Region 11: Math & Science Teacher Center Solving Equations

Region 11: Math & Science Teacher Center

Solving Equations

Where we’ve been…

Equality

Ratio & Proportion

Pattern Generalization

Patterns Review Patterns Review - Justify It!

3rd Grade MCA Practice Problem

1 4 7 10

Patterns Review Patterns Review - Justify It!

3rd Grade MCA Practice Problem

1 4 7 10

With a partner, talk about how a third grader would talk about this problem:

• Draw the next figure in the pattern

• How many dots will be in the next figure?

• Describe how make the pattern

Patterns ReviewPatterns Review

Adult Challenge: How many dots would be in the nth figure?

Patterns ReviewPatterns ReviewShare in Grade Level Groups:

• What strategies did you see kids use?

• What did the students find most challenging?

• What growth did you see?

• What surprised you?

Patterns ReviewPatterns ReviewWatch video and record:

Good questions Not-so-good questions

Patterns ReviewPatterns ReviewDiscuss video at your tables:

• What questions helped most to get your students to explain the explicit rule for different patterns?

• What did you learn about scaffolding questions?

•How do you know when you are done asking questions?

Solving EquationsSolving Equations

Goals:• Identify a developmental sequence for solving equations

• Structure “math talk “about expressions & equations

• Emphasize equivalence and how we communicate this with

students

• Discuss properties of equationsWork with a partner to solve problems on page

1

Solve.Solve.

Find all values that make the statement true.

xx +=+ 1753

Solve.Solve.

Find all values that make the statement true.

137 +=+++ bbbb

Solve.Solve.

Find all values that make the statement true.

92 =x

Solve.Solve.

Find all values that make the statement true.

7223 −≤ x

Solve.Solve.

Find all values that make the statement true.

5228 −−+ mm

Solve.Solve.

Find all values that make the statement true.

5=+ yx

What does solve mean?

Break

Think/Pair/Share…

• Define expression

• Define equation

Big Idea!

• We solve equations because we can make them true or false.

• We don’t solve expressions because we can’t make them either true or false.

Expression vs. EquationShare and discuss in your group as you work on page 2:

• How would you write the directions?

• How would you want/expect students to show their work?…What would you write on the board?

Expression vs. Equation

1345 += a

Expression vs. Equation

11753 +−− xx

Expression vs. Equation

)2(363 +=+ aa

Expression vs. Equation

€

1+ 2• 3

Expression vs. Equation

x=5

Expression vs. Equation

x840 =

€

5 = x

Expression vs. Equation

• Directions make a big difference!

• Directions depend on context and where you are in the curriculum

• Expressions are not equations

• No one “right way” to show work

• Most middle school textbook authors have thought carefully about what strategies to use to solve equations

CGI Algebra Video Clips

Benchmarks in Student Thinking

About The Equal SignDescription

1Children are asked to be specific about what they think the equal sign means

2Children accept as true some number sentence that is not of the form a + b = c

3

Children recognize that the equal sign represents a relation between two equal numbers, particularly through calculation (relational understanding of the equal sign)

4

Children are able to compare the mathematical expressions using relational thinking without carrying out the calculations (relational understanding across the equal sign)

Note: These benchmarks are a guide, not a firm sequence

CGI Algebra Video Clips

Solving the Equation

(4th grader, 2 minutes, 42 seconds)

1620 =−++ bbb

CGI Algebra Video Clips

Solving the Equation

(4th grader, 1 minutes, 51 seconds)

2013 +=++ kkk

CGI Algebra Video Clips

Solving the Equation

(4th grader, 39 seconds)

24+=++ eeee

CGI Algebra Video Clips

Solving the Equation

(4th grader, 51 seconds)

pp +=×824

Benchmarks in Student Thinking

About The Equal SignDescription

1Children are asked to be specific about what they think the equal sign means

2Children accept as true some number sentence that is not of the form a + b = c

3

Children recognize that the equal sign represents a relation between two equal numbers, particularly through calculation (relational understanding of the equal sign)

4

Children are able to compare the mathematical expressions using relational thinking without carrying out the calculations (relational understanding across the equal sign)

Note: These benchmarks are a guide, not a firm sequence

Lunch

Methods for solving linear Methods for solving linear equations of the form equations of the form axax ± ± bb = =

cxcx ± ± dd

Traditional Approach

vs.

