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Mehran University Newsletter Vol-X, Issue-I, 2024
Mehran University Newsletter Vol-X, Issue-I, 2024
Personalized Learning Bridges Middle School Math with a Geometric Approach
1.
Personalized Learning
Bridges Middle-School Math with a Geometric Approach Presented by Tracy Mittleider, MSEd and Kathleen Lawler Based on work of Joan A. Cotter, Ph.D. Aplus+ Conference October 24, 2012 © Activities for Learning, Inc., 2012
2.
Why a Geometric
Approach? © Activities for Learning, Inc., 2012
3.
Why a Geometric
Approach? Most students in “middle school” are visual learners. © Activities for Learning, Inc., 2012
4.
Why a Geometric
Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually). © Activities for Learning, Inc., 2012
5.
Why a Geometric
Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually). Therefore, it makes sense to teach them geometrically. © Activities for Learning, Inc., 2012
6.
Drawing Tools
Paper 11 in. by 8.5 in. © Activities for Learning, Inc., 2012
7.
Drawing Tools
Drawing board with paper © Activities for Learning, Inc., 2012
8.
Drawing Tools
Drawing board with paper © Activities for Learning, Inc., 2012
9.
Drawing Tools
T-square T-square © Activities for Learning, Inc., 2012
10.
Drawing Tools
T-square T-square © Activities for Learning, Inc., 2012
11.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
12.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
13.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
14.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
15.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
16.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
17.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
18.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
19.
Drawing Tools
T-square © Activities for Learning, Inc., 2012
20.
Drawing Tools
Pencil (mechanical) and eraser © Activities for Learning, Inc., 2012
21.
Drawing Tools
Pencil (mechanical) and eraser © Activities for Learning, Inc., 2012
22.
Drawing Tools
Pencil (mechanical) and eraser © Activities for Learning, Inc., 2012
23.
Drawing Tools
Pencil (mechanical) and eraser © Activities for Learning, Inc., 2012
24.
Drawing Tools
Pencil (mechanical) and eraser © Activities for Learning, Inc., 2012
25.
Drawing Tools
Pencil (mechanical) and eraser © Activities for Learning, Inc., 2012
26.
Drawing Tools
The 45 triangle 45° 45° 90° © Activities for Learning, Inc., 2012
27.
Drawing Tools
The 30-60 triangle (a set square) 30° 60° 90° © Activities for Learning, Inc., 2012
28.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
29.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
30.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
31.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
32.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
33.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
34.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
35.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
36.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
37.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
38.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
39.
Drawing Tools
The 30-60 triangle © Activities for Learning, Inc., 2012
40.
© Activities for
Learning, Inc., 2012
41.
Equilateral Triangles
Draw the base line © Activities for Learning, Inc., 2012
42.
Equilateral Triangles
Draw the base line © Activities for Learning, Inc., 2012
43.
Equilateral Triangles
Draw the sides © Activities for Learning, Inc., 2012
44.
Equilateral Triangles
Draw the sides © Activities for Learning, Inc., 2012
45.
Equilateral Triangles
Draw the sides © Activities for Learning, Inc., 2012
46.
Equilateral Triangles
Draw the sides © Activities for Learning, Inc., 2012
47.
Equilateral Triangles
Divide it into halves © Activities for Learning, Inc., 2012
48.
Equilateral Triangles
Divide it into halves © Activities for Learning, Inc., 2012
49.
Equilateral Triangles
Divide it into halves © Activities for Learning, Inc., 2012
50.
Equilateral Triangles
Divide it into halves 1 1 2 2 © Activities for Learning, Inc., 2012
51.
Equilateral Triangles
Divide it into halves © Activities for Learning, Inc., 2012
52.
Equilateral Triangles
Divide it into halves © Activities for Learning, Inc., 2012
53.
Equilateral Triangles
Divide it into halves © Activities for Learning, Inc., 2012
54.
Equilateral Triangles
Divide it into halves © Activities for Learning, Inc., 2012
55.
Equilateral Triangles
Divide it into halves © Activities for Learning, Inc., 2012
56.
