Inicios in Mathematics

1. Inicios in Mathematics NCCEP - UT Austin

2. More Algebra with some geometry

3. Comparing Fractions ACTIVITY: • In Navigator, open Frac.act • Students submit fractions that are: • Between the given fractions • Closest to one of the fractions • Equivalent to a given fraction (line on top) • Repeat 1-2 with decimals • Discuss relationship between fractions and decimals • What is a decimal equivalent? • Do all fractions have decimal equivalents? • Do all decimals have fraction equivalents? • Compare the practical usefulness of fraction and decimal notation (when is one or the other easier or better to use?) • What are irrational numbers? Π / 9 is a number between 1/2 and 1/4. (a) Find fractions close to this value; (b) Find decimals close to this value; (c) These are approximations: Could we ever represent exactly how far “off” these approximations are in decimal form? In any notation? FEATURES/SETTINGS TO USE: ZOOM In (or out) to identify or examine submissions If you want to send the new setting to the calculators, STOP the activity and then START the activity again. If you DON’T want the students to lose their work, change the configuration settings to have students start with “Equations from Calculator” BEFORE START. Students can exit the Activity Center to use the calculator and then re-enter

4. Fractions & Slope ACTIVITY: • In Navigator, open FracX.act • Students submit lines that are: • Between the given lines (with the same intercept) • Closest to one of the line • Lines with equivalent slopes to a given line (lines on top) • Repeat 1-2 with decimals • Discuss using function ideas • Regions where one line [f(x)]is “above” another [f(x)>g(x)], Is it always above? Where are the two lines =? [f(x) = g(x)] • Zoom out some. Turn on Grid. • Ask if lines with slopes like -1/2 will ever go through a location where a horizontal and vertical grid line will cross? Can they predict some of these? You could go to these locations by adjusting the settings for the axes. • Return to the original scale and ask if a line with a slope of π/9 would ever touch one of grid intersections. • Ask them to enter other lines with slopes between the given lines but that will not touch grid intersections (ever!). May want to first ask for other important irrational numbers (sqrt 2, e, etc.) FEATURES/SETTINGS TO USE: ZOOM In (or out) to identify or examine submissions If you want to send the new setting to the calculators, STOP the activity and then START the activity again. If you DON’T want the students to lose their work, change the configuration settings to have students start with “Equations from Calculator” BEFORE START. Students can exit the Activity Center to use the calculator and then re-enter

5. Points on the Grid ACTIVITY: • Open GridwLines.act • Have students enter the x and y locations (into L1 and L2) of points from a given linear function [e.g. Y=3/4x] that will be on the intersection of an x and y grid line (see white line below) • Now turn on or enter a function to go through the points. • More challenging: • Linear functions with a y-interceopt (Y1=3/4x +2) • Use negative slopes (Y1=-3/4x) • Change the grid lines to count by 0.5 (preferences) FEATURES/SETTINGS TO USE: List-Graph Tab ZOOM Fit to get all the points

6. Parallel & Perpendicular ACTIVITY: • Open Para.act • Parallel • Send lines that are parallel to a given line • Discuss patterns or holes in student responses • Could they express the slope of a parallel line using decimals? • Perpendicular - Open Perp.act • Send lines that are perpendicular to a give line • Discuss patterns • Could they express the slope of a perpendicular line using decimals? Other integers • What is the slope of a line perpendicular to one of their perpendicular lines? FEATURES/SETTINGS TO USE: Graph-Equations Tab ZOOM to get all the equations

7. Parallelogram & Rectangle ACTIVITY: • Open PGram.act • Parallelogram • Have students use parallel lines (not horizontal) to construct and submit a parallelogram. [Note: turn on each students lines 4 at a time] • Now construct a parallelogram in the third quadrant • Rectangle • Have students use parallel lines (not horizontal) to construct a rectangle. • Now construct a rectangle in the second quadrant • Could they make a square? FEATURES/SETTINGS TO USE: Hide all functions and then turn on (show) Y1 through Y4 for a given student. Zoom in or out to find extreme examples

8. Solving (Equals) ACTIVITY: • Open Solving.act • Beginning with the following 2x - 1 = x + 2 set Y1 = 2x - 1 (the left side) Y2 = x + 2 (the right side) • Now have them do a next step toward solving (2x -1 ) + 1 = (x + 2) + 1 2x = x + 3 Enter each “side” in for Y3 and Y4: Y3 = 2x Y4 = x + 3 • Continue until “solved” Y5 = x Y6 = 3 (5) Turn on a vertical line at x = 3, what do we notice? (6) Highlight groups of 4 (7) Challenge: *Have students create other equations to start with that have the same solution (x is three) as the original system> • How did they get these other equations. • Have them submit Y1-Y6 like above. What is similar? Are there patterns across examples? FEATURES/SETTINGS TO USE: Use the x= input to draw vertical lines Use Hide and then highlight to look at a group of lines.

