The geoboard just might be my all-time favorite math manipulative. There are so many interesting questions that can be explored with this easy to use math tool. When I first introduce students to their geoboards, I encourage open-ended exploration. At this phase, students usually create various shapes without consideration of each shape's properties. Once they're comfortable with this, I then engage my students with specific questions:

- Can you make a rectangle whose perimeter is 10?

- How many shapes can you make whose area is 12?

- Can you make a shape whose perimeter is larger than its area?

- How about a shape whose area is larger than its perimeter?

After my students understand the difference between perimeter and area, I then focus on patterns and relationships and provide even greater challenges:

- Build a rectangle and a square with equal areas. What do you notice about the perimeters? Is this always true? Can you find a counterexample?

- How can you find the area of a right triangle? What about other types of triangles?

Geoboards can be used throughout our students' study of math. Preschoolers can simply design various shapes while middle school students can explore advanced topics like Pick's Theorem . In addition to geometry, the geoboard can also serve as a tool for exploring fractions and algebraic thinking.

Are geoboards part of your math program? How do use this manipulative to promote mathematical reasoning in your students?

## Monday, October 29, 2012

## Sunday, October 21, 2012

### Versatile Pattern Blocks

Pattern blocks have many uses and ours are
as quiet as a mouse! Use them to explore
transformations, discover symmetry, compose and decompose shapes,
investigate fractions, introduce algebraic thinking, create patterns, and engage students
in authentic problem solving. These colorful shapes can provide learning opportunities for students throughout elementary and middle school.

One of my favorite ways to use pattern blocks is to enhance my students' conceptual understanding of fractions. The various shapes enable my students to move beyond traditional models for fractions, pizza pie circles and candy bar rectangles, to more elaborate structures. This leads to greater flexibility in my students' ability to visualize fractions which improves their problem solving skills. We begin by defining a particular combination of shapes as one whole. From there, we challenge students to build various fractions of the whole such as 1/3 or 1/4. When they are comfortable and familiar with these tasks, we proceed to the next level. What does it mean to build a shape that is 4/3 or 5/4 the size of the original whole? How about 5/3 or 7/4? Ultimately, we explore fraction algorithms visually. What does 4 ÷ 3/4 look like with pattern blocks? What does such an expression mean? When might we need to divide by fractional numbers in the real world? Modeling algorithms in this way creates a deeper understanding of division with non-whole numbers.

Do you use pattern blocks with your students? Please share your favorite activity in the comments.

Labels:
fractions,
math manipulatives,
pattern blocks,
problem solving,
teaching

## Tuesday, October 16, 2012

### Escape from Fraction Manor

Would your
students be able to create and order fractions if doing so meant they
could help Cleo the Cat escape from the spooky and dangerous Fraction
Manor?

In
this fun problem solving game, students collect cards as they journey
through three levels of Dr. Fractionstein's castle. Watch out for the monsters! They will try to prevent students from finding all of the cards. When each
level is completed they are presented with a series of math puzzles. The
cards that the students have collected contain digits that must be
arranged into a series of fractions in a given order. The puzzles
increase in difficulty at each
level.

Escape from Fraction Manor addresses both the Common Core State Standards for Mathematics and the Principles and Standards for School Mathematics.

**Common Core State Standards for Mathematics**

4.NF.2: Compare two fractions
with different numerators and different denominators, e.g., by
creating common denominators or numerators, or by comparing to a
benchmark fraction such as 1/2. Recognize that comparisons are valid
only when the two fractions refer to the same whole. Record the
results of comparisons with symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.

**Principles and Standards for School Mathematics**

Math - grades Grades 3-5: recognize and generate equivalent forms of commonly used fractions, decimals, and percents

Math - grades Grades 6-8: compare and order fractions, decimals and percents efficiently and find their approximate locations on a number line

## Friday, October 12, 2012

### Real World Math

Year after year, students make the steady ascent
along the rocky trails of Math Mountain. Arithmetic gives way to
algebra. Polygons lead to polyhedra. Functions progress from linear to
quadratic to exponential. But what's at the summit? What will students
do with all this knowledge?

When will we ever have to use this stuff?

Math Apprentice
hopes to answer that question. Designed for students in grades 4+,
Math Apprentice invites students to play the role of an intern at one of
eight businesses that use math. Students are given an overview of the
math by an animated, virtual employee. They may then choose to freely
explore math concepts or solve a specific problem.

The
math in the activities is a mix of grade appropriate concepts and
advanced mathematics. I think it's important for students to interact
with math beyond the standards. This is often where the real joy of math
can be found. Even young students can access difficult concepts if they
are presented in a meaningful and engaging way.

There are eight careers to explore:

- At the Sweet Treat Cafe, students analyze graphs, scale up recipes, and find the best buy.
- Students learn about ratios and conversion factors at the Wheel Works Bike Shop.
- At Game Pro, students use the Pythagorean Theorem to find the distance between the villain and the hero.
- Students become computer animators at Trigon Studios. They use sine and cosine function to manipulate characters and props in a movie scene.
- While interning at Doodles, students use various functions to create works of art.
- At Space Logic, students match robot speeds to distance vs time graphs and program a space rover to reach its destination.
- At Builders Inc, students must create room shapes whose dimensions meet the customer's specifications.
- While working at Adventure Rides, students determine the height of a roller coaster hill that will give the speed that is needed.

Laura Rose has written a comprehensive summary of Math Apprentice for Connexions in which she describes how Math Apprentice can be used in a middle
school classroom. She suggests the site could be the cornerstone of a
semester long project about math in the real world. It's my hope that students will spend time with Math Apprentice and internalize its
underlying message: math is the path to anything you want to be.

## Monday, October 8, 2012

### Spirograph Math

For our first real world project of the year, I introduced my
precalculus students to a Spirograph* toy. I passed out a variety of
gears and asked half of the group to rotate a gear around the outside of
another fixed gear while the others rotated gears around the inside. Stunning
images appeared from both groups.

We
compared the two processes and looked for patterns. The images depended
on three variables: the radius of the fixed circle, the radius of the
moving circle, and the placement of the pen. Would it be possible to
derive equations for the position of the pen? If so, could we then write
our own Spirograph* program? Much to our delight, we could. You may try our version here.

Even young students can appreciate the intricate beauty of the curves generated by mathematical equations, if not the math itself. Give your students time to explore this app. Have them vary the radius of each circle and the distance from the pen to the center. What connections can they make? They may not have the mathematical language or concepts to accurately describe what's happening but they will gain an appreciation for the power and complexity of math - something that could spark a life-long interest.

*Spirograph is a registered trademark of Hasbro, Inc.

## Thursday, October 4, 2012

### Problem Solving with Fractions

This fraction game introduces students to a character with an interesting dilemma.
Walker wants desperately to get home but the road has gaps that prevent him from
reaching his destination. At each gap, Walker is presented with
increasingly challenging tasks involving fraction pieces. Students must
help Walker solve these problems in order to move him closer to home. In the process, students apply their knowledge of fractions to an engaging problem within a meaningful context. Students work primarily with unit fractions at first but are later shown how to build any fraction needed.

Fraction concepts covered in this game include:

- unit fractions
- equivalent fractions
- comparing fractions
- adding and subtracting fractions
- multiplying and dividing fractions

This game is best presented as a whole group activity before students explore on their own. An interesting component of the game is a tool called the combinator. After solving simple challenges with unit fractions, students are introduced to higher level fractions such as 2/3 and 3/4. Students build these fractions by physically dragging unit fractions to the combinator. This tool provides a powerful visual for students who are beginning to develop a deeper understanding of fractions and how they are composed.

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