Activities in Spatial Thinking

THE INTERSECTING POLYGONS PROBLEM

Which polygons can you create by intersecting a triangle and a quadrilateral?

The Intersecting Polygons Problem

HOW MANY TRIANGLES?

This is a nice puzzle to do with a group.   It is easy to find some triangles but challenging to find all.  It also leads to wonderful conversations about congruence, transformations, measurement when students try to determine whether two triangles are really the "same."  The problem comes from the book 1000 Playthinks.

How Many Triangles?

TOOTHPICK POLYGONS

This is an open-ended puzzle with multiple solutions.  It is nice to do with a group of students so that students can all contribute.  The idea for the puzzle comes from Kordemsky's The Moscow Puzzles.

• Toothpick Squares  and  solutions
• Toothpick Triangles and solutions

COUNTING SQUARES

(An Introduction to Area)

The first of these exercises can be given to very young children, and the last ones are quite hard.  The exercises build intuition about area without the need for formal definitions.  After working through some of the exercises, the teacher should lead a class discussion about what strategies emerged for counting the squares.  Often students will discover on their own the "rules" for finding the areas of rectangles and triangles, and they will discover the strategy of relating a figure to an equivalent figure that is easier to measure.

Easiest version.  Shows the grid lines.

• Counting Squares1 

Harder.  Does not show grid lines.  All figures composed of discrete whole and half unit squares.  The last one in this series includes a figure with a hole (negative area).

• Counting Squares2.1
• Counting Squares2.2
• Counting Squares2.3

Harder still.  Students can't count directly and have to reason that the number of unit squares inside a triangle is half that of its related rectangle.

• Counting Squares3

These require even more abstraction - students have to relate irregular figures to rectangles and then subtract "negative space."

• Counting Squares4.1
• Counting Squares4.2

WHICH FIGURES ARE HALF SHADED?

This is a very simple activity that helps build a deeper and more flexible understanding of what "half" means.  Begin by asking students what what is meant by "half shaded" and ask students to explain how they know whether each figure is or is not half shaded.

• Which figures are half-shaded?
• Which squares are half-shaded?
• How many ways can you color half of a square?

COUNTING CUBES

These exercises give students practice with mental rotations and encourage counting by multiplication and other grouping strategies.  Counting exercises are good also because they require students to organize their thoughts so that they don't undercount or double-count.

• Counting Cubes (very easy)
• Counting Cubes #1
• Counting Cubes #2
• Counting Cubes #3
• Counting Cubes #4
• Counting Cubes #5
• Counting Cubes #6
• Cube counting / skip-counting by 3s
  
  

COUNTING EMBEDDED FIGURES

These counting exercises are more difficult than the others because of the way the figures overlap.

• Counting Triangles
• Counting triangles #2
• Counting triangles #3

CONGRUENT FIGURES

In these exercises students use mental rotations to identify congruent figures.

• Congruent Figures

THE TEN CHAIRS PROBLEM

This Problem comes from Kordemsky's The Moscow Puzzles.  Its simplicity makes it appropriate as a problem-solving exercise for young children.

Ten Chairs

THE BALLOON PROBLEM

The Balloon Problem:  This wonderful problem also comes from Kordemsky's The Moscow Puzzles.
It is similar to but more difficult than the Ten Chairs Problem above.  There are many solutions, so it is nice for an entire class of students to work on the problem and then share their solutions.

• The Balloon Problem

KEEP IT EVEN

Another good problem for young children.  This puzzle requires only an understanding of the concepts of even/odd.  It comes from The Moscow Puzzles.

Keeping it Even

THE SIX SQUARES PUZZLE

This is a rich puzzle.  (The idea is not mine, but I don't know its origin.)  The shapes can be enlarged and copied onto colorful card stock.  Student try to use all 20 shapes to make exactly 6 squares of the same size.

The Six Squares Puzzle

KNOTS

These exercises show illustrations of various loops of ribbon.  Some are simple loops and some are knots.  Students use mental rotation to figure out which is which.  This can be a quick exercise, or it could be the used to jumpstart a longer investigation.
Things to consider:  What strategies did you use to analyze the knots?  How many different kinds of knots are there?  How can you tell if two knots are really the same?  (In other words, how can you tell whether one can be changed into the other without breaking the ribbon?)

• Which are knots?
• Which are knots? 2

THE CARROT PROBLEM

In this problem children reason about the relative sizes of two squares.

• The Carrot Problem

HOLE-PUNCHING PUZZLES

These are challenging spatial puzzles that were inspired by problems on a standardized test for admission to dental schools. 

• Hole-Punching
• Hole-Punching (harder)