Web viewEarly level*/** Activities to develop knowledge: forwards and backwards number word...

Knowledge activities - ideas for each stage from NZ maths

Early level*/**

Activities to develop knowledge: forwards and backwards number word sequences

Rote counting is a valuable activity that can be done in many ways:

Sit in a circle and count together. Clapping or other rhythmic movements can be used to emphasise each number in the counting sequence.

Count in different voices "Let’s all count to 8 in a whisper voice, let’s all count backwards from 14 using a giant voice."

Use every opportunity to count together. Count the number of library books as they are returned, the number of children that are lined up waiting or the number of paint brushes needed for the class.

Students in small groups pass an object around the circle. As each student receives the object they say the next counting number. What is the highest number the group can count to? Reverse the direction the object is passed and count backwards.

Activities to develop knowledge: numeral identification

Writing Numbers

Get students to practice writing numbers in many different ways. For example:

painting numbers: "Let’s all paint some eights. Can you paint a small eight? Can you paint a large eight? Can you paint a red eight?"

tracing numbers in trays of sand making numerals out of clay include numbers in your handwriting programme. "Today we’re working on letter L and

the number 4" use calculators "Press number 3, now press number 9."

Lily Pads

Place numeral cards around the room, tell the students the numbers are lily pads and they are frogs. They need to find the number you call out and jump on it. "Jump on number 7, jump on number 6".

Number fans

Model number fans and saw “show me number two, five, three….” Start with numbers up to five, and move on as knowledge grows. Or say, “what number is this?” Or child can be teacher and ask you!

Lucky Dip

Have the students take turns drawing numeral cards from a large bag. As each student takes a "lucky dip" ask them to tell the others what number they have got and then all practice writing the number in the air together. "What number have you got Sam? Seven. Let’s all draw seven in the air. Across and down, that’s how we write a seven."


Activities to develop knowledge: grouping/ place value

Finding Fingers

Ask students to hold up a certain number of fingers. "Show me 3 fingers, show me 5 fingers." Encourage students to use the right number of fingers without having to count. Work first on numbers to 5, then numbers to 10. When focusing on numbers to ten identify both five and doubles based patterns. For example, eight can be recognised as five and three, or four and four.

Dice Patterns

Play simple dice games. Encourage students to recognise the number that has been rolled on the dice without counting the number of dots.

Five Frames and Ten Frames

Introduce children to five and ten frames. Make patterns with paint, playdough or in sand. Count dots and tap on table or in the air. Talk about what they see.

Early***/First *

This video shows teaching ideas for counting from one, counting on and imaging:

http://www2.nzmaths.co.nz/frames/ONPD/M7/08.aspx

Activities to develop knowledge: number sequence and order

Once students can count to 10, work on the numbers to 20. Extending the range of numbers in the counting sequence develops students’ knowledge.

Providing a visual cue for counting helps support the counting sequence and develop number identification skills. Visual supports for counting include the hundreds board, Slavonic abacus, number lines, and flip strips.

Counting together

Use a 100s board, preferably with flip-capability. Get the students to clap as they count in ones. Flip over the numbers on the hundreds boards as they are said. Practice the number sequence forward and backwards, and vary the starting point for the counting. Ask students to identify individual numbers on the hundreds board from the sequence they have just counted.

Try cutting up a Hundreds Board into horizontal strips from 0 -10, 11 – 20 etc.. Lay out strips from smallest to biggest in one long line. Count sequences and discuss patterns and connections we can see. Then put back together into a hundreds board, and discuss how the numbers are the same but arranged in a different way.



Counting out loud

Count in different voices "Let’s all count to 19 in a whisper voice, let’s all count backwards from 16 using a giant voice."

Activities to develop knowledge: grouping/ place value

This video shows ways to develop knowledge of patterns to ten:


Finding Fingers

Ask students to hold up a certain number of fingers using one or BOTH hands. "Show me 4 fingers, using two hands. How many on each hand? Can you make 4 a different way?” This encourages knowledge of combining and partitioning numbers.

Dice Patterns

Flash dice and domino patterns – regular and irregular, showing the pattern for a shorter and shorter time, until they can recognise the pattern when flashed for half a second.

Five and Ten Frames

Introduce students to five and ten frames. Flash ten frames cards and ask students to call out the number of dots on each card. Students may need to count the dots at first, but as recognition of the patterns grows encourage quick responses rather than counting. Vary the activity by asking students to call out the number of spaces on each card.

