Level D Lesson 9
Multiplication and Division Equations
In lesson 9 the objective is, the student will determine the unknown whole number in a multiplication or division equation using the related multiplication or division fact.
The skills students should have in order to help them in this lesson include, multiplication and division facts 0 through 9 families; vocabulary for multiplication and division including, product, quotient, factor, divisor, and dividend. And use of properties in multiplication of whole numbers.
We will have three essential questions that will be guiding our lesson. Number 1, what is the relationship between multiplication and division? Number 2, how do I use multiplication to solve a division equation? And number 3, how do I use division to solve a multiplication equation?
The SOLVE problem for this lesson is, DI’s grandmother is making 6 pies for a county fair. She has 24 eggs in her refrigerator. She will use the same number of eggs for each pie. She will use all her eggs. How many eggs will Di’s grandmother need for each pie? Set up an equation to show your answer.
We will start by Studying the Problem. First we want to identify where the question is located within the problem, and we will underline the question. How many eggs will Di’s grandmother need for each pie? Now we want to take this question and put it in our own words in the form of a statement. This problem is asking me to find the number of eggs in each pie.
During this lesson we will learn how to solve multiplication and division equations using whole numbers. We will 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.
In the first activity in this lesson we will be investigating facts with arrays. Each pair of students will need a set of centimeter cubes and two colored pencils to complete this activity.
Let’s start by looking at the first example. Here we are given two grids and two facts. The fact for Grid A is 3 times 4. And the fact for Grid B is 4 times 3. We will use centimeter cubes to create an array on Grid A to show 3 times 4. This fact means 3 groups with 4 items in each group. Now we will use centimeter cubes to create an array on Grid B to show the fact 4 times 3. This fact means 4 groups with 3 items in each group. Let’s take our concrete representation using the centimeter cubes and turn it into a pictorial representation. We can do this by removing 1 cube at a time from Grid A and shading the array using a colored pencil. Next we will remove 1 cube at a time from Grid B and shade the array using a different colored pencil. The product for the number fact 3 times 4 on Grid A is 12. And the product for the number fact 4 times 3 on Grid B is 12. We know this because there are 12 squares shaded on each grid.
Now let’s take a look at Grids C and D. The fact for Grid C is 12 divided by 4. And the fact for Grid D is 12 divided by 3. We will use centimeter cubes to place 12 cubes on Grid C with 4 cubes in each of the first 3 rows. Now using the centimeter cubes we will place 12 cubes on Grid D with 3 cubes in each of the first 4 rows. Let’s take this concrete representation using centimeter cubes and move it to a pictorial representation by removing 1 cube at a time from Grid C and shading in each grid square using a colored pencil. Let’s do the same on Grid D removing 1 cube at a time and shading it using a different colored pencil. The fact 12 divided by 4 on Grid C means 12 total items with 4 items in each group. We will draw a box around each group of 4 items. There are 3 groups of 4 items shown on Grid C. On Grid D the fact 12 divided by 3 means 12 total items with 3 items in each group. We will draw a box around each group of 3 items. There are 4 groups of 3 items shown on Grid D. The quotient for Grid C is 3 as there are 12 total items with 4 items in 3 groups. The quotient for Grid D is 4 as thee are 12 total items with 3 items in 4 groups.
Let’s now go back and compare Grids A and B. What do we know about 3 times 4 and 4 times 3? They have the same product, or equal number of centimeter cubes, but the arrays differ. If we know that 3 times 4 equals 12 and 4 times 3 equals 12, then we know that 12 is 3 times as many as 4 and 12 is 4 times as many as 3.
Now let’s compare Grids C and D. What do we know about 12 divided by 4 and 12 divided by 3? They have the same number of total items, but the number of items circled differs.
Looking at Grids A, B, C and D, what three numbers do all the facts have in common? 3 times 4 equals 12, 4 times 3 equals 12, 12 divided by 4 equals 3, and 12 divided by 3 equals 4. The three numbers that all of these facts have in common are 3, 4, and 12. We call this a fact family. The numbers can be used as factors and a product to create two multiplication sentences. A dividend, divisor and quotient create two division number sentences.
The grids in this next example have already been shaded in for us. Grid A shows 4 groups with 5 items in each group. Grid B shows 5 groups with 4 items in each group. The number sentence for Grid A is 4 times 5 equals 20. And the number sentence for Grid B is 5 times 4 equals 20. What do we know about Grids A and B? They have the same product, or equal number of shaded squares, but the arrays differ. If we know that 4 times 5 equals 20 and 5 times 4 equals 20, then we know that 20 is 4 times as many as 5 and 20 is 5 times as many as 4.
Let’s look at Grids C and D. Grid C shows 20 total items with 5 items in each of 4 groups. Grid D shows 20 total items with 4 items in each of 5 groups. The number sentence for Grid C is 20 divided by 5 equals 4. And the number sentence for Grid D is 20 divided by 4 equals 5. What do we know about Grids C and D? They have the same product, or equal number of shaded squares, but the arrays differ.
