A Math Tutoring Report With Practical Suggestions

Over the next while, I will be posting samples from actual reports I have made to parents of my tutoring students. Today’s topic is some practical suggestions for a child who is struggling with primary-level math. Hope you’ll find some helpful tips here!

Addition of Double-Digit (or Larger) Numbers

J was sure he wouldn’t be able to add double-digit numbers. So I showed him some practical ways to make it easier. We started with double-digit numbers that don’t require carrying (numbers added don’t add to more than 9 in any column). I emphasized that it is important to line up the numbers carefully so as not to become confused. To help him, we used cm graph paper (the squares are large enough for his handwriting of each number to fit in each square; there are different sizes of graph paper available, so choose a size that fits the child’s handwriting size). Then I had him add–starting with the ones column, then the tens column … and then I put down hundreds … and even thousands and ten thousands columns. He was very impressed that he could successfully add such large numbers! Building confidence is so important. Once he feels very confident with adding this way, we’ll talk about place value, and then introduce carrying.

Using coins

 

J also doesn’t seem to clearly understand how easy it is to add 5 to 10s. For example: 20 + 5 = 25. You can help him practice this at home by using dimes and nickles. For example, have him count 5 dimes and then add a nickle. This usually helps children catch onto this concept easily.

We practised counting coins — pennies, nickels, dimes and quarters. But while J can count them, it is much more difficult for him to try and figure out what coins to use for a given price. But, as with most children, he likes using coins, so this is something you can easily do with him for practice. Just write down a price–or “play store” and have him choose what he’d like to buy for the prices you’ve marked on items (for example, 38 cents or 55 cents), and have him figure out which coins to use. Let him figure out different combinations. For example: for 38 cents he could use 25 + 10 + 3 pennies OR 10 + 10 + 5 + 5 + 5 + 3 pennies. I realize that officially we don’t use pennies anymore in Canada, but we still need to be able to figure out whether to round up or down, so if you have pennies, it’s good to use them … and besides …

Rounding

J also needs practice with rounding in math. Using pennies, nickles and dimes is a great way to practice rounding! For example, 18 cents: dime, nickel, 3 pennies. To round, we want to just end up with dimes. So count the pennies and nickel. That’s “5 or more” so we round up and it becomes 20 cents (2 dimes). But if it was 13 cents (dime, 3 pennies), we can count the pennies and since they are “less than 5” we round down to 10.

Please also note: When rounding, always ask if the number to be rounded is “less than 5” or “5 or more.” Be careful NOT to say, “less than 5 or more than 5” as that leaves out the 5!

Numeral Order

 

J can generally do “> greater than”, “< less than”, and “= equal to”. However, I noticed that he said 7685 = 7865. I asked him to explain it, and he said that both numbers each contained a 7, a 6, an 8, and a 5. He didn’t seem to realize that just because the numbers contain the same digits, doesn’t mean that they are the same numbers if the digits are in a different order. So we talked about that and did some practice. This is something you’ll want to keep an eye on as you work with him. Really emphasize the “ones, tens, hundreds…” Use the graph paper, as discussed above, so he can clearly see the difference. Have him write the digits in different orders and read them aloud. If he is having difficulties reading the numbers, write abbreviations at the top of each column: th (thousands), h (hundreds), t (tens), o (ones).

What Calculation Method Should We Use?

In the workbook you are using with J, a wide variety of methods are presented for doing certain kinds of calculations, for example, double-digit addition. This can be confusing OR it can be helpful (for parents as well as for the student). My feeling is this:

  • First try the simplest/standard method.
  • If your child is having difficulty with the simple method, try one (or more, if needed) of the other methods, until you find one that your child “gets”–and then do lots of practice with that. Once your child understands the overall concept, it is wise to go back and learn the simple/standard method, as the child will no doubt run into it at various points later in math–and in “real life.” Note: You may discover that the method that helps your child understand the concept is different than the method that you think is easiest/most helpful. We all learn in different ways–so you as a parent may very well end up learning some surprising new ways to deal with a concept!
  • If your child “gets” the simple/standard method, but finds the concept very intriguing, encourage “exploration” of some of the other methods. Learning a variety of approaches can actually help to prepare the child for more complex mathematical concepts in future, as well as using them in “real life” applications.

Counting Backwards

J seems to have some difficulty with counting backwards … and especially counting backwards “over decade breaks” (for example: 62, 61, 60, 59, 58). This is a good thing to practice by using “real” learning opportunities. For example, as you walk along a sidewalk, count each square, starting with a number like 33 or 54 or whatever, and count backwards from square to square, making sure to go back into the next “decade.” You can also use things like “100 number charts” (with numbers in rows of 10), and count backwards, pointing at each number as you say it. Or you can use dot-to-dot pictures that go up to at least 50, but start at the end number and draw the dot-to-dot line backward to the beginning, counting aloud from number to number. Or you can use calendars and count backwards from the end of the month (eg 31) to the beginning (1). There are lots of different fun ways to learn to count backwards–just keep your eyes open for ideas!

Writing Numbers Backwards

J tends to write some numbers backwards. As he tends to be a “hands-on” kind of learner, try having him “write” these numbers with his finger on a textured surface (such as a piece of sandpaper or a textured fabric or on a sidewalk) so he can “feel” the number–or he could write in sand, or in finger paints, or in shaving cream. Or have him “trace” magnetic or foam or plastic numbers, first with his eyes open, and then with his eyes closed. Another idea is to have him close his eyes and you can “write” the number on the palm of his hand with your fingertip (or you could “write” it on his back) so he can get the feel of the number. Also, have him “write” it in the air (or on a window or other surface that has condensation on it–kids love this!) with his finger.

Learning Arithmetic Facts

J really needs to learn his number facts, and it would be great if he could learn his “doubles” first, as he can then use them to easily calculate other number facts. Some methods to learn them: Repeat them over and over (aloud and/or written). Say them as you do an activity, such as bouncing or tossing a ball, climbing stairs, jumping on the trampoline, etc. “Sing” or “chant” them. There are CDs available that have songs or chants for number facts, or you can make up your own. There are lots of free number fact apps available for your smartphone, tablet or cell phone–and many of them have a “game” approach which really appeals to lots of children. Of course, there are also flash cards available very inexpensively at dollar stores or thrift stores–or you can use an ordinary deck of cards (just use the number cards) to practice addition, subtraction or multiplication facts; just draw two cards at a time. Dice also work well, as do dominoes.

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What practical math tips have worked well with your child/ren? Why not share them with us in the comments?  Thank you!

 

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A Sample First Tutoring Session Report

Do you wonder what kinds of things might happen in a first tutoring session, and what kind of a report you might receive from the tutor? Here is an example of an actual report from an initial tutoring session with one of my students, whom I would be helping with math. B was an 8-year-old student, homeschooling through a DL (Distributed Learning) school. He had been struggling with math, and his parents had decided to see if some tutoring could help him. He was nervous at first, but by the end of this first session, both he and his dad were excited about their upcoming math tutoring lessons!

