Geometry and Measurement 7 th Grade Math Michael Hepola Henning Public School mhepola@henning.k12.mn.us
Executive Summary This 12-day unit is constructed with the idea of teaching this geometry section to students in the 7 th grade. Each instructional period is 45-minutes in length. My typical class size will be somewhere between 15 and 30 students. This unit starts out with the basic introduction of geometric shapes and their angles and then moves on to the calculating of the area of different geometric shapes. Minnesota Academic Standards 7.3.1.1 through 7.3.2.4 will be addressed in this unit. The unit will be opened with a Pretest in which we will measure the student s prior knowledge of some of the beginning stages of geometry and then we will conclude the unit with a Posttest. We will then compare the results of the Pretest and Posttest to analyze the effectiveness of this unit. Students will be led through traditional book lessons as well as putting those lessons to work using dynamic geometry tools such as Geometer s Sketchpad. Students should be able to answer the sample MCA questions found at the end of this report. 2
Table of Contents Day 2 Lines, Segments and Rays Page 4 Day 3-Angles Page 5 Day 4-Measuring Angles and Describing Them Page 6 Day 5-Classifying Triangles Page 7 Day 6-Classifying Polygons Page 8 Day 7-Identifying Parts of a Circle Page 9 Day 8-Finding the Area of Parallelograms and Triangles Page 10 Day 9-Finding the Area of Trapezoids and Irregular Figures Page 11 Day 10-Finding the Circumference and Area of a Circle Page 12 Sample MCA Questions for this Unit Page 13 Sample Pre-Test and Post-Test Questions Page 16 Source Citations Page 23 3
Day 2-(Day 1 will be Introduction and Pre-Test Standard No 7 th Grade Geometry and Measurement standard will be addressed today. Objective The students will be able to identify segments, rays and lines Launch Students will be directed to look up at the white board on which there will be a variety of drawings. These drawings will represent our vocabulary words for the day s lesson as well as get us started on defining what makes each of these unique. The different drawings will include a line, a point, a ray, a segment, a plane, 2 Intersecting lines and 2 Parallel lines. After studying these drawings for a couple minutes the students will then be asked some questions about these drawings. One sample question would be-(while pointing at the Segment and the Line) What do these two drawings have in common and what do they have not in common?, one possible answer we will be looking for is they both have 2 points so that is common but the one has arrows at the end and the other doesn t. We will then put the title of each drawing up on the board and go over what makes each of them unique. We will also go over how to name the segments, rays and lines. Explore Students will now be put into groups and be given a worksheet that includes different drawings from which they ll have to define what that drawing is by giving it the proper name. They will also be asked to draw some figures themselves by being given the name of a segment, ray or line. Students will be asked to look around the classroom and see if they can find some different parallel lines as well as some intersecting lines. The students will be asked to write a few of these down for their group. Share After a couple minutes or so the teacher will ask for some examples and will lead this class discussion. Hopefully the students will come up with several different lines so others can see examples other than theirs. Summarize Towards the end of class the teacher will ask the students to help discuss what was learned in today s lesson. The teacher will guide the students through this discussion to make sure they get the main points of the lesson. The outcome we ll be looking for is that they can distinguish between segments, rays, and lines; they must know how to label each of these and know what a point is. They will also be reminded that they learned about two different types of line relationships-- parallel and intersecting. 4
Day 3 Standard Geometry and Measurement: Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures. 7.3.1.1 Objective The students will be able to describe angles Launch Students will be asked a little bit about yesterday s lesson in which we learned about points, lines segments and rays. This will then lead us into a discussion about combining some of these items and what results from that. The teacher will then draw two rays with a common endpoint and describe to the students how this makes an angle. The sign for an angle will then be introduced and the vertex of the two rays meeting will be labeled along with a point on each ray. The students will then be taught how to name an angle using these three points. After seeing this drawing students will be introduced to the 4 different types of angles; acute, right, obtuse and straight and what defines each of these. Explore The students will then be given a handout with one section having several different drawings on it from which they must label the angle using proper labeling and also tell whether that angle is obtuse, acute, right or straight just by looking at it. The second section will have the students draw different scenarios out that are given to them, for example; draw Right Angle ACB. Share After having the students hand in their assignment we will have a few volunteer students go up to the white board and draw out some different scenarios given to them by the class. Summarize The teacher will remind the students that today s main objective was to distinguish different angles by looking at them and being able to define them as being right, obtuse, acute or straight angles based upon visual recognition. 5
