My learning progression is on converting percents, decimals, and fractions. It will lead to comparing ratios and using fractions and percents for ratio lengths. I will cover how to convert from percent to decimal to fraction and vise versa. I will start off with a hinge question then go in with my lesson. I plan to have the students in groups of four and they will be handed a worksheet of conversion.

This learning progression will be taught in a 10th grade High School Geometry classroom and the following three Common Core State Standards will be used as goals: HSG.CO.A.3, HSG.CO.A.4 and HSG.CO.A.5. Additionally, the mathematical practices that align with this learning progression are the following: MP5 Use appropriate tools strategically, MP7 Look for and make use of structure, and MP8 Look for and express regularity in repeated reasoning. The textbook this class uses is CORD Geometry 1 Learning in Context 4th edition. Specifically, I will be using chapter 4 on polynomials and factors.

Students have previously been introduced to patterns and relationships between two things. For example, in previous classes they have discussed series of geometric functions and more recently similarities in triangles. Specifically, students have worked with distances on the coordinate plane is relation to the origin and a point and how this constant equidistance moved around the origin would rotate a point which was a topic discussed in chapter 3. In this learning progression students will be once again looking at these rotations of points with the larger and focused picture of rotations, reflections, and translations of various geometric figures. The first skill and portion of this lesson will be to examine how to manipulate an object/shape on the coordinate plane through reflections that will maintain its original form. The various reflections, which will depend on the geometric object used at the time, will include over the y-axis, x-axis, y=x and y= -x. The students will practice this process with a variety of shapes and discover which shapes can or cannot use one of the above mentioned reflections. For example, a square can be both reflected across the y= x and y= -x, but a rectangle cannot. The second lesson will then discuss rotations of the same geometric objects and once again help students investigate which rotations certain shapes can and cannot rotated x number of degrees. The third lesson in this progression will be to consider translations of objects so as to still maintain shape, size, and angles. The third lesson will also help clarify the previous topics with practice with all of the movements as well as help students understand notation for the actions, like T(2, 3) referring to a translation of all vertices of an object to the right by 2 and vertically by 3.

Thus, this learning progressions is all about helping students see both with physical pictures and in their minds eye the mirrors, rotations, and translations found everywhere in mathematics and life. As can bee seen from the pictures above reflections are as common as looking into a pool of water. Therefore, modeling such actions with mathematics can help students understand its importance to our everyday life and aspiring to teach this beautiful concept as a guided lesson and at a pace that will not overwhelm any of our students.

The following curriculum is a Geometry curriculum based on and designed for alternative, second year, high school math students. The curriculum will following along with the Common Core State Standards (CCSS) for Washington State. The curriculum course will begin with a basic algebra review, to make sure all the students are at the same basic level of math, then each standard section will have an 8^{th} grade CCSS that will lead the students up to the high school standard. For each CCSS cluster, abstracts of lessons, lesson plan, worksheets, and types of assessments will be provided. In some areas we will also provide learning progressions to be used as examples for how learning progressions may be put together.

We will first start with an overview of what Common Core Standards will be used, then we will go into more detail of each cluster with what the standard states, how the lesson will be taught and what assessments will be given. For some of the lessons 8^{th} grade standards and high school standards will be aligned together and taught as one lesson. The overview points will be from the high school geometry common core standards, unless marked otherwise.

Resources we will be using for ours lessons will come from Kutasoftware.com, Math-Aids.com, and Illustrative Mathematics. Kutasoftware and Math-Aids are both websites for teachers to go and use free worksheets for their classrooms, both also have memberships available for more worksheet options. Illustrative Mathematics is an academic website that contains lessons which align with the CCSS.

After working with these types of students we have learned they very much enjoy hands on activities, as well as group work/project. We highly recommend finding ways to make each assignment hands on, if possible.

Geometry Overview

Congruence

HS: Experiment with transformations in the plane. Tie in 8^{th}: Understand congruence and similarity using physical models, transparencies, or geometry software.

Understand congruence in term of rigid motions

8^{th} Grade standard: Understand and apply the Pythagorean Theorem

Prove geometric theorems

Make geometric constructions

Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations

Prove theorems involving similarity

Define trigonometric ratios and solve problems involving right triangles

Apply trigonometry to general triangles

Circles

Understand and apply theorems about circles

Find arc lengths and areas of sectors of circles

Geometric Measurement and Dimension

8^{th} Grade standard: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres

Explain volume formulas and use them to solve problems

Visualize relationships between two-dimensional and three dimensional objects.

For this curriculum we have taken out the section Expressing Geometric Properties with Equations because this will be used in an Algebra 2 setting.

Congruence G-CO

Experiment with transformations in the plane

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Lesson Abstract: Students will be shown objects, such as an angle, perpendicular lines, parallel lines, etc. and be given precise definitions of those objects. Students can either take notes or be given papers with a picture of the object and its definition. The worksheets will be used to implant the definitions into the student’s brains to be used later on.

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Lesson Abstract: Starting out with a refresher for the students, we will refresh students on parallelograms, rectangles, squares, etc. to lead up into the use of Geogebra (which is a free, downloadable, mathematics software.)

