1. Students play in pairs, each with their own grid.
2. Each player rolls the dice and colours in an area on the grid indicated by the dice.  For example if they roll a 2 and a three they colour in any 2x3 rectangle.
3. The students should write the number of squares in the rectangle to indicate the product of the two sides.
4. The first player to colour in all the squares in their grid wins.
5. As the grids fill up players will roll totals that will not fit on the grid, you can allow them to break up the factors if you choose.  For example a student might identify that 6x4 is the same as 2x4 and 4x4.  This implicitly reinforces the distributive law of multiplication.

The size of the grid will determine the length of the game.  Players could draw the grid in their maths book, or use pre-drawn and photocopied grids provided by the teacher.

Getting Started

In this session students investigate the possible rectangular arrays for given whole numbers. They record the appropriate factors for each array.

1. Give each student 12 cubes and ask them to form a rectangle using all 12 cubes. 
   
   What size is the rectangle you have made? (Discuss the description of rectangles using rows and columns.)
   Have we found all the rectangles? How do you know?  (Expect the students to check each of the numbers to 12 although some may realise that you only need to check until the factors start repeating, e.g. 3 x 4 and 4 x 3. Highlight that the factor pairs repeat after 3 (Square root of 12 = 3.46).
2. As a class record each of the rectangles using squared paper.
3. Attach these rectangles to an A3 page headed with a 12.  Record the expression with each rectangle. Organise the rectangles from 1x12 to 12x1.  (This will allow for more systematic comparison with the factors of other numbers.)
   Are each of the rectangles unique? (‘Unique’ means unlike any others such as one array cannot be rotated or reflected to make another)
   If we remove any rectangles that are copies, how many unique rectangles are there? (1 x 12, 2 x 6, 3 x 4).
   What name do we give to numbers that multiply like 3 and 4, to give a number, like 12? (Factors of 12)
   You might record the findings using formal notation:
   Factors of 12 = {1, 2, 3, 4, 6, 12}
   It is also interesting to consider why whole number are not factors of 12. For example, 5 is not a factor because all factors of 5 end in either 0 or 5 (in the ones place).
4. Give each pair of students a number in the range 10 to 24) and ask them to form as many unique rectangles as they can using that number of cubes.  As they form the rectangles, first with cubes and then on squared paper, ask questions to draw their attention to the factors of the number in a systematic way, such as arranging the first factor from lowest to highest.
   How many rectangles have you found for your number?
   How do you know you have found them all?
   Why do some numbers have more rectangles than others?
   Are there any numbers that form only one rectangle?
   Is it possible to predict these ‘one rectangle’ numbers?
5. Ask each pair of students to glue their paper rectangles to the ‘page’ for their number. Alongside each rectangle, a corresponding multiplication should be written.
   For example, 4 x 3 or 3 x 4 for this array:
   
6. Share the number pages as a class.
   Which number has the most rectangles? (24) Why? Students may think that 24 has the most factors because it is the largest number. You might investigate 25 that has only 3 factors.
   Which numbers have only one rectangle (two factors)? (The primes 11, 13, 17, 19, 23)
   Which numbers have only 2 rectangles (four factors)? (4, 6, 8, 9, 10, 14)
   Can a number have 3 rectangles (six factors)? (Yes, 12 does)
   Can a number have 4 rectangles (eight factors)? (Yes, 24 does)
   Note that students may not recognise squares as a special class of rectangles. For example, they may not include 4 x 4 in their search for arrays with 16 cubes. It is important to discuss this point as it has implications for geometry as well.
7. Formally list sets of factors, e.g. Factors of 10 = {1, 2, 5, 10} and Factors of 11 = {1,11}.
   What happens when one of the rectangles is a square?
   These numbers are known as square numbers and they have an odd number of factors, e.g. Factors of 9 = {1, 3, 9}.
   Numbers that have more than two factors are called composite numbers. What numbers in the range 10-24 are composite? (10, 12, 14, 15, 16, 18, 20, 21, 22, 24)
   Numbers that have only two factors are called prime numbers. What numbers in the range 10-24 are prime? (11, 13, 17, 19, 23)
   Can a number be neither (not) composite or prime? (Usually this is an ‘either or’ choice except for the special case of one)
   How many rectangles can be made with one square? (only a 1 x 1 rectangle is possible, so 1 is a special case, neither composite or prime but it is a square number.)
8. Add these key points to a class chart, or ask your student pairs to create a one-page display that demonstrates and explains what factors are, giving examples of one square number, one composite number, one prime number, and one number that is not a composite or prime (1). They should use cube drawings and written equations to explain these key points. Roam and support students to understand each of these key concepts as necessary. 
    

