## Read an Excerpt

#### Big Ideas for Growing Mathematicians

#### Exploring Elementary Math with 20 Ready-to-Go Activities

**By Ann Kajander** **Chicago Review Press Incorporated**

** Copyright © 2007 Ann Kajander**

All rights reserved.

ISBN: 978-1-56976-212-7

CHAPTER 1**One in a Million**

**The BIG Idea**

Just how big is a million really?

**Content Areas in This Activity**

[check] Numeracy

[check] Estimation

[check] Basic probability

**Process Skills Used in This Activity**

[check] Creativity

[check] Reasoning

**Prerequisite Knowledge and Skills**

[check] Multiplication

[check] Place value for large numbers

**Age Appropriateness**

This activity is appropriate for all ages.

**The Mathematical Idea**

While babies can distinguish small numbers — for example, they can tell one object from two — older children are able to perceive larger numbers, such as distinguishing three from five or ten from twenty. Very large numbers, however, are much harder. Just how far does our numerical judgment go?

Lotteries, population figures, and high finance are but a few of the applications of math that use dazzlingly large numbers. It can be difficult to develop any appreciation of these very large numbers. How big is a thousand? How big is a million? They both sound big, but a million is much larger than a thousand — one thousand times larger in fact! So to have the same chance of winning a lottery with one ticket in a million as you would with one ticket out of a thousand, you would need 1,000 tickets from the first lottery! You can verify this as follows: Assume your chances of winning are the number of tickets you own out of the number of tickets sold. So if you own 1,000 tickets in a 1,000,000-ticket lottery, your chances of winning are:

1,000/1,000,000 = 1/1,000 = 1 ticket owned/1,000 sold

In this activity students investigate concretely just how big one million is and explore the chances of winning a one-in-a-million draw.

**Making It Work**

**Objectives**

Students investigate the chances of picking one object out of a million by attempting to model one million concretely.

**Materials**

Materials will vary depending on students' choices of model.

[check] a small bag of rice and a scale or measuring spoons may be useful to introduce the activity

[check] calculators (optional but helpful)

**Preparation**

Counting out and measuring or weighing in advance 100 grains of rice may save time but is not necessary.

**Procedure**

**1.** Challenge students to think about the question of how big 1,000,000 really is and to come up with a tangible and concrete but affordable model of 1,000,000 objects.

**2.** Have students discuss their ideas in pairs or small groups.

**3.** Ask students to estimate the amount of various materials required to model their ideas.

For example, if they suggest grains of rice, have them do some quick counting and estimating that will show this idea to be unrealistic — depending on the size, it can take close to 30 bags of rice to make 1,000,000 grains!

Many ideas (such as rice) work well for 10,000 or 100,000 objects, but the number 1,000,000 makes things just that much more difficult.

**4.** Encourage students to make their models both affordable and transportable.

**Suggestions**

Since a main point of the activity is for students to devise and calculate the amount needed for various substances, suggestions of suitable materials should be withheld as much as possible. The investigation of unsuitable materials is fruitful in itself. When I first did the investigation, I was staggered at just how much rice 1,000,000 grains really was! However, here are a few suggestions to illustrate possibilities if really needed:

* Use a computer to print out a page of *o's* or any other character on the smallest possible printer font. Count the number of lines of print vertically, and the number of characters across, and use a calculator to multiply length times width of the characters to determine the number of characters on each page. This number can be divided into 1,000,000 to determine the number of such pages needed. Then, color in one *o* to show the "one" in one million. I keep a display like this in my classroom and vary the location of the "one" colored in; students love to search for the "one."

* Use Internet research to estimate the number of hairs the average person has on his or her head. (It's about 100,000.) Assemble 10 people — 10 times 100,000 is 1,000,000. Ask one person to volunteer one hair to serve as the "one."

**Assessment**

Students' models should be reasonably accurate but affordable and portable. I also like to see the entire 1,000,000 — as opposed to students saying, "Well, if you had 10 of these ..." — because it makes the activity more challenging. The "one" should be identifiable.

CHAPTER 2**Math Magic**

**The BIG Idea**

Lots of cool puzzles have math as their basis.

**Content Areas in This Activity**

[check] Numeracy

**Process Skills Used in This Activity**

[check] Problem solving

[check] Reasoning

**Prerequisite Knowledge and Skills**

[check] Place value for large numbers

[check] Basic algebra (to solve Puzzle 2)

**Age Appropriateness**

Children of all ages can enjoy doing the puzzles. Puzzle 1 is solvable with knowledge of multiplying by 1,000. Puzzle 2 may require algebraic equation solving to explain fully.

