Download Unit 1 Geometry: Coordinates

Unit 1 Geometry: Coordinates
Introduction
In this unit, students will locate points on a coordinate plane and identify
coordinates of points. They will also use coordinate planes to compare
numeric patterns.
Materials
You will need a pre-drawn grid on the board for all lessons in this unit. If
you do not have such a grid, you can photocopy BLM 1 cm Grid Paper
(p. S-1) onto a transparency and project it onto the board. This will allow you
to draw and erase shapes and lines on the grid without erasing the grid itself.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
To save time, students can use ready-made coordinate grids on BLM
Coordinate Grids (p. S-3), but be sure that students are also able to draw
the grids.
Geometry
L-1
G5-1 Columns and Rows
Pages 1–2
STANDARDS
5.G.A.1
VOCABULARY
array
column
coordinates
ordered pair
row
c
row
l
u
m
n
r
column
w
Goals
Students will identify coordinates of points and draw points according to
their coordinates in an array of dots.
PRIOR KNOWLEDGE REQUIRED
Knows that arrays consist of columns of equal numbers of objects and
rows of equal numbers of objects
MATERIALS
deck of cards (NOTE: if available, use magnetic cards with a cookie
sheet or other metal board for demonstrations)
dividers, such as binders
Review the terms “column” and “row.” Draw an array of dots on the
board. Remind students that columns are vertical and rows are horizontal.
Use the picture in the margin as a reminder for students. Point out that the
shape looks like a cross. It wouldn’t look like a cross if we wrote “column”
across instead of down. (Briefly show the second picture in the margin on the
board, but erase it quickly.) Point out that if students ever forget which word
goes across, they can write the words both ways and choose the one that
looks like a cross. Have students draw the correct version in their notebooks.
Using coordinate systems to make it easy to find data. To illustrate the
idea of a coordinate system, start with a card trick:
1.Deal out 9 cards—face up—in an array of 3 cards × 3 cards.
3.Gather up the cards column by column—first one column that was not
selected, then the other that was not selected, and finally the column
that was selected on the top of the deck. Show clearly how to make
sure the 3 cards of the selected column are on the top of the deck.
x
4.Deal the cards face up in another 3 × 3 array so that the top 3 cards of
the deck are in the top row of the array.
5.Ask the volunteer to tell you what column the chosen card is in now.
The chosen card will always be the top card in that column.
Repeat the trick several times and ask students to try to figure out how
it works. Ask students to watch how you place the cards. You might also
repeat the trick with a 2 × 2 array.
L-2
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
2.Ask a student to select a card in the array and tell you what column that
card is in but not identify the card in any other way (e.g., by number or
by suit).
x
(MP.7)
When students understand how the trick works, ASK: Would there be
any point to the trick if the volunteer told me both the column number and
the row number of the selected card? (no) Trace a column and a row to
demonstrate that there is only one card that is in both the chosen column
and the chosen row. Explain that two pieces of information are enough to
clearly identify a position in an array or on a grid. Point out that labeling the
columns and rows creates a structure that allows us to describe a location
in a very effective way.
ASK: Will the trick work with a larger array? Have students try the trick with
a 4 × 4 array. They should see that the trick works for any square array of
cards that has the same number of columns and rows. (It can be modified
to fit a rectangular array—see Extension 1.)
ASK: Would the trick work if you switched the words “columns” and “rows”?
Have volunteers show how to perform the new trick.
Labeling columns and rows in arrays. Draw an array of 3 columns and
3 rows on the board and number the columns and rows as shown in the
margin. Stress that we label the columns from left to right and the rows
from bottom to top. Have students copy the picture.
3
2
1
123
Identifying dots in an array. Draw a random array on the board. Ask
students to trace specific columns or rows (e.g., the second column, the
third row) and then to circle specific dots (e.g., the dot in the third row and
the first column). Encourage students who have trouble locating a particular
dot to join the dots in the given row and column prior to circling the dot
itself. (The dot they are looking for will be at the intersection of the lines.)
