Picture Books
As a Reading teacher, I find every opportunity I can to incorporate picture books, novels and short stories into my instruction. I have found several math related picture books that are perfect for any grade level. Yes, middle school and high school students enjoy picture books just as much as elementary students. For the upper grades, these books can be used as a listening skill practice as well as a math skill practice. For lower grades, a read aloud coupled with demonstrations of problems on the chalkboard will have optimal results for the students. In the end we are talking about reading math. So it is only fitting that we take an actual story and apply our math skills.
Geometry
The following picture books really emphasize geometry and spatial reasoning. They would be great to use as a read-aloud while students are using manipulatives to solve the story.
Sir Cumference and the Sword in the Cone by Cindy Neuschwander
Neuschwander has written a series of books featuring "Sir Cumference". The main character of the series is Sir Cumference who is a knight in King Arthur's court. Other characters include Radius, Vertexand the Lady Di of Ameter. It does not take the reader long to realize that Neuschwander is using a play on words and incorporating geometry into the story. This particular story revolves around Vertex, who begins a quest to solve a riddle from the King. The one who solves the riddle and finds Edgecalibur will become the next king. The clue states,
Form the solids and find their places. How many edges, points, and faces? The shapes that make two will pass the test, but one that does not must be your quest. Three times as tall as its base is wide, the true King's future lies inside. (Neuschwander, 2003, p. 5)
Surrounding the edge of the parchment are diagrams of shapes. It is later realized these diagrams are solid shapes flattened out. Once this is recognized, Vertex and Radius begin cutting and folding the shapes. The quest now begins. They begin counting faces, points and edges of all the shapes. After many calculations and trial and error, Vertex understood that the cylinder and the cone are the only ones among the given figures that did not produce two when the numbers of faces and vertices were added together and the number of edges was subtracted. Now they know they are looking for an object in the shape of either a cone or a cylinder.
They find many cones under the stones in the path and now see that they must measure them. They are looking for a cone whose length is three times its base. Upon measuring many cones they find the one that is most likely to house Edgecalibur based on knowledge that the sword is approximately 48 inches long. Vertex releases the sword from the cone and takes it to King Arthur. King Arthur is very impressed with his math skills and dubs him Prince Vertex who will rule once Arthur steps down.
This book can be incorporated in many aspects of the curriculum. Just some of the connections are legend, reading, geometry, and problem solving. I maintain that picture books are rarely only for small children. A good picture book can span the grades from first to twelfth and offer a completely different perspective in each grade. This book is just such an example. In reference to word problems, it can be an introduction to a unit on geometric word problems. This book can be regarded as one long word problem, so it is a model example. I believe that using picture books to introduce concepts, on any subject, is always an attention grabber. The illustrations alone capture the attention of students, young and old alike. Specifically with this book, concepts of spatial reasoning, area, measurement, probability and of course geometry can be introduced or reinforced.
If this book interests you then the others in the series might also be of help. Sir Cumference and the Isle of Immeter covers the topics of area and perimeter. Sir Cumference and the First Round Table explores the topic of taking a rectangular table, which is too long and difficult for everyone seated to see and hear each other, and forming it into various shapes to find the ideal shape for the knights to sit and meet. Sir Cumference and the Dragon of Pi covers the number pi through a story in which Sir Cumference drinks a magic potion that turns him into a dragon and only the magic number pi can transform him back. In Sir Cumference and the Great Knight of Angleland, Radius uses a circular family medallion to discover angles and ultimately the use of a protractor.
A few word problems for Sir Cumference and the Sword in the Cone follow. More can be found in Appendix A.
The path of stones in the courtyard created a rectangular space with an area of 925 square feet. What is the width of the space if the length is 37 feet?
*A. 25 ft
B. 74 ft
C. 425.5 ft
D. 462.5 ft
King Arthur designed a flower garden in the courtyard in the shape of a square. He plans to build a walkway through the garden diagonally from one corner to the other. If the length of each side of the garden is 12 feet, which is closest to the length of the walkway?
A. 36 ft
B. 24 ft
*C. 17 ft
D. 13 ft
Vertex made a scale drawing of his room in the castle. The actual room has a width of 16 feet and a length of 24 feet. His drawing has a length of 3 inches. What is the width, in inches, of the scale drawing of the room? (2 in.)
Spaghetti and Meatballs for All! By Marilyn Burns
I think this book covers a lot of thinking about math in a fun story. The premise is that Mrs. and Mr. Comfort are throwing a dinner party for their family. They find out there will be 32 people. They arrange 8 tables with 4 chairs each. The problem arises when the families begin to arrive and want to sit together. They begin pushing the tables together to make one big table. As more and more guests arrive, the families continue to rearrange the tables so everyone can sit together. Of course throughout this time Mrs. Comfort is getting very distraught because she knows that if the tables are pushed together there will not be a seat for everyone. The story continues with more arrangements of the tables until eventually the tables start being pulled apart as the rest of the family arrives. In the end the tables again end up as 8 tables with 4 chairs each, just as Mrs. Comfort had originally set.
