Homework
help for fifth grade Investigations units
*Note: These
suggestions are written for the homework in the Additional Problems workbook, which
includes Student Sheets and games from the program as well as additional tasks
written for Stow-Munroe Falls Schools.
If teachers choose to send home other work, they may choose to send home
additional suggestions for helping your child.
Click here for explanations of
the math content in all of the fifth grade units.
Skip to Building on Numbers You Know
Skip to Between Never and Always
Skip to Data: Kids, Cats, and Ads
Mathematical Thinking at Grade 5 Download the two-page parent reference sheet and
letter for this unit
Investigation
1
Sessions
1, 2, and 3
(Session 1)
For SS 2, students
will be drawing rectangles for the factor pairs from each number, 1-25. A factor is a whole number that divides
evenly into the number given. For
example, for the number 6, students would draw a 1 by 6 rectangle and a 2 by 3
rectangle, since 1 x 6 = 6 and 2 x 3 = 6, and there are no other such
pairs. Some numbers have many
factors, and some only have two.
Be sure students label the dimensions of each rectangle and write the
factor pair in it (e.g., 2 x 3).
Then, on SS 2, students should record any observations they make about
the rectangles and factor pairs.
See the examples on the sheet.
(Session 2)
For #2-6, if
students do not remember the definitions of these words, they can use class
notes, a dictionary, or other reference (possibly online) to help.
(Session 3)
For SS 3, skip
counting is just counting by a chosen number. For example, to skip count by threes, we would say 3, 6, 9,
12, etc. For #8, help students to
think about patterns in the numbers they have said and written. For #9 and 10, if students do not
remember the definitions of these words, they can use class notes, a
dictionary, or other reference (possibly online) to help. Help students connect multiples to skip
counting if they are confused.
Sessions
4, 5, and 6
For all of these
tasks, help students to reason carefully about the numbers involved and to
double-check their answers. Some vocabulary
help: a factor is a whole number that divides evenly into the number
given. For example, the factors of
6 are 1, 2, 3, and 6. A multiple
represents groups of the given number; multiples of 6 are 6, 12, 18, 24,
etc. Prime numbers are only divisible
by themselves and 1; 7 is prime, for example. Composite numbers have more than two factors. A square number has a factor that can
be multiplied by itself to give the number; 9 is square because 3 x 3 = 9. In this case, 3 is the square
root. Even numbers can be divided
by 2 to give whole number answers; odd numbers cannot.
For SS 6, help
students to find as many answers for each puzzle as they can.
For SS 7, help
students to find as many numbers as they can that fit the first clue, and then
help students to check that the clue they write in #3 really makes the puzzle
impossible.
Building on Numbers You Know Download the two-page parent
reference sheet and letter for this unit soon
Investigation
1
Session
1
For all of these
tasks, students should use the most efficient strategies that make sense to
them. Do not impel your student to
use a method that he/she does not understand. For example, for the first task on SS 1, students might
consider how many 25s are in 100 and then use this to determine how many are in
300. Students should always try to
use number relationships that they know to solve these types of problems.
Session
2
For all of these
tasks, students should use the most efficient strategies that make sense to
them. Do not impel your student to
use a method that he/she does not understand. Students should always try to use number relationships that
they know to solve these types of problems. For example, for #4, students might find the difference
between 315 and 515 and then between 515 and 560, or they might find the
difference between 315 and 400 and the difference between 400 and 560 (then add
them).
Sessions
3 and 4
For all of these
tasks, students should use the most efficient strategies that make sense to
them. Do not impel your student to
use a method that he/she does not understand. Students should always try to use number relationships that
they know to solve these types of problems. For example, for the first task on SS 4, students can think
about one number they know they can use to skip count to 240, write out that
sequence of skip counting, then use that solution to find two more. For #7 above, students might consider
how many would be in 10 groups of 31 and then use that as a stepping stone to
the final answer.
Session
5
For SS 6, a
multiple tower is a sequence of numbers found by skip counting from 0. The goal of this activity is for
students to work on their number sense by reasoning about relationships among
numbers. Encourage your student to
use what he/she knows to find out what he/she does not know. If your student is struggling, have
him/her start with a small, manageable piece of the problem and use that as a
stepping stone to work toward the solution. Students will not benefit from using methods that are beyond
their understanding.
Sessions
6 and 7
For all of these
tasks, students should use the most efficient strategies that make sense to
them. Do not impel your student to
use a method that he/she does not understand. Students should always try to use number relationships that
they know to solve these types of problems. For example, in finding the difference between 1043 and 876,
students might find the difference between 876 and 900 and then between 900 and
1043 (then add them). Or, they
might find the difference between 1076 and 876 and then take off the difference
between 876 and 843. It is
important to learn to choose a method that is efficient for that particular
problem.
Session
8
For all of these
tasks, students should use the most efficient strategies that make sense to
them. Do not impel your student to
use a method that he/she does not understand. Students should always try to use number relationships that
they know to solve these types of problems. For example, in finding the difference between 784 and 1021,
students might find the difference between 784 and 800 and then between 800 and
1021 (then add them). Or, they
might find the difference between 784 and 1084 and then take off the difference
between 1084 and 1021. It is
important to learn to choose a method that is efficient for that particular
problem.
Investigation
2
Sessions
1 and 2
For all of these
tasks, students should use the most efficient strategies that make sense to
them. Do not impel your student to
use a method that he/she does not understand. Students should always try to use number relationships that
they know to solve these types of problems. For example, to find out how many 18s are in 400, a student
might think about how many 18s are in 180 (or another larger number) and use
that as a stepping stone to find the solution.
Session
3
For all of these
tasks, students should use the most efficient strategies that make sense to
them. Do not impel your student to
use a method that he/she does not understand. Students should always try to use number relationships that
they know to solve these types of problems (this builds number sense and
computational flexibility). For
example, to find out how many 43s are in 800, a student might think about how
much ten 43s would be, then how much five 43s would be, and use this
information as a stepping stone to find the solution.
Session
4
For all of these
tasks, students should use the most efficient strategies that make sense to
them. Do not impel your student to
use a method that he/she does not understand. Students should always try to use number relationships that
they know to solve these types of problems (this builds number sense and computational
flexibility). For example, to find
out how many 13s are in 168, a student might think about how many 13s would be
in 130 and use this information as a stepping stone to find the solution. If your student wishes to use the
traditional procedure, encourage him/her to also use a second strategy.
Sessions
5 and 6
For SS 17, a
multiple tower is a sequence of numbers found by skip counting from 0. The goal of this activity is for
students to work on their number sense by reasoning about relationships among
numbers. If your student is
struggling, have him/her start with a small, manageable piece of the problem
and use that as a stepping stone to work toward the solution. For SS 18, students should use the most
efficient strategies that make sense to them and that fit the problem. Do not impel your student to use a
method that he/she does not understand.