What three numbers do all the facts have in common? 4, 5 and 20. The numbers 4, 5 and 20 create a fact family. As these numbers can be used as factors and a product to create two multiplication number sentences or these numbers can be used as a dividend, divisor and quotient to create two division number sentences.
In this next example only Grid A has been shaded for us. Let’s determine the fact for Grid A. We have 6 groups of 5 items in each group. The fact is 6 times 5. The total number of shaded squares in the array is 30. So our number sentence for Grid A is 6 times 5 equals 30. Thinking about the fact families we worked with in the previous activities and the pictures they created let’s determine another multiplication fact for the fact family of 5, 6, and 30. This fact is 5 times 6 equals 30. We will shade Grid B to show 5 groups with 6 items in each group. What do we know about Grids A and B? They have the same product, or equal number of shaded squares, but the arrays differ. If we know that 6 times 5 equals 30 and 5 times 6 equals 30, then we know that 30 is 6 times as many as 5 and 30 is 5 times as many as 6.
Now let’s look at Grids C and D and create a division number fact using the numbers 30, 5 and 6. 30 divided by 5 equals 6. We will shade the squares on Grid C to show this fact. We have shaded a total of 30 items with 5 items in each of 6 groups. Now we want to create another division fact using the numbers 30, 5 and 6. 30 divided by 6 equals 5. We will now shade and box the squares on Grid D to show this fact. We have shaded a total of 30 squares. There are 6 in each of 5 groups. What do we know about Grids C and D? They have the same number of total items, but the number of items circled differs. What three numbers do all the facts have in common? These facts represent the fact family of 5, 6 and 30.
In this next example we will start by determining the fact for Grid A and Grid B. Grid A shows 3 groups with 6 items in each group. The fact for Grid A is 3 times 6. Grid B shows 6 groups with 3 items in each group. The fact for Grid B is 6 times 3. Let’s use the two grids to determine the third number in this fact family. The third number in this fact family is 18. We know this because 3 times 6 equals 18.and 6 times 3 equals 18. Both Grids A and B have a total of 18 shaded squares. What numbers are in the fact family? 3, 6, and 18.
Now let’s look at problem 2. The letter n represents the divisor or one of the numbers in the number sentence. We do not know what number is represented by n at this point. So it is called an unknown value. Mathematicians call a number sentence with an unknown value and an equal symbol an equation. We need to think about how this problem relates to the fact family for the first set of grids. It is a division equation that has two of the numbers in the fact family of 3, 6 and 18. What are we trying to find in this equation? We are looking for the divisor or the number to be divided into the 18 to find the quotient of 3. We can use what we know about fact families to solve this equation. I can multiply to find the product of 18. I know that one of the factors is 3. I can multiply 3 times 6 to find the product of 18. Using multiplication, we can find the missing divisor. The missing divisor is 6, so n our unknown value equals 6.
Let’s look at problem 3 and solve it using the same procedure as we did in problem 2. Again we have an unknown value represented by the letter n. Mathematicians call a number sentence with an unknown value and an equal symbol an equation. This is a division equation that has two of the numbers in the fact family of 3, 6, and 18. The divisor or the number to be divided into the 18 to find the quotient of 6 is what we are looking for. I can multiply to find the product of 18. I know that one of the factors is 6. I can multiply 6 times 3 to find the product of 18. Therefore using multiplication, we can find the missing divisor. The missing divisor is 3, n equals 3.
Let’s look at example 4. The fact for Grid A is 12 divided by 6 equals 2. As we have 12 total items divided into groups of 6 items which creates 2 groups. The fact for Grid B is 12 divided by 2 equals 6. As we have 12 shaded items divided into groups of 2 items for a total of 6 groups. What numbers are in this fact family? The numbers are 2, 6, and 12.
In problem 5 we are given the multiplication equation 2 times n equals 12. n is our unknown value. It represents one of the factors. It is the factor multiplied by 2 that will give you a product of 12. To find out what this fact is I can divide 12 by 2 to find the missing factor. 12 divided by 2 equals 6, so our missing factor or n equals 6.
We can use the same procedure to solve problem 6. 6 times n equals 12. N is our unknown value. We are looking for the factor multiplied by 6 that will give you a product of 12. I can divide 12 by 6 to find the missing factor. 12 divided by 6 equals 2. Our missing factor or n equals 2.
For problems 2 and 3 18 divided by n equals 6 and 18 divided by n equals 3. How do you solve for the unknown value? Using multiplication we can find the missing divisor. Multiplication is the opposite of division.
For problems 5 and 6 6 times n equals 12 and 2 times n equals 12. How do you solve for the unknown? Using division we can find the missing factor. Division is the opposite of multiplication.
