Pointillism and Density of Numbers | A Great STEAM Lesson for International Dot Day

International Dot Day

Happy International Dot Day! This year International Dot Day is September 15, 2025. It’s a global celebration of creativity, courage and collaboration. It’s also the perfect time for a STEAM lesson on the density of numbers, a definition of a point, color theory, and pointillism.

Essential Learning

Math Standards

This lesson may take several days or a week or two depending on how deep you want to dive into pointillism. I would recommend collaborating with an art teach to maximize the use of class time.

This lesson is aligned to Grade 6 standards but can easily be adapted for 4^th grade through high school with simple modifications. You can use these ideas to address the following Common Core Math Standards, but this lesson could be used in any grade where there is an emphasis on fractions or rational numbers.

• 4.G.1 Draw points, lines, line segments, rays, angles (right, acute, and obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
• 6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes.
• G.CO.1 Know precise definitions of ray, angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and arc length.

This STEAM lesson with student worksheets can be found on my Math Lessons or Fine Arts Lessons pages!

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Essential Understandings

This lesson aligns to NCTM’s Big Idea 3 about rational numbers: Any rational number can be represented in infinitely many equivalent symbolic forms. Specifically, it aligns to Essential Understanding 3b:

Between any two rational numbers there are infinitely many rational numbers.

A STEAM Lesson for Math and Art

Part 1: What is a Dot?  

Give students markers and a piece of paper. Tell students to draw a dot and sign their name. Then collect their work. Hang them on the wall. (Note: You may want to do this activity the day before starting the lesson, so you have time to hang the dots on the wall.)

Part 2: The Dot

Read The Dot by Peter H. Reyonlds aloud to the class. After reading, have students make another dot and add their new pictures to your “dot wall.”

Summary

In the book, Vashti, thinks that she can’t draw. That is until a teacher encourages her by stating, “Just make a mark and see where it takes you.” The teacher then proceeds to make her sign her name on her picture of a dot. To further encourage her, when Vashti walks into the classroom the next day she discovers her dot displayed as a work of art in a beautiful frame. This act unleashes Vashti’s confidence, and she starts creating all sorts of different types of dots until she can fill a gallery. She then encourages a classmate who can only draw squiggles to do the same.

Part 3: Point vs Dot

Now have students draw a point. Ask students what’s the difference between a dot and a point. If they have difficulty, have them look up the definitions.  Discuss how a dot has width, but a point does not. A point has 0 dimensions. Its purpose is to show position (on a number line or coordinate plane, etc.) Generally, in mathematics you label a point with a capital letter.

Part 4: Density of A Line Segment

This is the meat of the mathematics. Students should leave this activity understanding that there are infinitely many rational numbers between any two rational numbers. In others words, the number line is dense. Students may come to these understandings working with their partner, but definitely hit these ideas home during the post-discussion reflection questions.

Math Activity

1. Give students a piece of paper with an unmarked line segment on it. Ask them to figure out how many points are on it. Have them work independently for 3-5 minutes, and then discuss with a partner. Have the students explain their answers to the class. Make sure they explain their thinking. (Be neutral, don’t tell them if they are right or wrong at this point.)
2. Give students another line segment the same length as Part A, but this time the endpoints should be labeled 0 and 1. Repeat the same procedure as in Part A.
3. Give students another line segment the same length as Part A, but this time the endpoints should be labeled ¹/₄ and ³/₄. Have them find at least two number in between the two endpoints. (To extend this, challenge your high-ability students to find 3 or more numbers.)
4. Give students another line segment the same length as Part A, but this time the endpoints should be labeled ^-3/₁₀ and ^-4/₁₀. Follow the directions for Part C.
5. Give students another line segment the same length as Part A, but this time the endpoints should be labeled ³/₁₀ and ⁷/₂₀. Follow the directions for Part C.
6. Give students another line segment the same length as Part A, but this time the endpoints should be labeled 2 ³/₁₀ and 2 ⁸/₂₅. Follow the directions for Part C.

Reflection Questions

After completing the math activity above, give students the reflection questions to answer with a partner. After having time to discuss, bring the groups together for a class discussion.

