Addition and Subtraction Pick n Mix

Purpose

In this unit we look at a range of strategies for solving addition and subtraction problems with whole numbers with a view to students anticipating from the structure of a problem which strategies might be best suited.

Specific Learning Outcomes
  • mentally solve whole number addition and subtraction problems using:
  • compensation from tidy numbers including equal additions
  • place value
  • reversibility
  • use appropriate recording techniques
  • predict the usefulness of strategies for given problems
  • evaluate the effectiveness of their selected strategies
  • generalise the types of problems that are connected with particular strategies
Description of Mathematics

The strategy section of the New Zealand Number Framework consists of a sequence of global stages that students use to solve mental number problems. On this framework students working at different strategy stages use characteristic ways to solve problems. This unit of work is useful for students working at stage 6 of the Number Framework, Advanced Additive. Students at this stage select from a broad range of strategies to solve addition problems, subdividing and recombining numbers to simplify problems. The Number Framework also includes a knowledge component which details the knowledge students will need to develop in order to progress through the strategy stages of the framework. This unit draws on students' knowledge of compatible numbers to 10 and 100, and place value relationships to 1000.

The key teaching points are:

  • Some problems can be easier to solve in certain ways. Teachers should elicit strategy discussion around problems in order to get students to justify their decisions about strategy selection in terms of the usefulness of the strategy for the problem situation.
  • Tidy number strategies are useful when number(s) in an equation are close to an easier number to work from.
  • When applying tidy numbers to addition, the total or sum must remain unchanged.
  • When applying tidy numbers to subtraction, the difference between numbers must remain unchanged.
  • Place value strategies are useful when no renaming is needed.
  • Reversibility strategies are useful for subtraction problems where place value and numbers will be ineffective.
Required Resource Materials
Place value equipment
Activity

Getting Started

The purpose of this session is to explore the range of strategies that students have to solve addition and subtraction problems. This will enable you to elicit the strategies that students currently use to solve addition and subtraction problems, evaluate which strategies need to be focused on in greater depth and identify students in your group as "expert" in particular strategies. There are two problems given as examples for exploration. You may want to use further examples of your own.

Problem 1: Sarah has $288 in the bank. She then deposits her pay cheque for $127 from her part time job at PetCare. How much does she have now?

Ask the students to work out the answer in their heads. Give the students 2-3 minutes thinking time. Then ask them to share their solutions and how they solved it with their learning partner. The following are possible responses:

Place value:
288 + 127 is just like 288 + 100 +20 +7. So that’s 388… 408… 415.

Tidy numbers:
If I tidy 288 to 300 it would be easier. To do that I need to add 12 to 288, which means I have to take 12 off the 127. So that’s 300 plus 115. Easy!

As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the strategies that are elicited at this stage, recording them to reflect upon later in the unit (perhaps in a modelling book).

Problem 2 Sarah has $466 in her bank account and spends $178 on a new phone, how much money does she have left in her bank account?

Ask students to solve the problem mentally, giving them 2-3 minutes thinking time. Then ask them to share their solutions and how they solved it with their learning partner. Possible responses are:

Reversibility:
$466 - $178 is the same as saying how much do you need to add to $178 to get $466. $178 plus $22 makes $200, plus $200 more makes $400 plus $66 makes $466. If you add up $22 plus $200 plus $66 you get $288.
number line diagram.

Tidy numbers using equal additions:
You round the $178 to $200 by adding $22. $466 - $200 is $266. Then you put on $22 to keep the difference the same, so it’s $288.

There is a strong possibility that some students may misapply the addition tidy numbers strategy to subtraction problems. When using tidy numbers in subtraction the difference between the numbers in the equation must remain the same, so if you add an amount to one number, you must also add it to the other number. This is the opposite to addition, where the sum of the two numbers must remain unchanged when using a tidy numbers strategy, meaning that if you add an amount to one number you must subtract it from the other number. Some students may solve subtraction problems like 63 – 28 by tidying 28 to 30 (adding two), and 63 to 61 (subtracting two). The student has failed to keep the difference unchanged.

In this situation the students’ beliefs may need to be challenged by posing a conflict situation, using a simple example. For example, misapplying the tidy numbers strategy to 20 - 8 will leave the students with the problem 18 - 10. The situation can be modelled using Cuisenaire rods or a ruler.

As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the strategies that are elicited at this stage, recording them to reflect upon later in the unit (perhaps in a modelling book).

Ask students to reflect on the strategies that have been discussed in the session and evaluate which strategies that they personally need further work on, perhaps using thumb signals - thumbs up - confident and competent with the strategy, thumbs sideways - semi confident, thumbs down - not yet confident.

Use this information to plan for your subsequent teaching from the exploring section outlined below.

Exploring

Over the next two to three days, explore the following strategies making explicit the strategy you are concentrating on as the teacher and the reason for using the selected strategy

e.g. In the problem 357 + 189 tidy numbers would be a useful strategy because 189 is close to 200.

Each day follow a similar lesson structure to the introductory session, with students sharing their solutions to the initial questions and discussing why these questions lend themselves to the strategy being explicitly taught. Conclude each session by having students making some statements about when this strategy would be useful and why e.g. tidy numbers when one number is close to 100 or 1000, place value when no renaming is needed and reversibility when neither of the other two are helpful for subtraction. It is important to record these key ideas as they will be used for reflection at the end of the unit.

