Teacher Notes for Activities 1 to 3 - sapgrp.com.sg · 3 Solve Non-outine eal World Mathematics Problems Workbook 5 Singapore Asia Publishers Pte Ltd. All rights reserved. eproducible

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Teacher Notes for Activities 1 to 3


Learning ObjectiveTo use spatial visualisation and logical reasoning to solve problems

Materialscubes of 4 different colours, grid paper to draw views of the shapes and nets of cubes, digital camera to photograph the shapes

FocusThese activities explore arrangements of three-dimensional shapes in order to determine how particular outcomes are formed and to investigate the patterns they exhibit. Spatial visualisation and logical thinking as well as organisation are required as students investigate all likely arrangements to ensure the final forms match the given criteria or visualise a given shape in terms of its component parts.

Possible difficulties• Unable to visualise the shapes from the two-dimensional representations• Unable to keep track of the possibilities in any of the investigations

Extension• Investigate the pattern formed by the number of small cube faces on the outside of

the growing cubes: 6, 54, …• Build a shape from cubes, then photograph or draw the front, side and top views.

Ask another student to draw a grid plan from the pictures. Make the shape from the plan and compare with the original.

Teacher Notes.indd 1 2/2/2017 2:25:11 PM


Teacher Notes for Activity 4


Learning ObjectiveTo use logical reasoning and number sense to solve problems

Materialscounters, base 10 materials

FocusThis activity explores problems based on a conceptual understanding of whole numbers and fractions as well as an understanding of logical reasoning to take into account the problem’s context. Backtracking from the final position will assist in understanding and solving the problems, but counters could also be used to keep track of what is happening. Using a diagram or calculator are other ways to sort through the information while keeping the intent of the problem in mind.

Possible difficulties• Unable to see how the half egg, 10 more eggs or 2 extra chickens fit into the problems• Not thinking that a full loaf could be made up of 2 halves, 1 half and 2 quarters or 4

quarters to see how the 12 full loaves could be bought• Simply working on the basis of calculating the numbers in the problem to obtain

incorrect answers

Extension• Ask for the number of eggs or chickens each customer bought.• Change the numbers in the problems, but leave the problem statements the same: – more eggs at the end of the first problem (it must be an odd number) – fewer eggs at the second market (55, 85, 115, … eggs altogether) – more chickens available at the end of the day or 3, 4 or 5 more each time – the fourth problem is a very famous problem and would be difficult to change!• Change the problem’s context, but leave the numbers the same.• Have students make up problems like these of their own and challenge others to

solve them using diagrams or materials.





Learning ObjectiveTo read, interpret and analyse information

Materialscalculator, number expanders

FocusThese activities explore concepts of place value and number sense. The relationships among numbers and place value are analysed and students are encouraged to not only find possibilities but to also disregard numbers and combinations that are not possible. Place value and number sense are needed, rather than addition or multiplication.

Possible difficulties• Difficulty in understanding place value• Wanting to add, subtract or multiply rather than using place value or number sense• Not using all the criteria• Not using place value to solve the questions• Not taking into consideration the starting page of each chapter

Extension• Students could think up their own bookworm problems using different criteria.• Change the criteria involving the number of pages in a book and the page number

read to solve the problems based on the new criteria.• Students could make puzzle scroll cards where they use the existing question but

change the numbers or context to create a new problem.





Learning ObjectiveTo analyse and use information in word problems

Materialscalculator

FocusThese activities explore word problems that mostly require multiplication or division. Students need to determine what the problem is asking and in many cases carry out more than one step in order to find solutions. Analysis of the problems reveals that some questions contain additional information that is not needed. If necessary, a calculator can be used to assist with the calculation as these problems are about reading for information and determining what the problem is asking rather than computation or basic facts.

