Level D Lesson 7
Multiply Multi-digit Whole Numbers with Property Application
In lesson 7 the objective is, the student will multiply a whole number up to four digits by a one-digit whole number, multiply two two-digit numbers, and use strategies based on place value and the properties of operations.
The skills students should have in order to help them in this lesson include, multiplication facts for tables 0-9, and single-digit by single-digit multiplication using properties.
We will have three essential questions that will be guiding our lesson. Number 1, how can I use different strategies to multiply a four-digit number by a one-digit number? Number 2, how can I use different strategies to multiply two two-digit numbers? And number 3, can I use a model to show multiplication of multi-digit number?
The SOLVE problem for this lesson is, Fred has 4 problems for homework. The first problem is 32 times 23. Fred’s teacher wants the class to use the distributive property to show their work. Using the distributive property, how can Fred find the product?
We will begin by Studying the Problem. First, we want to identify where the question is located within the problem and underline the question. Using the distributive property, how can Fred find the product? Next, we will take this question and put it into our own words in the form of a statement. This problem is asking me to find the product of 32 times 23 using the distributive property.
During the lesson we will learn how to multiply 4-digit numbers by 1-digit numbers and 2-digit numbers by 2-digit numbers using the properties of the operations. 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.
We will begin this lesson by building multiplication arrays. Each pair of students will need a bag of centimeter cubes to help them with this activity. In problem 1 we are multiplying 6 by 12. What is problem 1 asking me to find? It’s asking me to find 6 groups of 12 items. Let’s first identify the factors. The factors are 6 and 12. 12 is the larger factor. Let’s see how we can break apart the factor of 12 into two addends that will be easier to multiply using tens and ones. The number 12 if we break it apart into tens and ones would give us the factors of 2 and 10. So to create two arrays the factors for the two arrays would be 6 times 2 and 6 times 10. We will first build a multiplication array using the centimeter cubes for 6 times 2. We have 6 groups with 2 items in each group. So we will place 6 groups of 2 centimeter cubes on our grid. Now let’s change our concrete representation into a pictorial representation by removing our centimeter cubes one at a time and shading in the space that was occupied by that centimeter cube with our colored pencil. We will label this array with the letter A. The product of the array is 12, because we have a total of 12 shaded squares. Next we want to show a multiplication array using centimeter cubes for 6 times 10. We will have 6 groups with 10 items in each group. Using our centimeter cubes we will place 6 groups of 10 centimeter cubes on the grid. Now let’s take our concrete representation of 6 groups of 10 and turn it into a pictorial representation by removing our centimeter cubes one at a time and shading in the space that is occupied by that centimeter cube. We will use a different color colored pencil to shade in our second fact on the grid. We will label our second array with the letter B. The product of 6 times 10 is 60. Because there are 60 shaded squares in this array. We can determine the product of 6 times 12 using the arrays by adding the two arrays together. First let’s draw a box around each of the arrays and we will label the fact for each of our arrays. The fact for array a is 6 times 2 equals 12. We have labeled the side of this array with the number 6 to represent the number of groups and the top of this array with the number 2 to represent the number of items in each group. Now let’s look at the fact for array B. This fact is 6 times 10 equals 60. We had already labeled our 6 on the side of the array to represent the number of groups, and we have labeled the top of our second array with the number 10 to represent the number of items in each group. This model to show multiplication is called an Area Model, because it covers area or space within the arrays.
Now let’s examine the box that is below the grid for this example. It is different from the grid because it is blank or open and has no grid lines. Let’s label each of the arrays that are contained within this box. The first array is the one on the left hand side that is smaller. This represents array A that we completed with our centimeter cubes and then shaded in. The larger of these boxes on the right hand side represents array B. The fact for array A was 6 times 2. So we will label the values of 6 and 2 on the outside of the array. The fact for array A was 6 times 2 equals 12. Array B represented the fact 6 times 10. We already have a 6 on the outside of our array and we will place a 10 at the top of the array, and label this array 6 times 10 equals 60. This is an Open Array. It shows the same information as the arrays on the grid. It is also an Area Model like our grid was because it covers area or space inside a picture or a model. Let’s add the arrays together to get the product of 6 and 12. Our first array had the fact 6 times 2 and our second array had the fact 6 times 10. We are going to add these two together. First we need to find the product of 6 times 2 and 6 times 10. 6 times 2 is 12 and 6 times 10 is 60. Now we will add the products together. The product of 6 and 12 is 72.
