of odd numbers (helpful)

• Multiplication, single digit (helpful)

**Age Appropriateness**

Children as young as seven can appreciate this activity if they do it slowly. For very young children, stick to a geometric understanding of sizes of squares, rather than the more numeric concept of square numbers.

**Mathematical Idea**

Square numbers are easy to illustrate geometrically. They are simply the numbers that make actual square shapes. We can build a 2 x 2 square shape to illustrate the square number 4, a 3 x 3 shape to illustrate 9, and so forth. The first part of this activity involves using small cubes or tiles to build squares of various dimensions. This will illustrate the concept of a square number in a visual way.

The first square number: 1 x 1 = 1

The second square number: 2 x 2 = 4

The third square number: 3 x 3 = 9

The fourth square number: 4 x 4 = 16

The fifth square number: 5 x 5 = 25

The second part of the activity explores an interesting idea (called a *theorem)* about odd numbers and square numbers: If you add a list of odd numbers starting at 1, you always get a square number! For example, 1 + 3 + 5 = 9, so the sum of the first three odd numbers is 9. Here's another one: 1 + 3 + 5 + 7 + 9 = 25 so the sum of the first five odd numbers is 25. Geometrically (that is, by using actual shapes), we can build each new square number by adding another odd number.

So we get the fifth square number (25) by adding the nine new cubes (or tiles) to the right of and below the 4 x 4 square — and 9 is the fifth odd number! We have a complicated-sounding mathematical theorem — the fact that the sum of the first n odd numbers is always n x n or n squared (n2) — illustrated in something as simple as an array of squares.

**Squares and Odd Numbers: Making It Work**

**Objectives**

• Children will construct square numbers geometrically.

• Children will experience a first taste of a geometric proof by exploring the notion that if we keep adding the next odd number, we always get a square number.

**Materials**

[check] at least 25 small cubes, such as sugar cubes or centicubes, for each child (a variety of colors is ideal), or the same number of square tiles per child

[check] photocopy of the Squares and Odd Numbers Activity Sheet (on page **11**) for each child

**Preparation**

None

**Procedure 1: Building a Square**

Older children who understand the concept of square numbers may want to skip to Procedure 2, opposite. You can focus on numerically defining square numbers through this first part of the activity, or you can simply define square numbers as the number of cubes (or tiles) needed to build a square, without explicitly listing all of the square numbers or getting into multiplication.

1. Hand out the activity sheet and cubes or tiles to each child. Have the children start with one cube (or tile). (If necessary, discuss the concept of 1 being a square number because the outline of one tile or one cube looks square; in other words, 1 x 1 = 1.)

2. Children should write down how many cubes it took to build the 1 x 1 square on the Square Numbers Chart (in the 1 x 1 row) on page **11**.

3. Next, have the children construct a 2 x 2 square and record on the chart the number of cubes or tiles they used.

4. Repeat the process for a 3 x 3, 4 x 4, then 5 x 5 square.

5. Explain to the children that the number of cubes they use for each square is a *square number* because they can build a square out of that number of cubes.

**Procedure 2: Adding Odd Numbers to Squares**

This next part of the activity involves adding odd numbers to squares. You may need to review the concept of odd and even numbers with younger children before continuing.

1. Have children start with one cube or tile. Ask how many pieces they need to add to create a 2 x 2 square, then have them build it.

2. Ask how many pieces they need to add to the 2 x 2 square to create a 3 x 3 square, then have them build it.

3. At this point, ask the children if they notice anything special about the numbers they are adding to each square to make the next highest size. (They should recognize 3 and 5 as odd numbers.)

4. Next ask the children how many cubes are in their current squares (9). They started with 1 cube, added 3, then added 5 for a total of 9 cubes: 1 + 3 + 5 = 9. Do they notice a pattern here? Guide the discussion as necessary to the realization that adding successive odd numbers together, starting with 1, will always create a square number. (Older kids can take this further to the realization that 9 is the *third* square number and that they added *three* odd numbers together to get 9.)

5. Have the children continue adding odd numbers to their squares to create the next largest square. Ask them if they think the idea will always be true no matter how big the square. Could there ever be an example where it wouldn't work? What would it take to be sure it will always happen? You could introduce the terms *theorem* and *proof* this point.

6. Finally, ask them to show how they can separate their largest square into odd numbers again.

**Suggestions**

• Building the squares with different colors — that is, using different colored cubes or tiles each time they add the next odd number — may make the idea clearer.

• Instead of handing out activity sheets and taking children through the above steps, you could pose the question: "Can you find a way to show that when you add up a list of odd numbers (starting with 1), you will always get a square number?" Allowing them to play with the idea is better than having them work through the directed activity, but of course this will work better with some children than with others.

