Unit 1 – Congruence and Similarity

  • You may not often take the time to think about how you move objects around as you go about daily life.  Many of the movements you make in a given day involve changing directions.  In the first half of this unit, you will review and investigate different kinds of motion on a coordinate graph.  You will focus on developing specific directions to slide, flip, turn, and stretch flat shapes.  You will also learn how to show where the shapes will be after a series of moves.
  • In the second half, you will use mathematics you already know to investigate how shapes change size.  You will also determine the unknown length of a side in a figure with given information about the lengths of other sides in the figure and in related figures.

In this chapter, you will learn:

  • Transform shapes by flipping, turning, and sliding them on a coordinate graph.
  • Describe movement on a graph using coordinates and expressions.
  • Compare shapes and use similarity to find missing side lengths of polygons, especially triangles.

These concepts and skills align with the following Common Core State Standards:

8th Grade Standards

  • CCSS.Math.Content.8.G.A.1
    Verify experimentally the properties of rotations, reflections, and translations:
  • Lines are taken to lines, and line segments to line segments of the same length.
  • CCSS.Math.Content.8.G.A.1.b
    Angles are taken to angles of the same measure.
    CCSS.Math.Content.8.G.A.1.c
    Parallel lines are taken to parallel lines.
    CCSS.Math.Content.8.G.A.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.
  • CCSS.Math.Content.8.G.A.3
    Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
  • CCSS.Math.Content.8.G.A.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.

High School Standards

  • CCSS.Math.Content.HSG.CO.A.1
    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.
  • CCSS.Math.Content.HSG.CO.A.2
    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).
  • CCSS.Math.Content.HSG.CO.A.3
    Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
  • CCSS.Math.Content.HSG.CO.A.4
    Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  • CCSS.Math.Content.HSG.CO.A.5
    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.

    Understand congruence in terms of rigid motions

  • CCSS.Math.Content.HSG.CO.B.6

    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.
  • CCSS.Math.Content.HSG.CO.B.7
    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.
  • CCSS.Math.Content.HSG.CO.B.8
    Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

 

Statistics!

Today we began our study of statistics with a review of dot plots, histograms, and box and whisker plots.

Please complete the Problem Set from Lesson 2 tonight for homework, focusing on the differences between Median and Mean.

More Completing the Square

First of all, I want to make something VERY, VERY clear.  You do not always need to use completing the square when solving a quadratic equation.  If you can factor it, that is A LOT easier.  If you can solve it with square roots, that is also A LOT easier.

Completing The Square is something you should only use when no other options are available.

Think about that when solving problems 28 – 33 on page 674.  (Even though these problems are from the next section, all of them can be solved using one of the methods mentioned above.)

Then go back to page 667 and solve problems 28, 30, 33, 36, and 37 using completing the square.

Graphs and Tables of Quadratics:

Today in class, Kyle reminded us of an important lesson:  The shape of a graph is not necessarily the path of the object.

Then Bibi, Kevin, and Ari showed us how to identify key features of parabolas in a table.

Please complete the entire problem set tonight for homework, beginning on page S59.