Identifying the column and the row for a location. Draw an array of dots
on the board and circle a dot. Ask students to write the coordinates of the
dot: Column
, Row
. Have students practice identifying the column
and the row using the following Activity.
ACTIVITY
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Students work in pairs and use grid paper. Partners use a divider, such
as a binder, to conceal their grids from each other. Each partner draws
a square array of dots on their grid starting with a 3 × 3 array and
increasing the challenge by increasing the number of columns and rows.
Partner 1 circles a dot in the array and tells Partner 2 the column and
the row it is in. Partner 2 identifies the dot on their own grid and circles it.
Partners switch roles a number of times before checking their answers.
Introduce ordered pairs. SAY: Imagine you have to write the coordinates
of 100 points. Would you like to write the words “column” and “row” 100
times? What could you do to shorten the information? Students might
suggest making a T-table or even writing a pair of numbers. ASK: How do
you know which is first, the column number or the row number? What if
you have to ask a partner to find a dot with a pair of numbers but do not
tell your partner which number identifies the column and which number
Geometry 5-1
L-3
3
2
1
123
(MP.6)
identifies the row? SAY: Here are two numbers: 2 and 3. I won’t tell you which
one is the column number and which one is the row number. How many
points can you find that could go with these two numbers? (2, see margin)
Explain that mathematicians around the world have agreed to give the
location of a point by two numbers in parentheses. The column number is
always on the left and the row number is always on the right: (column, row).
SAY: This means that the numbers in the pair have a specific order, so we
call them an ordered pair. Give students several ordered pairs and ask them
to identify the corresponding points in an array of dots. Explain that the
ordered pair of numbers can also be called the “coordinates of the point.”
Exercises
a)Draw a 4 × 4 array of dots. Label the columns with letters and the rows
with numbers, starting from the bottom left corner.
b)Mark the points (A, 4), (B, 2), (C, 3).
c)Mark two other points on the array and ask a partner to name them.
Extensions
2.
You will need 27 cards. Lay the cards out as shown in the margin—in
groups of threes in an array of 3 × 3. Ask a student to select a card and
tell you only what column it’s in. Collect the cards so that the 3 triples
in the chosen column are the 9 top cards on the deck. (You do not
need to keep the triples together.) Deal the cards again so that the top
9 cards form 3 triples in the top row of a new array. Ask the volunteer in
which column the card is now. The chosen card is now in the top triple
of that column. Gather the cards and deal again so that the 3 cards
from that triple become the bottom cards laid down for 3 new triples of
the top row. Finally, ask the volunteer which column the card is in now.
The first (bottom) card in the triple in the top row of that column will be
the chosen card.
(MP.1)
L-4
This version of the trick illustrates a powerful principle in science
and mathematics: when you are looking for a solution to a problem,
it is often possible to eliminate many possibilities by asking a wellformulated question. In the card trick, the card dealer can single out
1 of 27 possibilities by asking only 3 questions. Repeat the trick, asking
students how many possibilities were eliminated by the first question
(there were only 9 possible cards left, so 27 − 9 = 18 possibilities were
eliminated), by the second question (9 − 3 = 6 eliminated), and by the
third (3 − 1 = 2 eliminated).
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
1.Modify the card trick used to introduce the coordinate system for nonsquare arrays as follows. Deal out an array of 3 columns and 5 rows
cards. Have a student select a card and tell you what column it’s in.
Gather up the cards so that the 5 cards in the selected column are on
the top. Deal out the cards again as an array of 5 columns and 3 rows
and with the cards of the selected column now in the top row of the
new array. Ask the volunteer to tell you what column the chosen card
is in now. The top card in that column is the chosen card.
G5-2 Coordinate Grids
Pages 3–4
STANDARDS
5.G.A.1
VOCABULARY
axis/axes
column
coordinates
first, second coordinate
grid
intersection point
ordered pair
origin
row
x-axis, y-axis
Goals
Students will identify coordinates of points and draw points according to
their coordinates on a coordinate grid.