© Activities for
Learning, Inc., 2012
57.
Equilateral Triangles
Divide it into thirds © Activities for Learning, Inc., 2012
58.
Equilateral Triangles
Divide it into thirds © Activities for Learning, Inc., 2012
59.
Equilateral Triangles
Divide it into thirds © Activities for Learning, Inc., 2012
60.
Equilateral Triangles
Divide it into thirds © Activities for Learning, Inc., 2012
61.
Equilateral Triangles
Divide it into thirds © Activities for Learning, Inc., 2012
62.
Equilateral Triangles
Divide it into thirds © Activities for Learning, Inc., 2012
63.
Equilateral Triangles
Divide it into thirds © Activities for Learning, Inc., 2012
64.
Equilateral Triangles
Divide it into thirds 1 1 3 3 1 3 © Activities for Learning, Inc., 2012
65.
Equilateral Triangles
Divide it into thirds 1 3 1 1 3 3 © Activities for Learning, Inc., 2012
66.
© Activities for
Learning, Inc., 2012
67.
Equilateral Triangles
Divide it into fourths © Activities for Learning, Inc., 2012
68.
Equilateral Triangles
Divide it into fourths Find the center of the base line. © Activities for Learning, Inc., 2012
69.
Equilateral Triangles
Divide it into fourths tick mark © Activities for Learning, Inc., 2012
70.
Equilateral Triangles
Divide it into fourths © Activities for Learning, Inc., 2012
71.
Equilateral Triangles
Divide it into fourths © Activities for Learning, Inc., 2012
72.
Equilateral Triangles
Divide it into fourths © Activities for Learning, Inc., 2012
73.
Equilateral Triangles
Divide it into fourths © Activities for Learning, Inc., 2012
74.
Equilateral Triangles
Divide it into fourths © Activities for Learning, Inc., 2012
75.
Equilateral Triangles
Divide it into fourths 1 4 1 1 4 1 4 4 © Activities for Learning, Inc., 2012
76.
Equilateral Triangles
Divide it into fourths A B F C E D Name the rhombuses: © Activities for Learning, Inc., 2012
77.
Equilateral Triangles
Divide it into fourths A B F C E D Name the rhombuses: ABDF © Activities for Learning, Inc., 2012
78.
Equilateral Triangles
Divide it into fourths A B F C E D Name the rhombuses: ABDF, BCDF © Activities for Learning, Inc., 2012
79.
Equilateral Triangles
Divide it into fourths A B F C E D Name the rhombuses: ABDF, BCDF, © Activities for Learning, Inc., 2012 BDEF
80.
Equilateral Triangles
Divide it into fourths A B F C E D Name the trapezoids: © Activities for Learning, Inc., 2012
81.
Equilateral Triangles
Divide it into fourths A B F C E D Name the trapezoids: BCEF © Activities for Learning, Inc., 2012
82.
Equilateral Triangles
Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF © Activities for Learning, Inc., 2012
83.
Equilateral Triangles
Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF, © Activities for Learning, Inc., 2012 ABDE
84.
Equilateral Triangles
What if we cut this shape out…. © Activities for Learning, Inc., 2012
85.
Equilateral Triangles
…and folded it on the lines…. © Activities for Learning, Inc., 2012
86.
Equilateral Triangles © Activities
for Learning, Inc., 2012
87.
Equilateral Triangles © Activities
for Learning, Inc., 2012
88.
Equilateral Triangles © Activities
for Learning, Inc., 2012
89.
Equilateral Triangles © Activities
for Learning, Inc., 2012
90.
Equilateral Triangles © Activities
for Learning, Inc., 2012
91.
Equilateral Triangles
It’s a tetrahedron! © Activities for Learning, Inc., 2012
92.
Equilateral Triangles
Divide it into eighths © Activities for Learning, Inc., 2012
93.
Equilateral Triangles
Divide it into eighths © Activities for Learning, Inc., 2012
94.
Equilateral Triangles
Divide it into eighths © Activities for Learning, Inc., 2012
95.
