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Research on Students’ Mathematical

Thinking and the Strategies They Use to

Solve Mathematical Problems

At an Intermediate School

MSTE 502

Assignment 2

Fatimah Alsaleh

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Introduction

Students have been taught to solve mathematical problems using pencil and paper and other

traditional methods for many years, but with the development of teaching and learning

techniques, students now have access to sophisticated mental strategies to solve mathematical

problems. These include part-whole (partitioning) strategies such as standard place-value

partitioning, bridging-through-ten/decade, rounding and compensation, reversibility, equal

additions, equal multiplications and doubling and halving, to solve mathematical problems in

addition and subtraction; multiplication and division; proportion and ratio (Young-Loveridge,

2011). Students’ mathematical understanding can definitely be developed by using these

strategies.

Mathematical thinking is defined as figuring out a process for solving mathematical

numeracy tasks (Liu & Niess, 2006). Schoenfeld (1994), (as cited in Liu and Niess, 2006),

identifies it as creating a mathematical opinion, while according to Inprasitha and Loipha

(2007), students’ mathematical thinking means to “be able to think rationally and express

their thinking systematically and precisely” (p.256). Students’ capabilities for mathematical

thinking depend on quick calculations, correctness of answers, and the acquisition, choice

and use of mental strategies (Cai, 2003; Inprasitha & Loipha, 2007). Some researchers have

discussed how students’ mathematical thinking develops. Schliemann and Carraher (2002)

and Inprasitha and Loipha (2007) point out that students’ mathematical thinking could be

evolved from both their past and everyday experiences and reflections outside of school in

addition to the activities they are involved in at school. They also attribute the development

of mathematical understanding to social and cultural factors: if students participate in social,

cultural or commercial communication and interaction, their mathematical understanding will

evolve. In addition, Schliemann and Carraher (2002) indicate there are other factors

contributing to the development of mathematical understanding, such as students “increasing

reliance on introduced mathematical principles and representations” (p.242), “mathematical

procedures, notation, formal contents” (p.244), and “tools and representational systems”

(p.246). However, some students lack mathematical thinking because their mathematics

teachers do not have sufficient awareness themselves, of mathematical thinking. Therefore,

their teachers are unable to promote their students’ explicit expression of mathematical

understanding (Inprasitha & Loipha, 2007).

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Mathematical thinking is reflected in the students’ use of appropriate strategies. Students

solve mathematical problems in different ways as a result of different uses of various

strategies. Thompson (1999, 2000) illuminates and demonstrates some particular mental

strategies for addition and subtraction including rounding and compensation, balancing,

bridging-through-ten/decade and partitioning. Using the rounding and compensation strategy,

“a larger quantity is used initially, then subtraction or addition is used to compensate”

(Young-Loveridge, 2011, p. 53). Although this strategy is quite useful, it is not used as

commonly by children as it is by adults and it is hard for young children to invent it

(Thompson, 2000). According to Thompson (1999), in the balancing strategy, a difficult

number is split and part of it is added to the other number to create a multiple of ten.

Although researchers have rated it as an important mental strategy, only one participant in

Thompson’s (1999) study used the balancing strategy. A further strategy is bridging-through-

ten/decade. The difficulty of this as a concept reflects children’s struggles with equivalent

numbers, while it is a common calculation procedure for adults. Partitioning means the

numbers are split into tens and ones, and then operations are carried out on each group

separately (Thompson, 2000; Young-Loveridge, 2011). One major problem that students face

in using this strategy is “it breaks down with subtraction which necessitate regrouping”

(Thompson, 2000, p.24). The Ministry of Education number framework (2008) emphasises

that it is essential for students to progress in both the strategy section and the knowledge

section because each section is important for developing the other. The drawing below shows

the relationship between the strategy and knowledge sections:

Creates new knowledge through use

Provides the foundation for strategies

Source: Ministry of Education, Book 1: The number framework, 2008, p.1.

In terms of multiplicative thinking, some researchers define it as repeated addition, while

many other researchers have attempted to distinguish multiplicative thinking from additive

thinking and label it as a complex concept which is more than just doing repeated addition

(Clark & Kamii, 1996; Jacob & Willis, 2001; Young-Loveridge, 2011). There are some

differences between multiplicative and additive thinking: “the many-to-one correspondence

between three units of one and one unit of three; and the composition of inclusion relations

Strategy Knowledge

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on more than one level” (Clark & Kamii, 1996, p. 43). In addition, Young-Loveridge (2011)

states that another difference between them is their structure: while multiplication involves

proportional structure, addition involves part-whole structure. Groups or equal-sized parts are

always required when using multiplicative partitioning strategy, while additive partitioning

may include unequal-sized parts. While some students have the ability to undertake

multiplicative thinking, many students have difficulty solving multiplication problems (Clark

& Kamii, 1996).

