Number Sense Lessons
And Supplemental Activities
For Middle School Grades 7-8
By
Greg Gearey
Patrick McNallan
George Taus
Number Sense/Number Theory
Summer 2005
About Our Class
This lesson can be found at http://illuminations.nctm.org
Unit Overview
In the following lesson, students participate in an activity in which they focus on the
uses of numbers. The activity explores how students use numbers in school and
every day settings as a way for students and the teacher to get to know each other
at the beginning of the school year. The activity involves data collection and
interpretation, and probability and statistics. This activity also emphasizes a growthand-development theme.
Length: 2 periods
Grades: 6-8
Learning Objectives
Students will be able to:
*
*
*
collect and analyze data
determine the arithmetic mean
represent data using percent
Materials
*
One "About Our Class" activity sheet for each student
Instructional Plan
Background Information
In this activity, students collect and analyze data about a student's family and his or
her personal interests. Students determine the arithmetic mean and represent data
using percent.
Preparing the Investigation
Reproduce a copy of the activity sheet "About Our Class" for each student.
Structuring the Investigation
1.
Have the students complete survey questions a - g. Discuss the results as a
class. Ask students to think of a way to represent this information. Consider having
groups of students construct a table to display the class's data for all responses to
questions a - g.
2.
Have the students determine the mean number of people living in each
house. (Find the sum of the people in all the houses and divide this sum by the
number of houses.) Define range as the difference between the numbers of people in
the smallest and largest household, then ask for the range of the household size.
3.
Ask the class to report the total number and mean of the number of cars.
Then ask how students could determine the percent of families that have at least one
car. (Make a table showing the numbers of families with various numbers of cars,
and then determine the percent of families with at least one car by dividing the
number of families with one or more cars by the total number of families.)
4.
Have the class discuss the mean number of calculators in the families and the
percent of families that have at least one calculator in their home. Remind the
students that if they have at least one, they could have more than one.
5.
Ascertain the favorite activity for the class. Ask how this answer was
determined.
6.
Discuss the difference between the number of time the most common favorite
and the second-most-common-favorite TV shows were mentioned. The greater the
difference between the two numbers, the greater is the likelihood that the favorite
will still be the favorite in another sample.
7.
Discuss the completed graphs and questions from the class.
Extensions
Have students work in groups to create a way to organize the class's responses to
each of the survey questions. Ask the students to display their data to the class.
Standards and Expectations
Data Analysis & Probability 6-8
This lesson covers the following Data Analysis & Probability Standard Expectations:
*
use proportionality and a basic understanding of probability to make and test
conjectures about the results of experiments and simulations.
References
Dianne Bankard and Francis (Skip) Fennell.
1991. pp 26 - 33.
The Arithmetic Teacher. September,
Curcio, Frances R. Developing Graph Comprehension: Elementary and Middle
School Activities. Reston, Va.: National Council of Teachers of Mathematics, 1989.
Travel in the Solar System
This lesson can be found at http://illuminations.nctm.org
Unit Overview
In the following lesson, students participate in an activity in which they focus on the
uses of very large numbers.
Length: 2.5 periods
Grades: 6-8
Learning Objectives
Students will be able to:
• apply measurement and computation to gain insight into the large numbers
associated with distances in space
Materials
• Calculator
• Planets at a Glance data sheet
Instructional Plan
Mission
• To apply measurement and computation to gain insight into the large numbers
associated with distances in space
• To plan a trip to a planet in the solar system
This lesson focuses on human travel in space. One problem associated with traveling
in the solar system is the distance from on planet to another. But other problems
arise. A spacecraft needs fuel to make the long journeys in space, and humans need
food and water throughout the long journeys. The effects of microgravity, that is,
near zero gravity, on humans over time are unknown. The probability of collisions
with asteroids is uncertain, and many other aspects of long manned flights make the
task of space travel very complex.
However, publicity about unmanned flights to the planets continues to raise the
question of humans' traveling in space. Research is required to increase the
probability that prolonged space travel for humans can be accomplished safely. One
of the NASA projects that will move us closer to space travel is the International
Space Station, which will serve as a platform for many research agendas associated
with living and working in space for long periods of time.
Purpose
This lesson affords students the opportunity to think about two aspects of the time
required to complete space travel within the solar system. First, students consider
the amount of time that space travelers must spend on the journey. Second,
students think about what kinds of events might occur on Earth while the space
travelers are on their journey. Thinking about both situations improves students'
concept of time and distance as well as improves their understanding of the solar
system.
Getting Started
Begin the class by engaging students in a discussion about humans traveling through
the universe. In the movies and on television, students encounter science-fiction
stories about traveling at the speed of light and beyond to cross entire galaxies in a
matter of seconds. The activities in lesson 2 have reminded us that most of our
travel speeds are quite slow. We have not come close to traveling at the speed of
light. Although radio signals can return from Mars in a short time, it takes much
longer for a spacecraft from Earth to reach Mars.
