Notice and Wonder
You might try to notice
- • The quantities (known or unknown counts or measurements).
- • Relationships between quantities.
- • Information that is not given in the problem but that might be related or that the problem reminds you of.
- • Key words from the problem.
Your wonderings may include:
- • I wonder what will happen if …
- • I wonder what this word means …
- • I wonder if this pattern will continue…
- • What does this mean?
- • What do they want?
- • Does it have to be that way?
- • Do I need to figure that out?
- • How does this situation work?
- • Is there another way to think of it?
- • How will I know if this is true?
- • What is a good way to express that?
- • When is this true?
686: Duke is Missing
When Megan arrived home from school she found that someone, she suspected her mean brother Bobby, had left the yard gate open and her dog Duke had disappeared.She immediately called the local S.P.C.A., and the man on the phone asked her to describe Duke. Megan said, “He is a black-and-white spotted English Setter. His left eye is black and his right eye is white.”
At that moment Bobby grabbed the phone from her hand and said, “Duke’s head is 6 inches long, his tail is as long as his head plus half the length of his body, and his body is as long as the head and tail combined.”
How long is Duke?
1897: Video Viewing
My husband, Frank, and I enjoy watching television and then talking about what we’ve seen. Since Frank works several nights a week, we often tape shows to view later. This works well since we don’t particularly enjoy watching commercials–so we fast-forward right through them.We’ve found that a fourth to a third of an average television hour is actually commercials. We use an extended play option, so we can fit six hours of broadcasts on each tape.
With our busy schedules, we only watch a few hours of the recordings each week. Thus, we’ve fallen way behind (we’re just about to view Christmas episodes!), and we have eight full tapes waiting to be viewed.
On those eight tapes, what is the range of actual program time Frank and I will view?
2041: Jumping Rope
My jumping rope was cut in half
half was thrown away.
The other half was cut again
one third along the way.
The longer part (ten feet long)
is what I use to play.
How long was my jumping rope
when I began today?
10/29/08 Math Forum – Problem of the Week
4556: Pumpkin Carving for Charity*
Jerson is selling fruit at the Farmers’ Market. To attract customers to his booth, he has made up a contest! He has these photos of combinations of fruit on display. His contest is to find out how much each fruit weighs.
Note: Assume that both of the melons weigh the same, both of the pineapples weigh the same, and both of the limes weigh the same.
Melanie looks at his photos and tells him, “I know how much one melon weighs.”
How might Melanie have figured out how much one melon weighs? How much does each type of fruit weigh?
10/10/08 Women’s Walkway
Women’s Walkway* – posted October 6, 2008
On the brick “Women’s Walkway” from the intersection of 33rd and Chestnut to the intersection of 34th and Walnut in Philadelphia, I became fascinated with the pattern of the bricks.
Here is an area that I’m filling with the pattern:
I thought it was economical how some bricks were cut in half and some of the halves were also cut in half to neatly fit to the edges.
If the top faces of the uncut bricks are 2 units by 1 unit,
- how many bricks would it take to cover the full area of the pattern shown above?
- what is the area?
Math Forum – Problem of the Week
Two Heads Are Better Than One *
While Raul and Esteban were working on their uncle’s farm, their cousins Esperanza and Raquel were working on Dr. Dolittle’s farm. Raul tells his cousins the story of the Ostrich Llama Count, where they could figure out the number of ostriches and llamas based on the number of heads and the number of legs. Raquel replies,”That’s funny. Dr. Dolittle asked Esperanza and I to count his animals too, but we had to count ostriches and pushmi-pullyus. I counted 67 heads and Esperanza counted 134 legs, but Dr. Dolittle wasn’t sure whether we would be able to tell how many ostriches and how many pushmi-pullyus there are on his farm.” Will Raquel and Esperanza be able to know for sure how many ostriches and pushmi-pullyus are on the farm, based on the number of heads and the number of legs? Explain how you know.
Ostrich Llama Count
Raul said, “I counted 47 heads.”
Esteban added, “I counted 122 legs.”
“How many are ostriches? How many are llamas?” asked their uncle.
“It’s getting dark and I promised your mother I’d get you home for dinner. There’s no time to count again. You’ll have to figure out how many ostriches and how many llamas there are from that information when you get home. Can you give me a call after dinner and let me know your answer?”
How did Raul and Esteban figure out how many ostriches and how many llamas there were?
I was bored one day, staring at a window, and I started imagining a continuous series of diagonals, going across the panes of the window. It reminded me of the path a pool ball would make, if you shot it from one corner at a 45° angle.
- I always start the path in the upper left corner, and travel at a 45° diagonal across the grid.
- If the path hits an edge of the outer rectangle, it bounces off at a 45° angle and continues its travel.
- The path continues in this way until it hits a corner of the outer rectangle (where a corner pocket would be on a pool table).
Here are the paths generated by some of the windows in my house:
the 2×3 window
the 3×4 window
the 4×6 window
the 6×8 window
I noticed that sometimes the path goes through all of the little squares, and sometimes only some of them. I wondered how many of the square panes the path goes through for a window of a given size.
Question: How many square panes does the path go through in a 28 by 36 window?
Extra: What rule can you use for any size window?