This activity was originally designed as a multiple-choice worksheet where the students were prompted to match the definition in one column to the proper quadrilateral or polygon in the other column. After spending more time with these students, I realized using activities where they get up and interact with one another rather than handing them a worksheet that they can simply copy answers from their neighbors is far more reliable and useful. Getting the students to have to think individually has a higher rate of understanding than worksheets.

Though this is not a traditional assessment, you are able to assess what my students know and if they are able to identify different polygons and quadrilaterals, which is the central focus of this activity.

This activity shows that the students know the language and can connect definitions with meanings.

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Lesson Abstract: From the previous lesson students will what a parallelogram, rectangle, trapezoid, and a regular polygon is. From this students will be shown basic reflections, rotations, and translations. First students will be given the definition of each transformation and an example of how they would be used. Using real world items, such as a cell phone, will help students understand transformations better.

Having students split up into groups (preferably 3 depending on class size) give each group a different transformation to perform on an object, either one provided in class or drawn on graph paper, then have the students demonstrate to the whole class the step they took to perform the transformation given to them.

After each group has demonstrated their knowledge of their transformation the worksheets will be handed out for students to work on.

8^{th}: Understand congruence and similarity using physical models, transparencies, or geometry software.

Verify experimentally the properties of rotations, reflections, and translations:

a. Lines are taken to lines, and line segments to line segments of the same length.

b. Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two dimensional figures, describe a sequence that exhibits the similarity between them.

Lesson Abstract: As done in the previous lesson, this lesson is a continuum of that to help insure knowledge is being obtained.

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Lesson Abstract: At the beginning of the lesson the teacher will refresh the students on their knowledge of perpendicular lines, parallel lines, and angles. Then showing students how the degree measurement of a line is 180 degrees and if you cut a line with another line you create more angle measurement. From this students can be shown how the angles of perpendicular lines are 90 degrees and what happens between angles created by transversals cut onto parallel lines. Students will learn the terms corresponding angles, alternate interior and exterior angles, and same side interior and exterior angles.

Understand congruence in terms of rigid motions

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Lesson Abstract: This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern. The teacher may wish to spend time having the students visualize the pattern of seven hexagons instead of providing a picture, this is particularly important if the students are not familiar with this pattern.

This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square. This task uses the fact that the interior angles of a regular octagon measure 135∘ Students can also deduce this as part of the problem but this calculation is secondary to careful reasoning about reflections.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Lesson Abstract: The goal of this task is to understand how congruence of triangles, defined in terms of rigid motions, relates to the corresponding sides and angles of these triangles. In particular, there is a sequence of rigid motions mapping one triangle to another if and only if these two triangles have congruent corresponding sides and angles.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Lesson Abstract: During this lesson students given the definitions of triangular congruence, shown the different congruence’s, and given the opportunity to determine whether two given triangles are congruent or not. Students can also be shown one triangle transformed is still the same congruent triangle.

8^{th}: Understand and apply the Pythagorean Theorem.

Explain a proof of the Pythagorean Theorem and its converse.

Lesson Abstract: Students can be shown, via YouTube, different videos explaining the Pythagorean Theorem, and how it can be used to prove right triangles. Students will be given the equation to determine the Pythagorean Theorem and explained to that the legs of a right triangle squared equals the amount of the hypotenuse squared.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Lesson Abstract: Given two sides of a right triangle, students will be asked to use the Pythagorean Theorem to find the missing side. Using real world problems, such as given a building has such and such height and that the tip of the building shadow is such and such distance away, what is the hypotenuse of the right triangle create from the given information.

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Lesson Abstract: For this lesson students will be asked to recall graphing coordinate points. Given two points on a graph have the students create a right triangle, using the right triangle have students determine the Hypotenuse of that triangle to determine the distance between the two points.

Lesson Abstract: In previous lessons students have been show congruency and been working on certain theorems. From those this lesson can take their knowledge more in depth with proving theorems and congruencies. Students will be taught to make a two sided table, one side Statements, the other side Reasons. The Statements side will be what students know about what proves the theorems, and the Reason are why that’s true.

EX: Recall that vertical angles are a pair of opposite angles created by intersecting lines. Prove that vertical angles are congruent.

Solution: For this proof, you are not given a specific picture. When not given a picture, it helps to create a generic picture to reference in your proof. It’s important that the picture does not include any information that you cannot assume.

Lesson Abstract: This task is intended to help model a concrete situation with geometry. Placing seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?

Lesson Abstract: Triangle congruence criteria have been part of the geometry curriculum for centuries. For quadrilaterals, on the other hand, these nice tests seem to be lacking. This task addresses this issue for a specific class of quadrilaterals, namely parallelograms. This task is ideal for hands-on work or work with a computer to help visualize the possibilities. It turns out that knowing all four sides of two quadrilaterals are congruent is not enough to conclude that the quadrilaterals are congruent. Unlike with triangles, some information about angles is needed in order to conclude that two quadrilaterals are congruent.