Five Little Ducks

Sing the song "Five little ducks" Five little ducks went out one dayOver the hills and far awayMother Duck said "quack, quack, quack, quack"But only 4 little ducks came back.

Repeat, changing the numbers in each verse as fewer ducks come back each time. As students sing, have 5 students act out the song. As each duck gets lost, one student at a time moves away. After each verse, reinforce the fact of 5: "How many ducks are left? How many ducks have got lost? How many ducks were there to start with? That’s right we had 5 ducks, but 2 have got lost and now there’s 3 left, 2 and 3 make 5." This activity can be repeated with many other well known number songs working on facts to 5 and 10, for example Five Flying Saucers or Ten Fat Sausages.



Activities to develop knowledge: number identification

Number fans

“What number can you see? Can you make me a number….?” Combine numbers to make two digit numbers, or three digit numbers.

Peg that number

Give every student a number line and access to coloured pegs. Call out instructions such as "Put a red peg on the number 6, put a yellow peg on number 19". After 6 numbers have been called out students compare number lines with a partner to see if they have the same numbers pegged. For students that need more support, numeral cards can be held up as each number is called.

Bingo

Give each student a card with the numbers from 1 to 20 on. Provide 6 counters for each student and ask them to choose 6 numbers on their board and cover each of these with a counter. The teacher calls out numbers from 1 to 20, one at a time, and as each number is called students remove a counter if it is covered with the number called. The winner is the first student to remove all their counters.

The Number Bridge

Lay out numeral cards from 1 to 10. Ask one student to "walk the bridge" by standing on numerals slowly, one at a time, walking either up or down the "bridge." The remaining students call out the numbers as they are walked on.This activity can also be carried out using numbers from 10 to 20.

Use every opportunity to have students write numbers for themselves. For example:

as students solve number problems ask them to find the numeral card that represents the answer to the problem, or write down the answer

at newsboard time ask a student to write down the number that represents the date include numbers in your handwriting programme. "Today we’re working on letter L and

the number 4."

Activities to develop knowledge: basic facts

Cows in Paddocks

Tell the students Framer Brown has 5 cows and 2 paddocks and he wants to know how many different ways he can arrange the cows in the paddocks. Using plastic animals and paddocks drawn on paper, ask the students to arrange the cows in different ways. "If there are three cows in this paddock, how many are in the other paddock? What if we put 4 cows in this paddock?" Reinforce the facts of 5 as they are shown "2 cows and 3 cows make 5 cows altogether."This activity can be repeated using 10 animals to help students learn facts of 10.


Animal Strips

Animal strips are Material Master 5-2 , link:

http://nzmaths.co.nz/sites/default/files/Numeracy/2007matmas/Bk5/MM%205_2.pdf

This link from NZ maths shows how you can use animal strips to teach addition facts.

http://www.nzmaths.co.nz/sites/default/files/Animations/animal_strips3.swf

Activities to develop knowledge: written recording

As students solve number problems encourage them to record their working in a variety of ways including:

1. using pictures. For example to solve the problem "Tane has 3 apples and 4 bananas, how many pieces of fruit does he have altogether?" students may draw:

2. using diagrams. For example, to solve the problem "There are 12 cars in the car park and 4 drive away, how many cars are left?" students may draw:

http://www.nzmaths.co.nz/sites/default/files/Animations/animal_strips3.swf



3. using symbols:

4 and 3 is 7 4 plus 3 equals 7 4 + 3 = 7 Model these methods of recording for students and provide opportunities for them to

use them independently.

First*/**


These videos show counting activities using number lines and bead frames:



Counting

Count together, both forwards and backwards, starting from numbers other than 1. Focus students’ attention on numbers before and after "Let’s count from 35 to 79. What will be the first number we say after 35? What will be the last number we say before 79?"

Flip strip

This link:

http://www.nzmaths.co.nz/sites/default/files/Animations/flip_strip.swf

shows how Flip strips can be used to help students develop the ability to identify the number after any given number. Change the numbers in the strip, working with larger numbers as students become more confident.Flip strips are Material Master 4-2, link:


Lucky Dip

Use a set of numeral cards. Have students take a "lucky dip" to get a card then ask "What’s the number before this? What’s the number after this?" Identify the numbers in a hundreds board.

