Teaching Algebraic Reasoning, Graphing, and Connections with Geometry

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.

There are 12 in a dozen of eggsThere are four sides of 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 FORMULAS1,  DistanceVariables:d= distance traveledr= rate of travel (speed)t= time of travel Formula: d = rt2.  Perimeter of a rectangle Variables:p = perimeterl= lengthw= widthFormula: P = 2L + 2W 3.  Area of a rectangle Variables:A= area Formula: A= LW

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

The cost of a gallon of milk in the store is increased by 20 cents.  What is the cost of a gallon of milk that used to cost c dollars? What is the old cost of the old gallon of milk that now costs d dollars?Answer: c+0.20d-.20 (in dollars)

An equation is a mathematical expression starting 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.  As we know from filling our gas tank that day the amount is a function of the amount of gas purchased.  Letting the variable (output) represent the final bill in dollars and cents, then y is defined by the rule y=3.95s.  Machines help us compute this formula into a dollar amount"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.     

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.

A graph allows you to visualize data sets. 

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 are if they 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 through the center of the circle and end on the other side this would be the diameter.Using a circle and running around edge of the circle is called the circumference.

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 fromﾠhttp://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. http://www.ixl.com/math/grade-3/solve-for-the-variable-addition-and-subtraction-onlyﾠLong, C. et al. (2015). Mathematics Reasoning for Elementary Teachers. Boston, MA. PearsonMath 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-ruleReferences: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 fromhttp://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. Retrieved from http://www.ixl.com/math/grade-3/solve-for-the-variable-addition-and-subtraction-onlyMathematics. Long, C. et al. (2015). Mathematics Reasoning for Elementary Teachers. Boston, MA. PearsonMath 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-rule

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 (5,5) and (1,2) by using the formula.

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 

First before beginning any computation it is important to 1. Make sure you understand the problem.2. Devise a plan t help with the problem.3. Carry out the plan or compute the problem.4. Look back at how we solved the problem.  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 a not a linear function.- A quadratic function isf (x) = ax 2 + bx + c , where a, b, c if a≠ 0 . -Exponential function (see video)

Function tables are lists of possible values of a variable and the function's result.

Algebra, geometry are not tow separate areas.  Many of the same applications apply to both. 

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/Brain Pop. Gopher Game. Retrieved from https://www.brainpop.com/games/gameovergopher/Khan 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. 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

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 in the real world.