Day 4 Standard Geometry and Measurement: Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures. 7.3.1.1 Objective The students will be able to measure and describe angles The students will be able to work with pairs of angles Launch The teacher will start out the class period by reviewing yesterday s lesson on the different types of angles and their definitions. The students will then be told they will be using that knowledge to help them with today s lesson. The teacher will now hold up a protractor and ask the students if anyone could tell them what that is and possibly what it is used for. Depending on the class, different answers will be given and the teacher may have to define it for them. The students will then be given directions on how to use a protractor to measure different angles correctly. Students will then be introduced to the terms complementary and supplementary and taught that two angles are complementary if their sum is 90 and two angles are supplementary if their sum is 180. The teacher will then be introduced to adjacent, vertical and congruent angles and their definitions. Adjacent angles formed by two intersecting lines are supplementary; Vertical angles formed by two intersecting lines have equal measures which makes them congruent. Explore Students will now be given a protractor and asked to find the measurement of various angles found on a handout. They will then have to classify each angle as obtuse, right, acute or straight. The second part of the handout will give them the measurement of one angle and they will then be asked to give either that angles complement or supplement. Share After the students have finished their sheets we will share the information the students came up with. We will discuss what possibly went wrong if we have students with different angle measurements using the protractor. Summarize The teacher will make sure the students have a clear understanding of what complementary and supplementary angles are as well as what adjacent and vertical angles are. Students will also be reminded they must know how to use the protractor correctly and if any students had difficulty on the handout those issues will be addressed. 6
Day 5 Standard Geometry and Measurement: Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures. 7.3.1.1 Objective The students will be able to classify triangles by sides and angles The students will be able to find angle measures of a triangle Launch The students will be asked to take out a piece of paper and draw 3 different triangles on that sheet of paper. After completing this, the teacher will ask for some volunteers to go up to the board and draw their triangles. The teacher will expect most of the triangles to look like an equilateral triangle, but if not that will be a good indicator that the students have some understanding of the material already. After a few students have drawn their triangles the teacher will ask for someone to define what a triangle is. If no correct answer the teacher will provide it. The students will then be reminded of the term congruent and then it will be used in the definition of the different types of triangles; scalene (no congruent sides), isosceles (at least 2 congruent sides) and equilateral (3 congruent sides). Students will be reminded that the angles of a triangle will always add up to 180 and the definitions of right, acute and obtuse triangles will be introduced to the students as well. Explore Students will then have an opportunity to work in groups and will be given problems in which they have to classify triangles by their sides and by their angles. In some drawings the students will be given 2 of the angles of a triangle and they will need to find the value of the 3 rd angle. Share The teacher will now bring all the groups together and they will share the results of their work. Hopefully the groups will agree with each other as this will demonstrate a pretty solid understanding of the lesson. Summarize Towards the end of the period the teacher will ask the students for the main points of today s lesson and hopefully some of the responses will include; the angles of a triangle add up to 180 degrees, an isosceles triangle has at least 2 congruent sides, etc.. The teacher will also ask the students; How many angles of a triangle do you have to know in order to find the remaining angle or angles? 7
Day 6 Standard Geometry and Measurement: Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures. 7.3.1.1 Objective The students will be able to classify polygons The students will be able to identify special quadrilaterals Launch The students will be introduced to how polygons are classified using the number of sides. They will be shown examples of a triangle, quadrilateral, pentagon, hexagon, octagon and decagon. Regular polygons or those with all sides congruent and irregular polygons which are those with sides that are not all congruent will be defined and shown to the students. Students will then be introduced to some special quadrilaterals. A trapezoid has exactly one pair of parallel sides and a parallelogram has both pairs of opposite sides parallel. Within parallelograms we have a variety of types including rectangles which have 4 right angles, a rhombus which has 4 congruent sides, and a square which has 4 right angles and 4 congruent sides. Explore The teacher will then walk the students through an exercise in which they will use graph paper to draw a variety of shapes based on the definitions they ve been given. The teacher after handing out graph paper will ask the students to draw several different shapes including all of the different polygons for which they were given definitions. The scale of each students drawings can vary as long as the rules of each polygons definition is followed. Share The teacher will now ask for a couple volunteer students to show their drawings up on the document camera. As each shape is presented the teacher will direct the conversation around things such as, what makes this object a?, etc.. Summarize The students will be told that it is important they are able to tell the difference between the different polygons and that they will be exploring these polygons further in the near future. 8
Day 7 Standard Geometry and Measurement: Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures. 7.3.1.1 Objective The students will be able to identify the parts of a circle Launch The teacher will draw a circle up on the board and ask the students what the picture is. Hopefully the students will be able to say it s a circle. The teacher will then ask the students what else they know about circles. The discussion will then turn to some of the key vocabulary which includes the radius, diameter, central angle, a chord, semi-circle and arcs. Explore Students will then be given a variety of circles with spots labeled and they must use the proper label to describe the center, radius, chords, diameter and central angles of each circle. They then will be asked to label different arcs for each circle as well as being able to draw different circles based on descriptions given. Share The students will be asked to share some of their thoughts and ideas about circles. From this discussion the will be directed towards what did we learn about the different parts of a circle. Summarize The teacher will make sure the students have a clear understanding of each of the different terms relating to circles by putting a circle up on the board and having the students answer out loud each question about the different parts. 9
