# Math Art Part 2: Decomposing and composing squares and triangles

I wanted to show you another example of math art, this time using squares and triangles. This project also falls under the standards dealing with decomposing and composing shapes. With this project, students can create some unique designs while learning about squares, triangles, symmetry, fractions, and elements of art such as color and design. It would be a great project for first grade (using 2 squares) or for higher grades using 3 to 4 squares.

A great literature connection to this project is the book “The Greedy Triangle” by Marilyn Burns. (Click link to connect to Amazon.) The triangle in this book isn’t content with being 3-sided and transforms himself into other shapes (with the help of the Shapeshifter). Lots of great pictures showing real objects in the shape of triangles, squares, pentagons, hexagons, and more.

Marilyn Burns is a great math educator to check out, if you haven’t already. She has a company called Math Solutions (check out MathSolutions.com). Marilyn and her consultants have wonderful resources and advocate for constructivist views regarding math education. She is also the author of Number Talks and many math and literature lesson ideas.

### The 4 Triangle Investigation

Materials needed:

• Pre-cut squares 3″, 4″ or 5″ (I used brightly colored cardstock.)
• Scissors and glue
• Background paper to glue shapes to

Directions

1. Model how to cut a square in half (diagonally) to make two right triangles. (I advocate folding it first so that the two resulting triangles are as equivalent as possible.)
2. Guide students into showing different ways to put two triangles together to form another shape. Rule: Sides touching each other must be the same length. Let students practice making these shapes on their desk top (no gluing needed).
3. Help students realize they may need to use these actions:
• Slide the shape into place
• Flip it over to get a mirror image
• Rotate it around in a circular motion to align the edges
4. Students are then given 2 squares (to be cut into 4 triangles) and investigate different shapes they can make following the above rule. Here are some possibilities:
5. As the teacher,  you can decide how many creations you want each student to attempt.
6. These shapes can be glued onto construction paper (and cut out if desired).
7. As an extension, shapes can be sorted according to various attributes:
• # of sides
• symmetry
• # of angles
• regular polygons vs. irregular

# Math Art Part 1: Fraction circle art (3rd-5th)

Incorporating art with other subjects is a great way to engage students. In this post I will share one project which helps students gain hands-on experience with fractions. More to come in future posts.

## Fraction Circle Art

This project is inspired by Ed Emberley’s book “Picture Pie.” This is my favorite of his collection in which he shows dozens of ways to use fractions of circles to create almost anything. This book features mostly animals, flowers, and geometric designs. Students start with a circle (pre-cut with a circle cutting press is best, but you can also make nice circles by tracing around a can or drinking glass and cutting them out). Then the circle is folded and cut into these different fractional parts to create the design: halves, fourths, eighths, and sixteenths. These pieces are manuevered (think translations!!) and combined to make the desired art.

The pictured creations were made by 3rd and 4th graders during a session I conducted with them at Eisenhower Elementary. Here are a few of them.  So nice!!

# Fractions Part 6: Adding and Subtracting Fractions

Starting in 3rd grade, students start building understanding about adding and subtracting fractions by composing and decomposing simple fractions using concrete and visual models. Composing: 1/4 and 3/4 combine to make 4/4 (whole). Decomposing: 8/8 is made up of 2/8 and 6/8. In fourth grade, students begin to add and subtract fractions with like denominators, but should still be utilizing models, drawings, and number lines to illustrate and simplify. In fifth grade, students are expected to add and subtract fractions of all types (proper, improper, with unlike denominators, etc.).

With a firm foundation of composing and decomposing, partitioning, comparing, naming equivalent fractions, and understanding the relationship between certain fractions (such as halves / fourths / eighths / sixteenths; and thirds / ninths / sixths / twelfths; or fifths / tenths), then students are more prepared to perform operations with fractions. Here’s a great resource by Donna Boucher at Math Coach’s Corner: Composing and Decomposing Fractions activity on TPT (\$6)

Estimating: This is an important part of operations with fractions. Do you expect your answer to be less than 1/2, more than 1/2, more than 1? How do you know? If I was adding 8/9 + 11/12, my answer should be about _____? It should be slightly less than 2 because both of these fractions are almost 1.

If I am adding 4/6 and 6/8, my answer should be more than 1 because each of these fractions are greater than 1/2.

