This blog will connect Primary Gifted Program instructors for professional learning and problem-solving.
I was intrigued by the attention grabbing title of Chapter 13: What are These Strange Numbers? and the way the concept of negative numbers is explained without using a number line. I do like Einstein’s examples of how negative numbers are used to express money owed, ocean depth, and differences in temperature, but was very surprised when I didn’t see a number line. I chose Chapter 13, Einstein Level, Question 1, p. 130, as my sample problem and thought original +$8, -$17 owed, +$5 birthday credit, +$10 Grandmother credit for a grand total of $6. A number line to which positive and negative numbers are plotted could also be used as a visual. I wonder if the author purposefully didn’t introduce a number line because he felt that G/T students could mentally solve these types of problems or if he thought that those who need a visual would intuitively come up with one on their own.In Chapter 14, Einstein Level, Question 1, p. 140, the problem asks how 3 boys are going to share equal amounts of time at a lemonade stand during a 5 hour time period. I figured out how many minutes they each would work in an hour--> 60 minutes divided by 3 boys = 20 minutes per hour each and multiplied that by 5 ( representing the 5 hour time period) to get 100 minutes total for each boy. Students could also use the model at the beginning of the chapter of breaking down an hour into three 20 minute time parts --> 20 + 20 + 20 --> and adding 4 more time parts (representing the 4 remaining hours of the 5 hour total) of 20 minutes each below each of the original three 20s to get a total below of 100 minutes + 100 minutes + 100 minutes, which would represent the total equal amounts of time each of the 3 boys worked and also equals 300 minutes, which when divided by 60 minutes (1 hour) equals the 5 hour time period. Too bad we can’t use diagrams in a blog…
I did problem #1 on p. 140 (Let's share). I selected that chapter because kids are obsessed with sharing and things being "Fair". The book and I got the same answer but we didn't arrive at it by the same path. You had to divide 5 hours among 3 kids. I gave each kid 1 hour, added the last 2 hours together to find the # of min. & divided that by 3. I've always liked negative numbers (yeah, I know I'm strange) & I think kids would too -it's sort of a "secret" sort of math. I did problem #1. It's a bit tricky and the answer could vary by $1 depending on how the actual wording is interpreted.
Session 2-Question 3Two Einstein Problems:Chapter 14, Problem 1 Each child would have to work 1 hour and 40 minutes. Using Einstein’s example of breaking down an X-amount of dollars into individual coins before dividing helped to solve the time problem: break down hours into minutes before dividing the number of friends. Thanks Albert for your tip! Chapter 18, Problem 5 Dollars to doughnuts, Einstein came through again.The grandchildren were actually weighing their Tim Horton’s on their mother’s kitchen scales. FYI, Christian’s ¼ cake doughnut weighed more than Katerina’s Vanilla Dipped one (sprinkles galore)……this was definitely a family engaging problem. Anyway, convert pounds to ounces, divide by the number of doughnuts=’s the ounces per doughnut (4ozs.) and divide that # into four. Answer: ¼ of a doughnut weighs 1oz. Done! Calories zero!
I solved question #5 on page 131 for Chapter 13, Stange Numbers. Negative numbers seem to attract those gifted kiddos! I arrived at the author's answer -- it felt 43 degrees cooler to the child in the wind (the difference between 10 and -33 is 43 degrees).I also solved question #4 on page 165 for Chapter 16, "When are we going to get there?" Using the "machine method," divide 17.5 miles by 70 mph (speed of the cheetah) and you get .25 hours, or 15 minutes. If you use a calculator to solve, as I did, you'd have to spend some time helping the kids see that .25 hours = 15 minutes. I think students will like solving problems such as this one!