Equilateral Triangles
Divide it into eighths © Activities for Learning, Inc., 2012
96.
Equilateral Triangles
Divide it into eighths © Activities for Learning, Inc., 2012
97.
Equilateral Triangles
Divide it into sixths © Activities for Learning, Inc., 2012
98.
Equilateral Triangles
Divide it into sixths © Activities for Learning, Inc., 2012
99.
Equilateral Triangles
Divide it into sixths © Activities for Learning, Inc., 2012
100.
Equilateral Triangles
Divide it into sixths A B F G C E D Name an acute triangle: © Activities for Learning, Inc., 2012
101.
Equilateral Triangles
Divide it into sixths A B F G C E D Name an acute triangle: ACE © Activities for Learning, Inc., 2012
102.
Equilateral Triangles
Divide it into sixths A B F G C E D Name a right triangle: © Activities for Learning, Inc., 2012
103.
Equilateral Triangles
Divide it into sixths A B F G C E D Name a right triangle: ADE © Activities for Learning, Inc., 2012
104.
Equilateral Triangles
Divide it into sixths A B F G C E D Name a right triangle: ADE, ABE © Activities for Learning, Inc., 2012
105.
Equilateral Triangles
Divide it into sixths A B F G C E D Name a right triangle: ADE, ABE, ABG © Activities for Learning, Inc., 2012
106.
Equilateral Triangles
Divide it into sixths A B F G C E D Name an obtuse triangle: © Activities for Learning, Inc., 2012
107.
Equilateral Triangles
Divide it into sixths A B F G C E D Name an obtuse triangle: GCE © Activities for Learning, Inc., 2012
108.
Equilateral Triangles
Divide it into twelfths © Activities for Learning, Inc., 2012
109.
Equilateral Triangles
Divide it into twelfths © Activities for Learning, Inc., 2012
110.
Equilateral Triangles
Divide it into twelfths © Activities for Learning, Inc., 2012
111.
Equilateral Triangles
Divide it into twelfths © Activities for Learning, Inc., 2012
112.
Equilateral Triangles
Divide it into twelfths © Activities for Learning, Inc., 2012
113.
Equilateral Triangles
Divide it into twelfths © Activities for Learning, Inc., 2012
114.
Equilateral Triangles
Divide it into twelfths Can you see the cube? © Activities for Learning, Inc., 2012
115.
Equilateral Triangles
Divide it into twelfths Can you see the cube? © Activities for Learning, Inc., 2012
116.
Equilateral Triangles
Divide it into twelfths Do you see the hexagon? © Activities for Learning, Inc., 2012
117.
Equilateral Triangles
Divide it into twelfths Do you see the hexagon? © Activities for Learning, Inc., 2012
118.
Equilateral Triangles
Ninths © Activities for Learning, Inc., 2012
119.
Equilateral Triangles
Twenty-sevenths © Activities for Learning, Inc., 2012
120.
Equilateral Triangles
Twenty-sevenths © Activities for Learning, Inc., 2012
121.
Equilateral Triangles © Activities
for Learning, Inc., 2012
122.
Equilateral Triangles © Activities
for Learning, Inc., 2012
123.
Equilateral Triangles
Thirty-seconds © Activities for Learning, Inc., 2012
124.
Equilateral Triangles
Stephanie, 6, did 256! © Activities for Learning, Inc., 2012
125.
Equilateral Triangles © Activities
for Learning, Inc., 2012
126.
Equilateral Triangles © Activities
for Learning, Inc., 2012
127.
Equilateral Triangles © Activities
for Learning, Inc., 2012
128.
Equilateral Triangles © Activities
for Learning, Inc., 2012
129.
© Activities for
Learning, Inc., 2012
130.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
131.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
132.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
133.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
134.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
135.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
136.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
137.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
138.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
139.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
140.
Equilateral Triangles
Spiral © Activities for Learning, Inc., 2012
141.
Geometric Approach © Activities
for Learning, Inc., 2012
142.
Geometric Approach
• Basis of CAD (computer aided design). © Activities for Learning, Inc., 2012
143.
