Workshop 2: Solving Equa5ons NCTM Interac5ve Ins5tute, 2016 Name - - PowerPoint PPT Presentation

Workshop 2: Solving Equa5ons

NCTM Interac5ve Ins5tute, 2016

Name

Title/Posi5on Affilia5on Email Address

Warm Up

List these expressions from least to greatest: 2n 2n + 1 2(n + 1) 2n – 1 2(n – 1)

Reflec5on

What would students need to understand in

• rder to solve the warm up?

3

Common Core Standards

This session will address the following:

4

7.EE.1 Apply properGes of operaGons as strategies to add, subtract, factor, and expand linear expressions with raGonal coefficients. 7.EE.4 Use variables to represent quanGGes in a real-world

• r mathemaGcal problem, and construct simple

equaGons and inequaliGes by reasoning about the quanGGes.

Solving Equa5ons

Think about the instrucGonal sequence you use in teaching how to solve an equaGon. What do students do in the first lessons? What are criGcal benchmarks or ideas that students progress through in the instrucGonal sequence?

5

Algebra Magic

• Think of a number.
• MulGply the number by 3.
• Add 8 more than the
• riginal number.
• Divide by 4.
• Subtract the original

number. Compare your answer to

• thers at your table.

Why did this happen? Find 2 different ways to explain it.

6

Algebra Magic

What could be done to the steps in order to get the number you started with?

• Think of a number.
• MulGply the number by 3.
• Add 8 more than the original number.
• Divide by 4.
• Subtract the original number.

7

Wri5ng Expressions

• Enter the first three digits of your phone number.
• MulGply by 80.
• Add 1.
• MulGply by 250.
• Add the last four digits of your phone number.
• Repeat the above step.
• Subtract 250.
• Divide by 2.

Describe the number you have. How did the problem work?

8

Algebra Magic

Which of the following steps can you reverse without changing the result? Why? 1) Think of a number. 2) Subtract 7. 3) Add 3 more than the original number. 4) Add 4. 5) MulGply by 3. 6) Divide by 6.

9

Algebra Magic

The following trick is missing the last step.

• Think of a number.
• Take its opposite.
• MulGply by 2.
• Subtract 2.
• Divide by 2.
• ??????????

Decide what the last step should be for the given condiGon so final result is: a) One more than

• riginal number.

b) Opposite of original number. c) Always 0. d) Always -1.

10

Matching Expressions, Words, Tables, & Areas

Work collaboraGvely with your tablemates.

• Match cards to make a set with an expression,

words, table, and area card.

• If there is not a complete set, make a card for

the missing type(s) with one of the blank cards.

11

Matching Expressions, Words, Tables, & Areas

Large group discussion:

• Which, if any, of the groups of expressions are

equivalent to each other? How do you know?

• What will students learn as a result of this

acGvity?

• What challenges might student encounter

with this acGvity?

12

Expressions to Equa5ons

8 + 4 = + 7

What responses do students give for box?

13

A major misunderstanding

• Many students do not understand the equals

sign.

• They believe it signifies that the answer

comes next.

2x – 8 = 4x + 6

Equal Sign–Two Levels of Understanding

Opera5onal: Students see the equal sign as signaling something they must

“do” with the numbers such as “give me the answer.”

Rela5onal: Students see the equal sign as indicaGng two

quanGGes are equivalent, they represent the same

• amount. More advanced relaGonal thinking will lead to students

generalizing rather than actually compuGng the individual amounts. They see the equal sign as relaGng to “greater than,” “less than,” and “not equal to.”

Knuth, E. et. al (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13, 514–519.

Why is understanding the equal sign important?

Knuth, E. et. al (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13, 514–519.

Transi5oning to Rela5onal Thinking

True or False: 471 – 382 = 474 – 385 674 – 389 = 664 – 379 583 – 529 = 83 – 29 37 x 54 = 38 x 53 5 x 84 = 10 x 42 64 ÷ 14 = 32 ÷ 28 42 ÷ 16 = 84 ÷ 32

17

• No calculators – No computations
• Use relational thinking to justify answer.

