Longwood University Professional Development Seminar

Algebra, Number Sense, and Mathematical Connections

in Grades 3-5

ResourcesBlanton, Maria, et. al. Developing Essential

Understanding of Algebraic Thinking: Grades 3-5. Reston, Va.: National Council of Teachers of Mathematics, 2011

Jacob, Bill, and Catherine Twomey Fosnot. The California Frog-Jumping Contest: Algebra. Portsmouth, NH: Heinemann, 2007

Russell, Susan Jo, Deborah Schifter, & Virginia Bastable. Connecting Arithmetic to Algebra. Portsmouth, NH: Heinemann, 2011

ResourcesWickett, Maryann, et. al. Lessons for

Algebraic Thinking: Grades 3-5. Sausalito, CA: Math Solutions Publications, 2002.

Friel, Susan, et. al. Navigating through Algebra in Grades 6-8. Reston, VA: National Council of Teachers of Mathematics, ,2001.

Constructing Algebra

“Manipulation of numbers to produce an answer

can seem like a magic trick to learners if they haven’t constructed the implicit relations for themselves. The importance of this construction cannot be overemphasized, because it is precisely what enables learners to generalize”

(p. 15).

----Fosnot & Jacob (2010). Young mathematicians at work: Constructing Algebra. Portsmouth, NH: Heinemann.

The Magic TrickChoose any number (56)Add 3 (59)Multiply by 2 (118)Add 4 (122)Divide by 2 (61)Subtract the original number (61-56 = 5)The result is always 5

----Reimer (1992). Historical Connections in Mathematics: Resources for Using History of Mathematics in the Classroom. Aims Education Foundation.

Where is the magic?Choose a number (x)Add 3 (x + 3)Multiply by 2 2(x+3)Add 4 2(x+3) + 4Divide by 2 (x+3) + 2Subtract the number x + 3 + 2 –

xThe result is 5

Another trickChoose any number.Add 5. Double the result.Subtract 4.Divide by 2.Subtract the original number.Use algebra to show why the result is 3.

----Reimer (1992). Historical Connections in Mathematics: Resources for Using History of Mathematics in the Classroom. Aims Education Foundation.

Relational versus Instrumental Understanding

Big Idea 4Quantitative reasoning extends relationships between and among quantities to describe and generalize relationships among these quantities.

---Blanton, et al., 2011, p. 39.

Conceptual Foundation for Algebra“The experience of performing direct comparisons with physical materials and then representing them with algebraic [symbolic] statements builds a conceptual foundation that allows us to act on these symbolic statements. When symbolic or algebraic statements are introduced with no specific context, little meaning can be attached to them” (p. 43).

--Blanton et al., 2011

Essential Understanding 4b

“Known relationships between two quantities can be used as a basis for describing relationships with other quantities” (p. 42).

--Blanton et al., 2011

Algebraic SubstitutionSubstitution can be used to find a new

relationship between two quantities.

10 apples weigh the same as 2 melons.

How many melons would weigh the same as 25 apples? How do you know?

More SubstitutionSubstitution can be used to evaluate

an expression.

If M = 5, K = 8, and L = 14

Find the value of L + M – K

+ = 82 - 10 = 58

What is the value of ?How do you know?X + Y = 53Y – 14 = 36What is the value of x?How do you know?

More Substitutions

Transitive PropertyThe transitive property says that if a =b and

b = c, then a = c.

2 + = = 3 + 5 so…2 + = 3 + 5 so

What is the value of ? How do you know?

Open Sentences and the Transitive Property

14 +z= yy = 7 x 7

What is the value of z? How do you know?

“The flexibility in representing equality relationships sets the stage for solving equations” (Blanton, et al., p. 45)

Using a Double Open Numberline

Day Two: Jumping Buddies; Complete Appendix D (Jacob & Fosnot, 2007, pp. 21-26)

‘Jump and step’ on classroom number line

Draw ‘jumps and steps’ on a double open number line for Appendix D activity

Using double open number lines to represent & solve equations

Day Six & Day Seven: The Frog-Jumping Contest; Complete Appendix I (Jacob & Fosnot, 2007, pp. 45-53)

‘Jump and step’ on classroom number line

Draw ‘jumps and steps’ on a double open number line for 3 different frogs: Sunny, Cal, and Legs (see Appendix I activity)

Frog Jump OlympicsDivide into teams, with each team selecting one of the following activities and record your work on poster paper:

Appendix J: “Pairs Competition”

Appendix K: “Hitting the Mark”

Appendix L: “The Toads’ Three-jump, Two-step Event”

(Jacob & Fosnot, 2007, pp. 21-26)

Using Algebra Tiles Now that you have solved these

problems with the open number line try it with the Algebra tiles.

A representation-based proof connects the story to the symbolism. Using more than one representation strengthens student understanding of these connections.

Equivalence: Relationships Between Quantities

“These ideas [using a variety of representations or models] about the relationships of quantities are important steps in developing the ability to make sense of problems that have no story or real-world context… without this relational understanding of the equal sign, [students miss looking for and reasoning about]… the value that makes both quantities the same” (p. 45).

