A Developmental Continuum for Teaching Mathematical Understanding.


Developing Understanding in Addition.

It is very important that students make the connections between differently labelled concepts in maths, as with anything, and understand what they are doing and why, as well as developing a variety of efficient strategies that can be used to achieve the same goal. They should also be able to use multiple representations of the one solution (Eg., Model, Pictures (including Tables and Graphs), Number Sentences and Words (Worded Problems and Explanations)).

Regarding the Learning Intent, make it open enough to cover the range of strategies under development. We would have different success criteria for the range of students in our class however, which needs to be based on the collated data to date on where individual students are at regarding development of understanding of the different strategies.

While we are on the matter of teaching a range of strategies, it is important to note that the strategies are in a developmental sequence and students should not be exposed to the next strategy in the sequence until they are developmentally ready. This means you will have a range of students working on a range of strategies all in the one lesson.  Remember it is important that students always understand what they are doing and why, it needs to make sense to them. They will therefore always have a strategy to find a solution. Our role as a teacher is gradually move them to more efficient strategies along the continuum of strategies. This means while developing understanding of addition strategies you may have students in your class working on the following:

Counting all with models

Counting on by ones, (Number line)

Counting on by ones, (Hundreds Chart)

Counting on by tens and ones, (Hundreds Chart)

Addition using the Jump Strategy (Hundreds Chart supported) (Partial Sums),

Addition using the VVV Method (Partial Sums),

Column addition working Left to Right (Partial Sums).

Traditional Column Algorithm working Right to Left.

Developmental Continuum for Understanding Addition Student Record Tool.

Interspersed with these addition strategies are other concepts such as doubles, near doubles, friends of ten, decomposing numbers or bridging tens (7+5 = 7+3+2 = 12), etc. Students would also be working on, (in order), larger (1 to 2 to 3 to …. digit numbers) as well as working down to smaller numbers (money in cents (Tenths and Hundredths), volume and mass (to Thousandths), time (seconds to Thousandths), then other appropriate contexts.

You may also have students working out the difference (eg, change in money) after calculating the addition problem, while for other students finding the sum is enough. All students should develop a working understanding of Fact Families for Addition and Subtraction at an early stage, which includes the turn-around facts.

The whole time students are recording addition number sentences and are able to use as well as their developmentally appropriate current strategy, all strategies that came beforehand. Therefore all students have a range of strategies to solve addition problems and are able to use the strategy that best fits the problem under investigation.

Therefore as a teacher we need to provide opportunity for students to develop an understanding of the strategy next developmentally appropriate strategy in a lesson that is engaging and caters for the wide range of strategies under development. We probably would not have the whole range of strategies as listed above in the one class, but a narrower band that still contains at least 3 or 4 levels within the one classroom.

It is obviously very important to know your students as individual mathematicians if you are to know when it is appropriate to formally expose them to the next strategy. The students would have been informally exposed during the Share/Reflection to new strategies as students share their strategies in working out their individual solutions to the class task. Thorough record keeping is very important for tracking of student development and ensuring all students are progressing appropriately.

If students are working on the strategy developmentally appropriate to them as an individual mathematician, they will always be working in their Zone of Proximal Development, and therefore have a feeling of in control of their mathematics, remembering they will always have the previous strategies to support their working. This means students will never not understand and never not have a strategy they understand for the mathematics that they are working on. 

Regarding the activity for individual lessons, they should always be set in an authentic context (think where might the students use these strategies outside school), and allow students to select a range of numbers that are appropriate for their level of understanding and appropriate to the context under consideration. A good rule of thumb is never give students a number to work with unless you are giving that number for a particular reason. Allow students to Self Differentiate through choosing numbers that are developmentally appropriate to them as an individual learner. 