Make geometric constructions

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Lesson Abstract: For this lesson in the school year of 2014-2015 the students use Bulsa wood to construct geometric bridges, which we calculated how much weight each bridge could hold to determine structural stability. Materials such as glue, wood, straightedges, compasses, and razor cutter were provided by the teacher, students were not allowed to take supplies home, as to not lose them. A lesson such as this would be a great lesson for the students.

At the end of the congruence cluster possibility for Quiz.

Similarity, Right Triangles, and Trigonometry G-SRT

Understand similarity in terms of similarity transformations

Verify experimentally the properties of dilations given by a center and a scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Lesson Abstract: Standard G-SRT.1 asks students to “Verify experimentally” that a dilation takes a line that does not pass through the center to a line parallel to the original line, and that a dilation of a line segment (whether it passes through the center or not) is longer or shorter by the scale factor. This task gives students the opportunity to verify both of these facts for a specific example. It may be helpful to provide students with rulers so that they can measure and duplicate lengths without having to perform formal constructions. When working on this classroom task, students should be provided with a separate copy of the picture (or more space should be provided below the picture) so that they can draw the points A’, B’, and C’.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

Prove theorems about triangles.

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

SRT.2,3,4&5 Taught in CONGRUENCE, possible refresher worksheets

Define trigonometric ratios and solve problems involving right triangles

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Lesson Abstract: The purpose of this task is to use the notion of similarity to define the sine and cosine of an acute angle. When two triangles are similar triangles ratios of their corresponding side lengths are equivalent. In the special case when the triangles in question are right triangles and the ratio is that of a leg to the hypotenuse, these ratios depend, up to equivalence, only on the (acute) angle measures of the triangle. These equivalent ratios provide the definition of the sine and cosine of any acute angle.

Explain and use the relationship between the sine and cosine of complementary angles.

Lesson Abstract: The goal of this task is to provide a geometric explanation for the relationship between the sine and cosine of acute angles. In addition, students will need to think about the sine and cosine as functions of an angle in order to reason about when sina=cosa.

The picture is vital for both parts of this task. If done shortly after students have seen definitions of the sine and cosine students will hopefully quickly think about drawing and labeling an appropriate right triangle. If students are struggling to get started the teacher may need to push them in this direction.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

Lesson Abstract: This task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles (G-C.5), using trigonometric ratios to solve right triangles (G-SRT.8), and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found (MP.7). It also engages students in some of the steps of the modeling cycle, in particular:

(2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,

(3) analyzing and performing operations on these relationships to draw conclusions, and

(4) interpreting the results of the mathematics in terms of the original situation.

Lesson Abstract: The standard G-C.1 asks us to prove that all circles are similar. This means that given any two circles there is a sequence of transformations of the plane (reflections, rotations, translations, and dilations) transforming one to the other. If we give the plane a coordinate system, then the circles have equations and the transformations can be described explicitly with equations. The goal of this task is to work on showing that all circles are similar using these two different methods, the first visual and the second algebraic.

Identify and describe relationships among inscribed angles, radii, and chords.

Lesson Abstract: This lesson is used to introduce circles and their associated parts. The students will be instructed to write definitions and equations on the left. On the right, students are to then draw the corresponding parts of the circles into the diagrams provided.

Generally, this lesson will come after learning about regular and irregular polygons. But before the unit wrap up.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

(+) Construct a tangent line from a point outside a given circle to the circle.

Lesson Abstract: From the previous lessons students should know what inscribed angles of a circle are. From this students can be shown how to construct inscribed and circumscribed circles of triangles. Geogebra the teacher can show the students how to prove properties of angles and students can use that knowledge to prove properties for quadrilaterals inscribed in a circle.

Find arc lengths and areas of sectors of circles

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Lesson Abstract: This is a challenging task which requires students to carefully divide up the picture into different pieces for which the area is known. In particular, the area of the three circles is readily computed and so the key is to determine the shaded area. This can be done by noticing that when combined with three sectors of the circles it forms an equilateral triangle. Thus, in addition to understanding how to calculate the area of a circle sector, students will need to know the formula for the area of an equilateral triangle.

Geometric Measurement and Dimension G-GMD

8^{th:} Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Lesson Abstract: Given the terms cones, cylinders, and spheres, students will be given equations to solve for the volumes of different shapes and sizes of the provided term. Teachers can provide real world object, such as ice cream cones, canned foods, and different shaped balls for students to solve for the volume of.

Explain volume formulas and use them to solve problems

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Lesson Abstract: This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere. Students who are given this task must be familiar with the formula for the volume of a cylinder, the formula for the volume of a cone, and Cavalieri’s principle.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

Lesson Abstract: From previous lessons students should have learned equations to finding volumes of objects, and possible surface area, if not give them a few example problems of each equations needed to solve real world problems of cylinders, pyramids, cones, and spheres.

Visualize relationships between two-dimensional and three-dimensional objects

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Lesson Abstract: This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a “double-naped cone” with vertex at the center of the sphere and bases equal to the bases of the cylinder.