Prior knowledge
Use count on strategies
Use their “10 and” facts
Background
Knowing the basic facts to 10, including the compatible numbers, is essential knowledge if students are to advance to becoming part-whole thinkers. For older students, one way to provide practice and reinforcement of these skills is for the students to play maths games – with two dice. (One at least of these should have a number rather than dots). Another way to prepare students for the part-whole leap is to develop their understanding around their facts to 10, so while these exercises are based around compatible numbers to ten, they include a number of other issues that are significant to the learning of students at this stage. This can introduce a level of challenge even for students who initially seem to know their compatible numbers to 10. Important learning is outlined below.
Use of the equals sign
There are a lot of misconceptions around the use of the equals sign. Some students seem to think that it means “work out the answer”. Consequently, no equals signs have been provided in exercises where simple additions are required. Students with such an understanding may also think that 4 + 6 = 10 is a correct way to write the sentence, but not 10 = 4 + 6. (Hence number 25 in exercise 1). Such a question may be posed as part of practice, then discussed at a later teaching session, or could be included as part of a lesson.
The second part of exercise 1 is provided to reinforce the use of the equals sign when two things are identical/equal, and that the things on the left and right balance. This meaning of equals should also be raised as part of the teaching around this topic.
Inequalities
It has been noted that in recent years there has not been the same emphasis on developing an understanding of inequalities in primary mathematics. This is not intended, but may mean that some students come through without the understandings they have had in the past. Consequently some teaching may need to be provided before students understand the signs and the concepts required by this exercise.
Start, change, result unknown
Students don't really know a fact until they can recognise and use it in all three formats. Exercise 3 not only provides practice in recognising the facts in these other formats, but also introduces students to lower level algebra. Mental computation (or recall of known facts) of such simple equations is most sensible method of solution.
Use of shapes as unknowns
Students should have been working with shapes as unknowns for quite some time before reaching secondary school, so should be conversant with what is expected in using a shape in a sentence/equation. In this example, however, the meaning of the unknown has changed. Firstly there are two different shapes – which traditionally would mean that they represent different numbers (though there is the special case where they are the same.) Students may need to discuss this before attempting the problem). However, the unknowns do not represent a single number in this context. This too may to be introduced – that there could be lots of possibilities for such an equation (though is likely to arise naturally if you ask them all to think of two numbers that add up to ten.
The link between addition and subtraction
A single addition fact should be able to be turned into related subtraction facts and simple subtractions should be able to be solved using knowledge of basic addition facts. However, for many students, subtraction understanding lags behind addition understanding. Making the link between addition and subtraction is thus essential teaching at this level.
Word problems
One issue with providing word problems in an exercise alongside simple number problems is that some students learn not to read the words, and simply to pull out the numbers “and do the same to them”. To address this problem, this exercise includes a variety of formats of problem. In fact, number one requires a subtraction with the numbers 6 and 4 – rather than an addition, while others include change unknown format – so could equally be an addition or a subtraction that relies on their compatible number knowledge. This exercise thus provides a good basis for a teaching session around “what words tell us that we should be adding the numbers…” In this teaching session, students could be encouraged to develop a list of words commonly used to indicate that the operations of addition and subtraction are to be used.
Discovery based on patterning
Students learn a lot of mathematics (things that are not necessarily directly taught – or intended to be taught) by identifying patterns. Often, the better we are at identifying patterns, the better we are at mathematics. These exercises look to harness patterning to help students realise that knowing these facts to 10 mean that they can also answer a whole load of other problems. Both exercises require follow-up discussion – and additional practice built around consolidating these discoveries. For example, students could make a poster showing how to use their facts to 10 to answer other problems. They should also do some practice work in using this new skill – in all 3 formats, start, change and result unknown.
Comments on the Exercises
Exercise 1
Asks students to identify equations that sum to 10.
Exercise 2
Asks students to identify if single digit additions are <, > or = to 10
Exercise 3
Asks students to complete compatible number equations.
Exercise 4
Asks students to identify compatible numbers to 10.
Exercise 5
Asks students to identify compatible numbers in subtraction equations.
Exercise 6
Asks students to use compatible numbers to solve addition problems, e.g. 7 + 9 + 3 =
Exercise 7
Asks students to use compatible numbers to solve word problems.