There is spatial reasoning involved because 2 tables of 4 will seat 8, however if those 2 tables are pushed together then suddenly they only seat 6. The spacing problems continue as more tables are pushed together and fewer seats become available. One listening activity for this book is to provide students with manipulatives of the tables. This can even be simple squares of paper. As the story is being read, either individually or in groups, the students can arrange the tables to match what is happening in the story. This can help the students visualize area and perimeter. It would be pertinent to note that the combined area of the tables is constant, since the number of tables is constant, but the perimeter, which is proportional to the number of free edges, changes from configuration to configuration.
There are numerous ways to use this book as a supplement to the math curriculum. Substituting and interchanging the words chair, seat and perimeter would help the students understand units of measurement. How many seats will there be in a row of 5 tables that are set end-to-end? The students can also be asked what other options the Comfort family had in regards to seating. This can include getting more tables and arranging in various ways. What about an octagonal or hexagonal table? This configuration would be a bit more difficult than square tables. How could this have worked into the plan? This book can be used as a whole class read-aloud or be put at a station for individual or small group work. The importance is that the students are putting together the words of a story with the calculations of math. This can help them when they are asked to solve individual word problems.
Following are a sample of word problems for this book. More word problems can be found in Appendix B.
Which of the following is NOT a valid formula for the perimeter of a square table with side length s?
F. s + s + s + s
G. 2s + 2s
H. 4s
*J. s x s
Which expression can be used to solve the problem below? The Comfort's considered throwing their party at a hotel. To cater a luncheon, a hotel charges $50 per hour for use of a dining room plus $24.50 per person. What is the total cost for a 2-hour luncheon for 45 people?
F. 2 x 50 + 24.50 + 45
*G. 2 x 50 + 24.50 x 45
H. 2 x 24.50 + 50 x 45
J. 2 x 45 + 50 x 24.50
Exponential Growth
The following picture books deal with the concept of exponential growth. American Heritage dictionary defines this term as, "Growth of a system in which the amount being added to the system is proportional to the amount already present: the bigger the system is, the greater the increase." There are vast possibilities with this type of math. Many students don't realize how quickly numbers can increase if done exponentially, even in a short period of sessions. These two books incorporate the concept of exponential notation into a storyline that is easy to understand and thereby help to make the concept more concrete. Upon reading the two books you will see that although they both incorporate exponential growth, the results of the stories are different. In Demi's book the growth is unfettered, adding grains each time and not removing any within the process. At the start of Anno's story the main character removes one seed each time a doubling growth is experienced. In addition there are other removals of various amounts as the story continues. So in this regard, the growth is not purely exponential in that every time growth occurs, so does removal of a seed. However, each story does teach exponential growth approaching the total from a different path.
Anno's Magic Seeds by Mitsumasa Anno
One day Jack is strolling along and comes across a wizard. The wizard gives Jack two golden seeds and explains that the seeds are magic. His instructions are, "Bake 1 seed in the oven until it is red and then eat it. You will not be hungry again for a whole year. Bury the other seed in the ground now and care for it well. I promise you it will grow and give you 2 more magic seeds in the fall" (Anno, 1995, p.3). Jack does as he is instructed and continues the cycle of eating one and burying one for several years. One day he realizes that this cycle can just go on and on forever always breaking even. He decides to bury both seeds and gets through the winter by eating something else. In the spring two sprouts came up producing four seeds. Jack baked and ate one seed and planted the other three. He continues to do this for six years, each year planting more and more seeds. In that year he meets a girl named Alice and the two of them each eat a seed and bury the rest. The next year Alice and Jack marry and give 2 magic seeds to each of their 5 wedding guests. That same year Jack and Alice store 16 seeds in a storehouse and bury the rest. They eventually begin selling the magic seeds at the town market, always planting more than they sold. In the tenth year a hurricane comes and the land floods. Jack is able to tie the house to a tree and he scoops up one small bag of seeds. When the storm was over his fields were empty and bare. However, Jack sees that he was able to save 10 seeds which are enough for Jack, Alice and their baby to eat and the rest were planted to start their new life.
A sample of word problems follows. More can be found in Appendix C.