When investigating children’s mathematical thinking, the clinical (diagnostic) interview is

one of a variety of different interview forms. According to Brown, Findley and Montfort

(2007), in a clinical interview “the interviewers continually assessed the students’ reasoning

and pursued new lines of questioning to test their assessments” (p. 24). Researchers can

examine participants’ misconceptions and understanding about specific topics, along with

students' confidence in their understanding by asking them open-ended conceptual questions.

In addition, the clinical interview allows researchers to investigate cultural factors that may

affect participants’ understanding or misconceptions. The value of the diagnostic interview is

that it assesses the reliability and validity of the participants' responses and can investigate a

wide range of students’ conceptual understanding (Numeracy Professional Development

Projects (NDP) Book 2: The diagnostic interview, Ministry of Education, 2008).

This survey aimed to explore students’ conceptual understanding within three strategy

domains: addition and subtraction; multiplication and division; proportion and ratio, looking

at the kind of strategies they used and the nature of any misconceptions as well as one

knowledge domain: basic facts. Finally, this study investigated the preferred way to solve

mathematical problems, whether by calculating purely mentally or by recording the steps in

solving these problems using pencil and paper.

Method

Participants

The sample consisted of eight students at one public school (St Andrews Middle School) who

were in Year 8. They ranged between 11 and 12 years of age (3 students were 11 years old

and 5 students were 12 years old). Four of them were ranked as top students by their teachers,

whereas the other 4 students were labelled as less able students. Half of the students were

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females (3 top students and 1 less able student) and another half were males (3 less able

students and 1 top student) and the gender choice was random.

Procedure

A clinical interview was used to assess students' understanding of mathematical problems.

The interview was conducted on May 1, 2012 in a private room at St Andrews Middle School

by working with the students individually on numeracy tasks over a period of almost four

hours. Each student had approximately 30 minutes except for one student who had only 15

minutes and whom I ranked as the most mathematically capable of the eight students. A

photocopiable item from NDP Book 2: The diagnostic interview (Ministry of Education,

2008, p. 8-9, 30, 40-44, 47) was used while working with students task by task. All the

interviews were audio-taped for later coding, and notes were taken of students’ responses to

the numeracy tasks. The interview contained 19 items in four categories including three

strategy domains (addition and subtraction; multiplication and division; proportion and ratio);

and one knowledge domain which included basic facts as well as three questions to explore

the affective domain (see Appendix A).

Results

In this section the main results of this study are presented by organising them in four groups:

addition and subtraction; multiplication and division; proportion and ratio; and basic facts,

with analysis of the similarities or differences between the two groups of students (top

students and less able students) in terms of correctness and the errors in their responses to the

tasks provided, and the types and efficiency of strategies they used in their responses to the

different tasks. In addition, three general questions regarding students’ attitudes to

mathematics are presented. These dealt with the areas of: like or dislike of maths; perceived

ease or difficulty of maths; and preferred ways of solving mathematical tasks. Finally, there is

a summarised evaluation of students’ abilities to solve mathematical problems.

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Table (a) shows students’ stages according to the number framework using Numeracy Project Assessment (NumPA) form C

0

2

4

6

8

ADDITION &

SUBTRACTION

MULTIPLICATION &

DIVISION

PROPORTION & RATIO BASIC FACTS

Top Students' Stages

Nazeefah

Serena

Deepel

Vili

02468

ADDITION &

SUBTRACTION

MULTIPLICATION &

DIVISION

PROPORTION & RATIO BASIC FACTS

Less able Students' Stages

James

Eli

Shirley

Tyrael

Addition and subtraction

• 53 – 26

Interestingly, all students from both groups (top and less able students), except for one less

able student, solved this task correctly and most of them had the ability to get the correct

answer quickly. In addition, both groups shared the use of similar advanced part-whole

additive strategies such as place-value partitioning, reversibility and rounding and

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compensation. However, James was the student who gave a wrong answer and he tried to solve

this question by using his fingers. Serena, who was the most efficient in using mental strategies

among the other students, used the reversibility strategy:

