IM Alg2.1.11 Lesson: Adding Up
Evaluate mentally.
[math]\frac{1}{2}+\frac{1}{4}[/math]
[math]\frac{1}{2}+\frac{1}{4}+\frac{1}{8}[/math]
[math]\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}[/math]
[math]\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\frac{3}{16}[/math]
Tyler has a piece of paper and is sharing it with Elena, Clare, and Andre.
[size=150]He cuts the paper to create four equal pieces, then hands one piece each to the others and keeps one for himself.[br][/size][br]What fraction of the original piece of paper does each person have? You may use the applet below to draw a diagram.
[size=150]Tyler then takes his remaining paper and does it again. He cuts the paper to create four equal pieces, then hands one piece each to the others and keeps one for himself.[/size][br]What fraction of the original piece of paper does each person have now?[br]
[size=150]Tyler then takes his remaining paper and does it again.[/size][br]What fraction of the original piece of paper does each person have now?
What happens after more steps of the same process?
Here is a geometric shape built in steps.
[list][*]Step 0 is an equilateral triangle.[/*][/list][img]data:image/png;base64,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[/img][br][list][*]To go from Step 0 to Step 1, take every edge of Step 0 and replace its middle third with an outward-facing equilateral triangle.[/*][/list][img]data:image/png;base64,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[/img][br][list][*]To go from Step 1 to Step 2, take every edge of Step 1 and replace its middle third with an outward-facing equilateral triangle.[br][/*][*]This process can continue to create any step of the design.[/*][/list][list=1][/list][br]Find an equation to represent function [math]S[/math], where [math]S\left(n\right)[/math] is the number of sides in Step [math]n[/math]. What is [math]S\left(2\right)[/math]?[br]You may use the applet below to help you.
[size=150]Consider a different function [math]T[/math], where [math]T\left(n\right)[/math] is the number of [i]new[/i] triangles added when drawing Step [math]n[/math]. Let [math]T\left(0\right)=1[/math]. [/size][br]How many new triangles are there in Steps 1, 2, and 3? Explain how you know.
What is the [i]total[/i] number of triangles used in building Step 3?
Suppose the Step 0 triangle has area 1 square unit. Complete the table.
What patterns do you notice?
IM Alg2.1.11 Practice: Adding Up
Find the sum of the sequence:
[math]\frac{1}{3}+\frac{1}{9}[/math]
[math]\frac{2}{3}+\frac{2}{9}[/math]
[math]\frac{1}{3}+\frac{1}{9}+\frac{1}{27}[/math]
[math]\frac{2}{3}+\frac{2}{9}+\frac{2}{27}[/math]
[math]\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}[/math]
[math]\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}[/math]
Priya is walking down a long hallway. She walks halfway and stops. Then, she walks half of the remaining distance, and stops again. She continues to stop every time she goes half of the remaining distance.
What fraction of the length of the hallway will Priya have covered after she starts and stops two times?[br]
What fraction of the length of the hallway will Priya have covered after she starts and stops four times?[br]
Will Priya ever reach the end of the hallway, repeatedly starting and stopping at half the remaining distance? Explain your thinking.[br]
A geometric sequence h starts with 10, 5, . . .
Explain how you would calculate the value of the 100th term.
Here is a graph of sequence r.
[img]data:image/png;base64,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[/img][br][br]Define [math]r[/math] recursively using function notation.
An unfolded piece of paper is 0.05 mm thick.
Complete the table with the thickness of the piece of paper [math]T\left(n\right)[/math] after it is folded in half [math]n[/math] times.[br]
Define [math]T[/math] for the [math]n^{th}[/math] term.[br]
What is a reasonable domain for the function [math]T[/math]? Explain how you know.[br]
A piece of paper is 0.05 mm thick.
Complete the table with the thickness of the paper [math]t\left(n\right)[/math], in mm, after it has been folded [math]n[/math] times.[br]
Does [math]t\left(0.5\right)[/math] make sense? Explain how you know.[br]
An arithmetic sequence a starts 84, 77, . . .
Define [math]a[/math] recursively.[br]
Define [math]a[/math] for the [math]n^{th}[/math] term.[br]
Here is a pattern of growing rectangles:
Describe how the rectangle grows from Step 0 to Step 2.
Write an equation for sequence [math]S[/math], so that [math]S\left(n\right)[/math] is the number of squares in Step [math]n[/math].[br]
Is [math]S[/math] a geometric sequence, an arithmetic sequence, or neither? Explain how you know.[br]