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Category Archives: Math: Teaching Tips

Back to School Homeschool: Math Teaching Tips: Increasing Understanding

Math Teaching Tips: Digging Deeper

Over the years of homeschooling, I have become better and better (I think) at getting out more of math work than the kids just getting the correct answer. I’ve learned techniques to help us dig deeper and check if there is true understanding or whether there was just rote memorization of a process that resulted in a correct answer but may not be of benefit if the process is forgotten in future or cannot be applied/usedeasily if the problem does not fit a certain format.

Below are some techniques or “checks” that I use when I give the kids mental math problems, but I also use for written work.

1.   Does my student understand what he is looking for?

Make sure that your student understands what he is being asked to find. Sometimes the kids may get a wrong answer and then we find out that they didn’t hear the problem correctly or mixed up the numbers. In other cases, they just may not be sure what they need to find. Have student repeat/or restate what they need to find.

2.   Is the answer reasonable?

After your student comes up with an answer, ask them if the answer is reasonable.  Do this before they give you their answer. If they need help or say yes, but it really isn’t, I try to guide them along by telling them to estimate the answer or predict what the answer should look like.

 

Example 1:   Student is asked to estimate 25 x 37. To check for reasonableness of an answer, I would suggest that student start with 25 x 10, since 10 is a rounded number. 25 x 10 = 250, so then student could compare his answer with that. Then, I would suggest that my student multiply 25 x 40 (but remember that 25 x 10 was 250, so 25×40 would be 4x as large since 40 is 4x as large as 10).

 

One problem we had recently was 4 ¼ divided by some number = 5. I asked, what should the answer look like?  Well, usually when you divide by a number, your answer is smaller than what you started with. But I pointed out the fact, in this case, the answer is larger.  So, in this case the number we are dividing by would have to be a fraction (less than one)  I had to guide my student with this, but I tried to stress this fact or generalization so it could be applied in the future, in sha Allah.

 

Try to be consistent with having student check for reasonableness to help student build the habit of learning to automatically do this on their own, in sha Allah.

 

3.   How did you get your answer?

 

Have student explain the process they used to get their answer. Do this regardless of whether the problem is incorrect or correct.  Sometimes after my student gives their response, I try to help them revise it to make it more concise.

I may also, then tell them toexplain (as if they were explaining to  someone)how to do a similar problem (in general, regardless of the numbers). This requires that they use math vocabulary that can be applied to any problem (e.g. if you want to tell someone how to multiply ¾ x ¼  you might say, multiply the 3 x 1 and then 4 x 4, but in general (for any problem), you might say, multiply the numerators and then multiply the denominators. Thus now you have an explanation for any such problem.

 

If student is stuck on a problem, I have them tell me what might their first step be, then what, then what……

 

4.   Give a related problem to help student see patterns.

After my student has worked out a problem, I might give a related problem to see if they can work out a pattern that can be applied. 

 

For example, we had a problem of 5 percent of 80. Once they got the answer, I asked them what 20 percent would be, what 40 percent is. I try to get them to see that 20 percent is 4 times 5 percent and that they could just multiply their initial answer by 4 and then to get 40 percent, multiply the initial answer by 8.  I also teach them to identify what 10 percent of a number is and then use that to find what is asked if it is a multiple or factor of 10 (5 percent,  20 percent, 60 percent, etc. I think that doing this consistently builds up their ability to find easy percents like this faster.

 

5.   Make the problem simpler.

One of my famous lines is to “make the problem simpler” before they get into calculations. So, for example, if they are dividing 40/20, I tell them to take off the zeroes and then they have 4/2.  Whenever I have them make the problem simpler, I try to make sure that they understand what is going on (that taking off the zeroes is like dividing both by 10). In another problem, they might have 1000/25. In this, I suggest a “tricky” shortcut and they have to be sure to correct for this.  So I say, take off one zero and now I have 100/25 which is 4 and they have to remember to put the zero back on on top and thus adding a 0 to end of their answer of 4 to get 40.

If the problem was 7 x 1/7, I would remind student to use the multiplication shortcut by cancelling out the 7s as they could think of the problem as 7/1 x 1/7 so we have 7 x 1/(1 x 7) but they can rearrange it so that 7 x 1/ (7 x1) and the 7s cancel out to make 1 and they are left with 1 x 1 = 1.

 

6.   I’m stuck!

When doing written work, when one of the kids doesn’t understand what to do, he writes “don’t understand.” I have started to ask him to write specifically what it is he doesn’t understand or how he is stuck. In this way, I can get him toattempt to go past the “looking at the problem and drawing a blank stage and just moving on.” 

 

7.   Perimeter and Area problems

In our review work (I use NC’s Week by Week Essentials regularly) they are often asked to find perimeter or area. I noticed that they would get the two mixed up (i.e. find perimeter when they needed to find area). In other cases, they would simply find the area of say a rectangle pool but what was asked was the area of a walkway around the pool. So, I told them, for any perimeter or area problem to ALWAYS do the following:

  1. Draw a picture
  2. Write a formula

and I kept drilling this simple routine into their heads each time and Alhamdulillah, now, they draw a picture and write a formula pretty automatically and their work of this type is more accurate now.