Greetings!

So nice to meet B and family. I had fun! Hope you did, too 🙂 It was good to have B’s dad sitting in for this session, so he can see the methods I’ll be using, and can use them with B whenever they fit with the math in the workbook he is using at home.

Goals of our Tutoring Sessions

As this was B’s first day, my goal was to find out what B already knows in math, and what “gaps” (like missing ladder rungs or missing jigsaw puzzle pieces) he might have that we can fill in, which will make more advanced work much easier.

Getting to Know My Student

B told me he really likes cursive writing and art. He also showed me some work he is doing in his math workbook at home, and he said he likes when there are pictures, but not work without pictures. This suggests that he is a “graphic” learner, and we can take advantage of that to help him with his math. Also, he’s a pretty active, “physical” boy, so we’ll take that into account too, and use methods that include “active” learning.

Exploring My Student’s Skills and Needs

I used a workbook of math skill builders for grades 1-2 math with B, starting out with very simple work and lots of pictures, etc., to make him feel comfortable and confident. Here are some things I found out:

  • he is able to count well
  • he is able to tell what numbers “come after” but struggles with what numbers “come before”
  • he needs help with ordinal numbers (first, second, third, etc.)
  • he can “skip count” easily by 5s and 10s, and by 2s up to 10 (but then has trouble with teens and beyond). He really struggles with counting by 3s.
  • he knows quite a lot of his basic addition facts, and for facts he doesn’t know automatically, he knows to start with the larger digit and “count up.” His preferred method is “counting up” in his head (rather than finger counting or other similar methods).
  • he struggles with basic subtraction facts.
  • he is fine with “greater than” and “less than” (aka “more” or “less”)
  • he understands “tens” and “ones” for double-digit numbers.
  • he is able to measure with inches and centimeters, though he may be a bit confused about which is which.
  • he is able to add three numbers (single digits).
  • he was nervous about trying to add double digit and triple digit numbers, but once I explained about 1s, 10s, 100s, 1000s columns (place value), he had no trouble to add larger numbers (without carrying). I also introduced very simple single digit + double digit adding with one “carry,” and I think he’s really starting to “get it.”

Word Problems:

We did some “word problems.” B was able to read the word problems, more or less (he seems to have a bit of trouble with reading more complex words, but that could have been because of nervousness in today’s first lesson). He also wanted to solve the problems “in his head” and was hesitant to say the equations before giving the answer. It is really important that he expresses the equation, as word problems start to become more complex in grade 3 math, and he needs to be able to work them out “step by step.” I went over a couple of tips with him:

  • circle or underline the key numbers in the problem.
  • also, circle or underline the “keywords” in the problem, so he knows what kind of problem it is. For example, “in all” is a signal to add; “how many are left?” is a signal to subtract.

Tricks and Games Can Make Learning Math Fun

For the items he was having trouble with, I introduced a variety of “tricks.” I think Dad had even more fun learning the tricks than B did 🙂 He said he wished math was this much fun when he was in school.  Now that he knows the tricks, he can use them with B whenever appropriate work comes up in the workbook.

I also mentioned that I have a wide variety of math games, and we will use them during lesson times, when Brett needs a fun break. He is also welcome to borrow games he likes to play at home.

Here are some of the “tricks” I taught today:

  • Subtraction is just “backwards adding.” So if B learns his addition facts, the subtraction facts will be easy.
  • In fact, B only needs to learn half of the addition facts, because addition is commutative, which means, for example: 3+5 is the same as 5+3.
  • Once B has learned one fact such as 5+3=8, he has actually learned 4 facts that are in the number family of (3,5,8): 3+5=8, 5+3=8, 8-3=5, 8-5=3
  • I noticed that for addition facts (and subtraction facts) that B has not memorized, he was trying to “count add” them in his head–and it took him a long time. It is good he already knows to start with the larger number. While we ideally want him to be able to do the facts automatically (memorized), it may help him to learn them by using more “hands on” methods for practice. Some people are afraid their children will spend the rest of their life “finger counting,” but actually, using physical methods can be a very helpful way to learn facts, especially if the child is taught to use a variety of methods, so he doesn’t end up just using one and making it a habit. Some methods I will be introducing to him are:
    • draw sketches.
    • use a number line.
    • finger counting or counting with objects such as pieces of macaroni.
    • “touch math” … which I introduced today. This combines both “physical” (touching) and “graphical” (dots) which appear to be ways that Brett learns more easily. I have given Dad a sheet with the “touch dots” for the digits 1 to 9, and I showed Brett (and had him practice) adding using the dots. To start with, he can put the dots on the smaller number (for example, with 8+5, he can put dots on the 5), and then use his finger, or a pointer such as a pencil, and count: 8…9, 10, 11, 12, 13. After awhile, he’ll be able to stop drawing the dots and just use the finger/pointer … and before long, he’ll know the facts.

Doubles are Handy!

I recommend that you practice the “doubles” facts with B, as once he’s really comfortable with them, he can quickly and easily use them to do all kinds of “tricks” for other facts. The doubles he needs to practice are: 1+1=2, 2+2=4, 3+3=6, 4+4=8, 5+5=10, 6+6=12, 7+7=14, 8+8=16, 9+9=18, 10+10=20

Most children don’t have much trouble learning doubles. If B is having difficulty, here are some suggestions:

  • have him repeat them as a “chant.” The rhythm makes it easy to remember.
  • have him clap or tap his feet or tap a tabletop (or drum) as he chants them.
  • make it physical. As he says the facts, have him jump up (or down) stairs; or make a “hopscotch” on the sidewalk with chalk and have him jump from square to square as he says the facts; or bounce a ball (or toss it back and forth) as he says the facts; or call out the facts as he jumps on a trampoline; etc.

Once he’s learned the doubles facts, here are some fun tricks (which I showed B today):

  • for numbers that are one apart: example: 6+7 : double the smaller number and add one: 6+7= 6+6+1= 12+1=13 or 7+8=7+7+1= 14+1 = 15
  • for numbers that are two apart: example: 6+8 : think of the “inbetween number” and double it: for example: 6+8 = 7+7 = 14 or 7+9 = 8+8 = 16

Learning the 9s Facts Are Easy With These Tricks:

Another set of tricks are the “9s” tricks. There are several fun 9s tricks, for addition, subtraction, multiplication and division. Here are some tricks I showed him today:

  • to add 9 to another single digit number, the answer will always be a “teen” and the digit in the 1s column will be one less than the other number: for example: 9+5=___ The answer is a “teen” and the number in the one column will be (5-1) = 4. So the answer is “4 teen” = 14.
  • an alternative is to change the 9 to a 10, add the other number, and then subtract 1 … but I find the extra steps can be confusing for some students. If you want to try it, it would work like this: 9+5=____ (9+1)+5= (10+5)-1 = 15-1 = 14.
  • to subtract 9 from a “teen” number, the answer is 1 more than the other number in the ones column. So, for example, 17-9=___ The number in the ones column is 7 so the answer is (7+1)=8. Another example: 15-9=___ The answer is (5-1)=4.