Exercises 8 & 9
Asks students to use compatible numbers to ten with two digit numbers, e.g. 47 + 3 = 50, 10 + 90 = 100
Exercise 10
Extension problems
Addition and Subtraction Pick n Mix
In this unit we look at a range of strategies for solving addition and subtraction problems with whole numbers. This supports students anticipating, from the structure of a problem, which strategies might be best suited to solving it.
Students at Level 3 of the New Zealand Curriculum select from a broad range of strategies to solve addition problems. This involves partitoning and recombining numbers to simplify problems and draws on students' knowledge of addition and subtraction facts, and knowledge of place value of whole numbers to at least 1000.
The key teaching point is that some problems can be easier to solve in certain ways. Teachers should elicit strategy discussion around problems to get students to justify their decisions about strategy selection in terms of the usefulness of the strategy for the problem. The following ideas support this decision making:
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
The three main types of addition and subtraction problem are applied in this unit: joining sets (addition), separating sets (subtraction), and finding differences (either addition or subtraction). Choose contexts that make links to other relevant curriculum areas, reflect the cultural backgrounds, identities and interests of your student, and might broaden students’ views of when mathematics is applied. Commonly used settings might involve money, points in sport or cultural pursuits, measurements, and collectable items. For consistency, you could choose one context in which all of the problems presented within this unit could be framed.
Te reo Māori kupu such as tāpiri (addition), tango (subtraction), and huatango (difference in subtraction) could be introduced in this unit and used throughout other mathematical learning
Getting Started
The purpose of this session is to explore the range of strategies already used by students to solve addition and subtraction problems. This lesson will enable you to evaluate which strategies need to be focused on in greater depth. In turn, you will 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. Consider adapting the contexts used in these problems to further engage your students.
Ask the students to work out the answer in their heads or by recording in some way. Give the students plenty of thinking and recording time. Ask the students to share their solutions and how they solved the problem with a peer. The following are possible responses:
Place value (mentally, possibly with the support of equations on an empty number line):
288 + 127 is just like 288 + 100 +20 +7. So that’s 388… 408… 415.
Tidy numbers (mentally, possibly with the support of equations on an empty number line):
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.
Algorithm (usually written):
Students may visualise or record a written algorithm like this:
Understanding is revealed by the language used to describe the strategy, such as, “8 plus 7 equals 15. I wrote 5 in the ones place and carried the extra ten into the tens place because 15 is made up of 5 and 10 and I can only record one digit in the ones place.”
As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the correct strategies that are elicited, recording the strategies to reflect upon later in the unit on the class T chart (under the addition heading). You might also ask students to model how their strategies work with place value materials. Note that folding back to justify a strategy is often more difficult than performing the strategy.
Ask students to solve the problem mentally, giving them plenty of thinking and recording time. Then ask students to share their solutions and how they solved the problem with a peer. Possible responses are:
Reversibility (adding on to find the difference with recording to ease memory load):
$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.
Subtracting a tidy number then compensating
$466 - $200 = $266. I took off $200 instead of $178 so I need to ‘pay back’ $22. $266 + $22 = $288.
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.
Algorithm (usually written):
Students may visualise or record a written algorithm like this:
Understanding is revealed by the language used to describe the strategy, such as, “6 minus 8 doesn’t work (ignoring integers) so I changed one ten from the tens column into ones to make 16. I wrote 16 in the ones place and took one ten off in the tens place…”
As different strategies arise, ask the students to explain why they chose to solve the problem in that way. Accept all the correct strategies that are elicited, and justified, recording the strategies to reflect upon later in the unit (under the ‘subtraction’ heading on the class T chart). You might also ask students to model how their strategies work with place value materials. Note that folding back to justify a strategy is often more difficult than performing the strategy.
Subtraction strategies tend to be more difficult to control than addition strategies, given comparable numbers. Look out for students compensating the wrong way (taking more off) in the tidy number strategy and making errors when using the algorithm.
Use your observations to plan for your subsequent teaching from the exploring section outlined below.
Exploring
Over the next two to three days, explore different strategies for addition and subtraction of whole numbers. Give the strategies a name so students can tell others which strategy they are preferencing for a given problem. Highlight when certain strategies are most efficient, for example, In the problem 357 + 189 tidy numbers would be a useful strategy because 189 is close to 200.