**The Mathematical Idea**

Puzzles 1 and 2 draw on the notion of place value. Puzzle 1 uses the fact that multiplying by 1,000 shifts the digits of a number left by three places. The numbers used in the "magic" of the puzzle have a product of 1,001 ... so multiplying by 7, then 11, and then 13 is essentially the same as multiplying by 1,001 (7 × 11 × 13 = 1,001). For example, 123 × 1,001 is the same as 1,000 × 123 plus 1 × 123, or 123, 123.

Puzzle 2 uses the fact that multiplying by 9 is the same as multiplying a number by 10, and then subtracting the number from the result: 9n = 10n — n.

Puzzle 3 uses the fact that binary numbers — numbers that are powers of 2, such as 1, 2, 4, 8, 16, etc. — can generate all other numbers by being added together. For example, 12 = 8 + 4 and 63 = 32 + 16 + 8 + 4 + 2 + 1.

One is a binary number because it is two to the power zero. (Recall that any number to the power zero is one.)

Details are found in the puzzle answers at the end of the Activity Sheets.

**Making It Work**

**Objectives**

Students are intrigued by problems that seem mysterious. They also enjoy tricking others with puzzles. These puzzles have been found to be quite self-motivating!

**Materials**

**Puzzle 1**

[check] basic calculator

**Puzzle 3**

[check] Magic Cards as on activity sheet, one set per player

[check] scissors

**Preparation**

Copy the activity sheets, one set per student.

**Procedure**

**Puzzle 1**

This puzzle can be done in an adult-led large group.

**1.** Ask students to enter any three-digit number on their calculator. (Everyone can use a different number.)

**2.** Tell students you are going to teach them a magic trick. Ask them to multiply their number by 7 and press Enter. (At this point everyone will have a different number.)

**3.** Then ask everyone to multiply the new number by 11 and press Enter.

**4.** Now comes the "magic" part. Tell the students you are going to make their original number reappear twice. Ask everyone to multiply the number they now have on the screen by 13, say "Abracadabra," and then press Enter a final time.

**5.** Have students compare the numbers on their screens to the numbers they started with, and ask what they see. Their original number should have reappeared — that is, if a student's original number was 123, the student should now have 123123 on his or her screen.

**Puzzle 2**

This is a great trick for students to try on a grown-up, such as another teacher. They can play this with an adult magician first, until they know how it works, and then they can learn how to be the magician themselves.

**1.** Ask everyone to choose a number between 10 and 99. (Or, for adults, just ask them to record their ages.)

**2.** Have the participants multiply the number they just chose by 10 and write that new number down. (For example, a person who chooses 27 should write down 270.)

**3.** Ask everyone to choose a second number between 1 and 9, multiply it by 9, and write down the resulting number. (For example, a person who chooses 4 should write down 36.)

**4.** Now, ask everyone to subtract the second number from the first. (270 - 36 = 234)

**5.** Have the participant tell the magician the final resulting number. (234)

**6.** The magician can now tell the player his or her original mystery number. Here's the magic method: Truncate the ones digit (4, in this example) and add it to the other two digits (23 + 4). The result (27) is the original number! Why?

**7.** Children can perform the magician role themselves once they are able to see the trick that the magician is doing. It is best to have them find the trick themselves by playing the game several times. The biggest challenge is to explain why the trick works.

**Puzzle 3**

**1.** Cut out the cards from page **9.**

**2.** The adult can be the magician for the first time through. Instruct children to choose any number from 1 to 31, keeping the number in their heads.

**3.** Place the five cards face-up on the table. Children now pick up all the cards that contain that number and give them to the magician. (If they make a mistake doing this, the trick will not work.)

**4.** In his or her head, the magician adds all the numbers in the top left corner of the chosen cards, then tells the child the sum. This will be the child's original number!

**5.** Children can be the magician once they know the trick. It's also a great game for practice in mental addition.

**Suggestions**

Students do not necessarily need to know how or why the mathematics makes these puzzles work in order to enjoy them. Resist the temptation to tell students why they work right away. The more intriguing the mystery, the more mathematical thinking will occur.

**Assessment**

Students could be asked to solve, or explain, the mathematical basis for as many of the puzzles as they can. Many students will be able to solve at least the first one, but the others are more challenging.

CHAPTER 3**Cooperative problems**

**The BIG Idea**

Working together in math can help us see all sides of the problem.

**Content Areas in This Activity**

[check] Spatial reasoning

[check] Geometric terms such as *edge* and *face*

**Process Skills Used in This Activity**

[check] Creativity

[check] Reasoning

[check] Collaborative problem solving

**Prerequisite Knowledge and Skills**

[check] None

**Age Appropriateness**

This activity is appropriate for all ages as long as children can read simple clue cards.

