Level DAppendix B
Fact Masters Division
In the Fact Masters lesson on division the objective is, the student will develop the concepts and skills for mastery of division facts 9 by 9.
The skills students should have in order to help them in this lesson include knowledge of the numbers 0 through 100.
We will have three essential questions that will be guiding our lesson. Number 1, how can division facts be modeled using manipulatives? Number 2, what techniques can be used to practice division facts? And number 3, why is it important to be fluent in division facts?
The SOLVE problem for this lesson is, Mario collects football cards. He has 72 cards in his collection. He purchases 1 or 2 cards every time he receives his allowance. He wants to share his collection equally with 7 of his friends. How many football cards will Mario and his friends each have?
Right now we will be completing Step S, Study the Problem. We will start by underlining the question. How many football cards will Mario and his friends each have? Next we want to take this question and put it in the form of a statement in our own words. This problem is asking me to find the number of football cards Mario and his friends will each have.
During this lesson we will learn how to model and practice division facts for mastery. We will then use this knowledge to complete this SOLVE problem at the end of the lesson.
Throughout this lesson students will be working together in cooperative pairs. All students should know their role as either partner A, or partner B before beginning this lesson. For this lesson pairs of students will need to have a set of 81 beans and a full set of the TI, I cards seen here. Students will use these cards to identify the total items and how man items are in each group. Students will find how many groups they can create by dividing by the items given. Each pair of students will also need a set of the 9 cards that are seen here. Each card represents a group. When we split the total items by the number of items, we can find how many groups we have created. The TI, I cards each contain two numbers. The first number is the TI number, or total items, this is also known as the dividend. The I number represents the number of items in each group. This is also called the divisor in a division problem. We will use this card to model our first example.
On recording sheet 1, we will begin by labeling each of the numbers that we will be using. Our first number represents our total items or the dividend. This tells us how many total items we are starting with. The second number represents our items. This tells us how many items will be in each group. The card that we were using has our total number of items as 12, and our number of items in each group as 4. Let’s model this example using our beans and our cards. Partner A will be in charge of the beans. And partner B will be in charge of the cards. The total number of items is 12. So we represent this with 12 beans. The number 4 under the I for items, tells us how many items are in each group. Let’s split our 12 total items into groups of 4 items. On each card we will place 4 of the beans from our group of 12 beans. We have a total of 3 groups. Back on our recording sheet in the last box we will place a 3 to represent the 3 groups. Our answer is also called the quotient in a division problem. 12 total items divided into groups of 4 items, equals a total of 3 groups.
On recording sheet 2, you will notice that we have replaced our boxes with blanks and words. We have blank total items, with blank items in each group, and our answer is the number of groups that we can make. Our first number represents our total items, so we will place a TI over the first number. Remember that this is also called the dividend. Our second number represents our items, or how many items are in each group. Remember that the second number is also called the divisor. We will use the card where the total number of items is 15, and the items is 5. Let’s model this example together. The total number of items is 15. We will represent this with 15 beans. The number 5 under I for items, tells us how many items are in each group. Let’s split our 15 total items into groups of 5 items. Each card will contain 5 beans. Here’s 1 group of 5 items, 2 groups of 5 items, 3 groups of 5 items. There are total of 3 groups. To record our answer we have a total of 3 groups. Our quotient is 3.
For our next example we will use the card that our total items is 24 and our items is 6. Let’s model this together. Since our total items is 24, we will start with 24 beans. Our number of items in each group is 6. So we will move 6 beans into each group. Here’s 1 group of 6 beans, 2 groups of 6 beans, 3 groups of 6 beans, and 4 groups of 6 beans. So we have a total of 4 groups that we can make using 24 beans with 6 beans in each group. We will record our answer as 4 groups. As the number 4 represents the number of groups that we were able to make. Remember that our answer is also called the quotient in a division problem.
In this next example we have replaced some of our words with the division symbol. We’re looking for blank total items divided by blank items equals the number of groups. Our first number still represents our total items, so we place a TI over the first column. Our second number represents our items, so we will place an I over the second column. In this example we will use the card where our total items is 36 and our items in each group is 6. We will model this example by starting with 36 beans. Since we want to have 6 beans in each group, let’s circle groups of 6 beans. Each circle will represent 1 group, as our card represented the groups in previous examples. We have 1 group of 6 items, 2 groups of 6 items, 3 groups of 6 items, 4 groups of 6 items, 5 groups of 6 items, and 6 groups of 6 items. So we have a total of 6 groups. When we take 36 total items and divide them into groups of 6 items, we end up with 6 total groups.
