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Objective

Common Core State Standards

■ 4.MD.6

■ 4.MD.7

4Measurement and Data

Tessellation AnglesA tessellation is a fascinating combination of geometry and art. It is an arrangement of shapes that cover a surface without gaps or overlapping. Tessellations also can be analyzed mathematically. In this activity, students will investigate the characteristics of and measure the angles found in tessellations.

Talk About ItDiscuss the Try It! activity.

■ Ask: What is true about the sum of the angle measures at each vertex? Are you surprised by this answer?

■ Say: With your finger, point to a vertex in any of your tessellations or designs. Circle around the vertex with your finger. Ask: How many degrees are there in a full circle?

■ Ask: What is the fewest number of shapes you can use to form a vertex with Pattern Blocks?

■ Ask: What is the greatest number of shapes you can use to form a vertex with Pattern Blocks?

Solve ItReread the problem with students. The sum of the angle measures at each vertex is 360°. Note that when one shape is used repeatedly, there may be more than one way to combine angles at a vertex. For example, with a blue rhombus, a vertex can be formed by 3 shapes (120° + 120° + 120°), 4 shapes (60° + 120° + 60° + 120°), 5 shapes (120° + 60° + 60° + 60° + 60°), or 6 shapes (60° + 60° + 60° + 60° + 60° + 60°). Other shapes, such as squares, can form a vertex in only one way.

More IdeasFor other ways to teach about tessellations—

■ Have students use Pattern Blocks to create various types of designs: (1) designs with rotational symmetry that start with a yellow hexagon in the center and continue outward to form a circular shape; (2) designs with horizontal or vertical line symmetry; and (3) designs with no symmetry.

■ Have students work in pairs to create a tessellation. One student begins by choosing any Pattern Block. Then the other adds a block. Students take turns placing blocks. Challenge them to create a pattern with symmetry.

Formative AssessmentHave students try the following problem.

What is the measure of angle X?

A . 30° B . 60° C . 90° D . 120°

150° 90°

60°

Measurement and Data

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Materials• Pattern Blocks (at least 6 of

each color block per group)• protractors (1 per group)• 4-Column Recording Chart

(BLM 6; 2 copies per group)• pencils (1 per group)

Try It! 30 minutes | Groups of 3 to 6

Here is a problem about tessellations.

Matt is gluing Pattern Blocks to a board to make a display for his room. He places

the blocks in a pattern so there are no gaps or overlaps. What must be true

about the sum of the angle measures at each vertex in the pattern?

Introduce the problem. Then have students do the activity to solve the problem. Distribute Pattern Blocks, protractors, recording charts, and pencils to students. Discuss the characteristics of a tessellation and make sure students understand the term vertex.

1. Have students create a tessellation for each Pattern Blocks shape. Say: Make sure you form at least one vertex for each tessellation and be sure your tessellations have no gaps or overlaps. Students use at least six of each of the following: yellow hexagon, red trapezoid, blue rhombus, orange square, green triangle, and tan rhombus.

3. Ask: Do you think the sum will be 360° if you use any combination of shapes? Say: Experiment with this question by creating a design that uses all six colors of Pattern Blocks. Find the sum of the angle measures at each vertex. Students can use 3 to 12 shapes to form a vertex. The sum is always 360°.

2. Say: Measure the interior angles for each type of Pattern Blocks piece. You will want to locate vertices and related angle sides. Have students start a table with these columns: Shape, Sketch, Sum (of Angle Measures at Vertex). Ask: What is the sum of the angle measures at each vertex? The sum is always 360°.

If students get a sum that is not 360°, have them check for errors in measurement or arithmetic. Encourage students to use their table showing the angle measures of each Pattern Block. For example, when a blue rhombus is part of a vertex, the angle measure is either 60° or 120°. Guide students to choose the correct measure.

Name Measurement and DataLesson

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Hands-On Standards, Common Core Edition

4

Use Pattern Blocks to model each tiling/tessellation. Select a vertex. What is the sum of the angles at that vertex? What is the sum of the angles at any vertex? Show work to justify your answer.

1.

one vertex ________________

any vertex _________________

2.

one vertex ________________

any vertex _________________

Using Pattern Blocks, model each tiling/tessellation. What is the sum of the angles at any vertex? Justify your answer.

3.

__________________________

__________________________

__________________________

4.

__________________________

__________________________

5.

__________________________

__________________________

Answer Key

Download student pages at hand2mind.com/hosstudent.

(Check students’ work.)

(Check students’ models.)

360˚ 360˚

360˚ 360˚

360˚; On the right side of a vertex are two angles that form a straight angle. On the left side of a vertex are two angles (90˚) that form a straight line. 180˚ + 180˚ = 360˚.

360˚; On the right side of a vertex are two angles that form a straight angle. On the left side of a vertex are two angles that form a straight line. 180˚ + 180˚ = 360˚.

360˚; On the right side of a vertex are two angles that form a straight angle. On the left side of a vertex are two angles that form a straight line. 180˚ + 180˚ = 360˚.

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Hands-On Standards, Common Core Edition

Challenge! Could you make a tiling/tessellation using three congruent squares and one equilateral triangle that has sides the same length as the sides of the square? Explain your answer using angle measures. Draw a picture to help.

Answer Key

Download student pages at hand2mind.com/hosstudent.

Challenge: (Sample) No; Where the vertices come together, there are three 90˚ angles and one 60˚ angle; 90˚ + 90˚ + 90˚ + 60˚ ≠ 360˚.

Name Measurement and DataLesson

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Hands-On Standards, Common Core Edition

4

Use Pattern Blocks to model each tiling/tessellation. Select a vertex. What is the sum of the angles at that vertex? What is the sum of the angles at any vertex? Show work to justify your answer.

1.

one vertex ________________

any vertex _________________

2.

one vertex ________________

any vertex _________________

Using Pattern Blocks, model each tiling/tessellation. What is the sum of the angles at any vertex? Justify your answer.

3.

__________________________

__________________________

__________________________

4.

__________________________

__________________________

5.

__________________________

__________________________

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Name

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Hands-On Standards, Common Core Editionwww.hand2mind.com

Challenge! Could you make a tiling/tessellation using three congruent squares and one equilateral triangle that has sides the same length as the sides of the square? Explain your answer using angle measures. Draw a picture to help.

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Name

153BLM 6 4-Column Recording Chart

BLM

64-Colum

n Recording Chart©

ETA

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