Numbers - BELIZEAN STUDIES - Home



538789320500-109220000Quality Assurance and Development ServicesMinistry of Education 2020National Curriculum for Primary Mathematics Advisory Content Examples and Teaching Points STANDARD TWOTABLE OF CONTENTS TOC \o "1-1" \p " " \h \z \u Numbers PAGEREF _Toc50119821 \h 1Measurement PAGEREF _Toc50119822 \h 4Addition and Subtraction PAGEREF _Toc50119823 \h 12Data PAGEREF _Toc50119824 \h 15Multiplication and Division A PAGEREF _Toc50119825 \h 19Fractions and Decimals A PAGEREF _Toc50119826 \h 24Patterns PAGEREF _Toc50119827 \h 26Multiplication and Division B PAGEREF _Toc50119828 \h 32Fractions and Decimals B PAGEREF _Toc50119829 \h 35Fractions and Decimals C PAGEREF _Toc50119830 \h 41Geometry B PAGEREF _Toc50119831 \h 46NumbersNUMBERS NUMBERS 1.31 Identify the value of a digit based on its position in a number up to 5 digits.Place value assigns each digit in a number system a greater or lesser value depending upon where it appears in a number. Each place in a number is ten times greater than the place to its immediate right.-228601238251 x 10 = 1010 x 10 = 100100 x 10 = 10001000 x 10 = 10,000-1714-345 Although the digit 0 adds no value to a number, it can act as a placeholder. When a 0 appears to the right of at least one non-zero digit, it’s a placeholder. Placeholders are important for giving digits their proper place value. In this example, the zero in the tens column is a place value.1.32 Apply numbers up to 100,000 using numerical symbols and words to real life situations. Large numbers occur in many everyday situations such as:(i) Buying expensive objects, houses and land – “This car sells for $95,000.”(ii) Financial transactions – “The store owner deposited $15,312 into his bank account.”(iii) Distances – “It is more than fourteen thousand kilometers from Belize to Taiwan.” (iv) Social media connections – “My friend posted a video and got 22,105 likes.” (vi) Statistics – “57,169 people lived in Belize City in 2010.”Students need to practice reading large numbers fluently. They should review the use of commas in writing long numbers and understand how the commas make it easier to read them. NUMBERS1.33 Sequence a set of non-consecutive numbers in ascending and descending order up to 100,000 using the place value system.Ascending order means increasing the order of a series, sequence or a pattern. In terms of numbers, the increasing order is written from the least value to the highest value. Series and sequences follow a pattern where the numbers are written in an increasing or decreasing order, based on the common difference between the terms. Suppose, 2, 4, 6, 8, 10 is a series which is written in ascending order with a common difference of 2. We can represent it as 2 < 4 < 6 < 8 < 10.Descending order is the opposite of ascending order. That means, it is the opposite process of writing the numbers in increasing order. In the case of descending order, for a given set of numbers, the highest valued number is written first and the lowest valued number is written at last. It is denoted by the symbol ‘>’.Example:?Write 3, 7, 8, 2, 10, 28, 15 in descending order.Solution: 28 > 15 > 10 > 8 > 7 > 3 > 2 is the descending order of the given set of numbers.Sequence the following numbers in ascending order:12, 415 41, 321 4, 892 92, 1344,892 < 12, 415 < 41, 321 < 92, 134 Sequence the following numbers in descending order:23, 415 12, 321 40, 892 22, 13440,892 > 23,415 > 22, 134 > 12, 3211.34 State, read and write numbers up to 100,000 in expanded form.Expanded form is a way of writing numbers to show the value of each digit.383286025527000In expanded form: 52, 378 = 50,000 + 2,000 + 300 + 70 + 8. This follows logically from the place value system: 50,000 is the same as 5 x 10,0002,000 is the same as 2 x 1000300 is the same as 3 x 10070 is the same as 7 x 108 is the same as 8 x 1Expanded form is useful in performing mental calculations. 22 x 7 can be quickly done as (20 x 7) + (2 x 7).NUMBERS1.35 Round whole numbers up to 100,000 to specific place values.Students should be familiar with place value, before being able to round off well. The steps are:Underline the rounding place value.Look to the right of the rounding place.If the number is 5 or greater, ROUND UP by adding 1 to the underlined digit!If the number is less than 5, it stays the same. Numbers to the right of the rounding place turn to zeroes. For example if 98,732 is rounded to the nearest thousand, it becomes 99,000. The 8 is the digit in the thousands column. The number to the right of the 8 is greater than 5, hence 1 is added to the rounded digit (8); it is rounded up to 9 and all digits to the right (732) are changed to zeroes. 