Skip Counting

Rote count with body percussion to help develop and learn skip counting sequences. For example students touch their knees and quietly whisper "one" then clap and say "two", and so on.


http://www.nzmaths.co.nz/sites/default/files/Animations/flip_strip.swf




Skip counting on the number line

Use a large number line and put pegs on every fifth number. Read the numbers with pegs on to learn the skip counting sequence in fives. Gradually remove the pegs as the skip counting sequence is learned.Repeat for other skip counting sequences, for example 2 and 10.

Bead Strings

This link:

http://www.nzmaths.co.nz/sites/default/files/Animations/bead_strings.swf

shows how bead strings can be used to develop and practice skip counting sequences.

Activities to develop knowledge: numeral identification

Guess My Number

Use a hundreds board and choose a "mystery number" between 1 and 100. Students ask questions, one at a time, to find the mystery number. Students should be encouraged to ask open questions, for example, "Is it in the twenties?", "Is it an even number?" or "Is it greater than 50?" Direct questions such as "Is it 24?" should be disallowed. As each question eliminates possibilities for the mystery number, these are flipped over on the hundreds boards, so the remaining possibilities can be clearly seen. Once the mystery number is identified, clearly identify it for all students.

Activities to develop knowledge: grouping and place value

This video shows ways to help teach teen/ty numbers:


Teen numbers

Seat the students in pairs. One partner shows 10 fingers. The other partner shows any number of fingers from one to nine, say six. The "ones" person says "six" and the other partner says "10", and together they say "is sixteen".

Place value

This video shows ways to help teach knowledge of place value:


Activities to develop knowledge: Written Recording

As students solve number problems ask them to record their work using number sentences such as 5+2=7 and 8-3=5. Provide students with number sentences and ask then to make up story problems to go with them.



http://www.nzmaths.co.nz/sites/default/files/Animations/bead_strings.swf


First**/***


Ten frames groupings

Ten frames cards can be used to help students learn the addition facts of ten and related subtraction facts. Show students one frame at a time, and ask questions about the numbers of dots and spaces on each card. "How many dots? How many spaces? How many altogether?" Students can also be asked to provide addition and subtraction facts for each card. For example, the card with 7 dots and 3 spaces shows the facts of 7+3=10, 3+7=10, 10-7=3 and 10-3=7.

Patterns to ten

Unifix cubes can be used to help students learn to recall pairs of numbers that add to ten. Group the students in pairs or small groups each with a stack of ten cubes. Without the others seeing, a student takes some cubes from the stack. They show the other students the remaining cubes and they must say how many cubes were taken.

Doubles

Show students one numeral card at a time, using the numbers 1- 9. As each number is shown, students call out the doubles fact for that number, e.g. when "6" is shown, students call out "12."

Packets of Five

Link multicubes together into groups of 5. Develop children’s knowledge of the multiplication facts of 5 by asking questions such as "if we have 3 groups of 5, how many will we have altogether?" Record the facts in equation form (for example, 3 x 5 = 15) for students to refer to.

Arrow cards

Follow this link to see how arrow cards can be used to reinforce groupings of ten in 2 digit numbers.

http://www.nzmaths.co.nz/sites/default/files/Animations/arrow_cards.swf

Beans

This equipment animation shows how beans can be used to develop students’ knowledge of the groupings of ten in two digit numbers.

http://www.nzmaths.co.nz/sites/default/files/Animations/beans.swf

http://www.nzmaths.co.nz/sites/default/files/Animations/beans.swf

http://www.nzmaths.co.nz/sites/default/files/Animations/arrow_cards.swf



Number fans

This equipment animation shows how number fans can be used to develop number identification.

http://www.nzmaths.co.nz/sites/default/files/Animations/number_fans.swf

Fraction Pieces

Get the students to sort similar fraction pieces and then use these to create whole circles. Record the words and symbols for each fraction on the board or modelling book.


Rocket

Each student draws a rocket like the one shown.

Students take it in turns to roll 2 standard dice, and make a 2 digit number from the numbers rolled. For example, if a 3 and a 5 are thrown the student can choose to make the number 35 or 53. The student that rolled the dice then writes the number into their rocket, choosing which space to put it in; smaller numbers must go below larger numbers. If a player cannot place the number they have rolled they miss a turn. Once a number is written it cannot be removed. The winner is the first player to fill every floor of the rocket.

