How did you learn to multiply decimals? Probably the same way I did. First “ignore” the decimal point and then multiply as a standard algorithm. Next, count the places after the decimal point in the factors and insert the decimal point that many places in my product. Did I understand why I put the decimal point in that position? No. It worked, but it was a trick that didn’t included estimation or reasonableness that I feel is vital for students to really understanding decimal computation.
Common Core State Standards tell me that my students need three facets to their mathematical education.
2)procedural skills and fluencies
3)problem solving skills
This is called rigor. What does rigor look like in the classroom? To me, a classroom with rigor has an environment where each student is expected to learn at high levels. It is a place where students are taught conceptual understanding and procedural skills along with problem solving. This classroom would include scaffolding, small group instruction and/or intervention work to reach these goals. These components indicate that rigor is more than what I teach, it’s how I teach.
Below are the lessons and strategies I use to teach multiplying decimals with rigor.
Lessons to develop Conceptual Understanding
- Estimation Strategy I simply put mixed decimal factors on the board and have students find the product using estimation. They show me their answer and we discuss. Quick and easy. John Van de Walle suggests that instruction on computation with decimals must start with estimating. If students can accurately estimate products and quotients, they are more likely to correctly place the decimal point. (And that is the greatest obstacle students struggle with in decimal computation.)
- Pattern’s R Us–Georgia Lesson. This is an eye opening activity where students use calculators to find the products and quotients of decimals and mixed numbers, I love when I hear a “what?” I know they have gotten to the problem of multiplying a decimal by a decimal, or dividing a decimal by a decimal.
- Area models-Students can visualize the multiplication of decimals using base ten blocks. The Georgia Base Ten Activity is great for this, but before I get to that point I have my students build arrays with base ten blocks. Starting with whole number factors helps students set up their arrays. Next, we do a whole number factor and a mixed decimal number factor, and finally a mixed decimal by a mixed decimal. I let the students struggle a bit setting these up. When they have their vertical and horizontal dimensions put correctly I will write those numbers on their desk with an expo marker and then outline in marker where they need to fill in their rectangle with tenths and hundredths with base ten blocks.
During this whole process we talk about our product in relation to our factors,(is the product greater/less than your factors?).
- Next we do a model of a multiplying a decimal by a decimal. This is easily done on graph paper with two crayons or colored pencils.
Lessons to develop procedural understanding
- Engage New York lesson 11 unit 1 Objective: Multiply a decimal fraction by single-digit whole numbers, using the area model and place value.
- Engage New York lesson 12 unit 1 Objective: Multiply a decimal fraction by single-digit whole numbers, including using estimation to confirm the placement of the decimal point.
- Engage New York lesson 10 unit 2 Objective: Multiply decimal fractions with tenths by multi-digit whole numbers using place value understanding.
- Engage New York lesson 11 unit 2 Objective: Multiply decimal fractions by multi-digit whole numbers through conversion to a whole number problem and reasoning about the placement of the decimal.
- Converting a decimal into a fraction and multiplying. (I have not found a lesson that meets this standard.) Whiteboards and markers will do for practice.
Lessons that develop problem solving
- Georgia Lesson- Do you see an error
- Georgia Lesson-Field Trip
- Georgia Lesson-Bargain Shopping
- North Carolina Window Task
I will do more lessons, but I have found that these are key to establishing conceptual understanding, procedural fluency and problem solving for my students. Also, applying these skills in our number talks has brought more confidence in my students’ computational skills. Students can perform the operations, but also understand what they’re doing.
When my students finish a decimal multiplication problem, they will be able to tell me where the decimal goes and why. And that makes me happy!