 

8.   Make a plan

In general, I now see how helpful it was back in school when my math teachers would have us write out the following for each problem:

 

  1. Find: (What I am asked to find)
  2. Given: (What I know)
  3. Steps:

 

Sometimes, I have the kids write up a little paragraph on the steps of how they solve a general problem (such as how to determine if a number is odd or even). For my kids that are not as skilled in expressing themselves well in writing, I might provide a frame:

 

How To Tell If a Number is Odd or Even

Numbers can be ____________________ or _________________. Here is how to tell if a number is _____________ or _______________.  A number is _____________ if __________________________. For example, _______ is __________ because _______________________________. A number is  _____________ if ____________________________. The number ______________ is ____________ because ___________________________. Now, you know how to tell if any number is ___________________ or ___________________.

 

 

 

 
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Posted by on August 26, 2011 in Back to School Homeschool, Math: Teaching Tips

 

Did you check your work? (Math)

Do your kids check each and every math problem they do to see that it is correct?

One area that I think teachers don’t get to spend enough time doing is reinforcing the habit of checking  work. Sure they have posters reminding students to do this and tell students repeatedly, but do they really have the time to actually inculcate this habit in their students?

One of my latest campaigns is to really focus on my kids checking their work.  I think that with homeschooling, you have the perfect opportunity to do this as you tend to have more face time with each of your students. 

That being said, I have made a poster (smile)  to remind the kids to check their work, but I have also taken active steps to try to help ensure that they learn this habit, in sha Allah. For example, two of the kids are on division, so instead of giving them a whole page of problems (for other reasons this may not be productive as well), I give them fewer problems and insist that every problem be checked (I can easily tell if they have done this because they multiply back out and arrive at the dividend that they started with. This way, they know that they go the answer right, and I know (by the way, this cuts down on me having to physically do the problems myself to check as I don’t have an answer key). They may not go onto the next problem until they see that the answer is correct. 

Another step that I am trying to take to get them to correct their work is to pull back helping them when they get stuck (as I am prone to do). I think in the beginning this is good so that they can see the steps I take to go back and check, but eventually I tell them to: 1. check their checking (if they multiplied to check the work, a problem may lie in the checked part) and 2. go back through the problem again, most of the time, they subtracted or multiplied incorrectly (underscoring the importance of them getting those math facts internalized). My point being that I am turning the ownership more over to them.

So here’s my little poster, maybe it will be of some help. One thing I hope to do is take the check list and walk through the list with my kids (while correcting their work) to see if they have done each step and point out where they haven’t.

 

Download “Checking Your Work” Poster

image

 
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Posted by on October 12, 2010 in Math: Teaching Tips

 

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Is Your Math Curriculum Effective?

I recently came across an article at Homeschool Math.net http://www.homeschoolmath.net/teaching/coherent-curriculum.php entitled “Is Your Curriculum Coherent?” Although it sort of has a plug for its products at the end, it raised some excellent points about Math curricula in the states. I definitely recommend checking it out especially if you are a little leary of the typical math curricula out there for purchase these days.

One of the points that the article made was that today’s US curricula (for public schools generally) is that they teach so many topics in one year, not giving enough focus to any one topic in depth. The author suggested that one look at consecutive grades and take the topics from that was presented in both books to give a more in depth treatment of the subject. I have done this before with my older son as he is “behind” in math and I wanted to “catch” him up. So for example, fractions were introduced in one book and I felt that he could handle even more exposure on the topic so I went to the next grade and continued with the subject instead of going to a totally new topic in the next chapter. This was very effective, masha Allah.

Some of the main points that the author brought up with problems with US math curricula:

(from the article)

  • not focused. No country in the world covers as many topics as US in their mathematics textbooks. For example, in Japan, eighth-grade textbooks have about 10 topics whereas US books have over 30 topics.
  • highly repetitive. The average duration of a topic in US is almost 6 years (!) versus about 3 years in the best-performing countries. Lots of spiraling and reviewing is done. Like Schmidt says, “We introduce topics early and then repeat them year after year. To make matters worse, very little depth is added each time the topic is addressed because each year we devote much of the time to reviewing the topic.”
  • not very demanding by international standards, especially in the middle-school. In the USA, students keep studying basic arithmetic till 7th and 8th grade, whereas other coutries change to beginning concepts in algebra and geometry.
  • incoherent. The math books are like a collection of arbitrary topics. Like Schmidt et al. say, “…in the United States, mathematics standards are long laundry lists of seemingly un- related, separate topics.”

The article discusses some of the things you can do attempt to fix some of these problems with your current math book/curriculum. I haven’t gotten a chance to look at the products, but the author talks about the products which are topic books, each book focusing on one topic to give a more “comprehensive, in depth learning experience of a few connected topics.” The whole package is $47 which doesn’t seem too bad, compared to the price you will pay for some curricula. If anyone has tried this product, I would love to hear what you thought.

