1. I believe the reason students have difficulty learning algebra and that most people have trouble doing math word problems is that although children are taught how to do calculations with numbers, they are not given problems or practice seeing logical relationships or patterns among numbers. Word problems, like many problems in general, are about logic, not just facts. They are about trying to deduce what you don’t know from things you do know that might be relevant. And any reasonable way to make the deduction, or any good logic you can use, is fair game and helpful.

2. A student one time asked me how to do a word problem that involved three consecutive whole numbers and he had no idea how to get started because it never occurred to him to think of the second number as one more than the first, and the third number as two more than the first.
I asked my seven year old grandson one time if he had a garden hose that could put out twice as much water per second as my garden hose does, how much more or less water he would use compared to me if he watered half the amount of time I did. He said he couldn’t do that problem because it didn’t have any numbers. But it didn’t need numbers to solve it, other than the concepts of half and double given in the question.
And even when a problem has numbers, many people don’t know what to do with them. The reason so many people, who can easily do mathematical calculations you give them, cannot do word problems at all, is that they do not know from the information given what kinds of calculations will help or are necessary.

3. So the following problems are all meant to give children and adults practice figuring out and thinking about relationships between numbers. I used to give these sorts of question to my children as we rode in the car. They would do them in their heads whenever they could. We started doing these when they were five or six years old, after they were able to count to 100 by ones, two’s, fives, and tens, which we practiced while driving places also. They also knew how to do some adding and subtracting of smaller numbers. As they developed their skills, I made the problems more difficult, but still within their ability to figure out the answers.
Some of the questions in this program are fairly easy, but some are difficult. Do them at your own pace, or help your child do them at his or her comfortable pace. But let them figure out the answers themselves; that is the point of these, not being told how to do them or what the answers are. If a question is too difficult for now, move on to a different one and just be thinking about the ones you can’t do in the back of your mind as you do other things. You may have to wait till you are older or have more experience to do some of them.
The narrations start automatically on some of the pages; if you don’t need or want them, mute your computer. Some of the narrations require clicking on the icon of the speaker. The narratives are for young children who may not be able to read well yet, but who can still think well about numbers and solve these problems.
I am not giving the answers to them because you should be able to check your answer against the information given in each problem and tell by yourself whether it is correct or not. If it does not meet all the conditions of the problem, it is not the right answer. If it does, it is.

4. Your bag has three more things in it than my bag does.
I have a bag and you have a bag.
Your bag has three more things in it than my bag does.
If I have 2 things in my bag, how many things are in your bag?

5. Your bag has three more things in it than my bag does.
I have a bag and you have a bag.
Your bag has three more things in it than my bag does.
If you have 15 things in your bag, how many things are in my bag?

6. Your bag has three more things in it than my bag does.
I have a bag and you have a bag.
Your bag has three more things in it than my bag does.
If you have twice as many things in your bag as I have in mine, how many do we each have?

7. Your bag has three more things in it than my bag does.
I have a bag and you have a bag.
Your bag has three more things in it than my bag does.
If together we have 7 things, how many do you have and how many do I have?
If together we have 11 things, how many do you have and how many do I have?
If together we have 83 things, how many do you have and how many do I have?

8. Your bag has twice as many things in it as my bag does.
I have a bag and you have a bag.
Your bag has twice as many things in it as my bag does.
If I have 5 things in my bag, how many things are in your bag?

9. Your bag has twice as many things in it as my bag does.
I have a bag and you have a bag.
Your bag has twice as many things in it as my bag does.
If you have 14 things in your bag, how many things do I have in my bag?

10. Your bag has twice as many things as mine does.
I have a bag and you have a bag.
Your bag has twice as many things as mine does.
But if you give me one of your things, we will each have the same amount. How many did you have to start with and how many did I have to start with?

11. You have three times as many things in your bag as I do.
I have a bag and you have a bag.
You have three times as many things in your bag as I do.
But if you give me one of your things, we will each have the same amount. So how many things did you have to start with and how many did I have?

12. Your bag has twice as many things in it as my bag does.
I have a bag and you have a bag.
Your bag has twice as many things in it as my bag does.
If you have 4 more things in your bag than I have in my bag, how many things do you have in your bag and how many do I have in my bag?
If you have 9 more things in your bag than I have in my bag, how many things do you have in your bag and how many do I have in my bag?
If you have 100 more things in your bag than I have in my bag, how many things do you have in your bag and how many do I have in my bag?

13. Your bag has twice as many things in it as my bag does.
I have a bag and you have a bag.
Your bag has twice as many things in it as my bag does.
But if you take out six from your bag and give them to someone else, we will each have the same amount. How many did I have to start with and how many did you have to start with? And how many do we each end up with?

14. Your bag has twice as many things in it as my bag does.
I have a bag and you have a bag.
Your bag has twice as many things in it as my bag does.
But if you take out six from your bag and, instead of giving them away to someone else, you put them in my bag, we will each have the same amount. How many did I have to start with and how many did you have to start with? And how many do we each end up with?

15. Your bag has 1 more than double what I have in my bag.
I have a bag and you have a bag.
Your bag has 1 more than double what I have in my bag.
So if I have 2 things in my bag, how many do you have in your bag?
If I have 8 things in my bag, how many do you have in your bag?
If you have 7 things in your bag, how many do I have in mine?