3671063365085Note, in this example, some steps require rounding up and some do not.Note, in this example, some steps require rounding up and some do not.One way of checking students’ understanding is to ask them to round the same number in various ways, for example:Round 52,825 to the nearest ten thousand, (50,000) Round 52,825 to the nearest thousand, (53,000)Round 52,825 to the nearest hundred, (52,800)Round 52,825 to the nearest ten, (52,830)MeasurementMEASUREMENT 8.26 Measure, compare and record the length of lines and the size of objects using metres, centimetres and millimetres.This is the first time students are introduced to metric measurements in this curriculum. In lower division, only customary measurements are used. In standard two, only metric measurements are used. At this stage, there is no requirement for students to compare the two systems or to convert from one to the other.337883537465A note on spelling:The international metric system using the following spellings: metre, litre. In the U.S.A. these words are spelt meter, liter. In the UK, the word gram is sometimes spelt gramme. For the most part, this text uses the international spellings: metre, litre & gram.0A note on spelling:The international metric system using the following spellings: metre, litre. In the U.S.A. these words are spelt meter, liter. In the UK, the word gram is sometimes spelt gramme. For the most part, this text uses the international spellings: metre, litre & gram.Students need to gain a sense of the size of each of the units. What kind of objects are about a centimetre wide? Students can measure the width of a finger, pencil eraser or staple. Most doorways are about a metre wide if the frame is included. Other objects that are about a metre wide are refrigerators and some flat screen televisions. A guitar is about one metre long. A millimetre is the smallest of the units used for this learning outcome. A social security card might be one millimetre thick. A sewing needle might be one millimetre wide. -5270510541000The key to accurate measurements is knowing how to measure accurately with a ruler. For example, to measure the length of an eraser in cm using a ruler:Line up the left edge of the eraser with the line that points to zero on the ruler. Often this is not the same as the end of the ruler. In the diagram, lining up the edge of the eraser with the end of the ruler would lead to an incorrect measurement.Look at the right edge of the eraser and line it up with the line it stops at on the ruler. In this case it stops at the 4 on the ruler. The length of the eraser is 4cm. Note the shorter lines on the ruler measure 1 millimetre. Ask students to count how many millimetres make up one centimetre.Students can measure their height in centimetres. They can measure the width of the classroom in metres. Students should learn that1 centimetre = 10 millimetres1 metre = 100 centimetres1 metre = 1000 millimetres MEASUREMENT 8.27 Compare the distances to and from various places using kilometres. A kilometre is equal to 1000 metres. Most people will walk a kilometre in 12-15 minutes. They will cycle a kilometre in 3-5 minutes.When we need to get from one place to another, we measure the distance using kilometres. The distance from one city to another or how far a plane travels can be measured using kilometres. Some road distances from Belize City are:Hattieville- 25kmPunta Gorda -270 kmBelmopan - 80kmMexico City – 1558kmCorozal Town – 138 kmLadyville – 14kmThe straight line distance from Belmopan to New York City is 2,962 km.The straight line distance from Belmopan to Taiwan is 14,298 km.8.28 Measure, compare and record the mass of various objects using kilograms and grams. Mass refers to how much matter is in an object. Students may be more familiar with the word “weight”. However, in mathematics the term mass is used. In the metric system, mass is measured using grams and kilograms.A paperclip weighs about 1 gram. Hold one small paperclip in your hand. Does that weigh a lot? No! A gram is very light. That is why you often see things measured in hundreds of grams. Grams are often written as g (for short), so "300 g" means "300 grams". 