Follow a similar lesson structure each day 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 asking students to make statements about when the strategy would be most useful and why the certain problem is appropriate, e.g., tidy numbers when one number is close to 100 or 1000, standard place value (hundreds, tens, and ones) when no renaming is needed and reversibility when neither of the other two numbers are easy for subtraction. It is important to record examples of strategies as they will be used for reflection at the end of the unit. Some strategies may require more teaching time, greater use of materials, and more scaffolded and individualised teaching. Ensure that students who demonstrate proficiency with the strategies early on in each session have adequate opportunities for practice, extension, and supporting their peers (if appropriate).
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. Consider adapting the contexts reflected in these problems to further engage your students.
Tidy numbers then compensating
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). If 200 is subtracted (434 – 200 = 234) then the result is 21 less than the answer. 234 + 21 = 255. An empty number line shows this strategy clearly:
For addition questions, one addend can be tidied by taking from the other addend. Alternatively, both addends might be tidied, and compensation used to adjust for the tidying.
Shifting six between the addends gives 739 + 294 = 733 + 300 = 1033.
Rounding 587 to 600 and 395 to 400, then compensating gives 587 + 395 = 600 + 400 – 13 – 5 = $982.
Rounding 898 to 900 gives 1623 – 898 = 1623 – 900 + 2 = 725
568 + 392
661 - 393
1287 + 589
1432 - 596
Place Value (written algorithm)
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
As an algorithm the calculation is represented as:
3221 + 348
4886 - 1654
613 + 372
784 – 473
Reversibility (adding one rather than subtracting to find the difference)
The reversibility strategy involves turning a subtraction problem into an addition one so the problem above becomes:
169 + ? = 438
Using tidy numbers to solve the problem makes calculation easier:
Or
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. Pose the problems in contexts that are relevant and engaging for your students. Ask students to discuss, with a peer or in small groups, which strategy they think will be most useful for each problem and justify their view. For some problems many strategies may be equally efficient. After students have solved the problems, engage in discussion about the effectiveness of their selected strategies.
Some students may have a favourite strategy that they use, sometimes to the exclusion of all others. The best approach is to pose problems where the preferred strategy may not be the most efficient. For example, 289 + 748 is most suited to using tidy numbers and compensation.
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 two numbers in them, such as:
721 – 373 - 89
663 - 61 - 88
63 + 422 + 49
42 + 781 + 121
84 + 343 - 89
Dear family and whānau,
This week we have been investigating several different ways of approaching addition and subtraction problems. With your child, decide on a problem involving 3 or 4 digit numbers and solve it together, asking your child if they can show you more than one way it can be worked out. Share your thinking as well and compare your strategies.
Put some problems in a container and play Clever Draw: each person draws out a problem, solves it in their head, using materials or with written working out and then has to show the other person how they worked it out using a diagram (a drawing of your thinking).
Partitions
This unit is about partitioning whole numbers. It focuses on partitioning numbers to “make a ten” or a decade when adding whole numbers, for example 8 + 6 can be solved as (8 + 2) + 4. The unit uses measurement as a context.
Students at Level Two should understand that numbers are counts that can be split in ways that make the operations of addition, subtraction, multiplication and division easier. From Level One students will understand that counting a set tells how many objects are in the set. At Level Two they are learning that the count of a set can be partitioned and that the count of each subset tells how many objects are in that subset. Students also need to understand that partitions of a count can be recombined. For example, a count of ten can be partitioned into 1 and 9, 2 and 8, 3 and 7, etc.
It is important that students understand that because there are many ways to partition a number and they will need to choice the best partition to suit the question. In this unit we focus on making partitions that allow numbers to make a ten. For example 8 + 6, the 6 can be partitioned to 2 + 4, the 2 then combines with 8 to make a ten 8 + 2, + 4. For this reason teachers should ensure that they select questions that are best solved using the "make a ten" strategy as opposed to partitioning to make doubles. For example, 8 + 7 also encourages the strategy 7 + 7 + 1 rather than 8 + 2 + 5.
This unit uses the context of measurement. Examples and problems should use like measures, and students should be encouraged to write the units when they give an answer, for example 8m + 5m = 13m.
Session 1
In this session students investigate the fact that whole numbers can be partitioned in a number of ways.
How many animals should I put in the first paddock? (Put that number (for example, 3) in that paddock and the rest in the second paddock)
How many animals are in the second paddock? Make sure the students understand there is still 8 animals.
Session 2
In this session students investigate that partitioning teen numbers using the number ten. It is easiest to start by making teens numbers as 10 + x.