Now that we know what each of our numbers in a division problem represent, we have removed all of the words from our example, and replace them with just the division symbol and an equal sign. We still have the total items as our first number, and our items in each group as our second number. Remember that our total items is also called the dividend, and our items in each group is also called the divisor. In this example we will use the card where our total items is 16, and our items in each group will be 2. We will start with 16 beans. We want to separate these beans into groups of 2 beans. Now that we have separated our beans into groups of 2 beans, let’s count how many groups we have, 1, 2, 3, 4, 5, 6, 7, 8. There are 8 groups of 2 in 16. 16 total items divided 2 items in each group equals 8 groups. Or 16 divided by 2 equals 8.
In this next example we are going to use our beans and a grid to form a picture. Let’s create a model of 12 divided by 2. We will start with our 12 beans, placing each bean in a separate gridded square. Next we want to divide our beans into groups of 2 by circling them. Now let’s count the number of groups of 2 that we have made 1, 2, 3, 4, 5, 6. We have created 6 groups of beans or a quotient of 6. Since we do not always have access to our beans, we want to have another way of representing this example. We will do so by drawing a picture of the example. In place of our beans we are going to shade in each grid section that represents where a bean would have been. Here we will shade in 12 grid sections. Next we will circle our groups of 2 items. Now let’s count the number of groups that we have created, 1, 2, 3, 4, 5, 6. There are 6 groups of 2 in 12. Our quotient is the number of groups we have made or 6. We can read this problem as 12 items divided into groups of 2 or 12 divided by 2 equals 6.
We want to place this example on a gridded index card. This card will be published to our Fact Master Curtain so that we can reference it later on as needed for division problems. The example is 12 items divided into groups of 2. Pictorially we draw this shading in 12 squares on our grid. We then divided our 12 squares or 12 items into groups of 2 by circling each group of 2 items. This gave us a total of 6 groups or our quotient of 6. We then write our example as 12 divided 2 equals 6. When we posted this example on our Fact Master Curtain we want to find the divisor, which in this problem is 2, along the numbers that are on the left hand side of the curtain. Once we are at 2 we’re going to move across the curtain until we get to the quotient along the top side of the curtain. The quotient in this example is 6. Where the 2 from the left hand side of the curtain and the 6 from the top of the curtain meets is where we will place the card 12 divided by 2 equals 6. As this card is placed on the square that represents the dividend for this problem. Teachers be sure to check the accuracy of each card before it is published to the Fact Master Curtain. Empty squares that are left over after each student has published their example can be completed as a centers activity, when there’s extra time available in class, or as a take home assignment. We have now completed the concrete, visual, pictorial, and abstract stages of division.
Now we are going to go back to the SOLVE problem from the beginning of the lesson. Mario collects football cards. He has 72 cards in his collection. He purchases 1 or 2 cards every time he receives his allowance. He wants to share his collection equally with 7 of his friends. How many football cards will Mario and his friends each have?
In Step S at the beginning of the lesson we Studied the Problem. We started by underlining the question. How many football cards will Mario and his friends each have? We then put this question into the form of a statement using our own words. This problem is asking me to find the number of football cards Mario and his friends will each have.
In Step O, we will Organize the Facts. We will start by identifying the facts. Mario collects football cards, fact. He has 72 cards in his collection, fact. He purchases 1 or 2 cards every time he receives his allowance, fact. He wants to share his collection equally with 7 of his friends, fact. How many football cards will Mario and his friends each have? Next we will eliminate the unnecessary facts. We will refer back to Step S to remind us of what the problem is asking us to find. We’re asked to find the number of football cards Mario and his friends will each have. So knowing that he collects football cards is not going to help us to find out the number of cards they each have, so we will eliminate this fact. He has 72 cards in his collection. Knowing the number of cards in his collection is going to help us to solve this problem, so we will keep this fact. He purchases 1 or 2 cards every time he receives his allowance. Knowing when he purchases the cards is not going to help us to find out the number of cards each friend has. So we will cross out this fact as well. He wants to share his collection equally with 7 of his friends. Knowing how many people he is going to share these cards with is going to be important. So we will keep this fact. Then we list the necessary facts, 72 football cards, Mario and 7 friends.
In Step L, we Line up a Plan. We start by choosing an operation or operations that are going to help us. Looking at Step O, we have 72 football cards, and we know that Mario is going to be sharing them with his 7 friends. So we need to add to find out how many people all together the cards will be split up by, knowing that it is Mario and his 7 friends. We also know that we will be dividing the cards among Mario and these 7 friends. So we will need to add and divide. Our operations will be addition and division. Next, we want to write in words what your plan of action will be. First, we will add Mario and his friends to find the total number of people receiving cards. Then we will divide the number of football cards by the total number of people. Remember in Step L, that we are not allowed to use any numbers.
