NUMBER SENSE
Elaine Erickson
Red Lake Middle School
[email protected]
Grades 6-8
Summer 2007
Overview:
These number sense lessons are meant to reinforce what
the students are learning in their 7th grade mathematics
course. These lessons are designed in a way that you
could pick and choose whichever one you want to
strengthen the material being taught. Some may be used
more than once a week or once a month, but continuing
throughout the school year. The integer lessons can be
expanded to fit over an extended period of time.
Contents:
Lesson
Lesson
5-6)
Lesson
Lesson
Lesson
1: Every Breath You Take (pages 3-4)
2: Hot and Cold Cubes Number Combinations (pages
3: Adding Hot and Cold Cubes (pages 7-8)
4: Subtracting Hot and Cold Cubes (pages 9-10)
5: Multiplying Hot and Cold Cubes (pages 11-12)
Minnesota Standards Covered- Number & Operation
• Multiply and divide decimals, fractions, and mixed
numbers; solve real world and mathematical
problems using arithmetic with positive rational
numbers.
• Calculate with positive and negative rational
numbers, and rational numbers with whole number
exponents, to solve real world and mathematical
problems
2
3
Every Breath You Take
MN Standard: 6.1.3.4 Solve real world and mathematical
problems requiring arithmetic with decimals, fraction,
and mixed numbers.
Launch: Ask students how many times they breathe in a
minute? an hour? a day? Put them in groups of two and
distribute “ Every Breath You Take ” worksheet.
Explore: Ask students to complete the “ Every Breath
You Take ” worksheet.
Share: Have students describe the guesses made in their
tables. Describe how they got their answers for
breaths in an hour? day? Record the range of values
on the board.
Summarize: To convert units we can do dimensional
analysis. For instance:
17 breaths x 60 minutes = breaths per hour (the units cancel and we are multiplying by
1 minute
1 hour
a value of one (same numerator and denominator).
4
Hot and Cold Cubes Number Combinations
MN Standard: 7.1.2.1 Add, subtract, multiply, and
divide integers.
Materials:
Blue paper
Red paper
Scissors
Launch: Show the students a red cube and a blue cube.
Thinking of the cubes as temperature, ask them what the
red cube represents. (Hot cube or the warm front). The
red cube will make the temperature rise one degree.
What does the blue cube represent? (cold cube or the
cold front) The blue cube will make the temperature
fall one degree. Put on you forehead to show that the
blue cube will cool you off on a hot day.
Explore:
Put the students in groups of two and have them cut a
blue and red piece of paper into equal parts. (make as
many squares as possible)
Have the students represent the numbers one through
nine using the red and blue cube.
Is there any other way to show the number 2? (see
notes)
You will repeat this with 3 & 4 until the students
understand that there are many different ways to
represent one number.
How would you represent a -2 with the hot and cold
cubes?
Share:
For every number you talk about have the students show
on the board all the different ways you can represent
that number.
Why are you able to represent a number in many
different ways?
Summarize:
Realizing that you can represent a number in many
different hot and cold cube arrangements will allow you
to learn how to add, subtract, and multiply integers.
5
Notes for Lesson Two
Student’s possible beginning arrangements:
2:
3:
4:
5:
6:
7:
8:
9:
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H H
H H H
H H H H
What would happen to the temperature if I put a hot and
cold cube together? (hopefully they will realize that
putting the two together cancels each other out and it
would create a zero)
I would call a hot and cold cube together a stationary
front. The temperature will neither increase nor
decrease.
Have students create examples of different variations
for the number 2, 3, 4.
Possibilities:
2: HH
H H H
C
3: H H H
H H H H and so on…
C C
H H H H
C C
H H H H H
and so on…
C C
6
Adding Hot and Cold Cubes
MN Standard: 7.1.2.1 Add, subtract, multiply, and
divide integers.
Materials:
Student bags of hot and cold cubes
Launch: Talk about yesterday’s lesson and ask some
students to write on the board different variations of
the numbers 1-9. Yesterday we talked about different
variations of the cubes for the digits on the board,
today we are going to learn to add hot and cold cubes.