Object
Length
Splits
Sellotape dispenser
10 cm + cm
Duster
Stapler
Notebook
etc
Session 3
In this session students solve addition problems by partitioning numbers. They use the strategy “make a ten”.
How many counters are there to add on? (2)
What does 10 and 2 make?
What two numbers did we split the 5 into? (3 +2)
Why did we chose 3, then 2? (To make it up to 10)
Ask the students:
How many counters would be need to get to 10? (2)
How many counters are there still to add on? (4)
What is 10 plus 4? (14)
Check the answer using counters and the number strip.
Start at the 7, ask the students:
How many jumps do we need to add on? (4)
How many jumps is it to get to 10? (3)
How many left over from the 4? (1)
What is 10 and 1? (11)
Session 4
In this session students solve addition problems by partitioning numbers. They will solve problems that involve adding a 1 digit number to a 2 digit number. The “make a ten” strategy is applied to decade numbers, for example 28 + 7 is solved by making it up to 30 and then adding the remaining 5.
Show the students how they can draw a number line to suit the question.
Ask the students: what numbers are we adding together? (17 and 5)
Draw a line and the number 17 underneath.
Ask the students: what number can be jump to from here? (20)
how many jumps is that? (3)
how many of the 5 are left? (2)
What is the answer? (22)
Session 5
In this session you may wish to continue to give the students opportunities to practise addition partitioning with the make a ten or make a decade strategy. Problems could focus around one measurement theme, for example length. Or problems could focus around a theme such as camping and involve more than one measurement context, for example, weight of packs, time spent on activities, capacity of shower water, length of washing lines, etc.
Alternatively, you may wish to use the formats of session 3 and 4 to show students how this partitioning strategy can be used to solve subtraction problems. For example, 44 – 7, it takes 4 jumps to get back to 40, then the remaining 3 jumps takes us back to 37.
Dear Parents and Whānau,
This week in maths your child has been learning to use a number line and to make jumps to 10 to work out addition questions. Ask them to show you how they jump to the 10 first to solve 8 + 5.
You could pose some other similar problems, like 7 + 4, 9 + 6 and have them use jumps to 10 to solve them. Thank you for your help.
Compatible numbers to ten
These exercises and activities are for students to use independently of the teacher to practice number properties. Some of these activities would be suitable for homework. Others require follow-up during teaching sessions.
Prior knowledge
Background
There are a lot of misconceptions around the use of the equals sign. Some students seem to think that it means “work out the answer”. Consequently, no equals signs have been provided in exercises where simple additions are required. Students with such an understanding may also think that 4 + 6 = 10 is a correct way to write the sentence, but not 10 = 4 + 6. (Hence number 25 in exercise 1). Such a question may be posed as part of practice, then discussed at a later teaching session, or could be included as part of a lesson.
It has been noted that in recent years there has not been the same emphasis on developing an understanding of inequalities in primary mathematics. This is not intended, but may mean that some students come through without the understandings they have had in the past. Consequently some teaching may need to be provided before students understand the signs and the concepts required by this exercise.
Students don't really know a fact until they can recognise and use it in all three formats. Exercise 3 not only provides practice in recognising the facts in these other formats, but also introduces students to lower level algebra. Mental computation (or recall of known facts) of such simple equations is most sensible method of solution.
Students should have been working with shapes as unknowns for quite some time before reaching secondary school, so should be conversant with what is expected in using a shape in a sentence/equation. In this example, however, the meaning of the unknown has changed. Firstly there are two different shapes – which traditionally would mean that they represent different numbers (though there is the special case where they are the same.) Students may need to discuss this before attempting the problem). However, the unknowns do not represent a single number in this context. This too may to be introduced – that there could be lots of possibilities for such an equation (though is likely to arise naturally if you ask them all to think of two numbers that add up to ten.
A single addition fact should be able to be turned into related subtraction facts and simple subtractions should be able to be solved using knowledge of basic addition facts. However, for many students, subtraction understanding lags behind addition understanding. Making the link between addition and subtraction is thus essential teaching at this level.
One issue with providing word problems in an exercise alongside simple number problems is that some students learn not to read the words, and simply to pull out the numbers “and do the same to them”. To address this problem, this exercise includes a variety of formats of problem. In fact, number one requires a subtraction with the numbers 6 and 4 – rather than an addition, while others include change unknown format – so could equally be an addition or a subtraction that relies on their compatible number knowledge. This exercise thus provides a good basis for a teaching session around “what words tell us that we should be adding the numbers…” In this teaching session, students could be encouraged to develop a list of words commonly used to indicate that the operations of addition and subtraction are to be used.
