NUMBER SENSE
8th Grade Activities
Jennifer McLachlan
Little Falls Middle School
jmclachlan@lfalls.k12.mn.us
Maureen Wilke
Heritage Middle School
wilkem@ISD197.k12.mn.us
Overview:
“ Number sense – an intuitive feel for numbers and their
relationships – develops when children solve problems for
themselves. ”
Source:
Univ. of North Carolina, School of
Education
Number Sense is not something that can be taught during a
short period of time. Because a student’s number sense
slowly grows over time, it must be part of student’s daily
experiences. The lessons we have gathered allow students to
further develop and strengthen their understanding of
numbers and operations.
We will be using these lessons throughout the school year.
The lessons are designed in such a way that you can pick one
lesson that fits with material that you are already covering
in the classroom. The lessons range from 20-minute games to
reinforce learned concepts, to weekly challenge problems
that the students will explore and discuss during the week.
Contents:
1. Multiplying and Dividing Exponents (pages 5-10)
2. Exponential Growth (pages 11-15)
3. Multiplying and Dividing Scientific Notation (pages
16-21)
4. Modeling our Solar System (pages 22-29)
5. Ordering Rational and Irrational Numbers (pages 3031)
6. Irrational Numbers can “ In-Spiral” You (pages 3238)
7. Fantastic Four (Order of Operations) (page 39)
8. Magic Number (pages 40-42)
9. Barbie Bungee (pages 43-48)
10. How High? (page 49)
2
This unit covers the following Minnesota Standards:
Number &
Operation
8.1.1.1
Classify real numbers as rational or irrational. Know that when a square root of a positive
integer is not an integer, then it is irrational. Know that the sum of a rational number and an
irrational number is irrational, and the product of a non-zero rational number and an irrational
number is irrational
8.1.1.2
Compare real numbers; locate real numbers on a number line. Identify the square root of a
positive integer as an integer, or if it is not an integer, locate it as a real number between two
consecutive positive integers.
8.1.1.3
Determine rational approximations for solutions to problems involving real numbers.
8.1.1.4
Know and apply the properties of positive and negative integer exponents to generate
equivalent numerical expressions.
8.1.1.5
Express approximations of very large and very small numbers using scientific notation;
understand how calculators display numbers in scientific notation. Multiply and divide
numbers expressed in scientific notation, express the answer in scientific notation, using the
correct number of significant digits when physical measurements are involved
Algebra
8.2.1.1
Understand that a function is a relationship between an independent variable and a dependent
variable in which the value of the independent variable determines the value of the dependent
variable. Use functional notation, such as f(x), to represent such relationships.
8.2.1.2
Use linear functions to represent relationships in which changing the input variable by some
amount leads to a change in the output variable that is a constant times that amount.
8.2.2.1
Represent linear functions with tables, verbal descriptions, symbols, equations and graphs;
translate from one representation to another.
8.2.2.2
Identify graphical properties of linear functions including slopes and intercepts. Know that the
slope equals the rate of change, and that the y-intercept is zero when the function represents a
proportional relationship.
8.2.3.2
Justify steps in generating equivalent expressions by identifying the properties used, including
the properties of algebra. Properties include the associative, commutative and distributive
laws, and the order of operations, including grouping symbols.
8.2.4.2
Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation
in terms of the other variables. Justify the steps by identifying the properties of equalities used.
8.2.4.4
Use linear inequalities to represent relationships in various contexts.
Geometry
3
9.3.1.5
Make reasonable estimates and judgments about the accuracy of values resulting from
calculations involving measurements.
4
Multiplying and Dividing Exponents
Minnesota Standard:
8.1.1.4
Know and apply the properties of positive and negative integer exponents to generate
equivalent numerical expressions.
Objective: Students will be able to multiply and divide positive and
negative exponents by using a daily warm up to practice their new
learned skill.
