Transcript K-8 Mathematics Standards

Welcome to Grades K-2 Mathematics
Content Sessions
The focus is Algebraic Thinking.
The goal is to help you understand this mathematics better
to support your implementation of the K-8 Mathematics
Standards.
2008 May 29
Algebra: Grades K-2: slide 1
Importance of Algebraic Thinking
Algebraic thinking includes understanding and use of
properties of numbers and relationships among numbers.
Algebraic Thinking was chosen as the content focus
because it lays the foundation for the learning of algebra
in middle school and high school.
It takes most students a long time to develop algebraic
thinking, so it is important to begin this work in the
primary grades.
2008 May 29
Algebra: Grades K-2: slide 2
Introduction of Facilitators
• <Insert Facilitator Names Here>
2008 May 29
Algebra: Grades K-2: slide 3
Introduction of Participants
In a minute or two:
1. Introduce yourself.
2. Describe an important moment in your life that
contributed to your becoming a mathematics
educator.
3. Describe a moment in which you hit a
“mathematical wall” and had to struggle with
learning.
2008 May 29
Algebra: Grades K-2: slide 4
Overview
Some of the problems may be appropriate for
students to complete, but other problems are
intended ONLY for you as teachers.
As you work the problems, think about how you
might adapt them for the students you teach.
Also, think about what Performance Expectations
these problems might exemplify.
2008 May 29
Algebra: Grades K-2: slide 5
Problem Set 1
The focus of Problem Set 1 is understanding
equality.
You may work alone or with colleagues to solve
these problems.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 6
Problem 1.1
What is the mathematics underlying the concept
of equality?
That is, what would you want students to say if
you asked, “What does it mean for two
things to be equal?”
2008 May 29
Algebra: Grades K-2: slide 7
Problem 1.2
What do we want students to understand about
the equal sign (=)?
2008 May 29
Algebra: Grades K-2: slide 8
Problem 1.3
Carpenter, Franke, and Levi (2003, p. 9) report
data (shown on the next slide) showing students’
responses to the question below.
What number would you put in the box to
make this a true number sentence?
8+4=+5
What do you notice in the data? What conclusions
can you draw?
2008 May 29
Algebra: Grades K-2: slide 9
Problem 1.3
What number would you put in the box to make
this a true number sentence?
8+4=+5
Grade
1 and 2
3 and 4
5 and 6
7
5
9
2
Response/Percent Responding
12
17
58
13
49
25
76
21
12 and 17
8
10
2
What do you notice in the data? What conclusions
can you draw?
2008 May 29
Algebra: Grades K-2: slide 10
Problem 1.4
Write two or three learning targets for equality
and the equal sign.
Be as precise as possible; that is, what do you
want students to know about the equal sign?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 11
Problem Set 2
The focus of Problem Set 2 is number
relationships.
You may work alone or with colleagues to
solve these problems.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 12
Problem 2.1
Which of these number sentences are true, and
which are false? Explain your thinking.
a. 8 + 7 = 15
c. 8 + 7 = 8 + 7
e. 8 + 7 = 9 + 6
g. 8 = 8
2008 May 29
b. 15 = 8 + 7
d. 8 + 7 = 7 + 8
f. 8 + 7 = 87
Algebra: Grades K-2: slide 13
Problem 2.2
What number(s) could go in the box to make each
number sentence true? Explain your thinking.
a. 8 + 7 = 
c.  + 7 = 8 + 7
e. 15 = 
g. 39 + 57 =  + 59
2008 May 29
b. 8 + 7 =  + 7
d.  + 8 = 8 + 7
f.  = 7
h.  + 82 = 143 + 89
Algebra: Grades K-2: slide 14
Problem 2.3
What number(s) could be substituted for N to
make each number sentence true? Explain
your thinking.
a. 8 + 7 = N
c. N + 7 = 8 + 7
e. 15 = N
g. 39 + 57 = N + 59
2008 May 29
b. 8 + 7 = N + 7
d. N + 8 = 8 + 7
f. N = 7
h. N + 82 = 143 + 89
Algebra: Grades K-2: slide 15
Problem 2.4
Design a sequence of true/false and/or open
number sentences that you might use to
engage your students in thinking about the
equal sign.