 

Some Recommendations to the Parents

Don’t try to teach all these things at once. Just watch for opportunities to use the different methods as suitable work comes up in his workbook.

Also, it is very helpful indeed to watch for opportunities in “real life” situations, to do addition and subtraction. For example, when you are taking a road trip, get a map that shows distances between towns and have him add up the distance from one town to another. Or if you are baking an apple pie and doubling the recipe, and it calls for 8 apples, ask him what 8+8 is and ask him to count out that many from the apple box. Or if you are having a plate of cookies for supper, and there are 8 cookies to start with, just casually ask him how many will be left after everyone in the family has had one–and if there will be enough for everyone to have a second cookie. These kinds of “real life” experiences come up all the time; just keep an eye out for them, and use them in a casual way. His little sister will also love the game, and it will get her started on math facts without any formal teaching.

Oh! One more thing. If you want to practice “number facts” in some of the ways I’ve suggested, it works much better to just do 10 minutes or so daily, rather than 30 minutes twice a week or 60 minutes once a week. Frequent, short practice sections move short-term memory into long-term more efficiently.

Let Me Know How I Can Help

For future lessons, if you want me to help B with any particular pages in his workbook, just let me know what you want me to cover. Or if you notice he’s having trouble with something, just let me know.

And don’t worry–my first report is always long as it covers all the things I’ve noticed, and lots of explanations. In future, reports generally will be shorter!

Do You Have Any Questions About Initial Tutoring Sessions?

If you do, please feel free to ask them in the comments section, and I’ll do my best to answer them–or you can contact me directly, if you prefer, by email.

You can also find out more by reading the following posts:

You can also find out lots more about tutoring by checking out the links at the Tutoring Topics page and all links to all kinds of specific learning tips on the Home Education Tips page.

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Spelling Memorization Tips

The other day I read an article in which the author said allowing children to just spell “the way it sounds” to them will work out just fine. The children’s spelling, she assured her readers, will gradually become more and more “correct,” until someday they’ll all be great spellers!

Well… that might work for some children, especially if they love to read and tend to “just pick up” the accepted spellings. But for other children, spelling needs to be taught very carefully through step-by-step phonics teaching and through learning spelling “rules.” And then, unfortunately, many words in English, including a large proportion of “sight words” (aka frequently used words), don’t even follow the rules, and this means memorization.

Again, some children can memorize spellings quickly and easily by repeating the word a few times (spoken and/or and written). But memorization is much more difficult for other children. This post will provide you with lots and lots of ways to help your child (and maybe you, too, mom or dad or other significant adults) memorize those difficult words. Every person learns a bit differently, so try out these different (and often entertaining) approaches. We’ll start with some traditional “writing” approaches–and then on to more adventuresome alternatives!

A Variety of Worksheet Type Approaches:

Use these suggestions to create worksheets and discussions that are made especially for your child’s personal spelling challenges:

  • Go over spelling “rules” and “sounds” together orally
  • Draw little sketches to go with spelling words. It can be especially helpful for children to make their own little sketches; the connection (association) is often very helpful, and the sketches don’t even have to be particularly accurate. Sometimes they can just be a funny shape or even a sketch of an “opposite.” Each sketch will give children a “hanger” for the word.
  • Practice handwriting. Trace the word; fill in letter shape boxes (even have the children make their own); copy the word; then write without looking.
  • Make a list of rhyming words that are in the same word family as the word being learned.
  • Create a story using spelling list words. The children use their list to fill in the blanks, either “in context” of the story, or you can create a little sketch under each blank as a hint. Then encourage the children to create their own stories with the list words (and their own little sketches for the words). Silly stories are good as they are fun and memorable
  • Create word pyramids. On the top row, write the first letter; on the second row write the first two letters; on the third row the first three letters, and so on.
  • Use the letter (or combination of letters/blends or spelling rule) being studied in words other than the list words. If the children can come up with some of those words (or find them in a dictionary–a rhyming dictionary can be helpful for this), all the better.
  • Write the list words in one list. In a list beside the first one, write (in mixed up order) a rhyming word for each list word. Have the child match them. In the beginning, use words that follow the same pattern (eg. pane/mane); then try using other patterns that make the same sound (eg. pane/gain); and finally, try homonyms (eg. pane/pain).
  • Create funny sentences that contain the list words (if the child can make the sentences up, all the better).
  • Use compound words with the desired sound/letter combination. For example, if the word is “rain,” use “rainfall.” This is a great way to introduce syllables.
  • Find and circle (or underline or highlight) list words and other similar words in a story, news article, etc.
  • Put the list words in alphabetical order.
  • Discuss different sounds for a letter or letter combination (for example, “long a” can be written with magic e (age), two vowels walking (pain, say), and “eigh” (eight, weigh).
  • If you’re doing multiple ways of spelling a certain sound, as in the “long a” example–or if you have a single spelling that has different sounds for the same letter combination (eg: oo –> book, zoo), first sort the list words, then think of and list more words for each sound category.
  • Create a word search or a crossword puzzle; if children create their own puzzle it can be even more effective, especially crossword puzzles.
  • Create a chart (for example, list words that start with a vowel in one column and with a consonant in another column; or divide the list words into columns by long and short vowels, or by different vowels).
  • Sort the words into sound categories by writing the words on small cards or slips of paper, then sort them. Encourage the children to make their own cards.
  • Write a series of list words run together and have the child separate them.
  • Scramble the letters in the list words and have the child write them in the correct order.

Lots of Different Kinds of Spelling Tips:

Put aside the paper and pencil and start a spelling adventure!