Geometric Approach
• Basis of CAD (computer aided design). • Integrates many concepts. © Activities for Learning, Inc., 2012
144.
Geometric Approach
• Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. © Activities for Learning, Inc., 2012
145.
Geometric Approach
• Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. © Activities for Learning, Inc., 2012
146.
Geometric Approach
• Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. • Excellent preparation for high school mathematics. © Activities for Learning, Inc., 2012
147.
Geometric Approach
Goals © Activities for Learning, Inc., 2012
148.
Geometric Approach
Goals • To use elementary math. © Activities for Learning, Inc., 2012
149.
Geometric Approach
Goals • To use elementary math. • To learn to read math texts. © Activities for Learning, Inc., 2012
150.
Geometric Approach
Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. © Activities for Learning, Inc., 2012
151.
Geometric Approach
Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world. © Activities for Learning, Inc., 2012
152.
Geometric Approach
Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world. • To enjoy mathematics. © Activities for Learning, Inc., 2012
153.
Why a Geometric
Approach? © Activities for Learning, Inc., 2012
154.
Why a Geometric
Approach? Most students in “middle school” are visual learners. © Activities for Learning, Inc., 2012
155.
Why a Geometric
Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually). © Activities for Learning, Inc., 2012
156.
Why a Geometric
Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually). Therefore, it makes sense to teach them geometrically. © Activities for Learning, Inc., 2012
157.
© Activities for
Learning, Inc., 2012
158.
Squares
The 45 triangle. © Activities for Learning, Inc., 2012
159.
Squares
The 45 triangle is half of a square. © Activities for Learning, Inc., 2012
160.
Squares © Activities for
Learning, Inc., 2012
161.
Squares © Activities for
Learning, Inc., 2012
162.
Squares © Activities for
Learning, Inc., 2012
163.
Squares © Activities for
Learning, Inc., 2012
164.
Squares © Activities for
Learning, Inc., 2012
165.
Squares © Activities for
Learning, Inc., 2012
166.
Squares © Activities for
Learning, Inc., 2012
167.
Squares © Activities for
Learning, Inc., 2012
168.
Squares © Activities for
Learning, Inc., 2012
169.
Squares
Halves © Activities for Learning, Inc., 2012
170.
Squares
Fourths © Activities for Learning, Inc., 2012
171.
Squares
Eighths © Activities for Learning, Inc., 2012
172.
Squares
Sixteenths © Activities for Learning, Inc., 2012
173.
© Activities for
Learning, Inc., 2012
174.
Squares
Halves We need to find the center. © Activities for Learning, Inc., 2012
175.
Squares
Halves We need to find the center. © Activities for Learning, Inc., 2012
176.
Squares
Halves We need to find the center. © Activities for Learning, Inc., 2012
177.
Squares
Halves © Activities for Learning, Inc., 2012
178.
Squares
Halves © Activities for Learning, Inc., 2012
179.
Squares
Fourths © Activities for Learning, Inc., 2012
180.
Squares
Fourths © Activities for Learning, Inc., 2012
181.
Squares
Eighths Find the center of a small square. © Activities for Learning, Inc., 2012
182.
Squares
Eighths Find the center of a small square. © Activities for Learning, Inc., 2012
183.
Squares
Eighths Find the center of a small square. © Activities for Learning, Inc., 2012
184.
Squares
Eighths Find the center of a small square. © Activities for Learning, Inc., 2012
185.
Squares
Eighths © Activities for Learning, Inc., 2012
186.
Squares
Eighths © Activities for Learning, Inc., 2012
187.
Squares
Sixteenths © Activities for Learning, Inc., 2012
188.
Squares
Sixteenths © Activities for Learning, Inc., 2012
189.
Squares
Sixteenths © Activities for Learning, Inc., 2012
190.
© Activities for
Learning, Inc., 2012
191.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
192.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
193.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
194.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
195.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
196.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
197.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
198.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
199.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
200.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
201.
Squares
Circumscribed squares © Activities for Learning, Inc., 2012
202.
© Activities for
Learning, Inc., 2012
203.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
204.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
205.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
206.