Materials:
Transparencies of warm up
Launch: Students will enter the room and see the transparency on the
overhead. They will get out their homework and notebook and start the
warm up in a civil manner.
Explore:
For one week students will complete one chart daily in their
notebooks.
Share:
Randomly choose students to come to the overhead to solve the chart.
Pick two of the problems and ask students how they found their answer.
Summarize:
When multiplying exponents you need to add the exponents.
When dividing exponents you need to subtract the exponents.
5
Warm up
Book 3
X
Exponents
3−2
32
34
3−2
3
32
6
Warm up
Book 3
X
Exponents
2−4
23
25
2−1
2
26
7
Warm up
Book 3
X
Exponents
a−5
a
a8
a−1
a6
a7
8
Warm up
Book 3
÷
Exponents
10 6
107
108
1011
1012
1015
9
Warm up
Book 3
÷
Exponents
d2
d3
d4
d8
d11
d 18
10
Exponential Growth
Minnesota Standard:
8.1.1.4
Know and apply the properties of positive and negative integer exponents to generate
equivalent numerical expressions.
Objective: Students will gain an intuitive understanding of basic
exponential growth patterns. Students will begin to recognize
exponential patterns in tables. Students will solve problems involving
exponential growth and express a number in exponential notation and
standard notation.
Materials:
40 pieces of construction paper.
Scissors of each group
(8) 2 transparencies for selected group of students to share with
class
Launch: The computers broke down at school today and we have to vote
for school president and vice president. We are going to have to go
back to the old way and make ballots for everyone in the school. Each
group will get a piece of paper and a scissors. Your job is to make
equal size ballots out of this piece of paper.
This idea was from Connected Mathematics “ Growing, growing, growing ”
1998.
Explore:
Put students into groups of two. Students will cut the same piece of
paper in half until they can no longer cut the pieces in half. They
will make a table keeping track of the number of cuts to the number of
pieces of paper.
Follow the attached lesson.
Share:
Have the two groups share their results. Compare with others. Are they
the same?
Use the attached lesson and start at problem 1.1 follow up. Questions
(1-6).
Summarize:
When you are multiplying by the same number multiple times, it is
better to write it in exponential form. Using a calculator for large
exponents it is faster to use the exponent key then multiplying the
same number multiple times.
11
12
13
14
15
Multiplying and Dividing Scientific Notation
Minnesota Standard:
8.1.1.5
Express approximations of very large and very small numbers using scientific notation;
understand how calculators display numbers in scientific notation. Multiply and divide
numbers expressed in scientific notation, express the answer in scientific notation, using
the correct number of significant digits when physical measurements are involved.
Objective: Students will be able to multiply and divide numbers in
scientific notation by using a daily warm up to practice their new
learned skill.
Materials:
Transparencies of warm up
Launch: Students will enter the room and see the transparency on the
overhead. They will get out their homework and notebook and start the
warm up in a civil manner.
Explore:
For one week students will complete one chart daily in their
notebooks.
Share:
Randomly choose students to come to the overhead to solve the chart.
Pick two of the problems and ask students how they found their answer.
Summarize:
When multiplying numbers in scientific notation you need to multiply
the coefficients and add the exponents.
When dividing numbers in scientific notation you need to divide the
coefficients and subtract the exponents.
16
Warm up
Book 3
Scientific Notation
X
−2 × 10−7
4 × 1012
−2 ×1012
6 ×10−20
8 ×105
17
Warm up
Book 3
X
Scientific Notation
2 ×10−3
5 ×10 8
−2 ×1012
6 ×10−20
8 ×105
18
Warm up
Book 3
X
Scientific Notation
3 ×10 9
6 ×10 7
1.5 ×10−8
2.5 ×106
7.8 ×103
19
Warm up
Book 3
÷
Scientific Notation
−2 ×107
2 ×10−6
−4 ×1012
8 ×10−10
8 ×10 3
20
Warm up
Book 3
÷
Scientific Notation
2 ×10−120
5 ×10 8
1.5 ×103
3 ×10−20
6 ×1014
21
Modeling our Solar System
Minnesota Standard:
8.1.1.5
Express approximations of very large and very small numbers using scientific notation;
understand how calculators display numbers in scientific notation. Multiply and divide
numbers expressed in scientific notation, express the answer in scientific notation, using the
correct number of significant digits when physical measurements are involved.