Possible difficulties• Confusion over the need to carry out more than one step to arrive at a solution• Using all the numbers listed in the problems rather than just the numbers needed• Difficulty with the concept of profit

Extension• Discuss how problems can have more than one answer depending on different

interpretations.• Students could write their own problems and give them to other students to solve.• Explore how many of these problems could be solved using the repeated addition

technique on the calculator; i.e. enter ‘273 + 273’ and then hit the ‘=’ key – keep hitting the key and you will keep adding 273 without having to keep entering the number.





Learning ObjectiveTo solve problems involving time and make decisions based on particular criteria

Materialsclock

FocusThis activity explores reading for information, obtaining information from a number of sources (the information about the plane, timetable and the shuttle bus) and using it to find solutions. The problems involve thinking about and working with time. Decisions about time being too early or too late are needed rather than an exact time.

Possible difficulties• Unfamiliarity with a timetable• Confusion with 24-hour time• Including taxi and waiting time information in their travel calculations• Thinking that an exact flight is needed rather than flights that are neither too early

nor too late

Extension• Use the information and timetable with other criteria; for example, if you need to be

in Uluru for a sunset tour or midday camel ride.




Learning ObjectiveTo use strategic thinking to solve problems

Materialsgrid paper, counters in several different colours

FocusThese activities explore more complex problems in which the most difficult step is to find a way of coming to terms with the problem and what the question is asking. Using materials to assist is one way this can be done. Another is to use a diagram to assist in thinking backwards or making trials and adjusting to find a solution that matches all of the conditions.

Possible difficulties• Not using a diagram or table to come to terms with the problem conditions• Unable to see how to connect the time cycled to the distance travelled• Considering only some aspects of the puzzle scrolls

Extension• Write problems based on those in the puzzle scrolls.• Use different speeds and times for the problems on page 25.






Learning ObjectiveTo organise data and use number understanding to solve problems

Materialscalculator, 0–99 chart

FocusThese activities explore problems that rely on an ability to carefully analyse the relationships among the data and use an understanding of numbers to organise the information to keep track of the possible answers. Putting the various interrelated pieces of information into a table is a very helpful way of approaching the problems and provides a systematic way of dealing with a range of problems that have several overlapping conditions.Note: Activities 15 and 16 should be used in conjunction with each other.

Possible difficulties• Confusing the page numbers with the number of digits to provide an answer of 687• Not taking into consideration the remainders when considering multiples

Extension• Extend the problems in Activities 15 and 16 to other situations where pages are

numbered.• Provide other combinations of photos and pages for the team book problems.





Learning ObjectiveTo use patterns and logical reasoning to determine numbers in spatial arrangements

Materialscounters in two different colours, calculator

FocusThis activity explores students’ understanding of numbers in order to discern patterns that allow larger numbers to be determined without laboriously writing or counting all of the numbers up to the point asked for. It also links to the number patterns seen in square numbers.

Possible difficulties• Thinking that writing out all of the numbers is the only way to be sure of a solution• Unable to see how the odd and square numbers link to the patterns• Unable to verbalise a mathematical description of how the odd numbers and square

numbers relate to the placement of the numbers in the triangular pattern

Extension• What would happen if only even numbers were placed in the triangular pattern?• Investigate other arrangements of counters to give numbers; e.g. the sum of all the

numbers gives the triangular numbers 1, 3, 6, 10, …• Challenge students to find a relationship between the triangular and square number

patterns.• Find some background information about Pythagoras, an Ancient Greek

mathematician who was interested in number patterns as well as other topics about number and geometry.





Learning ObjectiveTo identify and use number understandings

Materialsmaterials or a calculator

FocusThese activities explore solving problems involving number sense, magic squares and logic. Analysis of the problems to locate given information is necessary to find the magic number or the arrangement of numbers. Materials or a calculator can be used to assist as these problems focus on the concepts of number sense and number logic rather than basic facts.

Possible difficulties• With the magic squares, considering only rows or columns rather than rows, columns

and diagonals• Not thinking strategically when doing the sudoku or the alphametic puzzles

Extension• Investigate other magic squares, magic numbers and alphametic puzzles.• Explore sudoku games in magazines, newspapers and on the Internet that involve

6-by-6 grids as well as 9-by-9 grids.• Try writing alphametic puzzles for other students to solve.





