< All Reports | Print this Section | Back to Contents > Appendix A - Training and trialling materialsThis appendix provides samples of the activities undertaken during the five weeks of the Train a Maths Tutor Program. The following notes and activities will be translated to suitable training manuals for use in other schools. Week 1: NotesThe decimal number systemThere have been many number systems invented by man since 3000 years BC. All of these systems used names and symbols to indicate numbers and all of them had an additive property but no multiplicative property. Most countries today use the Decimal Number System, called “decimal” because it has 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and each side-by-side place is related by 10 (i.e., 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand; and so on). The word, decimal, came from the Roman word, decem, which means 10. (December used to be the 10th month of the year in early Roman times.)
The most important number in the system is 1 because it is the unit from which we make numbers larger or smaller. We can make numbers larger in two ways − by adding 1 more or by making groups (which is the basis of multiplying). In the Decimal Number System, we are mainly interested in making groups of 10 and groups of 1000
Unlike all other number systems, the Decimal Number System has place value. This means that the place where a digit is written gives that digit its value (see below).
Another feature of the Decimal Number System is that we can add all the place values to make the number. For example: 385 = 300 + 80 + 5 CountingRote countingWhen children can say the number names in the right order without worrying about whether they touch one object as they say each name. For example: one, two, three, four, five,(good enough for starting Year 1) … six, seven, eight, nine, ten (even better for starting Year 1) If the students cannot say the number names in order, then they need to sing a lot of nursery rhymes like, “One, Two, buckle my shoe; Three, Four, knock at the door”, and so on. Point counting
This is when students can point to (or touch) each object in a set and say one number name for each object, as shown below.
When counting, the student has to remember which objects she or he has counted, and which objects s/he still has to count. It is much more difficult to do this when the objects to be counted are scattered and most difficult when the objects are arranged in a circle.
The counting activities shown below are more difficult than the scattered arrangement or the circle because we are going to give the students either the starting or the finishing number that is not at the end of a line. Play games like Snakes and Ladders and Ludo and any other board games where they have to count. Play games where they have to take the number of strides you tell them or they work out (such as the number of times they have the letter “N” or “T” in their name) Counting tells us “how many”We count to find out how many things there are in a set (e.g., How many students in the class? How many horses in the paddock?) Students need to know that the last number name they say when counting tells how many altogether.
Counting materials
Young children should begin counting real world materials (e.g., biscuits, lollies, marbles, teddy bears) and then toy versions of the real world materials. When they are able to point count, the teacher starts to use concrete (“hands on”) materials such as counters and then Unifix cubes. Young counters should then be able to count things in pictures. This is a bit more difficult for some students as they cannot pick up or more the objects in a picture. Advanced counting skillsSubitising (Sight counting) Subitising is recognising how many objects through sight alone. The way objects are arranged affects the ability to subitise. A and B below are the most difficult to subitise even though A is the easiest for the child to count by touching.
Counting-onCounting on 1, 2 or 3 from a given number is an essential strategy for addition facts. Example: 5 + 3. The child should think: five, six, seven, eight.
Counting backCounting back 1, 2 or 3 from a given number is an essential strategy for subtraction facts. Example: 6 – 2. The child should think: six, five, four.
For older children, use play money & put $26 (or $478) in a paper bag & then either take out $1, $2, or $3 or add another $1, $2, or $3. GroupingGrouping leads to place value so it is very important that young students have lots of practice making groups of 2, 3, 4, etc and 10. Making groups of 10 is what we are interested in for place value (10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand; and so on). Because students need to know that numbers get larger in value as they “grow” towards the left, they should use a Place Value Chart to help them put the groups in the right place.
Teaching sequenceStart towards the end of Grade 1. Make different groups out of a number of loose ones. For example, make groups of 3 out of 8 loose ones. Materials: 8 counters per child; 1 Groups/Ones Board. Say: Put out 8 ones (counters). Do you have enough ones to make a group of three? [Students should say, “Yes”.] Show me your group of three. How many groups of three do you have? [1] How many loose ones left? [5] Repeat these questions until no more groups can be made. Repeat with other groups (such as groups of 5, 4, 8, 10). The whole aim of grouping is to lead the students to understand Tens and Ones in Grade 2. Reading and writing numbersThe diagram below is often called the Rathmell triangle after the man who first wrote about it. It is a very good guide for teaching all new mathematics. 1. Start with a real-world problem (“story”). 2. Use materials to show this problem. 3. Teach the number name.