In this next example we are asked to find the product of 3 and 161 or another way of saying this is that we are looking for 3 groups of 161 items. Our factors are 3 and 161. How can 161 be broken down into 3 parts to make it easier to multiply using hundreds, tens and ones? If we break down the number 161 into hundreds we have 1 hundred, and into tens we have 6 tens or 60, and into ones we have 1 one. So the factors are 3, 1, 60, and 100. We can create facts using the factors by multiplying 3 by each of the factors of 161. Our factors distributed are 3 times 1, 3 times 60, and 3 times 100. Let’s look at the open arrays in the Area Model below. The first array represents 3 times 1. The product of 3 times 1 is 3. Our second array represents the fact 3 times 60, 3 times 60 equals 180. And our third array represents the fact 3 times 100, 3 times 100 is 300. How can you determine the product using the area model? Add the product of the three open arrays together. Let’s do that now. We had 3 times 1 equals 3, 3 times 60 equals 180, and 3 times 100 equals 300. We want to add our products together. 3 plus 180 plus 300 equals 483.
We are going to continue building facts with Open Arrays. Our next example asks us to find the product of 13 and 16. Notice that with this problem we are multiplying a two-digit number by another two-digit number. This is different from the example we completed previously, where we multiplied a one-digit number by a two-digit number, and a one-digit number by a three-digit number. Let’s look at the problem and find its factors. The factors in this problem are 13 and 16. Both 13 and 16 can be written as two addends to make it easier to multiply using the tens and ones. These addends for the number 13 are 10 and 3. These addends for the number 16 are 10 and 6. So the factors are 3, 6, 10, and 10. Let’s create facts using the factors. We will multiply 3 times 6 and 3 times 10 and we will multiply 10 times 6 and 10 times 10. Now let’s look at the open arrays in the area model and identify the array for each of our facts. The first array is going to represent our fact 3 times 6. 3 times 6 is 18. Our fact 3 times 10 will be the array next to the one that represents 3 times 6. Since we already have the 3 labeled on the outside of our area model 3 times 10 equals 30. Now let’s identify the array for 10 times 6. Notice that our 6 is also already labeled on the outside of our area model. We will represent 10 times 6 in the bottom left array on our area model. 10 times 6 is 60. Our final array will represent the fact 10 times 10. 10 times 10 is 100. How can you determine the product using the area model? We can add the products of the four open arrays together. Our four open arrays were 3 times 6, 3 times 10, 10 times 6, and 10 times 10. 3 times 6 is 18 plus 3 times 10 is 30 plus 10 times 6 is 60 plus 10 times 10 is 100. So we add these products together we get 208. So 13 times 16 equals 208.
In this next example we are working with the problem 2 times 2,124. The area model for A means 2 groups of 2,124 items. The fact for this area model is 2 times 2,124. 2 times 2,124 equals 4,248. On grid B we are going to create another area model for this example, this time breaking down the factors of 2,124 into ones, tens, hundreds, and thousands. There are 2 groups of 4 plus 2 groups of 20 plus 2 groups of 100 plus 2 groups of 2,000. Our facts for each of these arrays are 2 times 4, 2 times 20, 2 times 100, and 2 times 2,000. We want to add these facts together. We will use parenthesis to separate each of the facts from one another. The quantity 2 times 4 plus the quantity 2 times 20 plus the quantity 2 times 100 plus the quantity 2 times 2,000 equals 4,248. So we can conclude that both sets of the arrays are equal. Both have the same product but B is separated into four different open arrays, while A is one open array. Therefore if the products are the same then the facts can be written as equal to each other. 2 times 2,124 is equal to the quantity 2 times 4 plus the quantity 2 times 20 plus the quantity 2 times 100 plus the quantity 2 times 2,000. This problem models the Distributive Property as the 2 is distributed over each of the place values.
