1. R4R Samples (6-8)

2. Routine: Capturing Quantities

3. Abe’s Jeep
During the week, Abe goes to the gas station to purchase fuel for his Jeep. He has traveled 157 miles since the last time he completely filled his tank. Once again he completely fills his tank and he finds it takes 9.54 gallons.
He wants to calculate his gas mileage so he divides 157 miles by 10 gallons and finds the rate of 15.7 miles per gallon.
Was his gas mileage better than 15.7 or worse than 15.7 mpg? How do you know?
Q: is this partitive or measurement division
This problem tends to make people a little unsure:
Tell folks not to do the actual division problem to solve this question. Use their understanding of division – because this is partitive we can think about understanding of fair sharing miles for each gallon of gas. You can use a ten frame to model this idea; dividing 15.7 miles into each box and then discussing how you would redistribute the miles from the .46 gallons we did not use.
The big idea is that often times the questions we use math to answer do not need exact answers, but we do need to understand what these answers we find mean given the context presented.

4. Ruler Task
Jane needs some string for a project she is making. She needs the length of her piece of string to be 3 ½ of the rulers shown below. What is the length (in inches) of the piece of string Jane needs for her project? 1 2 3
Inches

5. Luis mixed 6 ounces of cherry syrup with 53 ounces of water to make a cherry-flavored drink. Martin mixed 5 ounces of the same cherry syrup with 42 ounces of water. Who made the drink with the stronger cherry flavor? Give mathematical evidence to justify your answer. NAEP M070401, cited in Thompson & Saldnha, 2001

6. Mr. Van Gogh’s art class was preparing to do their final projects, but the students began to argue over various shades of blue. Which recipe produces a darker blue colored mixture: 34 parts white to 14 parts blue, or 9 parts blue to 22 parts white?   Use Mathematics to justify your answer
The ratios are purposely out of order for each recipe to assess if students are actually looking at the problem or just pulling numbers. The shades of blue will be close, but I like this example because most people have experience buying paint and looking at shades of color that are very close to each other. May be more relevant than taste, which could be more difficult to distinguish between.
Have teachers solve the problem any way they like. You could have them post their thinking or use the subsequent student work slides and just analyze all the student strategies and discuss these. In the 3-5, teachers do not necessarily teach this content any longer and thus it may be less necessary to have them spending time going through the process of discussing and making connections. I use this problem as a “Smith and Stein” experience much like I use the Fruit by the Foot task.
Just hide the slides you do not want to use.

7. A recipe calls for 6 oz. chocolate per 3 cups of cream
A recipe calls for 6 oz. chocolate per 3 cups of cream. The mixture is too runny so I find a better recipe. The second (better) recipe calls for 5 oz. chocolate per 2 cups of cream. I’m out of cream, how much chocolate should I add to the original mixture to match the ratio of the second recipe?
Adapted from Kalman, Fixing ganache: Another real-life use for algebra, Mathematics Teacher, Feb

8. Two Machines, One Job
Ron’s Recycle Shop was started when Ron bought a used paper-shredding machine. Business was good, so Ron bought a new shredding machine. The old machine could shred a truckload of paper in 4 hours. The new machine could shred the same truckload in only 2 hours. How long will it take to shred a truckload of paper if Ron runs both shredders at the same time?
There are multiple ways to do this problem – enactive, iconic, and symbolic models are all possible.
OLD
NEW
Ron’s Truck
Van de Walle, 2007, p.15

9. Routine: Connecting Representations

10. Today’s Thinking Goals
Think like mathematicians
Look at and make sense of expressions
Connect expressions to visual models

12. 7×3 + 4×6
3×3 + 4×1 + 4×4 + 4×4
3×3 + 4×9
3×3 + 4×1 +2× 4×4

13. Whole Rectangle – Gray = Purple

14. Today’s Thinking Goals
Think like mathematicians
Look at and make sense of expressions
Connect expressions to visual models

15. Thinking Questions Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?
How many thinking questions should we give kids at one time? Do we develop these as we go?
How is this situation behaving?
What kind of problem is this?
Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?

17. 1) 2) 3) 4)
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

18. 1) 2× 6×2 + 10×2 12×2 + 4×2 + 6×2 6×2 + 10×2 + 6×2 14×2 + 4×2 + 4×21)
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

19. 1) 2) 2× 6×2 + 10×2 12×2 + 4×2 + 6×2 6×2 + 10×2 + 6×2 14×2 + 4×2 + 4×21) 2)
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

20. 1) 2) 3)
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

21. 1) 2) 3) 4)
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

22. 5) 2× 6×2 + 8×2 12×2 + 4×2 + 6×2 6×2 + 8×2 + 6×2 14×2 + 4×2 + 4×2
“I noticed ________,
so I __________. “
“They noticed ________,
so they __________. “ 5)
2× 6×2 + 8×2
12×2 + 4×2 + 6×2
6×2 + 8×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

23. Reflection
“When interpreting a mathematical representation, I learned to pay attention to _______________.”

24. Today’s Thinking Goals
Think like mathematicians
Look at and make sense of expressions
Connect expressions to visual models

25. 1 2 3 4 a) (2 x 4) + (1 x 4) + (2 x 2) b) 3 2 + 4×1 +(5×1)
c) 5 2 −[ 3×2 +1] 3
d) [(3 x 5) -1] + 4 4
e) (3 X 4) + (4 x 1) + 2

26. a) (2 x 4) + (1 x 4) + (2 x 2)
b) ×1 +(5×1)
c) 5 2 −[ 3×2 +1]
d) [(3 x 5) -1] + 4
e) (3 X 4) + (4 x 1) + 2

27. Today’s Thinking Goals
Think like mathematicians
Look at and make sense of expressions
Connect expressions to visual models

28. 1) Daily operating costs for the theater)
2) Number of tickets that must be sold for the theater to have a profit of $500.
3) Daily “break-even point” for the movie theate]
4) The rate of change in the relationship

29. Routine: Recognizing Repetition

30. Step 1
Step 2
Step 3
Step 4
Pattern #18, from Justin Lanier, Squares in step 43 = 129, Perimeter in step 43 = 260

31. Pattern #86, from Michael Fenton, Circles in step 43 = 2197

32. Pattern #149, from Matt Owen, Cubes/Squares in step 43 = 2027

34. Student 3
Student 4
Student 5
Student 1
Student 2
Student 11
Student 8
Student 6
Student 7
Student 9
Student 12
Student 10
Student 13

35. Pattern #142, from Don Steward, Cubes in step 43 = 3700

36. Pattern #200, Squares in step 43 = 1937

37. Pattern #000, Circles in step 43 = 132

38. Pattern #20, Helmets in step 43 = 3741

39. Pattern #158, from Ben Graber, Cubes in step 43 = 81,399

40. Step 1
Step 2
Step 3
Step 4
Pattern #162, from Paule Rodrigue, Bacteria in step 43 = x 10^13

41. Step 1
Step 2
Step 3
Step 4
Pattern #79, Yellow triangles in step 43 = e+20, Green triangles in step 43 = e+19

42. Pattern #146, from Andrew Stadel, Candy corns in step 43 = 946

43. Pattern #187, from Eric Appleton, squares in step 43 = 1937

44. Step 1
Step 2
Step 3
Pattern #218, from Ilona Vashchyshyn's student, Segments in step 43 = 510

45. Pattern #153, from Jacob Siehler, Hexagons in step 43 = 4,945

46. Pattern #163, from Katie Gates, Bunnies in step 43 = 1.364242 x 10^30