Functions Approach

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

1) Use of number facts(solve mentally)

Example:

3 + x = 7

Not-so-good for:

3x + 7 = 5x – 14

2) Generate and evaluate (“guess and check” or “trial and error

substitution”)

Example:

2x + 3 = 4x – 7

Not-so-good for:

3x – 7 = 10 – 4x

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

3) a. Undoing (or working backwards)Example:

20 = 3x – 424 = 3 • x

8 = xNot-so-good for:

3x + 7 = 5x – 14

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

3) b. Undoing

17 = 3p – 1x 3 - 1

+ 1÷ 3

p 3p 3p – 117186

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

4) Cover-upExample:

k + k + 13 = k + 20k + k + 13 = k + 13 + 7

k = 7Not-so-good for:

3x + 7 = 25 – 5x

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

5) Transposing (change side-change sign)

Example:

3x = 85x = 15 x = 3

Not-so-good for:

– 7 – 2x+ 2x7 +

203

2=x

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

6) Equivalent equations (performing the same operation on both

sides)

Example: 17 = 3x – 7

17 + 7 = 3x – 7 + 724 = 3x

24/3 = 3x/38 = x

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

Group Task #1

• Break into six table groups

• Write one or two good problems for each method on the table

• When the bell rings, move to the next table

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

Group Task #2

• Move to the table where you started• Work the problems using that

particular method• Star any problems that can’t be

solved easily with the method• Determine the minimum benchmark

level needed to solve these problems

• Be ready to report out

Traditional approach for Traditional approach for solving linear equations of the solving linear equations of the

form form axax ± ± bb = = cxcx ± ± dd

Functions approach for solving Functions approach for solving linear equations of the form linear equations of the form axax

± ± bb = = cxcx ± ± dd1) Table using graphing

calculator (similar to guess and check)

Example:3x – 4 = x + 6

x = 5

Y1 Y2

Functions approach for solving Functions approach for solving linear equations of the form linear equations of the form axax

± ± bb = = cxcx ± ± dd2) GraphingExample:3x – 4 = x + 6

x = 5

Y1 Y2

Why standards?Why standards? “By viewing algebra as a strand in the curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for more-sophisticated work in algebra in the middle grades and high school.”

- NCTM, 2000, p.37

True or FalseTrue or FalseYour textbook determines the algebra concepts Your textbook determines the algebra concepts and skills that you should cover at a and skills that you should cover at a particular grade level.particular grade level.

False: In a standards-based system, the focus is shifted from what is TAUGHT to what is LEARNED.

The standards tell us what students should know and be able to do.

TRUE FALSE don’t know

True or FalseTrue or False Algebra content has been shifted Algebra content has been shifted down and now starts in the middle down and now starts in the middle grades.grades.

False: Algebra and algebraic thinking are integrated across K-11 in the state standards. Every teacher has to do his/her part to give students the opportunity to learn the grade-level content.

TRUE FALSE don’t know

Sort the Standards Involving Solving

Equations

In your groups,

• Look through individual standards• Classify for grades 1 - 8• Ask for an answer key when finished

• Take time to reflect on standards

with your group members

BIG IDEA

TRADITIONAL ALGEBRA I

TRADITIONAL ALGEBRA I

8th GRADE STANDARDS

BIG IDEA

Break

Equivalence - Equivalence - ExpressionsExpressions

 On page 7: • State directions for each problem

• Simplify each expression or equation one operation at a time, leaving a trail down of equivalent expressions or equivalent equations.

