Samar opens the lesson by asking whether anyone has lost (to death) a member of the family recently. When Adam responds that his grandmother died, Samar asks, “Do you know anything about her inheritance when she passed away?” Realizing that Adam doesn’t understand the term inheritance, Samar clarifies by elaborating.
Samar: You know, her money or property? Did she give anything to her husband when she died?
Adam: I think she had some money.
Samar: Does anyone have an example of when someone in your family passed away, and there
was talk of property or money being left to the relatives?
Wadi: My grandpa died.
Samar: Was that your grandpa on your father’s side or your mother’s side?
Wadi: My father’s side.
Samar: Did your father get any money or land or valuables from your grandpa when he died?
Wadi: I think he got some money.
Having set the context of the lesson with two examples from the students’ lives, Samar tells the class, “In today’s lesson we’re going to use simple cases of inheritance just to get you familiar with the system of Islam.”3 She points to Adam and explains that when his grandmother died, her husband would have taken a definite fraction of her money. Because there were children, he would take one-fourth of her money, and the children would distribute the rest. If there had been no children, he would have taken half of her money, and the rest of it would depend on other relatives and whoever else was going to inherit from her.
To establish additional background for the lesson, Samar gives the class some historical background. As she writes on the board the name “Al-Khwarizmi,” both in English and in Arabic, she says:
Al-Khwarizmi4 lived at the time of the caliph Ma’mun,5who asked Al-Khwarizmi to write a book about the different cases of inheritance. Figuring out these inheritance rules takes a lot of thinking—someone’s taking one quarter of your money, someone else is taking one third, and so forth. It takes a lot of calculation, a lot of math to figure out all these rules. The Muslims wanted to seek more knowledge because of their religion, so that they could pray properly and figure out how to get their inheritance. Now in figuring out how to pray properly, Muslims had to figure out the position of the sun. That is how Muslims developed trigonometry—to find the right angles with the sun and so forth.
Returning to the board, Samar writes, “Al-Jabr Wal-Muqubalah,” both in English and in Arabic,6 and continues:
This is the name of the book written by Al-Khwarizmi. The second half of this book was all about inheritance, cases, fractions, and how to divide up one’s property. People who study his book say he was influenced by three civilizations: the Greek, the Mesopotamian, and the Hindu. We are all influenced by other cultures, aren’t we? He was able to get elements from each of these cultures to develop the science of algebra.
To be sure that her students are familiar with this branch of mathematics, Samar writes the word algebra on the board and asks them whether they have heard of the term algebra before? The students respond in the affirmative. Circling the words, algebra and Al-Jabr, Samar continues the lesson.
Samar: Do you notice the similarity of the words? Now which of these words do you think came
first, the English word or the Arabic word? Let’s see what Nadah thinks.
Nadah: I think the Arabic came first.
Samar: The Arabic word came first, and it entered the European culture and became the word
for the science of algebra; the science of unknowns—you don’t know how much this person is getting in inheritance. The putting together of unknowns is what the science of algebra is all about.
Because the students will be inventing their own algorithms for multiplying fractions in the lesson, Samar next establishes a context for understanding the term algorithm with more historical background: “Another contribution of Al-Khwarizmi was the word algorithm.” Samar writes “algorithm” and provides an example, which she also writes on the board.
Samar: If you want to simplify this fraction twelve-sixteenths, what do you do?
Ayya: You divide it by four?
Samar: Why four? Why did you choose four?
Yasser: Four is a common factor.
Returning to the board, Samar writes the equation, “(12 - 4)/(16 - 4) = % ”
Samar: That’s your algorithm. You’re simplifying the fraction. You found an algorithm to help
you do that, you followed a rule—that’s an algorithm. Where do you think the term algorithm came from?
She encourages the students to look at the words on the board and does a brief review The students easily make the connection between Al-Khwarizmi and the word algorithm.7 Samar then finishes establishing the basis for the lesson.
Samar: Today you are going to create your own algorithm for finding a fraction of a fraction. So,
I’m writing this key word, algorithm, on the board so we remember that our lesson today is about algorithms.