• How many fractions are between any two numbers?
• How many points are a number line?  
• How many points are a line segment? Is your conclusion true for even a very tiny short line segment? Explain.
• Does every line segment contain every number? Explain why or why not.
• Look at Segment AB and Segment CD. Are they the same length? How many points are on each? Explain how that can be.
• If a point has 0 dimensions, how can something such as a line segment contain infinitely many zero length points and still exist?
• Are fractions numbers? How do you know?
• A fraction has at least 2 numerals (one in the numerator and one in the denominator), is it one number or two numbers? Explain.

Note: You’ll find the answers to these questions and more information in the lesson plan found on the Math Lessons pages.

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Modifying the Lesson

To hit grade level concepts, I intentionally included negative fractions, fractions with unlike denominators and mixed numbers; however, this activity could easily be leveled up or down.  For younger students, have students use paper strips and have them find one fraction in between each number. For example, have students find one fraction between 0 & 1, 0 & ½, 0 & ¼, 0 & 1/8 etc. You could also modify this to focus on decimal fractions by using fractions whose denominator are multiples of 10, for example 1/10 & 2/10, 1/100 & 2/100 etc., and then have them show the equivalent fractions using decimal notation. For more advanced students, you could include more difficult numbers and even discuss how irrationals cause holes in the number line even though the number line appears solid.

Note: For students who struggle with understanding 0 and infinity, see my lessons and blog posts on A Mathematical Journey through Zero & Infinity-Part 1: A Mystery Box, Some Books, and a Contest or Engaging Math Lesson: Zero Zebras and Infinity.

You can find a more detailed lesson plan with a student handout on my Math Lessons and Fine Arts Lessons pages.

Part 5: Density of a Plane

Ask students how many points are on a plane. Then follow up by asking them what the difference between a line and a plane is? After discussion students should come to the conclusion that a plane is a system of two number lines (a horizontal axis typically labeled x, and a vertical axis typically labeled y.) Ask students if there is an infinite number of points in a plane. Then ask them if a plane like a number line is also dense.

Part 6: Building Background on Pointillism

Introduce students to pointillism.

Building Background

What is Pointillism?

Pointillism is a painting technique where the artist applies small dots of pure color to the canvas. When standing far away the eyes of the viewer, not the painter, mix the colors in his/her brain. It mimics the way that light works. At the time, this approach was much more scientific in nature than the impressionists who preceded the pointillists.

The Artists

George Seurat (1859-1891) is the inventor of this technique. His most famous painting is Sunday Afternoon on the Ile de La Grande Jatte. Paul Signac (1863-1935) was Seurat’s student and continued developing pointillism after Seurat’s death. Pointillism, although short lived, was very influential and inspired many different artists and art movements within art history. Other famous artists who experimented with pointillism are Camille Pissarro (1830-1903), Vincent van Gogh (1853-1890), Charles Angrand (1854-1906), Henry Edmund Cross (1856-1910), and Maximilien Luce (1858-1941).

Examples

Here are several websites to read more about pointillism:

• What Is Pointillism? Pointillist Landscapes | DailyArt Magazine
• Pointillism | Impressionism, Divisionism, Neo-Impressionism | Britannica
• The Art of Seurat: Science and Pointillism | madame scientist’s not-so-mad musings
• 7 things to know about Pointillism – Art Shortlist
• Pointillism – The Neo-Impressionist Dot Painting Technique (artincontext.org)

Show students some paintings that use pointillism. Then print out some pictures using a 4-tone color printer (cyan, magenta, yellow, and black), or if you can obtain a Sunday newspaper the colored comics section also works. Allow students to look at these pictures through a magnifying glass, so that they can see the dots. Tell student that they will be creating a pointillism painting after they learn a little about color theory.

Michel Eugène Chevreul and Color Theory

Michel Eugène Chevreul was a French chemist who developed a book titled The Laws and Contrasts of Color. He discussed how adjacent colors can affect the intensity of color. He noticed that the brain exaggerates colors when they are paired. He called this affect the exaggeration of differences. For example, a grey bar appears even lighter when placed next to a darker shade and the darker grey bar appears darker when placed next to a lighter hue—especially around the border.