This all leads to a situation where students are able to support other students in the development of strategies, including developing class anchor charts, but extending to making instructional videos, written explanations, mathematical dictionaries, student developed Picture Story Books based on an existing mentor texts (Eg., The Doorbell Rang), class Problem Of The Week where each strategy is used to find the same result (useful for NAPLAN), etc.


Teaching Strategy: Addition is Combining / (Counting Forwards).

Model adding two one digit numbers on a number line and/or Hundreds Chart.

E.g.,  I had 5 donuts and bought 3 more.

Circle 5 and move forward 3 jumps. Lands on 8. Circle 8.

 

                                                                                          +3
Write using Open Number Line:                             5               8

 

Model how to write as a Number Sentence:                      5 + 3 = 8

 

Discuss what other Number Sentence you can make using the same three numbers

Turn Around facts:                  3 + 5 = 8

And the Fact Family members:       8 – 3 = 5           and                8 – 5 = 3


Students then use their own Number Lines or Hundreds Charts to find solutions to own problems. Ensure students give a context to the numbers. I had 3 fish in one fish bowl and 5 in another bowl. How many fish did I have?

Model with manipulatives and on the Number Line  / Hundreds Chart.

Circle the appropriate numbers and show the movement in jumps (adding or subtracting)

Write the Number Sentence.

Find the Fact Family members (including Turn Around facts)

Write the problem solve in a written story and /or picture representing the problem.

 

In Summary:

Students begin counting by 1's using individual Number Lines (1-10 then 0-20) then move onto individual Hundreds Charts.

Progress students to count by 10's and 1's on their Hundreds Charts.

Next is 100's, 10's and 1's using their Hundreds Charts for support

For each of these encourage students writing the Open Number Line summary, Number Sentence, Fact Families and a picture or sentence describing problem solved for each.

Students are then ready to progress through understanding the Partial Sums approach to addition. This emphasises use of Place Value and Mental Computational strategies.

 Begin with two digit numbers then move onto three digit.

Strategy: Pulling both numbers apart into Place Value components:

     123 + 678 =

            100 + 600 = 700

                20 + 70 = 90

                    3 + 8 = 11

                700 + 90 + 11 = 801

 

Strategy:  Pulling one number into Place Value components:

   123 + 567 =

                123 + 500 = 623

                    623 + 60 = 683

                        693 + 7 = 690


This counting on method is excellent for adding time. E.g., 7.45am + 2 hours and 35 minutes.

 

 

OR

 

                             123    +    678

    (  100 + 20 + 3        +      600 + 70 + 8 )

Combine the hundreds, then tens, then ones. 

                   700   +      90    +        11      = 801

 

These lead into a formal algorithm for addition. Note working left to right as useful for approximation of solution.

   123

+678

   700

  + 90

  + 11

   801


Before finally teaching the traditional Right to Left algorithm. Note that this is the last strategy to teach students as it does not develop place value understanding and is not a mental computational strategy. 

                 1   1

                123

+678

   801



Real contexts for helping students develop understanding of addition:

  • Using Counters to represent items.
  • Use Catalogues (with dollars only) to buy two (or more) items and find total.
  •  Use Menus to purchase two (or more) items and find total.
  • Make the money amounts using Place Value denominations only ($100 notes, $10 notes, $1 coins)and add together using the Hundreds Chart.
  • Roll Die and keep adding to total. (good Warm-Up activity).
  • Making a Magic Potion. Give students supermarket catalogues and cut and add volumes of ingredients to make 4.5 litres of magic potion. E.g, one can Coke (375ml) plus one bottle shampoo (600ml) = 975 ml etc.
  • Catching Fish. Select one fish at a time and record weight and place into catch tank. Only able to keep three fish at any time. Continue fishing and keeping heaviest three fish, throwing back the lightest each time. After a designated time find total weight of three kept fish and find class fishing champion.  

 

Developing Understanding in Subtraction.

Teaching Strategy: Subtraction is Counting Backwards.

             Subtraction is the Difference between two numbers.

