Homeschool Math Blog

It's time to talk about WORD PROBLEMS! Why is it that many children have such trouble with them? Image by RadioFlyer007 (black border removed) Licensed with CC BY-NC-ND 2.0 I've written some thoughts about this issue... I feel it is fairly comprehensive article , and hopefully helpful! Let me know what you think. Here's the outline of the article: The problem The solution Problems to ponder/solve Student's Misguide to Problem Solving Resources And here's the beginning: The problem Have you ever noticed this kind of "recipe" for math lessons in many math books? LESSON X Explanation and examples. Numerical exercises. A few word problems. In other words, the word problems are usually in the END of the lesson, and just a few. But worse... if the lesson is about topic X, then the word problems are usually about the topic X too! Children might be learning about multi-digit multiplication, or subtraction, or dividing decimal...

Someone asked for help to explain the concepts of multiplicative word problems involving "as many as", like the ones below: 1) Haley had four times as many dollars as her sister. Together they had $60. How much money does Haley have? 2) Rachel had 5 times as many dollars as her sister, Nora. They had a total of $90. How much money did each of them have? The BAR MODEL is an excellent tool for helping children understand what is going on in these types of word problems. In (1), draw a bar for Haley and another for her sister. Divide Haley's bar into four parts, and make the other bar just one such part long. Haley |---|---|---|---| Sister |---| Now you will see that the TOTAL needs divided into FIVE equal parts — and from then on it is easy-peasy. Additionally, you can use Thinking Blocks website to build such bar models INTERACTIVELY. For problems like (1) and (2) above, choose the fourth model from the left (rightmost) in the top row that is b...

Someone asked me about the BAR MODELS in the problem solving lessons in Math Mammoth grade 5. This person said that they had just completed every problem in the lesson Problem Solving with Bar Models 1, but her son is still having a hard time knowing how to make the bar model after reading the problem, and knowing what to do with the fractional part once it is figured out. She was wondering if there was something else for supplemental practice , and if she should look for mastery in this area at this point. MY ANSWER : I feel it would be good to achieve some degree of mastery at this point (in 5th grade). I suggest  using the website Thinking Blocks for additional practice, in particular the "Multiplication and Division" and "Fractions" subsections. Also, if the problem solving lessons are difficult, consider spreading out the study of these lessons in Math Mammoth by starting the child simultaneously on the next chapter. For example, one could s...

I have used Math Stars problem solving sets with my girls for several years, and they have always greatly enjoyed them. Math Stars include various puzzles and challenging math problems. They  come as PDF files (free and ready to download & use) in sets for grades 1-8. They're great to use for summer math  or for some fun problem solving at any time. I tend to use the problems from one grade level below the grade the student is in. One reason I like these so much is the variety of the problems - geometrical puzzles, number puzzles, logical thinking, etc. --  all of it is included! Another reason is that the sets go all the way from grade 1 upward. There exist actually lots of problem solving resources but not so many for the early grades. Each set per grade has 10 two-page "newsletters", and each "newsletter" has 8-10 problems. The number of stars in each problem marks the level of difficulty. Answers are included. Math Stars Problem Solving ...

Someone recently asked me about showing work in math word problems and I thought others might enjoy hearing about this topic also. Personally, in the lower grades, I'd ask the child to EXPLAIN their thought processes orally, and then gradually teach them to write something on paper. The main thing students in grades 1-3 need to write is the actual calculations they did, not only the final answer. For example, if they added 23 and 87 to get the answer, they should write 23 + 87 = 110 and include the units of whatever it was, such as $23 + $87 = $110 or 23 cm + 87 cm = 110 cm. In the upper elementary grades (4-6) I'd like to see students write sentences and/or words in addition to the calculations so that another person can follow their solution. I'll give some examples. (from 4th grade) Mr. Jefferson travels from Paducah to Lexington and back, three times a month. What is his total mileage? (A map shows that the distance in question is 255 miles.) An example...

Someone asked me recently, I have had great success with the Grade 4 Light Blue Series with my twin daughters and we have been breezing through the curriculum. ... For the most part, they get how to find the fractional part but we did have some trouble on problems 4 on page 59 Worktext 4-B and problem 7 on page 60 in Worktext 4-B. They have to do with finding the total number of something when you are given a fractional part. I had a hard time teaching the concept of "working backwards" and was wondering if you have any additional resources or problems we could use to practice. Working backwards is of course perfectly FINE but I feel using the BAR MODEL is easier for children who have not worked through these kinds of problems before. The model shows them clearly HOW we "work backwards" to get the total. Here's an example from the previous page (page 58 in Math Mammoth 4-B ): This is the problem #4 they had trouble with One cold day, 1/8 of ...