In this next example we want to find the product of 22 times 13. The open array for A shows 22 groups of 13 items. The fact is 22 times 13. 22 times 13 equals 286. When looking at the area model B, we can see that this area model has been broken up into four separate arrays. We will break apart the factors of 22 and 13 into ones and tens on this area model. There are 2 groups of 3 plus 2 groups of 10 plus 20 groups of 3 plus 20 groups of 10. We have recorded the number fact on each array. We will now add these number facts together. The quantity 2 times 3 plus the quantity 2 times 10 plus the quantity 20 times 3 plus the quantity 20 times 10 equals 286. So we can conclude that both sets of arrays are equal. They both have the same product but B is separated into four open arrays while A shows one open array. If the products are the same then the facts can be written as equal to each other. 22 times 13 is equal to the quantity 2 times 3 plus the quantity 2 times 10 plus the quantity 20 times 3 plus the quantity 20 times 10. This problem models the Distributive Property as 22 is distributed over each of the place values of 13. 22 is distributed by being broken apart into ones and tens.
In this next example the sets of arrays in the area models have already been labeled for us. We need to identify the fact for each of these arrays. On area model A the fact is 3 times 2,213. On area model B we will have four facts. As our first factor 3 is distributed over each of the place values of our second factor 2,213. Let’s label each of the arrays for grid B. The first array represents the fact 3 times 3. The second array represents the fact 3 times 10. The third array represents the fact 3 times 200. And the fourth array represents the fact 3 times 2,000. Let’s take area model B and complete the section Way to Multiply. We are going to draw an arrow to each of our facts. We had four facts. So we will draw one line to represent each one of those facts and we will record the facts below each of those lines. The quantity 3 times 3 plus the quantity 3 times 10 plus the quantity 3 times 200 plus the quantity 3 times 2,000. Next we need to find the product of each of these facts. 3 times 3 is 9, 3 times 10 is 30, 3 times 200 is 600, and 3 times 2,000 is 6,000. We want to add these products together. When we add the product of each fact together we get our final product 6,639. Let’s take a look at the Way to Multiply section for model A. Here we have two factors, so we will draw 2 arrows to the factors of 3 and 2,213. The product of 3 and 2,213 is 6,639. This is the same product we got when we completed the way to multiply section for area model B. So we can say that 3 times 2,213 is equal to the quantity 3 times 3 plus the quantity 3 times 10 plus the quantity 3 times 200 plus the quantity 3 times 2,000. 3 times 2,213 equals 6,639.
In this next example the sets of arrays and the area models have again already been labeled for us. We need to identify the fact for model A first. We have array A’s fact is 34 times 26 or 34 groups of 26 items. Now let’s identify array B’s fact. We have 4 times 6 plus 4 times 20 plus 30 times 6 plus 30 times 20. We will find the product for each of our area models by using the Way to Multiply section. First let’s look at our area model B. There were four separate facts, 4 times 6 plus 4 times 20 plus 30 times 6 plus 30 times 20. Remember that we use parentheses to separate our facts from each other. Let’s find the product of each fact. 4 times 6 is 24 plus 4 times 20 is 80 plus 30 times 6 is 180 plus 30 times 20 is 600. We need to add together the products of these facts to find the final product for our area model. 24 plus 80 plus 180 plus 600 equals 884. Now let’s complete the way to multiply for area model A. Since we have two factors we will draw 2 arrows. We are multiplying 34 by 26. We can use a calculator if available to help us in checking our answer. The product is 884. So let’s summarize what we have completed with these area models. 34 times 26 is equal to the quantity 4 times 6 plus the quantity 4 times 20 plus the quantity 30 times 6 plus the quantity 30 times 20 and the quantity 4 times 6 plus the quantity 4 times 20 plus the quantity 30 times 6 plus the quantity 30 times 20 is equals to 884, then 34 times 26 is equal to 884.