9. Un-Solving ACTIVITY: • Open UnSolv.act • Option One: Have students work with just the calulator (exit Navigator) and enter Y1 = x - 1 Y2 = -2x + 8 Now Press 2nd + Quit to get to the home screen. Press 2nd + PRGM, go to 4:Vertical and press ENTER. Now type 3 and ENTER. Now do something creative to “both sides” and enter these Y3 = x * (x - 1) Y4 = x * (-2x + 8) Return to the home screen and redraw the vertical line, what do you notice? Now log onto the network and SEND Y1 through Y4. Discuss Option Two: Same as above but instead of drawing vertical line, students submit Y1 through Y4. Teacher draws vertical line and examines groups of lines (Y1 -> Y4) from students. (3) Why does “doing the same thing to both sides” work? Are the new functions the SAME as the original functions? What DOES stay the same? FEATURES/SETTINGS TO USE: X = input ZOOM to find intersections at x = 3

10. One-Side Solving ACTIVITY: Open Solv3-2.act Ask students to find the x and y value (solve) where this comparison is true: x - 1 = -2x + 8 (there are lots of ways to get this point) Ask students to make a “star” of lines at the intersection (3,2) Now enter either x - 1 or -2x + 8 into Y1. Choose any of the other lines through 3,2 to enter for Y2. Now solve the system (see Solving Inicio) [Y1 through Y6] BUT by changing ONLY ONE of the lines we did NOT do the same thing to both sides? Do we have to do the same thing to both sides? How does this work? Why was it okay no to do the same thing to both sides of the equation? FEATURES/SETTINGS TO USE: Draw a vertical line using X= input. ZOOM Fit to get all the points

11. Aiming for Roots ACTIVITY: Open Roots.act In Y1 through Y5 students enter linear functions that go through the point (-4,0) and Y6 through Y10. (Test by graphing). Now they send in their functions. Discuss patterns and fill in open regions. Ask them to write down at least 5 new linear functions for each side. FEATURES/SETTINGS TO USE: Sort by equation Name (Y1, Y2, etc.) Highlight the Y1 through Y5’s and hide others. Repeat for Y6 through Y10.

12. Quadratic Roots ACTIVITY: Open (or continue with) Roots.act Using the lines generated from Aiming for Roots, find combinations of lines that when multiplied give the following: [a] Concave up parabola touching the axis at only the left point [b] Concave down parabola touching touching the axis at only left point [c] Concave up parabola touching touching the axis at only right point [d] Concave down parabola touching touching the axis at only right point [e] Concave up parabola touching both the left and right points [f] Concave down parabola touching both the left and right points. Enter the original lines into Y1 & Y2; Then enter the lines multiplied (using parentheses) in Y3. Now simplify Y3 to a quadratic (in standard form) and enter it into Y4. Y1 = x + 4 Y2 = -x + 3 Y3 = (x + 4) (-x + 3) Y4 = -x^2 - x + 12 Send these to the front. If you’ve studied the quadratic formula, have students use it on the quadratic in Y4. What do they notice? FEATURES/SETTINGS TO USE: Change the scale of the vertical axis to better see the parabolas. ZOOM

13. Geometry (Cabri Jr) and Navigator Activities Using both Activity Center and Screen Capture

14. Area Invariance for Triangles, Parallelism and More (A Dynamic Geometry Interpretation of 1/2 b * h)

15. task device navigator ACTIVITY 1: TRIANIM0 (1) Moving only ONE vertex to find x,y coordinates where area stays constant (2) Submit points (on second calculator) to activity center (3) What do we notice? (4) How could we test? (e.g., Use equation feature of Nav to see if lines are parallel [same slope]) ACTIVITY 2: TRIANIM1 (1) Move only ONE point OTHER than the animated point until area stays constant (2) Use CLASS screen capture capability of Nav to proke conversation (3) How could we test? (e.g., Locally use Cabri Jr to measure alternate interior angles, etc. OR Make background in activity center and test slopes). Parallel? - How might we prove this? (slope; measure alternate interiors;…) ACTIVITY 3: TRIANIM2 (1) Move ONE or more non-animated points until area constant then STOP the animation when Area/Perimeter * 10 value is greatest (2) What do we notice (3) How test? What do we notice? How could we test this/these? (measure sides; perpendicular to median)

16. More Area and Perimeter Invariance Sum of the Parts Activities

17. Sum of the Parts Activities (6) ACTIVITY 0: (1) Open New (2) Draw a segment, triangle and quad. (3)Clear All (4) Draw a segment and the draw a triangle with one of its sides this segment (5) Now draw a quad with one of its sides a side of the quadrilateral (6) Optional: Measure an Area and a Length ACTIVITY 1: (1) Open PARTS1 and note the Area and Perimeter (2) Draw a SEGMENT from somewhere on the triangle somewhere else on the triangle (not along an edge) (3) Measure PARTS of this figure that SUM to the value for either AREA or PERIMETER (4) Move points Find Parts that Sum to one of these Values To find AREA of parts you’ll first have to draw the parts (a Triangle and Quad in this case) To find SUM of lengths you may need to repeat the additionoperation* (This Example is PARTS3) (This Example is PARTS2)

18. How to Calculate (i.e. Sum) Press Enter (1) POINT to the first NUMBER (2) Press Enter Press the KEY on the calculator for the OPERATION you want ( + in this case) (1) POINT to the 2nd NUMBER (2) Press Enter (1) DRAG the Sum to where you want it (2) Press Enter (3) Press Clear You may need to repeat this to get the SUM of the Parts