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.

**1)conceptual understanding**

**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!

### Patty Reed

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Alisa Louie says

Thank you for your post! Conceptual understanding is not only important, but essential. Sometimes with upper elementary kids, I notice teachers forget to use the great strategies we use with younger learners, even though they’re so effective! Base ten blocks belong in 4th and 5th grade too! :) What lucky students you have! I want to come watch this lesson in person!

Francis Jequinto says

Tons of great information here – I remember in my math methods course during certification we emphasized use of manipulatives like the blocks and they seem like such an obvious fit, but gathering from the comments it’s a fit that many people overlook.

Tom White says

I love your use of BT blocks! I use them to teach multiplication without decimals, but never would have thought to use them with decimals. Thanks!!

Aaron Brecek says

As a fellow math teacher, I believe estimation is a highly under-utilized tool. Most teachers don’t use it unless it is part of an estimation lesson, but without it how can students build number sense? How can a student determine if their answers are reasonable? It’s amazing how many students I get the just plug numbers into a calculator and trust whatever comes out and can’t even recognize if there was an error in their process.

Thank you for continuing to emphasize estimation as a standard practice. It pays huge dividends later.

Carina Stillman says

This post is fascinating to me as I am one of those parents who needs this kind of support. I always did well in math, probably because I am good at memorizing facts and rules. 1.3 x 2.4 = 52 + 260 = 312 = 3.12 when I add in two decimal points. I certainly couldn’t picture or explain why that was so, but the area model makes so much sense!

We just had our Spring Conferences and my son’s teacher took a few minutes to show us how she supports students in conceptualizing multiplication so that we can better help him at home; my husband I both left feeling like we had learned something. Your ideas for parent outreach are spot-on–take any and all opportunities to create understanding of these new ways.

Patricia Gustin says

Great resources, especially the photos. This strategy will work with algebra students–we can take them back to multiplying decimal numbers before asking them to multiply (x + 1) by (x + 7)

Estimating is such a valuable skill. My high school science students go straight for their calculators and bad things happen. In addition to upping the level of rigor in our classrooms, Common Core Math is bringing common sense back into common usage.

Patty Reed says

Thanks for your comments, I agree, estimation is a critical skill for our students. I don’t recall any estimation strategies taught in our previous curriculum. I love your quote, “Common Core Math is bringing common sense back into common usage.

Chris Gustafson says

Those of us who just learned the algorithm feel jealous of the students in your class! However, it also highlights the difficulty parents have with the math work their student bring home, since it does not resemble the way they were taught. What have you found to help parents support this type of math instruction?

Patty Reed says

Good question, and I don’t have a great answer for that. It’s a work in progress. Understand and finding these strategies has been a long road in itself, teaching has been a pleasure, but your right, next step would be getting them into the hands of the parents.

Here is what I have tried:

1. Parent night last year, but only a handful of parents attended, I would however like to try that again.

2. At each open house I gave handouts with the strategies on them.

3. Letting parents know my door (or computer) is always open to questions.

3. I have thought about emailing home strategies at each new unit. (Haven’t done that yet.)

4. Homework is only used as review.

5. Students with fluency so they can teach their parents. (I would enter a smiley face here if I could!)