Next Samar passes out four strips of paper (about 3" x 8" ) and a page of inheritance tasks (see Figure 9.1) to each student. Together, the class reads the first problem that Samar has simplified from the original as it appeared in Al-Jabr Wal-Muqubalah. The somewhat puzzled looks on the students’ faces prompt Samar to ask, “Does everyone know what the word capital means?” Following Ehsan’s reply, “Like the capital of Louisiana?” Samar clarifies the term by explaining that in the problem, capital means the man’s money, his property, or his inheritance.
Samar: What do we know from the problem?
Wadi: One-third of his capital went to strangers.
Samar: Do you think this is important? If a person dies, and he says I want to give one-third of
my money to strangers, can you divide up his money before giving the part to strangers? In Islam, you can’t. The person gave orders in his will that after he dies, one-third should go to a particular person.
Yasser: The stranger comes and takes the one-third of the money?
Samar: We should give it to him because we honor the wishes of the dead person.
Next, she holds up one strip of paper and directs the students to find a way to fold it into three equal parts. Some students immediately find a way, others seek help from a peer, and a few require
FIGURE 9.1 Inheritance Word Problems
Samar’s assistance to complete the task. After making sure that all the students have made all the parts equal, she returns to the problem.
Samar: If the paper represents the capital, how much did the stranger get?
A chorus of students: One-third.
Samar: And what’s left if he gives away one-third?
A chorus of students: Two-thirds.
Samar directs the students to use colored pencils to shade in the parts of the paper that are left for the sons. After they color two of the three parts on their papers, she asks them how much they think each of the sons will get from the rest of his money, the remainder. Various answers are offered, but Samar keeps pressing her question. The students eventually realize that each of the sons will get one-half of what is left. She returns to the board and says as she writes, “Each son will get one-half of the money that is left, which is two-thirds, or M> of 2/з. Now let’s find out the answer.” Reminding the students that the part already given away is not a focus right now, she folds back the part of the paper representing the capital given to the stranger and holds up the 2/з portion. She asks, “Now how should we fold the paper so we can see how much each of the two sons will get?” Jimha holds up his paper and folds it in half lengthwise, prompting Samar to ask whether anyone has a different way Several students fold their papers in half widthwise. “Will the answer be the same if we fold the paper this way?” she asks. Once students see that the results are the same, she reassures them both ways are correct.
After everyone’s papers are folded, Samar directs the students to shade one of the halves a different color from the previous shading.
Samar: This will represent what one of the sons is getting. This part is half of the two-thirds.
Now open the whole paper to see what part of the original capital, what fraction of the father’s money, the son is getting. Before you give an answer, look at the whole paper to see how many parts you have in the whole.
While Samar counts aloud with the students, the class realizes that the whole has been cut into six parts.
Samar: So, how many parts is the son taking?
Using her model, Samar points to the two parts, shaded with two different colors that represent each son’s inheritance, and asks another question.
Samar: So what fraction is that?
A chorus of students: Two-sixths.
Samar: Turn your attention to the equation on the board. So one-half of two-thirds
Satisfied that her students are not guessing at the answer but have physical evidence of the amount, she writes, “12 of 2/з = 2/б” and says, “Now let’s do another one.”
Ahmed reads the second problem aloud (see Figure 9.1). Samar questions on the board: “What do we know?” As the students volunteer the information from the problem, Samar writes, “Daughter gets Vi ; mother gets 1/з of the remainder.” Then, as she instructs the students to take a new piece of paper, she asks, “How should we fold this piece?” Most of the students reply, “One-half.” Samar directs the students’ attention back to the problem and reads aloud the part that supports the choice of one-half. Encouraging her students, she says, “Now that fold is easy, right?” As some students fold the paper in half lengthwise, she asks whether there will be a difference if they fold it another way and says, “Ahmad, try folding it one way and Yassen, you fold it the other way.” Again students visually see the same results.
Samar: Next you should color in one of the halves so that you can see the part of the money the
daughter is not getting. You might want to write “daughter” on the other part so you remember that part that is given to her. The mother is getting one-third of what?
She presses the students until they tell her that the mother’s share is one-third of what is left after the daughter has gotten her part; the mother gets one-third of one-half. Writing “1/з of M> ,” she asks them what they are going to do.
Samar: How are you going to fold the half paper?
A chorus of students: Three.
Samar: How do you know?
Uri: Because the mother got one-third of the half that is left.