Different strategies: There are many “tricks” or shortcuts available to show students how to quickly add, subtract, or multiply fractions. I believe these shortcuts are only useful after a students has a strong understanding of why and how to find a common denominator and equivalent fraction. These shortcuts do not help build conceptual understanding of fractions.  I will focus on ways to understand the why using visual and pictorial models. Get your FREE copy of the following guides by clicking HERE. Continue reading

# Fractions Part 5: Equivalent Fractions

This is part 5 of a series of fractions posts. Thanks for sticking around! Through explorations with fraction manipulatives, pictures, and drawings, we hope students begin to notice there may be different ways to express the same area using fractional terms. To cut a sandwich into halves and eat one of the halves is the same as cutting the same sandwich into fourths and eating two of the fourths. Read on for several freebies about equivalent fractions.

Students can gain experience finding equivalent fractions using models in several ways: Fraction strips, area models, set models, bar / length models, and number lines. Then with a strong understanding using concrete and pictorial models, the student is ready to apply paper-pencil methods to name equivalent fractions. But remember to use the same size whole: As in this picture using pattern blocks, 2/3 of a trapezoid is NOT the same as 2/3 of a hexagon.

Fraction Strips: If you don’t have sets of fraction strips, here is a free resource Fraction, decimal and percent strips charts via Kim Tran (TPT).  Commercially available strips are also nice. But, with some 1″ strips of construction paper, students can create their own and probably learn a lot about the relationship between halves, fourths, and eighths as well as thirds, sixths, and twelfths in the process of partitioning and cutting them.

TIP:  Make sure students know how to read a fraction strip chart. A couple of years ago I was working with a third grade class and assumed they could readily see that 5/10 was equivalent to 1/2. But after confusing looks, I realized that I needed to physically show them how to follow a line vertically down the chart to find other fractions that were in line (by placing a ruler or long pencil along the vertical line). Another example.  “To find another fraction equivalent to 1/3, find the line at the end of the 1/3 section and trace it vertically down the page to see if there are other fractions that stop along that same line. You should see in the sixth’s line that 2/6 lines up, and in the ninth’s line that 3/9 lines up and in the twelfths line that 4/12 lines up.”

After cutting and labeling strips, then explore equivalent fractions (those with the same size length). Students should be able to generalize that different fractions can used to represent the same area.

Area models:  By covering or partitioning shapes, students should notice that even though the same area is covered or shaded,  the number and size of the parts can change. Continue reading

# Fractions Part 4: Compare Fractions

Starting in 3rd grade, students start using words and symbols to read and write fractions (Oklahoma Academic Standards OAS 3.N.3.1), construct fractions (3.N.3.2), compose and decompose them (3.N.3.3), and order and compare them using models and number lines (3.N.3.4).  Fourth and Fifth graders continue to refine these skills. In this post, I will address different ways to compare fractions (keeping in mind the concrete-pictorial-abstract progression) by comparing numerators, comparing denominators, comparing to half, and utilizing knowledge of unit fractions. Students should have extensive experience utilizing models such as fraction strips, fraction circles, pattern blocks, number lines, pictures, and drawings to help build the concepts of fractional parts before being asked to put a <, >, or = sign between two fractions. See the end for a FREE comparing fractions guide.

In my opinion, determining if (or how) two fractions are equivalent is also a very important step when comparing fractions. However, regarding the OAS, students are not asked to represent or rename equivalent fractions until 4th grade (4.N.2.1). I will address equivalent fractions in the next post – just know that sometimes this skill goes hand in hand with comparing fractions. AND keep in mind that most of the standards for fractions through 4th grade stipulate “using concrete and pictorial models, fraction strips, number lines.” Students in 4th grade should not be expected to do abstract paper-pencil steps to simplify or “reduce” fractions to simplest terms, nor cross multiply to compare, etc. They need hands-on experience to more fully understand the concepts about fractions that are so difficult to grasp abstractly. Then in 5th grade students should have enough visual pictures in their head to solve operational problems with fractions. OK, that’s my soapbox. Don’t make it harder than it should be.

Materials to use: pattern blocks, fraction strips, fraction circles, cubes, tiles, two-color counters, Cuisenaire rods, number lines, paper plates, graham crackers, candy (m and m’s, skittles, etc.)