Students should always try to use number relationships that they know to
solve these types of problems. For
example, to solve 45 x 123, students might use ten groups of 123 as a step
toward finding out how many are in 45 groups of 123.
Session
7
For these tasks,
students should use the most efficient strategies that make sense to them and
that fit the problem. Do not impel
your student to use a method that he/she does not understand. Students should always try to use
number relationships that they know to solve these types of problems. For example, to solve 17 x 241,
students might use ten groups of 241 (and then perhaps five groups) as a step
toward finding out how many are in 17 groups of 241.
Investigation
3
Sessions
1, 2, and 3
Sets of cluster problems are designed to help students
build fluency with the distributive property. For instance, for the problem 16 x 37, problems in the
cluster might be 10 x 37, 5 x 37, 2 x 37, 16 x 20, 16 x 10, etc. There is no right or wrong set of
problems for the cluster; the goal is to help students find easier problems
that will help them solve the main problem at hand. In other words, students are learning to decompose (break
apart) the numbers in the problem in order to make the problem easier to solve,
whether mentally (more challenging) or on paper. Do not impel your student to use a method that he/she does
not understand; start with small pieces of the problem if your child struggles.
For #6, if
students are not sure, have them refer to class notes or (if the notes are not
helpful) leave this question blank.
For #7 and 8, students need to think about the order of operations,
which dictates that we solve problems insides parentheses first, then do any
further multiplying and/or dividing, then do any further adding and/or
subtracting.
Sessions
4, 5, and 6
Sets of cluster problems are designed to help students
build fluency with the distributive property. For instance, for the problem 263 ¸ 17, problems in the cluster might be 10 x
17, 5 x 17, 2 x 17, 170 ¸
17, 85 ¸ 17, etc. There is no right or wrong set of problems for the cluster;
the goal is to help students find easier problems that will help them solve the
main problem at hand. In other
words, students are learning to decompose (break apart) the numbers in the
problem in order to make the problem easier to solve, whether mentally (more
challenging) or on paper. Do not
impel your student to use a method that he/she does not understand; start with
small pieces of the problem if your child struggles. With division, very often a strategy that helps is to have
students take out groups
from the larger number. With the task above, this might mean taking 10 groups of
17 (or 170) from 263, finding that 93 remain, taking 5 groups of 17 (85) from
93, and seeing that 8 remain. So
the answer would be 15 (10 and 5 groups) with 8 left over.
Sessions
7, 8, and 9
The essential goal
of these sessions is to help students become efficient, flexible, and accurate
in multiplication and division. It
is important for students to be able to choose efficient methods for use in
different problems. Do not impel
your student to use a method that he/she does not understand. Students should always try to use
number relationships that they know to solve these types of problems. For multiplication, a good strategy is
often to take smaller groups of the numbers being multiplied and then add them
together. For division, a good
strategy is often to take smaller groups of the number by which we are dividing
away from the larger number (e.g., 263 ¸ 17: take 10 groups of 17 (or 170) from 263; find that 93
remain; take 5 groups of 17 (85) from 93; and see that 8 remain; so the answer
would be 15 (10 and 5 groups) with 8 left over).
Session
10
Do not impel your
student to use a method that he/she does not understand. Students should always try to use
number relationships that they know to solve these types of problems. For multiplication, a good strategy is
often to take smaller groups of the numbers being multiplied and then add them
together. For division, a good
strategy is often to take smaller groups of the number by which we are dividing
away from the larger number (e.g., 263 ¸ 17: take 10 groups of 17 (or 170) from 263; find that 93
remain; take 5 groups of 17 (85) from 93; and see that 8 remain; so the answer
would be 15 (10 and 5 groups) with 8 left over).
Investigation
4
Session
1
For SS 32, encourage
your student to look for patterns that allow him/her to do most of the work
mentally. For #3, a good strategy
is often to take smaller groups of the numbers being multiplied and then add
them together (e.g. 10 groups of 105, 2 groups of 105, etc.).
Session
2
For SS 33,
encourage your child to use relationships among the numbers involved to answer
the questions. For #2, a good
strategy is often to take smaller groups of the number by which we are dividing
away from the larger number (e.g., 263 ¸ 17: take 10 groups of 17 (or 170) from 263; find that 93
remain; take 5 groups of 17 (85) from 93; and see that 8 remain; so the answer
would be 15 (10 and 5 groups) with 8 left over). Some students will be able to do this more efficiently than
others at first.
Investigation
5
Sessions
1 and 2
For the game, note
that there are possible variations on the rule sheet, and also note that you
should decide before starting how remainders will be handled (see the bottom of
SS 34). For #3-4, encourage students
to use their knowledge of number relationships in efficient methods that are
appropriate for these problems.
Session
3
Encourage students
to use their knowledge of number relationships in efficient methods that are
appropriate for these problems.
Sessions
4, 5, and 6
For all of these
tasks, encourage students to use their knowledge of number relationships in
efficient methods that are appropriate for these problems. It is important for students to be able
to choose methods that are most efficient for certain problems.
Session
7
For these tasks,
encourage students to use their knowledge of number relationships in efficient
methods that are appropriate for these problems. It is important for students to be able to choose methods
that are most efficient for certain problems.
Additional
tasks for this unit
For all of these
tasks, if students are not sure about the answers, have them refer to class
notes or (if the notes are not helpful) leave questions blank as needed. For #2, students need to think about
the order of operations, which dictates that we solve problems insides
parentheses first, then do any further multiplying and/or dividing, then do any
further adding and/or subtracting.
For #3, students should pay attention to the different operations used
in the two problems.
Picturing Polygons *Note:
it will be helpful for students to have graph paper for this unit.
Download the two-page parent reference sheet and
letter for this unit
Investigation
1
Session
1
For SS 2 and #3, a
polygon is a closed 2-D figure with straight sides (that do not overlap). When students are creating their own
picture on SS 2, they can think of something in real life (e.g. a person) and try
to draw this by only using polygons, not by using any curves.
Session
2
For SS 3, the
picture does not have to be very complex; it can have just one polygon, or it
can have more than one. See above
for the definition of a polygon.
#2-4 are to help students brainstorm what they know about these figures
and terms already, so students should just write/draw as much as they can for
each question. If they wish, they
can use a reference book such as a dictionary for help.
Session
3
SS 4 basically is
a connect-the-dots activity. To
label/identify a point, a student should count the number of units right or
left from the origin (the center), then count the number of units up or
down. An example would be (3, -5)
for 3 right, 5 down. For #2, see
Session 1 above for the definition of a polygon. The x- and y-axes are the horizontal and vertical lines that
divide a graph into four sections (quadrants).
Session
4
For SS 5, if
students cannot remember the names of these figures, they should fill in as
many as possible. Notes from class
or other reference materials (dictionaries, internet) may be helpful. For #2, the x- and y-axes are the
horizontal and vertical lines that divide a graph into four sections
(quadrants). To label/identify a
point, a student should count the number of units right or left from the origin
(the center), then count the number of units up or down. An example would be (3, -5) for 3
right, 5 down.
