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 : 2 n 2 n + 1 2( n + 1) 2 n – 1 2( n – 1)

Reflec5on What would students need to understand in order to solve the warm up? 3

Common Core Standards This session will address the following: 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 or mathemaGcal problem, and construct simple equaGons and inequaliGes by reasoning about the quanGGes. 4

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 original number. • Divide by 4. • Subtract the original number. Compare your answer to others 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 Decide what the last step The following trick is should be for the given missing the last step. condiGon so final result is: • Think of a number. a) One more than • Take its opposite. original number. • MulGply by 2. b) Opposite of original • Subtract 2. number. • Divide by 2. 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. 2 x – 8 = 4 x + 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 : S tudents 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 • No calculators – No computations • Use relational thinking to justify answer. 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

Transi5oning to Rela5onal Thinking • No calculators – No computations • Use relational thinking to justify answer. 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

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. (2 nd 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 other related equaGons you could write? 5 + x = 12

Solving Equa5ons When you see an equaGon like this, what are 3 other 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 3 x + 2 = 4 x – 3

Solving Equa5ons Graph 3 x + 2 = 4 x – 3 Use your graphing calculator to graph the two expressions. How would you idenGfy the soluGon?

Solving Equa5ons Graphing 3 x + 2 = 4 x – 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 3 x + 2 = 4 x – 3 using algebraic steps?

Solving Equa5ons 3 x + 2 = 4 x – 3 3 x + 2 + 3 = 4 x – 3 + 3 3 x + 5 = 4 x 3 x – 3 x + 5 = 4 x – 3 x 5 = x 33

Solving Equa5ons 3 x + 2 = 4 x – 3 3 x + 5 = 4x A 3 5 = x S 3 x

Solving Equa5ons A: Add S: Subtract M: MulGply D: Divide CLT: Combine Like Terms DPMA: DistribuGve Property of MulGplicaGon over 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 – 3 x b. 3 + x = –( x + 3) c. –2 x = 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 – 3 x b. 3 + x = –( x + 3) c. –2 x = 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 of 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 offering 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