1000 grams - 1 kilogram. In the metric system, the word kilo always means 1,000.Kitchen scales that4709795-635 show grams and kilograms can be used for weighing everyday objects.A dictionary has a mass of about one kilogram. A small bag of sugar might be around 400 grams. An egg weighs around 20 grams.Some bathroom scales measure our mass using kilograms. An adult might have a mass of about 70 kg. Students can find their own mass and that of their friends.Kilograms are often written as kg (that is a "k" for "kilo" and a "g" for "gram), so "10 kg" means "10 kilograms".8.29 Measure, compare and record the capacity of a container using litre and millilitre.The volume of a container is a measure of how much the container will hold. The volume of a container that is filled with a liquid is often called its capacity. Capacity is the amount a container can hold. Liquids such as water, soft drinks, and fuel are often measured in litres. Smaller amounts of liquids are often measured in milliliters. In the metric system the prefix milli- means one thousandth. (Centi- means one hundredth)Show students several containers whose labels give the capacity in milliliters (mL).Demonstrate that there are 1,000 millilitres in 1 litre. Measure 250 millilitres of water with a measuring cup, and pour it into the 1-litre pitcher. Do this four times. Show students that the water reaches the 1-litre level. Since 4 x 250 mL = 1,000 mL, then 1 litre = 1,000 milliliters.Show students the 1-litre bottle, and remind them that the litre (L) is a metric unit of capacity. Ask volunteers to define capacity. Note that one litre of water has a mass of one kilogram. One litre of water fits into a cube measuring 10cm x 10cm x 10cm (10 x 10 x 10 = 1,000). This tells us that one millilitre of water fits into a cube measuring 1cm x 1cm x 1cm.Medicine dosages are often measured in millilitres. 1 teaspoon is about 5 millilitres. 8.30 Record temperature using a thermometer with a Celsius scale.The Celsius temperature scale is a common System Internationale (SI) temperature scale. It is based on a unit defined by assigning the temperatures of 0°C and 100°C to the freezing and boiling points of water. One degree Celsius is one hundredth of the range between these two points.-5446110031600 Students can:Record the classroom temperature every day.Record the classroom temperature at different times of the day.Move a thermometer from the shade to direct sunlight to observe what happens.In Belize, the daily weather forecast gives the air temperature in Fahrenheit so students are more likely to be familiar with this scale. On the other hand, most a/c units display the temperature in Celsius. Most thermometers available in stores have both scales. It is not necessary to convert between the two systems at this point. However, it might be useful to compare them directly, for example:Freezing point of water – 0°CBoiling point of water – 100°CRoom temperature on a warm day 26°CTemperature of a hot oven 220°CMEASUREMENT8.31 Record measures of time using minutes, seconds and hour.When students can measure the duration of events using non-standard units, they are ready to move to the use of standard units. Measuring with standard units involves the introduction of minutes, hours and seconds in addition to reading and recoding time on analogue and digital clocks. Recording and telling time enable students to develop an understanding of the size of the units of time. As the students become familiar with the size of a minute they should be given opportunities to estimate before measuring. Minutes need to be linked to the movement of the minute hand on the analogue clock and the digits on digital displays.8.32 Convert time from minutes to hours, hours to days and days to weeks.There are 60 minutes in one hour, 24 hours in one day and 7 days in one week. Converting between them requires multiplication or division.Sample problems.Convert 6 hours into minutes - 1 hour = 60 minutes. Therefore, 6 hours = 6 × 60 minutes = 360 minutes. How many days are there in 96 hours? – 1 day – 24 hours, therefore 96 hours = 96/24 hours = 4 days.Real World Examples1. A running race took Darryl 2 minutes and 43 seconds. The same race took Ian 170 seconds. Who was fastest?A: Darryl. He took 163 seconds. 2 minutes = 120 seconds, therefore 2 min 43 secs = 120+43 = 163. 2. The battery on a laptop was advertised to last 330 minutes. How long is this in hours?A: 5 ? hours. 330/60 = 5.5. This can be done by repeatedly adding 60 – 60+60+60+60+60=300 so 330/60 is 5 remainder 30. GEOMETRY7.20 Identify 2-D shapes with up to 10 sides. 2D shapes are actually flat shapes that has two dimensions, length and width. Finding shapes in real life: 542479217969100-1905179705Ask students to find out how many sides a dollar coin has. Ask them to find out the name of this shape.Many footballs are stitched together from a combination of pentagon and hexagon shaped panels. You will also see a hexagon if you look at the end of a pencil. Honeycombs in beehives are also hexagonal. Hexagons also appear on some turtle shells. Another source of 2D shapes in real life are road signs. 