The conceptual understanding of multiplication is really important. Developing the conceptual understanding takes a variety of carefully planned lessons, plenty of opportunities with manipulative, and many opportunities to discuss and share student thinking.
Building Understanding with Arrays
2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
In grade 2, students are drawing arrays, counting with arrays, and using repeated addition. Consider pairing the array with a related skip count that students are working on or successful with (like 2's, 5's Students can make connections between skip counting and arrays. Have students share their thinking about how these strategies are similar. How is skip counting similar to a 2x3 array? Where do you see 2, 4, 6 in the array? How could we write this as a repeated addition problem? Do you see any other patterns?
Include a variety of activities that allow students opportunities to think about and explain how skip counting, repeated addition and arrays are similar and different. Making connections among strategies and representations strengthens students ability to transfer information and retain concepts.
Students will also make observations about the "other way" to see the array. Consistently building in opportunities to talk about the Commutative Property of Multiplication is important. Doing one lesson on properties may not be enough for students to develop the understanding need to create meaning for the students and to remember what it means.
A solid understanding of the Commutative Property of Multiplication will shorten a student's list of multiplication facts needed to be memorized. In Grade 3, as you develop meaning of multiplication and begin putting those facts to memory, consider keeping tracks of facts you know on a multiplication table ... and don't forget to mark the other fact.
Lesson: What I know: Multiplication Table 10x10
The first facts that students learn are there facts for: 1, 2, 5, & 10. Have students begin highlighting the facts that they know.
Prior to this lesson, students will have experience working with skip counting, arrays, and groups. As you notice students beginning to recognize patterns, or just knowing what it will be, and the meaning of multiplication, you can have students start to work on the multiplication facts.
This activity can be a key component of your multiplication fact practice. By the end of the lesson students will be able to see how many facts are left. Continue to use this throughout the year as a way of seeing progress. This can be also be a great tool to differentiate the multiplication facts students are working on.
We know the 1 x (Any Number) = The Number. Go through the facts. Review that everyone knows them.
Next, highlight all of the x2 Facts. Discuss as you go the facts.
Students also quickly learn the x5 and x10 facts. Highlight both the row and column for these facts.
Once the multiplication facts for 1s, 2s, 5s, and 10s are highlighted, the facts that are left are the ones to work on. There are 36 facts left. BUT WAIT.... THERE'S MORE...
Remember how every fact has a partner fact because of the Commutative Property of Multiplication?
That means there are fewer than 36 facts. Let's take a look at how many:
Outlined in Pink are the facts that are Perfect Squares (middle school term) - 3x3, 4x4, 6x6, 7x7, 8x8, and 9x9. These multiplication facts only appear once, since the Commutative Property of Multiplication is the same multiplication fact.
Consider the pink line of facts as a line of symmetry. The facts that are "above" the line will have a partner fact "below" the line because of the Commutative Property of Multiplication.
So, let's count... we have 6 perfect square facts, and 1 + 8 + 6 other facts:
6 + 1 + 8 + 6
= 12 + 8 + 1 (adding my doubles 6+6)
= 21 (adding 12 + 8 to get 20, then +1)
There are only 21 facts to work on!
If you like this lesson, but students' skills are all over the map, you may consider doing this activity as part of a small group or one-on-one conversation. Students will only highlight the facts that they know. If there are students struggling with facts that other students in your class know, carefully handle the situation knowing your students' needs and the community in your classroom.
Have students keep this Multiplication Table Chart in their folder. As students work on their multiplication facts, they can highlight the facts that they learned.
This does not mean that students won't forget a fact. It happens. Continue to use activities and games that build students understanding, flexibility, speed and accuracy with multiplication facts.
Using Arrays to Create Meaning of Multiplication
With the formal work on multiplication beginning in grade 3, building from the understandings of repeated addition, arrays, and skip counting, students learn in 2nd grade is essential.
Using array cards or cutting out arrays from grid paper can be useful for many learning activities.
Consider the math lesson, Exploring Multiplication With Rectangles, found in Marilyn Burns book, "A Collection of Math Lessons From Grades 3 Through 6".
In this lesson, students approach the concept of multiplication through a geometry perspective - through arrays. Making connections between the rote procedures or memorization with geometry provides a visual to enhance their conceptual understanding and meaning of multiplication.
To model the activity, begin by using square tiles. Students have 12 tiles they need to arrange in a rectangle - with all spaces filled in (not an outline). Using all 12 tiles, students will create possible variations of 6x2, 3x4, and maybe even 1x12. This beginning conversation can be the place to support students in describing their rectangles (horizontal & vertical), what does it mean to have a "different" rectangle, and what does it mean to be "12" - in this case.
Students then record their rectangles on the their grid paper.
Next have students work in groups to find ALL possible rectangles for 16 tiles. Students should record their solutions on their grid paper, writing the number 16 in the middle of each rectangle.
Once this task is completed and debriefed, students may be ready for the group task.
Students will work in their groups to find "all the different rectangles there are for each of the numbers from 1 to 25. Use tiles to help. Draw each rectangle you find on the squared paper, write the number on it, and cut it out" (page 74).
Students will need to find a way to keep organized. Have groups take a couple of minutes to discuss how they plan on approaching the problem and organizing themselves. Groups will get started on the activity on this first day and possibly take an entire second day to complete the task.
On day three, students will discuss things they noticed about the rectangles they cut out and begin to make some conclusions about the rectangles and multiplication.
As groups finished on day two, a list of questions can be made available for students to begin to discussion that can be used on day three to highlight important concepts from the activity.
Here is the sample list of questions given in the book:
The last activity makes connections from the rectangles to the multiplication table. This requires that you have grid paper the coordinates with the size of the rectangles that were cut out earlier in the exploration.
