Math Spotlight: Multiplication of Larger Numbers

During 2nd quarter, we learned multiplication of larger numbers with a focus on different methods that can be used. In this post, my goal is to share with you how I have taught different multiplication methods so that you will have a better awareness of the ways you may see your child completing math problems and to provide you with a greater understanding of grade-level expectations. Students may also use this post to help themselves recall what was discussed in class.

Math Vocabulary: How do we talk about multiplication?

 

factor: the numbers being multiplied together
product: the result of multiplying numbers (the answer)
divisor: the number doing the dividing; the number going into the
dividend: the number being divided;
quotient: the result of dividing the dividend by the divisor (the answer)

What does below, on, and above grade level expectations look like for multiplication? I have decided to call these three levels of mastery "Building Blocks," "Goals," and "Gold." These levels are marked with dot symbols (2 dots, 3 dots, and 4 dots) Take note of the 3rd grade standard "Know from memory all products of two 1 digit numbers." I was surprised to see this directly stated in 3rd grade standards. Many of us are still working on multiplication fact mastery, but this made me feel that it is even more critical for students to leave 4th grade with solid recall of multiplication facts. 

Two Methods for Multiplication:

As we work with larger numbers, we use the rectangle method and the traditional method.


Rectangle method- This method splits the factors into place value parts (ones, tens, hundreds) and uses a rectangle to represent splitting the numbers apart. A two digit number would be split into two parts (tens and ones) and therefore needs two boxes. When you multiply a two digit by a two digit, you end up with four boxes. Notice how this looks like an rectangular area problem---you now multiply the “length” and “width” to find the partial products (or partial answer). You add all the parts to get the final product. (Recall that you also have to add in the “standard” multiplication method.)




= 740


+

= 148

= 888


With the rectangle method, students can visualize the idea that all parts of one number multiply all parts of the other number. The key here is that students understand you do not just multiply the tens in each factor and the ones in each factor (this is a common misconception), but the tens AND ones in each number must multiply the tens AND ones in the other number. So, a student who misconceives multiplication would think “I multiply 20 x 30 and 4 x 7. Add those together and I am finished.” (Nope!!!)

Shortcut/Standard Method: This method is how most of us were taught to multiply. The issue with going straight to this method is that it often creates students who don’t really understand multiplication, but know how to follow the steps. At this age, we are lucky we have the time to ensure students understand what they are doing. Common misconceptions/mistakes with the standard method include not shifting the second line over (which you do because when you multiply 2 x 7, you are really multiplying 20 x 7, therefore need to shift the second partial products over.) Often students are taught to “put a magic zero there” but don’t always know why they are doing so. I resist calling it a magic zero and always try to explain "It's because you are really multiplying by a ten." Note that being able to follow the standard algorithm is actually a 5th grade expectation, but I have taught it for mastery this year.

I know this post is packed with info, but I hope you enjoyed taking a closer look at the multiplication standards and methods we used in class. Eventually, I would like to start sending home skills traces like the one pictured above so that you have concrete information about grade level expectations. I am working to build assessments based off of these skill levels where a student knows he/she is moving through "Building Blocks," "Goal," and "Gold" type questions. Look for these in the future!

How does this connect to homework? You will see some above grade level (gold) questions on homework. This is because many students are ready for taking it to the next level and "gold" questions provide enough challenge to maintain a rigorous curriculum for all children. Now that you know what "GOALs" 4th graders have, you should expect that your child can solve those problems accurately and independently. I may even be able note the levels on homework now by using the 2, 3, and 4 dot system! How exciting!

Math Spotlight: Levels of Understanding

This week's post focuses on "Levels of Understanding."

Typically, before introducing students to the traditional way of solving math problems, I use manipulatives (ones cubes, tens sticks, hundreds blocks, etc) or printed models (pictures) to help students think about and understand what the computation really means. Using hands-on activities and guiding students through different types of questions helps me assess the depth of their understanding. My goal is not to simply teach students to memorize steps and procedures, but to ensure that they understand why they are doing what they are doing in each step.

When we think about learning skills and concepts, we should imagine those skills and concepts on a continuum of learning with children at different levels of readiness. Given a specific concept, your child may be at different levels at different times. I found the following descriptors from Kathy Richardson (she's a math guru that develops materials for assessing students' true understanding and misconceptions):

Ready to Apply (A) – The student can already do a particular task and is ready to use this skill in other settings. (This student receives more challenging work).
Needs Practice (P) – The student can do a particular task with some level of effort but still needs more experiences to develop facility and consistency. (This student typically receives work at grade level that increases in difficulty as his/her readiness increases.)
Needs Instruction (I) – The student has some idea of what a task is about but needs support. (This student receives direct instruction that begins with the lowest level of their individual understanding and builds up to problems with increased difficulty to help meet grade-level expectations.)
Needs Prerequisite (N) – The student does not yet understand the concept and needs to work with
mathematical ideas that precede the concept being assessed.

When I discovered these descriptors, I really wanted to jump up and down--I was excited because these categories of learning really capture how I think about individual students' understanding of a math concept and how I choose materials and create groups for focused instruction.

It is also important to consider the size of the number(s) when placing students at these levels for a given concept--the size of the number with which they are independently successful needs to be taken into consideration. I often find that concepts that students may seem to have "mastered" are merely in the process of truly being understood when they are presented with larger numbers. As we know, accuracy also becomes a larger issue as the size of numbers increase. To instruct students having difficulty (say learning the procedures for how to multiply 23 x 67), I begin by taking a step back and instructing them on how we would solve a simpler problem, like 23 x 7. Once the student has consistently demonstrated that they are able to complete problems at this level, we move on to adding a digit in the tens place for the second factor.

Next up...
I'm going to share examples of problems that are below, on, and above grade level based on NC state standards for math. When you see your child's math work, you can use these levels to have a better understanding of what they have accomplished. If they are successfully completing below and on grade level problems, but having difficulty with above grade level, this means that they are where they are supposed to be and that they are being challenged to push beyond the average 4th grade expectations. If they are having difficulty with "on grade level" problems, rest-assured that they are being served in a small group that meets them where they are and works to help them build up to solid grade-level abilities. 

By the way, I'm ready to reveal a secret...I'm a bit of a "math nerd" and I think that's cool!