Developing the Activity
Use the data about the Space Shuttle below to determine its speed in miles per hour.
Remind students that the Shuttle is not designed for travel among the planets. It is
designed for Earth-orbit tasks. However, its speed is helpful in judging the speeds for
twentieth century spacecraft. After students have done the calculations, come to
some agreement on an approximate speed for interplanetary travel. Assume that the
agreement is about 50,000 miles per hour. This figure gives us a reasonable speed
to use in thinking about space travel today. In the future, speeds will undoubtedly
increase.
Space Shuttle Data with Speed Computation
Space Shuttle Data:
Distance traveled by Shuttle = 4,164,183 miles
Time to travel given distance = 9d 23h 30 min
Speed Computation:
Using the data sheet The Planets at a Glance, students can determine the distance
from Earth to each of the other planets. This task is not trivial. The distances in the
chart are given in millions of miles. To facilitate computation and estimation,
students need to translate 67.2 million miles, the distance from the Sun to Venus,
into its full numeric form, 67,200,000. Before thinking about traveling to Venus,
students must remember that Earth is about 93,000,000 miles from the Sun. Use the
mean distance from the Sun to specified planets to calculate each distance from
Earth to the targeted planet.
To make this exploration manageable for middle school students, we are assuming
that the planets are aligned at their mean distances. Teachers should explain to
students that in actuality, this greatly simplified situation is unlikely to occur.
The students' next task is to calculate the time required to travel to each planet on
spacecraft that travel from 10,000 to 100,000 miles per hour.
Group the students into their mission teams of four students. Ask them to complete
a chart for travel to all the planets of the solar system at the speeds shown below.
They should use the mean distance of each planet from Earth.
In a typical class, students groan at the prospect of completing a chart with eighty
entries. The groans provide the opportunity to challenge the teams to think of
strategies for reducing the amount of calculations required to complete the chart.
When patterns are used to complete entries, the teams should record them. All
students in the teams should make the chart.
Discuss the patterns that teams used to complete the chart. List the patterns on the
chalkboard. After all team members have shared how their patterns helped reduce
their workload, ask each team how many different patterns they used.
The class now has a complete chart for travel to the solar system's planets
computed in hours of travel. It is time to develop some better notions of what these
times mean. Pose questions that require students to think about the practicality of
space travel:
• If we travel at the approximate speed of the Space Shuttle, which planets can we
reach in less than 10 years?
• How fast must we be able to travel to reach Jupiter in less than 10 years?
• Traveling to some of the planets at some of the indicated speeds would take more
than a lifetime, which is about 75 years. Which planets are too far away to be
reached in a lifetime?
• We would like to make round trips. Traveling at 10,000 per hour, to which planets
could we make a round trip in our lifetime?
These questions should be posed to the mission teams. The teams should discuss
the questions and agree on a team response. After students begin thinking about
time questions, have each mission team make up questions for the class to solve.
Closing the Activity
Pose selected mission team questions to the entire class. The quality of the questions
and the responses should indicate which students are gaining an understanding
about the time required for space travel.
Standards and Expectations
Number & Operations 6-8
This lesson covers the following Number & Operations Standard Expectations:
understand and use ratios and proportions to represent quantitative relationships.
develop an understanding of large numbers and recognize and appropriately use
exponential, scientific, and calculator notation.
understand the meaning and effects of arithmetic operations with fractions,
decimals, and integers.
References
Adapted from Travel in the Solar System in Mission Mathematics, Linking Aerospace
and the NCTM Standards, 5-8, a NASA/NCTM project, NCTM 1997.
Additional activities for this lesson can be found at http://illuminations.nctm.org
Integer Product Game
Objective:
Students will practice multiplying integers and determining factors for
numbers.
Materials:
Game sheet, 2 paper clips, handful of chips or beans or something to cover
products on game sheet.
Instructional Plan:
-We suggest playing a practice game on the overhead against a student volunteer.
-Hand out the rules for the game found in the book and make sure rules are clear.
-Partner students up to play the game.
-Hand out 2 paperclips per pair and some markers of some sort to cover products as kids
play.
-Encourage students to write down any strategies they are using or thinking about.
-After students have played several times, have them discuss their strategies, and then
you will likely want to re-pair them to play against a different opponent.
Reference: This game is part of Lesson 4.3 from 7th grade CMP, the book
“Accentuate the Negative”, pages 57—58.