7.21 Identify lines of symmetry in plane figures. If in a shape or image, you draw a line down the centre and notice that the left side is a reflection of the right side then the image or shape is said to have symmetry. A good way of investigating symmetry is by folding shapes made from paper. A line of symmetry is formed if the paper can be folded so that the two halves match up. This can help students discover that shapes that have sides of equal length have more lines of symmetry than sides with different length sides.324231036703000A square has more lines of symmetry than a rectangle. An equilateral triangle has more lines of symmetry than an isosceles triangle. 4512945654050015976602413000520705143500Human faces are symmetrical. Students can see this by looking in a mirror. A small mirror can also be used to investigate symmetry in shapes by placing it on the line of symmetry and looking at the image. Care should be taken to hold the mirror perpendicular to the image to prevent distortion.Students can also look for symmetry in nature. Most animals have symmetrical bodies and many have symmetrical patterns. Many flowers and leaves are symmetrical. A fun activity is to paint one half of a piece of paper with half a butterfly shape before folding it so that the wet paint creates a symmetrical shape.232410034290004236357.22 Draw circles of various sizes using a compass. 16598906540500Sample ActivitiesPractice sketching circles freehand.Draw two more circles that are smaller than your first, without using compass.Draw a circle by drawing round an object such as a coin or bottle.Draw a circle by tying a pencil to a piece of string.Draw several circles using a compass, varying the distance between the pencil point and the compass point.Draw a series of overlapping circles.GEOMETRY7.23 Identify the centre, radius, diameter and circumference of a circle. 449897520574000The centre (or center) of a circle is the point exactly in the middle. The distance from the centre to the edge of the circle is the same in all directions.Any straight line from the centre to the edge is a radius. The plural of radius is radii. This is because the term radius originally came from the ancient Greek language. For a given circle, all radii are exactly the same length.A line drawn from a point on the edge of the circle through the centre point to the opposite edge is a diameter. It is the distance across the circle through the centre. The diameter of a circle is always twice its radius.The perimeter of a circle is called the circumference. Suggested ActivitiesUsing the compass, draw a circle with a 4 cm radius – measure the distance between the pencil and compass points before drawing.Draw a circle with a diameter of 6cm – to get a diameter of 6cm, the distance between the pencil and compass points must be 3cm.Explore real world applications of this knowledge:Bicycle tires are measured in diameter. A 26 inch tire is for a wheel with a 26 inch diameter. Adult mountain bikes have 26 inch wheels. The diameter of the wheels of bikes for children is smaller. The centre circle of a football field has a diameter of 10 yards or 9.15 metres. Many restaurants sell pizza according to diameter. The diameter of the lens of a camera affects the quality of the photo it can take. GEOMETRY7.24 Calculate the perimeter of common shapes such as triangles, squares and rectangles by adding the lengths of all sides using metric units. To calculate the perimeter of a shape, we can add the lengths of each side. Note, this learning outcome does not require the use of formulas such as P=4l14922563500The perimeter of this square is 2cm+2cm+2cm+2cmThe perimeter of the square is 8cm.-2540109432The perimeter of this rectangle is 2cm+4cm+2cm+4cmThe perimeter of the rectangle is 12cm.-317526860500This method works for all shapes, including irregular ones.The perimeter of this triangle is the sum of its three sides: 2cm+5cm+6cm.The perimeter is 13cm.