53 – 26=

26 + 4 = 30, 30 + 20 = 50

50 + 3 = 53

20 + 4 + 3 = 27

In contrast, although Eli quickly got the right answer, he was not confident about his answer

because he got confused when he explained the strategy he used to subtract 26 people from 53

people and took a long time to remember the steps of his strategy (rounding and

compensation):

Less able student: Eli

(×) 53 – 20 = 40

(×) 53 – 20 = 33, 33 – 3 = 30, 30 – 6 = 24

� 53– 20 = 33, 33 – 3 = 30, 30 – 3 = 27

• 394 + 79

All students except for James were able to add 79 stamps to 394 stamps correctly. In terms of

strategies, place-value partitioning was the most common strategy used by both top and less

able students, in particular by the less able students. Another strategy used once by one top

student (Serena) was bridging-through-ten/ decade. Moreover, two top students (Nazeefah and

Deepel) and one less able student (Tyrael imagined using a standard written method such as

solving the problem vertically. For example, Serena was unique in solving this additive

problem by using the bridging-through-ten/ decade strategy:

Top student: Serena

I took 6 from 79 and added it to 394 to make 400

And then I took 73, which remained from 79, and added it to 400

So I got 473 stamps altogether

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Less able student

James, who struggled with this problem, was confused about whether he needed to add or

subtract while using rounding and compensation strategy:

He used a round number (400) instead of 394, then added 79 to 400 and got 475, which was

the wrong answer. He then forgot to subtract what he had added to 394 to make it 400.

• 5.3 – 2.89

All eight students had difficulty solving addition of decimals and only two students got the

right answer: Serena (top student) and Tyrael (Less able student). Serena solved this problem

using the reversibility strategy:

2 .89 + 0.11 = 3 metres

3 + 2 = 5, 5 + 0.3 = 5.3

2 + 0.11 + 0.3 = 2.41

Whereas the way Tyrael got the answer was not clear:

Putting 8 on 5.3 onto that one and taking 2 from 5.3

And that makes that number (2.41)

When he was asked “how did you do it this way?” He said, “My mom taught me to do it that

way”, and he could not solve this problem using another way. It may be that he was familiar

with this task and he memorised the answer for this question.

Shirley used place-value partitioning and did all the steps correctly, but at the end she got the

wrong answer:

5 – 2 = 3, 3 = 2 +1, 1 = 1.00

1.00 – 0.89 = 0.11, 2 + 0.11 = 2.11

0.11 + 0.3 = 0.14

2.11 + 0. 14 = 2.14

While Shirley was able to know how many tenths in one, she was unable to solve 0.11 + 0.3

because she was thinking in whole numbers not in decimals: 11 + 3 = 14.

Other examples of responses:

Eli: I never have done points before.

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Vili: I forget how to do this.

Multiplication and division

• 5 × 8 / 15 ÷ 5 (Trees)

All students from both groups were able to solve the two parts of this multiplication problem,

although they solved them in different ways. Top students used known multiplication facts and

division facts to solve the two parts of this multiplication problem. The less able students used

advanced counting strategy by using a combination of skip-counting and counting in ones, for

example, 5, 10, 15, 20, .…..40., excluding James who counted all the objects (trees) and then

used his fingers to get 3 more rows of 5 trees from 15 trees.

• (A) 5× 8 = 40 so 5 × 16 =? / (B) 3 × 20 = 60 so 3 × 18 =?

The majority of students reached the correct solutions for these problems, but top students used

different strategies from those used by the less able group:

Top students

The four students used advanced additive-early multiplicative part-whole because they were able

to solve problems (A) and (B) by deriving from multiplication facts and by doubling and

halving, for example:

(A) : Nazeefah

16 ÷ 2 = 8

And double 40 is 80

(B): Deepel

18 + 2 = 20, 2 × 3 = 6 so 60 – 6 = 54

All four students used exactly the same steps as Nazeefah and Deepel, except for Vili who

solved part (B) by using repeated addition.

Less able students

In contrast, only Shirley and Tyrael solved part (A) correctly, and they did not derive the answer

from known multiplication facts, but used repeated addition: 5 + 5 + 5+…..+5 = 80. The

problem with this question was using multiplication facts, as James said, “I am not good at

times, can I skip this question?”