  • Make associations between the spelling of the word and a picture or object. I’ve already suggested drawing little sketches. But you can use other sensory associations. For example, if learning to spell the name of a food, tasting it, smelling it and feeling it while practising the spelling can be very effective. Likewise, if learning a “sound” word (like “whistle”) listen to the sound and/or make the sound while learning the spelling.-
  • Practice in a variety of ways: flash cards, books that contain the word in context, writing the word in the context of a practical kind of writing rather than a regular spelling assignment (eg an email or letter to a friend or grandparent), etc.
  • Play games that strengthen vocabulary and word retrieval: Scrabble, Spill and Spell, Boggle, hangman, crosswords and word searches.
  • Play dice games like Snakes and Ladders–but before taking a turn you have to spell a word. Start with the easiest ones and work toward harder ones. If you spell it incorrectly, you can only move 1 place or half the places indicated on the dice. (Avoid having to miss a turn or other similar “punishment”).
  • Visualize: create a mind’s eye picture. Imagine some particularly memorable aspect of the word. Concentrate on getting a “flash” of that element. For example, if learning to spell “shoelace,” have the child close her eyes and picture a particularly bright, colourful, shiny shoelace on a favourite pair of shoes or boots. Then have her close her eyes and visualize the spelling of the word. Then even try to visualize them together.
  • Association is very helpful in memorizing any sequence of data (including spelling). For example, you are memorizing a list of spelling words: “banana, gloves, guitar, flashlight, midnight” (they don’t even need to have much in common). As you practice each word, visualize it or draw a sketch. Then link those pictures (and spelling) together. Imagine putting the gloves on your banana-sticky hands before you play your new guitar by flashlight at midnight. Use any kind of pictures and any linking story that pops into your mind. Crazy and fun is memorable!
  • Use a variety of writing tools when drawing or writing. Try pencil crayons, markers, sidewalk chalk, paints, black/white boards, or “write” with fingertips on textured surfaces (fabric, sand, finger paints, shaving cream, sandpaper, etc.).
  • Study the word for 15 to 20 seconds. Don’t just read the letters themselves, but look at the shape of the word, the shapes of the letters. Discuss them–think of it as an adventure with the word. Close eyes and recall as much as possible. Then open eyes and take in more detail. Close eyes and add new observations to the original mental picture. Repeat until you can’t come up with any more details. Then write the word with eyes closed (on a large sheet of paper or a whiteboard, etc.), drawing the “word picture” from your mind.
  • Write the word with different kinds of letters–eg. manuscript, cursive, different fonts (you can also do this typing on a word processor like Word), different sizes of letters, uppercase, lowercase, different colours, highlighted with different colours.
  • You can even “decorate” difficult-to-remember letters and letter combinations. For example, for the word “between,” in order to remember the “ee” long vowel combination, the child could sketch little eyes in the circles of the “e” letters, and emphasize the “smiley” bottom part of the letters in order to make an emogi funny face to put the “ee” into his memory.
  • Say or sing the word; set up a beat (tap, bounce a ball); say a syllable (or letter) for each beat.
  • Set up a pattern for the word: listen to a helper spell it aloud, spell it aloud yourself, write it in the air with giant imaginary letters, close eyes and visualize it, write it on paper, type it on the computer, write it in a sentence…. Figure out a pattern that works well for you.
  • Make up catchy rhymes or songs of the material to be memorized (or search a poetry or songbook for poetry/lyrics that repeat that word’s spelling sound or pattern–children’s poetry and songs like Mother Goose are especially good).
  • Have a family discussion related to the word. Talk about what it means, how it is spelled, any spelling rules, similar words–and how the word is used in practical ways.  Then find it used on cereal boxes, how-to instructions, recipes, newspaper articles, etc.
  • Have a helper spell a list word (start with the simplest one). The child repeats it. Then the helper spells that word and the next easiest one, and the child repeats both. See how far you can go with this. If a word is very difficult, start with one letter, then add a second, then a third, and so on. Or do it by syllables and/or by vowel combinations, consonant blends, etc.
  • Use repetitive, rhythmic physical actions/activities while doing spelling practice. For example: spell the word while playing hopscotch, climbing stairs, jumping on a trampoline, bouncing a ball, skipping with a jump rope, playing catch with a friend.
  • When using the word in writing assignments or practical writing, don’t be afraid to whisper or speak the spelling aloud; use the sketch you developed; close your eyes and visualize, etc. These associations and actions will bring the word back to your memory.
  • Make up funny acronyms of words you need to remember. For example, for “because”: Bunnies Eat Carrots And Usually See Everything.
  • Use a puppet. Have the puppet repeat or act out the spelling of the word, use it in a sentence, etc.
  • Teach someone else how to spell the word in as many ways as possible. Teaching a newly learned fact or concept is one of the best ways to retain it.
  • On small cards or slips of paper, make a collection of individual letters, vowel combinations, consonant blends, etc. (Make at least 2 or 3 each of the frequently used ones). Spread them out on the table or floor. Instead of writing the words, find the correct letters/combinations and put them together side by side. Then have a helper remove one or two important letters (without the child looking) and say what letters were removed.
  • Create a “bingo” game, but use spelling words instead of numbers or letters.
  • Act out the word with gestures or role-playing.
  • Play charades and similar traditional parlour games (use your spelling list words) with family and friends. Personal interaction can add greatly to memorization.
  • Spell the word(s) with another person. Take turns adding letters until the word is complete.

Long-term memory strategies:

  •  “Store” new words in memory categories with words you already know how to spell, that have similar attributes.
  • Use rote drill. But do short daily practice sessions (10 minutes or so) for 6 days in a row, then take a day off. Then practice the same word list again a week later, then a month later, then 6 months later. This is much more effective than long sessions a couple times a week.
  • For rule-based learning (as in spelling), combine practice and discussion. Give examples of words which follow the rules, and similar words which break that same rule. The learner identifies which is the “broken rule word” and which is the “rule word” and then explains the rule.
  • Have the helper spell the word to the child (either written or spoken). The child decides whether the helper spelled the word correctly or incorrectly (the helper decides which way to go before spelling it, but doesn’t tell the child). If the helper spelled it incorrectly, the child tells what the helper did wrong (for example, what rule was broken, or what letter is missing or should have been used instead), and asks the helper to try again. Is the helper correct now?
  • Knowledge is often best consolidated right before sleep. Read, practice or review material for 5 to 10 minutes before dozing off–then immediately go to sleep. The brain will continue to “practice.” Do a little “test” the next morning.

More Memorization Tips–for Math Facts, Tests, and More:

For lots more great memorization ideas–for spelling and all kinds of other memorization needs–I have developed 3 booklets with “Memory and Learning Strategies.” One is a general overview, the second is specific tips and tricks (from which the above ideas are drawn) and the third is for use with classrooms or groups of learners. You can get them here for just $2 each!

There are also lots of other great spelling tips in this blog (for FREE, of course). You can find links to a list of them in the “Tips to Tutor Your Child at Home–Reading” section of the Home Education Tips page.

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Ways to Relax and Overcome Math Anxiety

What Can We Do About our Child’s Math Anxiety?

I often have parents sign up their children for math tutoring due to concerns over the child’s math anxiety–which of course has a negative effect on math learning. Today I’ll share some of the advice I’ve given in these situations to parents and children

Take a Break!

