# Tag Archives: Hands-on Learning

### Partitioning Shapes: Is it Geometry or Fractions?

How early should we teach words like half, thirds, and fourths to children? I know that I have often heard that we give young children things they are not developmentally ready for, and I agree. But when it comes to having language identify a concrete experience, I think children can handle it. I was measuring lace and ribbon for a project, and my youngest daughter, Lilly, wanted to help. I gave her my measuring tape, and she proceeded to take it and “measure things.” I recognized words like two, half, five, and more in the midst of her conversation. She is only two and a half years old. She longs to be just like her brother and sister, so she loves to use words she hears them say. I’m not intentionally trying to teach her these concepts, just exposing her to a lot of language. The concept of “half” will come. Click here to check out a previous post about Bethany’s experience (my 1st grade daughter) with words like half and a fantastic book that helped her explore fractions.

The geometry domain for K-2 teachers uses the language and develops a foundation for the concepts of fractions. In those clusters students are going to partition shapes, decompose/compose shapes, and reason with shapes.

I’ve typed up the geometry standards that are foundational to fractions. Click on the image to download it. It is interesting to me that the writers continue to label the cluster “Reason with shapes and their attributes.” So what does reasoning look like for students? This is where the mathematical practices are found. When you reason you are “making sense,” moving from “quantitative” (concrete) to the “abstract,” “constructing viable arguments and critiquing the reasoning of others,” and “looking for structure.” Since students need to be communicating about this, I’m working on some sentence frames that I’ll blog about soon.

I was training some fantastic teachers at Kepler Neighborhood School in downtown Fresno, and we composed shapes using pipe cleaners and wikki sticks and then partitioned them.

The wikki sticks are preferred because they can stick to the table, and they are easy to cut when you need to make a partition.

You can also fold or build a half to make a visual that proves it is one-half of the whole. Geoboards are another tool students could use to compose shapes and partition them.

As students manipulate and explore these shapes, they are preparing for fraction concepts. Students need to build an expertise in partitioning shapes and communicating about them using the language of fractions. These experiences of decomposing/composing and partitioning with shapes will support students’ development of fractions in the future and are highly engaging and hands-on. So lets keep student’s hands and minds engaged in mathematics.

### How to Equip Your Students to Better Understand Multiplication, Part Three

I never liked word problems as a student. It was difficult for me to figure out which procedure to use, but I really didn’t like problems like this: Robert is three times as old as his younger brother Mark. Mark is 7 years old. How old is Robert? As I reflect on my experience, I… Continue Reading

### How to Equip Your Students to Better Understand Multiplication, Part Two

Using arrays has become much more prominent in the classroom. At first glance arrays seem very straightforward and simple for students. But what are the connections that are essential for students to build understanding of the concept of multiplication through arrays? Arrays are a model of multiplication. Just because your students can build an array… Continue Reading

### Five Squares, One Square: Solution

In two previous blog posts I talked about a puzzle made up of five two by two squares, where each square was cut in two along a line from a vertex to the midpoint of a side. The challenge, which I gave in the first post, was to put the ten pieces together to form… Continue Reading

### Five Squares, One Square: Some Hints and an Extension

In an earlier blog post I proposed a puzzle made of five 2 by 2 squares, each of which had been cut along a line from a corner to the midpoint of an opposite side so as to form a right triangle and a trapezoid with two right angles. The challenge of the puzzle was… Continue Reading

### Five Squares, One Square

A while back I posted a five triangle puzzle that involved putting together five 30-60-90 triangles to form a single triangle. Of all of the dozens of puzzles that I own and have made over the years, that is one of my favorites because of the opportunities it provides for students to think about important… Continue Reading

### The Five Triangle Puzzle: Solutions

The Five Triangle Puzzle was the subject of a post back on February 11. I’m hopeful that some of you will have downloaded the pieces and solved the puzzle. The challenge was to put all five pieces together to form a triangle and then to determine if there were other triangles that could be formed… Continue Reading

### Using Sentence Frames to Talk about Math

As I talked about in my earlier posts, I am really interested in learning about the Japanese methods of teaching math concepts. As I was exploring a 1st grade Japanese textbook that Phil Daro recommended, I noticed something I wanted to field test. The textbook had the students placing counters on top of the pictures… Continue Reading

### The Five Triangle Puzzle: Composing and Decomposing

Beginning in first grade and continuing through the grades, the Common Core Math Standards emphasize composing and decomposing shapes. One way to give students experience with composing shapes is through put-together puzzles. Actually, working at put-together puzzles involves lots of composing and decomposing. Sometimes what we compose is a solution; sometimes it’s not. While the… Continue Reading

### Frank Lloyd Wright, Froebel Geometric Gifts, and Hands-On Learning

This post is a continuation of the story of Froebel’s geometric gifts that was introduced in my previous post. I ended with a promise to tell a story about the famous architect, Frank Lloyd Wright. In 1876, when Wright was eight or nine years old, his mother attended the Philadelphia Centennial Exposition. Wright describes in… Continue Reading