Extension: The game can be extended to develop students’ ability to order 3 digit numbers by using 3 dice and placing resulting numbers between 110 and 667.

Number Line Flips (Number Sequence and Order domain)

Give the students an empty number line that shows only the numbers at the start and end. The other numbers are shown on the reverse side as a reference. Have the students place a peg on the number line and ask their partner to guess which number the peg is on. If the students have difficulty to start with, their partner can help by pointing to and saying reference numbers close to the pegged number. For example, "I am pointing to 15. What number is the peg on?"

Number Lines (Written Recording domain)

As students solve number problems ask them to use number lines to record their solution strategies. For example: 26 + 7 can be solved as (26 + 4) + 3, and recorded as

http://www.nzmaths.co.nz/sites/default/files/Animations/number_fans.swf


Second Level*


Students will by this stage know the basic addition and subtraction facts. It is important to ensure they can recall these basic facts

Pocket Facts

On small pocket sized pieces of card write the addition and subtraction facts so each card has one fact on it. Give each student a card with a new fact for them to learn during the day and to take home. When a student is able to recall their pocket fact they can swap it for another unknown fact. Keep track of the facts each student’s able to recall

Facts Recall

The students sit in a circle. One student has a tennis ball or koosh ball and they call a single digit number and toss the ball to someone else in the circle. The second person calls a single digit number and tosses the ball to a third person. The third person adds the numbers together and calls the answer, then calls a new number to start over

Students will by this stage know the basic facts for the 2, 5, and 10 times tables and are learning the other multiplication basic facts

Deriving facts from known facts

This video shows ways to teach children how to derive new facts from known facts:


Bead Frame

This equipment animation shows how to derive facts from known facts by sliding groups of beads on the bead frame. For example, how to solve 6 x 5 by using 5 x 5.

http://www.nzmaths.co.nz/sites/default/files/Animations/beadframe.swf

Happy Hundred

This equipment animation shows how to derive facts from known facts using the ten by ten array.

http://www.nzmaths.co.nz/sites/default/files/Animations/happy_hundred.swf

http://www.nzmaths.co.nz/sites/default/files/Animations/happy_hundred.swf

http://www.nzmaths.co.nz/sites/default/files/Animations/beadframe.swf



Multiply Target

This is a game to help develop recall of the 6, 7, 8, and 9 times tables. Draw a circle divided into quarters with the numbers 6, 7, 8 and 9 marked in the quarters.

This circle is a target. Get a student to toss a koosh ball at the target twice and multiply the two numbers together.

Activities to develop knowledge: grouping / place value

Students are learning how many 10s and 100s are in whole numbers so they can understand and operate with larger numbers.

Bead Frame

Show the students a bead frame and tell them each bead represents a hundred. Slide 8 beads across a wire and ask the students how much that represents. Slide 23 beads and ask the students how much that represents. Repeat this type of question and record the results for the students on a whiteboard. .

Number of beads Number represented8 80023 230057 5700

Encourage the students to see how the two columns relate to each other. Vary the question and ask how many beads are needed to represent 5700. This activity can be adapted so each bead is worth 10.


Play Money

Using play money students can count the number of ten or hundred pound notes to make thousands numbers. For example, tell the students to count out £1600 using the hundred pound notes and ask how many notes they used. Repeat this type of question. Ask the students to record their answers on a table.

Amount of money Number of $100 notes£1600 16

£2500

£3100

Ask the students to look at their result and now predict how many £100 notes are in £4600, they can check their prediction using the play money. The same activity can be done with £10 notes.

Students are learning how many groups of 2, 3 and 5 are in numbers up to 100, and finding the resulting remainders. For example, there are 7 groups of 5 in 37 with 2 remainders.

Tens Frame

Students are learning how many groups of 2, 3 and 5 are in numbers up to 100, and finding the resulting remainders. For example, there are 7 groups of 5 in 37 with 2 remainders.

Using tens frames show the students a number less than 100. For example, 43 is shown as:

Ask the students “How many groups of five are in this number?” and then “how many are left over?” Encourage the students to skip count in 5s to the number and count the remainders.


Students are learning to count forwards and backwards in tens and hundreds, and learning to give a number ten, and hundred after a number. This helps students develop an understanding of the place value of larger numbers.