KHarrell – because my sister just returned from South Africa – I decided to do Chapter 16 then there was a problem about a cheetah….all I heard about was the animals they saw on Safari…so I tackled this one…#4 – Cheetahs run at a speed of 70 miles per hour for only short distances – so if it could run for 17.5 at 70 miles per hour how long would it take? 70 miles is one hour…half of that is 35 miles in 30 minutes so half of 35 miles is 17.5 and half of 30 is 15 so it would take 15 minutes to run that. (I have to admit…I was real excited to figure this out with little frustration). Problem 2 – the fox started running at noon and ran 1 mile @ 5 miles per hour what time did he finish? Need to know how fast he can run 1 mile. Divide a mile into fifths. How much time is a fifth of an hour….12 minutes. (When I checked the back of the book – I have the right answer…but I didn’t figure out the same way!) I am using some of what I’ve learned in Singapore math…drawing pictures….it helps me out a lot!Chapter 18 Problem 3 – how much does a fluid once of milk weigh if a gallon weighs 16 pounds. At the beginning of the chapter you learn that 16 ounces are in 1 pound …this means that you have to multiply 16 x 16 to get 256 ounces to get total weight. We know that 128 fluid ounces is in one gallon so 1 fluid ounce equals 2 ounces of weight. (I had to look at a measuring to cup to keep up with the difference of ounce and fluid ounce…thank goodness for being near the kitchen.)Problem #5 – If a dozen donuts weights 3 pounds, how many ounces does ¼ of donut weigh? Have to convert pounds to ounces – 12 donuts equal 48 ounces – 48 divided by 12 equals 4 ounces per donut. Divide a donut into four pieces – each piece weighs 1 ounce. Had to keep the labeling going. Without labeling…I got lost in the math forgot what I was converting or trying to answer.
I completed #6 from Chapter 16 (p. 165) about how fast a fighter jet flies. It reminds me of my son and all his military-related knowledge! First I had to figure out how far a jet can fly in one minute, so I divided 60 minutes by 800 miles. The answer is 0.075 miles per minute. Then I multiplied 0.075 by 160 and came up with 12 minutes. I'm not sure this is how they came up with it in the book, but this is how it made sense to me and the answer is correct, so I'm going with it!The second problem I completed is #3 from chapter 13 "What are these strange numbers?". The question asks how far the temperature drops from the boiling point to the lowest possible temperature. First, I figured that the temperature had to drop 212 degrees from boiling to 0. Then it had to drop another 459.6 degrees to the lowest temperature. So I added 212 and 459.6 and came up with 671.6 degrees. I had to write it out so that it made sense to me before I checked to make sure I was correct. I guess I will need a math journal!
@ KHarrell... There was a problem that I figured out a little differently than the book suggests, too. I think it's okay and I think it will present the opportunity for discussion about solving problems. There is usually more than one way to get the right answer. I also think it's great for kids to see us drawing pictures or using other strategies to figure these things out. Not everyone can do these in their heads!@ Of Life... I like the idea of negative numbers as "secrets"! The kids will love that! Definitely going to steal that idea.
I enjoyed working through Chapter 12 - Magic of Math: Making Smart Guesses (p. 121-122). As a young child, there was nothing magic about math for me. It was just something I didn't enjoy. However, as an adult, I've had to embrace math as a part of every day life, and of course, as a part of being a teacher and a librarian. I loved working with this chapter because it took me back to my "guessing" days... I loved the estimating and the guessing. Gifted students, in my experience, love to guess and take risks, if they feel comfortable in the teaching environment... and this chapter will be great. I even called my Dad (a math whiz) and shared the Einstein level questions with him (the ones we could do w/o seeing the book). I also enjoyed Chapter 11: How Tall is it? (p. 111-112). I would really love to read Jack and the Beanstalk (one of many versions) and expand on the story by answering #4 on p. 112. It would be so much fun, and I was compelled to mark this page to do with the students, for sure. I also enjoyed thinking about #5 on p. 112 which deals with shadows. What fun it would be to go look at our shadows and then come back in and work a problem such as this one.
Chapter 18 #2 page 187 intersting how I felt I needed to visually see/draw the 1x1in square on the 1x1ft cardboard picture. Then it just helped me feel more confident 12 inches in a foot and 12inx12in=144ounces since 1x1in was 1 ounce.There are 16 ounces in a pound, page 99.Answer 9 pounds.Chapter 15 #2 Page 151 I am so visual.. Loved the cartoon of Einstein RIGHT THERE to remind me.. Drew my line.. Drew a line and write 5 for the first mile and then another line and wrote 5 for the next mile and then had 2 1/2 which is half of 5(1 mile) so 2 1/2 miles.