Subtraction

Use the relationship between addition and Subtraction. Addition is more easily solved mentally than subtraction using counting up strategies.

E.g., Change from $5.00 for a $3.75 coffee, mentally most people count up.

3.75 + .25 = 4.00              4.00 + 1.00 = 5.00           Change = 1.25

No renaming, no borrowing, no crossing out the 5 making it 4, changing the 0 to 10, etc.


Uses Part/Part/Whole thinking.

Part

Part

Whole

E.g., 100 - 37 = ?

100(Whole) – 37(Part) = ? (Part)      

Rearrange as      37(Part) + ? (Part)= 100 (Whole)

Using the 100 chart;

37 +3 = 40

40 +60  = 100

Therefore:    37 + 63 = 100 and 100 - 37 = 63

Encourage students once they are competent with this strategy on the 100 chart to practice working out mentally first. 

 

 

Solve 813- 586 = ?

Rearrange to 586 + ? = 813

586 + 4      >   590 + 10        >  600 + 200       >    800 + 13        >   813

             4                      +10                       +200                        +13   = 227.

This is a Mental Computational strategy, I.e., a strategy that we are able to perform in our heads, as well as developing a working understanding of Place Value. 

 

This Counting Up strategy is excellent for Elapsed Time. (whereas traditional algorithms do not work for time bridging different units. 

E.g., 3.12pm to 5.05 am

3.12pm. to 3.12am. is 12 hours.

3.12am to 4.00am. is 48 minutes.

4.00am to 5.05am is 1 hour and 5 minutes.

Total elapsed time 12+1 hours and 48+5 minutes = 13 hours & 53 minutes.

 

Counting Back:

Model finding solution to simple problem on hundreds chart such as “I had 10 donuts and ate 4. How many are left?” Circle 10 and count back 4 jumps. Lands on 6. Circle 6.

 

                                                                                                    -4

Write using Open Number Line:                             6        moving backwards       10

                                                                                             (Represent with an arrow)


Model how to write mathematically:                      10 – 4 = 6 

 

Discuss Fact families:                                 10 - 6 = 4,        6 + 4 = 10     and         4 + 6 = 10

 

Counting On (Finding the Difference):

Model finding solution to simple problem such as “I had 10 donuts and ate 4. How many are left?” Circle 10 and Circle 6 on a number line or 100 chart. How many jumps from 6 to 10?


                                                                                           +4

Write using Open Number Line:                             6              10

 

Model how to write mathematically:                      6 + 4 = 10

 

Discuss turn around fact:                                         10 – 4 = 6

 

 

Students use Hundreds Chart to find solutions to own problems.

  •  Using Counters to represent donuts (I had donuts but Dad ate some. How many were left?)
  •  Use Catalogues (with dollars only) to buy two (or more) items and find total. How much change from $50?
  • Use Menus to purchase two (or more) items and find total. How much change from $50.
  • Roll Die and keep subtracting from total. (Good Warm-Up activity)
  •  I had 200 minutes to watch a movie at the cinema. Which movie will I watch and how much time will I have spare? (Use newspapers or internet to find movie lengths).
  • I had 1 metre (100cm.) of liquorish. I ate the same amount each day. How long was it after each day? How long did it last? (Repeated subtraction).

 Developmental Continuum for Understanding Subtraction Student Record Tool.

 


Developing Understanding in Place Value.

Teaching Strategy: To see the ten-ness in numbers.

Use M.A.B. 100’s, 10’s and 1’s to make a picture. Draw and record the value of the picture. Have children swap and work out each others values. It is very important the students can visualise the numbers they are working with, which is very well supported by making the numbers using MAB. 