Investigation
2
Sessions
1, 2, and 3
(Session
1)
For SS 6, students
may want to refer to the definition of polygon (see Inv. 1, Session 1 note
above). For #2-4, notes from class
or other reference materials (dictionaries, internet) may be helpful. The types of triangles are
distinguished by the relationships among their sides and/or angles.
(Session
2)
For SS 7, students
will have to consider very carefully what they know about all rectangles and
all squares. This is to assess
their understanding of the class activity. For #2, notes from class or other reference materials
(dictionaries, internet) may be helpful.
For #3 and 4, students will need to think about what makes one shape
different from the other, and again, class notes or other reference sources may
be useful.
(Session
3)
For the game, play
as directed on SS 8. Be sure
students write the two statements required. A non-geometry example would be: Some (but not all)
apples are red. For #2-3, students will first need to
think about the properties of the shapes mentioned – how is a rhombus
defined? How is an equilateral
triangle defined? Then, they
will need to consider whether the statement could be true as written.
Sessions
4 and 5
(Session
4) – It would be helpful
to have pictures of the Power Polygons to complete this.
Students will have
begun these sheets in class. Here
is an explanation for each column on these sheets: Name of shape – Tell what type of triangle (SS 9)
or quadrilateral (SS 10) has the sides and angles given (to the left) IF there
is such a figure. If not, students
can write impossible. Set xy points – Students are to try to create the
described figure on a grid (to create the grid itself, they should just draw
the x and y axes, which are the horizontal and vertical lines that are the
basis for the grid, on the graph paper; they do not have to draw all of the
other lines since they are there).
Then, they list the ordered pairs on the grid that are the vertices
(corners) of the shape that they create (if it is possible). When locating a point on a grid, the
first number represents the move left or right, and the second number
represents the move up or down.
For the triangles, students should list 3 ordered pairs, and for the
quadrilaterals, 4 ordered pairs. Sketch
of Power Polygons –
Students should sketch how they could use Power Polygons to create this figure
(if it is possible). Power
Polygons are the plastic shapes that students have been using in class.
(Session
5)
For SS 11, a
parallelogram is a four-sided figure with two sets of parallel (and opposite)
sides. Vertices are the corners of
the figure. For all of these
tasks, to label/identify a point, a student should count the number of units
right or left from the origin (the center), then count the number of units up
or down. An example would be (3,
-5) for 3 right, 5 down.
Sessions
6 and 7
For SS 12,
students will need to use geometric reasoning to answer these questions. Help them to think about the
descriptions of the shapes at the top of SS 12 and use logic to decide how to
answer each question. Class notes
may also be helpful. For SS 13,
students will need to consider their work with the Geo-Logo software. The commands listed help students to
draw the shapes shown. Hint: rt indicates an angle (or a turn to the
right, e.g. rt 90 is a 90-degree angle), and fd indicates a move forward, which would give
the length of a side (e.g. fd 20 means a length of 20 pixels on the computer
screen). So students can look at
the sides and angles on the figures shown and compare them to each list of
commands to determine which set of commands matches each shape. For #3-6, students will need to
consider what they know about the given shapes; class notes or other reference
materials may be helpful. For
example, in #5, students will need to think about what all parallelograms have
in common, and they will need to draw an example that is not a rectangle (but
is still a parallelogram).
Session
8
SS 15 and #2 can
be done with a protractor if the student has not brought the Turtle Turner
home. If your child does have the
Turtle Turner, it essentially works like a protractor but is designed to
replicate the experience the students have with the software they are
using. To measure an angle,
students place the Turtle Turner on the angle so that the double-line with the
arrow is on one of the rays of the angle, with the vertex (point) of the angle
underneath the Turtle. Then they
can read the measure (estimating in between lines if needed). Be sure they check the reasonableness
of their answers, as students often tend to read the wrong measurements on
angle-measuring tools. Narrow
(acute) angles should be less than 90 degrees; wide (obtuse) angles should be
greater than 90 degrees. In #2,
the vertex (point) of an angle is labeled just with the letter that identifies
the point, and the rays (the lines that border the angle) are identified with
two letters, for instance, BC, with the first being the vertex and the second
being a point on the ray. Also, we
generally draw a small arrow over these two letters with the arrow pointing
right (away from the name of the vertex), like this: --> .
Session
9
For SS 17, be sure
that students answer each question fully.
For part 4, help students to think about the center of a circle and how
many of these angles it would take to completely surround the center, with all
the vertices (points) of the angles at the center. Then, help them to think about a
straight line segment with a point in it and how many of these angles it would
take to fill in the space from one side of the segment to the other, with the
vertices of the angles at the point.
For #2-3, students will need their Turtle Turners or protractors. Be sure students are reading the angle
measures correctly (it is good to compare them to a 90-degree angle, i.e., a
square corner). The interior of an
angle is the space between the rays, and the exterior is the space outside the
two rays (assuming they could extend forever).
Investigation
3
Sessions
1 and 2
For SS 18,
students can use their work from class or the reference sheet of regular polygons
(part of SS 18) to complete the chart.
They will need their Turtle Turners or protractors to measure any angles
that they did not measure in class.
When looking for patterns, one particular one they should identify
should be the one in the last column.
#2-3 can be answered using the information in the table.
Session
3
For SS 20, if
students did not circle to which group they were assigned, have them work with
group C. The directions are fairly
clear; be sure students complete each part. Remember that a non-regular polygon does not have sides that
are all equal or angles that are all equal.
Session
4
For SS 22, if
students are having trouble, have them start with an example square with side
lengths labeled (for each part), then draw the new figure that is
described. This will help them see
the relationships. For #2-3,
students will probably need to draw pictures to start, based on the information
in the problem. Then, they can
change them or create new pictures based on the changes described in the
tasks. For #2, keep in mind that
the side lengths of the first square can be found by considering that the
length and width must be the same, and they multiply to give 64 square meters.
Sessions
5 and 6
For SS 25, similar in mathematics means that the sides of the
figures are proportional to each other.
For example, a 2 x 6 rectangle would be similar to a 5 x 15 rectangle
because the sides of the second one are 2.5 times longer than the sides of the
first. (Also, there is a 1 to 3
ratio in each pair of sides.) A 2
x 6 rectangle would not be similar to a 3 x 7 because 2 times 1.5 is 3, but 6
times 1.5 is not 7. For #2-3, use
this explanation of similar to solve these problems – pictures will
definitely help. For #4, students
should find the area of each table in #3 and then describe how the areas are
related (one is how many times larger than the other?).