Objective: Students will engage in a class activity that will use
their knowledge of scientific notation to represent relative distances
between objects in our solar system.
Materials:
Find a location in the commons for students to represent the planets
Masking tape for marking distances of the planets.
40 sheets of butcher paper
Markers
Ruler
Copy master 10
Camera (optional)
This idea came from Impact Mathematics “ C ourse 3 ” 2001
Launch: Show the table of the
the sun. Imagine lining up the
Pluto on the other end so each
the sun. How would the planets
prediction:
Mercury
Pluto
Venus
Earth
Mars
average distance
planets with the
planet is at its
be spaced? Here
Jupiter
of the planets from
sun on one end and
average distance from
is one student’s
Saturn
Uranus
Neptune
Now make your own prediction. Without using your calculator sketch a
scale version of the planets lined up in a straight line from the sun.
Don’t worry about the sizes of the planets.
Explore:
Students will work together to put the planets in order based on their
distance from the sun. They will be able to look at large numbers in
scientific notation and they can compare the distances of the planets
from the sun.
See attached lesson starting at Create a Model.
Share:
On attached lesson go through questions 7-15 (skip 13) with students.
Have students write the scientific notation to standard notation.
Summarize:
When working with very small and very large numbers it is easier to
compare numbers using scientific notation instead of standard
notation.
22
23
24
25
26
27
28
29
Ordering Rational and Irrational Numbers
MINNESOTA Standards:
Compare real numbers; locate real numbers on a number line. Identify the square root of a
8.1.1.2 positive integer as an integer, or if it is not an integer, locate it as a real number between two
consecutive positive integers.
Objective:
Students will compare and order real numbers including numbers
expressed in scientific notation, in words, as powers of 10, as
radicals, or in standard form.
Materials:
Student Worksheet
Tag Board (for numbers)
Magnetic Tape / Squares (to place on the back of the tag board)
This Idea came from Britannica Math in Context
Launch:
Teacher will provide a line on the board. The number line will
only have zero marked. Provide magnetic numbers for students to place
on the number line.
Invite students to the board to order the numbers when finished
with assignments through out the week. The student exploration will
begin after students have received instruction on scientific notation,
exponents and square roots.
Explore:
Students will work with a partner to order the numbers from least
to greatest. Students should be allowed to use calculators to help
them compare numbers.
Share:
What did you notice about some of the numbers? Have students
discuss which numbers had the same value but were written differently.
Is zero larger or smaller than a decimal number?
Have students offer solutions to the worksheet.
Summarize:
Even though numbers look different, they can have the same value.
Another way to look at it is that 100 and 102 occupy the place on the
number line.
30
Each circle contains a number. The numbers are expressed in scientific notation, in
words, as powers of 10, as radicals, or in standard form.
1) Draw lines to connect the numbers that are the same.
2) In the column below, write the numbers in order from largest to smallest.
103
100
100
.0001
10,000
1
One
hundred
thousand
0
5 x 105
10 -3
Half a
Million
-4
10
Irrational Numbers can “ In-Spiral”
You
Minnesota Standard:
500,000
.1
31
8.1.1.1
8.1.1.2
8.1.1.3
Classify real numbers as rational or irrational. Know that when a square root of a positive
integer is not an integer, then it is irrational. Know that the sum of a rational number and an
irrational number is irrational, and the product of a non-zero rational number and an irrational
number is irrational.
Compare real numbers; locate real numbers on a number line. Identify the square root
of a positive integer as an integer, or if it is not an integer, locate it as a real number
between two consecutive positive integers.