Learning ObjectiveTo analyse and calculate information in written numerical problems

Materialsmaterials, calculator

FocusThese activities explore word problems that require a number of operations, including division. The wording has been kept fairly simple to help with the problem-solving process. Students need to determine what the problem is asking and, in many cases, carry out more than one operation in order to find solutions. If necessary, materials can be used to assist with the calculations as these problems are about reading for information and determining what the problems are asking rather than computation or basic facts.

Possible difficulties• Confusion over the need to carry out more than one step to arrive at a solution• Using all the numbers listed in the problem, rather than just the numbers needed• Not thinking in terms of the problem and writing solutions such as 6.04 trains• Difficulty with the concepts of tonnes, discount, percent and profit

Extension• Students could write their own problems and give them to other students to solve.





Learning Objective To use strategic thinking to solve problems

Materialscalculator

FocusThis activity explores problems that may have several answers and further analysis of the data is needed to determine if this is the case or if there is a unique solution. Knowledge of simple percentage is also required. A process of ‘try and adjust’ can also be used; however, reasoning logically about the possibilities and using a table or diagram to organise them will be more productive. These ways of thinking can then be generalised to other complex problems.

Possible difficulties• Not using a table or diagram to manage the data when using ‘try and adjust’• Only keeping one condition in mind when there are two aspects to be considered• Unable to see how to change the given prices to find sums and differences in order

to get the price of two of an item

Extension:• Discuss the various methods used by students to solve the problems. Include the

examples already discussed. Ask them to solve each problem using a different method from the one they used or tried first.

• Encourage students to use an algebraic way of thinking about the relationships among the information. Some students may be able to diagrammatise this or use symbols.

• Challenge students to change the numbers in the problems while still keeping the same solutions.

• Students could also write problems using different contexts and larger numbers for others in the class to try.





Learning Objective To use spatial visualisation and measurement to solve problems

Materialspaper to make and fold squares and equilateral triangles

FocusThese activities explore ideas of perimeter and area, using knowledge of squares and triangles to visualise shapes and to determine the lengths of sides and the areas from which they are composed. Spatial and logical thinking, as well as numerical reasoning and organisation, are required as students investigate the relationships among the shapes to determine the required lengths and areas.

Possible difficulties• Unable to visualise smaller shapes within the larger shapes• Cannot see how areas and perimeters are formed from those of the smaller shapes

Extension• Have students create similar questions based on finding shapes within shapes.





Learning Objective To interpret and organise information in a series of interrelated statements and to use logical thinking to find solutions

Materialscalculator

Focus These activities explore the concepts of averages, distance and payments and require students to interpret information located within a series of interrelated statements. Students need to read the problems carefully in order to take into consideration a number of different criteria. Tables and lists can be used to help manage the various criteria.

Possible difficulties• Not using a table or list to manage data• Not understanding average• Confusion when dealing with approximate times and distances

Extension• Write other problems using the same form of complex reasoning for other students

to solve.





Learning Objective To use logical reasoning and measurement to solve problems

Materialspaper to draw diagrams or tables and record times, calculator

FocusThis activity investigates distance and time expressed as fractions of the distance around a running track. Logical thinking and organisation are needed to see how the runners progress, keep track of their positions and determine when they will coincide.

Possible difficulties• Unable to visualise the distance each person jogs or the time they take• Unable to organise the information to keep track of the various criteria• Does not coordinate the different times people take to get to the start• Unable to convert metres to kilometres and vice versa

Extension• Find out how far each of the children would jog if they trained for one hour each day.• How far would they have jogged if they began training for one hour per day, four

weeks before the cross-country meet was scheduled?• If each girl wanted to jog for one minute and finish together at the original starting

line, where would they each have to start running? • If each boy wanted to jog 2 km, how many minutes apart would they each finish?• Challenge students to come up with similar problems of their own involving other

activities.