**The Mathematical Idea**

Working on mathematical problem-solving tasks in small groups takes a bit of practice and requires cooperative skills. Although working mathematicians often solve problems by talking about them with colleagues, sometimes children are asked to work on mathematics problems alone, which fails to get them used to working effectively with others. The activities in this chapter help reacquaint children with the skills they need to share their ideas and work collaboratively, and they remind us explicitly that sometimes everyone's input is needed to solve a problem.

Here is a story that might motivate this activity. An old man approaches a boy in a grocery store. He asks the boy, "Do you have a minute?" The boy assumes he wants help, but instead the old man asks him a question. "How many sides do you see?" asks the old man, holding a box up in front of the boy's face.

"I see one," says the boy.

The old man moves back and again holds up the box. "Now how many?"

"Now I see three sides," says the boy.

The old man puts the box on its edge on a shopping cart and asks the boy to step back from it. The old man stands back from the box on the other side. He says to the boy "Now the two of us working together can see all six sides."

**Making It Work**

**Objectives**

Students learn to work effectively together on spatial reasoning tasks by sharing their individual pieces of information about a shape's characteristics.

**Materials**

(for each group of three to four children)

**Activity 1**

[check] 2 yellow cubes

[check] 1 blue cube

[check] 1 red cube

[check] set of Activity 1 clue cards (see page **13**)

[check] ziplock bag

**Activity 2**

[check] 2 yellow cubes

[check] 2 red cubes

[check] 1 green cube

[check] set of Activity 2 clue cards (see page **13**-14)

[check] ziplock bag

**Activity 3**

[check] 2 red cubes

[check] 2 green cubes

[check] 1 yellow cube

[check] 1 blue cube

[check] set of Activity 3 clue cards (see page **14**)

[check] ziplock bag

**Note:***Cube colors may be changed but must be changed correspondingly on the clue cards!*

**Preparation**

Place one set of the required cubes and one set of clue cards (individually cut out from Activity Sheet) in an individual bag, such as a small ziplock bag, for each group of four students.

**Procedure**

**1.** Students work in groups of three or four. Each group should receive one prepared bag of materials.

**2.** Within each group, students should distribute the clue cards reasonably equally. Depending on the activity and the group size, they may get one, two, or three cards each. (Not everyone will have the same number of cards. The adult may choose to give out the cards if control is desired over who gets more cards due to reading level.)

**3.** Place the cubes in the center of the group so everyone can reach them.

**4.** One at a time, moving in a clockwise direction, each group member should read his or her clue card aloud, then move the blocks to reflect that card's clue, if possible.

**5.** Students continue reading their cards and moving the blocks until one shape has been created that satisfies the clues on every single clue card without changing the shape each time. Some students will have to reread their cards aloud and move the blocks again. The information on *all* the clue cards is required for a correct shape.

**Suggestions**

* Sometimes children need to be reminded to doublecheck that the final shape satisfies all the cards.

* One typical stumbling block is to forget that the cubes can also be stacked. After a suitable interval, a question such as "Is there any other way to arrange the cubes?" may prompt further thinking.

**Assessment**

The group has been successful when everyone has participated and the shape has satisfied all the clue cards. The group members are ready to move on to the next activity.

CHAPTER 4**Lots of Dots**

**The BIG Idea**

Students must work together effectively to solve this problem. That's an important skill in mathematics.

**Content Areas in This Activity**

[check] Spatial reasoning

[check] Terms such as *row, column, rectangle,* etc.

**Process Skills Used in This Activity**

[check] Communication

[check] Reasoning

**Prerequisite Knowledge and Skills**

[check] None

**Age Appropriateness**

This activity is appropriate for all ages.

**The Mathematical Idea**

As we saw in the tasks in chapter **3**, "Cooperative Problems," children need to develop skills to share their ideas and work collaboratively, and sometimes everyone's input is needed to solve a problem.

This activity builds on the tasks in chapter **3**. However, students must now communicate their reasoning by using words, rather than by moving blocks. This can be challenging at the beginning. They must also make decisions about which pieces of information can be assumed and which are crucial to describe. This is very like the work of a mathematician in that they are deciding which aspects of a problem to attend to and which to assume.

**Making It Work**

**Objectives**

Students learn to communicate their visualization and reasoning verbally so that others can understand. They also learn to follow others' descriptions and pose effective questions for clarification.

**Materials**

(for each group of four)

[check] full set of 24 cards (see page **19**-22), copied onto card stock and cut out for each group of students

**Preparation**

One player shuffles the set of cards, leaving them face down.

*(Continues...)*

Excerpted from **Big Ideas for Growing Mathematicians** by **Ann 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.

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