Additional
tasks for this unit
#1: The vertex
(point) of an angle is labeled just with the letter that identifies the point,
and the rays (the lines that border the angle) are identified with two letters,
for instance, BC, with the first being the vertex and the second being a point
on the ray. Also, we generally
draw a small arrow over these two letters with the arrow pointing right (away
from the name of the vertex), like this: --> . The interior of an angle is the space between the rays, and
the exterior is the space outside the two rays (assuming they could extend
forever).
#2-3: Students may
need to use class notes or other reference sources if they do not remember what
congruence is. Drawing pictures
may also help.
#4: Students may
need to use class notes or other reference sources if they do not remember the
vocabulary.
#5: It may help
for students to actually look at two clocks or draw pictures of what a small
clock and a large clock would look like.
#6: Students
should think about what the degree measure of an angle tells us.
Name That Portion Download the three-page parent reference sheet and
letter for this unit
Investigation
1
Session
1
1: Even if you
cannot find examples to cut out, you can list examples. 2: Are the 26 students part of the
class or the whole? 3: In a
fraction, the numerator (top number) is the part, and the denominator (bottom
number) is the whole. 4: Think
about 68 cents, and think about 0.5 as a part of a dollar (be careful!). 5: What is a whole number that is close
to 22 and of which you can easily find 1/3?
Session
2
1: An example
might be: There are 5 people in my house, and 3 wear glasses (3/5). 3 out of 5 people wear glasses. 2: Is the main group of people a part or the whole? 3: An example might be: The ratio of
people who wear glasses to those who do not is 3 to 2. 4: A percent is a fraction out of _____. 5: What is 1/5 of 100? 6: Think of 2/5 as 2 out of 5 –
is this closer to 1/2 or to nothing?
Sessions
3 and 4
1: If you colored
in 30 squares, the fraction could be 30/100 or 3/10, which is the same as
30%. 2: Could you use a 10 by 10
grid? 3: If you think of 35% as 35
out of 100, is 100 the whole or a part?
4: What percent is equal to 1/2?
What percent is equal to 1 whole?
To which is 80% closer? 5:
If you are not sure how to start (even after looking at your grids), write the
given percent as a fraction out of 100, and find other fractions that are equal
to this. 6: Write 60% as a
fraction, and then find at least one fraction that is equal to it. Use a 10 by 10 grid to help you decide
what decimal 60% equals (or look back at your fractions). 7: Think of 3/8 as 3 out of 8
items. Is this close to 0, 1/2, or
1 whole?
Sessions
5 and 6
1: Use your
percent strips to help if you are not sure which number is larger, but also try
to think about these without the strips since that should be your goal. Remember that a good strategy can be to
draw a picture of each fraction, but the wholes must be the same size. 2: These explanations should focus on
ways to place cards without using the percent strips – how did you think
about the numbers? 3: Think of 3/5
as 3 out of 5 items. Is this close
to 0, 1/2, or 1 whole? 4: Drawing
pictures is one way to solve this problem, but if you use this strategy, your
wholes must all be the same size.
You can also consider what percents these fractions equal. 5: Use your percent strips to help if
you are not sure which number is larger, but also try to think about these
without the strips since that should be your goal. Remember that a good strategy can be to draw a picture of
each fraction, but the wholes must be the same size. 6: These explanations should focus on ways to place cards
without using the percent strips – how did you think about the numbers?
7: Drawing pictures is one way to solve this problem, but if you use this
strategy, your wholes must all be the same size. You can also consider what percents these fractions equal.
Session
7
1: See the hint in
your workbook, and also think about how much is left out of 1 whole with each
fraction. It might help to draw a
picture, but this cannot count for one of your statements. 2: Think about different sized sets of
items that can be divided in half (the sets, that is). In each set, what fraction of the set
would be close to half but not quite equal? 3: Drawing pictures is one way to solve this problem, but if
you use this strategy, your wholes must all be the same size. You can also consider what percents these
fractions equal. 4: You can think
about 1/9 and what percent it would approximately equal; then multiply this by
5. Or, you can consider how close
to 1/2 this fraction is and estimate the percent based on this.
Investigation
2
Sessions
1 and 2
1-9: The clock
model is based on a regular face clock (non-digital). We think of the 12 on the clock as 0, or the starting point
as we move clockwise around it.
The 12 steps around the clock allow us to break the clock into halves,
thirds, fourths, sixths, and twelfths, and if we use the minutes, we can break
the clock down even further. For
instance, 1/6 of the way around the clock would be 2 steps because 2 is 1/6 of
12. 1/2 of the way around would be
6 steps. For #6-9 in the workbook,
think of the fractions in terms of hours – what is 1/3 of an hour? What is 1/4 of an hour? How do we show these on the clock?
Session
3
1-3: To use the
fraction strips, you will need to be sure that you have accurate strips and
that you line them up very carefully when solving problems. For instance, for #2 in the workbook,
you will want to line up the end of 1/2 with the end of 1/3 and see how much is
left out of the 1/2 (that the 1/3 does not overlap). You should be able to find this length on one of the other
strips. This strategy is difficult
to use on many problems, but it is a good way to model some basic fraction
computation and to show that answers make sense. 4-5: You might choose to use pictures or equivalent
fractions. The problem with using
decimals or percents in some of these problems is that you may not get an exact
answer.
Sessions
4 and 5
1: Use your
percent strips to help if you are not sure which number is larger, but also try
to think about these without the strips since that should be your goal. Remember that a good strategy can be to
draw a picture of each fraction, but the wholes must be the same size. 2: A mixed number includes a whole
number and a fractional part. 8/8
is 1 whole, so how would you write 9/8?
3: How many thirds are in 1 1/3?
Use this to decide what the numerator of your fraction should be. 4: You could use a picture, or you
could think about how much more than 1 whole each number includes. 5: You might want to draw a picture of
2/3 in a rectangle and then divide it further into more equal pieces to find
equivalent fractions. 6: To
compare fractions, you can draw pictures (but both wholes must be the same
size); you can think of the fractions as decimals or percents; you can compare
each of the fractions to 1/2 or 1 whole and see if this helps to decide which
is larger; etc. 7: Think of
fractions that are equal to 1 whole, and then list fractions that are very
close to these. Remember to list
one that is greater than 1 whole.
8: Which is closer to 1 whole?
Draw a picture if it helps.
Session
6
1: This game is
easier to explain with the board actually in front of you, but see the hint in
the workbook and the directions sheet.
One very good move might be this: I drew the fraction 4/5. I know that this equals 8/10, and 8/10
equals 5/10 + 3/10. 5/10 is 1/2,
so I moved forward 1/2 on the halves track and forward 3/10 on the 10ths
track. Of course, you could also
just move 4/5 forward on the 4/5 track, but this might not be possible
depending on where the markers are, and moving two markers is usually better strategy
than moving just one. 2-4: You
want to break each fraction into at least two others; see the example above for
4/5.