PRIOR KNOWLEDGE REQUIRED
Knows that arrays consist of columns of equal numbers of objects and
rows of equal numbers of objects
Can identify coordinates of points in an array
Can identify a point in an array given the coordinates
Can identify and plot points on horizontal and vertical number lines,
including fractions and decimals
MATERIALS
yardstick
BLM Grid with Decimals (p. L-17)
dividers, such as binders
Introduce grids. Review how to identify a dot given a pair of coordinates
in a labeled array of dots. Remind students that, in an ordered pair, the
column is always written first.
4
3
2
1
A B C D
Explain that grids are much more common than arrays of dots. Draw a
4 × 4 grid square and explain that there are two ways to label a grid. One
way is commonly used on maps: the squares on the grid are numbered or
lettered. When you are looking for an object on a map, it is described as
being located within a square, so your school might be described as “in
A4.” This means the school is located in the square that is in column A, in
the 4th row. Show the picture in the margin to illustrate what you mean, and
explain that the dot shows the approximate location of the school.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
This type of labeling is used in a common game, Battleship.
5
4
3
2
1
If available, show a real map such as a page from a city atlas. Tell students
that a certain landmark (for example, your school) is located in a specific
square (for example, B3). Have students find the location on the map. Point
out that many other objects are located within that square, so this way of
identifying the location is not very precise.
A BCDE
Geometry 5-2
Explain that, in mathematics, the grids are usually marked differently
because greater precision is needed. The numbers or letters mark the lines
of the grid themselves, and a point such as (A, 4) will be an intersection
point on the grid. Show what you mean using the picture in the margin.
The dot shows the position (A, 4).
L-5
Draw a 4 × 4 grid square on the board and number the lines on the grid
from 1 to 5, as shown in the margin. Point to various positions on the
grid and have students identify the coordinates. To see the answers of all
students at the same time, write a pair of brackets with blanks, ( ,
)
for each point, and have students signal the numbers that should appear in
each blank.
12 3 4 5
Have students draw a 5 × 5 square on grid paper and number the lines on
the grid starting from 0. Remind students that columns are labeled from left
to right, and rows are labeled from bottom to top. ASK: Which corner is
(0, 0)? (bottom left) What are the coordinates of the top right corner? (5, 5)
The top left corner? (0, 5) The bottom right corner? (5, 0)
Introduce the axes and origin. On the board, draw a coordinate grid as
shown in the margin. Point out the axes and mention that axes is the plural
of axis. The horizontal line is called the x-axis, and the vertical line is called
the y-axis. Tell students that the point at which the two axes intersect is
called the origin. Label the axes and the origin.
y-axis
5
4
3
2
1
origin
Label the axes with numbers and point out that the axes are actually
number lines, both starting at 0. Explain that the origin is the intersection of
two lines, so its coordinates are (0, 0).
x-axis
Repeat the Activity from Lesson G5-1 using the grids but providing the
coordinates as ordered pairs instead of column number and row number.
Introduce the x-coordinate and y-coordinate. On a grid on the board,
mark the point (4, 3) and have students identify its coordinates. Explain that
mathematicians say that the point (4, 3) has x = 4; trace your finger down
from the point, and show that it is directly above the number 4 on the x-axis.
Then trace with your finger from the point (4, 3) and left to the y-axis to look
at the number on the y-axis. SAY: Mathematicians say that this point has
y = 3. Have students rewrite the coordinates of the following points as
x=
,y=
: (2, 1), (1, 3), (4, 2), (3, 4). Then mark several points on
the grid and have students identify the x and the y for these points.
y
Review locating halves and quarters on a number line. Draw a number
line divided into quarters from 0 to 5 and ask students where the point 1/2
is located. Invite a volunteer to show the location. Repeat with 1/4, 2 1/2,
3 3/4, and 4 1/2.
