Teaching Algebraic Reasoning, Graphing, and Connections with Geometry

Why is it important to teach students algebraic reasoning, and graphing? Why do student's need to make connections with geometry? Forty years from now, four plus four will still equal eight. Students need critical thinking skills so they can think through an issue, and reason logically to solve problems outside the classroom in real world.

Lets start with algebraic reasoning. Here are some topics we will cover.• What are variables?• How to form algebraic expressions involving variables, numbers, and operation symbols?• What is the definition and visualization of functions?

A constant is a non-changing value that represents a fixed number. Here are several examples of a constant3, 10, π, 1/2.

One dozen eggs equals 12.There are four sides to a rectangleand2 + 5 = 7

Variables used in algebra are letters such as x, y, f, or any letter to represent unknown numbers.

Example:a(b+C) = ab+ac

Variables express relationships, here are some examples. J=K-6 ("Addy is six years younger than Kendra")K=J+6 ("Sara is six years older than Addy.")K-J=6 ("The difference in age between Sara and her younger sister Addy is six years.")

Find a number to place in the blank that makes the sentence true.__ + 7 = 9or(3x-6)(x-2)=0

Commonly used formulas

Algebraic expression is a representation that involves variables, numbers, and operations symbols.

An equation is a mathematical expression, stating that two algebraic expressions have the same value.

The definition of function requires that a single value y be assigned to each x-value in the domain.Example,We are at the gas station, filling the vehicle with gas which is priced at $3.95 per gallon. "How much does 15 gallons cost?" The answer is $59.25. This is the only cost of the gas because there is only one value for y for each x. Machines were invented to make these calucations fast and error free for us.

We can solve equations by using visual functions such as "What's my rule, function tables, arrow diagrams, machines, and graphs."

A math function table is a table used to plot possible outcomes.

Math is integrated into many different areas such as geometry, probability, and algebra. They will continue to keep building on one another.

Parallel lines are parallel if they are always the same distance apart and will never meet. Therefore, their slopes are always the same.Perpendicular lines meet at a right angle (90°).

Picture a circle and you are standing in the center of it. If you were to run from the center to the outside edge in a straight line that distance is the radius. Using the circle analogy again, if you were to run in a straight line, starting on one side through the center of the circle and end on the other side, this would be the diameter.If you were to run in a perfect circle, the distance that you covered would be the circumference of that circle.

References:Ehow. How to Use Functions to Teach Algebra. Retrieved from http://www.bing.com/videos/search?q=how+to+use+functions+tables+youtube&FORM=VIRE7#view=detail&mid=D23955324C10AFD5B81ED23955324C10AFD5B81EGraph Game. Retrieved from http://www.mathxl.com/Student/MediaPopup.aspx?type=assignedmedia&assignmentId=283516755&loc=3&flush=yes&record=1&OnOpenerViewed=doMedia_onViewedGraphic. Retrieved from http://www.jackanderica.com/math/images/plot007.jpgIXL. Retreived from http://www.ixl.com/math/grade-3/solve-for-the-variable-addition-and-subtraction-only Long, C. et al. (2015). Mathematics Reasoning for Elementary Teachers. Boston, MA. PearsonMath Antics. Circles, Circumference And Area. Retrieved from https://www.youtube.com/watch?v=O-cawByg2aAMath Help. Mapping Diagrams. Retrieved from http://www.bing.com/videos/search?q=functions+as+arrow+diagrams+video+algebra&FORM=VIRE5#view=detail&mid=49B4EB35DDBD9F4C138949B4EB35DDBD9F4C1389Math Plant. Retrieved from http://www.mathplanet.com/education/algebra-1/discovering-expressions,-equations-and-functions/expressions-and-variablesMath XL. Retrieved from http://www.mathxl.com/Student/MediaPopup.aspx?type=assignedmedia&assignmentId=283516755&loc=4&flush=yes&record=1&OnOpenerViewed=doMedia_onViewedSolving Equations. Retrieved from http://www.sosmath.com/algebra/solve/solve0/solve0.htmlThink Math. Retrieved from http://thinkmath.edc.org/resource/guess-my-ruleWho Wants to a Hundredaire. (n.d). Retrieved from http://www.math-play.com/Algebraic-Expressions-Millionaire/algebraic-expressions-game.html

Quadrant I both coordinates are positive.Quadrant II first coordinate is negative and second coordinate is positive.Quadrant III both coordinates are negative.Quadrant IV first coordinate is positive and second coordinate is negative.

We can use this formula to determine the distance between the two points (4,5) and (1,2).