I recently mentioned about changing math problems from closed to open , and gave a few examples. Just a few days ago my 5th grader was facing a page in Math Mammoth 5 filled with word problems that had to do with conversions between measuring units. So... I asked her to choose and solve 3 problems from that page, and THEN write her own story problem where the solver will need to convert measuring units. This is what she came up with. I guess I should have guessed the topic. :^) Photo by http://www.flickr.com/photos/stradablog/ There are three horses in a paddock, and each weighs a different amount. The first horse weighs 13,760 ounces, the second 1/2 ton, and the last one 1,100 pounds. What is the weight of all the horses together, in pounds? No calculator allowed. She needed some help, because in her original version, the weights of the horses were not realistic -- but it was good to have a discussion on that, as well. Try it yourself! Ask your kids to make story pro...

Want to do some problem solving? MathCounts Handbook for 2012-2013 not only includes information about their programs, but also creative and fun 300 problems to work! Answers are included as well. You could use these problems as challenging word problems or for problem solving workouts, not only for grades 6-8, but I'm sure for high school as well, since they are (mostly) not your typical math book problems but require some thinking. Here's the direct link to the Handbook: http://mathcounts.org/document.doc?id=886 If that doesn't work, go to the program's home page and scroll down to see the link to MathCounts Handbook.

Have you ever used this neat way of teaching math word problems to elementary grade students? Let them be like stories, and let the child create some too! Denise explores this idea in her recent post: Tell Me a (Math) Story Worth checking out!

Someone (a student perhaps?) sent me this problem From a horizontal distance of 8.5km a pilot observes that the angle of depression of the top and base of a central tower are 30 degrees and 33 degrees respectively.  Calculate  a. the shortest distance between the pilot and the base of the central tower b. the height of the central tower. Actually, a good part of this problem is to draw the correct diagram. Once you have the image right, then the math should be fairly obvious. But I'm going to help you out! Here's the image for this situation: (not to scale) The solution is posted here .

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Someone sent me a nice word problem to solve:   Image courtesy of Wednesday Elf - Mountainside Crochet Bob went to the zoo last week and was captivated by the lovely baby elephants. Some were pink and some were blue. When he counted the elephants, he found that the number of pink elephants was 2 1/2 times the number of blue elephants. What fraction of the elephants was blue? First of all, think: which ones are less, blue or pink? Clearly, the blue ones are less. Make those kind represented by ONE BLOCK or one unit. So, let's let the BLUE elephants to be |----|  (one block). Then, the pink ones would be |----|----|--| (2 1/2 times as many) Now, to get the fraction asked, I could use those little dashed lines in my blocks... See, I made each block have 4 little dashes ---- and the half-block has two. So, the Blue elephants are "four dashes" and the pink ones are "ten dashes". I know that in reality we don't know how many blue or pink elephan...

Someone sent me this problem to solve, so here goes: Photo courtesy of Brajeshwar Square polystyrene tiles, 50 cm by 50 cm, are used to cover the ceiling of a classroom measuring 7.4 m by 4.5 m 1. Find the number of tiles that are needed. 2. Find the total cost if one tile costs $130. Now, consider the room that is 7.4 m by 4.5 m. The tiles will not fit exactly along the 7.4-meter side, so we will need to cut some of the tiles. So we might as well consider it to be 7.5-meter side for the purpose of figuring how many tiles we need. Along the 7.5 meter side we would need 15 tiles, each 0.5 m. Why? Because 0.5 fits into 7.5 exactly 15 times. It's essentially a division problem... 7.5 ÷ 0.5 but you can solve it mentally by thinking that each meter takes 2 tiles, so 7.5 meters takes 7.5 x 2 = 15 tiles. Similarly, along the 4.5 meter side you would need 9 tiles, each 0.5 m. Therefore in total we need 15 x 9 = 135 tiles. Some of those will need cut! The cost will be 1...

These videos show you examples of how to use the bar or block model in solving math word problems. The examples are all about 5th grade level. Enjoy! I hope they are of help! First I solve the following word problem using a bar model (Singapore math style), taken from Math Mammoth grade 5 curriculum : Brenda and Lily shared the cost of a $11.70 lunch so that Brenda paid two times as much as Lily. Find their shares. Next problem is this: One rake is $5.60 more than the other, and together they cost $22.70. How much does the cheaper rake cost? And lastly, John spent 3/10 of his money, and had $45.57 left. How much did he have initially?

