# Lesson 2 Pythagorean Theorem

## Math

The Geometry of a Ktunaxa Fish Trap

Teacher Note: The Pythagorean Theorem will be discussed in this lesson and should not be used as an introduction to a unit on Pythagorean Theorem. Students will need a basic understanding of right triangles and the Pythagorean Theorem.

Learning Outcomes

The students will:

• Demonstrate an understanding of the Pythagorean relationship by calculating the measure of the third side of a right triangle, given the two other sides.

CONNECT

Goals:

The students will:

• Translate a 3-D shape into a 2-D shape drawing amenable to geometric calculations.
• Recognize and then calculate the height of a right triangle given the hypotenuse and the other two sides.

Students generate word problems using pictures, words, and symbols relating to a video example.

Activate Prior Knowledge:

1. View drawings of Pacific Northwest Basket Traps.
2. Watch the following video on the Ktunaxa Fish Trap.

Reminder: It is important to stop throughout the story and give students (A/B partners) opportunity to talk or respond to the story.

1. Point out to students that the length of the sticks used in the fishtrap, including the sticks used to create the opening, were about the same. So, if overlap is included for the stick used to form the overlap, the length of each stick would be about 1 metre.
2. Review the following information Pythagorean geometry:
• A right triangle has a 90 degree angle called a right angle.
• The side opposite the right angle is called the hypotenuse.
• The sides of a right triangle are often labelled a, b, and c, with c being the hypotenuse.
• The Pythagorean relationship of the sides in a right triangle states that a2 + b2 = c2

PROCESS

Predict and Question:

Given the previous information, ask the students what is the approximate height of a fishtrap. Students discuss with A/B partners the different height possibilities.

Procedure:

Algebra can be used to rearrange the following relationship, for the times that the length of the hypotenuse is known and the length of one of the other sides is known.

Information given for dimensions of a fishtrap:

• Radius = 15cm. This is one side of the triangle (a)
• Hypotenuse = 100cm. This side is (c)

Have the students draw and label a triangle to match the problem.

Using ‘h’ as the unknown, have students write a formula to match the problem.

h2 = 1002 – 152

Given the above formula, have the students solve the equation and then write a sentence to answer the problem. (Round to the nearest tenth of a centimeter)

TRANSFORM

Students generate word problems using pictures, words, and symbols relating to the video example. For example, a variety of word problems can be created by using the same formula as above and changing the data.

A/B Partners – Student partner groups trade their word problems with other partner groups and solve problems.

REFLECT

When can you use the formula a2 + b2 = c2 to solve a problem in real application? Use examples in your explanation.

Extend learning or next lesson

Students can find examples of right triangles and right triangle shaped objects in the local community to present to the class. The presentation can take many forms such as collages/videos/ photographs/drawings, etc. Students use the visual connection in order to apply the formula a2 + b2 = c2.

# Lesson 1 Circles radius, diameter, and circumference.

Learning Outcomes

The students will:

• Demonstrate an understanding of circles by describing the relationships among radius, diameter, and circumference of circles.
• Demonstrate an understanding of circles by relating circumference to pi.

CONNECT

Goals:

The students will:

• Estimate the ratio between the diameter of a circular shape and its circumference.
• Observe the approximate ratio between the diameter of a circular shape and its circumference.
• Calculate the radius given the diameter.

Students generate word problems using pictures, words, and symbols relating to a video example.

Materials:

• A Video of a Ktunaxa Fish Trap
• A variety of small (15-20cm), medium (25-30cm), and large (40-50cm) flat circular shapes collected by students (ie. plastic lids, hub caps)
• 1-2 cm wide masking tape
• Length measurement tools (ie. rulers, metre sticks, tape measures)

Activate Prior Knowledge:

1. View drawings of Pacific Northwest Basket Traps.
2. Watch the following video on the Ktunaxa Fish Trap.

Reminder: It is important to stop throughout the story and give students (A/B partners) opportunity to talk or respond to the story.

1. Point out to students that the aboriginal fisherman who designed and built these traps would know the approximate size of the opening, because they would know the approximate size of the fish that they wanted to catch. So, given the size of the opening, they would have to estimate the length of the stick that was needed to form the opening.
2. Ask the students to name the geometric term for the distance across the opening (diameter), from the middle of the opening to the edge (radius), and the distance around the opening (circumference).
3. Tell the students that the relationship between the distance across the opening and the distance around the opening will be investigated.

PROCESS

Predict and Question:

Prior to this lesson, as mentioned in the materials section above, students should have collected a variety of flat, circular shapes to measure. The teacher should have some extra shapes on hand for those students who have not brought their own.

Ask students to hold up their shapes and pose the question “How many diameters are there in the circumference?” Collect students guesses with a show of hands.

Procedure

1. Distribute pieces of masking tape to each student or student pair.
2. While holding their shapes, the students take one piece of masking tape and stick it around the entire outer edge of their shape. Then, they peel off the piece of tape and stick it to a flat surface such as their desk, a wall, or floor; keeping the tape in a straight line.
3. Next, students take a second piece of masking tape to measure the diameter of their shape – ensuring the tape sticks across the middle of the circular shape. Then, after tearing the tape off at the outside edge, students place the smaller piece underneath/above the long circumference piece.
4. Once Each student has completed this activity, they should measure the length of both pieces of tape and write the lengths in cm on each piece. Then, in groups of 2-4 students, have them observe each others tapes and ask them to determine the relationship between diameter and circumference. What do they notice about their own data when compared to their groups’ data? What is similar?
5. Ask the students about how many diameters equal one circumference. Notice that three short pieces are a little shorter than the single long circumference piece. Using calculators, students divide the length of the circumference by the length of the diameter and see what answers they come up with. Compare these answers with other students and look for similarities.
6. On the board, write the relationship: C/D=p(3.14). The actual answer is an “irrational number”. It cannot be written as a fraction, or as a terminating or repeating decimal. 3.14 is an estimate.
7. The opening of the fishtrap was usually about 30 cm. To the nearest hundredth place, have students calculate the circumference of the fishtrap; showing all the steps in their calculations: C=pd (C=p(30 cm))

TRANSFORM

Students generate word problems using pictures, words, and symbols relating to the video example. For example, a variety of word problems can be created by using the same formula as above and changing the data.

A/B Partners – Student partner groups trade their word problems with other partner groups and solve problems.

REFLECT

When can you use the formula C=pd to solve a problem in real application? Use examples in your explanation.

Extend learning or next lesson

Students can find examples of circles and circular shaped objects in the local community to present to the class. The presentation can take many forms such as collages/videos/ photographs/drawings, etc. Students use the visual connection in order to apply the formula C=pd.