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By Jason Caldwell B.Ed,Dip.Tch. Introduction Why do some students seem to find Maths difficult? I believe this question can be answered for 95% of students who are underachieving in mathematics very simply. Students who struggle in mathematics usually : Lack a strong relational understanding of place value (1s,10s,100s…) Lack a relational understanding of the four basic operations (+, -, x, /) Or have missing links or keys that underpin the new concepts that they are trying to work through. These three problems create maths anxiety, as students find themselves working through problems by following prescribed steps or formulas but not having a clue how they work and what it is all for.
Strategies for Building Strong Relational Understandings. Counting and number games. Make a game of counting anything that surrounds you. Activities such as counting buttons -counting 1 at a time then two, three, four or five at a time… help children get a picture of a “lot of” or a “set of” a number. Using a “stop watch” strategy adds a fun dimension and turns the activity into a game e.g. “count as many buttons as you can in thirty seconds…”
If a student can count beautifully to 100 but cannot count 6 objects, are they really counting? Obviously “rote counting” is important, but just the first step. Activities in counting that use concrete or semi-abstract materials (drawing) help students to get a picture in their heads of what numbers mean. Knowledge of how numbers work will naturally follow. The ability to relate a number name e.g. 5, to 5 objects and know that when the arrangement of the objects is changed -the number stays the same is called Conservation of number. Another fun activity is to count steps, jumps or actions. Adding the physical dimension helps make counting meaningful. You may create your own dance sequence together with different numbers of steps or actions. “Finger Flash” e.g. “how many fingers am I holding up?” Change “Now?”… As they are fast with up to ten fingers, I start to add more fingers as I see they are about to answer. Thus turning number recognition and counting into additon. once they have this down i start subtracting fingers as they are about to answer... "Rainbow Maths"
Just a little diagram to help show numbers and the relationship of numbers adding to ten. This is a good little diagram to use with little ones. Show them that when you add 0+10 it equals 10, then 1+9=10, following the rainbow line each time you show them the addition. Here is an idea, even make your own with them and color it in! Other Ideas: Blow and count bubbles, make a tower out of blocks and count the levels, head counts, Extension: Measuring using a ruler…
Place value It is important to use place value blocks in conjunction with a simplified table (with columns for ones, tens and hundred) when students are learning to add and subtract up to three decimal place numbers. Thus students model the algorithm or sum with the blocks to gain a relational understanding.
Students can also draw the blocks on a chart to model amounts or draw them in picture form… These days the easiest way to teach this is to use a Go Maths Student text for Grades 1-3. For a very reasonable price ($15.95) these texts cover the modeling and use of place value blocks and table thoroughly. The place value blocks can also be purchased at a reasonable cost (http://stores.ebay.com.au/SMART-SHOPPING $39 plus $10 Postage).
A game that I have found very helpful is the “Place Value Game”. You tell the students that you are going to throw a dice five times and each time they can decide to make the number worth “ones”, “ten” “hundreds” or “thousands”. They then have to add their numbers together to get as close as they can to one thousand. Have the students work in columns to write and add their numbers together. For Example if the first throw is 5, they can write down and add 5 or 50 or 500… If the next number is 2, they can write down and add 2 or 20 or 200. You play the game with them modeling and the winner is the person who gets the closest to 1000 without busting. As they get good at this you can increase the number of throws and the target number, I have had classes working towards a million in ten throws. There are several great place value games and activities at this site: http://www.theteacherscafe.com/Math/Place_Value.php
The 4 Main Operations -Addition, Subtraction, Multiplication and DivisionAddition = + = adding = increase = plus… If you can add all single digit numbers easily then you can add anything… It is fine to use your fingers so long as they are helping and useful not just a security blanket. You can also use a number line and place value chart… I like to use “finger flash” and start holding up two lots of fingers, one just after the other… Just as with counting, the use of concrete materials is important for students to see and understand the relationship of adding numbers… To build speed and knowledge of patterns I like to use 10x10 grids. Use random numbers down the side and numbers across the top that encourage students to use patterns and systems… Have students work down the columns… E.g. Addition Blaster! (Great name – come on let's sell it!)
The Easy Way Students enjoy learning addition patterns that take advantage of the decimal system and build strong mental Maths skills in an activity I call “The Easy Way”. I would play this game once a week. E.g. 1+9+2+8+7+3+4+6…=____ -sets of ten. 4+8+8+6+4+7=___ -requires splitting the 4 to add onto the 8’s to make two sets of ten… 3+4+6+7+9+3+1= ___ mixed up pairs and have leftovers. 13+7+44+6+9+5+3=___ and just keep going… This activity is timed and should never drag on -students try and finish as many sums as they can in 5 to 7 minutes and then mark together looking at the strategies used, which is more the focus than whether they got things wrong or right. Another great activity is to write a dozen numbers in a box all mixed up. E.g. 1 8 6 15 0 4 9 5 2 3 7 11 Allow students 30 seconds to find as many pairs of numbers that equal 10 or 20 or 50 or 100 that they can in 30 seconds. And don’t show the student the numbers until you start timing.