--Blanton et al., 2011

Big Idea 5Functional thinking includes generalizing relationships between covarying

quantities, expressing those relationships in words, symbols, tables, or graphs, and reasoning with these various representations to analyze function behavior.

---Blanton, et al., 2011, p. 47.

From NCTM Principles and Standards: The Teaching Principle

“Worthwhile tasks should be intriguing, with a level of challenge that invites speculation and hard work.

Such tasks often can be approached in more than one way, such as using an arithmetic counting approach, drawing a geometric diagram and enumerating possibilities… tasks accessible to students with varied prior knowledge and experience” (p. 19).

--- NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

Algebraic Thinking

“Algebraic thinking encompasses the set of

understandings that are needed to interpret

the world by translating information or events

into the

language of mathematics in order to explain

and predict phenomena…”

--- Lawrence & Hennessy (2002). Lessons for algebraic thinking: Grades 6-8. Sausalito, CA: Math Solutions Publications.

Algebraic Thinking

Applying these understandings effectively often

requires the following components:

Using or setting up a mathematical model, if needed

Gathering and recording data, if neededOrganizing data and looking for patternsDescribing and extending those patternsGeneralizing findings, often into a ruleUsing findings, including any rules, to make

predictions” (p. xi).--- Lawrence & Hennessy (2002). Lessons for algebraic

thinking: Grades 6-8. Sausalito, CA: Math Solutions Publications.

Essential Understanding 5a

“A function is a special mathematical relationship between two sets, where each element from one set, called the domain, is related uniquely to an element of the second set” (the range) (p. 48).

---Blanton, et al., 2011, p. 48.

Are these functions?1. {(father, children)} 2. {(Parker, his age)} 3. {(Jill, her height)}

Create your own set that is a function. Then create your own set that is not a function.

--Collins, A. and Dacey, L. (2011). The Xs and Whys of Algebra: Key Ideas and Common Misconceptions. Portland, Maine: Stenhouse.

Piles of Tiles: A growing pattern that is a function too!

Pile 1 Pile 2 Pile 3

6 blocks 10 blocks 14 blocks

Wickett, et. al., 2002, p. 197

Essential Understanding 5d“In working with functions, several important types of patterns or relationships might be observed among quantities that vary in relation to each other: recursive patterns, co-variational relationships, and correspondence rules” (p. 51).

---Blanton, et al., 2011

Two of EverythingWritten Lily Hong

Recursive Patterns2, 6, 10, 14, 18, ….

What type of pattern is this? Growing pattern

What is the pattern? +4 pattern

This is a recursive pattern because we use the previous value to determine the next value in the pattern.

Recursive Patterns with a Magic Pot

In Out

1 3

2 5

3 7

4 9

5

+2

+2

+2

+2

1010??Wickett, et. al., 2002, p. 12

Recognizing Co-variationwith the same Magic Pot

In Out

1 3

2 5

3 7

4 9

5

+2

+2

+2

+2

1010??

+1

+1

+1

+1

Wickett, et. al., 2002, p. 12

How would you describe this pattern using recursive thinking? Using co-variation?

In Out

10

8

84

2 4

6 12

20

16

It is still difficult to predict the number of coins that will come out when 100 coins are put in.

Wickett, et. al., 2002, p. 8

Shifting from Repeating Patterns Toward Constructing Generalizations

Tabular (T-chart) approach: Numeric a. Describe patterns in recursive

relationships: how outputs are related to one another (look vertically, how change occurs from one stage to the next stage); also called an iterative relationship

b. Describe patterns in functional relationships: how each input is related to the corresponding output (look horizontally, find a rule based on the stage or term); also called an explicit relationship

---Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6-10. Portsmouth, NH: Heinemann.

Correspondence and Two of Everything

In Out

5

4

42

1 2

3 6

10

8

Wickett, et. al., 2002, p. 8

Can you describe this pattern as a correspondence between the In and Out values?

In Out

5

4

52

1 3

3 7

11

9

How many coins come out of this magic pot if 100 coins are put in?

How would you describe this pattern using correspondence?

In Out

10

8

84

2 4

6 12

20

16

This kind of thinking makes it much easier to predict how many coins will come out of the pot when 100 are put in the pot.

Make your own magic pot ruleMake your own table of values for a magic

pot.

Describe your pattern as a correspondence (also called a function).

Switch your pattern with a partner and see if they can describe your pattern as a function. Wickett, et. al., 2002, pp. 21-22

Essential Understanding 5e

“Functions can be represented in a variety of forms, including words, symbols, tables, and graphs” (p. 55).

---Blanton, et al., 2011

Task: Do the Following for all three Banquet Table Patterns

Examine each table pattern and extend the pattern to 4 table arrangements.

Describe the pattern between the # of tables and # of people in words.

Make a table with the # of tables and # of people.Write an algebraic equation for the

correspondence between the number of tables and the number of people

Use the correspondence to find the number of people who can be seated around 50 triangular tables, 40 square tables, and 35 hexagon tables.

