Let’s start by understanding what does it mean that two things are similar? Is it the same as being congruent? Identical?

Look at the elements above, and think of which of them look similar. Surely, the first thing that comes to mind is the bike wheel, the pizza, and the clock! They look all pretty “similar”, because they all have the shape of a circle. In fact, if we put the pizza and the clock together, or one on top of the other, we will see that they have the same size. However, the wheel is bigger than the other two.  

Then, there has to be a difference here, right? Yes, some of them are similar and others congruent.

We say that two or more figures are congruent if they have the same shape and size. We say that they are similar if they have the same shape, but not necessarily the same size.

Therefore, it is pretty clear now that the wheel, the pizza, and the clock are all similar, but only the pizza and the clock are congruent. Not identical! They are different objects that have the same size and shape.

With this definition, kids can surely realize that the kite and the diamond, that maybe at the beginning looked similar, are not. They clearly do not have the same shape because the kite has four sides, and the diamond has five.

The main questions to ask kids, in order to determine similarities between objects, are: do they have the same shape? Do they have the same size? Then, if they are not sure, you can let them overlap the objects to compare them more closely. Notice that to do so, we may need to turn or flip the objects around. Therefore, our definition of similarity and congruence do not take into account the location of the objects in space. Also, when comparing objects at home or at school with kids, you should let them know that we are comparing the main shape of each object. In our group, for example, we are forgetting about the leaf’s stem, and the kite’s tail.

Let's come back to our group of objects. We have two other pairs of elements there that look similar: the drop and the leaf, the envelope, and the chocolate bar. Let’s take each pair of these objects and overlap them to answer the questions above.  

We see that the leaf and the drop do have the same shape. In fact, if we could enlarge the drop through a lens, we would see that their shapes match. Since they don’t have the same size, they are similar. On the other hand, when we overlap the envelope and the chocolate bar, we notice that the length of one of their sides is the same, whereas the other is not. This means that they don’t have the same shape, because one is longer than the other. Sometimes, it isn’t clear for kids why the chocolate bar and the envelope are not similar. For this reason, we need to improve our definition of similarity using ratios.

For the rest of the lesson, we will focus on triangles. There are plenty of triangular items around us that we can use to explain triangle's similarity to kids. Consider, for example, the following triangular objects. Are they similar? Congruent?  

From the introduction, kids had probably acquired enough familiarity with the concepts to be sure that they will not find any similarity with the hanger or the fairy hat, because their shapes are very particular, and different from the others. And they would also think that the traffic sign and the tent are similar, as well as the napkin and the ruler square. But, how to be sure?

There are three basic criteria to determine whether two triangles are similar, and they are easy to remember for their acronym.

AA, stands for “angle and angle”, is the first one. It says that if two triangles have two of their angles equal, the triangles are similar.

Kids may learn how to use this criterion, even if they don’t know how to measure angles with degrees. First, let them know that an angle is the part determined by two of the sides of the triangle. Then, they just need to compare the angles by overlapping the triangles.

Take a napkin and a ruler square, for example, and put one on top of the other, so that the square angles match. Then, repite the procedure to compare two more angles. If they match, that's it, the two triangles are similar!  

Our second criterion, called SAS for “side, angle, side”, says that if two triangles have two pairs of sides proportional, and the angles determined by those sides are equal, then the triangles are similar.

To try this criterion with kids, cut two triangles from colorful paper, or out boxes painted with different colors. Cut them to be similar, overlapping the triangles as in the image from the left. Flip or turn one of the triangles, and place them on a table, so that they look as in the image from the middle (not similar at first sight). Kids may try to use AAA here and get that the triangles are similar. But instead, we should try to teach them how to use SAS. First, let them find the way of overlapping the smaller triangle on top of the bigger one (as in the image from the left), and then proceed to color the corresponding sides as in the picture from the right. This is a nice way of teaching them the notion of the correspondence of sides. Then, ask them to measure the sides with a ruler, and to write down the numbers on pieces of paper, as is shown in the image from the right.  

Here, they will be ready to take radios and verify that the corresponding sides are proportional. If they chose to overlap the triangles as in the figure from the left, they will have to use sides marked black and magenta to make the radios. So, the corresponding ratios will be 6/4 (using the black sides), and 9/6 (using the magenta sides). Since both ratios (simplified) are the same number 3/2, the triangles have two pairs of proportional sides; and since the angles determined by those sides are equal, the triangles are similar.

There is no need for a deep understanding of proportionality to use this criterion. In fact, in some cases, this method can be applied even if we don’t know the length of the sides in the triangles. For example, let us consider the tent and the traffic sign from above. As before, we overlap the figures to see if the angles are the same. Then, we determine how many times a side of the sign fits in a side of the tent. In our example below, it fits four times. Thus, the base sides of these triangles are in proportion 1/4. If we repeat the procedure with the left sides, we will see that they are also in proportion 1/4. Therefore, two pairs of sides are proportional, and the angles between them are equal. By SAS, the two triangles are similar.  

Finally, we have criterion SSS, an acronym of “side, side, side”. It says that if the corresponding sides of two triangles have the same proportion, then the triangles are similar.

We can explain this method to kids, by using the previous argument with the green and violet triangles. We had already verified that the black sides and the magenta sides are in the same proportion, thus it remains to verify if the third pair is also. Notice that their proportion is 3/2. Thus, by SSS, the triangles are similar. Of course, we knew this from before, but this is just another way of showing kids that the triangles are similar.

To correctly use these three criteria, it is very important making clear what correspondence means. First, by making the angles coincide, and then by marking the sides as we explained in the example before. That way, we will avoid taking wrong ratios.

We hope this article helps to teach kids about similarities. Below there is a link to interactive practices for similar triangles, or use worksheets below to have your students learn the old fashion way which is proved to be working.