When black and white are adjacent to a color, they appear tinted with the original color’s complementary color. For example, if you put red next to black it will appear slightly green. If you put two complementary colors together such as blue and orange, the blue will appear bluer, and the green will appear greener. Therefore, when two complementary colors are juxtaposed, they enhance each other.

Note: Mixing light and mixing paint pigments have different effectives (one is additive and the other is subtractive). Critics said that Chevreul seemed to blend the two methods at times; although others disagreed.

Part 7: Introducing Color Theory

Give students three strips of paper with six 1-inch adjacent boxes on each.

1. On the first strip, label the far-left box black and the far-right box white. Have students cover the black box densely with dots. Then working from left to front have them make the dots less and less dense as they progress through the squares. Once they are finished, hang them on the wall. Then have the students step back and look at them. Ask them why some of the boxes appear different shades of gray even though they used black paint. Ask students if they think their results would be different if they completed the exercise on black paper.
2. On the second strip tell each student to choose one paint color. Have students cover the far-left box densely with the color of their choice. Then working from left to right have them make the dots less and less dense as they progress through the squares. Make sure they keep the far-right box white. Once they are finished hang them on the wall, and have the students step back and look at them. Ask them why some of the boxes appear different shades even though they used only one color of paint. Ask students if they think their results would be different if they completed the exercise on colored paper.
3. On the third strip have them color Box 1 with red dots, Box 3 with yellow dots and Box 5 with blue dots. Tell them not to overlap the dots. Then have them place red and yellow dots in Box 2 (non-overlapping), yellow and blue dots in Box 4 (non-overlapping), and blue and red dots in Box 6 (non-overlapping). Once they are finished hang them on the wall, and have the students step back and look at them. Ask them why boxes appear orange, green, and purple even though they never mixed the paint. Ask students how this connects with the density of the number line and plane in their previous math lesson.
4. Then introduce students to the color wheel and the appropriate vocabulary: primary, secondary, monochromatic, complementary, analogous, triad, warm colors, cool colors, hue, intensity, tint, shade, and value.

Part 8: Using the Law of Simultaneous Contrast

Michel Chevreul’s Law of Simultaneous Contrast states how light from one color with affect how we perceive a nearby color. Two colors close together will take on the hue of the complement of the adjacent color. For example a dark color next to a light one make both appear brighter. Two analogous colors will make a color appear duller.

An artist can use complementary colors on the edges of objects to make something appear more intense and vibrant. It may also be helpful to know that warmer colors seem warmer when contrasted with cooler colors and vice versa.

Application

Have students practice painting a piece of monochromatic fruit such as a red apple, an orange, or a lemon using pointillism and the Law of Simultaneous Contrast. Students will need a pencil, a piece of paper, Q-tips, and paint. Allow students to use a wide palette of colors to paint their chosen fruit or just the primary colors depending on the result that you want. However, do not allow them to use black.

Show them how capturing the shadows on the apple using contrasting colors help create the depth and roundness of an object. It may be helpful to take a picture of an apple and zoom in and show students the photograph’s pixels.

Explain to students that negative space also matters and artists carefully choose the color of the paper that they use. Discuss the pros and cons of painting on a black paper vs a white paper vs colored paper. Explain that they can use the negative space (the white of the paper) to capture the light reflected on the fruit.

Post student work and special draw attention to student work that successfully uses the Law of Simultaneous color.

Step 9: Tying Everything Together

Compare and contrast pointillism and the density of numbers (both on a line and plane.) It may be helpful to use a Venn diagram. This will give you an opportunity to review the big ideas from both art and math. The key idea that addresses both art and math is that a plane (and a line) contain an infinite number of points. We can’t see the individual point because our eyes perceive the little dots/points as one object.

Step 10: Pointillism Painting

Have students create their own pointillism painting using a reference picture of their choosing. You may choose to skip this step depending on how much time you want to devote to pointillism in the curriculum. You may also want to integrate technology and have students create a pointillism painting using computer technology.

If you like this lesson, you might like my other STEAM lesson titled A Mathematical Journey Through Zero and Infinity Part 2: The Zeroth Dimension and Vanishing Points | a STEAM Lesson. Check out my other lessons on my Math Lessons page or if you want some STEAM lessons check out my Fine Arts Lessons page.

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I’d love to hear how you implemented this lesson in your classroom!