Learning Objective To use logical reasoning and measurement to solve problems involving a balance, money or a calendar

Materialscounters, calculator, calendar

FocusThese activities explore different ways of solving problem situations and analyse the different aspects that contribute to a final solution. Logical reasoning, as well as an understanding of measurement when using money or a calendar, is required. In each situation, diagrams can be used to organise, sort and explore the data.

Possible difficulties• Thinking that each bag of nails has to be weighed one at a time• Not realising that 28 days is a multiple of 7 days• Difficulty in keeping track of numbers in the strategic thinking questions

Extension• Find some of the famous problems that use a balance to deal with different amounts

of counterfeit coins or ‘fool’s gold’.• If the café staff start work together on a Saturday, when will they next work together?• Change the roster of days the staff work.• Have students make up their own problems based on the calendar and days in a

week.



Teacher Notes for Activities 36 and 37

Teacher Notes for Activities 36 and 37

Learning Objective To use spatial visualisation, logical reasoning and an understanding of average and measurement to solve problems

Materialscalculator

Focus These activities explore different ways of solving extended problems, including the analysis of different criteria. Logical reasoning is needed, as well as an understanding of measurement (length, perimeter, area and direction). In each situation, materials, diagrams or tables can be used to organise, sort and explore the data.

Possible difficulties• Unable to visualise the storeys up and down the tall building• Errors with the cricketing averages by dealing with each score and not the totals• Unable to draw or interpret diagrams to see the mathematical relationships when

dealing with distances, perimeter, length and area

Extension• Change the position of the storeys where the people work or add further storeys and

conditions to the problem.• Students could make up their own problems involving lifts and deliveries and

challenge each other to solve them.• Students could write more complex problems involving clocks which lose or gain

time, or about the area, length and perimeter of different slopes, such as pentagons, hexagons or other polygons.





Learning Objective To analyse information and use proportional and logical reasoning to solve problems

Materialscalculator

FocusThis page explores problems that require an ability to carefully analyse the relationships among several different items of data and use an understanding of proportional reasoning and numbers to organise the information solved to keep track of the possible answers. The problems on page 79 introduce the idea of arranging the various interrelated numbers into a diagram that allows all of the overlapping conditions to be visualised and dealt with in a systematic way.

Possible difficulties• Not seeing how proportion applies to the problems on pages 75 and 76• Unable to sort out the layers of overlapping information on page 79

Extension• Use the Internet to seek out other problems using Venn diagrams.• Have students investigate the history of logic problems and the works of John Venn

and Lewis Carroll (mathematician and author of Alice in Wonderland).





Learning Objective To use logical reasoning and an ability to visualise a sequence of events so as to then use number patterns to solve problems

Materialscalculator

FocusThis activity explores problems that involve a large amount of data that needs both computation and patterning to determine solutions.

Possible difficulties• Not understanding the way in which the shells are distributed from the starting line• Only using the distances to the shells• Not attempting a solution because of the complexity of the steps or calculations

Extension• Use smaller or larger number of shells for the race and vary the distance between

shells.• Ask students to research the mathematician Karl Gauss and find out more about his

famous solution to the sum of the first 50 numbers.





Learning Objective To use spatial visualisation and logical reasoning to solve problems

Materialsdice, cubes, grid paper to draw views of the shapes and nets of cubes, digital camera to photograph the figures

FocusThese activities explore arrangements and dissections of three-dimensional figures in order to determine how particular outcomes are formed. Spatial and logical thinking as well as organisation are required as students investigate all likely arrangements to ensure the final figures match the given criteria or visualise a figure in terms of its component parts.

Possible difficulties• Unable to visualise the figures from the two-dimensional representations• Only considering cubes that can be readily seen on the outside of the figures• Unable to keep track of the possibilities in any of the investigations• Unable to visualise the opposite sides from the net of the cube

Extension• Students make stacks using different arrangements of cubes and ask other students

to investigate what would happen if all of the outside faces were painted.• Have students write their own name (if it has six letters) or a six-letter word on a

cube and then challenge a partner to draw a net of the cube.• Ask students to write a six-letter word on a net and then challenge a partner to draw

them on a representation of the cube.