Sessions
7 and 8
1: Since you have
to look for decimals, you can always use money if you cannot find other
examples. To write each decimal as
a fraction, you will need to read the decimal out loud (using place value
language) and use a denominator of 10, 100, 1000, or perhaps a larger number
depending on the decimal you find.
Then, you can write equivalent fractions and the equivalent percent
(which will be the equivalent fraction out of 100). 2: Remember that you can multiply or divide the numerator
and denominator by the same number to get equivalent fractions, and you can
prove that they are equal with pictures.
3: Use the simplest fraction that is equal to 18/24, and write the
familiar percent that you know for this fraction. 4: This percent can be written as a fraction out of 100
(then simplified). 5-6: You can
draw pictures or convert these fractions to decimals or percents in order to
check. 7-8: You want to break each
fraction into at least two others; see the example in Session 6 above for
4/5. 9-11: You might use a problem
that involves cooking, measuring, or foods like pizza, cake, or candy bars. You can draw pictures to solve, or use
equivalent fractions if you are sure you know how to do this.
Session
9
1: This game is
easier to explain with the board actually in front of you, but see the hint in
the workbook and the directions sheet.
One very good move might be this: I drew the fraction 4/5. I know that this equals 8/10, and 8/10
equals 5/10 + 3/10. 5/10 is 1/2,
so I moved forward 1/2 on the halves track and forward 3/10 on the 10ths
track. Of course, you could also
just move 4/5 forward on the 4/5 track, but this might not be possible
depending on where the markers are, and moving two markers is usually better
strategy than moving just one.
2-4: You can draw pictures to solve, or use equivalent fractions if you
are sure you know how to do this.
If you use pictures, be sure that the wholes that you draw are the same
size and that you divide each whole into equal pieces.
Investigation
3
Session
1
1-6: Remember that
in the decimal 0.48, 4 represents 4 tenths and 8 represents 8 hundredths (or,
48 represents 48 hundredths). In
this case, we could write the fraction 48/100 because this is exactly 48
hundredths. Remember that a
percent is a fraction out of 100, and if the fraction is out of 10, it is easy
to get the percent by finding an equivalent fraction that is out of 100.
Session
2
1-2: Remember that
if you draw a decimal like 0.6, this means 6/10 of the whole grid, and 1/10 is
a row or column. A decimal like
0.32 means 32/100 of the whole grid, and since there are 100 squares, this
should be easy. 3: Use the grid to
help; shade 1/10 or 10/100 of it, and think about how the other fractions are
equal to this amount. 4: Read 0.4
and 0.04 in place value language if you are not sure – are these the same
amount? 5: How would each of these
look on the grid? 6: Think of how
0.72 would look on the grid – what number of tenths is closest to this?
Sessions
3 and 4
1: Remember that
if you draw a decimal like 0.6, this means 6/10 of the whole grid, and 1/10 is
a row or column. A decimal like
0.32 means 32/100 of the whole grid, and since there are 100 squares, this
should be easy. 2-6: Remember that
in the decimal 0.482, 4 represents 4 tenths; 8 represents 8 hundredths, and 2
represents 2 thousandths (or, 482 represents 482 thousandths). In this case, we could write the
fraction 482/1000 because this is exactly 48 thousandths. Remember that a percent is a fraction
out of 100, and if the fraction is out of 10, it is easy to get the percent by
finding an equivalent fraction that is out of 100. 7: See the previous note for #2-6, and keep in mind that
tenths are larger than hundredths, which are smaller than thousandths, so we
must compare the tenths first, followed by the hundredths and thousandths. 8-11: Picture these on the grid if it
helps, and see the previous notes.
Sessions
5 and 6
1-2: It may help
to think of these in terms of money, or picture them on the grid. Also remember that you can include more
digits to the right in your answer if needed. 3-4: Picture these on the grids, and remember which column
represents which place value (see the note above). 5: You may find it easier to write this as a percent first
and then as a fraction, or perhaps the other way around. Reading the decimal with place value
language may help. 6: If you know
the fraction, you can find an equivalent fraction out of 100, which can help
you to write the percent and decimal.
If you know the decimal, you can write it as a fraction out of 10, 100
(which will be the percent), or 1000.
If you know the percent, you can write it as a fraction out of 100 and
then write this as a decimal (use what you know about hundredths). 7: Use the table on SS 21. 8: Read this fraction out loud how many wholes are in 13/4? How many parts left over? How can you write the part left over as
a decimal? 9: Use any of the
reference sheets you have to help.
10-11: Picture these on the grid if it helps, and see the previous notes
about place value (to which you must pay attention!).
Session
7
1: For your
numbers, you could use decimals or percents and/or think about how close to 1
whole each of these is. You can
use your references from earlier lessons to help. 2: It may help to change 0.6 to a percent (see previous
notes). 3: If you had to split
each of these between 3 people as equally as possible, how would you do
it? 4: Pay attention to what place
value each digit represents, and do not be fooled. You must compare the same place values in each decimal. 5: Picture these on the grid if it
helps, and see the previous notes about place value.
Session
8
1: For your
numbers, you could use decimals or percents and/or think about how close to 1
whole each of these is. You can
use your references from earlier lessons to help. 2: Picture these on the grid if it helps, and see the
previous notes about place value (to which you must pay attention!). 3: Which place represents
hundredths? Is this closer to that
number of hundredths or the next higher one? 4: Since this is greater than 1 whole, it will be more than
100% and a fraction greater than 1.
Hint: how many tenths are in 1.4, total?
Investigation 4
will be posted later in the year since we will not do it until later.
Between Never and Always Download the two-page parent reference sheet and
letter for this unit
Investigation
1
Sessions
1 and 2
1: For the
likelihood line, you can write the events you think of below the line, where
there is room on the page. For
questions 1 and 2, it is asking you what goes through your mind when something
that is unlikely happens, and what goes through your mind when something likely
happens. 2-3: A standard number
cube has the numbers 1-6 on its sides.
Outcomes are the things that could happen. The probability in #2 will be a fraction: how many outcomes
out of the total are 4ıs? Write
this as a fraction. 4: An
impossible event has no chance of happening (what number represents
this?). A certain event will
definitely happen; for example, when you flip a coin, the probability of
getting heads OR tails is 2/2, which equals what? 5: Think about how many outcomes are possible in each case
and how many of the total outcomes in each case would be the event listed in
the question. You may need to
write fractions to help compare.
6: An adult at school can also do this if no one is available at
home. Be sure to give real-life
examples of probability. 7: This
is asking for a fraction or percent that would represent a very low
probability. 8: Write a lower
probability, and show a way to compare it to the one you wrote in #7. 9: How many Bs are there in the word,
and how many total letters? Write
a fraction. Notice that the
denominator is very close to a number which would make the fraction easy to
convert to a percent. Give this
percent.
Sessions
3 and 4
1: 70 percent is
70 out of ___ (write this as a fraction).
