Foundations of Algebraic Thinking:
Place Value K-2
Name
Counte
glass
rs
blobs
Numb
Date
er Wo
rd
Tens
Single
button
s
s
Tens
square
Single
tile
s
Tens
Single
toothp
icks
s
Tens
lima bea
ns
Single
s
Tens
Single
s
Building Instructional
Leadership Across Oregon
“Developing Algebraic Thinking”
Session 3
March 2010
K-2 Goal:
Students use number sense, place value, and patterning to
construct concrete models for thinking algebraically. I can
describe how students begin to represent their thinking K2 and how this builds more abstract thinking, models, and
representations through grade 8.
Foundations of Algebraic Thinking: Place Value K-2
March Follow Up K-8 Math Institute
Foundations for Algebraic Thinking
Vocabulary
3 Activities
Activity Descriptions, Note Pages
5 Participant Activity Planner
9 Number Talk: Ten Frame Flash A. Compatible Pairs B. Groups of Ten C. Three Other Ways D. The Other Part of 100 E. Close to 100, Close to 1000 F. Solve Two Ways 10 11 20 24 29 31 34
2
Addition, Subtraction, Patterning K-2
Vocabulary
Properties
Associative Property : (Addition, Multiplication) For a given number sentence that combines three quantities (2 at a time); the initial pairing of the quantities is arbitrary. The way quantities associate will not change the result of the operation. (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c). Commutative Property: (Addition, multiplication) The order of factors or multiples doesn’t change the sum or product. Identity Property: (Addition: 0, Multiplication: 1) Adding 0 to another number (or multiplying by 1) cannot change the final quantity. Strategies: Addition & Subtraction
Compensation: Turning a number into a ‘friendly’ number to perform an operation, then adjusting for this change. Students often use compensation intuitively once they have a good understanding of quantity and number order. Example:
Compose/decompose: Numbers can be broken down into their smaller parts to allow children to have greater facility with numbers and operations. Students may use place value or landmark (friendly, benchmark) numbers to break numbers apart, depending on the numbers and context. This will transfer to understandings with larger numbers in subsequent grades. 3
Comparison (subtraction): Some contexts do not lend themselves to modeling with a “take‐away” model. How many more? How many less? Both of these call for comparing to arrive at a difference. The following comparison reflects the original question as no one is taking away or removing cars. Begin with young students using concrete models. In this example, young students would be using small toy cars, line them up, compare and count how many to see the difference. As children develop mathematically, we move them to more abstract models for understanding operations (example below with number lines). The ultimate goal is to move toward fluency with abstract numbers. Sam has 12 cars. Josie has 16 cars. How many more does Josie Have?
Take Away (subtraction): Some contexts will involve ‘taking away’ a specified amount and ask about the remaining amount. A Note on Fluency: Fluency does not relate to a specific number of math facts correctly
performed in a specific amount of time. (In fact, research continues to show the practice of ‘timed
tests’ most often detrimental to student learning and understanding!) Students need to have
conceptual understanding of operations. Once they have this understanding teachers can help them
learn various ways of remembering these basic facts. Why does it work that way? What does the
student already know that will help them remember new facts? Building a connection between
concrete models and representations to more abstract numbers provides the foundation for
conceptual understanding and computational fluency. Look for strategies embedded in teaching
materials to help identify how students work toward fluency.
4
Number Talk: Ten Frame Flash: Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.47 Materials
• Mini Ten Frames • Chart Paper •
Download blacklines at www.ablongman.com/vandewalleseries Activity This activity helps children see groups of ten. This Math Talk provides a wonderful setting for checking for understanding, sharing ideas, and multiple strategies. Recording different ways of “seeing” and thinking (as we do in class) facilitates students’ understanding of a wider range of ideas and makes individual thinking available to others. When we don’t take the time to record the ideas, we lose out on these wonderful opportunities. Keeping the Math Talk posters will also give teachers ways to connect new knowledge and concepts to prior learning. Pull out a previous poster, “Remember when we…?” __________________________________________________________________________________________
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____________________________________________________________________________________ Activity A: Compatible Pairs Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.148 Materials
• Direction Cards • Compatible Pairs Record Sheet • Compatible Pairs Number Sheet Activity Students work with a partner, take turns, and find a combination of numbers that equals 50, 100, 1000, or some other choice. Students justify their choice to their partner, and then cover the two numbers with a cube if they’re correct. Players continue until they’ve matched all the pairs. __________________________________________________________________________________________
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Activity B: Groups of Ten Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.130 Materials
• Containers of Counters • Number word sheet • Recording sheet • Counting Mat Activity Students get a recording sheet and choose a container of counters. The student dumps out the contents, records the amount as a number word, and groups the objects in as many tens as possible. The final count is then recorded on the recording sheet. Children collect their objects and choose another container. __________________________________________________________________________________________