A seed, given a favorable growth medium, doubles in number every 6.5 hours. Given that there were approximately 100 seeds to start with, how many seeds will there be in a day and a half? (approx. 4,648)
Jack is researching a newly-discovered species of seed. At time t = 0 hours, he puts one hundred seeds into what he has determined to be a favorable growth medium. Six hours later, he measures 450 seeds. Assuming exponential growth, what is the growth constant "k" for the seeds? (Round k to two decimal places.) (The growth constant is 0.25/hour)
One Grain of Rice: A Mathematical Folktale by Demi
The premise of the story is that an Indian raja made all the peasants give him most of their rice. They were able to keep just enough to get by. In the meantime he stored all of this rice in a royal storehouse for himself. Although he promises them that everyone would have rice to eat even in times of famine, the raja did not keep his promise. When there was a drought one year the peasants begged the raja to give them some of the rice that was stored in the royal storehouse. He refused saying, "No! How do I know how long the famine may last? I must have the rice for myself. Promise or no promise, a raja must not go hungry!" (Demi, 1997, p.6). One day a village girl name Rani saw that a bit of rice was falling from an elephant carrying a basket on his way to the palace. The girl carefully picked up each grain and filled her skirt. She took the grain to the raja to return it. As an acknowledgment of her good deed the raja told Rani she could ask for anything and receive it. The only thing she asked for is to give her a single grain of rice and then each day for thirty days double the amount given the day before. The raja thinks this is a meager reward but did as she wished. As the days went on it is not long before the raja realizes the intelligence of the girl. On the thirtieth day she is presented with 523,870,912 grains. Adding to the total grains from each of the previous 29 days, Rani ends up with a total of 1,073,741,823 grains of rice. Rani shared it will all the hungry people of the village and left a basket of rice for the raja. She told him to only take what grain he needed from now on. The raja promised to do so and, "for the rest of his days, the raja was truly wise and fair, as a raja should be" (Demi, 1997, p.30).
The following word problems can be implemented as an additional activity with One Grain of Rice. It is up to you how and when you implement them into the lesson, but it is important to show how the book is one big word problem and there are smaller word problems that can come from the story that take a turn different from the original story line. Additional word problems for this book are found in Appendix C.
Four handfuls of rice equal two thousand and forty-eight grains. How many grains are in one handful? (512) How many grains are in 6 handfuls? (3,072) How many handfuls would it take to hold 6,144 grains? (12)
In looking at patterns of the exponential growth table for this story we can see that the numbers can be found through different questions. For example: What is the product of the number or grains Rani got on the 5th day with the number she got on the 10th day? (32,768). We can see that this number equals the number of grains she got on the 15th day. How many grains of rice did Rani get on the 20th day? (1,048,576). What is the product of the number or grains of rice Rani got on the 10th day with itself? (Again, 1,048,576). A discussion of the reason for the equalities can generate a higher order level thinking skill for students.
The book has a chart in the back showing the progression of grains from day one to day thirty. This chart can be copied and certain days can be eliminated. The students can then calculate the correct number that would go in that day. In addition, the students could also calculate the amount of grains for days beyond thirty.
Money
Money seems to be a topic that most students relate to. It is one of the few topics that students have an interaction with outside of the classroom. Many math teachers say that using money to teach decimals really makes the subject concrete for most students. Money can be used to teach addition, subtraction, comparison and percent when dealing with interest rates. The following story uses money as the basis for the storyline. Although the story is somewhat elementary for middle school, if used with the right questions and challenges, students will embrace it more as a word problem challenge, than a story to be comprehended.
Pigs Will Be Pigs: Fun With Math and Money by Amy Axelrod
The story tells of a pig family who has nothing left to eat in the refrigerator. They decide to go out to eat but then realize that among them they only have one dollar. The family decides to search the house for money in order to come up with enough for the four of them to dine at a restaurant. The story gives us an account of the coins and bills each pig finds. In this way the story lends itself to listening skills practice because the students will have to keep track of all the money that is found and add up the coins to find total amounts. This book incorporates the theme of money with the math skill of addition. There is also a comparison and evaluation when the pigs go to a restaurant and have to decide what they can order based on the amount of money they have.
Here are some word problems that can be used as supplement to the story. Some questions include the menu that is part of the story. A copy of the menu should be provided to the students along with the word problem questions.
The piglets found six dimes, one dollar bill and two hundred pennies. Mrs. Pig found two nickels, five pennies, one quarter and a five-dollar bill. How much did they collect together? Who collected more money? ($9, Mrs. Pig) What is the correct order of steps to complete the previous two questions? (Add up piglets' money, add up Mrs. Pig's money, add the totals together, look at the figures and decide which is larger)
Mr. Pig found $23, Mrs. Pig found $6.90, the piglets found $4.77. What percent of the total did each pig find? Round to the tenths place. (Mr. Pig 66.3%, Mrs. Pig 19.9%, piglets 13.8%)
Write an equation(s) that represents Mr. and Mrs. Pig receiving 2/3 of the total money and the piglets receiving 1/3 of the total amount found of $34.67. Solve the equation. (22.88, 11.44)
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