2-Digit Array Multiplication

Lesson Content

• Lesson Plan Template:

  General Lesson Plan

• Learning Objectives: What should students know and be able to do as a result of this lesson?

  
  

  Students will understand the process for multiplying a 2-digit number times a 2-digit number. They will show this understanding by illustrating and explaining the strategies used to find an answer for a 2-digit x 2-digit multiplication problem by working from arrays.

• Prior Knowledge: What prior knowledge should students have for this lesson?

  
  
  1. Student should be able to represent the value of digits in 2-digit numbers.
  2. Students should be able to decompose numbers in multiple ways, but certainly by place value.
  3. Students should understand and be able to explain why the Distributive property works when multiplying single digit numbers. (MAFS.3.OA.2.5)
  4. The students should be able to apply conceptual understanding of a single digit x a double digit multiplication problem (array models for 4 groups of 12).
  5. Students should be able to fluently multiply one-digit whole numbers by multiples of 10 in the range 10-90. (MAFS.3.NBT.1.3) This is an essential skill. For example, 4 groups of 2 tens (20) is 8 tens or 80.

• Guiding Questions: What are the guiding questions for this lesson?

  
  
  • What looks familiar in this problem? What do you already know that will help you?
  • How many groups are there? How many are in each group?
  • Why does this answer make (not make) sense?
  • What action is happening?
  • How can you decompose this number by place value?
  • Does your array make a rectangle if you slide all the pieces together? (It should. All arrays make rectangles if done properly.)
  • What does this part of the array represent? How do you know?
  • Why do you add, not multiply, the partial products?
  • What is the same about these arrays? What is different?

• Teaching Phase: How will the teacher present the concept or skill to students?

  
  

  The use of place value manipulatives are essential for building understanding. Although I have used the ActiveBoard to show the arrays, I prefer to use magnetic place value strips and my regular board. In this way, we can represent the first problem on the board then create the arrays for more complex numbers but leave each array on the board. The children can then see the connections between the simpler problems and the more complex ones.

  Once you get to the Lakeshore site type in magnetic place value.

  Students should have large individual white boards with an array outlined with painter's tape and a bank of place value blocks.

  If white boards are not available, create the arrays on the students' desks. This can be seen in a later picture of the array.

  Give the My Favorite No, Formative Assessment. See the Formative Assessment phase for further directions.

  1. Give each student an index card and ask them to individually answer the following problem. Create an array for 4 x 12 and then show the math sentences that could be used to solve it.
  2. Each student responds on an index card.
  3. Quickly collect the cards and sort by Yes (correct) and No (incorrect) answers. Don't let the students see how others did but announce each yes and no. Select an incorrect response that can promote a strong discussion about the problem.
  4. Copy the problem on the board or on another card under the document camera.
  5. Ask the children to find what is right about what this person did. Discuss.
  6. Ask the children what did this person do incorrectly? Discuss. Both discussions should form a review of the meaning of multiplication, as well as give you further information on who may struggle with this problem.

  My Favorite No Example
  
  Ask students for an example of when they might need to multiply two 2-digit numbers? (I need 134 apples for a school event. I have 11 bags of apples. Each bag has 12 apples in it. Do I have enough apples?)

  Tell students that today they will multiply two 2-digit numbers using arrays. Explain to the students how you want them to understand how math works, as well as be able to do it. For that reason, we are going to go nice and easy, nice and slow, and build understanding of multiplying double-digit numbers. They will do that by creating representations and explaining their solution strategies.

