Transcript Chapter 3.2.1

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Vocabulary:
Integers - The set of numbers {…, –3, –2, –1, 0, 1, 2, 3, …}.
In this section, you will work to solve problems that involve distances and directions using diagrams
and numbers. This work will begin your investigation of integers (positive and negative whole
numbers and zero).
In the frog-jumping contest described in Chapter 1, the final measurement represents the distance
the frog moves away from a starting pad after three separate hops. The three hops of each frog in
the contest can be different lengths and go in different directions. However, for today’s lesson,
assume that frogs always hop along a straight line.
89. GETTING THERE
Elliott has been watching Dr. Frog take practice jumps all
day. The frog keeps landing 15 feet from the starting pad
after making three hops. Answer the questions below to
consider ways that Dr. Frog can travel 15 feet in
three hops.
a. Find three combinations of hops to move Dr. Frog
15 feet from where he started? Show your work
with pictures, words, numbers, or symbols.
b. Can the frog move 15 feet in three equal hops?
c. If two of the frog’s hops are each 10 feet long, how
could you describe the third hop so that he still
lands 15 feet away from the starting pad? Is there
more than one possibility?
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90. Elliott is so interested in the frogs that he is developing a video game about
them. In his game, a frog starts on a number line like the one above. The frog
can hop to the left and to the right.
For each part below, the game starts with the frog sitting at the number 3 on
the number line. (Use your Lesson 3.2.1 Resource Page)
a. If the frog starts at 3, hops to the right 4 units, to the left 7 units, and then to
the right 6 units, where will the frog end up?
b. If the frog makes three hops to the right and lands on 10, list the lengths of
two possible combinations of hops that will get it from 3 to 10.
c. Could the frog land on a positive number if it makes three hops to the
left? Use an example to show your thinking.
d. Additional Challenge: The frog made two hops of the same length to the
right and then hopped 6 units to the left. If the frog ended up at 11 on the
number line, how long were the first two hops?
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91. OPPOSITES
Find the Math Notes box titled “Opposites,” on page 31 of your toolkit. After
reading about opposites in the Math Notes box, answer the questions below.
a. Elliott played the frog video game twice. In each game, the frog
made one hop. After the second game, the frog ended up the same
distance from zero as after the first game, but on the opposite side of
zero.
Describe the two games: How could the frog start at 3, hop once, and
then in the second game again start at 3 and end up the same distance
from zero, but in opposite direction as the first game? Give a possible set
of hop directions, lengths, and ending points.
b. How would the frog hop to meet the following requirements?
* From 3, make one hop and land on 6.
* From 6, make one hop and land on the opposite of 6.
* From the opposite of 6, make one hop to land on the opposite of the
opposite of 6.
c. How could you write “the opposite of the opposite of 6” using math
symbols? Where would this be on the number line?
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92. While designing his video game, Elliott decided to replace some of his long
sentences describing hops with symbols. For example, to represent where the
frog traveled and landed in part (a) of problem #90, he wrote:
3+4−7+6
a. How does Elliott’s expression represent the words in part (a) of problem
#90? Where did the 3 come from? Why is one number subtracted?
b. If another set of hops was represented by 5 − 10 + 2 + 1, describe the frog’s
movements. Where did the frog start and where did it end up?
c. One game used the expression −5 + 10. Where does the frog start? Where
does it end up? What is special about the ending point?
d. Another game had a frog start at the opposite of –3, hop 5 units to the left,
9 units to the right, and then hop to its opposite position. Write an expression
to represent the frog’s motion on the number line. Where did the frog end up?
e. Where does the frog start in the expression −(−2) + 6? Where does it land?
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93. Another frog starts at −3 and hops four times. Its hops are listed at right.
Hop Lengths
Right 2 units
Left 7 units
Right 10 units
Left 3 units
a. If you have not done so already, write an expression (adding and subtracting)
for the frog’s movements. Where does the frog end up?
b. Is it possible for the frog to finish at 2 on the number line if it makes the same
hops in a different order?
c. Does the frog land in the same place no matter which hop the frog takes first,
second, etc.?
d. Kamille says that for the frog to end up at –1, she can ignore the last three
hops on the list. She says she only needs to move the frog to the right
2 units. Is she correct? Why?
e. Give another set of four hops that would have the frog end up where it started
at –3. Make the hops different lengths from one another. What needs to be
true about the frog’s four hops? Are they somehow related to each other?
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Tonight’s homework is…
3.2.1 Review & Preview, problems #94 – 98
• Label your assignment with your name and
Lesson number in the upper right hand corner
of a piece of notebook paper.
(Lesson 3.2.1)
•Show all work and justify your answers for full
credit.
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