Ways to Compare (when using same size wholes – you can’t compare 3/4 of a donut with 1/2 of a birthday cake):

• Using unit fractions:  If the fraction is a unit fraction, it has a 1 as a numerator. This should form the first type of comparison:  1/2 > 1/3 and 1/5 < 1/4 and 1/6 > 1/10, etc. This type of comparison is critical to fractional understanding.
• Same denominator: When the denominators are the same, then compare the numerators. 2/4 > 1/4.
• Same numerator:  When the numerators are the same, compare the denominators. For example: When comparing 2/5 with 2/10, since fifths are larger parts than tenths, 2/5 will be larger than 2/10. This is hard for some students to think about, because the smaller the number designated for the denominator, the larger the part (when comparing the same size whole).
• Unit fractions one away from the whole:  These are fractions in which there is one unit to be added to make it a whole (1). The numerators of these fractions will be one less than the denominator.  11/12 is 1/12 away from the whole (1). 7/8 is 1/8 away from the whole (1). Example: To compare 3/4 with 5/6, use manipulatives or a number line to see that 3/4 is 1/4 away from 1, while 5/6 is 1/6 away from 1.  Since 1/4 is a bigger part than 1/6, then 3/4 < 5/6.
• Less than half? More than half?  Learn all of the fractions that equal half. While this might sound simple, students often have misconceptions that 1/2 is the only way to describe half, or that a 5 must be in the fraction to be half (because 5 is the midpoint when used on a number line for rounding). I ask students to recall their addition facts dealing with doubles from 2nd grade. Since 2 + 2 = 4, two is half of 4, and 2/4 = 1/2. Repeat that with other forms of 1/2. Students should learn that finding half of an even-numbered denominator should be figured quickly (7/14, 9/18, 25/50, 50/100, etc.). Then use knowledge of half to determine if a fraction is less than half or more than half. Since 7/14 = 1/2, then I know that 6/14 < 1/2 and 9/14 > 1/2.

# Fractions Part 3: Misconceptions

The fractions focus today will be on some basic concepts that students should understand before they work to compare them, determine equivalent fractions, simplify them, use mixed fractions, or add / subtract them.  I am including a FREE copy of my Fraction Basics reference guide (click here), along with a photo of an anchor chart I made for a fourth grade class.

I have been rereading a book I love about fractions called “Beyond Pizzas and Pies, 1st Edition.” It has great examples of children’s misconceptions about fractions and lessons on how to try to remediate them. A recurring theme in the book is that while kids can learn “tricks” to help them solve fraction problems, they often do little to help students conceptualize what fractions are. Here’s a link to Math Solutions regarding this book: Beyond Pizzas and Pies (2nd Edition) Following are five  examples from the book that made an impact on me and my teaching (which I will go into more detail about on future posts). Continue reading

# Fractions Part 2: Constructing and Drawing

The standards (CCSS or any state) use varied verbs to describe what students are to do regarding fractions: form, compose, construct, model, partition, draw, decompose, share, identify,  read, write, describe, order, and compare. Satisfying these standards can often be accomplished through use of concrete methods (manipulatives) and pictorial models (drawings). Remember the best understanding of concepts usually follows the concrete, pictorial, abstract progression (CPA). In other words, “Let’s make it, draw it, and then use numbers to represent it.”

Through constructing and drawing, students will  be prepared for further work with fractions, and they begin to conceptualize the relationship between the size of denominators, the numerators, and the whole. Click here for a FREE copy of the pictures you will see below (3 page pdf).

Form / Compose / Construct / Model:  Use smaller shapes to form or compose larger shapes (which is also a geometry std. in KG and 1st). Put together fraction pieces or puzzles to make the whole shape (circle, rectangle, hexagon, etc.). Use fraction pieces to demonstrate understanding by constructing models of area, set, and length.

These pictures show different ways students can use manipulatives to form, compose, construct, and model fractional parts (pattern blocks, fraction circles, tangrams, linking cubes, color tiles, fraction bars, Cuisenaire rods, two-color counters):

Partition / Draw / Decompose / Share: Split larger shapes into smaller fractional parts (halves, thirds, fourths, etc.). Divide (fair share) objects into equal groups. Use models to decompose a fraction in more than one way. Represent fractions on a number line.