2: What is left out of 1 whole when you take out 1/4? 3: Think about the chance of getting
tails when you flip it once. What
would this mean for your results if you flipped the coin 40 times? 4: About what percent would 31/40
be? Is this close to what you
would expect to get? 5: Think
about the way that the boxes might be sitting on the shelf or the way they
might have been delivered from the factory. 6: Write the probabilities as fractions; the number of times
you will get what you want out of the total number of possibilities. Use what you know about the probability
to make the predictions. 7: Recall
that you had a spinner that had 4 equal sections, 3 of which were one color and
1 unshaded. An outcome is what
could happen. Write the
probabilities as fractions. 8: The
shaded part was 3/4 of the spinner; use this to make your prediction –
one way to think about it is how many times out of 50 you think you would get
the fourth that is not shaded. 9:
Theoretical probability is what we expect to happen; experimental probability
is what does happen. Both are
written as fractions. 10: Think
about how many times you would expect to get 2 and each of the other numbers if
you rolled the cube 6 or 12 times.
Session
5
1: To create a
line plot, draw a number line that includes at least all of the values in the
set of data. Then, above the line,
draw an X above each number each time it appears in the set, so that you create
columns of Xs above the numbers on the line. 2: The range of the data is the difference between the
largest value and the smallest value.
The mode of the data is the value(s) that occur(s) most often. The median of the data is the middle
value when all values are in order from least to greatest (repeats
included). If there is an even
number of values, the median is the number between the two middle values. 3: Count how many values are above and
below the median, and write each of these numbers in a fraction out of the total
number of values. 4: Think about
the experiment: would this be likely to happen? Why or why not?
5: Consider how many reds most people got and how many marbles are in
the bag.
Session
6
1: In a fair coin,
does it matter how many heads you have gotten before? 2: A fair number cube would have a 1/6 chance of getting
each number from 1-6. About how
many out of 70 do you think would be 2s?
3: Look at two coins, and think about all of the combinations of heads
and tails you could get if you tossed them at the same time. 4: The mean of the data is what we
typically know as the average. It
is found by adding all of the values (repeats included) and dividing by the
total number of values. 5: What do
you notice about all of the numbers that she rolled at first? How does this change slightly in the
next set of rolls?
Session
7
1: Refer to the
scoring options and spinners on SS 6 or 7 for examples. The probabilities will be fractions
showing the total number of times you can score each option out of the total. You will play the game at school with
someone. 2: Factors are the whole
numbers that divide evenly into a number.
The factors of 12 are 1, 2, 3, 4, 6, 12. 3: Multiples represent groups of a number. The first four multiples of 12 are 12,
24, 36, 48. 4-10: This is
referring to SS 6, page 1.
4: Outcomes are what you can get – what can happen. 5: How many parts of the spinner are
there, and how many are labeled with each number? Write a fraction for each outcome (number on the
spinner). 6: How many out of the
total have a 5? How many have a
25? 7: Out of 8, count the total
number of outcomes that are two-digit numbers or multiples of 4 (could be
both). 8: Out of 8, count the
total number of outcomes that are two-digit numbers AND multiples of 4 (at the
same time). 9: Prime numbers have
exactly two factors (see above for definition of a factor): themselves and
1. How many numbers do not fit
this description out of the 8 that are there?
Though
Investigation 2 is in the workbook, we will not be using it during 2006-2007.
Containers and Cubes Download the two-page parent reference sheet and
letter for this unit
Investigation
1
Sessions
1 and 2
Complete
SS 3 as directed. For #1, you will
need graph paper, and you will need to create a box that this package would
fill exactly. You will need to use
your reasoning skills to decide how to do this without having cubes to work
with. For #2, you do not have to
create a box, only a prediction.
Assume that where you see no cubes, there are no cubes all the way down
to the bottom.
*Parents: For SS
3, PLEASE DO NOT give your students the formula for volume of a box. The major goal of this unit is for
students to develop and understand this formula, because most students who are
simply given the formula really do not understand why it makes sense. Students should use spatial reasoning
skills to determine how many cubes they believe are there; this will be easier
for some students than others at this point in the unit because many students
do not yet fully grasp the idea of rows, columns, and layers. The graph paper box should exactly fit
the package in the first task.
Assume that there are no gaps inside the packages.
Sessions
3 and 4
Complete
SS 5 as directed. For the second
part, you can draw either a 3-D box or a net of this box. For the fourth part, the question is
asking if you think there are other boxes that will work and why or why not.
*Parents: For
this, you may need to know that a net is a picture of what the box would look like if it were cut
on enough of the edges to be laid flat.
By these sessions, students are starting to develop the formula for volume
of a box, but they still need to be very aware of where that formula comes from
and how it relates to the structure of the box. This is why a drawing is required – only relying on
numeric explanations removes geometric and measurement reasoning from the
activity.
Investigation
2
Sessions
1 and 2
Complete
SS 7 as directed. You will need
3/4-inch graph paper.
*Parents:
Encourage students to use logical thinking skills to make their
predictions. There are many valid
ways to predict these answers. The
graph paper box should exactly fit the package in the first task.
Sessions
3 and 4
Complete
SS 9 as directed. You will want to
have your completed SS 6 for reference.
You should show how you know that your solutions will work with pictures
and words.
*Parents: The box
in the middle of the page is box 1.
See the example below the box for one possible answer.
Session
5
No homework was
assigned by the authors in this session since there is an assessment in
class. Teachers may choose to assign
other tasks.
Investigation
3
Sessions
1 and 2
Complete
SS 13 as directed. Remember that
you are looking for measurements that indicate how much something holds,
not how long or heavy it is. Be
sure to look for containers to take to school (in the second part).
*Parents: See the
directions on the sheet and the note above. For the first task, the word cubic in the measurement is key.
Session
3
Complete
SS 14 as directed. For the part
where you are to describe and justify your method, you will need to show how
you know that the method you used makes sense (a picture may help).
*Parents: Help
your students measure the dimensions of a room with the meter tape and then to
determine the volume using those dimensions. Be sure that students do not just write, I used a
formula, and I know it works because we learned it in class.
Session
4
No homework was
assigned by the authors in this session.
Teachers may choose to assign other tasks.
Data: Kids, Cats, and Ads Download the two-page parent
reference sheet and letter for this unit
Investigation
1
Session 1
1: For SS 2, you
will need SS 1 to have the complete rules from class. 2: The answer is either numerical or categorical –
what are you collecting? 3: What
question was asked to start this experiment? 4: Were you surprised at all? If so, why? If
not, why not? 5: Remember that the
median is the middle number when all data are in order from least to
greatest. 6: Remember that the
mode is the piece of data that occurs the most often.
Sessions 2 and
3
1: In each
statement, you will need to compare the information given for students and
adults. Is one fraction larger
than the other? If the median is
higher, what does that tell you?