-2655-346The perimeter of this shape is the sum of its sides: 3cm+1cm+3cm+4cm+2cm=13cmA tip for teaching perimeter of irregular shapes is to cross off each side length as you have added it. A common mistake for children calculating perimeter is that they miss a side out completely or count a side more than once.You can use a geo-board, palletta sticks or string to show that many different shapes can have the same perimeter.Addition and SubtractionADDITION AND SUBTRACTION3.25 Add and subtract positive numbers up to 5-digits, with and without regrouping. Regrouping during addition and subtraction was covered in standard one with smaller numbers. However, it is still advisable to practice adding and subtracting without regrouping first. The standard one toolkit shows how to use base ten blocks to teach regrouping.A strong foundational understanding of base-ten number values and the place value system is required before students can successfully add larger numbers together. Review these concepts at the beginning of the unit using place value charts, base ten blocks, number lines, ten frames, and any other hands-on or visual tools. TThThHTO73,381+25,30898,689 Addition without regrouping. Note that the sum of the digits in each column is less than 10 so no regrouping is required.TThThHTO42,974-30,63112,343 Subtraction without regrouping.Note that each number in the top row is larger than the number below it, so no regrouping is required.Regrouping in math is borrowing or carrying a digit to aid in a math operation. During addition, If the sum of the digits in one column is greater than 9, then regrouping is required. 44080599233 Teaching subtraction with regrouping strategies is most effectively done with the use of manipulatives such as base ten blocks and subtraction place value cards. It is also useful to review place value and expanded form.The following poem can help students remember the steps. More on top? No need to stop. More on the floor? Go next door. Get ten more! Numbers the same? Zeroes the game!2115709351900 3.26 Complete number sentences using mixed operations of addition and subtraction. A number sentence is an arrangement of numbers and symbols such as23 + 7 = 3017 – 4 = 13This learning outcome requires the completion of number sentences using both addition and subtraction, for example:15 + 4 – 3 = 162,911 – 11 + 100 = 3,000.Note: When both addition and subtraction operations appear in a number sentence, the operations are done in order that they appear from left to right. ge3.27 Explain the commutative property of addition.2392565400685004539961437515005393171287424005123006368588+00+4013778368589=00=3320819435322002982769400570+00+When two numbers are added, the sum is the same even if you switch the order of the numbers being added. The numbers being added are addends. That is, 1575 + 25 = 1600352488555245Imagine you are paying for something using cash. Does the order that you give the bank notes to the cashier matter? If you give a $50 note then a $10 note it is the same as if you give the $10 first, then the $50.0Imagine you are paying for something using cash. Does the order that you give the bank notes to the cashier matter? If you give a $50 note then a $10 note it is the same as if you give the $10 first, then the $50. 25 + 1575 =1600This can be demonstrated using groups of real objects. The commutative property will never apply to subtraction. 25 – 15 = 3015 – 25 = -10.3.28 Solve problems using the commutative property of addition. Solving problems using commutative property of addition allows students to arrange number to be added in an easier and more understandable way. Given, 9 + 3 +10, students can rearrange these numbers from least to greatest and add them. The answer will be the same as if they add them in the original order. In real life situation we can think of ourselves getting dressed. It doesn't matter if you put on your shirt before your pants or your pants before your shirt. It doesn't matter which we put on first, our shirt or our pants because the end result is the same. DataDATA 11.13 Represent data collected by students on a bar graph and a pictograph. A pictograph can be a time-consuming representation of data if the data are large in number. To overcome this problem, bar graphs are used. A bar chart is the simplest representation of data. It is a graphical representation of the data in the form of rectangular bars or columns of equal width.-2349535242500Pictograph VS Bar Graph Note, this learning outcome requires students to collect, organize and count data themselves. It is not fully achieved just by drawing a bar graph using data supplied by the teacher.11.14 Represent data collected by students on a dot plot.A Dot Plot displays data in a chart using dots compiled in rows or columns. Unless it has been grouped, each dot represents a single piece of data.Example: Minutes to Eat BreakfastA survey of "How long does it take you to eat breakfast?" has these results:Minutes:0123456789101112People:6235250023741Which means that 6 people take 0 minutes to eat breakfast (they probably had no breakfast!), 2 people say they only spend 1 minute having breakfast, etc.And here is the dot plot:DATA11.15 Determine the median for a given set of data