Learning Objective To read, interpret and analyse information

Materialscalculator, number expanders

FocusThese activities explore concepts of place value and number sense. The relationships among numbers and place value is analysed and students are encouraged to not only find possibilities but to also disregard numbers and combinations that are not possible. Place value and number sense are needed rather than addition or multiplication.

Possible difficulties• Poor understanding of place value• Wanting to add, subtract or multiply rather than using place value or number sense• Not using all of the criteria• Not using place value to solve the questions• Not taking into consideration the starting page of each chapter

Extension• Change the criteria involving the number of pages in a book and the page number

read to and explore the problems again based on the new criteria.• Students could think up their own calculator problems and write the criteria to match.• Work out the different possibilities for two children to have three boxes, each with

different totals.





Learning Objective To interpret and organise information contained in a series of interrelated statements and to use logical thinking to find solutions

FocusThese activities explore the concepts of averages, distance, payments and involve students carefully reading interrelated statements within problem situations. Students need to read the stories carefully in order to take into consideration a number of different criteria. Tables and lists can be used to help manage the various criteria.

Possible difficulties• Not using a table or list to manage the data• Not understanding ‘average’• Confusion when dealing with approximate times and distances

Extension• Construct a table to show the running distance and how it varies from month to

month.• Write other problems, using the same form of complex reasoning, for other students

to solve.





Learning Objective To use strategic thinking to solve problems

Materialscalculator

FocusThis activity explores problems that may have several answers and further analysis of the data is needed to determine whether this is the case or if there is a unique solution. A process of ‘try and adjust’ could be used; however, logical reasoning of the possibilities and the use of a table or diagram is more productive. These ways of thinking provide a means to solve other complex problems.

Possible difficulties• Not using a table or diagram to manage the data• Not considering all of the possible answers when only information about the tyres is

given for the cars and motorcycles – there are 19 possibilities!• Only considering one condition when there are two to be considered

Extension• Discuss the various methods available to solve the problems. Include those

discussed above. Ask students to solve each problem using a different method from the one that they used or tried first. Encourage them to use a table to organise their approach.

• Change the conditions for the first problem; e.g. 650 motorcycle and truck tyres, 650 motorcycle and car tyres, different numbers of vehicles or tyres.

• With the second problem, use an odd number of trucks rather than motorcycles.• Ask the students to make problems, based on Problems 3 and 4, for other students

in the class to answer.





Learning Objective To use spatial visualisation and measurement to solve problems

FocusThese activities explore students’ ideas of area and perimeter, using their knowledge of squares and triangles to visualise shapes and to determine the areas of smaller shapes or lengths from which the squares and triangles are composed. Spatial and logical thinking, as well as numerical reasoning and organisation, are involved as students investigate the interrelationship of area and perimeter.

Possible difficulties• Uncertain of area and perimeter• May confuse area and perimeter because of a reliance on rules in place of

understanding• Does not understand that sides of a square or equilateral triangle are of equal length• Unable to visualise the smaller shapes within the large shapes• Cannot keep track of the number of sides that need to be used to determine perimeter

Extension• Have students investigate shapes made from overlapping equilateral triangles in the

same way as those made from overlapping squares.• Ask students to create their own shapes made up of squares or equilateral triangles

to determine areas and perimeters.• Investigate parks made up of paths and triangular-shaped lawns or garden beds

with different dimensions for the sides of the garden, path or triangular areas.• Investigate paths and garden beds within different shaped parks.


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Teacher Notes for Activities 1 to 3 - sapgrp.com.sg · 3 Solve Non-outine eal World Mathematics Problems Workbook 5 Singapore Asia Publishers Pte Ltd. All rights reserved. eproducible