I enjoy teaching children how to partition common shapes into fractional parts – because it involves drawing. Too often, if I just tell them to divide a rectangle or circle into fourths or sixths, I get something like this: Continue reading

# Fractions Part I: Basics KG-2nd grade

This is the first post of several I will devote to fractions, starting with basic understanding in first grade and moving up toward operations with fractions in 5th and 6th grade. I would love to hear from you about your students successes and/or difficulties with fractions so I can be sure to address this topic to meet your needs. Free resources below.

What is a fraction?  A fraction represents a part of a whole. It consists of a numerator (which tells how many parts we are describing) and a denominator (how many parts the whole is divided into).

Some basics:

• Fractional parts must be equal. This is a concept introduced in first grade. (See some lesson plan ideas below.)

From Pinterest

• The larger the denominator, the smaller the parts AND the smaller the denominator, the larger the parts — when comparing identical sized objects. You can’t compare 1/4 of a cookie with 1/4 of a cake. This is one of the hardest concepts to grasp – so lots of hands-on experience is needed.
• Helpful manipulatives to use with fractions: pattern blocks, color tiles, Cuisenaire rods, fraction circles, fraction strips, fraction bars, graham crackers.
• Be careful about always referring to fractions as “the shaded part.” While this might be true with pictures on worksheets, fractions can be described in these ways also: What fraction of the students are boys? What fraction of the pizza was eaten? What fraction of the candy bar is left? What fraction would belong here on the number line?
• Lines do not necessarily define the fractional part. On the picture shown, the left shows 1/4 shaded. The right also shows 1/4 shaded, but students are likely to say 1/3. Why? Because they count the parts shown (3) and the shaded parts (1) and put that together as a fraction. A way to show this is still 1/4 is to show that the shaded part will fit into the whole shape 4 times.
• A unit fraction: This is a fraction with 1 as the numerator (1/4, 1/8, etc.). It is one unit of the whole.
• A fraction that is one unit away from a whole has a numerator one less than the denominator. Examples: 2/3, 3/4, 7/8, 11/12, etc. This is helpful to conceptualize when comparing fractions.
• When reading a fraction number line, compare it to a bar model. Then it is easier to see it is the spaces are the focus, not the tick marks.

# Illuminations NCTM Interactives

Resource – http://illuminations.nctm.org

This is a math resource I absolutely love! It is a product of the National Council for the Teachers of Mathematics (NCTM). This site includes lesson plans and interactive activities. Search in several ways: by topic, by standard, or by grade level. Need some strategy games? Check out “Calculation Nation” (some of which can be played against other players), and “Brain Teasers.” I have just added this link to my Resources page (on my blog home page).

Many of the lessons connect to exploration projects and literature. The interactive features are outstanding!! These are perfect for the smartboard, on laptops, or tablets. Once you are on the home screen, click the Interactives box (right side) and then the desired grade level. There are dozens of great applets, but here are a few you might really like. I have linked them for easy reference, so just click on the  title and you’ll be there:

Dynamic Paper: Customize graph paper, number lines, spinners, nets, number grids, shapes (to include pattern blocks, color tiles, and attribute blocks), and tessellations. You can also choose inches or cm. These can be customized, saved and printed as jpeg or pdf. I created the spinner shown here.

Five Frame and Ten Frame tools: Geat activities to build number sense using five or ten frames. These may take 1-2 minutes to load.

Cubes: Build a rectangular prism one cube, or row, or layer at a time and then compute the volume or surface area.

Coin Box: Drag and exchange coins. There is also a feature I like (the grid at the bottom right corner), which puts coins in blocks (by 1s for pennies, 5s for nickels, 10s for dimes, and 25s for quarters). This really helps see the value of the coins. Want more info about coin blocks? Once on the Coin Box page, click on the “Related Resources” tab.

Equivalent Fractions: Build different fractions in circular or rectangular format. Compare them and see them on a number line. You can manipulate the numerators and denominators to see fractions change right before your eyes! Others for fractions: Fraction Models (which includes decimal and percent equivalencies) and the Fraction Game.

Geometric Solids: Create a shape (either transparent or solid) and swivel it around to see all of the faces, vertices, and edges.

For your graphing needs, check out the Bar Grapher, Circle Grapher, and Data Grapher. With these tools you can create graphs using any of your own data. Some of these need Java installed.

Enjoy these and so many more!!! Let us know if there are others you recommend.