4. Teach the number symbol. The pictures below show a couple of examples.
Material -->Language activities Put out 4 teddy bears (or counters, Unifix cubes) and say: How many bears are there? Repeat this many times with different numbers until the student can say all the correct number names. Language -->Material activities Say: Richard has seven cars. Show me this number of cars with the counters. Repeat this many times with different numbers until the student can put out the correct number of counters. Material--> activities Put out 4 teddy bears (or counters, Unifix cubes) and say: Write the number that shows how many bears there are. Repeat this many times with different number names until the student shows that s/he can connect the correct number symbol to the number of objects shown. Symbol --> Material activities Write a number such as 8. Say: Richard has this many cars. Show me this number of cars with the counters. Repeat this many times with different numbers until the student can connect the number symbol to the correct number of counters. Language --> Symbol activities Say: Margaret has five apples. Write the number that shows how many. Repeat this many times with different number names until the student shows that s/he can connect the correct number symbol to the number name. Symbol --> Material activities Write a number such as 8. Say: Val has this many headbands. How many headbands does he have? [Student should say, “eight”.] Repeat this many times with different numbers until the student can connect the number name to the correct number symbol. Place valueAs said earlier, the Decimal Number System has a place value feature as shown by the figure below.
Place value materials sequenceTo help the students understand place value, we use the materials shown below and in the order shown. Real world materials (e.g., biscuits, lollies, marbles, teddy bears) Concrete materials (e.g., counters, Unifix cubes, bundling sticks or straws, MAB blocks).
Virtual (computer) images of the materials. Pictures of the materials. Teaching action sequence
Week 1: Monday trainingCounting (see Notes)Playing games to develop the notion of “counting the jumps” when counting on a number line (see Ludo and Snakes and Ladders). Tutors were asked to develop a Snakes and Ladders Game that didn’t start at 1.
Sample modified Snakes and Ladders game
Week 2: Trial 1 materialActivity 1 – 100 Board jigsawMaterials100 Board; Blank 100 Board; 100 Board Jigsaw (a 100 Board cut into at least 4 pieces). Directions
Questions/Hints[If students are having difficulty placing the pieces on the 100 Board.] Find the piece that matches the numbers on the 100 Board! Find the piece that has a corner (has the shaded square, the 9, the 90 or the 99). [When a piece is placed correctly.] Tell me how you worked that out? Why does the piece match? What are the tens? What are the ones? How are the numbers next to each other alike? How are they different? [If having difficulty placing the pieces on the blank 100 Board.] Remember the 100 Board? Have a look where your piece fits. Now place it on the blank Board. [If still having trouble.] What numbers are on the left of the 100 Board? What number is in the top left corner? Can you find these numbers? [If having difficulty placing the middle piece on the blank 100 Board.] Remember the 100 Board? Think where your piece fits. Now place it on the blank Board. [If still having trouble.] Look at the actual 100 Board. Count down from 0 and to the right to find a corner number in your piece. Repeat on the blank Board. Now place the piece. Complete the 100 board
Blank 100 board
100 Board Jigsaw
Activity 2 – Complete the jigsawMaterialsBlank 100 Board (laminated); 3-piece Jigsaw; 2-piece jigsaw Directions
Questions/Hints[If student is completing the jigsaw correctly.] Tell me how you worked that out? Why did you place the jigsaw piece so this number is in this position [pointing]? If the student is having trouble.] Remember the 100 Board? Look at the blank squares beside the numbers. Can you see a pattern in the numbers that will help you work out the number to write? [If still having trouble.] Look at the pattern in the tens down the columns! Look at the pattern in the ones across the rows! 100 Board jigsaw 2-Piece jigsaw Activity 3 – “Three in a row”Materials100 Board; Counters – one colour for each player; Card deck (Ace is 1, 10 is zero, no Jacks, Queens or Kings) Directions