Counting number sequences

As a whole class, or in a small group, count together forward and backwards from any number in tens and hundreds. For example, 488, 498, 508, …


Forward and backwards with 10s and 100s

Make two spinners, one with 10 and 100 marked on it, the other with + and – marked on it.

Choose a number to start from and write it on the whiteboard for the students to see. Ask for a volunteer to spin both spinners. The students then write the new number on a piece of paper. Check they are correct. For example, start with 4790, if the spinners showed 100 and +, then the new number is 4890. Starting from the new number, then repeat this activity by spinning the spinners. Check that students can say as well as write the new number.

Continue a fraction sequence

Make copies of the fraction pieces sheet, available from http://www.nzmaths.co.nz/sites/default/files/Numeracy/2007matmas/Bk4/MM%204_19.pdf

Each child will need two copies of the sheet. Ask the children to cut out the circles that are divided into thirds. Ask them to show one third, then add a third to show two thirds, then add a third to make three thirds and complete the circle. Ask them how many thirds it takes to complete the circle and record this on the whiteboard. Continue to count the thirds; four thirds, five thirds and six thirds. Record the sequence on the whiteboard:

1 third1/3

2 thirds2/3

3 thirds3/31

4 thirds4/31 1/3

5 thirds5/31 2/3

6 thirds6/32

Ask the students to count the sequence backwards as they take the circles apart. Repeat this activity with the other fraction pieces.

Ordering unit fractions

Give lots of opportunities to order numeral cards with unit fractions from smallest to biggest.

http://www.nzmaths.co.nz/sites/default/files/Numeracy/2007matmas/Bk4/MM%204_19.pdf



Students are developing their knowledge and understanding of decimals so they can operate with decimals at later stages.

Recording decimals

Ask students to record decimals in different ways. Decimals can be expressed in words, fractions or decimals in expanded or compact form. For example, write the decimal 4.63 on the whiteboard and ask the students to write it in different ways. Then write their suggestions on the whiteboard. They may include expressions such as:4 ones, 6 tenths and 3 hundredths, 4 ones and 63 hundredths, 4 + 0.6 + 0.03. 4 + 6/10 + 3/100 etcRepeat the activity with a different decimal and challenge the students to come up with as many expressions as they can.

Representing decimals

Use these decimats to show children how decimal fractions can be represented visually.


Children can make decimals from symbols and vice versa and order decimal fractions:



Second Level**/***


Students at this stage are learning to find equivalent fractions for halves, thirds, quarters, fifths, and tenths. They are developing an understanding that fractional equivalents are different names for the same fractional amount. This knowledge is useful when comparing fractions and for converting between fractions, decimals and percentages. Knowing equivalent fractions also makes it easier to add and subtract fractions.

Fraction Wall

This equipment animation: http://www.nzmaths.co.nz/sites/default/files/Animations/fraction_wall.swf

demonstrates how bar fractions can be used to illustrate equivalent fractions. For example, 1/3 = 2/6.

Finding equivalent fractions with sets

This equipment animation: http://www.nzmaths.co.nz/sites/default/files/Animations/beans_2.swf

demonstrates how to find equivalent fractions using sets. For example,

http://www.nzmaths.co.nz/sites/default/files/Animations/beans_2.swf

http://www.nzmaths.co.nz/sites/default/files/Animations/fraction_wall.swf



Using multiplication and division to find equivalent fractions

Students will use their knowledge of multiplication and division facts to find equivalents for fractions with denominators up to 100 (and 1000). For example, when calculating ¾ of 60, students can find one quarter of 60 is 15, then multiply this by three to get three quarters: 3 x 15 = 60. This can be shown conceptually on a double number line:

l

This link takes you to the Material Master 8-9 from NZ maths, and asks students to explain if two fractions are equivalent or not.


Students at this stage can order unit fractions for halves, thirds, quarters, fifths, and tenths. Ordering fractions beyond unit fractions requires a good understanding of the size of each fraction.

Comparing fractions

Using strips of paper students can compare the relative size of fractions. Take a strip of paper and fold it into parts, for example, quarters and make the lines in quarters.

Take another strip of paper the same length and fold it into thirds and mark the lines.