Squares
Inscribed squares Compare the area of the squares. © Activities for Learning, Inc., 2012
207.
Squares
Inscribed squares Compare the area of the squares. © Activities for Learning, Inc., 2012
208.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
209.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
210.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
211.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
212.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
213.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
214.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
215.
Squares
Inscribed squares © Activities for Learning, Inc., 2012
216.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
217.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
218.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
219.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
220.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
221.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
222.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
223.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
224.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
225.
Squares
Right triangle spiral © Activities for Learning, Inc., 2012
226.
© Activities for
Learning, Inc., 2012
227.
Hexagons
Start with a small equilateral triangle. © Activities for Learning, Inc., 2012
228.
Hexagons © Activities for
Learning, Inc., 2012
229.
Hexagons © Activities for
Learning, Inc., 2012
230.
Hexagons © Activities for
Learning, Inc., 2012
231.
Hexagons © Activities for
Learning, Inc., 2012
232.
Hexagons © Activities for
Learning, Inc., 2012
233.
Hexagons © Activities for
Learning, Inc., 2012
234.
Hexagons © Activities for
Learning, Inc., 2012
235.
Hexagons © Activities for
Learning, Inc., 2012
236.
Hexagons
Draw all the diagonals. © Activities for Learning, Inc., 2012
237.
Hexagons
Draw all the diagonals. © Activities for Learning, Inc., 2012
238.
Hexagons
Draw all the diagonals. © Activities for Learning, Inc., 2012
239.
Hexagons
Draw all the diagonals. © Activities for Learning, Inc., 2012
240.
Hexagons
Draw all the diagonals. © Activities for Learning, Inc., 2012
241.
Hexagons
Draw all the diagonals. © Activities for Learning, Inc., 2012
242.
Hexagons
Do you see another hexagon? © Activities for Learning, Inc., 2012
243.
Hexagons
Find the figures Do you see another hexagon? © Activities for Learning, Inc., 2012
244.
Hexagons
Find the figures 2 non-congruent rectangles? © Activities for Learning, Inc., 2012
245.
Hexagons
Find the figures 2 non-congruent rectangles? © Activities for Learning, Inc., 2012
246.
Hexagons
Find the figures 2 non-congruent rectangles? © Activities for Learning, Inc., 2012
247.
Hexagons
Find the figures 3 non-congruent right triangles? © Activities for Learning, Inc., 2012
248.
Hexagons
Find the figures 3 non-congruent right triangles? © Activities for Learning, Inc., 2012
249.
Hexagons
Find the figures 3 non-congruent right triangles? © Activities for Learning, Inc., 2012
250.
Hexagons
Find the figures 3 non-congruent right triangles? © Activities for Learning, Inc., 2012
251.
Hexagons
Find the figures 2 non-congruent rhombuses? © Activities for Learning, Inc., 2012
252.
Hexagons
Find the figures 2 non-congruent rhombuses? © Activities for Learning, Inc., 2012
253.
Hexagons
Find the figures 2 non-congruent rhombuses? © Activities for Learning, Inc., 2012
254.
Hexagons
Find the figures 3 similar equilateral triangles? © Activities for Learning, Inc., 2012
255.
Hexagons
Find the figures 3 similar equilateral triangles? © Activities for Learning, Inc., 2012
256.
Hexagons
Find the figures 3 similar equilateral triangles? © Activities for Learning, Inc., 2012
257.
Hexagons
Find the figures 3 similar obtuse triangles? © Activities for Learning, Inc., 2012
258.
Hexagons
Find the figures 3 similar obtuse triangles? © Activities for Learning, Inc., 2012
259.
Hexagons
Find the figures 3 similar obtuse triangles? © Activities for Learning, Inc., 2012
260.
Hexagons
Find the figures 3 similar obtuse triangles? © Activities for Learning, Inc., 2012
261.
Hexagons
Find the figures 3 non-congruent trapezoids? © Activities for Learning, Inc., 2012
262.
Hexagons
Find the figures 3 non-congruent trapezoids? © Activities for Learning, Inc., 2012
263.