Questions/Hints[When two cards are dealt.] What number do you want to cover? Which card will be tens? Which card will be ones? Where is this number on the 100 Board? [When covering a number.] What numbers are next to the number you just covered? Above it? To the right? To the left? Below it? Diagonally? [Just before dealing.] What number do you want? Why? Will this enable you to win the game? What cards do you need to be dealt? Week 2: Trial 2 materialActivity 1 – Getting to know the 100 BoardMaterials100 Board (0-99) Language Rows (go across); Columns (go down) DirectionsAsk the students to look at the 0-99 Board and identify the number patterns in the rows and columns and how they relate to tens and ones. Questions/Hints
Activity 2 - More or lessMaterials100 Board (0-99); Counters; Calculator. 100 Board (100-199 - if students can do the first activity easily). Directions
If students can do these easily, repeat for 132 and 175 on the 100 Board for 100-199. Questions/Hints[If students have difficulty covering the number that is 1 more.] Count on one more from the number. Use your calculator to prove that you covered the right number. [When student has covered the correct number.] Where is it in relation to 34? [On the right] What digit changed − the tens or the ones? [Ones] [If having difficulty covering the number that is 1 less.] Count back one from the number. Use your calculator to prove that you covered the right number. [When student has covered the correct number.] Where is it in relation to 34? [Left] What digit changed − the tens or the ones? [Ones] [If having difficulty covering the number that is 10 more.] Add 10 to the number. [If students cannot add 10.] Count 10 spaces on from 34. Use your calculator to prove that you covered the right number. [When student has covered the correct number.] Where is it in relation to 34? [Below] What digit changed − the tens or the ones? [Tens] [If having difficulty covering the number that is 10 less.] Subtract 10 from the number. [If students cannot subtract 10.] Count 10 spaces back from 34. Use your calculator to prove that you covered the right number. [When student has covered the correct number.] Where is it in relation to 34? [Above] What digit changed − the tens or the ones? [Tens] 100 Board (100-199)
Activity 3 - More or Less WindowsMaterials100 Board (0-99), Window, Calculator, Water-soluble pens 400-499 and 600-699 100 Boards (400-499 and 600-699 - if students can easily do the first activity.) Directions
Questions/Hints[If student cannot write in the numbers.] Look at the Board. Look at number 37. What is the number to the right? [38] How is this related to 37? [Ones increased by one.] What is the number to the left? [36] How is this related to 37? [Ones decreased by one.] What is the number above? [27] How is this related to 37? [Tens decreased by one.] What is the number below? [47] How is this related to 37? [Tens increased by one.] [If student cannot see the relationships.] Use the calculator to add and subtract 1 and 10. Count on 1 and 10; count back 1 and 10. [If student has difficulty with 97.] Place 100-199 Board under 0-99 Board. Use calculator to add 10 to 97. [If students can’t do three digits.] Look at the Board (440-499 or 600-699 depending on question). Look at this number. Look at the numbers to the right, left, above and below. Do the hundreds change? What does change? [Ones for left and right; tens for above and below.] 100 Board (400-499)
100 Board (600-699)
Activity 4 - Fill the spacesMaterialsFill the Space 1; 100 Board, Calculator Puzzle Page; Fill the Space 2 (for proficient students only) Directions
Questions/Hints[If students have problems with Fill the Space 1 activities.] Look at the 100 Board. Look at 56. What numbers are on the right and left of 56? How are they related to 56? [Ones increase and decrease.] What numbers are above and below 56? How are they related to 56? [Tens increase and decrease.] Use this to work out the missing numbers. [If students continue to have problems.] Find the given number on the 100 Board. Use a calculator to add and subtract one and to add and subtract ten. How are these numbers relate to the given number. Use this to write in the numbers beside the given number. [If still having problems.] Do the first one for them, describing how you did it. [If students having problems with Puzzle Page.] Look at the 100 Board. What is relationship between a number and one more, one less, ten more and ten less? Is this the same as the relationship between numbers that are beside, above and below other numbers? [Ones the same vertically. Tens the same horizontally.] Work backwards in puzzle. [If students having problems with 3 digits.] Look at 400-499 and 600-699 Boards. What changes with numbers beside, above and below other numbers? [Hundreds do not change. Ones the same vertically. Tens the same horizontally.]