The students question which way they should fold next, and Samar again invites them to try both ways. She models a lengthwise fold because it is easier for most of the students to do accurately. She circulates around the room and reminds the students to look at the shaded part of the paper as they decide how much of the money will go to the mother. To the class, she says, “Now change your color and shade in the part the mother is getting. How many of the three parts will that be?” She is pleased by the response, “One part.” To make the connection between the problem on the board and the folded paper, she rereads “1/з of M> = ?” as she colors one piece on her model. She then asks, “Now what fraction is this double-shaded part? Let’s figure this out.”
What had been a very teacher-directed lesson to students sitting in straight rows takes off in a new direction as the students get up out of their seats, cluster together in little groups, fold and unfold as they chat, question each other, and explain their conjectures. Samar circulates among the groups, repeating the question, listening to the discussions, and giving hints where needed. Then she brings the class back to attention.
Samar: So what is one-third of one-half?
Chorus of students: One-sixth.
Samar: So the equation is, one-third of one-half equals one-sixth. And what is the
Samar directs the students to look at their folded, shaded papers once again as she asks them what is left when the daughter took her one-half and how much of this half the is father getting. Writing on the board, “2/з of M> = ?” she tells the students to shade the parts of the paper that represents the father’s share and write “father” on these parts. Unfolding her model paper, Samar asks the students to count along with her as she points to the parts of the paper that go to the daughter, to the mother, and to the father.
Samar: Does everybody see that the father is getting two-thirds of the half that didn’t go to the
daughter, or two-sixths of the whole inheritance?
Wadi: “The daughter takes the most, then the father, and then the mother.
Samar: Yes, maybe she was lucky, I guess, that she doesn’t have a sister or a brother. If she had a
sister or a brother, she would have to share her half.
Waad asks, “Can we do number three?” Pleased by this question, Samar reminds the students that, after the next problem, they will be developing an algorithm for solving such problems. Before moving on, she makes a list on the board of all of the equations derived from the first two problems:
Nadah reads the third problem (see Figure 9.1). Samar helps students construct the process for a solution similar to the other problems.
Samar notices that Nadia has been quietly folding and unfolding her paper and writing fractions in her notebook. Confident that the shy Nadia has figured out the solution, Samar calls her to the front of the room to explain what she has calculated.
Nadia, who frequently gets eclipsed by the more assertive classmates, timidly steps to the front of the class. Using her paper to demonstrate, she says:
Well, first I folded the paper in four equal parts because that’s what the denominator of three- fourths told me to do. Then I turned the paper the other way and folded it in three equal parts because we had to figure two-thirds for the next part of the problem. The denominator this time was three. Then I noticed I had 12 parts when I opened up the paper. So, now I’m figuring out the two-thirds of the three-fourths.
Surprising herself that she said so much in front of the room, Nadia stops abruptly and, holding her head high, walks to the back of the room. The girls give her a high-five as, smiling, she takes her seat.
Samar thanks Nadia and decides to repeat her steps aloud to the class, adding, “Go back to your whole paper and open your paper. I want you to tell me how many parts you have in your whole.” Some students count by inspection; a couple count each part to themselves. In unison she hears, “Twelve.” “Now, here is my next question, how many parts is the father getting out of the twelve.” Almost immediately, the response, “Six!” thunders through the room. To the list on the board, Samar writes, “2/з of % = 6/i2. ”
Samar instructs the students to look at the list of equations on the board (see Figure 9.2) and to justify how they came up with these answers. She tells them to take a few minutes to look at the fraction sentences.
Samar: Look at the results. What is happening? What is going on? We already know that we
have the answers because we folded the papers. We tried it and it works. Now, if I have a fraction of another fraction, and I don’t have a paper to fold, how can I find my answer?
FIGURE 9.2 Organizing Board Work for Discovery of Patterns
She pauses before going on and reads aloud the four equations on the board.
Samar: What’s going on?
Abdullah: One times two is two; two times three is six; one times one is one . . .” (Samar interrupts him.)
Samar: Let’s try what Abdullah is doing. He seems to be doing some kind of multiplication.
Waddi: He’s multiplying the numerators, and then he’s multiplying the bottom numbers.
Eshan: The denominators.
Pointing to the list of equations, Samar and the students continue reading where Abdullah left off.
She presses the students to tell her the algorithm.
Yasser: You trade the fractions. I mean, you switch the second one upside down. (He realizes the
error and corrects himself.) Oh. No. That’s division. Here you just multiply
Samar suggests that they go over Abdullah’s algorithm again and asks the students to explain
what the algorithm is.