Hexagons
Find the figures 3 non-congruent trapezoids? © Activities for Learning, Inc., 2012
264.
Hexagons
Find the figures 3 non-congruent trapezoids? © Activities for Learning, Inc., 2012
265.
Hexagons
Find the figures 2 similar kites? © Activities for Learning, Inc., 2012
266.
Hexagons
Find the figures 2 similar kites? © Activities for Learning, Inc., 2012
267.
Hexagons
Find the figures 2 similar kites? © Activities for Learning, Inc., 2012
268.
© Activities for
Learning, Inc., 2012
269.
Hexagons
Constructing the small star Apothem: perpendicular line from side to center. © Activities for Learning, Inc., 2012
270.
Hexagons
Constructing the small star Apothem: perpendicular line from side to center. © Activities for Learning, Inc., 2012
271.
Hexagons
Constructing the small star Apothem: perpendicular line from side to center. © Activities for Learning, Inc., 2012
272.
Hexagons
Constructing the small star © Activities for Learning, Inc., 2012
273.
Hexagons
Constructing the small star © Activities for Learning, Inc., 2012
274.
Hexagons
Constructing the small star © Activities for Learning, Inc., 2012
275.
Hexagons
Constructing the small star © Activities for Learning, Inc., 2012
276.
Hexagons
Constructing the small star Ratio of the areas of star and hexagon? © Activities for Learning, Inc., 2012
277.
Hexagons
Constructing the small star Ratio of the areas of star and hexagon? © Activities for Learning, Inc., 2012
278.
Hexagons
Constructing the large star © Activities for Learning, Inc., 2012
279.
Hexagons
Constructing the large star © Activities for Learning, Inc., 2012
280.
Hexagons
Constructing the large star © Activities for Learning, Inc., 2012
281.
Hexagons
Constructing the large star © Activities for Learning, Inc., 2012
282.
Hexagons
Constructing the large star © Activities for Learning, Inc., 2012
283.
Hexagons
Constructing the large star © Activities for Learning, Inc., 2012
284.
Hexagons
Constructing the large star © Activities for Learning, Inc., 2012
285.
Hexagons
Constructing the large star © Activities for Learning, Inc., 2012
286.
The Clock
11 12 1 10 2 9 3 8 4 7 6 5 © Activities for Learning, Inc., 2012
287.
The Clock
11 12 1 10 2 9 3 8 4 7 6 5 © Activities for Learning, Inc., 2012
288.
The Clock
11 12 1 10 2 9 3 8 4 7 6 5 © Activities for Learning, Inc., 2012
289.
The Clock
11 12 1 10 2 9 3 8 4 7 6 5 © Activities for Learning, Inc., 2012
290.
The Clock
11 12 1 10 2 9 3 8 4 7 6 5 © Activities for Learning, Inc., 2012
291.
The Clock
11 12 1 10 2 9 3 8 4 7 6 5 11 12 1 10 2 9 3 8 4 7 6 5 © Activities for Learning, Inc., 2012
292.
Right Triangle Design ©
Activities for Learning, Inc., 2012
293.
Right Triangle Design ©
Activities for Learning, Inc., 2012
294.
Right Triangle Design ©
Activities for Learning, Inc., 2012
295.
Right Triangle Design ©
Activities for Learning, Inc., 2012
296.
Right Triangle Design ©
Activities for Learning, Inc., 2012
297.
Right Triangle Design ©
Activities for Learning, Inc., 2012
298.
Right Triangle Design ©
Activities for Learning, Inc., 2012
299.
Right Triangle Design ©
Activities for Learning, Inc., 2012
300.
Right Triangle Design ©
Activities for Learning, Inc., 2012
301.
Right Triangle Design ©
Activities for Learning, Inc., 2012
302.
© Activities for
Learning, Inc., 2012
303.
Tangrams © Activities for
Learning, Inc., 2012
304.
Tangrams © Activities for
Learning, Inc., 2012
305.
Tangrams © Activities for
Learning, Inc., 2012
306.
