Building Schema in Preparation for Division of Decimals

As you may know if you read the August blog entry Introducing Equations with Multiple Variables, I have been very interested in increasing my understanding of how children learn fractions, decimals, and percents, particularly how they learn to use these numbers in operations. Often students cite their 5th/6th grade years when these concepts were introduced as the beginning of their mathematical woes. The topic has been of particular personal interest to me because my oldest daughter is in 5th grade and I am homeschooling her in mathematics this year. As mentioned previously, we are using a Japanese curriculum as a primary guide for content. The Japanese curriculum introduces division of decimals in 4th grade, unlike the U.S. where it usually occurs in late 5th or 6th grade. As I looked at the text, I became concerned that Kylie was not yet ready for the manner in which the text approached the topic, given her limited schema for the topic. This seems like a typical problem that teachers face--approaching a new topic for which students have potentially insufficient background knowledge. This blog entry explores how I am dealing with this problem in the context of division of decimals. (I'm sure there are multiple strategies that might be effective here above and beyond those that I describe. I would love to hear them!) Instead of abandoning the topic, I decided to plan an approach that would involve skipping ahead to a geometry unit for our core topic, and use our minilessons (5-10 minute lesson starters) and our "something different" Friday classes to build schema in preparation for the unit.

I chose minilessons that would focus on building her whole number division skills. This was really essential because I don't want division of decimals to seem like a "brand new" concept to her. She needs to anchor new knowledge on previous schema. Whole number division will play a pivotal role in the process as she extends it to decimal numbers. Therefore, I wanted to extend her in this area. She has only used invented algorithms to solve division. These algorithms have gained considerable efficiency since 3rd grade, but I wanted her to make greater use of the distributive and associative properties. I also wanted her to develop her notation skills. This goal involved both algebraic notation and standard representation. Although this developed over a period of some weeks, I will give you an example from her recent work. I gave her the following string of exercises and asked her to solve them mentally and then write a number sentence which would illustrate her mental strategy. She did not have to write a number sentence if she already had that fact in her schema. The actual string was written with a division symbol, but I'm finding that symbol hard to recreate in this post.

The string and her number sentences:
28/7 = 4 ; (14 +14) / 7 = 4
70/7 = 10
98/7 = 14 ; (70 + 28) / 7 = 14
168/7 = 24 ; (70 + 98) /7 = 24
170/17 = 7
187/17 = 11 ; (170 + 17) /17 = 11
340/17 = 20 ; (170 + 170) / 17 = 20
357/17 = 21 ; (340 + 17) / 17 = 21

Initially, I introduced standard representation of division by taking problems she had already solved mentally and then represented her own strategy in that format. I should point out at this point that I am discussing standard representation, not the standard algorithm of long division. At first glance, there are similarities, but long division has a set procedure, so all students who carry out the procedure will essentially show the exact same work on their pages. Standard representation looks similar, but incorporates students own strategies and they use facts that they possess in their schema. Another difference is that long division does not maintain place value, but Kylie's representation does. Note that she solved 187 / 17 by decomposing 187 into 170 and 17, two numbers that she could easily divide by 17. Therefore, her standard representation looks like that shown below. The answer 11 is placed on top as her final step when she calculates how many 17s she used. (I apologize for the appearance of this--it should look like long division, but I can't make the notation look right on the blog).


187
-170 (10)
17
-17 (1)
0

Up to this point you see my strategy is to ensure that she understands division with whole numbers, has developed efficient strategies that make use of her schema, and that she has appropriate systems of notation. However, I simultaneously want to build her understanding of division of decimal numbers. I am taking a two-pronged approach with this. First, I want her to develop an understanding of situations where division of decimals are required. To this end, I give her occasional homework problems with fairly easy division of decimals that do not require sophisticated algorithms to solve. An example is shown below.

You have 7.5 cups of sugar in the cupboard. One of your favorite recipes takes 3 cups of sugar to make (very sweet!). How many batches can you make?

My second goal is to build Kylie's schema related to place value patterns present in decimal division. She has been exposed to decimal numbers in 4th grade and has had a previous unit this school year on multiplication of decimal numbers. At the beginning of the school year, she explored the impact on numbers if they were multiplied or divided by factors of 10. I decided to extend these ideas by having her carry out some calculator investigations that would allow her to learn how to manipulate decimal numbers in formulating strategies for division of decimal numbers. Shown below is the first calculator investigation I designed, along with Kylie's answers:

Investigation A--Decimal position in the divisor
420 / 28 = 15
420 / 2.8 = 150
420 / 0.28 = 1500
420 / 0.028 = 15,000 <--- 1st Predict;

Patterns you noticed: I notice when the decimal moves to the left on the divisor the number gets bigger--10 times bigger. ;

Investigation B--Decimal position in the dividend
420 / 28 = 15
42 / 28 = 1.5
4.2 / 28 = 0.15
0.42 / 28 = 0.015 <--- 1st Predict;

Patterns you notice: I noticed when the decimal point on the dividend moves to the left, the quotient gets smaller--10 times smaller. Then I gave her 2 additional problems and challenged her to use the patterns she had just discovered to solve them: 42 / 2.8 and 4.2 / 2.8 Notice that the calculator allowed numerous calculations to be carried out with ease, allowing her to focus her attention on unfolding patterns. She actually really enjoyed this investigation and was thrilled that she was able to successfully predict answers and solve the challenge problems. The next calculator investigation will build on this one, but focus her attention a bit more on these patterns. It is shown below:


Calculator Investigations A and B:
Complete each number sentence by filling in blanks with x10, x100, x1000, etc. OR /10, /100, /1000, etc. After each investigation, write what you learned.

Calculator investigation A--Different decimal positions in dividend
3650 /25 = 14.6 ________
365 / 25 = 14.6
36.5 /25 = 14.6 _________
3.65 / 25 = 14.6 ________ <--- 1st Predict;

Calculator investigation B--Different decimal positions in divisor
365 / 2500 = 14.6 ________
365 / 250 = 14.6 _________
365 / 25 = 14.6
365 / 2.5 = 14.6 _________
365 / 0.25 = 14.6 ________ <--- 1st Predict;

Calculator Investigation C--What happens when both the dividend and the divisor increase or decrease by the same factor of 10?
25 /5 =
(25 x 10) /(5 x 10) = 250 / 50 =
(25 x 100)/(5 x 100) = 2500/500 =
(25/10)/(5/10) = 2.5/0.5 =
(25/100)/5/100) = 0.25/0.05 =

What happens to the quotient?
Use what you learned in this investigation to turn the following problem into a whole number operation you can easily solve: 2.4/0.12

It is possible to build schema for division of decimal numbers, allowing students to learn with comprehension and develop flexible strategies which personally make sense.