Subtraction = - = takeaway = minus I never teach subtraction until students are strong with their addition. I have never meet a student who had good addition skills who struggled with subtraction. This is because subtraction is the same as addition, it is just backwards. “Backwards Addition” E.g. 4+5=9 and 9-4=5 (See the relationship?) When it comes to building speed in subtraction grids are great as you can draw out the patterns and as students try to improve their times they will be forced to use the patterns. E.g.
Patterns -1 (just count back), -10 (take 1 from the 10’s column), -9 = (- 10+1), -11= (-10-1), -15=(-5-10)…
Multiplication = x = lots of = repeated Addition = Crossover Multiplication I talk with so many people who rate a primary student's mathematics ability on how well they have rote learnt their basic times tables. Is this not just an assessment of time spent memorizing words or the ability to memorise? Sometimes I have a bit of fun with this. When a parent is adamant that their student knows all their “times tables” and I can tell they are rote learnt, I ask the student a couple of basic facts and of course they are correct and then I throw “8x16” in. “That’s not a times table!” they reply. It certainly is a basic multiplication fact to anyone who knows how their tables work as they would simply break it down and add 8x10 add 6x8 =80+48=128.
Remember the importance of sequencing students learning:
So I would read multiplication equations like "4x5" as "4 lots of 5". E.g. "3x4" said "3 lots of 4" =
This leads to a great trick I find students love. It’s called “Crossover Multiplication” and involves drawing lines that cross over to find the answer. E.g. To work out 3 lots of 4, just draw 3 lines then 4 lines that cross the three and count the crossovers… When teaching tables or basic facts I like to teach the patterns and systems! 1’s -any number times one is the same number! 2’s -are just doubles and you can practice adding double for fun in your head… 3’s -are your 2’s plus one more lot of… 4’s -are double your 2’s 5’s -to do your fives learn the 10’s and half them or counting in 5’s and see the pattern 5,0,5,0,5,0… in the one’s column. The first pattern in powerful e.g. 264x5= (half 264) 132 x (put on a zero) 10. 6’s -are double your 3’s 7’s -to do your sevens you use a trick that I call “break downs” E.g. 7x4= 5x4+2x4 and we break it down to 5 and 2 lots of because our 2’s and 5’s are simple. 8’s are just double your 4’s 9’s - = x10-one lot or use the 9’s finger calculator God gave you. E.g. 9x6 -by holding up all your fingers and putting down your sixth finger you see the number of tens on the left and the number of ones on the right. 10’s -It is easy to multiply by ten, you just add zero after the number! 100’s just add 2 zeros, 100’s 3 zeros… 11’s -= 10’s +1 more lot of… 12’s -we use “break downs” –10’s +2’s.
Another system I call “do the table add the zeros” is shown in the following example: 2x3=6 2x30=60 2x300=600 and 20x30=600 or 2,000x3,000=6,000,000
If we know these patterns and systems our brain becomes a very powerful calculator and not just a “parrot”
More Advanced Strategies e.g. 75 X 18 = 75 X 20 - (2 X 75) = 1500 -150 = 1350 or 15 X 93 = 15 X 100 - 15 X 7 = 1500 - 105 = 1395 There are several other strategies that are very powerful. One called "Doulbe and Half" and is based of the principle that 1X2 is the same as 2X1 or doulbe one side and halve the other and the total will be the same. So 12 X 45 = 6 X 90 = 6 X 9 add the 0 = 540
Or 24 X 73 =12 X 146 = 6 X 292 = 3 x (600-16) = 3 X 584 = 1500 + 240 + 12 = 1752
Division = Sharing I never teach division until students are strong in their multiplication. Just as subtraction is the same as addition, it is just backwards, Division is “Backwards Multiplication” E.g. 4x5=20 ---20 / 4=5 E.g. 20 cakes shared to 4 people =5 cakes each.
I was explaining this to a class once when a teacher who had been teaching for longer than I had been alive said, “I wish someone would have told me that before…” This teacher could even “teach” students how to do short division but had never really understood the relationship. I use this sharing relationship and knowledge of multiplication facts to teach division. The important thing to teach students is the relationship. If students understand what division means they can work out any whole number division problems. As the equation gets large it does not get harder –it just may take a little longer. It is important to teach long division before short division and even to encourage students to draw the problem out. Short division to most students is just a magic trick that makes little sense. Isn't that how it should be?
1 Answer. The numerator is 1 and the denominator is 2. That’s great but what does “denommmonarator” mean? So, the bottom number is what you cut the pie into. E.g. you cut the pie into 2. and the – is the division symbol and means “divided by”. E.g. 1 divided into 2. Drawing out fractions is essential… Use concrete examples to understand the meaning of fractions, and compare and order them. I get students making their own labeled fractions kits out of even length strips of light card. An A4 page strip of light card is good for this. E.g. This site has information on fractions:
Additional Mathematics Keys: If you are stuck: Don’t stress or panic. Go back over the instructions and examples. Reread the question. Simplify the numbers in the problem. Draw the problem out or model it with concrete materials. Ask for help don’t sit there and get stressed. Express an equation in monetary terms –this seems to make things more meaningful to students! With Ace Maths Paces have students keep their own copy of the keys and formulas. Jason’s Grid template
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Students label each piece with the word name, fraction and decimal name and percentage...