Tangrams © Activities for
Learning, Inc., 2012
307.
Tangrams © Activities for
Learning, Inc., 2012
308.
Tangrams © Activities for
Learning, Inc., 2012
309.
Tangrams © Activities for
Learning, Inc., 2012
310.
Tangrams
Are the 3 green figures are equal in © Activities for Learning, Inc., 2012 area?
311.
Tangrams © Activities for
Learning, Inc., 2012
312.
Tangrams
The 3 green figures ARE equal in area. © Activities for Learning, Inc., 2012
313.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
314.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
315.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
316.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
317.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
318.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
319.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
320.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
321.
Tangrams
Constructing a tangram arrangement. © Activities for Learning, Inc., 2012
322.
Tangrams © Activities for
Learning, Inc., 2012
323.
Tangram Design Symmetry
Start at the left side of the paper. © Activities for Learning, Inc., 2012
324.
Tangram Design Symmetry
Start at the left side of the paper. © Activities for Learning, Inc., 2012
325.
Tangram Design Symmetry
Reflect the image vertically across the © Activities for Learning, Inc., 2012 line.
326.
Tangram Design Symmetry
Reflect the image vertically. © Activities for Learning, Inc., 2012
327.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
328.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
329.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
330.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
331.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
332.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
333.
Tangram Design Symmetry
Reflect the image horizontally. © Activities for Learning, Inc., 2012
334.
Tangram Design Symmetry
Reflect the image horizontally. © Activities for Learning, Inc., 2012
335.
Tangram Design Symmetry
The blue rotated 180° is the yellow. © Activities for Learning, Inc., 2012
336.
Tangram Design Symmetry
Reflect the image vertically. © Activities for Learning, Inc., 2012
337.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
338.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
339.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
340.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
341.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
342.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
343.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
344.
Tangram Design Symmetry ©
Activities for Learning, Inc., 2012
345.
Tangram Design Symmetry
Two horizontal reflections is the original. © Activities for Learning, Inc., 2012
346.
Tangram Design Symmetry
• Such constructions require thinking ahead. © Activities for Learning, Inc., 2012
347.
Tangram Design Symmetry
• Such constructions require thinking ahead. • Each step must be justified; no guessing. © Activities for Learning, Inc., 2012
348.
Tangram Design Symmetry
• Such constructions require thinking ahead. • Each step must be justified; no guessing. • Symmetry is in everyday life. © Activities for Learning, Inc., 2012
349.
Tangram Design Symmetry
• Such constructions require thinking ahead. • Each step must be justified; no guessing. • Symmetry is in everyday life. • Symmetry is often on tests. © Activities for Learning, Inc., 2012
350.
© Activities for
Learning, Inc., 2012
351.
Pythagorean Theorem
First construct a right triangle. © Activities for Learning, Inc., 2012
352.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
353.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
354.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
355.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
356.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
357.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
358.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
359.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
360.
Pythagorean Theorem
Area of blue squares = yellow square. © Activities for Learning, Inc., 2012
361.
Pythagorean Theorem
Area of blue squares = yellow square. © Activities for Learning, Inc., 2012
362.
Pythagorean Theorem
Draw a right triangle. © Activities for Learning, Inc., 2012
363.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
364.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
365.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
366.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
367.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
368.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
369.
Pythagorean Theorem
Turn the t-square upside down. © Activities for Learning, Inc., 2012
370.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
371.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
372.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
373.
Pythagorean Theorem
Do not move the t-square. © Activities for Learning, Inc., 2012
374.
Pythagorean Theorem
Do not move the t-square. © Activities for Learning, Inc., 2012
375.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
376.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
377.
Pythagorean Theorem © Activities
for Learning, Inc., 2012
378.
Pythagorean Theorem
Area of blue squares = yellow square. © Activities for Learning, Inc., 2012
379.
Triangle Area
A B C © Activities for Learning, Inc., 2012
380.
Triangle Area
A B C Draw the altitude. © Activities for Learning, Inc., 2012
381.
Triangle Area
A B C Area is half of enclosing rectangle. © Activities for Learning, Inc., 2012
382.