2
1
0
12
L-6
x
Halves and quarters on a coordinate grid. Draw a large coordinate grid
where the axes are divided into quarters, as shown in the margin, and mark
the point (1/2, 1 1/4) on it. Using a yardstick, draw dashed lines from the
point to the axes and have students identify the coordinates of the point.
Repeat with several other points.
Draw the point (1 1/8, 0) and ask students to identify the coordinates.
Emphasize that points can be between the increments on the number line.
Repeat with points (0, 1 3/8) and (1 7/8, 1 7/8).
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Tell students that the x-coordinate is often called the first coordinate, because
it is written first, and the y-coordinate is called the second coordinate.
Review marking decimal tenths on a number line. Draw a number line
divided into tenths, from 0 to 3.5, but label only the whole numbers. Point to
a few locations on the line and have students identify the decimal for each
location. Write several decimals and point at locations on the number line.
Have students signal with thumbs up and thumbs down to indicate whether
the location you are pointing at is the decimal in question.
Decimal tenths on a coordinate grid. Display BLM Grid with Decimals.
Use the point A (1.8, 2.4) to show students how to determine the coordinates
of the point, and then invite volunteers to draw lines from other points to
the axes. Then have students match the points to the coordinates on the
BLM. Point to the locations on the grid for each coordinate pair and have
students signal thumbs up or thumbs down to indicate whether this point is
the one given by the coordinates.
Extensions
(MP.1, MP.3)
1.
Predict which point is lower on the y-axis: (0, 1.8) or (0, 1 3/4). Explain
your prediction. Plot the points and check your answer.
Answer: (0, 1 3/4), because 1 3/4 is less than 1.8
2.Draw the following points on a coordinate grid:
(2, 6), (4, 4), (5, 7), (7, 8), (5, 2), (3, 4), (2, 1), (0, 0)
Then join the points in the order you drew them. Join the first point to
the last point. What letter did you make?
Answer: N
3.Players will need a divider to conceal the coordinate grids they are
working on from partners. Player 1 draws a square or a rectangle on the
coordinate grid and tells Player 2 the coordinates of the vertices. Player
2 tries to visualize the shape and guess what kind it is before plotting
the vertices. Player 2 plots the vertices and checks the answer.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Advanced: Use other shapes, such as rhombuses, parallelograms,
and trapezoids.
Geometry 5-2
L-7
G5-3 Sliding Points
Page 5
Goals
STANDARDS
5.G.A.2
Students will slide points on a grid and describe slides.
VOCABULARY
axis/axes
coordinates
origin
x-axis, y-axis
PRIOR KNOWLEDGE REQUIRED
Can distinguish right from left
Can locate points on a coordinate grid
Can identify coordinates of points on a coordinate grid
MATERIALS
magnetic boards with a grid on them and small circular
magnets (optional)
ball
BLM Coordinate Grids (p. S-3)
A
B
The point slid from
position A to position B
3 units right.
Sliding dots on a grid. If available, give each student a magnetic board
with grid paper on it (e.g., a cookie pan) and a small circular magnet to
represent a point. (If unavailable, have students draw points on grid paper.)
Demonstrate how to count how many units a dot slides in one direction.
Have students practice sliding points right and left, and then up and down.
Students should be able to identify how far a point slides in a particular
direction and be able to slide a point a given distance. If students have
difficulty distinguishing between left and right, write the letters “L” and “R”
on the left and right sides of the board.
Once students have demonstrated that they can slide a dot in a given
direction, show them how to slide a dot in a combination of directions. You
might also draw a basketball court on a magnetic grid and invite volunteers
to move a small circular magnet as if they were passing a basketball.
Exercises
a) 3 units right
b) 5 units left
c) 7 units down
d) 2 units up
e) 2 units left and 5 units up
2.Position several small figures of players on grid intersections around
the “court” and number the players. NOTE: the underlined numbers
and directions are only suggestions and can be varied as you wish.
a)Player A passes the ball 5 units right and 2 units up. Who receives
the pass?
b)Player B wants to pass the ball to Player C. How many units left and
how many units down should the ball go?