I just wanted to share this article as I found it very interesting and made me take a new look at math and how to more effectively teach my children, insha Allah. I currently use older textbooks and just am not happy with how material is presented.

Another math site that is really helpful is www.aaamath.com. They have lessons from K – 8 grades with interactive activities. When I first looked at this site I thought, gee I could really just use this for lessons, then I looked at it and wondered what kind of order to present the material, it was grouped by subjects, not arranged haphazardly like a lot of math textbooks where a topic is introduced and then something other may come in between and then an that initial topic is revisited, just a little more advanced. In light of reading the article above, I realized that you just follow the sequence given, working on one concept at a time. Sometimes I find in a math book that even within a chapter that seems to be on one topic, they will throw in other unrelated topics . So I highly recommend checking this out if you don’t have a math program, its free. There is even a progress report where you can record your student’s progress. They have simple interactive practice at the computer (no fluff but gets the job done) and is probably great for those students who don’t like to do a lot of pencil and paper math. If you fill student needs more written practice, you can probably find plenty of free worksheets online for the topic you are studying.

Tip: If you use a program such as Teleport Pro (and there are many programs like it) you can grab the sites and save them to your hard drive or flash disk and view them off line. I found that the interactive activities still work offline. I did the same with the science curriculum see Sweet Science post, so if you have limited internet access, this is a BIG help.

And lastly, I highly recommend checking out North Carolina Department of Education Math resources. http://community.learnnc.org/dpi/math/archives/instructional_resources/

You can find links to the materials on my homeschooling helps by the grade pages at TJ. I find that giving my kids these problems/activities really stimulates them to do more out of the box thinking. They have 36 weeks of activities and a weekly printable game and review sheet as well as weekly mental math activities. Its been just about my favorite math resource since I discovered it. You can’t really use it as your math program I think, but as a supplement. It also has blackline masters for printing and packets that give you activity ideas to provide practice with concepts for your students/children.

 
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Posted by on November 8, 2007 in Math: Teaching Tips

 

Math Teaching Helpers – Using a picture to solve a problem

 

I am a very process oriented person and I like to jot down tips/points that I use to explain concepts to my kids so that when the next one comes along to a concept I don’t have to start from scratch.

 

 

Grade 3

 
 

Using a picture to solve a problem

 
 

 
 

We’ve recently gone over this in our textbook and I jotted down some notes so that when the next 3rd grader comes along, insha Allah, I could refer to them.

 
 

  • Explain to student that sometimes it is helpful to draw pictures to help solve problems. I explained that I sometimes use pictures to help me solve everyday problems.

     
     

  • Read the whole problem through. Have student determine what it is he is looking for/trying to find.

     
     

  • Go back and read ONE sentence at a time and draw only that information before going on to the next part.

     
     

  • Tell student that pictures do not have to be artistic quality and that simple shapes or symbols will do. Emphasize to student that details are not important.

     
     

    • Example: To find the number of rose bushes, a student might simply make an “r” instead of drawing detailed roses. Emphasize to student that details are not important.

     
     

  • After student draws the picture, have him go back and see if the picture matches the problem.

     
     

  • Have student solve the problem. Encourage him to look for patterns, and shortcuts.

     
     

  • Have student check answer and see if it makes sense.

     
     

    Example problems:

     
     

  • At the masjid, there are four rows of prayer rugs with seven rugs in each row.

    Behind those rows, there are two rows of five prayer rugs. How many prayer rugs are there in all?

     
     

  • At the masjid, there are 4 long shelves for shoes. On the first shelf, there are six pairs of shoes. On the second there are seven pairs of shoes, on the third shelf there are nine pairs of shoes, and on the fourth shelf there are two pairs of shoes. How many pairs of shoes are there in all?

     
     

  • A building has six stories. On the first and third stories, there are 4 windows. On the second and fourth stories, there are five windows and on the fifth and sixth stories, there are 3 windows. How many windows are there in all?

     
     

  • At the masjid, there are many copies of the Quraan in a book case. On one shelf, there are four copies. On another shelf, there are 5 copies. On the last shelf, there are six copies. How many copies of the Quraan are there in all?

     
     

  • At a clothing store, there are four racks of women’s clothes. On one rack, there are three black hijabs. On the second rack, there are six green hijabs. On the third rack, there are two brown hijabs, and on the fourth rack there are 3 blue hijabs. How many hijabs are there in all?

     
     

  • At an Eid party, there were yummy goodies served on three trays. On the first tray, there were six crescent shaped cookies and five masjid shaped cookies. On the second tray, there were four prayer rug shaped cookies and six star shaped cookies. On the last tray, there were eight crescent shaped cookies and four kabah shaped cupcakes. How many treats were there in all?

     
     

     
     

 
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Posted by on May 31, 2007 in Math: Teaching Tips