So we can conclude that in order to solve for a missing unknown in a multiplication equation we can break it apart and use division to solve. We can also conclude that in order solve for a missing unknown in a division equation we can break it apart and use multiplication to solve.
Let’s apply what we have just learned to a couple of problems. The first problem is n divided by 8 equals 6. We know that there are three numbers in a fact family. In this division equation we are looking for the dividend. The numbers that we already know in this fact family are 8 and 6. The operation that is being used in this equation is division. The opposite operation is multiplication. We will use the opposite operation to solve for the unknown value. 6 times 8 equals 48. 48 is the unknown value. Let’s test the equation by placing the number 48 in as the dividend in the problem. 48 divided by 8 equals 6, so we have found the unknown value.
Now let’s look at the problem 4 times n equals 36. What is missing in the equation is a factor. We know that there are three numbers in a fact family. And we already know two of the numbers in this fact family. They are 4 and 36. The operation used in this equation is multiplication. The opposite operation is used to find the unknown value in an equation. The opposite operation of multiplication is division. We can determine the third number in this fact family by dividing our known numbers. 36 divided by 4 equals 9. So our missing factor for the equation is 9. Let’s test the equation. 4 times 9 equals 36. So we have found the correct unknown value.
Now let’s record the steps to solve an equation. First, we identify what the question is asking me to find. Second, identify the numbers I already know. Third, identify the operation. Fourth, we identify the opposite operation. Fifth, we perform the operation. And finally, we rewrite the equation filling in the unknown value.
Let’s apply these steps to one more example. We will complete this example without the graphic organizer, n divided by 9 equals 8. We need to find the total number of items. That is our dividend. The numbers that are known are 9 and 8. The operation in this equation is division. The opposite operation is multiplication. We will take our known numbers and multiply them together. 9 times 8 equals 72. So our unknown number or our dividend is 72. Let’s place 72 into our equation as the dividend. 72 divided by 9 equals 8.
We are now going to go back to the SOLVE problem from the beginning of the lesson. Di’s grandmother is make 6 pies for a county fair. She has 24 eggs in her refrigerator. She will use the same number of eggs for each pie. She will use all her eggs. How many eggs will Di’s grandmother need for each pie? Set up an equation to show your answer.
In Step S, we Studied the Problem. We underlined the question. How many eggs will Di’s grandmother need for each pie? And then we put this question in our own words in the form of a statement. This problem is asking me to find the number of eggs in each pie.
In Step O, we will Organize the Facts. We will start by identifying the facts. Di’s grandmother is making 6 pies for a county fair, fact. She has 24 eggs, fact in her refrigerator, fact. She will use the same number of eggs for each pie, fact. She will use all her eggs, fact. How many eggs will Di’s grandmother need for each pie? Set up an equation to show your answer. Now we will eliminate the unnecessary facts or those facts that do not help us to find the number of eggs in each pie. Di’s grandmother is making 6 pies for a county fair. We need to know how many pies she is making as she is going to divide the eggs among those pies. So let’s keep that fact. She has 24 eggs. This is also necessary in finding out the number of eggs in each pie, in her refrigerator. Knowing where the eggs are is not going to help us to solve this problem so we will eliminate this fact. She will use the same number of eggs for each pie. This is going to be important. And knowing that she will use all of the eggs is also going to be important. So we will keep these facts. Now let’s list the necessary facts. 24 eggs, 6 pies, same number of eggs in each pie.
In Step L, we will Line up a Plan. First, we need to choose an operation or operations to help us to solve the problem. We will need to use multiplication to solve this problem. Next we will write in words what your plan of action will be. Set up an equation showing the total number of eggs as the product and the number of pies as one factor.
In Step V, we Verify Your Plan with Action. First estimate your answer. We can estimate about 3 or 4 eggs will be in each pie. Next we will carry out your plan. Our plan was to set up an equation showing the total number of eggs which is 24 as the product and the number of pies as one factor. So we will use n to represent one of our factors, n times 6 equals 24. In order to solve for our missing number we can use the opposite operation of multiplication. We will divide 24 by 6. 24 divided by 6 equals 4. We will place 4 as our unknown factor in our original equation. 4 times 6 equals 24. So we have found that there will be 4 eggs in each pie.
In Step E, we Examine Your Results. First, we ask ourselves, does your answer make sense? Here we’re comparing your answer to the question. Yes, the answer makes sense, because I am looking for the number of eggs in each pie. Next, is your answer reasonable? Here you want to compare your answer to the estimate. Yes, the answer is reasonable, because it is close to my estimate of 3 or 4 eggs. And our third questions is, is your answer accurate? Here you want to check your work. Double check to see that when you divide 24 by 6 you get 4. You can also double check by solving the original equation. 4 times 6 equals 24. So yes, our answer is accurate. Finally, write your answer in a complete sentence. There are 4 eggs in each pie.