Students learn a lot of mathematics (things that are not necessarily directly taught – or intended to be taught) by identifying patterns. Often, the better we are at identifying patterns, the better we are at mathematics. These exercises look to harness patterning to help students realise that knowing these facts to 10 mean that they can also answer a whole load of other problems. Both exercises require follow-up discussion – and additional practice built around consolidating these discoveries. For example, students could make a poster showing how to use their facts to 10 to answer other problems. They should also do some practice work in using this new skill – in all 3 formats, start, change and result unknown.
Comments on the Exercises
Book 4: “Tens frames again” and “Patterns to 10”
Compatible numbers to 20
These exercises and activities are for students to use independently of the teacher to practice number properties. Some of these activities would be suitable for homework. Others require follow-up during teaching sessions.
Prior knowledge
Background
Many of these activities parallel those in “compatible numbers to 10”. They are designed for students who already have their facts to 10, but need to extend these to 20. However, these students may or may not have had exposure to the algebra aspects of those activities, so they have largely been repeated here. The teaching notes for “compatible numbers to 10” contain information about some significant teaching points embedded in both sets of exercises, so will not be repeated here, but should be accessed and read prior to using these activities.
Comments on the Exercises
When reviewing this exercise with students it can be useful to ask students “why” some of the problems are false, or do not equal 20. For example, students may come up with reasons like “as both of the numbers are less than 10, then the total must also be less than 20”.
This exercise is in two parts, the second part again reverses the sense of the problems so the sum is on the right of the symbol. This is more challenging and will need to be discussed as part of relevant teaching before setting these problems. In this second set of problems students are doing the reverse of comparing a sum to 20. Rather, they are comparing 20 to the results of their calculation. This reverse sense can be problematic as the students are using inequalities, so may want to use the incorrect sign. Reading the sentence in reverse can alleviate this problem.
A little bit more/A little bit less
These exercises and activities are for students to use independently of the teacher to practise number properties.
solve multiplication problems by rounding up to a tidy number and then compensating
solve multiplication problems by rounding down to a tidy number and then compensating
Multiplication and Division, AM (Stage 7)
Practice exercises with answers (PDF or Word)
Prior knowledge
Multiply basic facts with tens, hundreds and thousands eg 3 x 60, 5 x 200
Recall addition and subtraction basic facts
Background
This activity uses the multiplication strategy of rounding to a tidy number and then compensating. Numbers can be rounded up or down to the nearest tidy number. Rounding down to the nearest tidy ten number is effectively using the place value strategy, for example 2 x 73 is (2 x 70) + (2 x 3) = 146. When rounding up to a tidy number the compensation involves subtraction. For example 8 x 29 = (8 x 30) - (8 x 1). Students need to be careful to subtract 8 x 1 and not 29 x 1. Later examples in the activity involve rounding to 15 or 25
Number generalisation:
Ideas that can be developed from this activity:
Eg.
3 x 28 = 3 x 30 – 3 x 2
3 x a = 3 x (a + b) – 3 x b
3 x 31 = 3 x 30 + 3 x 1
3 x a = 3 x (a-b) + 3 x b
Comments on the Exercises
The exercises have been set up in the following way.
Exercises 1 – 7, Student has been given a tidy number statement eg 5 x 20 = 100 and they use this given fact to find the answer to eg 5 x 19 or 5 x 22 (by compensation – add on a little or subtract a little).
Exercise 2
Exercise 3
Asks students to multiply by a “ten number”. Add on a little, value added could be more than ten.
Exercise 4
Asks students to multiply by a “ten number”. Subtract a little, value subtracted could be more than ten.
Exercise 5
Exercise 6
Asks students to multiply by a “hundred number”. Subtract a little, value subtracted could be more than ten.
Exercise 7
Exercise 8
Asks students to make up their own problems that use the same strategy.
Book 6: A Little Bit More/ALittle Bit Less
Loopy Sheets: PDF (96KB) or Word (132KB)
Loopy master: PDF (19KB) or Word (45KB)
Figure It Out References
Number Link 2, Year 7-8, Planting with the Whanau, page 6