If you have picked up this book, I am going to assume that you want to introduce your students to a mathematics that is beautiful and creative and that encourages them to explore and be curious. After many years of working in mathematics education, researching and teaching different groups of students, I have come to realize that there are two completely different versions of mathematics. For many people in the world, mathematics is a subject of rules and procedures. But the mathematics we will explore together with your students is one that is open, creative, and beautiful. When your students meet this mathematics, I predict that they will be engaged and inspired, and they will learn the mathematical ideas deeply.
In this big idea, we help students understand two concepts that they will use for many years in mathematics and that are often misunderstood by students. The first is the meaning of equality; the second is the meaning of a fraction.
In students’ mathematical journeys, they will meet the idea of “equals” often and in many different ways; in this activity, we introduce the students to the idea of equality between shapes. Researchers have identified that many problems occur for students when they develop the idea that equals means “do something,” a kind of instruction or operation (Powell, 2012). This comes about when students see expressions such as 4 + 5 = ? They think the equals sign is telling them to add the numbers. But the equals sign is not an instruction; it is a relational symbol, showing that the numbers or shapes on either side of the symbol are in an equal relationship. This is an important and big idea for students. In the Visualize activity, students will discuss the meaning of equals as they consider different shapes.
As students compare shapes, they will also get an opportunity to combat a common misconception in fraction learning. Many students believe—incorrectly—that for fractions to be equal, they have to have the same shape. This misconception comes about because students are so often shown fractions that have been partitioned into shapes that are equal in area, as well as congruent. But when we think about equal fractions, we only need to think about the area; the shape of the fraction is not relevant. In this activity, students will see that different shapes fill a half of an object, as they have the same area even when the shape is different. As students play with spatial puzzles, and think about their qualities, they will learn that shapes, like numbers, can be understood flexibly.
In the Visualize activity, students consider the question, What does it mean for two or more shapes to be equal? They also think about ways to partition shapes into equal parts. Students will look at squares and cut squares to decide and then explain how they know that parts are equal. They will begin to use the language of fractions, and they will be naming shapes as well and justifying how they know whether they are equal or not.
The Play activity builds off the Visualize activity as students work to partition squares into fourths in as many different ways as possible, folding and cutting. Students are encouraged to find equal areas that are not congruent. Students should play with the shapes, rotating and flipping them, realizing that they can transform shapes and make new ones.
In our Investigate activity, students partition a rectangle into rows and columns where the pieces of the partition are equal. This activity starts to build toward a visual representation of multiplication and should be connected to area. Students investigate how to partition rectangles into rows and columns of square units, which should be challenging, giving students lots of opportunity for brain strengthening. It will be helpful and interesting to collect the different strategies they use.