Two Banquet table patterns: Triangles and Squares

Wickett, et. al., 2002, p. 224-238

One More Banquet table pattern: Hexagons

Wickett, et. al., 2002, p. 239

Tables for Table PatternsTriangle Table

Pattern

# of Tables

# of People

1 3

2 4

3 5

4 6

Square Table Pattern

# of Tables

# of People

1 4

2 6

3 8

4 10

Hexagon Table Pattern

# of Tables

# of People

1 6

2 10

3 14

4 18

Reflecting on Table PatternsCompare your patterns. How are they

alike? How are they different?

What if we created a new table to help us with our comparison?

Why is there always a +2 in the pattern?

Concluding the Table Pattern

Correspondence for number of sides and number of

peopleP = (s– 2)n +2

Number of sides on a single table (s)

Number of People (p) where n is the number of tables

3 n + 24 2n + 26 4n + 2

Task: Building with Toothpicks1. Build each shape listed on the handout. Use a ‘T-chart’ (table) to

record the perimeter of each shape. 2. Next, predict the perimeter of the 5th shape in the sequence.

Explain your prediction. Build the shape to verify your prediction. 3. What stays the same (constant) moving from shape to shape (or,

‘stage to stage,’ or ‘term to term’)? What changes (variable)?4. Write at least one rule to find the perimeter for any shape n and

justify your reasoning.5. What is the perimeter for shape 30?6. How did you use additive thinking? Or, did you use multiplicative

thinking? Both? Describe in words your thinking.7. Rebuild the shapes making only the perimeter with toothpicks.

What changes? What stays the same? Describe any new insights for the problem.

---Friel, et. al., 2002, p. 75

Mathematize the ‘Building with Toothpicks’ problem

What are the ‘big mathematics ideas’ of this problem?

Identify different ‘models’ that can be used to solve this problem.

What ‘strategies’ may be used to solve this problem?

What specific VA Standards of Learning are connected to this problem? Make a list.

Be ready to share in whole-group discussion.

Teaching Task: Analyzing Student Work

Examine each of the students’ responses.

Describe the thinking of each student … What relationships did each student find? Are they correct? Why or Why not? Any surprises? As a teacher, what would you do with each student to follow-up on his/her specific work?

---Friel, et. al., 2002, pp. 14-16

Our Focus: 5 Big Ideas The properties of arithmetic used with the four operations are

the same for algebra.

An equation uses an ‘equal sign’ to show that two quantities are equivalent.

A variable is a tool that describes relationships in many different ways.

Quantitative reasoning extends relationships between and among quantities.

Functional relationships are expressed in words, symbols, tables, or graphs, and reasoning occurs between these representations.

1. ---Blanton, et al., 2011, pp. 12-14.

What is mathematics?“Mathematics is a way of thinking that involves studying patterns, making conjectures, looking for underlying structure and regularity, identifying and describing relationships, and developing mathematical arguments to show when and why these relationships hold.”

---Russell, Schifter, & Bastable, (2011), p. 2.

Classroom Talk… Promotes Student Understanding

Classroom dialogue may provide direct access to ideas, relationships among those ideas, strategies, procedures, facts, mathematical history, and more…

Classroom dialogue also supports student learning indirectly, through the building of a social environment—a community—that encourages learning… students are encouraged to treat one another as equal partners in thinking, conjecturing, exploring, and sharing ideas.

---Chapin, S.H., O’Connor, C., & Anderson, N.C. (2009). Classroom Discussions: Using Math Talk To Help Students Learn, Grades 1–6, p. 6.

Five Productive Teacher Talk Moves

1. Revoicing: To help with the lack of clarity in a student’s verbalization of his/her ideas; e.g., “So you’re saying that it’s an odd number?”

2. Repeating: Asking students to restate someone else’s reasoning; e.g., “Can you repeat what he just said in your own words?”

3. Reasoning: Asking students to apply their own reasoning to someone else’s reasoning; e.g., “Do you agree or disagree, and why?”

Five Productive Teacher Talk Moves4. Adding On: Prompting students for

further participation; e.g., “Would someone like to add something more to this?” or, Who would like to add something more to that response?

5. Waiting: Using wait time; e.g., “Take your time… we’ll wait…”

---Chapin, S.H., O’Connor, C., & Anderson, N.C. (2009). Classroom Discussions: Using Math Talk

To Help Students Learn, Grades 1–6, p. 13.

Three Productive Talk Formats

Whole-Class Discussion

Small-Group Discussion Partner Talk

---Chapin, S.H., O’Connor, C., & Anderson, N.C. (2009). Classroom Discussions: Using Math Talk To Help Students Learn, Grades 1–6, p. 18.

24 ÷ 8 Partitive (Sharing): Put 24 things into 8 groups; how many in each

group?

24 ÷ 8 Measurement or Quotative(Grouping): Make

groups of 8; how many groups are there?

24 ÷ 8

The divisor is the number of groups. How many in each group?

Story context:

I made 24 cookies to share among 8 people. How many cookies will each person receive?

24 ÷ 8

The divisor is the number in each group.

How many groups?

Story context:

I have 24 cookies to put in bags of 8.

How many bags will I fill?

24 ÷ 8

24 ÷ 3

24 ÷ 1/2

24 ÷ 1/4