For the later tasks, you may want to draw a rectangle that represents
the whole group and then fill in the parts that are mentioned in the task. Then, you can decide whether students
or adults were better overall. For
#6, you can make up the number of students and adults in your line plots. It may take some trial and error
(perhaps not), but your finished line plot should reflect the information
given. Remember that a line plot
is like a number line that has columns of Xs above the numbers (in this case,
the numbers along the line would represent seconds, and the Xs would represent
people who balanced for that number of seconds). 2: What question was asked to start this experiment? 3: A conclusion is something that
cannot be seen directly from the data – it is something that you think
based on what you see in the data.
Be sure to support your idea with details from the data. 4: To find the range, subtract the
lowest value in the set from the highest.
5: For SS 5, use SS 3 and your comparisons from class to write two to
four statements comparing the time that students and adults typically can
balance. 6: Do you think the data
would be very different? Very
similar? Why? 7: Remember that the median is the
middle number when all data are in order from least to greatest. 8: Remember that the mode is the piece
of data that occurs the most often, and the mean is the average.
Session 4
1: Remember that a
line plot is like a number line that has columns of Xs above the numbers (in
this case, the numbers along the line would represent seconds, and the Xs would
represent the number of times that each of these values occurred. 2: Remember that the mode is the piece
of data that occurs the most often; the median is the middle number when all
data are in order from least to greatest, and the mean is the average. The range is the difference between the
highest and lowest values. 3:
Which of these is most like most of the values in the set? 4: What might have been the experiment
or survey in which this data was collected?
Investigation
2
Session 1
1: If you were
assigned SS 8, be very careful as you gather this information about one or two
cats. If you were assigned SS 9,
along with your final answers, you need to record the numbers that you tried
and the difference between each of these numbers and the given target
numbers. Describe your thinking
strategies at the bottom. 2: Use
the root word category
here to help you. 3: Use the root
word number here to
help you. 4: What question was
asked to start this project? 5: Remember
that a frequency table has two columns, where one column lists the items in the
survey or experiment and one lists the number of times that each occurred.
Session 2
1: For SS 10,
along with your final answers, you need to record the numbers that you tried
and the difference between each of these numbers and the given target
numbers. Describe your thinking
strategies at the bottom. 2: After
having studied the cat data, what are two things about cats that you think are
not connected? For example, what
would probably not be
related to tail length? 3: The
first blank is either numerical or categorical. For the second blank, think about line graphs that you have
seen and what usually is represented on the horizontal (bottom) axis. 4: Remember that a frequency table has
two columns, where one column lists the items in the survey or experiment and
one lists the number of times that each occurred.
Session 3
1: A line graph
has a vertical and horizontal axis.
Typically, in this case, the day numbers would be placed along the
bottom, and a scale for inches would be created going up the left side (in
equal intervals – like 1s, halves, 2s, etc.) – look at the numbers
to choose a good interval for the scale.
Then, put a point on the graph for each day at the height shown, and
connect the points. 2: Is this
data in numbers or categories? 3:
Where might this data have come from?
4: Make a prediction based on the data and what the graph looks like;
give a height for each day.
Investigation
3
Session 1
1: To write the
fraction for each line, write the number who gave that answer out of the
total. To give a familiar
fraction, you may want to use your data strips, but find a fraction that is
close to your fraction such as 1/2, 1/4, 3/4, 1/3, 2/3, 2/5, 5/8, etc. Sometimes it helps if the numerator or
denominator of a familiar fraction is only 1 away from that of your
fraction. 2: To write the fraction
for each, write the number who gave that answer out of the total. Then, decide whether there is a number that
you can divide both the numerator and denominator by, and if so, do this until
you cannot do it anymore. 3:
Remember that a percent is the number out of 100 that is equal to your fraction
(approximately, in this case). It
might help to think about 50%, 25%, or 75%. 4: You can find a better approximation by dividing the
numerator by the denominator. 5:
Use the notes from #3 and #4 here.
Sessions 2 and
3
1: A sample is a
part of a whole group of people that was surveyed. For instance, if a survey is taken about favorite movies
last weekend, generally not all people who saw movies are surveyed; it is only
a sample. 2: See #1; also think
about what is done with the data from the sample once it is found. 3: Are all surveys good representations
of the whole group of people? Why
or why not? 4: Use the
relationship between 31 and 95 to help you (also, what is 95 close to that
could help you solve this?). 5:
Think about your own school and class to help you.
Session 4
1: See SS 16 (you will
need SS 15). A representative
sample is a sample that accurately represents the data for the whole group of
people, not just the people who were surveyed. Does the class data match the national data? 2: Are we collecting numbers
when we ask these questions? 3:
Think of the types of graphs that you know and have used this year and in other
years. 4: The whole circle has to
represent 30 people, and there are 360 degrees in a circle. So how many degrees would represent 1
person? Use this to answer the
questions. 5: A double bar graph
looks like a regular bar graph except that there are pairs of bars instead of
single ones, in this case, one for each question. Do not forget to give conclusions about the results.
Investigation
4
Session 1
1: For SS 17, you
will want to find the area of the whole page (length x width) and the area of
any rectangle that is an ad. Then,
write a fraction with the total area in ads out of the total area. You can measure in inches or
centimeters. 2: Think about the fact
that area is flat. Which of these
types of units best represents flat space? 3: Use the formula noted above. 4: What is 240/336 close to? Can you change one of these numbers slightly to make the
fraction equal to a familiar fraction?
5: You can change a fraction to a percent by dividing the numerator by
the denominator and multiplying by 100.
Session 2
1: If you are
finishing any class work at home, you will want to find the area of the whole
page (length x width) and the area of any rectangle that is an ad. Then, write a fraction with the total
area in ads out of the total area.
You can measure in inches or centimeters. 2: What common denominator is divisible by both 8 and
3? Find equivalent fractions for
1/8 and 1/3 that have this denominator; then add. 3: In this case, you can find a simpler fraction that is
equal to this one and then use it to find the fraction out of 100. You can also change a fraction to a
percent by dividing the numerator by the denominator and multiplying by
100. 4: Use your answer to #3, or
think about the fact that 95 is very close to 100. 5: How can you divide a whole to get both halves and
fifths? In other words, what
common denominator is divisible by both 2 and 5? Find equivalent fractions (perhaps using models) for these;
then add.
Session 3
1-3: For each
task, how can you divide a whole to get each of the denominators that are
given? In other words, what common
denominator is divisible by each of the denominators given? Find equivalent fractions (perhaps using
models) for these; then add or subtract.
4: It will help to start by comparing each of these to 1 whole and
1/2. Then, it may help to think
about how close to 1 whole each one is.
5: You can change a fraction to a decimal by dividing the numerator by
the denominator and to a percent by then multiplying by 100.
