9.3.4C Algebra in Geometry
Solve real-world and mathematical geometric problems using algebraic methods.
Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure.
Overview
Standard 9.3.4 Essential Understandings
Students focusing on geometry have typically already learned and used a significant amount of algebra in their mathematics course work. This might mean they have completed at least one separate course in algebra or have studied algebra as part of an integrated approach to mathematics. Either way, they must be able to recognize the close, integral relationship between algebra and geometry and be able to use this interconnectedness to solve mathematical problems. Activities in this standard will help students do this by showing them places in their course work and also in real life where geometry and algebra are used together in order to solve problems.
In this standard, students will connect algebra and geometry in a number of ways. They will calculate distances and angle measures using trigonometric ratios. They will look at slope of a line, midpoint of a segment and length of a segment using formulas derived from coordinate geometry. They will calculate geometric transformations through the use of the Cartesian plane. In addition, they will graph diameter and circumference of round objects, in order to see a visual interpretation of the number pi.
All Standard Benchmarks
9.3.4.1
Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle.
9.3.4.2
Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios.
9.3.4.3
Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts.
9.3.4.4
Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments.
9.3.4.5
Know the equation for the graph of a circle with radius r and center (h, k), (x - h)2 + (y - k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations.
9.3.4.6
Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90˚, to solve problems involving figures on a coordinate grid.
9.3.4.7
Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure.
9.3.4 Benchmark Group C - Algebra in Geometric Problems
9.3.4.7
Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Relate algebra with geometry in specific problem solving situations;
- Apply algebraic formulas and/or equations to geometric settings.
Work from previous grades that supports this new learning includes:
- Solve algebraic equations;
- Connect algebra with geometry.
NCTM Standards
Geometry
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:
- Analyze properties and determine attributes of two- and three-dimensional objects;
- Use trigonometric relationships to determine lengths and angle measures.
Specify locations and describe spatial relationships using coordinate geometry and other representational systems:
- Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations;
- Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.
Apply transformations and use symmetry to analyze mathematical situations:
- Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices;
- Use various representations to help understand the effects of simple transformations and their compositions.
Common Core State Standards (CCSS)
HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Define trigonometric ratios and solve problems involving right triangles.
- HS.G-SRT.8. 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.
HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Apply trigonometry to general triangles.
- HS.G-SRT.9. Explain and use the relationship between the sine and cosine of complementary angles.
- HS.G-SRT.10. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
HS.G-GPE (Expressing Geometric Properties with Equations) Translate between the geometric description and the equation for a conic section.
- HS.G-GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
HS.G-GPE (Expressing Geometric Properties with Equations) Use coordinates to prove simple geometric theorems algebraically.
- HS.G-GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
- HS.G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
HS.G-CO (Congruence) Experiment with transformations in the plane.
- HS.G-CO.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).
- HS.G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
- HS.G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
- HS.G-CO.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.
HS.A-CED (Creating Equations) Create equations that describe numbers or relationships.
- HS.A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Misconceptions
Student Misconceptions and Common Errors
- Students might interchange the terms circumference and diameter.
- Students might interchange the terms perimeter (or circumference) with area.
- Students might apply formulas incorrectly.
Vignette
In the Classroom
Teacher: Today, as part of the work for this unit, we're going to work with an activity from the National Council of Teachers of Mathematics. It will involve measuring the distance around some round objects, and also measuring the distance across the objects.
What's another word for distance around a circle?
Student 3: Circumference.
Teacher: Right. What's another word for distance across a circle?
Student 4: Diameter, if it's a chord that goes through the circle's center.
Teacher: Yes, it is. OK, now you can get started.
This activity comes from the National Council of Teachers of Mathematics (NCTM). It is titled "Pi Line" and is found on NCTM's Illuminations website. The activity requires students to measure, as accurately as possible, circumference and diameter of "round" objects. See the notes on the website below for specific materials needed for this activity.http://illuminations.nctm.org/LessonDetail.aspx?id=L575
Resources
Teacher Notes
- Emphasize the definitions of terms.
- Check at regular intervals that students are applying formulas correctly.
- Have a large sheet in the classroom with definitions and/or formulas on it.
Additional Instructional Resources
National Council of Teachers of Mathematics (NCTM). (2010). Focus in High School Mathematics: Reasoning and Sense Making in Geometry. Reston, VA: National Council of Teachers of Mathematics.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- What other instructional strategies can be used to engage students' study of angles associated with parallel and perpendicular lines?
- How can manipulatives be used to help students visualize these abstract concepts?
- How can the instruction be scaffolded for students?
- What additional scaffolding is needed to provide ELL students?
- Do the tasks that have been designed connect to underlying concepts or focus on memorization?
- How can it be determined if students have reached this learning goal?
- How can the lesson be differentiated?
Assessment
Teacher: Today, as part of the assessment for this unit, we're going to work with an activity from the National Council of Teachers of Mathematics. One of the questions I'd like you to think about as you do the activity is this:
When will a square's perimeter be equal to its area?
Student 1: I don't think the perimeter will ever be equal to its area.
Teacher: Can you explain your thinking?
Student 1: Well, the perimeter measures distance, usually in linear units, and the area measures surface, usually in square units. Since these are two different concepts, the two will never be equal.
Teacher: That's right. How might you re-phrase the question?
Student 1: When will a square's perimeter have the same numerical value as its area?
Teacher: That sounds good. This is one of the questions you should think about as you work on this activity. You'll be working with one partner in the lab, and I'll switch you halfway through. In other words, if you spent the first half manipulating the computer, then you'll spend the second half writing and recording, and vice versa. OK, let's go to the computer lab.
This activity comes from the National Council of Teachers of Mathematics (NCTM). It is titled "All in the Family" and is found on NCTM's Illuminations website. It does require Internet access, either for the teacher in a "demonstration" mode or for the students at computers in a lab setting, depending on availability. To find this activity on the Internet, click on the link below.
Differentiation
- Focus on one topic/activity per class period.
- For the Pi Line activity listed in the Instructional Resources section, help students stay organized by printing a chart for them to record object number, circumference, diameter and ratio of circumference to diameter.
- In a computer lab setting, pair up an ELL student with a student who is not an ELL student.
- Have definitions and formulas printed, with diagrams where possible.
- Have directions printed for the Vignette activities, especially for the computer activities.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
graphing concepts relative to a square (side length, perimeter, area, etc.). | providing materials for measurement of round objects in the Vignette activity. |
relating algebra with geometry by using the above concepts in equations. | supervising students in a computer lab setting or demonstrating computer applications. |
measuring circumference and diameter of round objects. | observing and helping as necessary. |
Parent Resources
- Sophia geometry lessons
This social learning community website includes lessons on multiple topics. Most of these lessons were developed by teachers and reviewed.
- Khan Academy geometry lessons
This website offers a library of videos and practice exercises on multiple topics. - Teacher Tube, YouTube
These websites include multiple uploaded lessons on most school topics.
- Mathematics textbooks
Many textbook publishers have websites with additional resources and tutorials. Check your child's textbook for a weblink.