A Conceptual Model for Solving Percent
Problems
Length: 1.5 period(s)
Grades: 6-8
Number & Operations
This lesson is found at Illuminations on the NCTM website.
http://illuminations.nctm.org
In this grades 5-8 lesson, students will use a 10 x 10 grid, which
is a common model for visualizing percents to solve various types
of percent problems. This model offers a means of representing
the given information as well as suggesting different approaches
for finding a solution. This NCTM Publication-Based Lesson Plan is
adapted from "A Conceptual Model for Solving Percent Problems"
by Albert B. Bennett, Jr., and L. Ted Nelson. The article appeared
in Mathematics Teaching in the Middle School Vol.1, No.1
(April 1994) Pp. 20-25.
Learning Objectives
Students will be able to:
*
shade a 10 x 10 grid to represent a given percent
*
determine the percent for a given shaded amount on a
10 x 10 grid
*
determine the value of one of the small squares in a
10 x 10 grid
*
determine the value of the 10 x 10 grid given the value of
one of the small squares
*
use the 10 x 10 grid model to solve problems
Materials
*
*
10 x 10 Grids
Dry-erase, water-based, or grease markers.
Instructional Plan
Applications of percents are often taught by solving proportions
that require variables and some familiarity with algebra. Such
formal approaches to teaching percents have not been successful
for many junior high and high school students (Wiebe 1986). This
article presents an alternative method that focuses on the basic
concept of percent, that of "parts per hundred." A 10 x 10 grid,
which is a common model for visualizing percents, is extended in
the following examples to solve various types of percent
problems. This model offers a means of representing the given
information as well as suggesting different approaches for finding
a solution.
Standards and Expectations
Number & Operations 6-8
*
compare and order fractions, decimals, and percents
efficiently and find their approximate locations on a number line.
*
develop meaning for integers and represent and compare
quantities with them.
*
work flexibly with fractions, decimals, and percents to solve
problems.
References
*
Bennett, Albert B., and L. Ted Nelson. Mathematics for
Elementary Teachers: An Activity Approach. 3d ed. Dubuque, Ia.:
Wm. C. Brown Publishers, 1992.
*
Dye, David, chair of committee for Minnesota Department of
Education. "Position Paper; Teaching and Learning Percent." St.
Paul, Minn.: Department of Education, 1981.
*
Wiebe, James H. "Manipulating Percentages." Arithmetic Teacher
33 (January 1986):23-26.
*
Bennett, Albert B. and L. Ted Nelson. "A Conceptual Model
for Solving Percent Problems." Mathematics Teaching in the
Middle School 1 (April 1994): 20-25.
*
_____, Mathematics for Elementary Teachers: A Conceptual
Approach. 3d ed. Dubuque, Ia: Wm. C. Brown Publishers, 1992.
*
National Council of Teachers of Mathematics. Curriculum and
Evaluation Standards for School Mathematics. Reston, Va.: The
Council, 1989.
Having Fun with Baseball Statistics
Length: 2 period(s)
Grades:
3-5
6-8
Communication
Number Sense
This lesson can be found at Illuminations on the NCTM website
http://illuminations.nctm.org
The following grades 5-8 activities allow students to explore
statistics surrounding baseball. They are exposed to connections
between various mathematical concepts and see where this
mathematics is used in areas with which they are familiar. This
lesson plan appeared in the May 1996 edition of Mathematics
Teaching in the Middle School Journal.
Learning Objectives
Students will
*
develop skills in mathematical reasoning and computations
and apply those skills to everyday life
Materials
*
Game-Card Worksheets
*
Baseball cards
*
Calculators
*
Paper clips
*
Overhead projector and sample transparencies are helpful
but are not necessary.
Instructional Plan
How often have middle school teachers had to ask students to
put away their baseball cards? The lesson described in this article
capitalizes on students' interest in sports while providing valuable
instruction on a variety of appropriate mathematical topics.
Specifically, this lesson meets the recommendations of the
NCTM's Curriculum and Evaluation Standards for School
Mathematics (1989) by integrating the study of fractions,
decimals, percents, rounding, Cartesian coordinates, probability,
and statistics in a cooperative setting that allows students to
have fun as they learn.
Overview
The lesson involves the following four phases: (1) students
familiarize themselves with the meanings of the statistics found
on baseball cards; (2) students complete the worksheet portion
of the game-card worksheet, by converting information from
their baseball cards into appropriate decimals, fractions, and
percents; (3) students complete the game-card portion of the
worksheet based on these data; (4) students play a simulated
game of baseball. Approximately ninety minutes is required to
complete this activity. It can, therefore, be scheduled in a single
extended time block, often available in middle school schedules,
or split over two forty-five-minute class periods. I have found
that in the latter, phases 1 and 2 can be completed during the
first class session. The teacher can then collect and check the
students' worksheets before they use this information to create
the 10-by-10 array on their game cards for phase 3.
References
*
National Council of Teachers of Mathematics. Curriculum and
Evaluation Standards for School Mathematics. Reston, Va.: The
Council, 1989.
*
Written by Robert J. Quinn
Mathematics Teaching in the Middle School Journal, May 1996.