What is the slope? A real world example is walking up or down a flight of stairs. The slope of a line defines the steepness of the line. It allows us to know if the line is rising or falling.Slope = Rise/Run

We need to study the problem to have a better understanding of what we are being asked. We need to decide how to solve it. Then it is important to being to solve it. Then reflect and look back at the solution.What Theorem needs to be used. 1. The point-slope form is will be used when we are given a point (x1, y1) and a slope m. Then the numbers are plugged into the formula:y – y1 = m(x – x1) 2. The slope-intercept form is used where m is the slope and b is the y-intercept.y=mx+b3 For the two-point form of the equation of a line we would use the formula y – y1 = m(x – x1)The slope and (x1, y1) is a point on the line. Point-slope is the form used most often when finding the equation of a line.Functions:-A linear function is a straight-line graph.

- A non-linear function is not a linear function.- A quadratic function isf (x) = ax 2 + bx + c , where a, b, c if a≠ 0 . -Exponential function (see video)

Additional website resources for teaching algebraic reasoning, graphing, along with connections in geometry.1. Alex Learning Lessons (Investigating and Discovering Input & Output Patterns).http://alex.state.al.us/lesson_view.php?id=263382. Function Tableshttp://www.prometheanplanet.com/en/Search/resources/language/english/?Keywords=function+tables3. Teacher Channel Video (learn how Ms. Bottomley uses a function machine to have students predict the rule). https://www.teachingchannel.org/videos/algebra-lesson-planning4. Graphing Linear Functions Worksheethttp://www.mathworksheets4kids.com/function/graphing-linear1.pdf5. Graphing Two Variables to Express Algebraic Relationshiphttp://www.pbslearningmedia.org/resource/vtl07.math.algebra.var.lptwovaria/using-two-variables-to-express-algebraic-relationships/6. Modeling with Mathematics (Keys to Modeling with Mathematics in the K-8 Classroom).https://www.teachingchannel.org/blog/2013/05/29/3-keys-to-modeling-with-mathematics/7. Variableshttp://www.pbslearningmedia.org/resource/vtl07.math.algebra.var.lptwovaria/using-two-variables-to-express-algebraic-relationships/

References:Alex Lesson Plans. Alabama Learning Exchange. Retrieved fromhttp://alex.state.al.us/lesson_view.php?id=26338https://www.youtube.com/watch?v=k4KdGiJPbN8Basics About a Circle. http://www.areacircle.com/basics-about-area-of-a-circle.htmlBasic Algebra. Basic Mathematics. Retrieved from http://www.basic-mathematics.com/what-is-slope.htmlBetter Explained Math Insights with a Click. Retrieved from http://betterexplained.com/articles/techniques-for-adding-the-numbers-1-to-100/Bing Image. The Cartesian Coordinate Plane. Carhttp://www.plane-land.net/wp-content/uploads/2012/01/Cartesian-Coordinate-Plane.jpgBing Image. Fibonacci Sequence. Retrieved from https://nakawligaw.files.wordpress.com/2014/02/fibonacci-sequence.pngBing Image. Pie. Retrieved from http://3.bp.blogspot.com/_a1VB-fjgHlQ/TLXptDsnlYI/AAAAAAAAAFY/EHoN5473VVU/s1600/diameter,+radius,+circumference.jpgBing Image. Plotting Points. Retrieved from http://stufiles.sanjac.edu/THEA/THEA_Math_Review/Graphs_of_Number_Relationships/930.pngBrain Pop. Gopher Game. Retrieved from https://www.brainpop.com/games/gameovergopher/Ehow. Game for Parallel and Perpendicular lines. Retrieved from https://www.youtube.com/watch?t=17&v=hQ64ZoDvN_kKhan Academy. Coordinate Plane. Retrieved fromhttps://www.khanacademy.org/math/algebra-basics/core-algebra-graphing-lines-slope/core-algebra-graphing-coordinate-plan/v/plot-ordered-pairsMath Worksheets 4 Kids. Retrieved from http://www.mathworksheets4kids.com/function/graphing-linear1.pdfPromethean Planet. Teaching Resources. Prometheplanet. Retrieved from http://www.prometheanplanet.com/en/Search/resources/language/english/?Keywords=function+tablesPBS. Using Two Variables to Express Algebraic Equations. Retrieved from http://www.pbslearningmedia.org/resource/vtl07.math.algebra.var.lptwovaria/using-two-variables-to-express-algebraic-relationships/Purple Math. The Quadrants o the Cartesian Plane. http://www.purplemath.com/modules/plane3.htmTeaching Channel. Chessboard Algebra Function Machine. Retrieved fromhttps://www.teachingchannel.org/videos/algebra-lesson-planning Writing Equations of a Line. Algebra Lab. Retrieved from http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_LinearEqEquationsOfLines.xml