Really? Does that sound too easy? How is a child supposed to learn if you back off the pressure? Here are some thoughts:

  • If you are stuck on a math assignment, leave the math work, and go do something totally different. When you come back, you’ll often find that the difficult question is now reasonably easy. By leaving the work for a while, your brain will subconsciously keep working on the problem, but more important, by doing something totally different, you will relax and de-stress, and come back to the problem with a fresh outlook and much less anxiety.
  • When a child finds math difficult and becomes anxious, I would suggest doing only a few minutes at a time, then take a brain break (doing some short activity that is fun and relaxing, or some activity that you need to do anyway, that doesn’t take “brain work,” such as going for a walk around the block, doing some stretches or  exercises, tidying up, drawing a picture or coloring, having a snack, or taking a shower. Then come back to the work and repeat the procedure.
  • Try doing “math” for no more than 25 minutes at a time, then take a 5 to 10 minute break before returning to it. If still feeling stressed, shorten the time to 15 minutes math and a 5 minute break. Even though it may seem that the math homework time is being stretched out, you will actually be able to do the work more quickly and accurately while you are doing it, and you will feel much more relaxed and less stressed and will retain the information better.
  • If you have reached the point of feeling really stressed, take a longer break, such as 30 minutes or an hour, or even until the next morning (but don’t put it off much longer than that), and do something that really distracts your mind from the math. Then you can return for a fresh start. You may need to review the work you have already done, but that will only take a few minutes and your brain will quickly get into gear for the math again.
  • Sadly, one can’t usually get up and go do something else during an exam, but at least in the exam, you can still do the “easy” questions first, and they will often jog your mind for the tough questions. Even in an exam situation, you can close your eyes, stretch, rest your head on the desk for a couple minutes, draw a couple doodles, hum your favourite song in your mind, or whatever helps you relax. You might also ask the teacher to divide the test into sections and only give the child one section at a time, starting with the easiest. This will allow the child to get up and take a break before returning to do more since the child hasn’t yet seen the rest of the test and thus won’t have a chance to sneak off and get help or check the textbook.

Keep it Simple!

Some textbooks and/or curriculums and/or classroom teachers offer a variety of approaches to a particular concept. While this can be very helpful as it provides for different learning styles and background experience, and also allows children who are especially creative or independent learners to do lots of exploration, it can be frustrating for a child who is already anxious about math. My feeling is this: – try the simplest/standard method. – if the child is having difficulty with the simple method, try one or more of the other methods, find one that the child “gets” and do

My feeling is this: Try the method you think will be easiest for your child to handle–based on your knowledge of how your child learns best, in math of course but also in other subject areas. You may need to experiment a little. Try one method and watch carefully for how your child responds. If the child catches on quickly, great. If not, take a break to allow your child to relax, then try another method. Do this until you find one that the child “gets” and then do lots of practice with it.

Then, when the child understands and is comfortable with that approach, you can experiment with other approaches, especially if the child now finds the concept interesting or intriguing and wants to explore it. Note: It is wise to have your child at least learn the “standard” (most commonly used) method, as in the future, it will no doubt come up as a basic foundation for more advanced math, and will also come up in real life math.

Create Word Problems From the Start

It is also wise to introduce word problems right from the start. Even when introducing such basics as counting and simple math facts, think of them in terms of experiences in real life so that the concept makes clear, concrete sense to the child rather than being some difficult, hazy theory to which the child can’t relate. Furthermore, as math becomes more difficult, more and more word problems will be introduced, so by using them right from the start, the child will be comfortable with them. Even before the child can read, you can easily create word problems and discuss them with your child. Just relate them to your child’s own experience. For example: “If I gave you 3 cookies for dessert, and you ate two of them, how many would you have left for a snack later? … Hmm… How can we figure that out? Should we add or subtract? Why should we subtract? What method could we use? Finger counting? Use our ruler for a number line? Draw a picture? …”

If you are using written word problems, together circle or underline the key numbers/facts in the problem, Also circle or underline the “key words” in the problem, and discuss what kind of problem it is. For example, the phrase “in all” is a signal to add; “how many are left?” is a signal to subtract. If in doubt, draw little pictures to illustrate the problem–or even get some cookies and make it a real life situation–a very effective way to learn!

Encourage your child to create his own word problems, and guide you through them, too! Teaching something just learned is one of the most effective ways to ensure the learning will “stick” and it also shows you whether or not the child has really understood the process.

Also Create Equations From the Start

Once you’ve discussed a word problem, turn it into an equation. This approach can be much more effective than simply giving the child a list of equations to solve. When the child creates the equation (based on the word problem), his understanding and use of math will be much more effective. He also wanted to solve the problems “in his head” and was hesitant to say the equations before giving the answer. It is really important that he expresses the equation, as word problems start to become more complex in grade 3 math, and he needs to be able to work them out “step by step.” I went over a couple of things with him: – circle or underline the key numbers in the problem – also circle or underline the ”

Sometimes a child who is anxious will want to solve the problem “in her head” and be hesitant to say the equation before giving the answer. It is really important that she expresses the equation out loud, and also writes it down–even if she can’t read or write yet, she could write it as little sketches and you could introduce plus (+), minus (-) and equal (=) signs. Using as many senses as possible will really help the child learn. Also, as word problems start to become more complex in later math, she needs to be able to work them out “step by step” and starting right away, in the early stages, will make that easier and less stressful. If your child is already anxious and stressful, go back to the simplest arithmetic concepts and review them, using these methods, creating a new, strong foundation (it might also demonstrate some “gaps” in the child’s learning which you can resolve before moving on).

Math Fact Learning Suggestions

Some children can memorize and quickly write down the facts on a timed quiz, but when it comes to putting them into practical use in an equation or word problem, they find it difficult to remember the facts, as they are now working through a multiple-step process, which is more complicated. Other children have great difficulty just memorizing the facts, not to mention actually using them in practical ways.

It is important for children to practice basic math facts and concepts in equations and word problems rather than just in timed quizzes or flashcards. In fact, time quizzes can be very stressful for many children. I personally think timed quizzes should be reserved for children who enjoy them and who find memorization and repetition easy.

For other children, there are a variety of ways to help them learn the basics without so much stress. Some examples:

  • use of different sense methods such as “finger writing” on a textured surface;
  • close eyes and “picturing” the facts or spelling in the mind;
  • say the entire fact aloud (not just the answer);
  • listen to mom or someone else say the full fact aloud;
  • draw little sketches to go with facts being struggled with;
  • create entertaining/funny stories that involve the fact

Oh! One more thing. If you want to practice “number facts” in some of the ways I’ve suggested, it works much better to just do 10 minutes or so daily, rather than 30 minutes twice a week or 60 minutes once a week. Frequent, short practice sections move short term memory into long-term more efficiently. For lots more ideas on how to learn math facts (and concepts, too), check out my Easy to Learn math booklets.

And Don’t Forget Real-Life Math! It’s so Important!

It is very helpful indeed to watch for opportunities in “real life” situations, to do addition and subtraction (and then more advanced math, of course). For example:

  • when you are taking a road trip, get a map that shows distances between towns and add up the distance from one town to another.
  • if you are baking an apple pie and doubling the recipe, and it calls for 8 apples, ask what 8+8 (or 2×8) is, then ask your child to count out that many from the apple box.
  • if you are having a plate of cookies for supper, and there are 8 cookies to start with, just casually ask how many will be left after everyone in the family has had one–and if there will be enough for everyone to have a second cookie.