L-8
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
1. Pass the ball…
Bonus
Player D sent the ball 3 units up. How many units, and in which direction(s),
should Player E move to get the pass?
ACTIVITIES 1–2
1.Students stand in a grid format. Assign students to be points
on the grid. Give one student a ball and provide directions such
as “Slide the ball 3 units to the right.” The student with the ball
passes it to the correct “point on the grid.”
2.In the schoolyard, draw a grid on the ground. Students move a
certain number of units in various combinations of directions by
hopping from point to point on the grid.
X-coordinate and y-coordinate show distance along the axes. Draw a
coordinate grid on the board and remind students how to find points with
coordinates. Have them use BLM Coordinate Grids and locate the points
(4, 3), (2, 4), (5, 6), and (1, 0) on the grid.
Ask students to imagine that the coordinate grid lines are streets, and the
points identified by coordinate pairs are addresses. A truck starts at the
origin, point (0, 0), and drives to point (4, 3). It is a heavy truck, so the driver
prefers to make as few turns as possible. Invite a volunteer to trace a
possible route on the grid, and then ask students to describe the path.
(3 units up, 4 units right, or 4 units right, 3 units up) Discuss the distances
the truck has to travel in each direction. There are two possible routes.
Route 1: travel along the x-axis first for 4 units because the point is on
vertical line 4, and continue parallel to the y-axis for 3 units because the
point is on the horizontal line 3. Route 2: travel along the y-axis for 3 units,
and then continue parallel to the x-axis for 4 units. Point out that the numbers
of the streets are also the distances that need to be traveled in a certain
direction, parallel to an axis. Repeat the whole discussion with the truck
traveling to a different point or destination.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Exercises: How many units right and how many units up do you need to go
from the origin to get to each point?
a) (3, 2)
b) (5, 10)
c) (24, 18) d) (125, 987)
Extensions
(MP.1)
4
3
3
2
1
1. T
he grids below are treasure maps. The treasure is hidden at one of the
vertices of the grid. The other marked points are clues to the location of
the treasure. The numbers beside marked points give the total number
of units you need to slide from the point to the treasure. For example, in
the map in the margin, the treasure is at point (2, 2), and the number 3
beside the point (3, 4) shows the total number of units to slide from
(3, 4) to (2, 2): 2 units down and 1 unit left, 2 + 1 = 3.
0 1234
Geometry 5-3
L-9
Ask students to find the treasure in each map. NOTE: In some maps,
the clues lead to two possible locations, thus more information would
be needed to find the treasure; in another, more than enough is given.
Which ones are these?
i)
4
3
2
1
ii)4
3
2
2
1
4
iii)
4
3
2
1
1
2
1
2
0 1234
0 1234
0
vi)
v)
iv) 4
4
4
3
3
3
2
4
2
2
2
1
1
1
1
1
3
2
0
0 1234
0 1234
3
1234
2
2
3
1234
Answers: i) (1, 1); ii) (0, 1); iii) not enough information, could be (1, 1)
or (0, 2); iv) (0, 1); v) not enough information, could be (1, 1) or (2, 2);
vi) (2, 2), too much information
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
2.A game for two: Player 1 draws a 4 × 4 grid similar to those in
Extension 1 and picks a point at which to “bury” a treasure. Player 2
tries to guess the location (coordinates) of the treasure. Each time
Player 2 makes a guess, Player 1 writes the total number of units
needed to slide from the guessed point to the treasure. Player 2
uses this information to make an educated next guess. Players
switch roles when the treasure is found.
L-10
Teacher’s Guide for AP Book 5.2
G5-4 Maps
Pages 6–7
Goals
STANDARDS
5.G.A.2, 5.NF.B.6
Students will solve problems using coordinate grids.