Today, my daughter had to tackle this word problem in her 5th grade Math Mammoth (lesson Multiply Mixed Numbers): The square on the right measures 10 in. × 10 in. and the rectangle inside it measures 6 7/8 in. × 3 1/2 in. How many square inches is the colored area? This requires the student to multiply mixed numbers,  subtract mixed numbers,  understand about area, and  understand how to find the "colored" area by subtraction of areas.  So it is a multi-step word problem. She didn't understand it, she said. My first "help" was this: "Let's say we change those fractions to whole numbers 6 and 3. Can you mark those in the image? Would you be able to solve the problem now?" The strategy I used is: If you can't solve the problem at hand, change it and make it easier. Then try to solve the easier problem. She was able to mark 6 and 3 on the sides of the rectangle (that is inside the square). But she said she couldn't solve it....

Photo credit: leojmelsrub Ahmed sent me in this kind of word problem: A tea producer want to market mixed green tea leaves at $14 per pound. how many pounds of high mountain green tea leaves worth $20 per pound must be mixed with 90 pounds of regular green tea leaved worth $10 per pound? I have solve many problems like this one on my blog, but it never hurts to solve some more. This can be solved with algebra, using a chart. I've done that before for similar problems... so if you are reading this, and you feel a bit "rusty" in this area, try to make the chart yourself first, before you read further! For the chart, we also need to choose a variable or several. In this case it is easy:  the unknown is obviously what is asked, or the amount of high mountain green tea. Note also that the cost is always the price per pound times the amount. mountain green regular green the mixture tea leaves tea leaves -----------------------------------------------------------...

I have 2 questions on fractions which I can't solve. There were 3/5 as many adults as children on a bus. At the next bus stop, 6 adults and 6 children boarded the bus. As a result, there were 2/3 as many adults as children on the bus. How many people were on the bus at first? A solution using bar diagrams (Singapore style): children |----|----|----|----|----| adults |----|----|----| Then we have: children |----|----|----|----|----| +6 adults |----|----|----| +6 Here look at the difference of children and adults. There are two "blocks" more children than adults. We know the number of adults is 2/3 of the number of children... therefore the DIFFERENCE of two blocks must be 1/3 of the children. Now look: children |----|----|----|----|----| +6 Two of those blocks is 1/3 of the total... the other two blocks is another 1/3 of the total... so |----| +6 or one block and 6 must be 1/3 of the total. This means the +6 must be one block. Or, one block = 6. This no...

Today I had the opportunity to solve a real math problem involving a circle and a chord of known length in it. I had to find the radius . It wasn't a textbook problem or a puzzle on some website, but a math problem I needed to solve for my own needs. For a tiny while I thought I could find the answer online, but I didn't, so I'm writing it out in case someone else needs it -- they should be able to find this solution by searching the Internet. I wanted to make a kind of " moon-sliver shapes" in CorelDraw, to use as watermarks in my new books. I have the height and the width of the "sliver". Here's the problem mathematically: I have a chord of a circle, 17 mm in length in my example, and the other distance marked in the image is 5 mm. I need to find the radius of the circle, AND the angle measure of the arc of the circle that makes the sliver's rounded part. At first, like I said, I searched around if there was some theorem or ...

Here's a word problem that someone sent me recently: The total number of girls in the combined junior and senior classes is equal to the number of boys in those two classes. If the senior class has 400 students and the junior class has 300 students, and if the ratio of boys to girls in the senior class is 5:3, what is the ratio of boys to girls in junior class? This problem gives a lot of information, and it sounds like it can be solved many different ways. But the first task is to notice what we are given and what we are asked . We are asked about a ratio .  We're given one ratio, and all kinds of totals. There are boys and girls , senior and junior classes. In other words, there are four groups: senior boys, senior girls, junior boys and junior boys. This sounds like it can be solved by setting up some equations and using algebra. To start, you could for example notice the two facts given about the senior class: The senior class has 400 students and the ratio of...

I was asked just recently about word problems and WHAT operation to use in them. Can you put a better explanation of when to use addition, subtraction, multiplication, and division, because my son is having a hard time understanding. Many children have the same problem, and I have written about it in the past as well. In a nutshell, I feel the MAIN reason students have trouble is the way word problems are presented (or not presented) in math books . Typically (and I still see this approach), when a lesson in a math book is on some operation , then the word problems in that lesson are typically solved using that particular operation . So, any intelligent kid who notices this pattern won't take time to decipher what the problem says, but will just take the two numbers that appear in the problem, and apply the operation that the lesson is about! (For example, if the lesson is on subtraction, then the word problems are solved using subtraction.) I have tried to AVOID this...