Describe why you selected the problems you
did.
2008 May 29
Algebra: Grades K-2: slide 16
Reflection
What did you learn (or re-learn) from solving
these problems?
Where in the K-8 Mathematics Standards do
these ideas appear?
The Standards uses the word, equation, instead
of the phrase, number sentence. What is the
difference in these two terms?
2008 May 29
Algebra: Grades K-2: slide 17
Problem Set 3
The focus of Problem Set 3 is making number
sentences true.
You may work alone or with colleagues to
solve the problems in set 3.1.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 18
Problem 3.1
In each number sentence, what number(s) could
be substituted for the variable to make that
number sentence true? Explain your thinking.
a. 6 + 9 = 8 + 10 + d
b. 9 - 6 = 8 - 4 + g
c. 10 - 6 = 8 - 4 + a
d. 5 + 8 + d = 6 + 9 + d
2008 May 29
Algebra: Grades K-2: slide 19
Problem 3.1
In each number sentence, what number(s) could
be substituted for the variable to make that
number sentence true? Explain your thinking.
e. 5 + 8 - d = 6 + 9 - d
f. 5 + 8 + d + d = 6 + 9 + d
g. 5 + 8 - d - d = 6 + 9 - d
h. 5 + 8 - d = 6 + 9 - d - d
2008 May 29
Algebra: Grades K-2: slide 20
Problem 3.2
a. Solve: d + d + d - 20 = 16
b. Look at video 5.1 (Carpenter, et al., 2003).
Focus your attention on the student’s
strategy.
Does the student’s strategy illustrate
algebraic thinking?
2008 May 29
Algebra: Grades K-2: slide 21
Problem 3.3
a. Solve: k + k + 13 = k + 20
b. Look at video 5.2 (Carpenter, et al., 2003).
Focus your attention on the student’s
strategy.
Does the student’s strategy illustrate
algebraic thinking?
2008 May 29
Algebra: Grades K-2: slide 22
Problem 3.4
How are the strategies used in videos 5.1 and
5.2 alike?
How are they different?
2008 May 29
Algebra: Grades K-2: slide 23
Reflection
How might your students solve these or similar
problems? What strategies might they use?
How could you adapt these problems for your
teaching?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 24
Problem Set 4
The focus of Problem Set 4 is relational
thinking.
You may work alone or with colleagues to
solve these problems.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 25
Relational Thinking
Make an argument that this equation is true,
WITHOUT computing each sum.
25 + 17 = 24 + 18
Make an argument that this equation is false,
WITHOUT computing each difference.
25 - 17 = 24 - 18
2008 May 29
Algebra: Grades K-2: slide 26
Problem 4.1
Which number sentences are true and which are
false? Justify your answers.
a. 3,765 + 2,987 = 3,565 + 3,187
b. 4,013 – 2,333 = 4,043 – 2,363
c. 8,041 – 3,762 = 8,051 – 3,752
d. 5,328 + 3,933 = 8,328 + 933
e. 6,789 – 6,345 = 789 - 345
2008 May 29
Algebra: Grades K-2: slide 27
Problem 4.2
Rank the following problem from easiest to
most difficult (for students). Justify your
choices.
a. 73 + 56 = 71 + d
c. 68 + b = 57 + 69
e. 96 + 67 = 67 + p
g. 74 – 37 = 75 - q
2008 May 29
b. 92 – 57 = g - 56
d. 56 – 23 = f - 25
f. 87 + 45 = y + 46
Algebra: Grades K-2: slide 28
Problem 4.3
Decide whether each number sentence below is true or
false. Justify your choices. How do you think students
would justify the choices?
a. 56 = 50 + 6
b. 87 = 7 + 80
c. 93 = 9 + 30
d. 94 = 80 + 14
e. 94 = 70 + 24
f. 246 = 24 x 10 + 6
g. 47 + 38 = 40 + 30 + 7 + 8
h. 78 + 24 = 98 + 4
i. 63 – 28 = 60 – 20 – 3 – 8
j. 63 – 28 = 60 – 20 + 3 – 8
2008 May 29
Algebra: Grades K-2: slide 29
Problem 4.4
Create a set of problems that might encourage
students to use relational thinking.
