Algebraic ThinkingDecember 8th, 2010

SD Counts Cohort

Platte, SD

Roxane Dyk

• Share activities you created

• Equality/relational thinking/operations

What has happened since the last time we met, either inside/outside your classroom?

Do this Activity(Order of Operations)

• 1 + 20 x (6 + 2) / 2 =• 1 + 20 x 8 / 2 =• 1 + 160 / 2 =• 1 + 80 =

ComicExcuses

• Ten Commandments of Math

• 10 Excuses for Not Doing Homework

• Algebra Definitions

• Math Books for Sale

Ground Rules

. Honor private think time.Be prepared for sessions, everyone should have something to contribute.Listen carefully to take in another participant’s ideas.Cell phones off or on vibrate.Start and end each session on time.Take care of your needs.

Problem-solving Activity

• During Marie’s birthday party, Marie thinks about the ages of her parents and herself. Marie says, “Hey, Mom and Dad, together your ages add up to 100 years!” Her dad is surprised, “You are right,” he says, “and your age and mine total 64 years.” Marie replies, “And my age and Mom’s total 58.” How old are Mom, Dad and Marie?

• Explain how you got your answer

Outcomes for today’s session

• Celebrate 50 Years of Life

• Review equality, relational thinking, operations

• Begin to understand concepts of early algebra

Early Algebra

• In grades K-3 these are the concepts of Algebra that need to be covered and covered well– Equality– Relational thinking– Basic number properties

EqualityRelational Thinking

• Pull out the handout in your packets• Revisit “Big Ideas”• What are/have you been doing in your

classroom to “foster” these concepts?– Talk to your neighbor

• What did you find out about your students relational thinking skills?– Talk to your neighbor

• How have you continued to address each item?

Video Clip 1.6 EqualityTitle 6-Chapter 1

• This is a teacher working with a group of students

• Listen to what the teacher does before she allows the students to explain.

• Pay attention to the questions the teacher asks

• What did you notice?

Video Clip 2.5 Relational ThinkingTitle 11-Title 1

• Watch to the type of problem the teacher poses and listen to the students explain

Properties in Math

We must take the time to make sure our students know and

understand

Big Ideas of “Early Algebra”

• These are ideas that you want to know and understand about teaching algebra concepts in elementary school

• Read the handout individually

• Talk to your table and discuss what you believe each idea is representing

Early Algebra Concepts

#1-This will depend on your instructional materials.

Does your series have

-specific algebra units-taught

in isolation

-concepts embedded

-no algebra at all

Problem-solving situations

• #2

- Good examples are hard to find if you

don’t have them in your book. Once

you find good examples; just change

the numbers

*Sample Algebraic Sense Problems

How do we approach “algebra”

• #3– “Today we are going to use what we know

about numbers”

NOT– “Today we are doing algebra”

• Tell them they were just doing algebra when they are finished

Box to Variables

#4

Moving away from the

Mathematical Convention

• Because of the “mathematician's rule”..

When the same variable appears more than one time in a number sentence, the variable must be replaced by the same number throughout the number sentence.

Caged Mice

• Work with your table• Number yourselves of 1-2-3 (see posted

job descriptions)• Identify all of the combinations of mice in a

cage that you would find.• Be ready to prove to me how you know

you have found all of the combinations• Be ready to discuss the different ways to

record your answers

The Caged Mice Problem

• Ricky has 7 pet mice. He keeps them in two cages. One cage is red and the other is blue. Show all the ways that 7 mice can be in these two cages.

Combinations

• Does anyone know the number of combinations there would be for “n” mice and how do you know?

• What would the number sentence look like for this combination?

• Title III book has more activities that go with finding combinations

Early Algebra Number Sentences

• We are going to work through some early algebra number sentences.

• Work with a partner to solve these problems using ONLY your understanding of equality, relational thinking, basic number properties and mathematician’s rules. NO FORMAL ALGEBRA

• Solve these problems the way 2nd-6th graders would solve.

Number Sentence

What can we put in for s and t so that this number sentence is true:

s + t = 8

You must work with a lot of sentences helping your students understand letters as variables.

Read Page 71

Number Sentence

What if I have 2 letters that are the same?

s + s=10

Mathematician’s Rule-if variables are the same the number’s used are the same

Read Page 72

Solving Equations

Once students have learned the rule for dealing with number sentences with repeated variables, they can

attempt to solve number sentences that challenge their mathematical thinking.

The goal in giving students these types of number sentences is not to teach students efficient ways to solve

algebra equations; it is to engage them in thinking flexibly about number operations and relations.

d + d – 5 = 13

Read Page 73

Lets watch some students solve equations….

• Video 4.1-4th grader-2’42”

• Video 4.2-4th grader-1’51”

• Video 4.3-4th grader-’39”

• Video 4.4-4th grader-’51”

Challenge Problems

• Page 77

• Think about the different ways students will use their understanding of equality, relational thinking, basis number properties and the mathematician’s rule to solve these problems

Article

• “Algebraic Problem-Solving in the Primary Grades”

• Read individually

• Share with your partner

• Share large group

Next meeting

• January 18th 8:30-3:30• Conjectures and Justification