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Triangles and Parallelograms

I. Section Objectives

  • Understand the basic concepts of the meaning of area.
  • Use formulas to find the area of specific types of polygons.

II. Multiple Intelligences

  • Teach the material in this lesson, and then move on to the activity.
  • To complete this activity, you will need to prepare enough drawings for one- half of the students in the class. One- half of the class receives a drawing of a complex figure.
  • You will need to prepare the area measurements for this complex figure for the other half of the class.
  • Then hand out one to each student. Some students will receive drawings and some will receive measurements.
  • The students will need to figure out the measure of their figure and find the person in the room who has the correct area measurement for their figure.
  • The activity is complete when both persons are sure that they have been matched up correctly.
  • This is a noisy activity, but the students will have a lot of fun doing it. It also has a lot of movement in it which is excellent for kinesthetic learners.
  • Then repeat this activity. Be sure that those who received drawings get area measurements and those who had measurements receive drawings.
  • Intelligence- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Provide students with notes to refer to throughout the lesson and activity.
  • Area of rectangle = l \times w
  • Area of parallelogram= bh
  • Area of triangle = \frac{1}{2} bh
  • Write out the Congruent Area Postulate
  • Write out the Area of Whole is Sum of Parts Postulate.
  • Be sure that students copy these notes down in their notebooks.

IV. Alternative Assessment

  • Student assessment is done through the activity. Were the students able to find the correct “match- up?”
  • Assist students who have difficulty with the assignment.

Trapezoids, Rhombi and Kites

I. Section Objectives

  • Understand the relationships between the areas of two categories of quadrilaterals: basic quadrilaterals and special quadrilaterals.
  • Derive area formulas for trapezoids, rhombi and kites.
  • Apply the area formula for these special quadrilaterals.

II. Multiple Intelligences

  • Be sure to teach the material in this lesson in an interactive way. I would recommend going through the material without the text first. That way each student can explore the different characteristics of the three figures without using the information in the text as a guide.
  • Then go back to the text and go over the information in it. The students will see this in a new way because they will have already “discovered” it.
  • Here are some notes on each figure in the section. This will help the students to “break down” the content.
  • Trapezoid- quadrilateral with one pair of parallel sides.
  • Formula- \frac{1}{2} (b_1 + b_2)h
  • Finding the Area of a Rhombus
  • 1. Frame a rhombus in a rectangle.
  • 2. Notice all of the triangles.
  • 3. 4 triangles to fill in the rhombus
  • 4. 8 triangles fill in the rectangle.
  • 5. 4 is half of 8
  • 6. Area of rhombus = \frac{1}{2} area of a rectangle.
  • \mathrm{Formula}= A = \frac{1}{2} d_1 d_2
  • Finding the Area of a Kite
  • 1. Frame in a rectangle.
  • 2. Notice the similarities with the rhombus.
  • 3. Use the same formula as a rhombus.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal.

III. Special Needs/Modifications

  • Review finding the area of a rectangle.
  • Review finding the area of a parallelogram.
  • Review finding the area of a triangle.
  • Show how more than one figure can be combined together to create a new figure.
  • Example- a rectangle and a triangle together.
  • Review finding the area of such a figure.

IV. Alternative Assessment

  • Observe students as they work through the explorations.
  • Allow a lot of time for students to speculate and share their findings.

Area of Similar Polygons

I. Section Objectives

  • Understand the relationship between the scale factor of similar polygons and their areas.
  • Apply scale factors to solve problems about areas of similar polygons.
  • Use scale models or scale drawings.

II. Multiple Intelligences

  • There are several different components to this lesson.
  • First, we can start with the basic information to write the formula for finding the area of similar polygons. In working through this section, use an example on the board and take the students through each step in the text as you do the work out on the board. This will help them to “see” where the formula really comes from.
  • The next section is on scale drawings and scale models.
  • One of the best ways for the students to understand scale drawings is to complete one.
  • You could break the students off into pairs and have them create a scale drawing of the classroom. Allow students to use chart paper, rulers, tape measures, colored pencils and to create their own scale for the diagram.
  • Have students work in pairs to complete the table. You could also expand this activity and add Mt. Everest to the table.
  • The section on the giant can be fun. Ask the students to create a drawing to show how there aren’t any twelve foot giants. They can use the information in the text as a guide.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Review perimeter.
  • Review area of different figures.
  • Review Pythagorean Theorem.
  • Review finding the area of a rhombus.
  • Define squaring a number.

IV. Alternative Assessment

  • Use observation and student work product to assess student understanding.
  • You can collect work for a class work grade when students are finished.

Circumference and Arc Length

I. Section Objectives

  • Understand the basic idea of a limit.
  • Calculate the circumference of a circle.
  • Calculate the length of an arc of a circle.