Students can now make comparisons between quarters and thirds. For example, 2/3 > 1/2 and 3/4 > 2/3 etc. The Numeracy Professional Development Projects, Book 4 Teaching Number Knowledge has an activity “Who Gets More?” which uses squared paper to make comparisons. A piece of paper can be folded into quarters and three quarters coloured in. Using a second piece of paper the same size, fold it into thirds and colour in two thirds. By comparing the amount of paper coloured in students will see that three quarters is bigger than two thirds.

http://www.nzmaths.co.nz/sites/default/files/Numeracy/2008numPDFs/NumBk4.pdf



Students at this stage can read and write decimals. The students are learning to order decimal numbers. This requires a good understanding of how the position of the decimal point affects the relative size of the numbers. Students learning to work with decimals numbers commonly mistakenly assume that the more digits in a number, the larger it wil be, for example, assume 0.25 is greater than 0.4.

Using decimats

The decimat from Material Master 7-3, link: http://www.nzmaths.co.nz/sites/default/files/Numeracy/2007matmas/Bk7/MM%207_3.pdf

can be used to clearly illustrate decimals in the tenths and hundredths. Give the students copies of the decimat and ask them to shade in decimals. They can then compare decimals and order them. For example, 0.4 > 0.25 can be clearly shown as:

Comparing decimals with equipment

The Pipe Decimals equipment animation:

http://nzmaths.co.nz/sites/default/files/Animations/pipe_decimals.swf

shows how the size of two decimals can be compared using equipment.

Comparing decimals with Arrow Cards

The Arrow Cards 2 equipment animation:

http://nzmaths.co.nz/sites/default/files/Animations/arrow_cards2.swf

shows how using arrow cards can help students to order decimals as they consider firstly the tenths column, then the hundredths, then the thousandths column.


Students at this stage will know the basic multiplication facts up to 10 x 10. It is important that students can also recall the corresponding division facts.

http://nzmaths.co.nz/sites/default/files/Animations/arrow_cards2.swf

http://www2.nzmaths.co.nz/frames/Animations/arrow_cards2.html

http://nzmaths.co.nz/sites/default/files/Animations/pipe_decimals.swf

http://www2.nzmaths.co.nz/frames/Animations/pipe_decimals.html



Division Flash Cards

The Material Master 4-37, link:


has flash cards for practising instant recall of division facts. Cover one of the factors in the corners of the card with your thumb and ask the students to name the missing factor. Students can work in pairs to practise the division facts using the flash cards.

Pocket Facts

On small pocket sized pieces of card write the division facts for the 10 x 10 basic facts so each card has one fact on it. Give each student a card with a new fact for them to learn during the day and to take home. When a student is able to recall their pocket fact they can swap it for a new one. Keep track of the facts each student is able to recall

An important item of knowledge linked to the basic facts is the ability to identify factors of numbers to 100.

Factor trees

Factor trees can be used to find the factors of a number using prime factorisation. Each branch of the tree is divided until a prime factor is reached. Combinations of factors can then be found. For example:

Factor bingo

Each student has a strip with 10 numbers on it. Each number is the result of a multiplication basic fact. A pile of cards numbered 4 – 9 is placed between the players. As a card is turned over players write the number under any bingo number that it is a factor of. For example,

The first player to have a factor for each number calls bingo to win.

Students at this stage are learning to recall the benchmark conversions between fractions, decimals and percentages: halves, thirds, quarters, fifths, and tenths. Students will need to come to this stage with an understanding of the size of fractions and the meaning of the tenths and hundredths columns.



Using decimats

The decimat from Material Master 7-3, link:


can be used to help students to understand the equivalence of fractions, decimals, and percentages. Ask students to shade in one fifth of the decimat and record this as 1/5. Ask the students to use the tenths and hundredths divisions on the decimat to find the decimal equivalent and record this as 0.20. Finally ask the students to convert this to a percentage and record this as 20%. Students should work out conversions for halves, thirds, quarters, fifths, and tenths.

Recalling common conversions

Once students understand the conversion between fractions, decimals, and percentages they can begin to recall these conversions. Make sets of cards with the fraction, decimal and percentage for halves, thirds, quarters, fifths, and tenths marked on them. The cards can be used to play games such as Go Fish where students ask each other for cards to complete a set of three, or memory where the 3 matching cards are found.


Second*** and beyond….