Sessions Two, Three and Four

Over the next 2-3 days ask your students to create rectangle charts for each of the numbers from 1 to 100. Use the charts to develop their understanding of factors, multiples, and primes.

1. Look at the charts from the previous day.
   I could say that the factors of 12 are 1, 2, 3, 4, 6 and 12. What does the word factor mean?
   You might look up the internet to create a formal definition of the word “factor.”
   I could say that 12 is a multiple of 1, 2, 3, 4, 6, and 12. What does the word multiple mean? Create a formal definition of the word “multiple.”
2. Choose other numbers in the range 10-24 and invite your students to make statements that use the words factor/s and multiple/s. The characteristics of multiples of a given number are used to test for divisibility by that number. For example, multiples of 2 are {2, 4, 6, 8, 10, 12,…} so to test to see if a number has 2 as a factor we only need to check that it is even, so has 0, 2, 4, 6 or 8 in the ones place. The divisibility rule for 5 is also easy but for other factors like 3, 6, and 9 the rules involve digital sums.
3. Put the numbers from 24 to 100 in a hat or special container. One student in the pair picks a number from the hat and then the two students work together to construct all rectangular arrays for the number. They should record the rectangles on squared paper and then attach these rectangles to an A3 piece of paper.  As the students work ask questions that focus on their identification of the factors of a number.
   How many rectangles have you found for your number?
   How do you know you have found them all?
   What are the factors of your number?
   Is your number prime or composite? How do you know?
   Is your number a square number?
4. An interesting investigation is the relationship between square numbers and the square root function. For example, 64 is a square number since 8 x 8 is one possible array. The factors of 64 are {1, 2, 4, 8, 16, 32, 64}. An odd number of factors is one property of natural numbers that are squares. Show students that a calculator can be used to find that the square root of 64 is 8. 
   What does 8 as a factor have to do with an array of 64 cubes? (8 is the side length of a square with an area of 64 square units.)
   Using this knowledge, students could add the square root of their number (if it is a square number) to their chart. Ensure they use the correct notation and can explain the connection between the square root and their display of squares.
5. When the students have completed a number, they select another from the hat.
6. Roam ans support students to create their charts as needed. At the end of each session look at the developing display of rectangle charts.  Invite pairs of students to share their findings with the class.
7. You might use a hundred chart to colour code the composite and prime numbers. Provide students with their own chart and show a virtual hundreds board on a large screen. This may lead to a discussion about which kind of number is most common. Students may notice that primes become less frequent as the range is extended.
   
8. Note that primes tend to be located next to multiples of six. For example, both 5 and 7 are primes. However, being one more or less than a multiple of six does not guarantee that a number is prime. For example, 48 is a multiple of six. 47 is prime but 49 is a square composite.

Session Five

Today we look at our completed display of rectangle charts and create a newsletter for our families telling them about our findings.

