## Chessboard Challenge

### Susanna Vondeck

#### Description

This lesson has been created for use with the book [The Kings's Chessboard] by David Birch. The students predict and extend the numerical pattern of twice the day before's total (multiplying by 2 or doubling). They search for other patterns within their calculations and state rules for the relationships. They also check for measurement accuracy in the book by weighing grains of rice.

#### Objectives

The student describes, extends, creates, predicts, and generalizes numerical and geometric patterns using a variety of models (for example, lists, tables, graphs, charts, diagrams, calendar math).

#### Materials

-The book titled [The King's Chessboard] by David Birch (1988), New York: Puffin Pied Piper Books
-Sticky notes to cover specific parts of the book to be revealed after the students finish their calculations
-Two sheets of legal size (8 1/2" x 11") copy paper or two light colors of construction paper
-Tape or glue to attach papers
-Overhead transparency of chessboard pattern (8 inch by 8 inch square)
-A model chessboard with the answers written on each square for reference
-A 1 pound bag of rice
-A kitchen scale (or other type of scale used with a bag for the rice)

#### Preparations

1. Create the Chessboard Challenge workspace by making a 16 inch by 16 inch square out of two sheets of 8 1/2" x 17" paper. Divide the square into 2 inch squares to make 64 squares. You can make the workspace look like a chessboard by lightly shading every other square or by using 2 different light colors of construction paper. Label (in small print) each square with the numbers 1-64 in the top left corner. Circle each number. Make 1 chessboard for each student.
2. Make an 8 inch by 8 inch square on an overhead transparency for yourself. Divide it into 64 one inch squares.
3. Make a "cheat sheet" for the chessboard with the answers (they are outlined in the procedures section) so you will be able to refer to it while observing the students and checking their calculations.
4. Try to get a kitchen scale so it will be easier for the students to weigh the rice. If you cannot get a kitchen scale, have a plastic bag available to pour the rice into.
5. Preview the book so you will know when to pause so the students can figure out the next number in the pattern. Locate pages 20 and 21 in the book [The King's Chessboard]. You will need to cover the third paragraph on page 20 that begins "As the King . . ." and the picture on page 21. This gives the answer to the tons of rice the King would have to give the wise man, which is one of the questions posed to the students.
6. Students will need to be able to multiply numbers up to the billions period by 2. You may want to have any students experiencing difficulty with this work with a partner for assistance or they can double the numbers and add.
7. Gather all the materials for the lesson.

#### Procedures

1. Tell students that we will be reading a book titled The King's Chessboard. Say "This book poses a measurement problem that will cause us to explore patterns."

2. Pass out the chessboards to each student. Have your overhead chessboard pattern transparency, as well. Fill this in after the book has confirmed answers or when you want to model for the students how to fill in a square.

3. Tell the students that the book has a rule for a pattern we need to follow. Tell them each square on the chessboard is for a number in the pattern. Explain that we will be doing computations with numbers up to the hundred billions place and that they may want to take out some extra paper for their calculations. You may want to have the students do their calculations in pencil, but write on the chessboard in pen so you can see their mistakes and not have them worry so much about (and waste time on) making corrections. This will also make it easier to read their answers on the colored chessboard.

4. Begin reading the book. On page 4, the wise man starts giving examples of the pattern. Have students listen to this and guess the rule. Come to a class consensus or majority of what they think the rule is. Continue reading. The pattern rule is given in the last sentence of page 4 with the sentence "Thus for each square give me twice the number of grains of the square before it, and so on for every square of the chessboard." Discuss if the class was correct.

5. As you read page 8, the students should write the number 1 in the first square for the one grain of rice given to the wise man on the first day. Tell the students that each circled number in the top left corner of each square represents a day of the 64 days that the King must give the wise man rice.

6. Before page 10, they should be able to predict the next few squares without difficulty. Square 2=2, Square 3=4, Square 4=8, Square 5=16, Square 6=32, Square 7=64, Square 8=128 grains. Page 10 gives answers for the rice the wise man received for Squares 2 and 3 and page 11 skips ahead to Square 8. Give students time to fill in squares 2-8 before you read pages 10 and 11. Walk around the room and observe students.

7. Read pages 10-11 and discuss if students were correct about the numbers they wrote on their chessboard.

8. Have them calculate and fill in the squares for the 9th - 12th day while you observe. Read page 12 and discuss if they were correct. Square 9=256, Square 10=512, Square 11= 1,024, and Square 12= 2,048 grains of rice/1 ounce. At this time, ask the students about what the weigher did on the 12th day. (He began counting by ounces). Discuss why it is easier. They should write the number 1 (oz.) in the 12th square along with 2,048 grains. Have the students circle this square because this is where grains turn into ounces and a second pattern starts. They should keep in mind that this pattern HAS THE SAME RULE.

9. Tell the students they are going to check if the Weigher is correct about his estimate of 2,048 grains of rice being equal to 1 ounce after we complete the chessboard challenge.