Transi5oning to Rela5onal Thinking

What is the value of variable? 73 + 56 = 71 + d 67 – 49 = c – 46 234 + 578 = 234 + 576 + d 94 + 87 – 38 = 94 + 85 – 39 + f 92 – 57 = 94 – 56 + g 68 + 58 = 57 + 69 – b 56 – 23 = 59 – 25 – s

18

• No calculators – No computations
• Use relational thinking to justify answer.

Solving Equa5ons

An equaGon states that two expressions are equivalent for certain values of a variable. EquaGons become useful in invesGgaGng relaGonships between two expressions.

19

Solving Equa5ons

• Many curriculum materials begin with

equaGons like this: 14 – w = 9

Foegen, A. & Dougherty, B. J. (2013). Algebra screening and progress monitoring study.

Solving Equa5ons

14 – w = 9 48% of students (1615) got it correct. (2nd grade CCSSM standard)

Foegen, A. & Dougherty, B. J. (2013). Algebra screening and progress monitoring study.

Solving Equa5ons

Rather than starGng with ‘easy’ equaGons and applying algebraic manipulaGons, let’s consider a developmental approach.

5 + x = 12 5 – 5 + x = 12 – 5 x = 7

Solving Equa5ons

5 + x = 12 What number added to 5 equals 12? What basic fact do you know that could tell you the missing addend?

Solving Equa5ons

When you see an equaGon like this, what are 3

• ther related equaGons you could write?

5 + x = 12

Solving Equa5ons

When you see an equaGon like this, what are 3

• ther related equaGons you could write?

5 + x = 12

5 + x = 12 x + 5 = 12 12 – 5 = x 12 – x = 5

Solving Equa5on

Diagrams with manipulaGves are another way that can support students’ understanding of solving equaGons.

26

Solving Equa5ons

Work with a partner at your table to complete the lab. Be prepared to share your ideas.

27

Solving Equa5ons

3x + 2 = 4x – 3

Solving Equa5ons

Graph 3x + 2 = 4x – 3 Use your graphing calculator to graph the two expressions. How would you idenGfy the soluGon?

Solving Equa5ons

Graphing 3x + 2 = 4x – 3

Solving Equa5ons

• 1. Logical reasoning/inspecGon
• 2. Fact families/inverse operaGons
• 3. Physical materials/diagrams
• 4. Tables
• 5. Graphing

31

Solving Equa5ons

How would you solve 3x + 2 = 4x – 3 using algebraic steps?

Solving Equa5ons

3x + 2 = 4x – 3 3x + 2 + 3 = 4x – 3 + 3 3x + 5 = 4x 3x – 3x + 5 = 4x – 3x 5 = x

33

Solving Equa5ons

3x + 2 = 4x – 3 3x + 5 = 4x A3 5 = x S3x

Solving Equa5ons

A: Add S: Subtract M: MulGply D: Divide CLT: Combine Like Terms DPMA: DistribuGve Property of MulGplicaGon

• ver AddiGon

35

Why is it important to understand solving equa5ons

Dan challenged Amy to write an equaGon that has a soluGon of 3. Which equaGon could Amy have wriken?

• a. 4 – x = 10 – 3x
• b. 3 + x = –(x + 3)
• c. –2x = 6
• d. x + 2 = 3

Sample of student work

Dan challenged Amy to write an equaGon that has a soluGon of 3. Which equaGon could Amy have wriken?

• a. 4 – x = 10 – 3x
• b. 3 + x = –(x + 3)
• c. –2x = 6
• d. x + 2 = 3

Reflec5on

• What new idea(s) do you want to implement

into your classroom?

• What challenges did you encounter during this

session?

38

Reflec5on

(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 47)

Reflec5on

(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 48)

Disclaimer

The National Council of Teachers of Mathematics is a public voice

• f mathematics education, providing vision, leadership, and

professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all

• students. NCTM’s Institutes, an official professional development
• ffering of the National Council of Teachers of Mathematics,

supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of

• viewpoints. The views expressed or implied in the Institutes,

unless otherwise noted, should not be interpreted as official positions of the Council.

41

42