Triangle Area
A B C Another altitude. © Activities for Learning, Inc., 2012
383.
Triangle Area
A B C Find the area. © Activities for Learning, Inc., 2012
384.
Triangle Area
A B C Another altitude. © Activities for Learning, Inc., 2012
385.
Triangle Area
A B C Find the area. © Activities for Learning, Inc., 2012
386.
Triangle Area
A B C Altitudes intersect at the orthocenter. © Activities for Learning, Inc., 2012
387.
Triangle Area
A B C 3 areas slightly different. Why? © Activities for Learning, Inc., 2012
388.
Geometric Approach © Activities
for Learning, Inc., 2012
389.
Geometric Approach
• Basis of CAD (computer aided design). © Activities for Learning, Inc., 2012
390.
Geometric Approach
• Basis of CAD (computer aided design). • Integrates many concepts. © Activities for Learning, Inc., 2012
391.
Geometric Approach
• Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. © Activities for Learning, Inc., 2012
392.
Geometric Approach
• Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. © Activities for Learning, Inc., 2012
393.
Geometric Approach
• Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. • Excellent preparation for high school mathematics. © Activities for Learning, Inc., 2012
394.
Geometric Approach
Goals © Activities for Learning, Inc., 2012
395.
Geometric Approach
Goals • To use elementary math. © Activities for Learning, Inc., 2012
396.
Geometric Approach
Goals • To use elementary math. • To learn to read math texts. © Activities for Learning, Inc., 2012
397.
Geometric Approach
Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. © Activities for Learning, Inc., 2012
398.
Geometric Approach
Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world. © Activities for Learning, Inc., 2012
399.
Geometric Approach
Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world. • To enjoy mathematics. © Activities for Learning, Inc., 2012
400.
Personalized Learning
Bridges Middle-School Math with a Geometric Approach – Part 2 Presented by Tracy Mittleider, MSEd and Kathleen Lawler Based on work of Joan A. Cotter, Ph.D. Aplus+ Conference October 24, 2012 © Activities for Learning, Inc., 2012
Notes de l'éditeur
Right-handed user.
Hold pencil 2-3cm (1 in.)from tip.
Parallel lines.
Triangle must hug the t-square.
This doesn’t work!
Perpendicular lines.
Perpendicular lines.
Still more parallel lines.
Intersecting lines.
Intersecting lines.
Maintain a margin.
Move the t-square down.
Draw a side.
Flip the triangle and repeat.
Flip the triangle and repeat.
A bit of erasing.
The two halves are congruent.
Divide it in half another way.
Divide it in half another way.
Divide it in half another way.
Divide it in half another way.
A bit of erasing.
Flip the triangle and repeat.
Can you figure out how to do it?
Finding the center of the base line.
Be sure the tick mark intersects the line.
Flip the triangle over and repeat.
How do you draw the last line?
The smaller triangles are similar to the larger triangle.
Name the rhombuses:
Name the rhombuses: ABDF,
Name the rhombuses. ABDF, BCDF,
Name the rhombuses: ABDF, BCDF, BDEF.
Name the trapezoids:
Name the trapezoids: BCEF,
Name the trapezoids: BCEF, ACDF,
Name the trapezoids: BCEF, ACDF, ABDE.
Cut it out….
and fold on the lines.
and fold on the lines.
and fold on the dotted lines.
and fold on the dotted lines.
and fold on the dotted lines.
How could you draw eighths?
Not very esthetic.
Can you divide it into sixths?
Name an acute triangle:
Name an acute triangle: ACE.
Name a right triangle: ADE, ABE, ABG.
Draw the base line.
Move the t-square down.
Draw the sides.
Draw the diagonal…
or just a tick mark.
Draw the top.
A bit of erasing.
Outside square is twice the size of the inscribed square.
No tracing or measuring.
No tracing or measuring.
No tracing or measuring.
No tracing or measuring.
No tracing or measuring.
No tracing or measuring.
No tracing or measuring.
No tracing or measuring.
No tracing or measuring.
More examples to construct.
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