PRIOR KNOWLEDGE REQUIRED
VOCABULARY
axes
coordinates
Can plot points, including decimals, on a coordinate grid
Can slide points according to descriptions
Can describe a slide for a point
MATERIALS
BLM Washington and Oregon (p. L-18)
(MP.4)
Solving problems using coordinate grids. Use BLM Washington and
Oregon to review concepts learned to date. Answer the questions below as
a class and have students explain the solution.
Exercises
1.What are the coordinates of the following cities? Use decimal points as
necessary.
a) Spokane, WA b) Aberdeen, WA
d) Eugene, OR
e) Yakima, WA
c) Portland, OR
Answers: a) (7, 8), b) (1, 7), c) (2, 5), d) (1.5, 3), e) (4, 6.5)
2. What city is located at each point?
a) (2.4, 8)
Bonus: 1
b) (1.5, 1)
c) (2.2, 7.5)
2
1
,4
3
3
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) Seattle, WA; b) Grants Pass, OR; c) Tacoma, WA;
Bonus: Salem, OR
Solving problems using slides on maps. Explain that the directions north,
south, east, and west are marked on the map with letters N, S, E, and W.
Ask students to show with their fingers which direction on the map they
would move if they are looking for a point that is 1 square north (up),
100 miles west (left), 20 miles south (down), or 50 miles east (right).
Exercises: What is located at each point?
a) 2 squares south of Eugene, OR
b) 1 square west and 2 squares north of Portland, OR
c) 3.5 squares east of Grants Pass, OR
1
d)
square south and 3 squares east from Aberdeen, WA
2
Geometry 5-4
L-11
Bonus: If you go 6 squares west of Spokane, WA, you get to a large park.
What is it called? What are the coordinates of the park symbol (a tree) on
the map?
Answers: a) Grants Pass, OR; b) Aberdeen, WA; c) Hart Mountain National
Antelope Refuge; d) Yakima, WA; Bonus: Olympic National Park, (1.3, 8.3)
(MP.4)
Using the scale. Explain that each square on the map is 50 miles long and
50 miles wide. ASK: If you move 2 squares north, how many miles north are
you moving? (100 mi) If you move 150 miles south, how many squares are
you moving? (3 squares) If you move 1/2 square west, how many miles are
you moving? (25 mi) How do you know? (half of 50 miles is 25 miles) If you
move 2 1/2 squares west, how many miles are you moving? (125 mi) How
do you know? (2 squares = 100 mi, 1/2 square = 25 mi, so 2 1/2 squares
= 125 mi)
Exercises
1. What is located at each point?
a) 100 miles east of Aberdeen, WA
b) 50 miles west of Willamette National Forest
c) 100 miles south of Eugene, OR
d) 175 miles east of Grants Pass, OR
e)125 miles west and 100 miles north of Hart Mountain National
Antelope Refuge
Answers: a) Mt. Rainier National Forest; b) Eugene, OR; c) Grants
Pass, OR; d) Hart Mountain National Refuge; e) Willamette National
Forest
2.Describe a way to travel (in miles and direction) from one location to the
other. Note that directions should be given as north, south, east, and
west (rather than diagonal travel).
b) From Mt. Rainer National Forest to Portland, OR
c) From Portland, OR to Willamette National Forest
d)From Willamette National Forest to Hart Mountain National
Antelope Refuge
Answers: a) 100 miles east, b) 100 miles south and 50 miles west,
c) 100 miles south and 25 miles east, d) 100 miles south and
125 miles east
3.A helicopter set out from Grants Pass, OR, flew to Eugene, OR, and
then flew to Willamette National Forest. How far did it fly?
Answer: 150 miles
L-12
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a) From Aberdeen, WA to Mt. Rainer National Forest
G5-5 Graphing Sequences
Pages 8–9
STANDARDS
5.G.A.1, 5.OA.B.3
VOCABULARY
axes
coordinate grid
decrease
increase
operation
ordered pair
sequence
x-axis
x-coordinate, first
coordinate
y-axis
y-coordinate, second
coordinate
Goals
Students will graph pairs of sequences on a coordinate grid.