In regard to part (B), only one student (Eli) gave no response to this problem, while the other

three students differed in their approach to solving part (B): Shirley solved it by deriving as

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Deepel did, but Tyrael used repeated addition by counting on 3. However, James got the correct

answer by guessing 3 times:

(×) 3 lots 18 is 35

(×) 42, do I need go up

� Is it 54

• 24 × 6

Only two of the less able students (James and Tyrael) could not get an accurate solution. Top

students used advanced mental strategies such as place-value partitioning (Serena) and doubling

and halving (Nazeefah); and imagining using a standard written method such as solving the

problem vertically (Deepel). However, most students from less able group and one top student

(Vili) used early additive part-whole approaches because the most common methods they used

were repeated addition and/or forming the factors where they had a known multiplication fact.

Some examples of the difficulties facing students in solving this multiplication problem were:

1. Eli:

24 + 24 = 48 So 48 + 48 + 48

40 + 40 =80, 80 + 40 = 120

(×) 8 + 8 = 16, 16 + 8 =23, 120 + 23 = 143

� 8 + 8 = 16, 16 + 8 = 24, 120 + 24 = 144

2. James tried to use repeated addition, but he was not sure about his solution and at the end

he stopped thinking and said, “I do not know”:

20 in each basket

In 2 baskets 40, 40 + 80 +100

And count by his fingers 4 muffins for each basket

• 72 ÷ 4

Most students had difficulty working out a correct response. As Eli said, “this one is hard”. Only

3 top students could solve this problem by using advanced mental strategies such as reversibility

combined with deriving from known multiplication facts and reversibility with times table; and by

guessing, especially among less able students.

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E.g. Deepel:

20 × 4 = 80, 80 – 72 = 8, 8 = 2 × 4

20 – 2 = 18, 18 × 4 = 72

So 18 cars

Proportion and ratio:

• Which cake has been cut into thirds?

All students could identify symbols for unit fractions and observe equal sharing of objects

physically. In addition, seven students identified that cake number 4 was not cut into thirds

because the three pieces were not equal in size. However, they had trouble in solving the next

two questions.

• ¾ of 28

Only three top students were able to give an appropriate solution for this problem. They found

the fraction of the number mentally using known multiplication facts and a combination of

division facts and multiplication, for example:

Nazeefah and Deepel used the same way:

28 ÷ 4 = 7, 7 × 3 = 21

Serena:

7 × 4 = 28, 7 × 3 =21

Vili tried to guess, but he failed and said, “Is it 7 or 9?”

In contrast, some students from the less able students’ group gave no response and some gave

wrong responses. The problem with Eli was confusion between three quarters and a half:

Is ¾ a half?

½ of 28 is 14

While James said, “I am not very good at these”.

• 2/3 of what equals 12

This task was the most problematic because most students did not understand what the question

meant and took a long time to recognize what was required. Less able students were the most

confused and none of them got the right answer. Two of them gave no response and the other two

students tried to guess, but they failed to solve this problem correctly. However, all the four top

students correctly identified the number mentally by using advanced additive-early proportional

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part-whole strategies such as division and multiplication or division and repeated addition, for

example:

12 ÷ 2 = 6, 6 × 3 = 18

In addition early additive part-whole strategy such as using trial and improvement with addition

facts was used by one student (Vili): 6 + 6 = 12, 12 + 6 =18.

Basic facts

Basic facts questions were the easiest part of the interviews. All the students were able to recall

different types of the facts and the majority had the ability to remember known basic facts fluently.

Less able students except for Shirley became stuck on remembering division facts but only James

who could not recall both subtraction and multiplication facts.

Table (b) shows students’ responses to basic fact questions.

Basic facts Top students Less able students

Nazeefah Serena Deepel Vili James Eli Shirley Tyrael

Addition

6 + 9 � � � � � � � �

6 + 6 � � � � � � � �

6 + = 10 � � � � � � � �

Subtraction

17 – 9 � � � � � � � �

15 - 6 � � � � 10 � � �

Multiplication

5 × 7 � � � � 42 � � �

8 × 5 � � � � � � � �

Division

56 ÷ 7 � � � � NR NR � 12

63 ÷ 9 � � � � � NR � 50

NR = No Response

Students’ attitudes and abilities

Although only one student (Tyrael) disliked maths and most other students from both groups liked

it, or were neutral, only one top student, Nazeefah said, “Maths is easy”. In regard to preferred

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ways of solving mathematical problems, two of the top students, Nazeefah and Deeple preferred

think mentally while solving problems, whereas six students found it easier to record all the steps

on paper.

Deepel: I can figure in my head instead of on my paper

Nazeefah: If you record in a paper, it takes more time

Top students

Although Serena was the best student at thinking mentally, she found it easier to record on paper.