  • I only have (Hundred and ) Ten Dollar notes and Dollar coins in my purse/wallet. How could I pay for a book costing $32? Students select item from Catalogue (no cents), cut out and draw how they would pay for item. More able student show many ways of paying for item. (3 tens and 2 ones, 2 tens and 12 ones, 1 ten and 22 ones, 32 ones)
  • Continue above activity with students purchasing from catalogue or menu. Show how they would pay for each item using place value denominations. (Extension. How much did they spend?)
  • Students can self differentiate by choosing numbers they feel confident dealing with. Some may be adding two 1 digit prices, while others adding more than two 3 digit prices with cents as well, all within the one lesson, one activity. Just provide  with a variety of catalogues. I like to keep on a theme, all camping catalogues, all menus, all supermarket shopping, etc
  •  Play roll to 200. Each roll can be that number of ones or that number of tens. After ten rolls the player closest to 200 without busting (over 200) is the winner.

                                                                                0  + 3030 +  4 = 34 etc.

  • Assessment tip : Ask students if they were purchasing a toy and their parent offered to pay for one digit, which digit should they get their parent to pay for if the item cost $13.75. Students record and explain.



Developing Understanding in Multiplication.   

Teaching Strategy: Multiplication is Repeated Addition and Arrays.

                      Multiplication is best achieved through breaking numbers into their Place Value components and multiplying each component separately, then combining the products.

Begin with repeated addition. 2+2+2+2+2=10

Represent as Groups Of        5 x (groups of) 2 = 10

Represent as an Array            

     
     

            

Ensure students can make the Fact Family number sentences for the given multiplication. I.e., Given 5 x 2=10, therefore 2 x 5 = 10, 10 :/ 5 = 2, 10 :/ 2 = 5. 

Represent multiplication with MAB as an array   1 digit x 1 digit.

Represent multiplication with MAB  as an array 1 digit x 2 digit .

Represent multiplication on grid paper as an array 1 digit x 2 digit .


This then leads to the visual representation of the Partial Products Area model below. without MAB.


Partial Products using Area Model.

123 x 5

x

100

20

3

5

500

100

15

 

500 + 100 + 15 = 615

 

123 x 567

x

100

20

3

500

50000

10000

1500

60

6000

1200

180

7

700

140

21

 

Find sum of individual products. Encourage mental addition.


More simple products can be found mentally.

123 X 6  

100 x 6 = 600        20 x 6 = 120     3 x 6 = 18          600 + 120 + 18 = 738

 

This is the strategy we also encourage for algebraic multiplication.

(x+2)(x-3)

 

x

+2

x

X^2

+2x

-3

-3x

-6

 

X^2    +2x      -3x      -6 

=      x^2 - x - 6



This strategy will also work when multiplying different units.

4 lots of 3 hours, 27 minutes and 8 seconds

x

3 Hours

20 Minutes

7 Minutes

8 Seconds

4

12 Hours

80 Minutes

28 Minutes

32 Seconds

 

Convert and add. 12 Hours + 1 Hour and 20 Min. + 28 Min. + 32 Sec.

13 Hours, 48 Minutes and 32 Seconds. 



Multiplication using Skip Counting Patterns.

Run through the Counting Patterns with the students completing and look for patterns (in the ones place).

1

2

3

4

5

6

7

8

9

2

4

6

8

0

2

4

6

8

3

6

9

2

 

8

1

4

7

4

8

2

6

 

4

8

2

6

5

0

5

0

 

0

5

0

5

6

 

8

 

 

 

2

 

4

7

 

1

 

 

 

9

 

3

8

 

4

 

 

 

6

 

2

9

 

7

 

 

 

3

 

1

0

 

0

 

 

 

0

 

0

 

Note we know the 1’s, 5’s and 9’s patterns already.

Note 8’s is the reverse of 2’s, 7’s the reverse of 3’s and 6’s reverse of 4’s.

So if we write down the 2’s, 3’s and 4’s we have the 8’s, 7’s, and 6’s. So bold above are the only ones we need to remember.

This compliments the Area Partial Products method as using this strategy we only ever need 1 digit by 1 digit multiplication.