*
The Topps Company. Baseball cards, licensed by Major League
Baseball and the Major League Baseball Player’s Association,
1993.
Dilution
Learning Objectives:
*Students will get to work with small numbers that apply to everyday
life.
Materials:
*Red or blue food coloring ( 1 ounce per group);water (2 gallons per
group); 10 ounce transparent, graduated measuring containers;
stirring stick.
Instructional Plan:
*First pour 1 ounce of food coloring into a container and add 9 ounces
of water to it. This will make a 10-ounce solution. Stir the solution.
*Record the diluted solution into a chart in decimal form.
*Next pour 1-ounce of the new solution into another container and add
9 ounces of water to it.
*Record the new diluted solution.
*Continue this process to the millionths place.
*Leave all 6 solutions in the front of the room so students can see the
process.
*Students will describe what happens when a power of 10 is
repeatedly divided by 10.
*Students continue to solve problems about dilution that involve
powers of 10 using negative exponents.
Number Portion
of
of Food
Dilutions Coloring
1
0.1
2
0.01
3
4
5
6
Reference: Britannica, Mathematics in Context p. 76-81
Number Portion
of
of Food
Dilutions Coloring
1
0.1
2
3
4
5
6
Pick Up Sticks
Objective: Develop the concepts of odd and even in game play strategy.
Materials: Writing board, chalk or marker. May also be played with any 15 items such
as sticks, beans, paperclips, etc…
Activity:
- The goal is to force your opponent to pick up the last “stick”.
- Students should pair up into teams.
- There are 3 rows of five items, for a total of 15 items.
IIIII
IIIII
IIIII
-
You may pick 1, 2, 3, 4 or 5 sticks from any one row on your turn.
Alternate turns.
If you force your opponent to pick up the last stick you win.
Model the game with the students on the writing board.
Let the students play a few games to get a feel for the game.
Then challenge teams to develop a winning strategy and express it on paper.
Challenge the strategies to a real game and see if it wins.
Winning strategy: Go first, leave odd number of sticks with odd number of rows or even
number of matchsticks with even number of rows.
Four 4’s Project
Objective: Students will improve their number sense and improve
efficiency using correct order of operations.
Materials: Four 4’s sheet (optional)
Activity:
-The instructor will begin the activity with a challenge for the students---to
write a mathematical problem with an answer of 1 using exactly four fours,
and no other numbers. Have selected students show answers on the board.
(Hopefully they will generate several different solutions)
-Discuss answers and proper order of operations.
-Students will now work in groups to write 19 more mathematical equations
using exactly four fours in it with answers of 2, 3, 4, …..etc
-Compile students answers after they have had enough time to work on all
of the problems.
-Have students correct their peers work on the board.
Homework: 1-2-3-4 Challenge
-Students will use the numbers 1,2,3, and 4 only, to write math problems
with the answers of 1,2,3,4,5,6,7,8,9,10,11, and 12. Encourage them to
come up with more than one solution.
Bullseye
Objectives: Students will get to know each other and they will be
able to do the order of operations correctly.
Materials: Bullseye sheets
Activity:
-You will need to assemble students into group of 3 or 4.
-Students will collaborate as a group to come up with a set of
numbers.
-After groups have answered the set of questions they will use the
numbers in an equation to hit a target answer of ten using
mathematical operations. If students can not hit the number they
are to get as close as they can.
- They will continue the activity by using the same numbers in an
equation to hit an answer of a hundred.
- You could continue setting new target numbers for your
students to challenge their ability using order of operations.
Timber
Objective: Real life applications using division.
Materials: Timber worksheet, rulers, and scrap paper.
Activity:
-Students will need to draw lines that are 120mm, 160mm,
200mm, 240mm and 280mm.
- For each problem students will need to use a ruler to measure
distances to find the most affective use of building materials from
the lengths above.
The Right Price Game
Learning Objectives: Students will work on basic addition and
subtraction facts either mentally or paper pencil.
Materials: The right price Game sheet, or make up three items of
your own and a total price
:
Instructional Plan
*Find as many prices as you can that will work for each round, that
is the sum of the three items adds up to the total.
*You have 15 seconds per round.
*After each game list all solutions the class came up with.
*Play as many rounds as time permits.
Number Sense Assessment
Observational adjustments:
We will use in class observations of student progress to make adjustments to our
instructional plan when using game activities.
With games like Pick Up Sticks, the game will only hold a student’s interest until they
figure out the winning strategy. After that, given the rules of the game, the student’s will
lose interest in the activity. As teachers we will either modify the rules or move on to the
next activity when we observe that kids have mastered the winning strategy.
Formal assessments:
The lessons in our project that are more involved, and meet the state standards, fit into
our curriculum in a number of different units. We will be assessing the material from
these lessons by slightly modifying our current assessment tools.