The tables are provided for my sixth-graders to design their own fraction word problems. As the title suggests, students will then investigate what happens to the quotient when they change the divisor. This first table guides students through designing an initial set of problems. The second table provides a broader, less defined set of choices.
This post is my mental meandering about place value, base numbering systems, and their relationship. I am sorting through what and how to present place value and our base ten system of numbers to my sixth graders. There is much more here than I plan to present to them. This detail helps me sort through what to do. Before reading this post, I suggest that you write about how the position of a digit affects its value (place value) and base ten numbers. Remember that ours is a base ten place value system. Try to explain a base two system, although this system is difficult to explain. Other systems are easier to explain because the pattern becomes apparent. Explain a higher number systems such as base three or base four, then try to explain base twelve! I find this post difficult to write. This difficulty in writing reveals how confusing are the concepts of place value and base ten numbering system. It is important to have students work through many seemingly simple examples. Students must talk through the problems, explain the problems, and be able to write down those explanations.
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Hopefully, you have tried to write something. So, here goes …
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As a sixth-grade math teacher (or any math teacher fourth-grade and above), you have probably told students that the value of a digit is ten-times greater as you move each place value to the left. This is true: 5 ones becomes 50 ones (5 tens). I have said this to my students. But, as I write these posts and think carefully about my students’ difficulty in understanding math, I realize that this simple statement is complex. To conceptualize “ten-times greater as moving from ones to tens” is difficult. Even as I write this, I think, “C’mon Steve, this is silly; 5 ones to 5 tens is ten times more. It is not that hard!”.
Students will begin by discussing multiplication as equal groups of things. They will provide examples of “things” and “groups”. I will give kids some things to work with: centimeter cubes, pennies, beans; and, some containers in which to place those things: plastic bowls, coffee cans, circles on paper. Take the time to have kids do this. Sixth-graders need this time. Have them think about it – all aspects of it: things, groups, equal groups – write about it, and explain it. Students should include drawings in their writing – what are the images that they are building in their heads as they work through these simple examples?
This next school-year, I have an opportunity to teach math within my science and social studies classes. Our sixth-graders simply need more minutes to think about and work on mathematics. The students and teachers at my school are grouped into villages of 75 to 90 students. The students are divided into three groups who move together between their math, science, social studies, language arts, PE, and exploratory classes. This next year, I will be able to take some minutes from my science and social studies classes to teach math.
It's All About The Math 