1. Display the factor charts, in order, for the class to examine.
2. Encourage the students to look at the charts and write statements (in pairs) about their observations. The following questions may be used as prompts for the students. 
   Which number has the most factors?
   How many prime numbers are there less than 100?
   Is there a way to predict if a number, like 51 or 57, is prime?
   What number do you think is the most interesting? Why?
   Which decade has the most prime numbers? Why do you think it is the tens decade?
3. Share statements.
4. Investigate the method used by a mathematician from ancient Greece. His name was Eratosthenes, and he is most famous for estimating the circumference of the Earth. That was quite a feat for a person who lived from 276 - 194BC. Eratosthenes used a sieve method to leave behind only the primes. You can find videos online explaining the method. Your students might be interested.  
5. Use students’ statements to form the basis for the newsletter home. 
6. In addition to the class statements you may like to include the following challenging brainteaser. 
   
   Challenge: The Census Problem
   A census taker approaches a house and asks the person who answers the door.
   "How many children do you have, and what are their ages?"
   Person: I have three children.  The product of their ages is 36, the sum of their ages is equal to the address of the house next door."
   The census taker walks next door, comes back and says to the woman.
   "I need more information."
   Person: "I have to go.  My oldest child is sleeping upstairs."
   Census taker: "Thank you, I have everything I need."
   
   Question:  What are the ages of each of the three children?
   
   This is a very famous problem and the first reaction of people is often that insufficient information is given. However, it can be solved.
   First make a systematic is of the sets of three factors that have a product of 36, since the ages of the three children have a product of 36. Begin with one as a factor and increase the size of factors from there:
   1 x 1 x 36 (unlikely a child is 36 years old)
   1 x 2 x 18, 1 x 3 x 12, 1 x 4 x 9, 1 x 6 x 6
   2 x 2 x 9, 2 x 3 x 6,
   3 x 3 x 4
   
   Since the only combination possible with four is covered that is the end of the list. Next find the sums by adding the factors.
   1 + 1 + 36 = 38,
   1 + 2 + 18 = 21, 1 + 3 + 12 = 16, 1 + 4 + 9 = 14, 1 + 6 + 6 = 13,
   2 + 2 + 9 = 13, 2 + 3 + 6 = 11,
   3 + 3 + 4 = 10.
   
   The only way the census worker needs more information is the possibility that two or more sets of factors might have the same sum. In this case 1, 6, and 6 have the same sum and 2, 2, and 9.
   
   Knowing that there is an oldest child means that 2, 2, and 9 are the ages of the children.

Show two rows of three beads on the Slavonic abacus. Ask the students to show you three and three on their fingers. Many of them will know that this is six fingers altogether. Ask them what they will need to do in order to change their fingers to four and three (add one finger). Ask them what 4 + 3 must be. Record the equation on the board or modelling book. Ask the students to show you how they changed 3 + 3 into 4 + 3 on the abacus.

Ask the students to look at the beads and see if they can help work out the answer without counting each bead. Look for answers based on 10 + 4 = 14. Note that this requires knowledge of the teen numbers. Ask the students what they would do to change 7 + 7 into 8 + 7 (add one), and what the answer would be.

Pose similar problems. For example: 6 + 6 = 12 so 7 + 6 = ?, 9 + 9 = 18 so 8 + 9 = ?, 8 + 8 = 16, so 7 + 8 = ?.

Repeat these doubles with fingers, tens frames, and Fly Flip cards. For example, two students act like mirrors facing each other and show seven fingers. They match the full hands (10) and the two twos (four). Similarly two Fly Flip cards showing seven can be put down. Two fives will be visible and two twos will be behind their backs.

Activity – Another Way of Doubling

Another representation is to place two of the same tens frame cards side by side. Talk about the fives structure and point to the two rows of five. “How many dots is that? How many dots altogether?” For example, 6 + 6 = 12.

Activity

The students could play Double Somersaults Plus or Minus One to consolidate their doubles and related facts.

Each player rules up a column for “tens” and a column for “ones”. The aim of the game is to get a total as close to 100 as possible. The student tosses a dice and decides whether the number will be put in the ones or the tens place. For example, if a four is thrown, it could either be 40 or four. The dice is rolled a total of seven times. All seven numbers must be used. The total of all the columned numbers may exceed 100, but the students will need to decide which player has got closer to 100.