These kinds of “real life” experiences come up all the time; just keep an eye out for them, and use them in a casual way. Include the whole family in these math activities. Make them like a game. Older family members will enjoy it, and younger siblings will also love these games and they will get started on math without any formal teaching!

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Simple Tips to Solve Your Child’s Math Struggles


Why is math so hard? What can we do?

While some people are naturally “math wizards,” most of us need some help from time to time. Surprisingly, often the help we need is some simple tips, more likely to do with our approaches to the work than to anything actually “mathematical”!

When you are helping your child with math (or doing math yourself) and run into difficulties, here are some simple, helpful tips:

 

 

  • Know “math facts” accurately:

So many incorrect answers can be traced simply to an incorrect math fact. Ideally, of course, you’ll want to memorize those facts of addition, subtraction, multiplication and division (For lots of great ideas on how to learn math facts, check out my Easy to Learn Math Facts booklets. But meanwhile, go ahead and use a math facts chart or a calculator.

  • Learn and remember basic math concepts:

Many times, children have simply missed out on or have forgotten basic “rules.” Let’s use positive and negative numbers as an example of these kinds of “rules.” Do you remember these?

– When using a number line, subtraction goes to the left and addition goes to the right

– A negative number minus a negative number always makes an even “larger” negative number (-12 -14 = -26) just as a positive number plus a positive number makes a “larger” positive number (12 + 14 = 26)

– A negative plus a positive makes a smaller negative … or even a positive (-8 +2 = -6 OR -4 + 5 = 1). Using a number line to illustrate this will be very helpful to your child.

– A positive plus a negative always makes a smaller positive or even a negative [8 + (-3) = 8-3 = 5 OR 4 + (-6) = 4-6 = -2

– When multiplying, negative x negative = positive. -6 x -3 = 18

– When multiplying, negative x positive = negative. -6 x 3 = 18

– When multiplying, positive x positive = positive 6 x 3 = 18
So… if your child is struggling in a particular “unit” in math, set aside the questions for a few minutes and review the rules. Do some examples together, then try the questions again.

  • Math is logical … and it is also balanced.

An “equation” requires that the math on one side of the equation “is equal to” the math on the other side (2+4=6 SO 6=6). I had so much trouble with algebra when I was young until someone pointed that out–and then it was like a light went on!

When working out equations (whether simple or complex ones), remember that if you do something to one side, you must do the same to the other side. The equation must be kept balanced at every step.

And here’s a related and very important tip: Until you are very comfortable with working out equations, do ONE step at a time. Here’s an example from a simple algebraic equation:
5 – x = 2
-5       -5 *
-x = 2-5
-x = -3
x = 3 **
* To get the unknown (x) on one side of the equation, you need to put the number(s) on the other side. To do that, you do the opposite: so the opposite of 5 is -5. To keep the equation equal, do the same thing on both sides!
** When both sides are negative, the negatives cancel out and the answer is positive!
As you go along in math, the equations become more and more complicated. It is very important to write out each step clearly (neatly). Also, especially with the unknown “x”, use the written form rather than the printed form, as the printed form is too easy to confuse with the x that means “multiplied by.”

  • Use illustrations or videos (or real-life examples)!

Many people understand math concepts better with illustrations like pictures or videos–or of course with real-life examples. There are lots and lots of wonderful videos online that teach all kinds of math concepts. Just do a search for whatever topic you’re struggling with, and you’ll find some wonderful, colourful, entertaining help! Or take a look around your home and your daily activities. How can you use this math concept in real life?

  • Accuracy is always really important.

Especially when doing geometry, make sure you have your tools (your geometry set) handy, and use it very carefully. I see so many children “estimate” lengths, draw geometric figures “freehand”, try to plot (x,y) sets on a graph without using graph paper, or even if using graph paper be in a hurry and end up putting the dot in the wrong place. This also, of course, applies to using tables, graphs, etc.

This even applies to solving equations. It is amazing how many children (and adults) “forget” or don’t bother to write in the proper sign (+ – x /) when calculating in a column, fail to underline the list of numbers before solving, or scrawl down the numbers so that when it comes to solving the problem, they can’t read the numeral and see it as a “9” instead of a “4” or a “5” instead of a “6.”

  • Show your work!

A lot of people dislike spending time writing down all their work on a math problem because “it’s a waste of time” and “I can do it in my head.” But here are some good reasons to show your work:

– You won’t forget anything. It’s easy for simple little things to “slip out” of your mind. On the other hand, it’s always a good idea to practice doing “mental math” in case you find yourself in a situation (like comparing prices at a store) where you don’t have a paper and pen or calculator handy.

– After solving the problem, you can easily check your work and catch and fix any little errors–just remember to then re-do the rest of the equation because it will change, too.

– On a test, many teachers will allow at least “part marks” if they can see your work. If you only write the answer and it’s wrong–then you’ll get a “0” for that question.

– Even when you’re doing a whole bunch of similar equations, doing the work each time will get the process firmly embedded in your memory–and in future, when you come across these kinds of questions, you’ll remember how to do them.

– In future, you will come across more and more complicated math problems that use “easy basics” in them. If you have those easy basics firmly embedded in your memory, all you’ll need to learn is the new parts instead of having to learn everything all over again. Get those “foundational bricks” set in place, and the rest of the mathematical “building” will rise up much more easily!

One more tip: Some really simple issues:

Check to see if your child is wearing his/her glasses … has good lighting … is using a sharpened pencil and has a good eraser handy … is using appropriate math tools … is in a quiet, undistracted setting ….

So many times, a sheet of math work with many errors can be traced to one or more of these simple issues!

Watch for future posts on more simple math tips and tricks!

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Exam Study Tips

Goodbye, BC Provincials! Yes!

Phew! The BC Ministry of Education has decided to axe nearly all secondary school provincial exams (for the time being anyway–Ministry of Ed decisions are known to change quite frequently–if you doubt me, see below for an example of changes over the years). So no more “provincial exams” except for the grade 12 English courses, English 12, English 12 First Peoples, Communications 12, and the French language equivalents: Français langue première 12, Français langue seconde immersion, and 12 Évaluation de numératie.

I Don’t Need to Learn to Study for Big Exams Anymore–Right?

Awesome! Students don’t really need to learn to study for big exams anymore, right? Well, don’t get so excited just yet. To start with, those grade 12 English and/or French exams are often really important for getting into your college or university of choice. Then there are still teachers who design classroom exams that are similar in format to provincial exams and are just as difficult, if not more so. And without exam practice at the secondary school level (or homeschooling without exams, if that is your educational path), you may find yourself in shock when you face mid-terms and final exams in post-secondary educational institutions.