  1. Build from the Formative Assessment problem. To increase engagement, use the names of students in your class for these simple word problems. Say, "Pete bought 14 bags of goldfish with 12 goldfish in each bag. How many goldfish did Pete buy? Ask, "What are we trying to find out? (the amount of goldfish Pete bought) What information in the story will help us find that out? (14 bags and 12 goldfish in each bag) What operation should we use? (multiplication, 14 groups of 12)
  2. Tell the students to use what they know about arrays and figure out how to build an array to represent 14 groups of 12.
  3. Let the children struggle, talk with one another, and create some possibilities. You will need to assess the frustration in the room. Offer hints or stop the exploration and discuss as a class, if you feel the students are too confused.
  4. If you have students who built the correct array, let them come to the board, re-create the array with the magnetic pieces and explain their thinking. If no one can do this, then you can present the sequence below. The students should be watching and assisting, not working on their boards.
  5. Our problem is 14 groups of 12.
  • Where should I place the 14? (side for the number of rows) Where should I place the 12 (top of the array, how many are in each row.)
  • I need 14 rows of 12. Using the place value blocks, how would I make 1 row of 12 (ten rod and 2 ones, do this)
  • So, we decomposed the 12 into 10 + 2. Record this decomposition on the board's array.
  • Say, I need 14 rows of 12, but since I don't have a 14 manipulative, I will decompose the 14 by place value. What will that be? (10 + 4) Record this decomposition on the array.
  • Now, I'm ready to finish the array. Ten groups of ten. Point to these numbers and where you will place the pieces. Begin counting and placing on the board 1 ten, 2 tens, 3 tens...Your drama skills should go into action and you pretend this is boring and tiring. Ask the students for a shorter way to show the array 10 groups of 10. Students hopefully will say a 100 flat, and you take down the tens and put up the 100 flat. Reinforce that 10 groups of 10 is 100. Record 100 under the flat. The array should look like the following: Array 1 Picture
  • Now, we are ready to build the array, 10 groups of 2. Again, begin by placing and counting 1 group of 2, 2 groups of 2, 3 groups of 2... Pretend you are bored and fatigued. Ask for a shorter way to show 10 groups of 2. Hopefully, the students will suggest 2 ten rods. Place these in the array, so that the ten squares are vertical, not horizontal. It should be 10 rows of 2, not 2 rows of ten. 10 rows of 2 is 20. Write the 20 under the 2 ten rods. Array 2 Picture
  • Now, we are ready to build the array 4 groups of 10. Count and place, 1 x 10, 2 x 10, 3 x 10, 4 groups of 10. 10, 20, 30, 40, so 4 groups of 10 is 40. Record 40 under the 4 rows of 10. Array 3 Picture
  • Ask, have I finished, yet? What do I still need to do? (4 groups of 2) Create this and place 8 under it. Array 4 Picture
  • State, whew, now I am finished. My answer is 100, 20, 40, 8. Let the students tell you how silly you are. Ask, what do I still have to do? Let the students tell you to put all of the answers (partial products) together by adding. You can do this mentally or record the addition problem
  • Put the final multiplication answer underneath or above. Final Answer Ask the students if they can see the 4 smaller arrays within the big array. Discuss how this can help us stay organized.
  • Ask the students to compare this array with the 4 groups of 12. What is the same? Why? What is different? Why?

  Leave this problem on the board and remind the students they may refer back to it as you solve a few more problems together. Move to the Guided Practice.

• Guided Practice: What activities or exercises will the students complete with teacher guidance?

  
  

  You will now give the problem and let the students build the array and find the solution. Begin with only teen numbers. Once they have shown understanding, make one number in the 20s. Try to keep all digits less that 5, so they do not have to put out so many manipulatives.

  You can make up situations to go with the following numbers using the names of your students or write the problem on the board and let the students build the array. Give one problem, allow the students to build, then either have students present the array and the solution, or you do it on the board.

  As students begin to explain their strategies, encourage complete and specific explanations. For example, instead of always saying 14 times 12, say 14 groups of 12; instead of 4 times 1, say 4 times 1 ten or ten. Do not allow short cuts in vocabulary while students are still trying to understand the concept. The goal is not just the correct numerical answer, but an ability to explain the action of the operation.

  • 11 x 13
  • 13 x 12
  • 14 x 11
  • 12 x 21
  • 22 x 31

• Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?

  
  

  The Independent Practice may need to be completed on another day.

  As students begin to understand the procedure you can hand out the attached worksheet. Since it is a word document, change the names of the people in the problems to your students' names.

  Independent Practice

  Independent Practice Answer Key

  As students show they understand, you can let them draw the pieces rather than use the blocks. Gradually move to recording just the answer to the 4 arrays in the correct area of the array. This leads to use of the area model for multiplication.

• Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?

  
  

  Have a double digit array problem posted on the board to provide an example for the summary discussion.

  Select questions from the Guiding Questions to promote a math discussion multiplying double digit numbers. You might use the following:

  • What does this part of the array represent? How do you know?
  • Why do you add, not multiply, the partial products?
  • What is the same about these arrays? What is different?

  Using the place value blocks, post an array without writing the factors. Ask the students to see if they can figure out what the factors are. Example of Array without Factors

  This task reveals the depth of the student understanding. I would suggest doing this as a warm up for future lessons.

  When you feel students are ready, give the attached summative assessment.

  Summative Assessment

  Summative Assessment Answer Key

• Summative Assessment

  
  

  The Summative Assessment is found in the Closure phase of the lesson.

• Formative Assessment

  
  

  Use the Formative Assessment attached in the Teaching Phase to assess the students' prior knowledge.

  A video showing how to use the My Favorite No strategy is found on the Teaching Channel.

  Use the Guiding Questions to assess student thinking during the lesson. As you find students who are not understanding the task, allow them more time and practice with the manipulatives.

• Feedback to Students

  
  

  Use the Guiding Questions to probe and clarify student thinking. There are further responses to students found in the Teach and Guided Practice phases.