If you take all of the rectangles that represent the number 12, then one at a time, place them on the grid. The 3x4 rectangle can be placed on the grid with 3 rows and 4 columns. Lift up the corner and place the number 12 in the bottom right most square on the grid paper. Then place the rectangle in the other direction - 4 rows and 3 columns. Do the same in the bottom right most square on the grid paper. And continue modeling the process with the 2x6 and 1x12 rectangles.
To model the activity, begin by using square tiles. Students have 12 tiles they need to arrange in a rectangle - with all spaces filled in (not an outline). Using all 12 tiles, students will create possible variations of 6x2, 3x4, and maybe even 1x12. This beginning conversation can be the place to support students in describing their rectangles (horizontal & vertical), what does it mean to have a "different" rectangle, and what does it mean to be "12" - in this case.
Students then record their rectangles on the their grid paper.
Next have students work in groups to find ALL possible rectangles for 16 tiles. Students should record their solutions on their grid paper, writing the number 16 in the middle of each rectangle.
Once this task is completed and debriefed, students may be ready for the group task.
Students will work in their groups to find "all the different rectangles there are for each of the numbers from 1 to 25. Use tiles to help. Draw each rectangle you find on the squared paper, write the number on it, and cut it out" (page 74).
Students will need to find a way to keep organized. Have groups take a couple of minutes to discuss how they plan on approaching the problem and organizing themselves. Groups will get started on the activity on this first day and possibly take an entire second day to complete the task.
On day three, students will discuss things they noticed about the rectangles they cut out and begin to make some conclusions about the rectangles and multiplication.
As groups finished on day two, a list of questions can be made available for students to begin to discussion that can be used on day three to highlight important concepts from the activity.
Here is the sample list of questions given in the book:
- Which rectangles have a side with two squares on them? Write the numbers from smallest to largest.
- Which rectangles have a side with three squares on them? Write the numbers from smallest to largest.
- Do the same for rectangles with four squares on a side.
- Do the same for rectangles with five squares on a side.
- Which numbers have rectangles that are squares? List them from smallest to largest. How many squares with there be in the next largest square you could make?
- What is the smallest number that has two different rectangles? Three different rectangles? Four?
- Which numbers have only one rectangle? List them from smallest to largest.
The last activity makes connections from the rectangles to the multiplication table. This requires that you have grid paper the coordinates with the size of the rectangles that were cut out earlier in the exploration. If you take all of the rectangles that represent the number 12, then one at a time, place them on the grid. The 3x4 rectangle can be placed on the grid with 3 rows and 4 columns. Lift up the corner and place the number 12 in the bottom right most square on the grid paper. Then place the rectangle in the other direction - 4 rows and 3 columns. Do the same in the bottom right most square on the grid paper. And continue modeling the process with the 2x6 and 1x12 rectangles.
Students will continue this process with all of the other rectangles. Through this process they will be building a multiplication table.
Check out Marilyn Burns' book, A Collection of Math Lessons' from Grades 3 through 6 for more detailed instructions.
3.OA.1 Interpret products of whole numbers … as the total number of objects in a group
3.OA.3 Use multiplication & division within 100 to solve word problems…
3.MD.7 Relate area to the operations of multiplication and division.
On the SciMath MN website, there is a vignette the uses arrays to show how to use facts you know to find facts that are a challenge.
Students can consider which facts that they know and facts that the don't know or are having trouble consistently remembering.
Let's say we need to work on 6x7. Using grid paper I would cut out a 6x7 rectangle. Then I would think about facts I know that are within 6x7. I know 6x5. S I shade a rectangle that is 6x5. And 6x2. So now 2 facts that I do know can be added together to get the product of 6x7.
Check out the entire vignette for a more detailed description of this activity.
Students can consider which facts that they know and facts that the don't know or are having trouble consistently remembering.
Let's say we need to work on 6x7. Using grid paper I would cut out a 6x7 rectangle. Then I would think about facts I know that are within 6x7. I know 6x5. S I shade a rectangle that is 6x5. And 6x2. So now 2 facts that I do know can be added together to get the product of 6x7.
Check out the entire vignette for a more detailed description of this activity.
This lesson would work well as a strategy to use the facts you know to help you with facts you don't know. The lesson can also be used to support thinking around the Distributive Property.
Read more here.
Building Understanding with Equal Groups
Students may also be making connections between skip counting, repeated addition, and equal groups.
Students will come with some background information and experience using repeated addition and skip counting for 2's & 5's. This would be a great place to start bringing in the formal understanding and conceptual knowledge for multiplication.
3.OA.4 Determine the unknown number in a multiplication or division equation…
3.OA.7 Fluently multiply and divide with 100 using relationships… and properties … By the end of Grade 3, know from memory all products of two one-digit numbers.
3.NBT.3 Multiply 1-digit whole numbers by 10 in the range of 10-90 ...
Math & Literature
There are plenty of books to support multiplicative thinking. One fun book that you can incorporate during early multiplication explorations is the book, "The Grapes of Math" by Greg Tan. Each page in the book explores a math riddle and has students looking at objects in arrays and counting. The illustrations and riddles can lead to exciting conversation around mental math, addition strategies, multiplication, and arrays.
Consider the following riddle:
Students may be able to see an array that is rotated. If you count at an angle (askew), then you see a 4x4 array. Students can count ... 4 + 4 + 4 + 4 = 16 ... or 4, 8, 12, 16 or ... 4x4 = 16.
Doing one or two riddles at a time throughout the year would be an exciting and fun way to start math time. This would work well during a Number Talk Routine and would support evidenced based conversations in math class.
Check out more from Greg Tang at: http://gregtangmath.com/
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