She was able to solve all the tasks by using different advanced mental strategies in only 15 minutes,

while when she was asked if she like maths she said, “not really” and she found some subjects in

maths easy, but others not so easy. Nazeefah and Deepel had exactly the same ability of working

out different mathematical problems with different advanced mental part-whole strategies and both

them had difficulty solving addition using decimals. Their capability was reflected in their positive

attitude to maths as both Nazeefah and Deepel like maths. Interestingly, Vili, who was ranked as a

top student by his teacher, had only a medium capacity to respond to different tasks because he

used some advanced mental strategies and some early part-whole strategies. Also he was not able

to solve addition of decimals or to find fractions of numbers.

Less able students

While Shirley was labelled as a less able student, she was nearly as good at solving different parts

of the problems as the top students and she used at least two different advanced mental strategies,

but her problem was in solving addition of decimals, division problems and fractions. A similar

situation was observed with Tyrael, but he also could not deal with division facts. Eli was not able

to solve many problems. He was good at addition and subtraction problems using two different

advanced mental strategies, and he was also able to recall basic facts except for division facts.

James had the lowest ability to solve mathematical problems and he was only able to recall addition

facts. James used counting strategies to solve mathematical tasks, but in most tasks he only guessed

and used his fingers to count

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Table (c) shows correctness and examples of misconceptions in solving problems in three

operational domains.

NR = No Response

Discussion

The results from my clinical interviews support the number framework in terms of the

relationship between strategy and knowledge sections, as students used a broad range of

known facts to support their mental strategies. They also support the views about

multiplicative thinking cited in the literature: most of the less able students did not have the

ability to think multiplicatively. This was indicated by the substantial use of repeated addition

as a very simple means of solving multiplicative problems and other mathematical problems.

Other mathematical problems that students found quite hard and challenging were solving

addition of decimals and finding a fraction of a number. They also sometimes struggled with

explaining the strategies they used. Most them preferred to record their steps in solving

mathematical problems as a useful way to remember their steps.

Interestingly, the results were not consistent with the literature in terms of mental strategies

used by students. Place-value partitioning was the most common mental strategy in the

students’ responses and most them were experts in using this strategy. This result conflicted

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with Thompson’s (2000) argument that some students have difficulty using this strategy. The

least common strategies used by the interviewees were bridging and rounding and

compensation. This finding is similar to that of Thompson (2000 & 1999), who stated that

neither of these strategies were commonly used by children. An interesting point found in the

interviews was that none of the participants used the balancing strategy to solve mathematical

problems and this finding is similar to the result found in Thompson’s (1999) investigation,

which showed only one example of the use of this strategy.

One limitation of this study was that the room provided to conduct the interviews was not

suitable because it was not quiet and the participants could see outside the room, which

affected their focus with the tasks. It may have made them want to skip answering some

questions to finish quickly. Another weakness was that I am not familiar with using

‘mathematical mental strategies’ in English and I do not have sufficient knowledge about

teaching from the maths curriculum in New Zealand schools. Thus I could not trust a student

when he claimed that he had never done addition of decimals because I did not know if he

was being truthful or if he merely did not know the answer for that question and wanted to

skip the question.

A further limitation presented when many students were able to give accurate responses

quickly, but were unable to explain their strategies, and some students used guessing to

answer some tasks. It raised the question of whether these students were already familiar with

the interview tasks or whether they genuinely had the ability to do rapid mathematical

thinking. Another question was: if the top students were given hard tasks from stage 8

problems, would they be able to solve these harder tasks? The results of this study suggest

that further research is needed to compare teachers’ ways of teaching and students learning.

Investigating whether students use what they learn from school or whether they create

different ways of learning themselves or learn from their homes, is important.

I hope that this study helps teachers to be aware of which mental strategies are most and least

commonly used by students and to recognise students’ difficulties in maths, which could help

them to reassess their ways of teaching. Teachers need to have real understanding of

mathematical thinking in order to develop their students’ mathematical thinking. Finally,

because I have found that the teaching of maths in Saudi Arabia is different compared to New

Zealand, I suggest that a study of how teachers can support students’ mathematical

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understanding needs to be investigated in Saudi Arabia comparing it to western countries

which would be a valuable addition to mathematics education research.