Extension Activity

Use larger numbers and decimals for the target numbers. Vary the number of throws and what the thrown number can represent, such as:

Closest to 1 000: 10 throws of hundreds, tens, or ones.

Closest to 10: 10 throws of ones, tenths, or hundredths.

Closest to 1: 10 throws of tenths, hundredths, or thousandths
 

Using Materials

Problem: “Murray has $1,858 in the bank. His grandfather put some more money in Murray’s account. Now Murray has $5,683. How much money did Murray’s grandfather put in Murray’s account?”

Write 1 858 + ? = 5 683 on the board. Get the students to make two strips of paper of equal length and label them as shown:

Discuss why 1 858 off the second strip shows the answer is 5 683 – 1 858 and work out the answer with a calculator.

Problem: “Geraldine collects stamps. Her parents give her a packet of 355 stamps for her birthday. Altogether she now has 6 040 stamps. How many stamps did Geraldine have before her birthday?”

Write ? + 355 = 6 040 on the board. Repeat the method with strips shown above.

Examples: Word stories and recording for: 3 333 + ? = 4 141

? + 5 601 = 45 893
? + 7 928 = 30 281
$234.56 + ? = $789.40 ...

Using Imaging

Problem: “Miles has $345, and he wants to buy a mountain bike costing $601. How much more money does he have to  save?”

Write 345 + ?= 601 on the board. Ask the students to imagine the strips and solve the problem. Drop back to drawing the strips on the board if needed.

Examples: Word stories and recording for: 4 567 + ? = 6 012

? + 9 567 = 12 211
? + 6 443 = 14 601
$291.36 + ? = $1,089.43 ...

Using Number Properties

Examples: Use a calculator to work out these answers: 44 675.83 +  ? = 76 645.93

? + 345.67 = 5 678.09
444 1/3 +  ? = 800 1/3
484 609 + ? = 1 678 980
? + 34.78902 = 56.00912 ...

Understanding Number Properties:

a + ? = b where a and b stand for numbers. How would you work out the number that goes in the box? (Answer: Work out the answer to b – a.)

Create two arrays using animal cards to represent 6 _ 8 and 3 _ 8. Ask the students how many animals are in each array. Pose the problem of adding these sets of animals together, that is 48 + 24. Allow the students to present their strategies.

Some students may note that (6 x 8) + (3 x 8) = 9 x 8 = 72 (combining sets of eight).

Provide two similar examples, getting the students to make the arrays then solve theresulting addition problem. Suitable problems include:

(5 x 7) + (4 x 7) = 9 x 7                     (3x 6) + (6 x 6) = 9 x 6

(4 x 9) + (6 x 9) = 10 x 9                   (7 x 8) + (2 x 8) = 9 x 8

Ask the students to state what these problems have in common (both addends aremultiples of a common number).

Using Imaging

Pose problems for the students to image, beginning with the addends instead of the factors. If necessary, fold back to the materials. Possible examples include:

16 + 24 + 32 as (2 x 8) + (3 x 8) + (4 x 8) = 9 x 8

49 + 28 as (7 x 7) + (4 x 7) = 11 x 7

45 + 27 + 18 as (5 x 9) + (3 x 9) + (2 x 9) = 10 x 9

Extend the imaging to include subtraction problems:

81 – 27 as (9 x 9) – (3 x 9) = 6 x 9       64 – 48 as (8 x 8) – (6x 8) = 2 _ 8

63 – 35 as (9 x 7) – (5 x 7) = 4 x 7       90 – 36 as (10 x 9) – (4 x 9) = 6 _ 9

54 – 36 as (9 x 6) – (6 x 6) = 3 x 6 or (6 x 9) – (4 x 9) = 2 x 9

Using Number Properties

Pose addition and subtraction problems where it is helpful to identify a common factor.