Ayya: You multiply the numerators together, and you multiply the denominators together.
Those numbers are the answer as a fraction.
Samar: Very good, Ayya. That is the algorithm for multiplying fractions! But do you think it will
work all the time? Let’s pick some problems we have not done and see if that is true. If we use Abdullah’s algorithm for one-half of two-fourth, what should be the answer? (She writes “У2 of 2/” on the board.)
Samar then asks them to use paper folding to check it.
Waddi: That works.
Closing the lesson, Samar sums up what the students have accomplished that day and introduces the
next application using fractions.
Samar: Well, it looks like we do have a nice algorithm that can make our working with the
inheritance problems easier. Apply Abdullah’s algorithm to this problem. (She writes, “2/з of 3/5 =?”) What is the answer?
Hands shoot up all over the class. Samar calls on Nadah, who says, “Six-fifteenths.” Making sure the others know how Nadah got her answer, Samar invites Nadah to the board to point to the numbers she multiplied to get the numerator and denominator of the answer.
Samar: We can try one more. (She writes, 3/4 x 5/6 =?)
Magdi: That’s easy. That’s fifteen over twenty-four.
Samar: What is another name for Abdullah’s algorithm?
Chorus of students: Multiplication of fractions!
Samar: Today we figured out how to find the fraction that represents the part of the
inheritance that the mother or father or daughter is getting. What we will do next time is to find out the value of the fraction—how much money, for example—that fraction represents. So, if a person left behind $18, 000 and, from problem 2, the father is taking two-sixths of that, we will be figuring out what is two-sixths of $18,000.
Ayya: Can we do it?
Samar: OK. Let’s do that one. We’ll be using the same algorithm we discovered today.
(She writes on the board, “Son leaves $18,000. Father gets 2/б.”) How can we solve this problem? (She gets no response.) There really is a simpler way to solve this problem. Let’s start by using a simpler amount, $18, instead of the given, just so we can get a better sense of the problem. What are we now looking for?
Wadi: Two-sixths of eighteen.
Samar: We now have two-sixths of eighteen as our expression. (She writes, “2/б of 18"
on the board.) How can we find out what that is?
Wadi: We can divide by six.
Samar: OK. But let’s first see if we can use Abdullah’s algorithm to solve this equation.
(Samarpoints to the equation.) I want to write eighteen as a fraction, so my equation will have two fractions in it just like the ones we did earlier.
Several students: But eighteen is a whole number.
Samar: Yes, but if I want to make it into a fraction, I have to draw a line under it, like
this (She draws a line under “18.") What is the number I can put under 18 to make it into a fraction and not change its value?
Hearing many suggestions—“zero . . . six . . . eighteen . . . one . . . twenty”—and sensing her students’ confusion, Samar returns to the board and writes, “Vi.” After establishing that Vi means the same as 1, she writes, “2Л.” These examples help remind students that a whole number can be expressed as a fraction with a denominator of one and that the value of the whole number will not change.
Continuing, Samar goes back to the equation “V6 x 18/ = ?” and asks the students what they should do next. “We could multiply two times eighteen and six times one using Abdullah’s algorithm,” suggests Rushti. Jimha looks puzzled, so Samar goes back to the problems done earlier in the class and reminds him of how the class got those answers, first from the paper folding and then from Abdullah’s algorithm.
Mushan: The answer is 36 over six.
Nadia: And that’s really six.
Samar: So, if the father is getting six out of 18 dollars, how many dollars out of the
$18,000 is the father really getting?
Chorus of students: 6,000.
Before the lesson closes, one of the authors, who has been sitting in the back of the room with the girls, notices that Nadah has figured out another way to do the problem and alerts Samar that there’s another procedure to share with the class. Samar calls Nadah to the board. Nadah points to the equation: “52 of 2/з = 2/б” and explains: “The two-sixths is really one-third.” Acknowledging Nadah’s insight, Samar suggests that the class look again at the paper strip from the problem.
Samar: Remember how we counted up the parts that the father received—two out of six? Well,
look at the strip in a different way Can you really see it as three parts instead of six parts? (Nada refolds her paper model into three equal parts.) So we could do the problem using this fraction in lowest terms, too.
Samar ends the lesson by telling the students to write Abdullah’s algorithm in their math journals.