Knowledge being developed

At this stage it is important to continue developing students’ knowledge in the Number Sequence and Order, and Grouping / Place Value domains. This knowledge will enable them to use a range of proportional strategies to solve problems involving fractions, decimals, percentages and ratios which is characteristic of students working at the advanced proportional stage.

In the Number Sequence and Order domain students need to be able to:

know the number of tenths, hundredths, and thousands that are in numbers up to three decimal places

know what happens when a whole number or decimal is multiplied or divided by a power of 10

In the Grouping and Place Value domain students need to be able to:

order fractions, decimals, and percentages

Students will need to know these things in order to use advanced proportional strategies.

There are other areas of knowledge that students will continue to develop at Stage 8. More specifically students need to develop the ability to:

say word number sequences by thousandths, hundredths, tenths, ones etc starting from any decimal number (Number Sequence and Order)

recall fraction ↔ decimal ↔ percentage conversions for given fractions and decimals (Basic Facts)

identify common factors of numbers to 100, including the highest common factor (Basic Facts)

Knowledge being developed: number sequence and order

Students at this stage are learning to order fractions, decimals and percentages. Students should already be able to order sets of fractions and sets of decimals to three decimal places. They will also be able to recall conversions between benchmark fractions and decimals. At this stage students are learning to use this knowledge to order sets that include mixtures of fractions, decimals, and percentages.



Comparing and ordering fractions, decimals, and percentages

Use a strip of paper to make a triple number line that shows fractions, decimals, and percentages. Fold the strip of paper in half and ask the students what fraction, decimal and percent are on the half way mark.

Continue folding the paper to show quarters and ask the students what fraction, decimal and percentages are shown by the folded lines. Mark these numbers on the triple number line.Ask the students to order some fractions, decimals and percentages. For example, order 60%, 0.4 and 3/4 from smallest to biggest. Direct students thinking by asking questions such as:Which number would do you think would be the smallest? (0.3)Why? (It is the only one less than a half)How can you compare 60% and 3/4? (Change both to percentages)Which is bigger 60% or 75%? (75%)Students can check their ordering by marking the numbers on the number line. More numbers such as thirds, fifths and tenths can be marked on the number line for reference as required.


Students at this stage are extending their knowledge of decimal numbers. They are learning that the base ten nature of the number system extends to the decimal numbers. A foundational understanding is that the value of each place is 10 times larger than the one to its right. When you multiply by 10 each digit effectively moves one place to the left, as this makes the value of each digit ten times larger than it was previously. Just as there are 10 tens in a 100 and 10 hundreds in a 1000, there are 10 hundredths in a tenth and 10 tenths in 1. Students at this stage will learn what happens to whole and decimals numbers when they are multiplied or divided by 10 and they will be able to name the number of tenths, hundredths and thousandths in decimal numbers.


Multiplying whole numbers by a power of 10

Ask the students: What is 32 x 10?Encourage the students to think of moving the digits a place value column to the left by asking: What is 1 ten times ten? (1 hundred)32 is made of 3 tens, what is 3 tens times ten? (3 hundreds)What is 2 ones times by ten? (2 tens)Now combine the 3 hundreds and the 2 tens, what is the answer? (320)Continue with multiplying by 100 and then 1000.

Comparing decimals with equipment

The Pipe Decimals equipment animation, link:

http://www2.nzmaths.co.nz/frames/Animations/pipe_decimals.html


http://www.nzmaths.co.nz/sites/default/files/Animations/pipe_decimals.swf

shows how the size of two decimals can be compared using equipment.

Dividing whole numbers by a power of 10

Ask the students to divide £560 by ten. There are 50 tens in 500 and 6 tens in 60. If necessary show this using the play money. Ensure students understand that dividing by 10 is the same as asking how many tens are in £560. Continue with dividing by a 100 and then a 1000.


Multiplying decimal numbers by a power of 10

Show the students £6.20 using play money. Remind the students that £0.20 is 20 pence. Ask the students: What is 10 x £6.20? What is 10 x £6? What is 10 x 20p? (If necessary count this out with coins to check) Show the question on a place value table. Ensure that the students can relate this to multiplying whole numbers by ten and understand why multiplying by ten results in a shift to the left.

100s 10s 1s . 0.1s 0.01s 0.001s10 x 10 10 x 1 10 x 0.1 . 10 x 0.01 10 x 0.001 10 x 0.0001

6 . 2

6 2 .