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____________________________________________________________________________________ Activity C: Three Other Ways
Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.133 Materials
• Number strips with written numbers • glue • Recording sheets • Items to represent hundreds, tens & ones (toothpicks, dots, color tiles, etc.) Activity Students, working in groups or pairs, get a recording sheet and choose a number strip. They glue the number strip on their recording sheet. Using standard representations (dots for ones, toothpicks for tens, & squares for hundreds), students record their number one way. Then they find three other ways to record them. __________________________________________________________________________________________
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Activity D: The Other Part of 100
Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.147 Materials
• Mini ten‐frame cards Activity Two students work together with min ten‐frame cards. One student makes a two‐digit number. Then both students work mentally to determine what would go with the first number to make 100. Students justify their answer. Each person takes turns choosing the first number. __________________________________________________________________________________________
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____________________________________________________________________________________ Activity E: Close to 100, Close to 1000
Grade 3, Grade 4 Investigations, 1st Edition (Pearson) Materials
• Game sheet • Numeral Cards 0‐9 Activity Six cards are dealt to each player. Each player uses any four of these cards to make two numbers, that when added, come as close as possible to a total of 100. __________________________________________________________________________________________
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Activity F: Solve Two Ways
Materials
• Sample problems • glue • Blank paper Activity Choose a problem. Glue it down (top) of a blank page. (Add name/date!) Estimate an answer. Jot this down – students should have an idea of a reasonable answer. Solve ‐ Is this a reasonable answer? Solve another way – reinforces multiple strategies and gives students another way to solve to verify. Write a story (context) to match the original __________________________________________________________________________________________
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NAME
DATE
Work with some of the activities. Try to get to at least 3. Identify the skill targeted (can you
match this to a grade level or standard(s)?) Identify ways the activity can be modified (for
both accelerated learning and depth).
The Other Part of 100
Three Other Ways
Compatible Pairs
Groups of Ten
Close to 100, Close to 1000
Name
Counters
glass blobs
Date
Number Word
Tens
Singles
buttons
Solve Two Ways!
Tens
Singles
square tile
Tens
Singles
toothpicks
Tens
Singles
lima beans
Tens
Singles
print
9
Number Talk: Ten Frame Flash
What number properties and strategies are evident in this activity?
10
A. Compatible Pairs
This activity can be done as a worksheet activity or with the full class using the
overhead projector. Prepare a transparency or duplicate a page with a search
task.
For today you will select a mat to work with and follow the rules
below:
Markers of any type can be used to cover number pairs…
11
Name
Date
Compatible Pairs Record Sheet
Make __________
Represent your thinking
=
Represent your thinking
=
Represent your thinking
=
Represent your thinking
=
12
Name
Date
Make __________
Compatible Pairs Record Sheet
Represent your thinking
+
=
Represent your thinking
+
=
Represent your thinking
+
=
Represent your thinking
+
=
13
Compatible Pairs!
Make 50
14
Compatible Pairs!
Make 1000
15
-
Compatible Pairs!
Make
?
16
Compatible Pairs!
Make
?
17
18
19
B. Groups of Ten
Choose a container of counters. Get a record sheet and write
your name on it
Dump out and count the contents
Number Words
Record the amount as a number word
eleven
ten
one
twelve
twenty
two
thirteen
thirty
three
fourteen
forth
four
fifteen
fifty
five
sixteen
sixty
six
seventeen
seventy
seven
eighteen
eighty
eight
nineteen
ninety
nine
Group the counters in as many tens as possible (use a counting mat to
help you!)
Name
Counter
s
glass
blob s
Date
Number
Word
Tens
Sin gles
buttons
Record this on your paper.
Tens
Sin gles
square
tile
Tens
Sin gles
toothpic
ks
Tens
lima bea
ns
Sin gles
Tens
Sin gles
Return the counters to the container.
20
Number Words
eleven
ten
one
twelve
twenty
two
thirteen
thirty
three
fourteen
forth
four
fifteen
fifty
five
sixteen
sixty
six
seventeen
seventy
seven
eighteen
eighty
eight
nineteen
ninety
nine
Groups of Ten Reference Page
21
Name
Counters
Date
Number Word
glass blobs
Tens
Singles
buttons
Tens
Singles
square tile
Tens
Singles
toothpicks
Tens
Singles
lima beans
Tens
Singles
Groups of Ten Record Sheet
22
10-frame check mats
23
C.
Three Other Ways
Students work in groups or pairs.
or
1. Draw out a strip with a written number. Read it with your partner(s).
Five hund venty-five
sed sevent
red re
y-five
hund
Five
2. Glue the strip of paper down on your record sheet.
Five hundred seventy-five
3. Build the number using standard representation (with your model)
4. Find and record three more ways to show this number.
24
Name
Date
Three Other Ways Record Sheet (Tens, Units)
glue written number here
Tens ______
Ones ______
Tens ______
Ones ______
Tens ______
Ones ______
Tens ______
Ones ______
Build the number with standard representation. Find three more ways!
glue written number here
Tens ______
Ones ______
Tens ______
Ones ______
Tens ______
Ones ______
Tens ______
Ones ______
glue written number here
Tens ______
Ones ______
Tens ______
Ones ______
Tens ______
Ones ______
Tens ______
Ones ______
25
Name
Date
Three Other Ways Record Sheet (Hundreds, Tens, Units)
Write Numeral ____________ + _____________ + ______________ = _______________
Use standard Notation and build to match. Next, Find at least three other ways to build the same number.
