1. SEE Equations Through Tape Diagrams
Lisa Olin
56th NWMC
Oct. 2017

2. In case you haven’t used tape diagrams…
Tape diagrams are mentioned in CCSS as a recommended tool
Tape diagrams are made of rectangles (tapes)
Equal-sized boxes contain equal amounts
Students will draw boxes the same size better if rows are touching 3 9 12

3. Why I Love Tape Diagrams!
Visual
A visual that maintains mathematical relationships
Flexibility as a math tool
Reveal/review algebra notation
Helpful for ELL students & special needs students
Allow students to use intuition

4. Intuitive? 4 4 4 ?

5. Best part?
Students can SEE an equation as a relationship of equal quantities.
Isn’t that what they lose on problems like this?
7 + 3 = n *

6. My introduction CCSS -- named in the standards as a tool to use
First introduced to tape diagrams by Ruth Parker, of Number Talks fame, and her colleague, Patty Lofgren, both from MEC (Mathematics Education Collaborative). Debbie Olsen has joined.
Used tape diagrams for solving proportions first
Started to play with tape diagrams for equations

7. Thank you!
Owe thanks to secondary Shelton teachers for letting me practice on them.
I taught them about tape diagrams so that they themselves could use the tools that their students had learned in middle school and earlier grades.
The teachers questioned me and our journey continued.

8. Tape diagram work begins early
can be shown as
3 x 9 can be
15 ÷ 3 needs to start with the whole 4 5 ? 9 9 9 ? 15 ? ? ?

9. Reinforces fact families for some students
Subtraction seems a little more tricky to show with variables so students need a clear understanding of whole number subtraction.
10 - 2
Reinforces fact families for some students 10 2 ?

10. Don’t rush the basics
Be sure that students understand how to show operations with whole numbers before they begin to use variables in equations.
Practice with students but don’t use up lots of class time. Remember the intuitive part?
Do these tapes show 4(3) = t or t ÷ 3 = 4
Great place to talk about inverse operations. 4 4 4 t t 4 4 4

11. How do we begin today? Be the student…
Be the beginner, the confused child or adult…
Get out some paper. 
Beginning with the basics: y + 8 = ?
y + 8 = 13
How can we be sure that both sides are equivalent? Stack them! See equivalence?

12. Background… Beginning with the basics: y + 8 = ? y + 8 = 13
How can we be sure that both sides are equivalent? Stack them!
Do you see equivalence now? = y 8 13

13. Fold paper
Fold your paper so that the fold itself divides your paper in half vertically (aka “Hot dog” fold)
Tape diagrams on the left, please.

14. Start simple: Begin with a simple equation like x + 5 = 12
Think about a student’s point of view when drawing tape diagrams. 12 x 5 x 5 12 ? 5 ? 5 7 5 7 5

15. There are some rules: Don’t use graph paper Do connect the tapes
Don’t draw the tape diagrams for students
Let tape diagrams make sense! t + 12 = 12
Do think about size as you draw tape diagrams.
If you make terms look equally sized, kids may assume that their values should be equal
If you know that your variable will be smaller than the other term, draw it smaller (plan ahead)

16. More rules: For beginners, keep terms separate: NOT
Always work to reinforce mathematical properties & demonstrate connections with the structures of mathematics
When done, ask students to show where the original equation is shown and where the solution is shown x 5
x + 5

17. Your turn for addition:
w + 9 = = f + 8 or 14 + y + 5 = 23 Where is the equation and where do you see your solution? How do you know the solution is true?

18. Multiplication 4k = 32 or 4k = 132 or 27 = 3y
30 = 4k or 6y = 42 or 8y = 0.16
Where is the equation and where do you see your solution?

19. Division start with the whole
Where is the equation and where do you see your solution?

20. Subtraction start with the whole or students lose sight of the “minuend”.
45 – 10 =
Minuend – Subtrahend = Difference
Minuend
- Subtrahend
Difference

21. Subtraction: begin with the whole
f – 5 = = m – 11 Where is the equation? Where do you see your solution? Select a way to mark this.

22. Does the work we’ve done fit with what we know about equations?
Addition and multiplication were simpler. Why?
Commutative, multiplication is shortcut for addition
Subtraction and division needed to start with a whole. (not commutative, division is a shortcut…)
Expressions can be represented in different ways.
Do mathematical properties still hold true?

