[img]data:image/png;base64,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[/img][br]What do you notice? What do you wonder?
[list][*]Jada’s pet turtle walked 10 feet, and then half that length again.[/*][*]Jada’s baby brother walked 3 feet, and then half that length again.[/*][*]Jada’s hamster walked 4.5 feet, and then half that length again.[/*][*]Jada’s robot walked 1 foot, and then half that length again.[/*][*]A person walked feet and then half that length again.[/*][/list]
Explain how you computed the total distance in each case.
Two students each wrote an equation to represent the relationship between the initial distance walked ([math]x[/math]) and the total distance walked ([math]y[/math]).[br][list][*]Mai wrote [math]y=x+\frac{1}{2}\cdot x[/math].[/*][*]Kiran wrote [math]y=\frac{3}{2}\cdot x[/math].[br][/*][/list]Do you agree with either of them? Explain your reasoning.
[size=150]Zeno jumped 8 meters. Then he jumped half as far again (4 meters). Then he jumped half as far again (2 meters). So after 3 jumps, he was [math]8+4+2=14[/math] meters from his starting place.[/size][br][br]Zeno kept jumping half as far again. How far would he be after 4 jumps? 5 jumps? 6 jumps?
Before he started jumping, Zeno put a mark on the floor that was exactly 16 meters from his starting place. How close can Zeno get to the mark if he keeps jumping half as far again?
If you enjoyed thinking about this problem, consider researching Zeno's Paradox.
Write a story for the diagram that doesn't have a match in the above activity.
In the applet below, there is a set of cards that have proportional relationships represented three different ways: as descriptions, equations, and tables. [br][br][list=1][*]Take turns with a partner to match a description with an equation and a table.[br][list=1][*]For each match you find, explain to your partner how you know it’s a match.[/*][*]For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking.[/*][/list][/*][*]When you agree on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches.[/*][/list]