 


  • Practice Skip Counting patterns 2’s to 10’s. (See Skip Counting Wizard)
  • Play “Around the World”. Counting by one of the skip counting patterns one student saying each sequential number. Students who say a number ending in 0 (multiple of ten) receive a token. Students saying a hundred multiple receive 3 tokens. Player with the most  at end of allocated time wins.
  • Investigate the patterns in the ones place for each counting pattern.  I.e., Skip Counting by 4’s,  4,8,2,6,0,4,8,2,6,0,....
  • Use the skip counting patterns to solve open problems. “There were dogs running in the park. How many legs might there have been?”
  • Investigate the relationship between repeated addition and multiplication through activities. E.g., “My frog jumped the same distance every jump along a meter ruler. What numbers did he land on?”
  • Create a “My Zoo book.” Students write addition and multiplication number sentences using dice to tell number of enclosures and second roll the number of animals in each enclosure. Write the addition and multiplication number sentence for each. How many animals in your zoo?
  • Investigate Arrays, repeated addition and multiplication as they occur in students lives. Have students work both ways. Making the array from products and finding the products for an array. Make an array that shows 3x4. Make an array for 12.
  • The windows of the building in the city were in an array. What did the building look like. Write the number sentences and fact families for the array. 
  • Model Partial Products using Manipulatives. Have students make the money amount for one item. E.g., $12 is one $10 note and two $1 coins. If I want to buy 5 of these items I need 5 lots of this money, I.e., five lots of $10 and five lots of two $1. How much money is that?
  • Model Partial Products using visuals. How could I pay for $12? Draw one $10 and two $1. I want to buy 4 of these $12 items. Draw four lots of the one $10 and four lots of the two $1. How much money is that altogether?


Developing Understanding in Division.   

Teaching Strategy: Division is Sharing. Use multiplication (or repeated addition) knowledge to solve sharing problems.

How do I share these lollies with my friends. Investigate sharing 25 lollies between 2,3,4,5,... friends. What patterns are found? (As the number of friends increases the amount of lollies each gets is decreases)

Similar to how we use addition strategies to find the solution to subtraction problems, we here use multiplication to find the solution to division problems.

This also utilises Part/Part/Whole

Part

Part

Whole

 

124(Whole) ÷ 4 (Part) = ? (Part)

Rearrange to          4 (Part) x ?(Part) = 124(Whole)

It is much easier to think of multiplication than division

Mental strategies for division involve pulling apart the number into place value components and sharing of each part.

Use the understanding of Fact families to rearrange. 20 :/5 = ?  Rearrange to 5 x ? = 20, then use known facts. 5 x 4 = 20, so 20 :/5 = 4


Strategy 1.

126 shared between 3 is

                    100 shared between 3                    30            30            30      with 10 left over

                    Share that 10 between 3                 3               3                3    with 1 left over

                    We now have to share the remaining 1 and 26 (Ie. 27)

                     27 shared between 3                        9            9            9      

        Thats 42 in each group.                                42        42        42


Strategy 2.

794 shared between 3 is

            3 x 200 = 600        194 left over

            3 x 50   = 150        44 left over

            3 x 10   = 30            14 left over

            3 x 4     = 12               2 left over

        So 794 divided by 3 is 264 remainder 2,  or  264 and 2/3



Investigate sharing of

  • Different amounts of items using counters.
  • Money, including splitting the bill and sharing the cost of items. (Provide students with real menus)
  • Sharing time between tasks. If I want to play 1,2,3,4,5,... games in one (two, three,...) hour(s), how long can I spend on each game?”
  • Length. “I have 1 metre of liquorish. How much do we each get if I share this between 2,3,4,5,... friends?”
  • Share a number made with MAB between 2,3,4,5,... 
  • Share a number made with $100 notes, $10 notes and $1 coins between 2,3,4,5,... people.

 

Developing Understanding in Fractions.


 

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