In Step V, we will Verify Our Plan with Action. We will start by estimating your answer. Since there are 72 cards and we’re splitting them up among Mario and his 7 friends, we know the answer is going to be less than 10 cards to each friend and Mario together, so let’s estimate about 8 cards. Next we will carry out your plan. The first step of our plan was to add Mario and his friend to find the total number of people receiving the cards. Mario is 1 person plus his 7 friends will give us the sum or the total number of people who will be receiving cards. 1 plus 7 equals 8. Next, we will divide the number of football cards, which is 72, by the total number of people which was our sum, to give us the quotient, or the total number of cards that each of the friends and Mario will receive. We found that the sum was 8. So we’re going to divide 72 by 8. 72 divided by 8 equals 9 cards.
Now let’s Examine Your Results. First we check to see if our answer makes sense, by comparing it to the question. We found that Mario and his friends each received 9 cards. Since we were looking for the number of football cards Mario and his friends will each have we can say, yes are answer makes sense, because we are looking for how many football cards Mario and his friends will each have. Next, is your answer reasonable? Here we compare our answer to the estimate. Our estimate was about 8 cards. And our answer was 9 cards. So yes, our answer is reasonable, because it is close to the estimate of about 8 cards. Last, we want to check if your answer is accurate? Here you want to check your work. You can use a separate area of your paper, a calculator or discuss this with a partner. First we had to add 1 plus 7 which gave us 8, and then divide our cards, which was 72 by the 8 total people who would be receiving them. 72 divided by 8 equals 9. So our answer is yes, our answer is accurate. Finally we write your answer in a complete sentence. Mario and his friends will each have 9 football cards.
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, how can division facts be modeled using manipulatives? Building pictures, showing how items are divided into groups of items to read a fact correctly.
Number 2, what techniques can be used to practice division facts? Building the facts using manipulatives.
And number 3, why is it important to be fluent in division facts? To help when solving problems, especially word problems that involve division facts.
Now that we have completed the concrete, pictorial or visual, and abstract forms of division, we move into our daily practice drill. The daily practice drill is done in 2 to 3 minutes per day, 4 times per week. There are a total of 13 practice drills that are provided for you in the teacher’s edition. Each practice drill from days 1 through 13 gets progressively more challenging. Once you reach day 13, you will return back to day 1 and start over. Each drill can be completed using the CD or DVD that is provided with the teacher materials. This activity can also be student lead or teacher lead. Make it fun by having students create their own beat to say the facts to. As this drill is done 4 days out of the week, the 5th day of the week is provided for you to do the division fact quiz. Quizzes should be given once a week. It is recommended that the quizzes be given a Wednesday if possible. There are 2 forms of the quiz. There’s a quiz A and a quiz B which you can alternate between week to week. Whatever day and time you choose to give the Fact Master Quiz, you want to give it at the same time on the same day each week. Students should be provided with 2 1 / 2 minutes to complete the quiz. Coach students to complete the facts that they know first, then return to the facts they are unsure of or do not know. You will then review the quiz together having students highlight or circle incorrect items on their own quiz. These quizzes are not intended for a grade. Have students see how they are progressing over time, by shading in the number of questions that they answered correctly on each quiz. For those problems that student have more difficultly with, they can create a Fact Master Ring, holding up to 12 problems their struggling with at a time. These facts are each placed on their own index card, and are attached together with a paperclip so that students can take them with them and practice anytime. The front of their index card contains the facts without the answer. Students should say the problem 3 times without the answer. 48 divided by 8, 48 divided by 8 48 divided by 8. Then students will flip the card over for the answer. And say the problem 3 more times with the answer. 48 divided by 8 equals 6, 48 divided by 8 equals 6, 48 divided by 8 equals 6. Once the student has mastered this example on their Fact Master Quiz, they can take this fact off of their Fact Master Ring and replace it with another fact that they are struggling with. Remember, that students should not have any more than 10 facts on their Fact Master Ring at any one time. Once a fact is mastered have students shade in the facts that they have mastered on the Grid for Basic Division Facts. Once shaded in this fact can be asked by the teacher to that student at anytime and the student should be able to give the answer to that fact automatically. For students who complete a certain percent of mastery on their quizzes there is a Fact Masters Club. The teacher chooses to what percent of mastery students must achieve before they can become a member of the club. For instance a teacher may choose to have 90 percent accuracy on their fact master quiz for 2 weeks in a row. Once students receive their Fact Master Certificate each week that they complete the same percent of mastery or above a sticker, smiley face, or star or something of that nature can be added to the certificate to show their continued success. Once students receive 100 percent accuracy challenge students with the time that it takes for them to complete the quiz with 100 percent accuracy continually building so that students are completing the entire quiz with 100 percent accuracy in less and less time.