16. Your bag has 5 times what I have in my bag.
I have a bag and you have a bag.
Your bag has 5 times what I have in my bag.
If I have 4 things in my bag, how many things do you have in your bag?
If you have 20 things in your bag, how many do I have in my bag?

17. Again, you have twice as many things in your bag as I have in my bag.
I have a bag and you have a bag.
Again, you have twice as many things in your bag as I have in my bag.
But if I add two things to my bag, that makes us even. So how many things did you start with and how many things did I start with?
This time, however, if I add five things to my bag, that gives me two more things than you have. How many did I start with and how many did you start with? And how many do we each end up with?

18. What number comes next: 2, 4, 6, 8, _____?

19. There are two consecutive whole numbers (meaning two numbers in a row) that add up to 9. What are they?
There are two numbers in a row that add up to 15. What are they?
There are three numbers in a row that add up to 6. What are they?
There are three numbers in a row that add up to 12. What are they?
There are three numbers in a row that add up to 60. What are they?
There are three numbers in a row that add up to 96. What are they?

20. A square is a rectangle whose sides are all the same length as each other. All the sides are equal.
If this square is 4 feet all the way around, how long is each side?
If the square is 4 miles all the way around, how long is each side?
If the square is 16 inches all the way around, how long is each side?
If the square is 20 centimeters all the way around, how long is each side?
If you ride your bike around this square, the wheels go around 48 times (that is each wheel makes 48 full rotations). How many wheel rotations do they make on each side of the square?
If the square takes 1 hour to walk around, how many minutes does it take to walk one side?

21. A square is a rectangle whose sides are all the same length as each other. All the sides are equal.
If this square is 1 inch all the way around, how long is each side?
If the square is 1 mile all the way around, how long is each side?
If the square takes 2 hours to walk around, how many minutes does it take to walk one side?
If the square takes a whole day and night to walk around at an even pace, how many hours does it take to walk one side?

22. A square is a rectangle whose sides are all the same length as each other. All the sides are equal.
If it took 6 wheel rotations on each side, how many wheel rotations would it take to go all the way around this square?
If it took 2½ hours to walk each side, how long would it take you to walk all the way around this square?

23. A square is a rectangle whose sides are all the same length as each other. In other words, all the sides are equal in length to each other. D A C B
Suppose you rode your bicycle around this square, and it took you longer to go from side B to side D than it does to go in the opposite direction from side D to side B. What could cause that?
Suppose you rode your bicycle around this square, and it took you longer to go from side D to side B than it does to go from side B to side D. What could cause that?
Suppose you rode your bicycle around this square, and it took you longer to go from side A to side C than it does to go from side C to side A. What could cause that?

24. This rectangle is twice as long as it is high, meaning the longer sides are two times the length of the shorter sides.
If this rectangle is 6 feet all the way around, how long is each side?
If the rectangle is 12 miles all the way around, how long is each side?
If the rectangle is 18 feet all the way around, how long is each side?
If the rectangle is 24 inches all the way around, how long is each side?
If the rectangle is 48 centimeters all the way around, how long is each side?
If the rectangle takes 60 minutes to walk around, how long does it take to walk each side?
A “lightyear” is a measure of distance, not a length of time like it sounds it would be. One lightyear is the distance light travels in one year. If the rectangle is 90 lightyears all the way around, how many lightyears long is each side?

25. This rectangle is three times longer than it is tall, meaning the longer sides are three times the length of the shorter sides.
If this rectangle is 8 feet all the way around, how long is each side?
If the rectangle is 16 miles all the way around, how long is each side?
If the rectangle is 24 feet all the way around, how long is each side?
If the rectangle is 48 centimeters all the way around, how long is each side?
If the rectangle takes 64 minutes to walk around, how long does it take to walk each side?
One lightyear is the distance light travels in one year. If the rectangle is 80 lightyears all the way around, how many lightyears long is each side?

26. If it takes three hours to walk to your grandmother’s house, and you can bicycle twice as fast as you walk, how long will it take you to bicycle there?
If you can bicycle three times as fast as you can walk by peddling harder, how long would it take you to bicycle there?
If it takes a train 10 hours to go from one city to another at 60 miles per hour without stopping, and an airplane flies between them at 600 miles per hour, how long would it take the plane to fly from one to the other?
What if it took the train 50 hours? How long would it take the airplane?
Suppose your dog runs twice as fast as you do. How long would it take you to run 200 yards, if it takes your dog 12 seconds?

27. If your dog runs twice as fast as you do, how much of a head start do you need in order to at least tie if the dog runs 100 yards for the race? Where do you need to start at to be able to tie?
Suppose you and someone else drive at the same speed, and you plan to meet at a city that is part way between both of you -- but 100 miles from where you are and only 50 miles from where they are. If it will take them an hour to get there and they start at 4:00 in the afternoon, what time do you have to start in order to arrive at the same time they do?
Suppose, that in the above case, you could drive twice as fast as they do because you have freeway all the way and they have congested streets all the way. What time would you have to leave to arrive at the same time they do if they leave at 3:00 in the afternoon?