23. “This will be too easy….”
Increase rigor by
changing the size of given numbers
changing notation or order
by including fractions, decimals, etc.
increasing the difficulty of given expressions

24. Remember that early problem?
7 + 3 = n + 4
How could a student draw this as a tape diagram?

25. Time to play!
2c + 5 = 21 2x – 5 = 21 Be careful with this one!

26. Distributive Property
3(t + 2) = or 3(t + 2) = 27 – 6
18 = 2(r - 3) time to get creative! r 3
r - 3
r - 3 3 18 9 9

27. Turn it around!
What equation is shown with this tape diagram? Have students write a problem situation from the tape diagram.

28. Diagrams from problem situations…
Steve and his four friends paid a total of $80 for t-shirts and movie tickets to see Captain America. If the t-shirts sold for $9, how much did each movie ticket cost?
Write an equation to solve this. 5(c + 9) = 80 if c = the price of the ticket.
Draw a diagram. 80

29. Challenge yourself! 3(x + 4) = x + 30
Where do you see 3x + 12 from distributive?
x + 4
x + 4
x + 4
x + 30 x 30 x x x 4 4 4 18 12 d d d
On the second problem, start with 1/2k, subtract 1 on the line below in order to get 1/2k – 1.
Then I’d double the top tape by adding 1/2k to the right. That would give me k as a longer tape that I’d probably add to the very top. You can move in any direction when solving equations with tapes, as long as you maintain mathematical properties and equivalences.

30. Review: Great for introductory work
Visual representation of equality
Intuitive
Meaningful
Fun---Kids like tape diagrams!

31. You will reach a limit at which tape diagrams
become more complicated than standard
algorithms.
What then?
Look to the tape diagrams to demonstrate meaning for the algorithm.

32. Lets consider standard “steps”
Right side of paper.
Go back to the right side of our papers, solve equations using algorithms. Look for the equation and solution. Look for mathematical structures like Distributive Property or examples of Commutative Property, or Associative. Use the vocab, but probably don’t assess it.

33. 3x - 1 = 26 3x 1
3x - 1 1 26 27 x x x 9 9 9
Where do you see the original equation?
Where do you see adding 1?
Where do you see dividing by 3?
Where do you see the solution?

34. There are limitations.
Equations can get too complicated and time consuming.
(This could be a fun challenge/extension!)
Negative numbers can become a problem because negative spaces may be contradictory to what our students understand  we must make sense for our students.
c + 5 should not look the same as c + (-5), right?

35. Variables on both sides of = sign
d + 3 = 2d – Do subtraction first. 2d
Some problems seem very hard to diagram and it is easy to get stuck, but remember to look at the whole picture and keep in mind that you are looking to “isolate the variable”, a phrase I rarely use in middle school since it has little meaning for most of them. 4
2d - 4 4
d + 3 4 3 d 7 d d d

36. The trouble with negatives…
h – 6 = 3h + 4 negative values
Tapes can support reasoning but would have to tolerate use of negative spaces
(not for beginners)
Following procedures that I taught, the diagram on the right is correct but how can 3h be smaller than h? If h is negative, 3h is smaller but this can be confusing to students.
If you are willing to tolerate a space for -6, the tape on the right might work for your class. They must see that 2 (-5) is the only option to be equivalent to When working negatives, the standard steps start to be neater. However, you may have some intrigued students who want to continue with tape diagrams. It would be interesting! h
h - 6 6
h - 6
3h + 4 ? 6
3h + 4 h -6 10 3h h 2h 4
-10 4

37. Standard algorithm is needed, but try tape diagrams.
Many teachers resist tape diagrams and will teach equations what I consider backwards: Standard algorithm first, then tape diagrams. Seems like teaching students to add fractions before you draw any pictures of what fractions look like. However, tape diagrams will still be helpful.

38. Colleagues who had learned about tape diagrams complained about student understanding of equations.
I asked if they’d tried tape diagrams.
No.
Maybe you should try tape diagrams.
Guess the results?

39. Reflection time… Are tape diagrams an algorithm?
Hmmm…
Are tape diagrams an algorithm?
Do they communicate an understanding of equality?
Do they show mathematical relationships?
Could they make equations more meaningful for our students?
Story of my ELA colleague who cried when he first saw an equation that made sense in tape diagram form.

40. Not my goal to make you cry.

41. My goal is to give your students that moment when they really SEE equations as meaningful.

42. Thanks
Thank you for being a learner today.