Valentine's Day Math Activity: Heart Puzzle

Posted by Angie Seltzer on 1/30/2023 to K-5 Geometry

Fun Heart Puzzle Activity

By combining basic geometric shapes such as rectangles, triangles, circles, and squares in various ways, students can create more complex shapes. For example, two identical right triangles can be combined to form a kite shape, a parallelogram, or a rectangle. (One triangle may need to be flipped.)

In this free printable activity, students will make a heart shape and then use the same pieces to make other shapes. Click the image to view and download the three-page PDF, including the puzzle and answers.

What quadrilaterals can you make from the triangles in this puzzle?

The Challenge question asks what other shapes can be made from two or more puzzle pieces. Download the PDF to see possible answers. This activity gives students an opportunity to discuss several types of quadrilaterals, which are four-sided polygons.

A rectangle a four-sided shape with four right angles and opposite sides of equal length.
A parallelogram is a four-sided shape that has two pairs of opposite sides parallel and of equal length. The opposite angles are the same size.
A trapezoid is a four-sided shape with one pair of parallel sides. (Guideline vary as to whether a parallelogram should be considered a specific case of a trapezoid. That is, some definitions of trapezoid say "exactly one pair of parallel sides" while others say "at least one pair of parallel sides.) The side lengths and angles of a trapezoid can vary.
A kite is a four-sided shape that has two sets of adjacent sides of equal length. Also, a kite has one pair of opposite angles the same size.

What's special about the three triangles in the heart puzzle?

Have students compare and contrast the triangles. Here are things they may notice.

Each triangle has a right angle and two smaller (acute) angles.
If you match up the angles (corners), you will see that the angles are the same size. (The sides are different). For younger students, you may say that the angles have the same pointiness. Also, explain that sides lengths do not matter when you are comparing angles.
These three triangles are mathematically similar. In math, similar means having the same shape but not necessarily the same size. Think of being able to enlarge or reduce one triangle to fit exactly one top of another one. In common language (not math), similar can refer to various characteristics. For example, one type of cheese may have similar flavor to another type. Remind students that similar in math has a very specific meaning.

How can you use any rectangle to make a set of three similar triangles?

It's easy! Here are diagrams with an explanation below.

Cut a piece of paper to make any large rectangle.
Using a ruler, draw a diagonal line segment from opposite corners.
Cut the rectangle along the diagonal to make two triangles.
Position one triangle so that the longest side is the base, and draw a segment perpendicular to the base that connects to the opposite corner (the right angle). This segment is called an altitude. One way to make a perpendicular segment is to slide the edge of a sheet of paper along the base until the paper just touches the opposite corner or top point of the triangle. Mark the point along the base that is just below the top point.
Draw the altitude and cut along this line.

How can this activity be extended for middle school students?

With middle school students, take this activity a step further. Have them measure the sides of any pair of similar triangles and check whether the lengths are proportionate. The diagram below shows how to write three ratios for a pair of triangles. Any two of the ratios can form a proportion.

Have students write measurements instead of the letters a through f. Then divide the numerator by the denominator for each ratio. Since measurements are not exact, these ratios might not be exactly the same, but they should be close. Students can write ratios and proportions for the triangles in the heart puzzle. And, if they made three right triangles from a rectangle, they can check those ratios as well.

How does composing shapes help students?

An activity involving composing shapes helps students in two main ways.

The activity provides opportunities for students to discuss shape names and vocabulary.
It helps students improve their visual thinking skills. For example, students can predict what the composed shape will look like. Then, as students move the pieces, they can see if the result matches their prediction. If necessary, students can adjust the placement of pieces and try again.

Have a fun Valentine's Day!

Angie