with an odd number of elements.The median of a set of data is the middlemost number in the set when it is written out in a series from smallest to largest. Note this learning outcome is restricted to sets of data with an odd number of elements. An extra step is required when there is an even number of elements.To remember the definition of a median, just think of the median of a road, which is the middlemost part of the road. In the problem below, 12 is the median: it is the number that is halfway into the set. Problem: The Staine family has 5 children, aged 9, 12, 7, 16 and 13. What is the age of the middle child? Solution: Ordering the children’s ages from least to greatest, we get: 7, 9, 12, 13, 16 Answer: The age of the middle child is the middlemost number in the data set, which is 12. 11.16 Determine the range of a given set of data. The range of a data set is known as the difference between the largest and smallest values. All you have to do to find it is to arrange the set of numbers from smallest to largest and to subtract the smallest value from the largest.For example, if a data set is 7, 8, 25, 8, 4, 17 the range is found by:(a) arranging them in ascending order: 4, 7, 8, 8, 17, 25, then(b) subtracting the smallest value from the largest: 25 – 4.The range in this example is 21.One real world example of range is the difference between the highest and lowest temperature on the daily weather forecast. For example, if the highest temperature in a day is 92°F and the lowest temperature is 75°F, then the range is 92- 75=17°F. In Celsius this is 33°C - 24°C =9°C.Other examples:(1) The prices of jackets available at a store are $72, $45, $57, $89, $50.In ascending order, this data set is $45, $50, $57, $72, $89The range is the lowest value subtracted from the highest value: $89-$45 = $54. The range from the cheapest jacket to the most expensive is $54.(2) The shoe sizes of the boys in a class are 6, 8, 6?, 7, 7, 7?. In ascending order this is 6, 7, 7, 7?, 8.The range is 2. DATA11.17 Determine the mode of a given set of data.The mode of a data set is the number that occurs most frequently in the set. To easily find the mode, put the numbers in order from least to greatest and count how many times each number occurs. The number that occurs the most is the mode. To find the mode of 5, 4, 5, 4, 2, 9, 8, 5(a) rearrange numbers: in ascending order: 2, 4, 4, 5, 5, 5, 8, 9(b) identify the number that occurs the most. The mode is 5.If all numbers occur the same amount of times, there is no mode. If two numbers occur the most, then there are two modes. This data set has two modes: 2,2,4,5,7,7,9.The mode can tell us which item is the most common or most popular. For example, here is a list of the shoe sizes of the girls in a class listed in ascending order: 4, 4, 4?, 5, 5, 5, 5, 5?, 6, 8. The mode is 5 so we can say the most common shoe size in the class is 5. A store might use this information when buying stock. The store will want to have more size 5 shoes than any other size. 11.18 Identify the probability that an event will happen in a situation with a finite number of possible outcomes using the phrase “with a probability of x out of y.”Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true. A simple example is the tossing of a coin. Two outcomes, "heads" and "tails" are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is one out of two.Other ways of expressing probability, such as by fractions or percentages are covered in later grades. The focus on this unit is on the concept of probability as a way of identifying the chance or likelihood that an event will occur.136005146108What is the probability of getting a 3 with one shake of a dice?Students should shake a dice 12 times and record the results each time. They should compare these results with other students. They will find that not everyone got the same result.There is a difference between what actually happens and the probability that something will happen. It is possible to shake a dice 12 times and get a 3 every time. However, the probability of this is exceedingly low.In theory, though, there are 6 possible outcomes (1, 2, 3, 4, 5, 6) so the probability of getting a 3 is 1 out of 6. ActivityAsk the student to toss a coin 3 times and record how many “heads” they get. What is the possibility of getting 3 heads?Write down all the possible outcomes:1st Toss2nd Toss3rd Toss1HHH2HHT3HTT4HTH5THH6THT7TTH8TTT There are 8 different possible combinations. Only one of these give 3 heads, so the probability of getting 3 heads is 1 out of 8.4 of the possible combinations give 2 or more heads so the possibility of getting this