Activity 5 - More or less three in a rowMaterials100 Board; Counters – one colour for each player; Card deck (Ace is 1, no Tens, Jacks, Queens or Kings). Directions
Questions/Hints[When dealt two cards.] What number do you want to cover? Will it be 1 more, 1 less, 10 more or 10 less than the number shown in the cards? [Just before dealing.] What number do you want? Why? Will this enable you to win the game? What cards do you need to be dealt? Week 3: Trial 1 materialActivity 1 – Introducing place value charts & bundling strawsMaterialsTens-Ones Place Value Chart (PVC); bundling straws Language Ones, tens, hundreds; groups; place value. Directions
Questions/Hints[If students do not know the largest and smallest values.] Read the place names as I point to them (pointing from right to left). Would you rather have 10 dollars or 1 dollar? The tens are larger than the ones. Is the tens place on the left or right? What would be the smallest amount of money − 10 dollars or 1 dollar? The ones are the smallest place. Is it on the left or right? [When showing loose straws.] These are called ones. Where would you put them on the Place Value Chart? [Ones place] [When showing a bundle of ten straws.] These are called tens. Where would you put them on the Place Value Chart? [Tens place] [When making sure the student knows where to put the tens and ones.] Here are some ones. Where do they go? Put them on the PVC in the correct place. How many ones are there? [Repeat for other collections of straws.] Here are some bundles of ten. What are they called? Where do they go? Put them on the PVC in the correct place. How many tens are there? [Repeat for other bundles of 10 straws.] Tens-Ones place value chart
Activity 2 - Making groups of 10MaterialsTens-Ones Place Value Chart (PVC); bundling straws. Directions
Questions/Hints[When making groups of 10.] Here is a handful of ones. How many tens can you make from these ones. Do you have enough ones to make a group of 10? Show me. Put a rubber band around them to hold them together. Where will you put the 10 on the Place Value Chart - in the tens place or the ones place [pointing to the ones and tens places]? Do you have enough ones to make another ten? [Keep going until all possible bundles of tens have been formed.] [When all the bundles of tens have been made.] How many tens and how many ones? What number have you made? Count by tens first [ten, twenty, thirty, forty, etc)]. Activity 3 - Making numbersMaterialsTens-Ones Place Value Chart (PVC); bundling straws. Directions
Questions/Hints[If students have difficulties with the 2 tens and 7 ones.] Show me the 2 tens part. Show me the 7 ones part. How do we say the 2 tens part? [Twenty] How do we say 7 ones? [Seven] How do we say them together? [Twenty-seven] [If students need more reinforcement.] Place your left hand on the tens and say “2 tens”; move your hand to the ones and say “7 ones”. Now point from left to right, saying the whole number name [“twenty seven”]. [Repeat this for the other numbers.] [If students have difficulties with the twenty-five.] Look at the parts of twenty-five. Look at the twenty part. How many tens make twenty? Show me the 5 part. How many ones in this part? Place these tens and ones on the PVC. Read the numbers both ways (as “twenty-five” and as “2 tens and 5 ones”) using the hand movements of (1) above. [As students begin to be able to construct and read two-digit numbers.] Which is worth more − the twenty part or the 7 part? What position, left or right, is the part that is worth more? Activity 4 - Writing NumbersMaterialsTens-Ones Place Value Chart (PVC); bundling straws; worksheet with small Tens-Ones Boards drawn on it; pencil or pen. CARDS??? Directions
41)Repeat for six tens and 4 ones, 4 tens and 1 one, one ten and 4 ones, and 4 tens. Reverse the direction - start from saying the number, move to writing the number on the Worksheet, to putting straws on the PVC. Questions/Hints[Material to writing number.] How many tens? Where would you write the number to show these ones? How many loose ones? Where would you write the number to show these ones? [Reading the number.] What is another way to say 3 tens? (Thirty) How do you say 7 ones? How do you say it all together? [Thirty-seven] Tens-ones board worksheet
Activity 5 - Counting and trading ones for tensMaterialsPlace Value Chart; bundling straws; writing paper with Tens Ones; pencil; calculator Directions
Week 3: Trial 2 materialActivity 1 – Ungrouping tens to make onesMaterialsPlace Value Chart (PVC); bundling straws; dice Language Ones, tens; ungroup Directions