A Bit of History

When I was a secondary school student, I never wrote a provincial final exam, as I’d always received “recommends” due to sufficiently high marks in my classroom work (See? the rules keep changing). So I blithely signed up to write the “Provincial Scholarship Exams” that happened to be introduced in my grade 12 year. I chose History 12 and English Literature 12. And yes, I did get sufficiently high marks (just barely) to get the $200 scholarship money (which could be spent any way I wished–I bought my first car, a sweet little 1964 Valiant, which carried me to and from college classes very nicely). But I’d have received much higher marks if only I’d known how to study for those kinds of exams. And then my post-secondary exams (and marks) were even scarier–until I finally buckled down and learned the tips and tricks of the exam trade.  I sure wished I had learned them (and practised them) sooner.

So … Herewith Some Exam Study Tips…

Practice Exams:

If your teacher (and/or the Ministry of Ed) provides you with “practice exams,” you’ll want to take advantage of them. They will give you a really good sense of what to expect in the real exams, and point you in the direction of things you need to know (yes, course content, but also exam formats, tricky type questions, and more). So, with your practice exam in hand:

  • Go through the exam, and do all the questions, then mark the exam, using the marking system provided.
  • Go back and re-do the questions you got wrong the first time. Mark your 2nd effort.
  • If you still got some questions wrong the 2nd time, ask your math tutor or teacher for help, as well as studying the appropriate section in your textbook and doing lots of practice questions.
  • But if you got those difficult questions right the 2nd time, analyze what you did differently. Why did you get it wrong the first time and right the second time? For example:
    • Did you read it incorrectly?
    • Did you read it too fast?
    • Did you get the answer right in your “figuring” but then mark the incorrect multiple choice answer?
    • Did you remember how to do it the 2nd time, but didn’t remember the 1st time? What happened between times to help you remember? Or it might mean you DO know the process, but you need to practice it more so it is really memorized and automatic.
    • Watch for patterns–do you often just hurry too much? Slow down! Or?
  • Keep in mind, for the actual exam, what caused you problems with the practice exam, so you don’t make the same kinds of mistakes.

Now It’s Time to Study for the Exam (And Practice Again!):

  • Spread your study time over at least 4 or 5 days, rather than “cramming” at the last minute. The night before the exam, relax, get a good night’s sleep, and have a good breakfast. If you really feel you must do some studying the night before, just skim over your practice test answers one last time and/or the topics you have most difficulty with.
  • First focus on the questions you got wrong both times; get help, and practice, practice, practice.
  • Then focus on the questions you got wrong only the first time, and practice them too.
  • Some practice exams follow up with a separate section in which sample answers are given for the paragraph or essay questions in the practice exam, along with marks and the reasons for the marks. Study these carefully, and compare them with the answer you gave on the practice test. Figure out how you can follow the example of the “good” sample answers to improve your own written answers–and rewrite your answer(s) until you feel that your answer would get a good mark.
  • Go through your class notes, assignments, and the tests you took during the term. Look for and do extra study on:
    • topics you had trouble with in assignments or class tests
    • topics that are in the textbook or in the course outline but your teacher didn’t cover (or just skimmed over) in class–if they are in the course outline, they can be on the exam
    • the kinds of question formats you had most difficulty with: multiple choice? essay questions? “show your work” questions? Practice, practice, practice.
    • If you have difficulty with sentence, paragraph or essay writing questions, ask your teacher or another qualified person to read your “practice” answers and suggest ways you can improve your writing ability. Look back at old written assignments and written test questions, read the teacher’s comments carefully, and learn from them.
  • When you feel you have your “problems” figured out, take another practice exam. If not available, create your own practice exam, using the same format. In fact, this is often a superior method, as you are reviewing the material while creating the exam (and you’ll find out that creating a good exam is a tough job).
    • If there are still areas you’re having problems with, focus on those areas again.
    • Use the practice exam to get used to the format and timing.  If you finish well ahead of the provided exam time and yet made quite a few mistakes, maybe you are in too much of a hurry. Slow down; read each question twice to be sure you understand it before answering it.
  • For exams that are designed by the classroom teacher (or by the professor in college or university), keep on the lookout during class time and in handouts for “hints” about what the teacher might be wanting. It is important to read and listen to understand the kind of information the teacher gives, because often these things will turn up in the exam questions in some way.  For example, your teacher might give a handout with many terms, definitions, and examples–or refer to these kinds of things mentioned  frequently in class. Keep track of them, then test yourself on them as you study. There’s a good chance that these are things your teacher considers very important and will put on the exam.

During the Exam:

  •  If you aren’t sure of an answer, you can leave it blank, and go on to answer other questions; then, when you are done, go back to the questions you skipped. Often, doing the other questions will remind you of how to do the “skipped” questions. (Occasionally, you will be told you must answer every question; in this case, put in an answer, but if you have time later, go back and double-check the ones you were doubtful about).
  • Read all the questions carefully before answering: read once and answer in your head, then read a second time. Do you agree with your first answer? If so, write it down. If not, read a third time–then answer with the best answer you can give.
  • Don’t rush. Read every question carefully–and in multiple-choice type questions, read all the choices, so you won’t be tricked by an “almost right” answer. It’s better to answer the majority of the questions correctly and not quite finish the exam, than to rush through and get a lot of answers wrong.
  • Be careful with multiple-choice questions. You are looking for the “one right answer.” There may be several “almost” or “partly” right answers–but find the right one. Also watch out for “trick questions” that use “double negatives” or other methods to throw you off. Read the opening statement of the question twice to make sure you know what it is asking.
  • If you complete the exam and still have time left, go over all the questions again, to make sure you read each one correctly, and recorded the correct answer (look out for “silly mistakes” and “typos”).
  • If you had to “guess” at an answer, and still aren’t sure about it after re-reading it, it is usually best to leave your “first guess” as the answer.
  • Especially for exams like English or Social Studies, where there are essay questions as well as multiple choice questions, it is wise to skim through the test first, and note how much each section is worth; then make yourself a little “schedule.” If you are taking too long on one section, set it aside and do the other sections that have higher mark value; then if you have time, go back and finish the section on which you were taking too long.
  • But this “scheduling” also applies to math and other subjects.  Planning ahead only takes 5 to 10 minutes and can make a big difference in your results.
  • When you are skimming through before you start the exam, look for sections for which you are sure you know the answers, and make sure you answer them. Save for later the “tough” sections which you’re not sure about.
  • For questions you are not clear on, you may actually be having trouble understanding the question format, more than not knowing the answer. Break down the questions into all their parts; learn to examine the questions carefully and answer all parts of them.
  • A hint for essay (or even paragraph) questions: first, read the question very, very carefully–2 or 3 times. Then make yourself a good outline before writing your answer. Use arrows to show how different points are connected to each other. Then stick to the outline as you write. If you run out of time, you can put a little note at the end, referring the marking person to your outline; from what you have already written, they will know your writing style, organizational ability, and other writing skills, and they can quickly see, based on the outline, your knowledge of the topic and where you are going with the essay.
  • If you still have time once you have finished the exam and have checked your answers for accuracy of information, re-read your sentence, paragraph, and essay questions and correct spelling and other grammar issues. This can make a difference in your mark, especially in English exams. If some words are “scrawled” and difficult to read, cross them out and rewrite them neatly–all markers will be pleased.
  • Note that you do not have to have perfect grammar in written answers, but you do need to write in a “clear” and concise manner that answers the question accurately. If you are to give an opinion or perspective, make sure you back it up with good reasons. And don’t repeat yourself over and over, just trying to fill up space. A relatively short answer that is clear is much better than a rambling answer.
  • Even if you feel that “writing” is your strong suit and you’d rather start with the long-answer questions, doing the multiple choice type questions first (or at the least, skimming through them) can give you some good hints on ways to answer written questions.