Remember addition means to put things together.
Explore:
Put the students back into their paired groups from
yesterday.
Have them pull out their hot and cold cubes.
Put the addition problem 2 + 2 on the board.
Have students work with their partner to represent this
equation with their cubes.
Discuss as a class how you would solve this problem.
Put the addition problem 2 + -2 on the board
Have students work with their partner to represent this
equation with their cubes.
Discuss as a class how you would solve this problem.
Use the rest of the problems in the notes and have
students work on them in your groups.
Share:
Groups of students will share their results on the
board for the class to see.
Why can all these variations work?
Can we come up with some type of rules that might help
us to add hot and cold cubes?
Summarize:
Realizing that you can represent a number in many
different hot and cold cube arrangements allows us to
add different integers with the hot and cold cubes to
get an answer.
7
Notes for Lesson Three
Addition problems
1. 1 + 3
2. -3 + 5
3. -4 + -6
4. 2 + -4
Possible answers for the launch
2 + 2
HH
HH
HHH
HHH
C
2 + -2
HH
CC
HHHH
CCCC
CC
C
HH
8
Subtracting Hot and Cold Cubes
MN Standard: 7.1.2.1 Add, subtract, multiply, and
divide integers.
Materials:
Student bags of hot and cold cubes
Launch: Talk about yesterday’s lesson and ask some
students to write on the board their answers to some of
yesterday problems. Remind them that when we add we put
together the cubes. Today we are going to learn to take
away cubes or subtraction.
Explore:
Put the students back into their paired groups from
yesterday.
Have them pull out their hot and cold cubes.
Put the subtraction problem 2 - 2 on the board.
Discuss as a class how you would solve this problem.
Put the subtraction problem 1-2 on the board.
Discuss as a class how you would solve this problem.
Use the rest of the problems in the notes and have
students work on them in your groups.
Share:
Groups of students will share their results on the
board for the class to see.
Why can all these variations work?
Are their any problems that were harder than others?
What did you do to solve those problems?
Were their any special discoveries made during this
activity?
Can we come up with some type of rules that might help
us to subtract hot and cold cubes?
Summarize:
Realizing that you can represent a number in many
different hot and cold cube arrangements we are able to
subtract different integers with the hot and cold cubes
to get an answer.
9
Notes for Lesson Four
Subtraction problems
•
•
•
•
-2 - 1
-3 -5
4 - 6
6 – (-4)
Possible answers for the launch
-2 – 1
CCC
(Take away one hot to get
your answer)
H
10
Multiplying Hot and Cold Cubes
MN Standard: 7.1.2.1 Add, subtract, multiply, and
divide integers.
Materials:
Student bags of hot and cold cubes
Launch: Talk about yesterday’s lesson and ask some
students to write on the board their answers to some of
yesterday problems. Remind them that when we add we put
together the cubes. When we subtract we take away
cubes. Today we will learn to multiply the hot and cold
cubes.
Explore:
Put the students back into their paired groups from
yesterday.
Have them pull out their hot and cold cubes.
Put the multiplication problem 2 X 2 on the board.
Discuss as a class how you would solve this problem.
Put the Multiplication problem 3 X -2 on the board.
Discuss as a class how you would solve this problem.
Use the rest of the problems in the notes and have
students work on them in your groups.
Share:
Groups of students will share their results on the
board for the class to see.
Why can all these variations work?
Can you share the differences in multiplication verses
addition and subtraction?
Can we come up with some type of rules that might help
us to multiply hot and cold cubes?
Summarize:
Realizing that you can represent a number in many
different hot and cold cube arrangements we are able to
multiply different integers with the hot and cold cubes
to get an answer.
Notes for Lesson Five
Multiplication problems
11
•
•
•
•
2 X 1
-3 X 4
5 X -2
-6 X (-4)
Possible answers for the launch
2 X 2
HH
HH
3 X -2
CC
This is 3 groups of 2 cold
CC
CC
(remember this says take away 2 groups of hot)
HHH
Take away
CCC
Left with 6 cold
Take away
HHH
CCC
12