Investigation
5
Session 1
1: For SS 19,
think about whether you felt like the questions were fair and made sense. Did they provide the information that
you hoped they would? If not, how
might you change them? 2: When we
collect numbers, we get numerical data.
When we collect information in categories that are not numbers, this is
categorical data. 3: Remember that
a whole circle has 360 degrees in it, so you can write a fraction for each type
of pizza and then find an equivalent fraction with 360 in the denominator. 4: Your frequency table should have 2
columns, a row with titles, and three more rows, one for each type of pizza. Use tally marks to show how many people
liked each type.
Session 2
2: A line graph
has a vertical and horizontal axis.
Typically, in this case, the years would be placed along the bottom, and
a scale for the number of trees would be created going up the left side (in
equal intervals – like 1s, 5s, 2s, etc.) – look at the numbers to
choose a good interval for the scale.
Then, put a point on the graph for each year at the number shown, and
connect the points. 3: Where might
this data have been collected? 5:
Using the data so far, make a reasonable guess about how the number of trees
will change over the next three years.
Pay attention to increases and decreases.
Sessions 3, 4,
and 5
2: How do you say
the number 1.3 without using the word point?
(one and ________) Use this
to write a mixed number. For the
percent, remember that this is greater than 1 whole, so it will be greater than
100 percent. What percent is the
fraction part of the mixed number equal to? Write a fraction out of 100 that it equals in order to find
out. 3: How many do not like
fish? This is the other part; use
this and the first part to write a ratio.
4: Think of sets of objects that can be divided in half or almost
divided in half. Give examples of
fractions of these sets that are not quite half the set. 5: Can you change the numerator or
denominator very slightly to create a fraction that is close? Think about what mixed number 7/2 is
equal to in order to help. 6-7:
For each task, how can you divide a whole to get each of the denominators that
are given? In other words, what
common denominator is divisible by each of the denominators given? Find equivalent fractions (perhaps
using models) for these; then add or subtract. 8: What number close to 568 would make an easier fraction
out of 900 to consider? What
familiar fraction is this fraction equal to? 9: Which of these must you have numbers for?
Patterns of Change Download the
two-page parent reference sheet and letter for this unit
Investigation
1
Sessions 1 and
2
1 and 5: Starting at
the left side of the grid, draw columns of squares up from the bottom so that
it looks like you have a bar graph.
These columns should increase or decrease in a regular way, which, for
this unit, means by the same number each time. For the table, Step number is the number of the column in the tile
pattern (from the left). New
tiles (step size) is how
many tiles were added beyond those in the previous step (column in the
pattern). Total so far is how many tiles have been used altogether
through that step in the pattern.
For #5, to create these graphs, students will plot points (these are not
bar graphs). For instance, on a
graph of ³Step² and ³New tiles (step size),² graph each point by starting at
the bottom left corner of the graph, moving to the right to the Step number,
and moving up to the line for the ³New tiles² number. The point will go where the gridlines cross (not between
them, unless at some point one value is not a whole number). 2: For this rule, just write what
operation you perform to get from one number to the next. 3: The variable should be the missing
number. An equation is a symbolic
expression with an equal sign. 4:
How much does he earn in a week?
6: What do you have to do to each value n to get the corresponding value
t? Write an equation to show
this. 7: Use the equation you
wrote in #6. 8: What happens as
you move from column to column in the table (to the total)?
Sessions 3 and
4
1: The basic goal
is to create a pattern similar to one of those on SS 4 but not exactly the
same. It should grow in the same
way as the one you choose but probably should start in a slightly different
way. 2: What two things do you
have to do to each value n to get the corresponding value t? Write an equation to show this. 3: Use the equation you wrote in
#2. 4: What happens as you move
from column to column in the table (to the total)? 5: Coordinate graphs are the graphs where you plot points,
like on SS 3 and 4. What would you
see on the graph? 6-7: Note what
is happening from one number to the next.
8: Write what operation you use in the pattern (in symbols). 9) You will need a variable to
represent the number of posts and one for the number of nails. What do you have to do to the number of
posts to get the number of nails?
Investigation
3
Session 1
1: Think about
this carefully – perhaps consider a 5-year old and a 35-year old. 2: Coordinate graphs are the graphs
where you plot points, like on SS 3, 4, and 15 (where it says Graph). You might need to actually make the
graphs to answer this question, but first look at the distance for each person
in the same amount of time, and picture how this would look on the graph. 3: Your variables could be t and d in
each case. What do you have to do
in each case to t to get d? 4: Be
aware that the question asks about her distance in 20 seconds, not 10. 5: Use the table or your equation to
help answer this.
Session 2
1: Make up a story
about a boy and a girl, each traveling along a track below. Tell where they started, how many steps
they took in which directions, how large their steps were, and where they each
finished. Then mark these
movements on the tracks and fill in the tables. The time in seconds should go up steadily in the
tables. For the graph, see the
note above in Sessions 1-2 of Inv. 1 for help. 2: Use a two-column table with one column for f and one for
g. Choose 6 values for f, and
determine what g would be in each case; then fill them in. 3: What two things do you do, starting
with f, to get to g?
Session 3
Make up a story
about a boy and a girl, each traveling along a track below. Tell where they started, how many steps
they took in which directions, how large their steps were (the sizes should
change throughout), and where they each finished. Then mark these movements on the tracks and fill in the
tables. The time in seconds should
go up steadily in the tables. For
the graph, see the note above in Sessions 1-2 of Inv. 1 for help.
Session 4
1: What two things
do you have to do, starting with the number of months (m), to find out how much
money will be owed (c)? Write this
in an equation with the variables and math symbols. 2: Tell what you did for #1. 3: Use your equation from #1 (then check it another
way). 4: How much does the total
cost increase after 3 months have passed?
5: You might actually have to make the graph (see the notes above in
Session 1 and Sessions 1-2 of Inv. 1 for help), but also think about what the
cost would be at 0 months.
Sessions 5 and
6
1 and 2: What is the
difference between each two numbers?
3: Write in math symbols the operation you perform to get from one
number to the next. 4: Think of a
real-life situation that involves decimals that could decrease. 5 and 6: Think to yourself – what
was the step size at 0 seconds?
What could it be at 1 second?
2? (etc.) Then, think – what is the
position at 0 seconds? What could
it be at 1 second? 2? (etc.) There is not one right answer for these graphs. 7: How do the graphs look
different? Why? What are they each comparing? 8: Sit for 5 seconds – how much
did your distance from your chair increase? 9: What two things do you have to do, starting with x, to get
y in each case? Write this in an
equation with the variables and math symbols.
Session 7
1: Use two rows,
one for the day number (by 2s) and one for the height. Use the data given to complete the
table. 2: One way might take
longer than the other way. 3: What
do you have to do to the day number to get the height? Write this in an equation with d and
h. 4: Use an x and y axis (as on
SS 26). To plot a point, start at
the bottom left and move to the right for each day number, then up to the
height for that day.
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