**Assessment**

Determine understanding by asking the children to explain the ideas of squares and square numbers, as well as the ideas of proof and theorem, to the degree that you've discussed them.

CHAPTER 3**Activity 3**

CUBES in a ROOM

**The BIG Idea**

Verbally describing three-dimensional shapes requires careful communication.

**Content Areas in This Activity**

• Three-dimensional visualization

• Geometric terminology

**Process Skills Used in This Activity**

• Communication

• Creativity

**Prerequisite Knowledge and Skills**

None

**Age Appropriateness**

Children under the age of seven may have difficulty with this activity on their own, but most will be able to handle the activity with a grown-up such as a parent or older buddy observing and helping. Eight- to ten-year-olds should be able to work in pairs without a grown-up. In general, more intervention is needed with younger children to remind them that they must explain, not show or draw, the shape, and to help them select appropriate language.

**Mathematical Idea**

Communication is an important aspect of mathematical development. This activity encourages verbal communication, problem solving, and spatial reasoning. Children may develop an understanding of terminology such as *face, width, length, height, square, level,* and so forth as a natural outcome of this activity. They may also develop an appreciation of the need for precise mathematical language.

The activity involves creating a shape out of 27 or fewer cubes, a shape that must fit into the "room" — an open box that won't allow the shape to be larger than three cubes in any direction. After one child creates a shape, he or she must describe that shape using only words to a partner, who will try to reconstruct the shape based on the other child's words alone, without seeing the shape.

**Cubes in a Room: Making It Work**

**Objectives**

• Children will improve their verbal communication skills in mathematics, possibly learning new mathematics terminology.

• Children will strengthen their ability to visualize in three dimensions.

• Children will build teamwork skills.

**Materials**

[check] 27 small cubes, such as sugar cubes or centicubes, for each child

[check] sheet of paper for each pair (two boxes can be made from an 8 ½"x 11" sheet)

[check] scissors

[check] transparent tape

[check] file folder or book (to hide shapes-in-progress from partner's view) for each pair

4 photocopy of Cubes in a Room Activity Sheet (page **17**) for each child

**Preparation**

• Prepare the boxes ahead of time by first enlarging the template on page **15** to the size you need and making a photocopy for each child before cutting out the templates. This open-front paper box should be barely larger than the cube made with all 27 small cubes. This is the "room" for each structure.

If your class will use centicubes, enlarge so that each square is about 1 ¼" x 1 ¼" (3.125 cm x 3.125 cm), making it slightly larger than the maximum size for the shape (which can't exceed 1 1/5", or 3 cm, high or wide). If your class will use sugar cubes, measure a sugar cube, multiply that number by three, and add about ¼" (0.5 cm) to determine how big each square in the template should be.

• After you have photocopies of the template at the size you need, cut each one out and tape it into a box that will be open in the front and on the bottom. Double-check that the box is the right size for no larger than a three-cube by three-cube by three-cube structure.

**Procedure**

1. Have the children pair up, then explain that one of them will construct from the cubes a shape that must fit in the box. (You can tell them that it can't be bigger than three cubes in any direction, but often it is easier to explain by saying, "It must fit in the box.")

2. Have the children practice building shapes together at first to be sure the rules are clear: Shapes can be no bigger than three-cubes by three-cubes by three-cubes, and cubes must touch on edges or full faces. After completing a shape, each child should slide it into the box to make sure it fits.

3. After everyone understands the idea of creating the shape within size limits, have the children decide which of them will construct the shape first; then have them put up the divider between them.

4. The child who will build first makes a structure out of cubes as described above.

5. After finishing the structure, that child should slide it into the box to make sure it fits.

6. Then that child describes the shape orally so that the other can build it. Encourage the children to ask questions of the person who is describing the shape. Encourage the ones describing the shape to use correct mathematical terms, and not drawings or gestures. Discuss these terms as the need arises; for example, "We call the flat part a face."

7. After the children are able to duplicate their partners' shapes, repeat the process, but this time the other child in the pair creates and describes the shape.

Excerpted fromBig Ideas for Small MathematiciansbyAnn Kajander. Copyright © 2007 Ann Kajander. Excerpted by permission of Chicago Review Press Incorporated.

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

**Ann Kajander**, is the head of mathematics at a secondary Lakehead public school and runs the Kindermath Enrichment Project. She is also an adjunct professor in the mathematics department at Lakehead University. She has contributed articles to *Teaching Children Mathematics* and has presented at the National Council of Teachers of Mathematics conference.

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