Be ready to explain the grade-level of your
problems and justify your choices of
numbers.
2008 May 29
Algebra: Grades K-2: slide 30
Reflection
What is relational thinking?
Why is relational thinking important for
students to be able to do?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 31
Problem Set 5
The focus of Problem Set 5 is properties of
operations.
You may work alone or with colleagues to
solve the problems in set 5.1.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 32
Problem 5.1
Make four groups with each group exploring
one operation.
a. Explore the properties of addition.
b. Explore the properties of subtraction.
c. Explore the properties of multiplication.
d. Explore the properties of division.
2008 May 29
Algebra: Grades K-2: slide 33
Problem 5.2
Describe the commutative and associative
properties for addition.
Represent these properties using symbols. Do
the same for multiplication.
Can you do the same for subtraction and
division? Why or why not?
2008 May 29
Algebra: Grades K-2: slide 34
Problem 5.3
Read this equation aloud using words rather than
symbols:
a + b = (a + 1) + (b - 1)
What mathematical idea does this equation
represent?
Is the equation true or false? Explain your answer.
2008 May 29
Algebra: Grades K-2: slide 35
Problem 5.4
Look at video 3.3 (Carpenter, et al., 2003).
Focus your attention on the strategies the student
uses.
What, if anything, do you think the student learned
during this interview?
What problem would you pose to check your
hypothesis?
2008 May 29
Algebra: Grades K-2: slide 36
Reflection
What do students need to know about the
properties of operations?
How can you help students learn those
properties?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 37
Welcome to Grades K-2 Mathematics
Content Sessions
The focus is Algebraic Thinking.
The goal is to help you understand this mathematics better
to support your implementation of the K-8 Mathematics
Standards.
2008 May 29
Algebra: Grades K-2: slide 38
Problem Set 6
The focus of Problem Set 6 is justification and
proof.
You may work alone or with colleagues to
solve the problems in set 6.1.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 39
Problem 6.1
True or false: a - b - c = a - (b + c).
Justify your answer.
2008 May 29
Algebra: Grades K-2: slide 40
Problem 6.2
If you have 5 sodas and each person gets half a
soda, how many people will get to drink soda?
True or false: N ÷ 1/2 = 2 x N
Justify your answer.
True or false: N ÷ 1/3 = 3 x N
Justify your answer.
2008 May 29
Algebra: Grades K-2: slide 41
Problem 6.3
Look at video 7.2.
Focus your attention on the student’s
explanations.
What do you think this student understands
about proof?
How do the interviewer’s questions help reveal
what the student knows?
2008 May 29
Algebra: Grades K-2: slide 42
Reflection
What do you look for in a child’s argument
about whether something is true or false?
How sophisticated can you expect children’s
arguments to be?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 43
Problem Set 7
The focus of Problem Set 7 is what happens
and why.
You may work alone or with colleagues to
solve these problems.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 44
Problem 7.1
True or false? Explain your answers.
a. 87 ÷ 5 and 7 ÷ 5 have the same remainder
b. 876 ÷ 5 and 6 ÷ 5 have the same remainder
c. 895 ÷ 5 and 5 ÷ 5 have the same remainder
d. If abc represents a 3-digit number, abc ÷ 5 and
c ÷ 5 have the same remainder
e. What rule can you state for divisibility by 5?
Justify the rule.
2008 May 29
Algebra: Grades K-2: slide 45
Problem 7.2
True or false? Explain your answers.
a. 65 ÷ 3 and (6 + 5) ÷ 3 have the same remainder
b. 652 ÷ 3 and (6 + 5 + 2) ÷ 3 have the same
remainder
c. 651 ÷ 3 and (6 + 5 + 1) ÷ 3 have the same
remainder
d. If abc represents a 3-digit number, abc ÷ 3 and (a
+ b + c) ÷ 3 have the same remainder
e. What rule can you state for divisibility by 3? Justify
the rule.