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References

Brown, S., Montfort, D., & Findley, K. (2007). Using interviews to identify student

misconceptions in dynamics. Proceedings of Frontiers in Education Conference, Milwaukee,

WI. Retrieved from http://fie-conference.org/fie2007/papers/1260.pdf

Cai, J. (2003). Singaporean students’ mathematical thinking in problem solving and problem

posing: An exploratory study. International Journal of Mathematical Education in

Science and Technology, 34(5), 719-737. doi: 10.1080/00207390310001595401

Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in

grades 1-5. Journal for Research in Mathematics Education, 27(1), 41-51.

Inprasitha, M., & Loipha, S. (2007). Developing student’s mathematical thinking through

lesson study in Thailand. Progress report of the APEC project: Collaborative Studies

on Innovations for Teaching and Learning Mathematics in Different Cultures (II)-

Lesson Study Focusing on Mathematical Thinking. Center for Research on

International Cooperation in Educational Development, Japan. Retrieved from

http://www.criced.tsukuba.ac.jp/math/apec/apec2007/progress_report/specialists_sessi

on/Maitree_Inprasitha_&_Suladda_Loipha.pdf

Jacob, L., & Willis, S. (2001). Recognising the difference between additive and

multiplicative thinking in young children. In J. Bobis, B. Perry, & M. Mitchelmore

(Eds.), Numeracy and Beyond (pp. 306-313). Sydney, Australia: The Mathematics

Education Research Group of Australasia.

Liu, P., & Niess, M. L. (2006). An exploratory study college students’ view of mathematical

thinking in a historical approach calculus course. Mathematical Thinking and

Learning, 8(4), 373-406.

Ministry of Education. (2008a). Book 1: The number framework. Wellington, New Zealand:

Author.

Ministry of Education. (2008b). Book 2: The diagnostic interview. Wellington, New Zealand:

Author.

Schliemann, A.D., & Carraher, D.W. (2002). The evolution of mathematical understanding:

everyday versus idealized reasoning. Developmental Review, 22(2), 242-266.

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Thompson, I. (1999). Mental calculation strategies for addition and subtraction: Part 1.

Mathematics in School, 28(5), 2-4.

Thompson, I. (2000). Mental calculation strategies for addition and subtraction: Part 2.

Mathematics in School, 29(1), 24-26.

Young-Loveridge, J., & Mills, J. (2011). Supporting students' additive thinking: The use of

equal additions for subtraction. Set: Research Information for Teachers, 1(1), 51-60.

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Appendices

Appendix A

Interview Questions

A/ Do you like mathematics?

B/ Do find mathematics easy or difficult?

C/ strategy questions:

20

21

22

23

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D/ which one do you think is easier: Using a pencil and a paper to solve problems or think

mentally?

Appendix B

Research proposal

Project Title:

Research on Students’ Mathematical Thinking

Participants

Eight students from St Andrews Middle School who are in year 8, four of whom find

mathematics is easy, while the others find it difficult.

Materials

Audio-recorder, photocopiable materials.

NDP Book 2: The diagnostic interview (Ministry of Education, 2008, p. 8-9, 30, 40-44, 47).

Procedures

Interview questions:

1/Addition and Subtraction

Question 1

There are 53 people on the bus. 26 people get off. How many people are left on the bus?

25

Question 2

Sandra has 394 stamps. She gets 79 stamps from her brother. How many stamps does she

have then?

Question 3

Marija has a 5.3 metre length of fabric. She uses 2.89 metres of it to make a tracksuit. How

much fabric has she got left?

2/Multiplication and Division

Question 1

Here is a forest of trees. There are 5 trees in each row, and there are 8 rows.

How many trees are there in the forest altogether? If I planted 15 more trees, how many rows

of 5 would I have then altogether?

Question 2

5 × 8 = 40 So 5 ×16 =

Also 3×20= 60 So 3×12=

Question 3

There are 24 muffins in each basket. How many muffins are there altogether?

Question 4

At the car factory, they need 4 wheels to make each car. How many cars could they make

with 72 wheels?

3/Proportions and Ratios

Question 1

Which of these cakes has been cut into thirds?

Question 2

What is ¾ of 28?

Question 3

12 is 2/3 of a number. What is the number?

26

4/Basic facts

Question 1 Question 2

6 + 9 6 + 6

Question 3 Question 4

17 – 9 15 – 6

Question 5

6 + = 10

Question 6 Question 7

5× 7 8 × 5

Question 8 Question 9

56 ÷ 7 63 ÷ 9

Probes:

How did you get your answer?

How did you work that out?

What made you decide to do it that way?