Using Tape Diagrams and Variables to Develop Algebra Skills Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Thursday, September 11, 2014 4:30 – 7:30

Singapore Math: Simple or Complex Focus your discussion of this article around these two quotes: 1. “Using the bar model approach, Singapore textbooks enable students to solve difficult math problems and learn how to think symbolically.” 2.”It (bar models) communicates graphically and instantly the information that the learner already knows, and it shows the student how to use that information to solve the problem.”

Learning Intentions and Success Criteria We are learning to: understand how representing problems with tape diagrams lead to algebra skills. make explicit the connections between tape diagrams and symbols and equations. Success Criteria We will be successful when we can relate problems that use tape diagrams to symbols and equations used in algebra.

Use and Connect Mathematical Representations High-leverage Teaching Practice #3 Use and Connect Mathematical Representations Physical: Use concrete objects to show, study, act upon, or manipulate mathematical ideas (e.g., cubes, counters, paper strips). Visual: Illustrate, show, or work with mathematical ideas using diagrams, pictures, number lines, graphs, and other math drawings. Symbolic: Record or work with mathematical ideas using numerals, variables, tables, and other symbols. Contextual: Situate mathematical ideas in everyday, real-world, imaginary, or mathematical situations and contexts. Debrief reading of PtA on Mathematical Representations. Verbal: Use language (words) to interpret, state, define, or describe mathematical ideas.

Use and Connect Mathematical Representations High-leverage Teaching Practice #3 Use and Connect Mathematical Representations Now think about the arrows; what are some strategies you use that helps students make connections between the various representational modes? In particular, describe some ways you are helping students understand symbols, or as the Signapore Math article stated, to “think symbolically.”

Expressions and Equations (EE) Reason about and solve one-variable equations and inequalities. 6.EE.5 6.EE.6 6.EE.7 Represent and analyze quantitative relationships between dependent and independent variables. 6.EE.9 Solve real-world and mathematical problems using numerical and algebraic expressions and equations. 7.EE.3 7.EE.4 Count off in your groups from 1 to 6. Each person is assigned one of the standards. Read and highlight key terms and phrases in your assigned standard. Put a few key ideas or questions on your whiteboard related to the standard for discussion with your group. Take turns summarizing your assigned standard (as best you can) or asking questions of your group in order to better understand the standard. Try to get a gist of how key ideas related to expressions and equations are progressing in sophistication across these standards and these grades.

Sample Assessment Item (Gr 8) Leon plants 3 rows of tomatoes with n plants. He also plants 1 row of beans with 5 plants in the row. 1st row 2nd row 3rd row Row of beans From MCA III Item Sampler Gr 8 2012 Minnesota Common Assessment Which equation can be used to find t, the total number of plants planted? A. t = n + 8 C. t = 3n + 5 B. t = 3n + 1 D. t = 5n + 3

Ann sells bracelets for $4 each Ann sells bracelets for $4 each. Draw a diagram that shows x, which represents the total number of bracelets. . . . $4 bracelets x times From MCA III Item Samples Gr 8 2012

Ann sells bracelets for $4 each and necklaces for $8 each Ann sells bracelets for $4 each and necklaces for $8 each. Draw a diagram that shows x, the number of bracelets, and y, the number of necklaces that Ann sells. . . . $4 bracelets x times $8 necklaces y times From MCA III Item Samples Gr 8 2012 Write an equation for the total number of necklaces and bracelets sold? Write an equation for the total number of necklaces and bracelets sold?

Ann sells bracelets for $4 each and necklaces for $8 each Ann sells bracelets for $4 each and necklaces for $8 each. Draw a diagram that shows x, the number of bracelets, and y, the number of necklaces that Ann sells. . . . $4 bracelets x times $8 necklaces y times From MCA III Item Samples Gr 8 2012 Write an equation for the total number of necklaces and bracelets sold? Which inequality shows the number of bracelets and necklaces that Ann must see to make at least $100? A. 4x + 8y < 100 B. 8x + 4y < 100 C. 4x + 8y > 100 D. 8x + 4y >100

Using Variables with Tape Diagrams East Meadow School District October, 2013 Tape Diagrams Using Variables with Tape Diagrams Max had x brownies. He ate 4 brownies and shared the remaining brownies among his 6 friends equally. How many brownies did each friend receive? Express your answer in terms of x. x brownies “You do” www.CommonCore.org

Using Variables with Tape Diagrams East Meadow School District October, 2013 Tape Diagrams Using Variables with Tape Diagrams Max had x brownies. He ate 4 brownies and shared the remaining brownies among his 6 friends equally. How many brownies did each friend receive? Express your answer in terms of x. x brownies “You do” www.CommonCore.org

Using Variables with Tape Diagrams East Meadow School District October, 2013 Tape Diagrams Using Variables with Tape Diagrams Max had x brownies. He ate 4 brownies and shared the remaining brownies among his 6 friends equally. How many brownies did each friend receive? Express your answer in terms of x. x brownies If x = 34, how many brownies does each friend get? www.CommonCore.org

I’ve Got All My Sisters and Me The ages of my 2 sisters, Carolyn and Julie, and I are three consecutive integers. Represent this situation with a tape diagram. Let x = Julie’s age in years. x + x + 1 + x + 1 + 1 = 153 3x + 3 = 153 3x = 150 x = 50 Julie was 50 years old, Carolyn was 51 years old, Connie was 52 years old. Julie Carolyn Connie Ask participants to draw a tape diagram for three consecutive integers. What do they know, what can they label? If Connie is the oldest and Julie is the youngest, what do they know, what can they label? At some point in our lives, the sum of our ages was 153. How old was each sister? At some point in our lives, the sum of our ages was 153. How old is each sister?

One Last Thought “The bar modeling tool (tape diagrams) has taught students not only to solve math problems but also to represent them symbolically-the mainstay of algebraic reasoning.” Singapore Math: Simple or Complex. p. 28

Insights How do tape diagrams help establish the relationships among the numbers in a word problem? 2. How is this similar to or different from how you learned to do word problems or the strategies you have used to teach students to solve word problems? 3. What benefits do you see to using tape diagrams? 4. As you think about your own classroom, how might you begin to use tape diagrams with your students?

Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.