1. The Juicy Middle of Real Word Problems No
Resources Needed:
Hot spot
Laptop
The Juicy Middle of Real Word Problems 

2. Read it and deicide to skip
Skip it
Pick numbers randomly and apply an operation
Read the first line and decide to skip
Ask teacher for help (akaa “the answer”)
Don’t read it and guess the answer
If you are wondering if you chose the right session, let me help you make your decision real quick:
When your students see a task like this, do they:
Read it and decide to skip it?
Read the first line and decide to skip it?
Don’t read it at all and guess the answer?
Skip it?
Ask the teacher for help (aka wanting each step handed to them)?
Pick numbers randomly and apply an operation?
Ask teacher for help (akaa “the answer”)?
…and you are looking for ways to help student make sense of word problems and persevere in solving them…then this is the session for you.
If you have students who have done any of these and are looking for way to help students make sense of word problems and persevere in solving word problems, than you are in the right session and you will leave with applicable stragies that can be applied with any curriculum with minimum planning. BAMB!!! Lets go.
Ask teacher for help (aka giving each step)
@timsmccaffrey

3. Source: Chris Shore
@timsmccaffrey

4. The Math Club has chosen to make T-shirts with the school’s logo and sell them for a fundraiser. There are two local competitors.
Demo Lesson
Debrief
Task Source:

5. No Tailless Task Source: Transition to Algebra by Heinemann
Tailless Task
Source: Transition to Algebra by Heinemann
@timsmccaffrey

6. Tailless Problem

7. Tailless Problem @timsmccaffrey MAT.06.CR.2.000RP.A.096 Claim 2
I Do…We Do…You Do
@timsmccaffrey
MAT.06.CR.2.000RP.A.096 Claim 2

8. Tailless Problem @timsmccaffrey MAT.07.ER.2.0000G.A.295
I Do…We Do…You Do
@timsmccaffrey
MAT.07.ER G.A.295

9. Tailless Problem @timsmccaffrey MAT.08.CR.2.0000G.B.132 Claim 2
I Do…We Do…You Do
MAT.08.CR G.B.132 Claim 2
@timsmccaffrey

10. No Key Elements Present a task with no question(s)
S’s write what they can say FOR SURE
S’s share what they know and prove it to one another
S’s write down questions that make sense to ask given the information
Present student thinking and “yours” when necessary
Have extension questions ready for those who finish quickly
Discuss the power of tailless problems. Deconstruct the experience in a google doc. Ahead of time write the key factors and include them if they are not stated:
For example:
Listing what you can say “for sure” can help you discover things you didn’t realize you knew.
Focusing vs. funneling
Talk moves
Discussion Builders
Give information and have S’s write what they can say for sure
Give additional information and have S’s write again what they can say for sure (show examples from top of page 10)
Have student’s defend their “for sure” statements.
Turn “for sure” statements into questions?
@timsmccaffrey

11. No Headless Task Source: Transition to Algebra by Heinemann
On the contrary, we have headless problems
Headless Task
Source: Transition to Algebra by Heinemann
@timsmccaffrey

12. Two large storage tanks, T and W, contain water
Two large storage tanks, T and W, contain water. T starts losing water at the same time additional water starts flowing into W. When will the two tanks contain the same amount of water?
Give information about both tanks.
Notice a question has not been presented. I would like you to think deeply about the information given and write down what you can say for sure about the two storage containers.
For example,….
Tailless Problem…
S’s first work independently on what they can say for sure.
S’s share with others and continue working in groups
T Pose purposeful question and use focusing questioning from group to group. In order to really understand the thinking of the students, this is a perfect opportunity to focus and see what S’s are thinking and why.
S’s write down three questions they can ask that makes sense about this situation.
S’s share out and T record questions. Take a vote on interesting questions. Choose question.
T asks, “How is the information we have going to help us with our question at hand?” Also, is additional information needed to answer the question?
End lesson
S’s are now ready to extend their thinking and possibly do the “teacher problem” if it was not already asked.
@timsmccaffrey

13. Tank W
Tank T
Hours
Amount of Water 9
450 11
350 13
250

14. Two large storage tanks, T and W, contain water
Two large storage tanks, T and W, contain water. T starts losing water at the same time additional water starts flowing into W. The graph below shows the amount of water in each tank over a period of time. Assume that the rates of water loss and water gain continue as shown.
When will the two tanks contain the same amount of water? Explain how you found your answer and interpret your solution in terms of the problem.  
If you have not already done so, write an equation for each storage tank that can be used to determine the amount of water in the tank at any given number of hours.  
Explain what the different parts of each equation mean in terms of the problem.
Explain what the different parts of each equation mean in terms of the graph.
Give information about both tanks.
Notice a question has not been presented. I would like you to think deeply about the information given and write down what you can say for sure about the two storage containers.
For example,….
Tailless Problem…
S’s first work independently on what they can say for sure.
S’s share with others and continue working in groups
T Pose purposeful question and use focusing questioning from group to group. In order to really understand the thinking of the students, this is a perfect opportunity to focus and see what S’s are thinking and why.
S’s write down three questions they can ask that makes sense about this situation.
S’s share out and T record questions. Take a vote on interesting questions. Choose question.
T asks, “How is the information we have going to help us with our question at hand?” Also, is additional information needed to answer the question?
End lesson
S’s are now ready to extend their thinking and possibly do the “teacher problem” if it was not already asked.
Headless Task
@timsmccaffrey
Adapted from NAEP Released Items, M10 #13