PRIOR KNOWLEDGE REQUIRED
Can plot points on a coordinate grid
Can slide points according to descriptions
Can describe a slide for a point
MATERIALS
BLM Coordinate Grids (p. S-3)
Review extending sequences given a rule. Remind students that if they
have a rule such as “Start at 2. Add 3 each time,” they can produce a
sequence: 2, 5, 8, 11, …. The rules can use different operations, including
multiplication and division. We will only deal with sequences that are made
by the same operation.
Exercises: Write the first four numbers in the sequence using the rule.
a) Start at 3. Add 7 each time.
b) Start at 1. Multiply by 3 each time.
c) Start at 256. Divide by 2 each time.
d) Start at 256. Subtract 2 each time.
Answers: a) 3, 10, 17, 24; b) 1, 3, 9, 27; c) 256, 128, 64, 32;
d) 256, 254, 252, 250
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Add 4 Add 3
0
2
4
5
8
8
12
11
Geometry 5-5
Extending sequences in a table. Remind students that they also used
tables to extend two sequences together. Show the example in the margin
and have students tell you how to extend each sequence. (answers are
in italics) Have students draw tables with 2 columns and 5 rows for the
exercises below and leave room to add another column.
Exercises: Extend both sequences in the table.
a)
Add 2
0
Subtract 2 b) Add 3
10
0
Add 3
2
c)
Add 2
1
Subtract 2 d) Add 4
0
14
Add 5
0
e)
Subtract 3
15
Subtract 2 f) Add 3
1
15
Multiply by 2
2
L-13
Add 4 Add 3 Ordered pair
0
2
(0, 2)
4
5
(4, 5)
8
8
(8, 8)
12
11
(12, 11)
Converting pairs of sequences to ordered pairs. Explain that we can
create an ordered pair from each row in a table, and then draw a point
on a coordinate grid by using the ordered pair as the coordinates of the
point. Add a column to the table in the example and show how to write
the ordered pairs: (0, 2), (4, 5), (8, 8), and (12, 11). See the margin for the
answer. Have students add a column to each table and write the ordered
pairs in the last exercise.
Plotting pairs of sequences on coordinate grids. Show students how
to plot the point (4, 5) on a coordinate plane: go along the x-axis 4 units
because the x-coordinate is 4, and then go up (parallel to the y-axis) 5 units.
ASK: Why do we go up 5 units? (the y-coordinate is 5) Have students draw
a fresh coordinate grid and plot the point (4, 5) on it. As a class, plot the
rest of the points from the example above (see result below). Then have
students plot the points from the tables they produced earlier. To avoid
drawing multiple coordinate grids, students can either plot all the points
on one grid and use a separate color for each new table, or they can use
premade grids from BLM Coordinate Grids.
y
0 12 3456 7891011
(MP.7)
Add 2 Add 3
2
3
x
Pairs of sequences made by addition or subtraction make straight
lines. Ask students to join the points that come from the same table on
each graph. What do they notice? Students should see that the points in
all tables but one fall onto a straight line. ASK: Which table did not make a
straight line? (f) How is one of the rules that make this table different from
the other rules? (all the other rules are made with addition or subtraction,
but this table has one of the rules made by multiplication)
Present another pair of sequences in a table (see margin) and have
students predict whether the points will be on the same line. Have them
make the ordered pairs and plot them to check. ASK: Are the points on the
same line? (yes)
Then ask students to write two rules made by addition or subtraction, and
produce four pairs of numbers using the rules. Ask them to try to make the
rules so the numbers are smaller than 16 and so they can fit the points on a
grid on BLM Coordinate Grids.