Hundreds _______
Tens ______
Hundreds _______
Tens ______
Hundreds _______
Tens ______
Hundreds _______
Tens ______
Ones ______
Ones ______
Ones ______
Ones ______
Write Numeral ____________ + _____________ + ______________ = _______________
Use standard Notation and build to match. Next, Find at least three other ways to build the same number.
Hundreds _______
Tens ______
Ones ______
Hundreds _______
Tens ______
Ones ______
Hundreds _______
Tens ______
Ones ______
Hundreds _______
Tens ______
Ones ______
Write Numeral ____________ + _____________ + ______________ = _______________
Use standard Notation and build to match. Next, Find at least three other ways to build the same number.
Hundreds _______
Tens ______
Ones ______
Hundreds _______
Tens ______
Ones ______
Hundreds _______
Tens ______
Ones ______
Hundreds _______
Tens ______
Ones ______
26
one hundred forty-four
eight hundred twenty-two
five hundred seven
three hundred eleven
two hundred eighty-nine
six hundred nineteen
nine hundred twenty-one
five hundred forty-seven
seven hundred fifty-one
six hundred thirty-four
two hundred sixty-three
three hundred twenty-eight
two hundred seventy-six
six hundred thirty-four
one hundred eighty-nine
one hundred forty-six
two hundred sixty-two
four hundred thirty-five
one hundred six
three hundred sixteen
one hundred ten
five hundred sixty
eight hundred
three hundred eighty-two
three hundred fifteen
two hundred
three hundred forty
six hundred fifty-two
six hundred ninety-six
four hundred seventy
seven hundred fourteen
seven hundred seventy-eight
five hundred thirty-nine
five hundred twenty-four
five hundred seventy-seven
seven hundred twenty-nine
four hundred ninety-four
one hundred eighty-six
seven hundred fifty-five
three hundred fifty-eight
27
forty-four
twenty-two
seven
eleven
eighty-nine
nineteen
twenty-one
forty-seven
fifty-one
thirty-four
sixty-three
twenty-eight
seventy-six
thirty-four
eighty-nine
forty-six
sixty-two
thirty-five
six
sixteen
one hundred ten
sixty
eight hundred
eighty-two
fifteen
forty-eight
forty
fifty-two
ninety-six
seventy
fourteen
seventy-eight
thirty-nine
twenty-four
d seventy-seven
twenty-nine
ninety-four
eighty-six
fifty-five
fifty-eight
28
D. The Other Part of 100
Two students work together with a set of little ten-frame cards.
1. One student makes a two-digit number.
2. Then both students work mentally to
goes with the ten-frame to make 100.
determine what
3. They write their solutions on paper and then check by making the
other part with the cards to see if the total is 100.
4. Students take turns making the starting number.
29
Name
Date
The Other Part of 100 Record Sheet
1
Represent first number
What’s the other part of 100?
Write the numeral
Write the numeral
Represent first number
What’s the other part of 100?
Write the numeral
Write the numeral
Represent first number
What’s the other part of 100?
Write the numeral
Write the numeral
Represent first number
What’s the other part of 100?
Write the numeral
Write the numeral
Justify
2
Justify
3
Justify
4
Justify
30
Investigations, 1st Edition, Grade 3 (Pearson)
+
+
= ______
= ______
32
+
+
= ______
= ______
33
F. Solve Two Ways!
Choose a problem and stick it to the top of your work paper.
Estimate an answer… write it down
Solve 2 different ways
Hmmm…is my answer “reasonable?”
Write a story (give the problem a context) to match the original
problem.
Note: Print number sentences on return address labels or cut apart (to be glued on to blank
paper)
34
354 + 537 =
307 - 289 =
1025 - ? = 375
789 + 1038 =
6521 - 387 =
73 + ? = 1000
1578 + 199 =
1000 – 271 =
100 – ? = 33.75
325+ 327+ 324=
1001 – 552 =
6.75 – ? =3.21
2957 + 123 =
352 - 175
10.25 – ? = 6.82
55 + 175 + 350=
210 – 35 =
9974 – ? = 55
724 + 1268 =
149 – 94 =
?- 1,581 = 16,785
562+625+256=
333 – 39 =
? + 526 = 2583
253+ 257+ 255=
17,458 – 873=
? – 73.2 = 2052
642 + 658 =
2952- 766 =
? – 0.671 = 372.6
35