2008 May 29
Algebra: Grades K-2: slide 46
Problem 7.3
Take any 3 digits (not all the same!!) and make
the greatest and least 3-digit numbers.
Subtract the lesser from the greater to make the
high-low difference.
Repeat this process for that difference.
Keep on repeating the process.
What happens? Why?
2008 May 29
Algebra: Grades K-2: slide 47
Problem 7.4
Take a three-digit number (with digits not all the
same), reverse its digits, and subtract the lesser
from the greater.
Reverse the digits of the result and add these two
numbers.
132 becomes 231, and 231 - 132 = 99 = 099
099 becomes 990, and 099 + 990 = 1089
Try this process for several numbers.
What happens? Why?
2008 May 29
Algebra: Grades K-2: slide 48
Reflection
Why is knowledge of divisibility important for
students to know?
In K-2, the ideas of odd and even are important
for children to learn. How are those ideas
related to divisibility rules?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 49
Problem Set 8
The focus of Problem Set 8 is representations.
You may work alone or with colleagues to
solve the problems in set 8.1.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 50
Problem 8.1
Look at video 7.1 and video 8.1.
Focus your attention on students’ representations.
How are the two representations of odd numbers
alike? How are they different?
Which representation is more convincing?
What assumptions is each student making?
2008 May 29
Algebra: Grades K-2: slide 51
Problem 8.2
How could you use variables to represent an
even number?
An odd number?
A multiple of 5?
2008 May 29
Algebra: Grades K-2: slide 52
Problem 8.3
Suppose that when N is divided by 3 the
remainder is 1, and that when P is
divided by 3 the remainder is 2.
What is the remainder of N + P when you
divide it by 3?
What is the remainder of N - P when you
divide it by 3?
2008 May 29
Algebra: Grades K-2: slide 53
Reflection
How can representations help students reason?
How can we help students learn to use
representations to clarify their thinking?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 54
Problem Set 9
The focus of Problem Set 1 is patterns and
conjectures.
You may work alone or with colleagues to
solve these problems.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 55
Problem 9.1
What do you notice about each pair of
products below.
What happens? Why?
6 x 6 and 5 x 7
30 x 30 and 29 x 31
500 x 500 and 499 x 501
N x N and (N - 1) x (N + 1)
2008 May 29
Algebra: Grades K-2: slide 56
Problem 9.2
Guess these products. Then check your
guesses.
300 x 300 and 298 x 302
300 x 300 and 295 x 305
N x N and (N - a) x (N + a)
2008 May 29
Algebra: Grades K-2: slide 57
Reflection
What kinds of patterns might you ask K-2
students to explore?
Where in the K-8 Mathematics Standards do
these ideas appear?
2008 May 29
Algebra: Grades K-2: slide 58
Problem Set 10
The focus of Problem Set 1 is reflection on
your thinking.
You may work alone or with colleagues to
solve these problems.
When you are done, share your solutions with
others.
2008 May 29
Algebra: Grades K-2: slide 59
Problem 10.1
What did you learn (or re-learn) from working
on these problems?
How did the videos help you understand the
mathematics ideas?
2008 May 29
Algebra: Grades K-2: slide 60
Problem 10.2
Look back over the problems.
Which ones could you use directly with
children?
Which ones could you adapt for use with
children?
2008 May 29
Algebra: Grades K-2: slide 61
Problem 10.3
Where do these problems “fit” in the K-8
Mathematics Standards?
Which problems might you share with other
teachers in your school? Why would
those problems be important ones to
share?
2008 May 29
Algebra: Grades K-2: slide 62
Closing Comments
Implementing the K-8 Mathematics
Standards will require a deep focus of
mathematics ideas at each grade.
Personal understanding of these ideas will
make the implementation process easier.
2008 May 29
Algebra: Grades K-2: slide 63