Determine rational approximations for solutions to problems involving real numbers.
Objective: To investigate the difference between rational and
irrational numbers by visualizing, creating, and discussing a project
that displays rational and irrational numbers. To be able to determine
the approximations of irrational numbers. To be able to compare,
locate, and identify positive and negative square roots on a number
line.
Materials:
• White paper (one for pair of partners)
• Index card (one for pair of partners)
• Markers
• Number line -20 to 20
The idea for this lesson came from Mathematics Teaching in the Middle
School, April 2007.
Launch: Go around the room and select students and decide who will get
homework this evening and who won’t. Students who don’t get homework I
could say things such as “ I like your shoes ” , “ I like the color of
your hair, “ I like how you sit up in your desk,” etc. The students
who get homework I could say things such as “ I don’t like you tapping
your pencil ” , “ I don’t like the desk you are in ” , “ I don’t like
that you came in late ” , etc. Ask the students in the class what they
are thinking about my behavior. Write the words on the board. Could we
think of other words? If they can’t come up with irrational,
eventually say it. Discuss the difference between rational and
irrational numbers.
Explore: Students will create a right triangle and put them together
to create a wheel. Students will be solving radical numbers on the
hypotenuse to see if there is a pattern to the numbers and the wheel.
See attached lesson.
Extension:
After students complete their Wheel of Theodorus they need to find the
square roots of all the hypotenuse numbers.
On a number line from -20 to 20 put the hypotenuse numbers on the
number line.
Introduce negative square root and ask the students to put them in
order on the number line.
Share: Groups will pair share with another group to compare their
“ W heel of Theordorus ” and calculations. They will also compare their
32
rational and irrational numbers on the number line. Students may
notice that as the number under the radical sign grows so does the
value of the square root. May also notice the pattern of perfect
squares – between 1 and 4 are 3, between 4 and 9 are 5, between 9 and
16 are 7, … (consecutive odds). Students may look at odds, size of
the wheel, and evens or other patterns as well. Looking at our
“ W heel of Theordorus ” can you explain the difference between a
rational and an irrational number? When looking the hypotenuse sides,
what can you say about the square roots of those numbers?
Summarize: Display students’ work on
(pod or house). Looking at the wheel
square because it ends or repeats. It
irrational number is a number that is
You cannot write it as a fraction.
walls throughout the school area
a rational number is a perfect
can be written as a fraction. An
nonrepeating and nonterminating.
33
34
35
36
37
38
Fantastic Four
MINNESOTA Standards:
Justify steps in generating equivalent expressions by identifying the properties used,
8.2.3.2 including the properties of algebra. Properties include the associative, commutative and
distributive laws, and the order of operations, including grouping symbols.
Objective:
Students will apply properties of algebra and order of operations
to generate equivalent expressions.
Materials:
Set of Overhead Cards
Launch:
Write 2, 3, 4, 5 on the board. Ask students to spend a minute
trying to develop equations using all four numbers. The equations can
be equal to any number. Encourage them to use parenthesis, exponents
and radicals.
Invite students to the board to share equations. Then ask
students to see if they can arrange the numbers to equal 13. Ask for
possible solutions.
4 + 2 x 3 +5 = 13
Explore:
The teacher will turn over 5 cards. The sixth card will be a
target number. Teams will be given a predetermined amount of time to
work in their group to develop equations. Teams will earn points for
each equation. An equation using 2 numbers = 2 pts, 3 numbers = 3
pts, etc. Equations with exponents or radicals get 2 extra points.
Share:
After each round students will share solutions that earned more
than 3 pts. Class will discuss strategies and importance of order of
operations.
Summarize:
Discuss how many of the equations are similar, but because of how
numbers are grouped or the rules for order of operation, the solutions
are the same. The teacher should conclude you can make new equations
by using the associative and commutative properties.