15. You and two of your friends saved money for four weeks
You and two of your friends saved money for four weeks. Which of you saved the greatest amount of money and who saved the least amount of money?
Talk through the Headless question:
Pose Question
S’s write down the information needed to solve this problem
S’s also write down any questions they have about the situation.
When students ask for the information, mathematize it by giving the students different mathematical forms of the same information.
When you show the graph, you can ask, “How is the graph related to how much you earned in a week?” “How are you interpreting the graph?”
When you show the equation, ask, “What does this have to do with how much money you saved in a week, 2 weeks, 3 weeks?” What do the variables mean?
When you give the table, “how does the table tell us about how much you saved in a week, 2, or 3?”
You can always ask, “Is more information needed?”
The point I need to drive home is you have control over the manner in which you give the information. The students will ask for the information but you can mathematize it and help them make sense of it in the context of the problem and what they are asking for.
MAT.08.TE.2.000EE.C.202
@timsmccaffrey

16. How much did I save each week?
You
Antwan
Carla
What about my two other friends?
When students ask for the information, mathematize it by giving the students different mathematical forms of the same information. MTP: Connecting multiple representations.
When you show the graph, you can ask, “How is the graph related to how much you earned in a week?” “How are you interpreting the graph?”
When you show the equation, ask, “What does this have to do with how much money you saved in a week, 2 weeks, 3 weeks?” What do the variables mean?
When you give the table, “how does the table tell us about how much you saved in a week, 2, or 3?”
You can always ask, “Is more information needed?”
@timsmccaffrey
MAT.08.TE.2.000EE.C.202

17. You Antwan Carla $2/week $2.50/week $1.75/week MAT.08.TE.2.000EE.C.202
@timsmccaffrey

18. Headless Problem
You and two of your friends saved money for four weeks. Which of you saved the greatest amount of money each week and who saved the least amount of money each week?
Talk through the Headless question:
Pose Question
S’s write down the information needed to solve this problem
S’s also write down any questions they have about the situation.
When students ask for the information, mathematize it by giving the students different mathematical forms of the same information.
When you show the graph, you can ask, “How is the graph related to how much you earned in a week?” “How are you interpreting the graph?”
When you show the equation, ask, “What does this have to do with how much money you saved in a week, 2 weeks, 3 weeks?” What do the variables mean?
When you give the table, “how does the table tell us about how much you saved in a week, 2, or 3?”
You can always ask, “Is more information needed?”
The point I need to drive home is you have control over the manner in which you give the information. The students will ask for the information but you can mathematize it and help them make sense of it in the context of the problem and what they are asking for.
@timsmccaffrey
MAT.08.TE.2.000EE.C.202

19. Write 3 equivalent ratios of blue tiles to green tiles.
Headless Problem
@timsmccaffrey
MAT.06.SR.2.000RP.D.181 C1 TA

20. Headless Problem @timsmccaffrey MAT.07.CR.2.000EE.D.158
@timsmccaffrey
MAT.07.CR.2.000EE.D.158

21. Headless Problem
You are wanting to purchase many shirts from a T-shirt company. What is the cost of each shirt?
@timsmccaffrey
MAT.08.ER F.C.130 Claim 2

22. No No Key Elements Present a question with “just enough” information
S’s determine what information is needed
T asks how the information would help s’s with the problem
T mathematizes the information
@timsmccaffrey

23. Read it and deicide to skip
Skip it
Pick numbers randomly and apply an operation
Read the first line and decide to skip
Ask teacher for help (akaa “the answer”)
Don’t read it and guess the answer
If you are wondering if you chose the right session, let me help you make your decision real quick:
When your students see a task like this, do they:
Read it and decide to skip it?
Read the first line and decide to skip it?
Don’t read it at all and guess the answer?
Skip it?
Ask the teacher for help (aka wanting each step handed to them)?
Pick numbers randomly and apply an operation?
Ask teacher for help (akaa “the answer”)?
…and you are looking for ways to help student make sense of word problems and persevere in solving them…then this is the session for you.
If you have students who have done any of these and are looking for way to help students make sense of word problems and persevere in solving word problems, than you are in the right session and you will leave with applicable stragies that can be applied with any curriculum with minimum planning. BAMB!!! Lets go.
Ask teacher for help (aka giving each step)
@timsmccaffrey

24. any teacher. any curriculum. any lesson.
for all children.
@timsmccaffrey

25. Building Sense Making and Perseverance ThroughNo
Building Sense Making and Perseverance Through
Resources Needed:
Document Camera
Laptop
Water tank problem
Headless &
Tailless Tasks
@timsmccaffrey

26. Workshop
Create a number of headless and tailless math tasks that you can use next week in your classroom.
Headless Task
Think about how you will mathematize the information.
Tailless Task
Think about what questions to focus on.
Share out tasks using document camera.
@timsmccaffrey

27. Source: Chris Shore @timsmccaffrey
How will headless and tailless problems help students make sense of problems and persevere in solving them?
Source: Chris Shore
@timsmccaffrey

28. The Juicy Middle of Real Word Problems
Cool Idea came from “Transition to Algebra” by Heinemann No
The Juicy Middle of Real Word Problems 
Tim McCaffrey
Assistant Principal – Sequoia Middle School
aodmath.com/CMCS16
@timsmccaffrey

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