Ok...since I am on vacation, I decided to try the Chapter 17 on What is My Speed? I have to say the question #3 did not seem Einstein to me. Maybe it is because I think of this as a driver all of the time when travelling anywhere. It reads,"If a car is travelling 1 mile each minute, how fast is it travelling? Depending on where and in what state, the answer could make the drive susceptible to a ticket...(not in Michigan I am finding out) I just multiplied each number by the number of minutes in an hour to get 60 mph. I visited the Detroit Zoo with my granddaughter and daughter and was surprised to find some feisty turtles there, so I in Chapter 16 When are We Going to Get There, it tells the tortoise and the hare story about their great race. The problem reads, "If the race was 5 miles and the tortoise went at a speed of 1/4 mile per hour, how long did it take the tortoise to finish the race?" Of course if the rabbit takes a prolong nap, it doesn't matter. The tortoise is the ace. But to figure the math, first I figured how much time it would take to get 1 mile, which is 4 x 1/4. Then I multiplied 4 x 5 miles and got 20. 20 hours is a long nap...maybe the story ended differently, a good chance to add some creative writing there.
I was confused about how to solve Question #5 on page 103 that asked how long it would take three kids to paint one car until I read Einstein’s hint about thinking in terms of what fraction of the car each kid would be able to paint in an hour - then it all became clear! Melissa could paint ½ the car in one hour, Dave could paint ¼ of the car in an hour and Kate could paint ¼ of the car in one hour. Therefore the whole car could be painted by all three kids in one hour. Another problem, Question #2 on page 130, involves a format I’ve seen a million times and always have to stop and think about carefully to avoid being tricked (Einstein even warned that it is a “tricky problem”). This is one that I always get wrong if I try to do it in my head. I have to draw a picture of a vertical number line from 6 up to 0 and physically hop up 5 and down 4 to end up on 5 for the first day, then up 5 again the second day to see that I’m already out at 0 without having to go back down 4 – so the answer is that it only takes two days to get out of the hole. So many of the solutions in this book can be facilitated by a visualization of the factors involved. I also love the way the author throws in multi-step conversions by using inches/feet/yards all in one question in #5 on page 112. And don’t be fooled by the distance question about an echo on page 122 – you need to count the distance both ways – to and from – the mountain (I’m speaking from experience here!).
For my Einstein level questions I chose the problem on page 141, question 5 on Sammy’s bag of money. For some odd reason I enjoy working these backwards math questions. With these types of problems it is also helpful to draw pictures. Since I am a strong visual learner, I like to use these methods. The other problem I chose was the snake in the well problem on page 130, question 2. When I saw the phrase: be careful this is a trick problem, I had to defeat it. When those phrases appear, the problem is usually much simpler than it looks which was exactly the case. It would be easy to make a huge production out of this one, figuring how much the poor snake slips back every day if you do not pick up that he escapes by the second day.
kharrell - after reading all of the comments in this area, I see that perhaps we should have some kind of a journal for the students to write in as they solve. We all wrote the steps we did. If the students draw or talk their way a problem it might be nice to watch the process through a journal. Since I already do a writing journal with my students, we may mark half way through to be our "math journal." I think it would be interesting to get parents to work on some of these problems as well.
In response to KHarrell, I love your idea about journaling on their problem solving strategies. As is evident from many of the comments above, there are many ways to solve a single problem. It would be helpful for the students to share out their strategies and compare them with others so they might pick up some new ideas to "save for a rainy day"!
Chapter 10, Question 1 dealt with "How Much Will I Need?" For me in my old school math, the problem involved some division, multiplication, and addition. I say that because it amazes me how often a problem can be solved multiple ways.Chapter 13, Question 4 dealt with time and negative numbers. Love, love, love negative numbers so this was fun. A little multiplication, addition, and subtraction and I was good to go.BUT......honesty time. Not being a math teacher and being out of touch with the way concepts are currently being taught intimidates me. I think the book is excellent and challenging, but I'll have to rely heavily on sending the students back to the author's presentation of a concept and letting the students muck through the difficulties. Best not for me to confuse them. I feel better admitting this.Viking, I too enjoy working problems backwards. Or rather I often guess an answer and then work the problem to see if it works. Strange, but this probably validates why I should NOT teach math!MillieH and KHarrell, journaling their problem solving techniques is excellent. Comparing strategies and having those notes to do so would benefit all.