39
Mystery Number
MINNESOTA Standards:
8.2.4.2
Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation
in terms of the other variables. Justify the steps by identifying the properties of equalities used.
Objective:
Students will formulate generalized patterns out of specific
cases.
Students will develop a deeper understanding that a variable
represents a
range of possible values
Students will enhance their fluency in manipulating symbolic
expressions.
Materials:
Worksheet for each student
This idea came from Mathematics Teaching in the Middle School,
March 2007
Launch:
Say to the students “ I can read your minds. . . and I can prove
it! ” Ask a volunteer to write down 3 consecutive numbers. Have the
student add up the three numbers and tell you the total.
(Teacher will divide the number by 3 to find the middle number in
the series.) Repeat this process several times. Then tell the
students that they TOO can be mind readers.
Explore:
Students will work together in teams to develop a set of steps to
explain each scenario on the worksheet. Students should pick a number
and run through the directions for each situation to make sure it
works. Then they will identify what mathematical operation is
changing their number. Finally, they will write a generalized rule or
algebraic equation for each step.
** Students may need help with the last scenario. This case may
require that they look at the arithmetic and algebraic equations
first. They should see that when the “ answer ” is given after
completing the steps, there are still a few simple calculations that
the “ mind reader ” needs to do before the solution is obtained.
7) Take away 10 from the answer
8) Divide it by 10
9) Reveal
the favorite day!!
Share:
Allow students to share strategies to each of the problems.
Develop the rules/steps used to figure out the original situation (3
consecutive integers) as a whole class.
Summarize:
Conclude that variables help link simple arithmetic to algebra by
reasoning about generalized patterns. Variables are “ p lace holders ”
for a range of possible values. Justify each of the steps by
identifying the properties of equalities were used.
40
41
42
Barbie Bungee
MINNESOTA Standards:
Understand that a function is a relationship between an independent variable and a dependent
8.2.1.1 variable in which the value of the independent variable determines the value of the dependent
variable. Use functional notation, such as f(x), to represent such relationships.
8.2.2.1
Represent linear functions with tables, verbal descriptions, symbols, equations and graphs;
translate from one representation to another.
Identify graphical properties of linear functions including slopes and intercepts. Know that the
8.2.2.2 slope equals the rate of change, and that the y-intercept is zero when the function represents a
proportional relationship.
8.2.4.4 Use linear inequalities to represent relationships in various contexts.
Objective:
Students will explore relationship between tables and linear
graphs, to explain mathematical relationships.
Students will find the line of best fit and approximate and
interpret rate of change from graphical and numerical data.
Materials:
Rubber bands (all the same size and type)
Yardsticks or measuring tapes
Masking tape
Barbie® dolls (or similar)
Barbie Bungee Activity Sheet
*This lesson was found on the National Council of Teachers of
Mathematics website.
www.nctm.org
Launch:
Watch a video segment on bungee jumping. “ Do you think the
length of the cord and size of the person matters when bungee jumping.
Would it be smart to lie about your height and weight? ” Allow
students to offer suggestions as to why an accurate estimate of height
and weight would be important to conduct a safe bungee jump.
Explore:
After a brief introduction, set up the lesson by telling students
that they will be creating a bungee jump for a Barbie® doll. Their
objective is to give Barbie the greatest thrill while still ensuring
that she is safe. This means that she should come as close as possible
to the ground without hitting the floor.
Explain that students will conduct an experiment, collect data,
and then use the data to predict the maximum number of rubber bands
that should be used to give Barbie a safe jump from a height of 400
cm. (At the end of the lesson, students should test their conjectures
by dropping Barbie from this height. If you school does not have a
location that will allow such a drop, then you may wish to adjust the
height for this prediction.)
Hand-out the Barbie Bungee packet to each student. In addition,
give each group of 3-4 students a Barbie doll, 15-20 rubber bands, a
large piece of paper, some tape, and a measuring tool. Before
students begin, demonstrate how to create the double loop that
43
attaches to Barbies feet. Also show how a slipknot can be used to add
additional rubber bands. Then allow students enough time to complete
the experiment and record the results in the data table for Question
2.