Ask problems that include numbers with zeros as place holders, for example 4.05 x 10 = 40.5. Continue with multiplying decimal numbers by 100 and 1000. For example, 3.25 x 100 = 325.

100s 10s 1s . 0.1s 0.01s 0.001s

100 x 1 100 x 0.1 100 x 0.01 . 100 x 0.001 100 x 0.0001

3 . 2 5

3 2 5 .

http://www.nzmaths.co.nz/sites/default/files/Animations/pipe_decimals.swf


Multiplying decimal numbers using place value

Use place value equipment to show multiplication with decimals. Explain to the students that in this example the small cubes will represent tenths, the rods will represent ones and the flat blocks represent tens. Remind students that 10 tenths make 1 one and show this with the place value equipment. Ask the students: what is 4 tenths times by 10? Write this as 0.4 x 10Count out 4 tenths and make 10 sets of this. Then rearrange the 40 tenths to show they are the same as 4 rods or 4 ones.

The place value equipment can also be used to demonstrate problems using more than one unit, for example 1.4 x 3.

Dividing decimal numbers by a power of 10

Ask the students: What is $35.50 divided by 10?

o What is 30 divided by 10?o What is 5 divided by 10?o What is 50c divided by 10?

Show the question on a place value table and ask the students to notice the pattern of moving digits a column to the right 35 ÷ 10 = 3.55. Ensure students understand that dividing by ten will make a number ten times smaller. As the value of each place within a number is ten times smaller with each move to the right dividing a number by ten effectively moves all digits one column to the right. Ask problems that include numbers with zeros as place holders, for example, 10.75 ÷ 10 = 1.075. Continue with dividing decimal numbers by 100 and 1000. For example, 70.3 ÷ 100 = 0.703

Multiplying and dividing with decimals using calculators

Students need to understand that multiplying by ten will make a number ten times bigger and will result in the digits moving to the left. Dividing by ten will make a number ten times smaller and will result in the digits moving to the right. Using a calculator students can look for patterns when multiplying and dividing numbers by ten. Ask the students to use a calculator to solve 3.2 x 10. Ask them to explain the answer they get in terms of where the digits move to.


Relationship between tenths and hundredths

The Decimats 1 equipment animation, link:

http://nzmaths.co.nz/sites/default/files/Animations/decimats1.swf

shows how to use decimats to illustrate the relationship between tenths and hundredths.

Number of tenths, hundredths, thousandths in whole and decimal numbers

Place value tables can support students learning to name the number of tenths, hundredths, thousandths in whole and decimal numbers. Ask the students to write the number 4.752 on a place value table.

100s 10s 1s . 0.1s 0.01s 0.001s4 . 7 5 2

.

Ask the students how many hundredths are in 4.752Start at the hundredths column:How many hundredths are in the hundredths column? (5)How many hundredths are in 7 tenths? (7 x 10 = 70) How many hundredths are in 4 ones? (4 x 100 = 400)How many hundredths are in 4.752? (475)

Ask problems that include numbers with zeros as place holders. For example, how many tenths are in 60.32?Start at the tenths column:How many tenths are in the tenths column? (3)How many tenths are in 0 ones? (0 x 10 = 0)How many tenths are in 6 tens? (6 x 100 = 600)How many tenths are in 60.32? (603)

http://nzmaths.co.nz/sites/default/files/Animations/decimats1.swf

http://www2.nzmaths.co.nz/frames/Animations/decimats1.html


Using arrow cards to find the number of tenths, hundredths, thousandths

The decimal arrow cards from Material Master 7-2, link:


can be used to help students work out how many tenths, hundredths or thousandths are in decimal number. Use the arrow cards to show 2.436 with the 2, 0.4, 0.03 and 0.006 cards overlaying each other.Show the students how the number 2.436 can be expanded out into the separate arrow cards. Ask the students how many tenths are in 2.436?Start at the thousandths column:How many tenths are in the thousandths column? (0)How many tenths are in 3 hundredths? (0)How many tenths are in 4 tenths? (4)How many tenths are in 2 ones? (2 x 10 = 20) How many tenths are in 2.436? (24)


Web viewEarly level*/** Activities to develop knowledge: forwards and backwards number word...

Documents

Transcript of Web viewEarly level*/** Activities to develop knowledge: forwards and backwards number word...