Questions/Hints[When students are about to roll the die.] How many tens do you have? How many loose ones? Will you have enough loose ones to remove the ones when you roll the die? Will you have to ungroup a ten? [After students have rolled the die.] Do you have enough loose ones to take away the ones showed on the die? [If student says Yes, say: Show me. [I student says No, say: How can you get more ones from this number?] [Ungroup a 10 and put the 10 ones in the Ones place.] How many tens do you have now? How many ones? Activity 2 - Getting to know the MAB blocks; TradingMaterialsTens-Ones Place Value Chart (PVC); MAB ones and tens. Language Small blocks (ones); long blocks (tens); trading tens for ones. Directions
Questions/Hints[If students are unsure about the relations between MAB tens and ones.] Line up the small blocks beside the long block (see the picture below). How many ones equal one ten? [When students are about to roll the die.] How many tens do you have? How many loose ones? Will you have enough loose ones to remove the ones when you roll the die? Will you have to trade a ten – change a MAB ten for ten MAB ones? [After students have rolled the die.] Do you have enough loose ones to take away the ones showed on the die? [Yes] Show me. [No] How can you get more ones from this number? [Remove a MAB ten block and replace it with ten MAB one blocks – this is trading.] How many tens do you have now? How many ones? Activity 3 - Introducing hundredsMaterialsHundreds-Tens-Ones Place Value Chart (PVC); MAB hundreds, tens & ones LanguageHundreds; MAB hundreds blocks; trading hundreds for tens. Directions
Questions/Hints[If students have problems placing the 99.] Show me the ninety part. How many tens? Show me the nine part? How many ones? [When adding the first one.] How many ones do you have now? Do you have enough ones to trade for a ten? [Student should trade] How many tens do you have now? Do you have enough tens to trade for a hundred? Compare ten tens with one hundred (as in picture). What is the number?
[When adding the remaining ones.] Ask how many hundreds? How many tens? How many ones? Write the number. Say the number. Hundreds, tens, ones place value chart
Activity 4 - Trading gamesMaterialsHundreds-Tens-Ones Place Value Chart (PVC); MAB hundreds, tens & ones Directions
Questions/Hints[When students are about to roll the die.] How many hundreds do you have? How many loose tens do you have? How many loose ones? Will you have enough loose tens to make a hundred or to remove the tens when you roll the die? Will you have to trade – change ten MAB tens for a MAB hundred/MAB hundred for ten MAB tens? [After students have rolled the die.] Do you have enough loose tens to make a hundred or to remove the tens when you roll the die? [Yes] Show me. [No] How can you get more tens from this number? [Trade a MAB hundred block for ten MAB ten blocks.] How many hundreds do you have now? How many tens do you have now? How many ones? Say the number. Write the number. Activity 5 - Calculator countingMaterialsPlace Value Chart (PVC); MAB blocks; calculator; blank paper on which to write numbers. DirectionsSet up the session. Initially, the tutor uses the blocks on a large PVC and writes the numbers on a piece of paper (on which there is written Hundreds Tens Ones). The student uses the calculator. Counting by hundreds. Start with 364 on the PVC. Ask the students to read the number [three hundred and sixty-four]. Tutor to write the number on the paper. Say Add one hundred. [Tutor puts out one more Hundred and writes the new number under the starting number; the student adds 100 on the calculator.] Keep doing this until you have 964. Ask the students to read each calculator number aloud and to whether it matches the number shown by the blocks. Have the student read the list of numbers aloud and ask which place changed – the ones, the tens or the hundreds? Ask why the ones didn’t change and why the tens didn’t change. Repeat the above starting from 273 but with the tutor using the calculator and the students writing on paper. When the activity is finished, ask the students what place changed first when they were counting by 100s. Check the written numbers to see if only the 100s changed. Ask the student to enter 438 on the calculator and to read the number. You show the number on the PVC with MAB as well as record the number on paper. Say: If we count by 10s, which place will change first – the hundreds, the tens, or the ones? Add 10 to your calculator, what number do you have now? I’m going to add 1 ten to my MAB. Does my number match yours? Activity 6 - String counting (Just for Fun!)Materials: String counting cards (next page)A length of wool.