After all groups have completed the table, ask them to check
their data. They should look for numerical irregularities. If any
numbers in the table do not seem to fit, they may need to re-do the
experiment for JUST the numbers of rubber bands where the data appears
to be abnormal.
To create a graph of the data, you may wish to have the students
use the Illuminations website. You may use the Line of Best fit
Activity or allow them to enter the data in the Barbie Bungee
Spreadsheet.
At the end of the lesson, take students to a location where
Barbie can be dropped from a significant height. Possibilities include
a balcony, the top row of bleachers, or even standing on a ladder in
an area with a high ceiling. Allow students to test their conjecture
about the maximum number of centimeters that can be used for a jump of
400 centimeters.
Share:
How many rubber bands are needed for Barbie to safely jump from a
height of 400 cm?
[Answers will vary, but students should use the line of best fit and the
regression equation to determine an answer.]
What is the minimum height from which Barbie should jump if 25
rubber bands are used?
[Answers will vary, but students should use the line of best fit and the
regression equation to determine an answer.]
How do you think the type and width of the rubber band might
affect the results? Do you think age of the rubber bands would affect
the results--that is, what would happen if you used older rubber
bands?
[Rubber bands lose their elasticity with age or when exposed to extreme
temperatures. Students would probably choose not to jump from a bridge if the bungee
cord were old and brittle.]
If some weight were added to Barbie, would you need to use more or
fewer rubber bands to achieve the same results? Conjecture a
relationship between the amount of weight added and the change in the
number of rubber bands needed.
Summarize:
Slope means the rate at which the line changes and y-intercept is
the initial value. Describe slope and y-intercept as found
within the context of this problem.
Identify and define the dependent and independent variables.
Examine tables of data and discuss its relationship to the linear
equation.
44
45
46
47
48
How High?
MINNESOTA Standards:
9.3.1.5
Make reasonable estimates and judgments about the accuracy of values resulting from
calculations involving measurements.
8.2.4.2
Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation
in terms of the other variables. Justify the steps by identifying the properties of equalities used.
Objective:
Students will compare relationship among volumes of similar
objects.
Students will calculate the height of each object through
algebraic equations.
Students will develop their own conjecture about the volume of a
cone in relation to the volume of a cylinder with the same base.
Materials:
Computer Lab – National Library of Virtual Manipulatives
www.nlvm.usu.edu
Two cylinders of different height, the bases must be the same
size.
One cylinder that has a different size base.
Launch:
Partially fill one of the two similar cylinders with water. “ I f
I were to pour this water into this second cylinder, how high will the
liquid be? ” Let students ponder for a few moments. Pour the liquid
and discuss the results.
Then ask them to think about how high the liquid would be in the
cylinder that has a larger/smaller base. Give students more time to
predict the out come. Pour the water and discuss the results.
Explore:
Students will go to the computer lab and access the website for
the National Library of Virtual Manipulatives.
Under the Geometry
strand for 6-8 Grades the students will choose the activity “ How
High? ” . Students should be sure to bring paper, pencil and a
calculator to the lab.
Students will calculate the height of the
liquid before moving the slider to the appropriate height. The
teacher needs to circulate through the lab to make sure that students
are not just randomly guessing.
As students begin to understand the relationships and can
manipulate the formula for volume of a rectangular prism and a
cylinder, challenge them to find a rule (formula) for the relationship
of the cones volume to that of a cylinder with the same base.
Share:
Return to the classroom for large group discussion about the
activity. Ask for student’s rules or formula’s describing the
relationship of a cone and a cylinder with the same base.
Summarize:
49
Successfully made reasonable estimates about volume from
calculations. Conclude that the volume of a cone is one-third the
volume of a cylinder with the same base.
50