L-14
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
11
10
9
8
7
6
5
4
3
2
1
Have students work in pairs and make a table with the two sequences
they produced, make the ordered pairs, and plot them. ASK: Did everyone
get the same table? (no) Did everyone get a straight line? (yes) Have
several students share the tables they created and what the plotted pairs
of sequences look like. Look for a variety of pairs of sequences: different
combinations of operations, different starting points. ASK: Do you think
every pair of sequences will make a straight line? (no) Will every pair of
sequences made by addition or subtraction make a straight line? (yes)
(MP.3, MP.7)
Discuss why pairs of sequences made by addition or subtraction make
straight lines using the pair of sequences in the table below. ASK: What
slide do you make when you pass from point (2, 1) to point (4, 4)? (2 units
right, 3 units up) Repeat with the slides from (4, 4) to (6, 7) and from (6, 7)
to (8, 10). ASK: Did you get the same slide each time? (yes) Where does
the number of units you slide the point right appear in your table? (it is the
number you add in the first sequence) Where does the number of units you
slide the point up appear in your table? (it is the number you add in the
second sequence) Draw an arrow representing each slide as shown below.
Explain that, when the slides are the same, the arrows showing the slides
continue each other and make a straight line, so it makes sense that the
points will be on a straight line.
y
Add 2 Add 3 Ordered pair
2
1
(2, 1)
4
4
(4, 4)
6
7
(6, 7)
8
10
(8, 10)
10
9
8
7
6
5
4
3
2
1
0 12 3456 78910
x
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Direction of the slide is determined by the operations in the sequences.
Ask students to check that the pair of sequences they produced above
also result in slides that are exactly the same. ASK: Who has slides that go
right and up? What operation are you using for both sequences? (addition)
Who has slides that go left and up? What operation are you using in the
first sequence? (subtraction) In the second sequence? (addition) Repeat
with slides going right and down and slides going left and down. Explain
that, when a sequence is made by addition, the numbers in the sequence
increase (grow). This means you go right or up the axes, which are
number lines. ASK: What happens to the numbers if a sequence is made
by subtraction? (the numbers decrease) In this case, in which direction
do we go on a horizontal number line? (left) On the x-axis? (left) In which
direction do we go on a vertical number line? (down) On the y-axis? (down)
Geometry 5-5
L-15
y
Pairs of sequences where at least one sequence is made by
multiplication do not produce a straight line. Now have students
predict if the pair of sequences “Start at 0. Add 2 each time” and
“Start at 1. Multiply by 3 each time” will produce a straight line. (no)
Have students write the first three pairs and plot them to check. (See
margin for the result.) Again, ask students to describe the slides from
the first point to the second and from the second to the third. Are the
slides the same? (no) The slides are not the same, so if you draw the
line segments, they meet at an angle and do not make a straight line.
9
8
7
6
5
4
3
2
1
0 123456
x
Extensions
a) E
xtend the sequences to three rows of numbers and write a list of
ordered pairs for the table.
(MP.1)
i)
Add 3 Add 4 ii)Add 3 Add 4 iii)Add 3 Add 4
0
0
1
3
5
1
b)Plot the ordered pairs on the same grid and join the points with a line.
Use a different color for each table.
c) How are the rules in the tables the same? How are they different?
d) How are the lines the same? How are they different?
e)Write two rules “Start at
.
each time” to make a line
parallel to the other three lines. Plot the line to check your answer.
f)Write two rules “Start at
.
each time” to make a line
not parallel to the other four lines. Plot the line to check your answer.
Answers
y
b) See grid in margin.
11
10
9
8
7
6
5
4
3
2
1
c)The numbers added are the same for all tables, but the starting
numbers are different.
e)Start at
. Add 4 each time. Start at
numbers will vary.
0 1234567 891011
L-16
x
. Add 3 each time. Starting
f)Starting numbers will vary. Any pair of rules made by addition or
subtraction that does not have “Add 4” in the first rule and “Add 3”
in the second rule will work.
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
d)All three lines are parallel, but they pass through different points.