The speed of a freight train is 14km/h slower than the speed of a passenger train. The freight train travels 330 km, in the same time that it takes a passenger train to travel 360 km. Find the speed of each train.
Again, an algebra problem about speeds. Again, we will make a simple table about the two trains. The table will have columns for speed, distance, and time.
| distance | speed | time ----------------------------------------------- Freight train | 330 | v | ------------------------------------------------- Passenger train| 360 | v + 14 |
Notice the problem says "in the same time". Let's call that t.
| distance | speed | time ----------------------------------------------- Freight train | 330 | v | t ------------------------------------------------- Passenger train| 360 | v + 14 | t
Of course, the goal is to have an equation in a single variable, not in two variables (t and v).
Since the time is the same, we can build an equation of the time t = ... for both trains, and then set those expressions to be equal.
For the freight train, t = distance/speed = 330/v For the passenger train, t = 360/(v + 14)
Now let's make those equal:
330/v = 360/(v + 14)
Cross multiply:
360v = 330(v + 14)
360v = 330v + 4620
30v = 4620
v = 154
So, the speed of the freight train is 154 km/h and the speed of the passenger train is then 168 km/h.
Check: How long will it take for the freight train to travel 330 km? Well, 330/154 hours, or 2.142857143 hours.
How long will it take for the passenger train to travel 360 km? Well, 360/168 hours, or 2.142857143 hours. Seems to match.
What is "subitizing"? It's probably a new word for most. It basically means being able to recognize instantly how many objects you see, without counting.
We all do that with the dots on the face of a dice. People who play dominoes with the larger set also learn to do that with the dots on the dominoes (they can go up to 18).
But why would such be important? It is for children, because it promotes number sense, and also ties in with the concepts of addition, subtraction, and even place value.
Look at the dominoes in the picture, and especially the one in the front with turquoise dots on the left side and three orange dots on the right.
Looking at the turquoise dots, can you tell how many there are without counting?
Most of us realize that it is just one less than three rows of three, or nine. So therefore it has eight dots. That is "subitizing".
By learning to subitize a child is learning: • number sense • to associate a number with a whole collection • to visualize quantities and the associated number • relationships between numbers • to visualize subsets of a number • to use subsets of numbers to quickly make addition and subtraction equations • the foundation for quick and creative mental arithmetic • place value concepts
I recently had the opportunity to watch a video made specifically to address this math concept, called Subitize Me! It is about 28 minutes long and meant for children from PreK to 2nd or 3rd grade.
This is the trailer of the video:
In it, two children go through an adventure of sorts inside ancient ruins where they have to first be able to tell the number of gold coins before they can escape. The children get there through a series of seven "lessons", for example with dice, with a snowman, or spacecraft. In each lesson they learn more and more about subitizing, until finally they are able to see the large number of coins in groups of ten tens (hundred) and groups of tens.
My children liked the movie and wanted to watch it twice (thus far). It is actually recommeded to watch it repeatedly over time so children can cement the concepts shown in the video.
This is a commercial movie on a DVD. The DVD also has some additional exercises for children to watch and think through. You can also purchase (separately) a booklet for the teacher that has activities you can do and has a brief summary of the movie.
The video is well made and ties in with the math concepts very well. I would recommend it to elementary school teachers, libraries, co-ops, math clubs and any such places where many children have the opportunity to watch it.
I'm definitely not against home users (parents) purchasing it, either. This concept is beneficial for all children and teachers! However, I consider it a bit pricey, as compared to other math-related movies and DVDs that are available commercially, because it does not really cover that big a part of the typical math curriculum. So you need to consider your budget. Maybe you could resell it afterwards, which would reduce the cost.
I also hope they would make it available for watching online for a small fee, kind of like how you can "rent" (download) movies from Amazon for a few dollars. That would make it more accessible to larger numbers of parents.
Please see more information about the movie at Movie Makers.
Disclaimer: I received from Movie Makers company a link to be able to watch this movie for free, plus the PDF version of the teacher activity booklet. I did not receive any other compensation. The opinions are mine.
Photo courtesy of Ben Cooper