The questions provided are intended as examples for the promotion of the identified strategies. If the students are not secure with a strategy you may need to make up some of your own questions to address student needs.

Tidy numbers/Equal additions
Room 9 are selling muesli bars at lunchtime to raise money for their camp. They had 434 at the beginning of lunchtime and sold 179, how many did they have left to sell?

The tidy numbers strategy involves rounding a number in a question to make the question easier to solve. In the above question, 179 can be rounded to either 180 (by adding 1), or 200 (by adding 21). Tidying one number will affect the other numbers in an equation. For subtraction questions, it is important that the difference between numbers remains the same. In the question above, the number to be tidied is 179 (to 200). In order to do this, we add 21. To keep the difference between the numbers the same, we must add 21 to the number we are subtracting from. The net result of the equation is then:
(434 + 21) - (179 + 21) =
455 – 200 (the tidied number) =
255

For addition questions, the amount used to ‘tidy’ needs to be taken from the other number(s) in the equation. The net result of tidying the below question (739 + 294) is:
(294 + 6) + (739 – 6) =
300 (the tidied number) + 733 =
1033

Lead the students to discover the effect that tidying numbers has on the quantities you are dealing with. The following questions can be used to elicit discussion about the strategy.

  • What tidy number could you use that is close to one of the numbers in the problem?
  • What do you need to do to the other number if you tidy up this number? Why?
  • Why is this strategy useful for this problem?
  • What knowledge helps you to solve a problem like this?

If the students do not understand the tidy numbers concept, use place value equipment (for example a number line) to show the problems physically. Some students may find it useful to record and keep track of their thinking.

Use the following questions for further practice if required:
Ariana has scored 739 runs for her cricket club this season. Last season she scored 294, how many did she score in total in the last two seasons?
Farmer Dan has 1623 sheep and he sells 898 sheep at the local sale. How many sheep does he have left?
568 + 392
661 - 393
1287 + 589
1432 - 596

Note that the problems posed here are using a tidying up strategy rather than tidying down i.e. 103 down to 100 as in these situations place value tends to be a more useful strategy.

Place Value For the community hangi 356 potatoes had been peeled and there were 233 left to be peeled, how many potatoes will there be altogether?

The place value strategy involves adding the ones, tens, hundreds, and so on. In the above problem:
300 + 200 is added
Then 50 + 30
And finally 6 + 3

The following questions can be used to elicit discussion about the strategy:

  • How can you use your knowledge of place value to solve this problem?
  • Why is this strategy useful for this problem?

If the students do not understand the concept, use place value equipment (such as blocks) to show the problems physically. Some students may find it useful to record and keep track of their thinking .

Use the following questions for further practice if required:
Zac has $498 available on his eftpos card and spends $243 on a new BMX bike, how much money does he have left?
3221 + 348
4886 - 1654
613 + 372
784 - 473

Reversibility

Faloa is helping his Mum build a path. There were 438 bricks in the pile and they used 169 of them yesterday, how many bricks have they got left for today?

The reversibility strategy involves turning a subtraction problem into an addition one so the problem above becomes:
169 + ? = 438

Then using tidy numbers to solve
  number line diagram.
Or
number line diagram.

The following questions can be used to elicit discussion about the strategy:

  • How could we think of this as an addition equation?
  • What do you need to add to make it easier to solve?
  • How can you keep track of how much you have added altogether?
  • Why is this strategy useful for this problem?
  • What knowledge helps you to solve a problem like this?

If the students do not understand the concept, use a number line to show the problems physically. Some students may find it useful to record and keep track of their thinking using a number line.

Use the following questions for further practice if required:
At the mail sorting office there were 547 letters to be sorted, 268 of these were distributed to private boxes, how many were left to be delivered?
The school library has a total collection of 1034 books and 459 are issued at the moment, how many are on the shelves?
628 - 342
537 - 261
742 - 353
1521 - 754
1762 - 968
1656 - 867

Reflecting

As a conclusion to the week’s work, give the students the following five problems to solve asking them to discuss which strategy they think will be useful for each problem and why before they solve them. After they have solved the problems, engage in discussion about the effectiveness of their selected strategies for the problems.

There may be a few students who do not concur with the group about the usefulness of a particular strategy for a given problem. This is perfectly acceptable as long as they are able to justify their thinking. Many students will have a favourite strategy that they use, sometimes to the exclusion of all others. Most often this is a place value strategy. These students should not be discouraged from using place value to solve problems but should be exposed to problems where place value is an inefficient strategy, because of repeated renaming, for example 289 + 748, or 453 – 257. Explain that even though you may have a favourite strategy, it is worthwhile to practice other strategies and thereby have a broad range from which to select. It is importnat to encourage students to look carefully at the numbers and then make a decision about what strategy will be most efficient for working with these numbers. 

Problems for discussion
1318 - 747
763 - 194
433 + 452
1993 + 639
4729 - 1318

You might also like to also try some problems with more than 2 numbers in them
721 – 373 - 89
663 - 61 - 88
63 + 422 + 49
42 + 781 + 121
84 + 343 - 89

Discuss the different strategies explored during the week and ask students to explain in their own words what types of problem each strategy would be useful for solving, and what types of problem each strategy would not be useful for solving. Ask the students to draw a strategy ‘from a hat’ and write questions specific to that strategy for a partner.

Conclude the unit by showing the students the questions asked in the initial session again and discuss whether they would solve them in a different way now, and how their thinking has evolved.


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