Grade 8: Rational and Irrational Numbers

Objective

Students will understand that some lengths (like the diagonal of a unit square) cannot be represented by rational numbers. Using the "tilted square" construction, students will see how square roots of whole numbers arise geometrically as lengths, and will practice computing areas to find these square roots (e.g., √2, √5, √25).

Materials

• "Grade 8: Rational and Irrational Numbers" resource (Chapter 7, Section 7.1, including Figures 2, 3, and 4)
• Graph paper or plain paper and pencils (for students to draw square grids)
• Ruler (optional, for Example 1 with Figure 1)
• Whiteboard/chalkboard for teacher notes

Warm-up (~5 min)

1. Draw a horizontal number line on the board. Mark 0, 1, 2, and a few negative integers.
2. Ask students to recall (or read aloud from the resource) how a rational number like 1/3 or 2/3 is placed on the number line by dividing the unit interval into equal parts.
3. Pose the question from the resource: "Do all points on the number line correspond to a rational number?" Tell students this is the big question for today's lesson.

Main Activity (~25 min)

Work through the tilted square constructions from Section 7.1 as a class.

1. (5 min) Set up Example 2 (Figure 2): Have students draw a 2×2 grid of unit squares on their paper. Instruct them to connect the midpoints of the outer square's sides to form a "tilted" square inside it.
2. Ask: What is the area of the whole 2×2 square? (4 square units)
3. Explain (using the resource's reasoning) that the tilted square is made of triangles that together equal half the area of each unit square, so the tilted square's area is 2 square units.
4. Since area = side², the side length of the tilted square is a number whose square is 2 — this is √2.

1. (8 min) Work through Example 3 (Figure 3): Have students draw a 3×3 square grid and construct the tilted square as described (connecting points to form four right triangles with legs 1 and 2 outside the tilted square).
2. Guide students to calculate: total area = 9, each triangle has area 1, four triangles = 4, so tilted square area = 9 − 4 = 5.
3. Side length of tilted square = √5.

1. (7 min) Work through the second part of Example 3 (Figure 4): Describe or draw a 7×7 square with tilted square inside, where outer triangles have legs 3 and 4.
2. Calculate: total area = 49, each triangle area = 6, four triangles = 24, so tilted square area = 49 − 24 = 25.
3. Since 25 = 5², the side length is √25 = 5 (a perfect square — a whole number answer, unlike √2 and √5).

1. (5 min) Class discussion: Ask students to compare the three results:
2. √2 and √5 do not appear to be whole numbers or simple fractions — these are examples of irrational numbers.
3. √25 = 5 is a whole number, so 25 is called a perfect square.
4. Reinforce the definition from the resource: "A positive integer whose square root is a positive integer is called a perfect square."

Wrap-up / Exit Ticket (~10 min)

Have students answer the following on a half-sheet of paper (based directly on the lesson):

1. In your own words, explain what the symbol √A means (a number whose square is A).
2. Using the tilted-square method, if a large square has side length 4 (area = 16) and each of the four outside triangles has area 3, what is the area of the tilted square inside? What is the side length of the tilted square (write it using a square root symbol)?
3. True or False: Every whole number has a whole number as its square root. Explain briefly using an example from today's lesson (e.g., √5 vs. √25).

Collect the papers as students finish, or have them turn them in at the door.

If Time Remains

Have students try constructing their own tilted square using a 4×4 grid (area = 16 square units). Ask them to pick a triangle size for the four corner triangles (e.g., legs of 1 and 3), calculate the area of the tilted square, and identify what square root that area represents. Students can compare their results with a neighbor.