48 + 56 + 24 + 32 = 20 x 8      42 + 35 + 49 + 14 = 20 x 7

72 – 27 + 45 – 36 = 6 x 9        88 – 56 – 16 + 32 = 6 x 8

120 – 54 – 48 – 18 = 0 x 6      77 – 28 + 14 – 35 = 4 x 7

At this stage, offering students a regular daily dose of mental calculation is strongly recommended. It is a good idea to record only one problem on the board or in the modelling book at a time and not allow the students to use pencil and paper. Make sure they all have time to solve each problem. Don’t allow early finishers to call out the answer.

For example, take the problem 73 – 29 = ?.
Prompt the students to look carefully at the numbers before deciding how they might solve this. The following are possible strategies:

• Equal adjustments: solved by adding 1 to both numbers, so 74 – 30 = 44.
• Rounding and compensating: 73 – 29 becomes 73 – 30 = 43, then 43 + 1 = 44.
• Reversibility: adding up from 29 to 73, so 1+ 40 + 3 = 44.
• Place value: partitioning the 29, so 73 – 20 = 53 →53 – 3 = 50 →50 – 6 = 44.

When recording the strategies the students selected to use, ask: “What is it about the numbers that made you choose the strategy you used?” and “Which of these strategies is the most efficient? Why?” If the students have used place-value strategies to solve the problem, they may need to revise equal adjustments and rounding and compensating.

As well as presenting problems for the students to solve as equations, it is also important to present them as word problems – for example: “The children had to blow up 182 balloons to decorate the school hall. By playtime, they had blown up 26. How many more did they still need to blow up?”

Some Problem Sets

Record the problems on the board or modelling book in the horizontal form.

Set 1
45 + 58     
67 + □ = 121    
8 001 – 7 998    
26 + □ = 52    
81 – 67     
456 + 144   
789 – 85         
□ + 58 = 189     
33 + 809 + 67 + 91
 

Set 2
28 + 72       
191 + □ = 210       
7 001 – 21      
39 + □ = 77        
234 – 99         
6091 + 109  
2 782 – 15            
□ + 123 = 149       
616 + 407 – 16 + 93
 

Set 3
999 + 702        
287 + □ = 400        
2 067 – 999       
45 + □ = 91     
771 – 37 316 + 684        
709 – 70                
□ + 88 = 200        
7 898 – 6 000 – 98 – 100
 

Set 4
38 + 128          
14 + □ = 101          
9 000 – 8 985         
102 – □ = 34   
800 – 33 78 + 124       
4 444 – 145                
□ + 8 = 1 003          
4 700 – 498 + 200 –2
 

Set 5
405 + 58         
880 + □ = 921        
8789 – 7 678        
80 – □ = 41    
701 – 96  8 888 + 122     
781 – 45                 
□ + 48 = 789         
6 000 – 979 – 11 – 10

Preliminary Knowledge

The students need to know 10 hundreds make 1 thousand and vice versa, and 10

thousands make 1 ten thousand and vice versa.

Using Materials

Problem: The Bank of Mathematics has run out of $1000 notes. Alison wants to

withdraw $2315 in $1, $10 and $100 dollar. How many $100 notes does she get?

Examples. Repeat for: $2601, $3190, $1555, $1209, $2001, $1222, $2081….

Using Imaging

Problem: Tickets to a concert cost $100 each. How many tickets could you buy if you  have $3215?

Write $3215 on the board. Shield 3 one thousands, 2 one hundreds, 1 ten and 5 ones.

Ask the students what you can see. Discuss how many hundred dollar notes you could  get by exchanging the thousands. Discuss which notes are irrelevant (the ten and the ones).

Shielding and Imaging only: Examples. Find the number of hundreds in:

$1608, $2897, $2782, $3519, $3091, $4000….

Using the Number Properties

Examples. Find the number of hundreds in: 3459, 8012, 9090, 6088, 3280, 5823,

7721, 2083….

Challenging examples. Find the number of hundreds in: 13 409, 28 002, 78 370, 12

088, 45 290, 82 356, 21 344….

Find the number of tens in: 3709, 8002, 8579, 5208, 4829, 82 333, 12 897, 30

897, 89 000, 50 890