Three cards (Cards A, B and C) with numbers on the right and left margins of one side. The numbers follow an order given on the card, for example, counting in 10s. The lines follow the path the wool would follow as it wraps around the card when correctly joining the numbers in order on the other side. However, the lines do not form a pattern so the student cannot self-check his or her answers. This is “Just for Fun” activity. One is done for you (see below). Other string cardsSee next page. DirectionsDo the cards in the following order: Counting in 10s; Counting in 100s; Counting in 1s. Wrapping the wool around the card in order of the numbers and checking by turning the card over. Questions/Hints[If student is incorrect.] Ask the student to write the numbers in order in a column and then to follow this order of numbers with the wool.
String counting cards
Week 4: Trial 1 materialActivity 1 - Translating numbers from language to symbol formMaterials“Cross the River” worksheet and cards; markers; PVC with Hundreds, tens, Ones; MAB; deck of cards with the tens and picture cards removed. Directions
Questions/Hints
If the student reads the number correctly (e.g., two hundred and seventy-five) but covers the wrong stepping stone (e.g., 20075), then have him/her show 275 on a PVC with MAB and use a playing card to represent the digit in each place. Then ask him/her to find the stepping stone that has the same number as shown by the cards. Note difficulties that students have with: (a) reading numbers, (b) writing numbers, (c) knowing place values, and (d) relating place values to reading and writing numbers. Choose activities to help the students overcome these difficulties. “CROSS THE RIVER ” Workheet
Find a safe path across the river – don’t tread on a crocodile
Cut out the cards and turn them over in a stack. Activity 2 - Reading numbers (knowing place value positions)MaterialsPlace Value Chart (PVC); MAB Directions
Activity 3 - Writing numbers (in relation to reading & place value)MaterialsPlace Value Chart (PVC); MAB; digit cards; paper; pen Directions
Activity 4 - Comparing numbersMaterials“Comparing numbers” sheet for 2- and 3-digit numbers, card deck. Place Value Chart (PVC); MAB; digit cards; paper; pen DirectionsThe following games are for two players.
Questions/HintsDuring each play, tutor asks: What card have you got? Where do you think you should put it? Why? What number would you like to get? Why? Focus of tutor questioning is to get the students to see that large digits should be placed in leftmost positions (and small digits in the ones position). [If student still has difficulties.] Use MAB to put numbers on Board and look at which number has the most material, or place numbers on a number line.
Week 4: Trial 2 materialActivity 1 - Reading and writing numbers with teensMaterialsPlace Value Chart (PVC); MAB; digit cards; paper; pen Directions
Questions/Hints[If difficulty with 216.] Tutor says: Show me the two hundred part; show me the teen part; show me the six part. Discuss what the number should/could have been called: two hundred and “onety-six.” [If difficulty with comparing 216 with 260 and 261.] Tutor says: How many ones in two hundred and sixteen, how many in two hundred and sixty, and how many in two hundred and sixty one? Repeat for tens and hundreds. Activity 2 – Reading and writing numbers with zerosMaterialsPlace Value Chart (PVC); MAB, digit cards; paper; pen Directions
Questions/Hints[If problems with saying the number.] Tutor says: Show me the two hundred part; put a digit card underneath to show this part. Show me the tens left over (the teen part); put a digit card underneath to show this part. Show me the left over ones (the seven part); put a digit card underneath to show this part. Activity 3 - Relationship between place-value positionsMaterialsCalculators; digit cards; PVC with hundreds, tens, ones; MAB; paper; pen Directions
Digit Cards
Activity 4 – Game: “Close, Closer, Closest”MaterialsCalculators; playing cards with 10s and picture cards removed; cards; paper; pen; Place Value Chart (PVC); MAB Directions
Block B: Powerpoint training activities
Block B: Excel training activities
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