10. Now have students predict the pattern in ounces for the 13th - 16th days. They should have 13th=2 oz.,14th=4 oz.,15th=8 oz., and 16th=16 oz. See if any students make a prediction about what happens at 16 oz.

11. Read page 13 and have them write 1 lb. on the 16th square along with 16 oz. if they have not already done so. They should also put a triangle around the 16th square to indicate that this is where ounces turned into pounds and a third pattern starts with the same rule. Have them start thinking about what will happen in a few days in the book. Have them extend the pounds out for a few days. This would be a good place to stop for the day.

12. Before reading any further, make sure students have entries on their squares up to at least the 30th day. Have them decide when they think we will start a fourth pattern. You want them to conclude that on the 27th day, we will have 2,048 pounds of rice which is a little over a ton (2,000 lbs). Continue reading pages 14-16. They should check their patterns. Square 17=2 lbs., Square 18=4, Square 19=8, Square 20=16, Square 21=32, Square 22=64, Square 23=128 lbs. or 1 sack, Square 24=256 or 2 sacks, Square 25=512 or 4 sacks, Square 26=1,024 or 8 sacks, and Square 27=2,048 lbs. or 16 sacks which we are now estimating to 1 ton.

13. At this point, they should write 1 ton on the 27th Square and put a rhombus around this square to show that this is where pounds became tons.

14. Have students continue the pattern in tons until the 32nd square. They will be halfway through their chessboard. 28th=2 tons, 29th=4, 30=8, 31=16, and 32=32 tons.

15. Read pages 17-21. Remember that you have covered the answer to "the total of all the rice that was to be sent for all of the sixty-four days." Tell students we will be solving that after we get all our chessboard information filled in.

16. Finish reading the book as the students begin working on the calculations to extend the pattern from the 33rd day to the 64th day. They can continue doing this for the rest of this session and the beginning of the next one. You may also need to assign it for homework. Observe them as they complete the chessboard.

17. Here are the rest of the squares: 33=64 tons, 34=128, 35=256, 36=512,
37= 1,024 T, 38= 2,048 T, 39= 4,096 , 40= 8,192 , 41= 16,384 , 42= 32,768 ,
43= 65,536 , 44= 131,072 , 45= 262,144 , 46= 524,288 , 47= 1,048,576 ,
48= 2,097,152 , 49= 4,194,304 , 50= 8,388,608 , 51= 16,777,216 ,
52= 33,554,432 , 53= 67,108,864 , 54= 134,217,728 , 55= 268,435,456 ,
56= 536,870,912 , 57= 1,073,741,824 , 58= 2,147,483,648 , 59= 4,294,967,296 ,
60= 8,589,934,592 , 61= 17,179,869,184 , 62= 34,359,738,368 ,
63= 68,719,476,736 , 64= 137,438,953,572 tons.

18. When the students have completed their chessboards, you will check all calculations on the overhead calculator. Have students model how to do some of the written calculations and calculator checking.

19. After students finish checking their chessboards, they should search for other patterns that occur with the numbers they hsve written on each square. They should write down the patterns in their math journals. They will need to analyze the patterns and write a rule for the relationships (i.e. multiply by 4 or 8). They can share the patterns with their peers and have them guess the rule. They can also show the first three numbers of the pattern and have other students extend the pattern and state the rule.

#### Assessments

Observe the students as they complete the Chessboard Challenge. Students who meet the standards for Algebraic Thinking will be able to predict and extend each pattern that emerges from the information in the book. The pattern of twice the day before's rice stays constant throughout the book. However, the measurement unit changes. Are your students able to predict the pattern of grains of rice, ounces of rice, pounds of rice, and tons of rice? Are they filling in the chessboard squares correctly? Are students able to explain how and why the grains changed into ounces, ounces into pounds, and pounds into tons? They should also be able to explain why this is easier to do than to keep counting individual grains of rice. Are students able to participate in checking and confirming answers (especially the final question about total tons of rice)? Check the accuracy of the other patterns they found and wrote in their math journals. Observe if they are able to guess each other's rules and extend their patterns.

#### Extensions

1. After they complete their chessboards, they are going to figure out the answer for the total of all the rice the King would have to give the wise man.
Have students suggest how to do this (Add each days' rice to each other). Explain that they will need to find the answer in tons, so you will need to model how to add days 1-11 to get 1 oz. Add days 1-15 to get 1 lb. Add days 1-26 to get 1 T. Then add the 1 ton to all the other tons on days 27-64. Do the students get the answer that is in the book? (274,877,906,944 tons)
2. Students will work in teams to check if the weigher's estimate was correct. Have each team count out 2,048 grains of rice and way them to confirm that it is one ounce. You could also have them count 1,024 grains and see if they weigh 1/2 an ounce. Discuss why they think he made a good estimate or not.
3. They can also continue looking for patterns in everyday life and writing them down in their journal. They can work with their peers to have them guess the rule or extend the pattern. One pattern that they observe every day at school is calendar math activities.