Day 8 Standard Geometry and Measurement: Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures. 7.3.1.1 Objective: The students will be able to find the area of a parallelogram The students will be able to find the area of a triangle Launch The teacher will review with the students what a parallelogram and what a triangle are. The students will then be introduced to new vocabulary including; height of a parallelogram and base of a parallelogram. The formula for finding the area of a parallelogram will be given; Area (A)=Base (b) * Height (h) or A=bh. The area of a rectangle which is a special parallelogram is simply; Length (l) * Width (w) or A=lh. The students will then be introduced to the Area formula for a triangle which is Area (A)=1/2 * Base (b)*height (h) or A=1/2 bh. Explore Students will be asked to find the area of a variety of different parallelograms and triangles. Some of them will be given as drawings with the proper measurements labeled and others will simply give the measurements without a picture. Share Students will be asked to share what they ve learned. The teacher will also ask the students to get into groups and to grab a meter stick and find an example or either a parallelogram or triangle somewhere in the classroom and they are to measure the appropriate things to find the area of that object. Some possible examples would be the whiteboard, a computer monitor, cabinet door, etc Summarize The teacher will summarize the day s activities by reminding the students of the formulas and giving them a heads-up, that these formulas will be handy in determining the area of other objects as well. 10
Day 9 Standard Geometry and Measurement: Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures. 7.3.1.1 Objective: The students will be able to find the area of a trapezoid The students will be able to find the areas of irregular figures Launch The teacher will review the Area formulas for a triangle and parallelogram and then introduce the students to trapezoids and irregular figures. The students will be given the formula for finding the area of a trapezoid which is A=1/2h(b 1 +b 2 ) where A=Area, h=height and b=bases. After giving the students a moment to soak that formula in, the teacher then draws an irregular figure up on the board. The teacher then asks the students if any of the Area formulas they have been given would work for that object. After some discussion the teacher will direct them towards coming to the conclusion that you can divide that object up into smaller familiar looking objects and we can then find the area of those smaller objects and then add up the results of those. Explore The teacher will then have the students work through a variety of different figures of which they need to find the area. Some of them will be regular figures others will require the student to divide an irregular figure into smaller regular figures and then add the areas together. Share The teacher will then ask the students to get into some groups chosen by the teacher. The teacher will then give the students a couple real-life examples of irregular figures. These figures represent two rooms in a house. The students are to figure out the area of this irregular figure because the owner wants to replace the flooring in the room and with the cost of flooring does not want to spend any more than he/she needs to. Summarize The teacher will try and get the students to come up with some other examples of real-life application of the area formulas for different shapes. The teacher should use this time to make sure the students have a comfortable understanding of the material covered in Days 8 and 9. 11
Day 10 Standard Geometry and Measurement: Use reasoning with proportions and ratios to determine measurements, justify formulas and solve real-world and mathematical problems involving circles and related geometric figures. 7.3.1.1 Objective: The students will be able to find the circumference of a circle The students will be able to find the area of a circle Launch The teacher will introduce the students to the concept of the circumference of a circle by showing them a quarter and explaining to them the distance the quarter rolls until it is facing exactly the same direction is equal to is circumference which is the distance around a circle. Pi is introduced in regards to it being the ratio of the circle s circumference to its diameter. The circumference formula of C= where C=circumference, d=diameter and r=radius. Pi is explained as being represented by 22/7 or 3.14 or if using calculators the Pi key is pointed out to the students. It is also explained that their figures may vary slightly depending on what Pi approximation they use. The students will then be given the Area of a Circle formula which is A=πr 2. Explore The students are then walked through some examples of circles and they are to find both the circumference and the area of each of these circles. After going through these examples the teacher will use Geometer s Sketchpad and walk the students through an activity in which they draw a circle with 3 points labeled. The students are then shown how to write a formula using these three points to figure out the Area of the circle as well as the circumference of the circle. After writing their own formulas the students are shown that there is a way of calculating both the Area and Circumference built into Sketchpad. The students then are asked to animate their circles and keep an eye on the two formula results to ensure they have completed the correct formula. Share The teacher may ask a couple of the students to come up and share their Geometer Sketchpad documents with the class. Summarize The teacher will now make sure the students grasp the concept of Pi and its relationship to the Area and Circumference of a circle. 12
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7 th Grade Geometry Pre-Test Match each definition with the Letter of the drawing below that corresponds best. A B 1. Ray 2. Line 3. Point 4. Segment C D 16
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Sources Prentice Hall Mathematics Course 2. Pearson/Prentice Hall. Charles, Branch- Boyd, Illingworth, Mills, Reeves. 2004 Math Worksheets 4 Kids. http://www.mathworksheets4kids.com/ Minnesota Comprehensive Assessments-Series II, Mathematics Item Sampler Grade 7. http://connect.spps.org/uploads/grade7_math_mcaii_item_sampler.pdf 23