Equivalence - Equivalence - ExpressionsExpressions

 Directions:

)352(37 ×+−

Equivalence - Equivalence - ExpressionsExpressions

 Directions:

2)573(23 ++−− xx

Equivalence - Equivalence - ExpressionsExpressions

 Directions:  

Check:

mmm 5)5(2143 +−=+

Equivalence - Equivalence - ExpressionsExpressions

Expressions are equivalent…

…if every line has the same value for the same value of x

Equivalence - Equivalence - ExpressionsExpressions

Equations are equivalent…

…if they have the same solution set (but each line has a different value for a given value of x).

Equivalence - Equivalence - ExpressionsExpressions

 Can I add 5? … 10?

 78133 −=+ xx

STRETCH

Balance metaphor for equationsBalance metaphor for equations

Same weight on both sides

page 8 of the handout

Balance metaphor for equationsBalance metaphor for equations

Build a balance to solve:

4 x + 4 = 12

Balance metaphor for equationsBalance metaphor for equations

Build a balance to solve:

x + y = 7

Balance metaphor for equationsBalance metaphor for equations

Build a balance to solve:

2 x + 4 = 2 x + 9

Problems with balance metaphorProblems with balance metaphor

Negatives

Problems with balance metaphorsProblems with balance metaphors

Subtraction

15653 −=+− xxx

Metaphors for equations – Metaphors for equations – Balance (same weight on both sides)Balance (same weight on both sides)

1. Open Number sentences (CGI)

Metaphors for equations – Metaphors for equations – Balance (same weight on both sides)Balance (same weight on both sides)

ii.Algebra Tiles (McDougal Littell) or Bags of Gold (CMP)

=

Algebra Tiles – McDougal Littell (Course 2)

x

x

x

CMP – Moving Straight Ahead (7th grade)

Metaphors for equations – Metaphors for equations – Balance (same weight on both sides)Balance (same weight on both sides)

3. Equation Mat (CPM – Algebra Connections)

xx

x

x

x

- -

++

= -1

= +1

xx 3312 −−=−

STRETCH

Baseline Assessment

Baseline AssessmentBaseline Assessment

1. Find the value of m that makes the number sentence below true.

12 = 4 • m

Baseline AssessmentBaseline Assessment

2. Find the value of b that makes the number sentence below true.

15 + 3b = 42

Baseline AssessmentBaseline Assessment

3. Find the value of x that makes the number sentence below true.

12x - 10 = 6x + 32

Baseline AssessmentBaseline Assessment

4. Find the value of n that makes the number sentence below true. Show your steps to demonstrate how you solved the problem.

4 + n - 2 + 5 = 11 + 3 + 5

Baseline AssessmentBaseline Assessment

5. Balance A is balanced. The amount on the left side of Balance A is tripled for Balance B. Draw in what should appear on the right side of Balance B to be balanced.

Explain why your answer works.

?

Balance A Balance B

Baseline AssessmentBaseline Assessment

6. Determine which of the equations below are equivalent to: 3b – 4 = b + 6 Circle yes, no or do not know for each part.

Equivalent to 3b – 4 = b + 6

a) 3b + 4 = b – 6 yes no do not know

b) 3b – 4 + 7 = b + 6 + 7 yes no do not know

c) 2b – 4 = 6 yes no do not know

d) 4b – 3 = 6 + b yes no do not know

Properties of EquationsProperties of Equations

Addition Property of Equality

Words

Adding the same number to each side of an equation produces an equivalent equation

Algebra

from McDougal Littell Math Course 2

Properties of EquationsProperties of Equations

What other properties maintain equivalence?

Words

Algebra

Big Ideas

We want to:

• Learn the different methods that children use to solve

linear equations

• Learn how to select examples to push kids from less sophisticated methods (i.e. guess and check) to more algebraic

methods

• Learn different metaphors for helping students solve equations

Evaluation

On an index card, please record:

P: one positive from today’s work

M: one ‘minus’ or concern from today’s work

I: something that you found interesting or intriguing from today’s work.

Thanks for your feedback.