II. Multiple Intelligences

  • When working through this lesson, it is a good idea to begin by reviewing previously learned information about a circle.
  • Have students brainstorm a list and then write them on the board.
  • These include the labels for radius, diameter, center angle, arc, interior, etc.
  • Also review that there are 360^\circ in a circle.
  • Then move on to the measurement for pi and having students understand the measurement for pi.
  • Use the exploration in the text for this.
  • The activity to differentiate this lesson comes in the example where the circle is inscribed inside the square on the graph paper.
  • The students can count the units to figure out that the length of the side of the square is also the diameter of the square.
  • Here is the activity.
  • 1. Have students draw their own circle inscribed in a square.
  • 2. Exchange papers with a partner.
  • 3. Each student must label the length of the diameter.
  • 4. Find the circumference of the circle.
  • Allow time for sharing when students have finished.
  • Walk through the section on how to find the arc measures. Be sure that the students understand the \frac{60}{360} ratio and how it makes sense to multiply the diameter with the measure of the arc to find the arc measure.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Write formulas for finding the diameter and the radius of a circle on the board.
  • You can even do a few examples to have students practice finding these measures.
  • Write the formula for circumference on the board.
  • Allow time for student questions.

IV. Alternative Assessment

  • Observe students as they work on the circle dilemma.
  • Offer assistance when needed.
  • Listen for how the students solved the dilemmas when the students are sharing their work after the activity.

Circles and Sectors

I. Section Objectives

  • Calculate the area of a circle.
  • Calculate the area of a sector.
  • Expand understanding of the limit concept.

II. Multiple Intelligences

  • Teach the material in this lesson and then differentiate it with the following activity.
  • Activity- have students work in pairs.
  • Students begin by drawing a square on graph paper and then inscribing a circle within the square. The students can decide how much of the square is taken up by the circle.
  • Have students shade in the area of the square around the circle.
  • Exchange papers.
  • Students work with each other’s papers.
  • They need to find the area of each circle.
  • Then they need to find the area of the shaded region of each circle.
  • Request that students write out the steps that they did to complete this assignment.
  • Allow time for students to share their work at the end of the activity.
  • When working with the sectors, be sure that the students understand what is meant by a sector.
  • Use a diagram to show students a sector in a circle.
  • Then show how it has an arc measure and how it also has a measure of the area of the circle.
  • This will help students to make sense of the formula to find the area of the sector.
  • Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Review finding the circumference of the circle.
  • Review pi.
  • Review the concept of the limit and how it leads to pi.
  • Write out the formulas on the board. Request students write these notes in their notebooks.

IV. Alternative Assessment

  • Collect the student worksheet from the activity.
  • Read all of the steps that the students wrote and check their process.
  • Is there anything missing? Do the students understand where each part of the formula comes from?
  • Is there higher level thinking here or are students just “using” the formula?

Regular Polygons

I. Section Objectives

  • Recognize and use the terms involved in developing formulas for regular polygons.
  • Calculate the area and perimeter of a regular polygon.
  • Relate area and perimeter formulas for regular polygons to the limit process in prior lessons.

II. Multiple Intelligences

  • When working through this lesson, be sure to explain each formula and how it was arrived at slowly and with detail.
  • I recommend beginning the lesson without the text.
  • Use the text as a teacher guide and break down the information in it for the students.
  • Use the board/overhead to show each step.
  • Begin by labeling the regular polygon and its parts in different colors.
  • You can use these colors to track through to the formulas.
  • For example, if you used red for the n in the diagram, then whenever the n is presented in a formula, you can put it in red.
  • Color will help the students to track the information from the diagram to the examples and back again.
  • Allow plenty of time for student questions and repeat material as necessary.
  • Intelligences- linguistic, logical- mathematical, visual- spatial

III. Special Needs/Modifications

  • Go to the simplest version of each of these formulas for the students to make sense of this unit.
  • Because there is so much processing in this lesson, special needs students will have difficulty following all of the different possible options.
  • Simplify it as much as possible.
  • You want the students to understand the core concepts involved.
  • Using color, as suggested above, will help special needs students.

IV. Alternative Assessment

  • Ask questions and answer a lot of questions in this lesson.
  • Since most of this lesson is about student process, be sure that the students are following the lesson.

Geometric Probability

I. Section Objectives

  • Identify favorable outcomes and total outcomes.
  • Express geometric situations in probability terms.
  • Interpret probabilities in terms of lengths and areas.

II. Multiple Intelligences

  • Review the basics of probability.
  • Review the ratio for probability.
  • Activity 1- Basic Probability- have the students use two number cubes and figure out what the probability would be to roll an even number.
  • Students can work in pairs during this activity.
  • Allow time for them to share their work when finished.
  • Are there any surprising results?
  • How did the students arrive at their answers?
  • Activity 2- Geometric Probability
  • Students work in pairs again.
  • The students work together to design their own problem for determining geometric probability.
  • Have the students write their problems out, use diagrams and not solve the problems.
  • You can use these problems at a later date.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Provide the students with some notes on probability.
  • Review how to convert fractions, decimals and percentages.

IV. Alternative Assessment

  • Collect student problems.
  • Check them for accuracy.
  • Reassign them to students for a homework or classwork assignment.
  • They could also be used for an extra credit problem.
  • Have students explain their answers and write them out in words not just show a solution.

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