What other exam study tips would you add?

Please share them in the comments! Thank you!

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Fun Hands-On Math Activities Part 4

This is Part 4 of a series on fun math activities you can do at home. You can find a list of all the posts in this series in the “Fun Math Activities” section on the Home Education Tips page.

Learn to Use an Abacus

It’s true that the abacus was developed long ago in China, but you might be surprised to learn that it appears to have been developed independently in several parts of the world, including by the Maya in North America. Did you know that in contests between abacus and calculator users, abacus users are often able to solve complicated equations faster? This would be a fun experiment to try with family members and friends. In many parts of the world, the abacus is still used daily in markets, banking, and other practical mathematics. Encourage your children (and adults in your family, too) to use an abacus for practical math around the home.  Playing with an abacus is a great way for children to discover counting and arithmetic concepts on their own, and a hands-on way to memorize and understand math facts. You can purchase an abacus quite inexpensively at toy stores, even though they really aren’t a “toy” but a useful calculation device–and they’re often available at thrift stores, too.

Bowling and Other Sports Are a Great Way to Practice Math

There’s nothing like keeping score at a bowling alley to practice those addition facts and skills–and have fun while you’re at it. Unfortunately, many bowling alleys now have electronic score-keeping devices, but usually, they’ll have some old-fashioned scoring forms tucked away under the counter if you just ask. You can find a thorough explanation of bowling scoring here and you can download a printable score sheet here Besides keeping score, sports like bowling are a great way to develop strategy skills and even use probability to predict scores for games based on past statistics, also important aspects of mathematics. This can be developed into graphing and other math skills, too. If you have family or friends who are enthusiastic bowlers–or love baseball, football, hockey, tennis and other sports that require score-keeping and strategy, why not ask them to take your young sports enthusiast under their wing, and teach them these skills?

Math for Young Artists

“Copy and Colour” books are present drawings in grids, with blank grids on the facing pages. Show your child how to make accurate drawings by following the grid drawings. Alternatively, you can choose a sketch or even a photo. Overlay it with tracing paper, and draw dots at important points. Then your child can connect the dots with lines, referring to the original. Or, of course, you can number the dots (or just use a dot-to-dot book). Another slightly more complex method is to start with the tracing paper dot overlay–but then instead of giving the child the paper, you can first lay it over a geometric grid and write down the (x,y) points of the dots of a piece of paper. The child can then locate the positions of the dots and put them on the grid–and then, using the original picture as a guide, along with the dots, recreate the picture.

Another slightly more complex method is to start with the tracing paper dot overlay–but then instead of giving the child the paper, you can first lay it over a geometric grid and write down the (x,y) points of the dots of a piece of paper. The child can then locate the positions of the dots and put them on the grid–and then, using the original picture as a guide, along with the dots, recreate the picture. If you need a review on plotting on (x,y) coordinates, you and your child will have lots of fun watching and learning from this video!

Build and Use a Geoboard

Geoboards can be quite expensive to buy–but they’re really inexpensive to create and in the process involve using geometric skills. The picture at the top of this page shows a geoboard made of a board and some nails, with coloured elastic bands. The geoboard I made used a cast-off piece of pegboard and some leftover screws and bolts I dug out from a bucket in my hubby’s workshop.  (Here’s a good set of instructions).

There are so many ways you can use a geoboard, and they’re almost all mathematical–just use your imagination, or better yet, let your children use their imaginations! For example, with my 25 point (5×5) geoboard grid, I drew 4 designs on graph paper marked with 25 point grids. My child chose two of them – the first a cross, and the second his initials. He used elastics to reproduce them on the geoboard. Next, I had him create his own design on the geoboard, and then reproduce it onto the graph paper. Finally, I had him create another design onto the graph paper, and then reproduce it himself onto the geoboard. He really enjoyed doing all these things, and even started “experimenting” by using combinations of different coloured elastics, stretching elastics to different lengths, using multiple elastics versus just one, and so on.

Then I put grid paper marked with 25 dots (which match the posts on the geoboard) set inside clear acetate. I drew a series of exercises/designs for the geoboard, one at a time, on the acetate/grid paper with an erasable marker, and then my child re-created the design on the geoboard. Then he created and reproduced his own designs. With the eraeable grid, endless designs can be worked out. If you’re having trouble thinking of some, check out these excellent ideas on a Pinterest board. (Scroll down to see more ideas on that page).

Also, I explained “right angles, acute angles, and obtuse angles” with sketches on the acetate/grid paper, and with angles of elastics on the geoboard. I showed my child how to use a protractor, and he was soon easily naming the degrees of different angles, and naming which of the 3 types of angle each was.

Probability Experiments

Probability experiments are always fun. Here are some examples:

  • Draw a crayon from a bag that contains 1 brown, 1 red, 1 yellow, and 2 blue crayons (or colour combinations of your choice. Have your child predict what will be chosen more often and why (eg. blue because there more blues–you can also create a probability “tree” if you like). Do about 30 or 40 draws, and see if the prediction is correct.
  • Roll two dice together and record (in a table or on a graph) how many times each possible combination comes up (1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 2/2, 2/3, 2/4, 2/5, 2/6, 3/3, 3/4, 3/5, 3/6, 4/4, 4/5, 4/6, 5/5, 5/6, 6/6). Before rolling the dice, predict which combination(s) will come up most, or whether they will be the same. Do the rolls match up to the prediction? Why or why not? (There may be several suggestions for unexpected results). Then discuss how this information could be used in a dice game such as Yahtzee to plan strategy.
  • Travel tips: predict what proportion of licence plates from different provinces/states will be seen and why; then graph how frequently different plates are seen. Discuss. You could do similar predictions, recording and graphing for a combination of highway stops such as rest stops, fast food restaurants and gas stations. For a long road trip, you could keep track of gas mileage between each fill-up. Theoretically, using the same vehicle, the children might predict similar gas mileage each time–but likely, the results will be somewhat different each time. Discuss the factors that might make a difference (eg straight highway vs towns that have stops and starts; weight in the vehicle after buying–or consuming–groceries; driving uphill compared to downhill or on flat stretches, etc.).

 

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