outcome is 4 out of 8 or 1 out of 2.Real world examples can be explored. For example, a class has 10 boys and 15 girls. The teacher put all their names in a bag and pulls one out. What is the probability that a boy’s name will be selected?There are ten boys out of a total of 25 students so the probability is 10 out of 25. This can be reduced to 2 out of 5. Multiplication and Division AMULTIPLICATION AND DIVISION A4.17 Multiply, mentally, numbers from 0 to 12 with automaticity.Before teaching this learning outcome, review the idea that multiplication is repeated addition.3x4 = 3+3+3+3 =12 (3 is added to 3 a total of 4 times.)5x6 = 5+5+5+5+5+5 = 30 (5 is added to 5 a total of 6 times)Automaticity is the ability to perform skilled tasks quickly and effortlessly without occupying the mind with the low-level details required to do it. Automaticity is attained through learning, repetition, and practice. In math, students have attained automaticity (also known as math fact fluency) when they can easily retrieve basic facts from their long-term memory in all four operations (+, ?, ×, ÷) without conscious effort or attention. In developing automaticity, there needs to be a shift away from rote memorization of math facts and toward a strategy-based approach for learning math facts. There’s a big difference between memorizing and understanding. While we want children to develop automaticity with their facts, it is important that fluency is rooted in number sense–an understanding of how numbers are related.A few suggestions for developing automaticity:Provide concrete or pictorial support. The ability to create mental images of numbers and facts helps students make sense of numbers. Tasks that either use pictures to represent facts or have students draw representations for facts help them develop that ability to form mental images.Focus on strategies, not facts. When learning addition facts, strategies like make a 10 and Using Doubles are very powerful. Use of flash cards, playing cards, tic tac toe, dice can be great ways to develop automaticity. 4.18 Multiply two 2-digit numbers together without regrouping. Before teaching this learning outcome, teachers should review place value and expanded form.Before doing calculations on paper, students can practice mentally multiplying 2 digit by 1 digit numbers by expanding them. For example, 12x3Since 12 can be expanded to 10 + 2. 12 x 3 can be calculated as 10 x 3 + 2 x 3 = 30+6 = 36.Gradually increasing the size of the numbers increases the difficulty level. It can be practiced for a few minutes every day until it becomes automatic.23 x 5 = 20 x 5 + 3 x 5 = 100+15 = 11547 x 7 = 40 x 7 + 7 x 7 = 280 + 49 = 329 (Note 40 x 7 can be calculated as 4 x 7 x 10, which may be easier)Using pen and paper using the column method can be done without regrouping using the methods outlined below. The teacher should first review multiplying a 2-digit number by a 1-digit number in unit columns.31x3931x3=3 so a 3 is written in the ones column3x3=9 so a 9 is written in the tens column Relating this to expanded form can help students understand.31 is 3 tens and 1 one, or 30+1. Therefore 31x3 is the same as 30x3 + 1x3 – that is 90+3 = 93The problem could be done as follows31x33+9093164145599777Step 1: Multiply the numbers in the ones column – 1 x 3 = 3Step 2: Multiply the top number in the tens column (3 which represents 30) by the bottom number in the ones column (3) = 30 x 3 = 90Step 3:Add these numbers together – 3 + 90 = 93.0Step 1: Multiply the numbers in the ones column – 1 x 3 = 3Step 2: Multiply the top number in the tens column (3 which represents 30) by the bottom number in the ones column (3) = 30 x 3 = 90Step 3:Add these numbers together – 3 + 90 = 93.Multiplying two 2-digit numbers can be done with a similar process but with some additional steps. Students need to be able to explain why these steps are used.One example is as follows21x2336020400483 80390910985500Step 1: Multiply the numbers in the ones column – 3 x 1 = 31292860440055Step 3: Multiply the top number in the ones column (1) by the bottom number in the tens column (20) = 20x1 = 2000Step 3: Multiply the top number in the ones column (1) by the bottom number in the tens column (20) = 20x1 = 20726440443230008039096477000Step 2: Multiply the top number in the tens column (2 which represents 20) by the bottom number in the ones column (3) =3x20 = 60727709244475001296035220345Step 4: Multiply the top number in the tens column (20) by the bottom number in the tens column (20) = 20x20 = 40000Step 4: Multiply the top number in the tens column (20) by the bottom number in the tens column (20) = 20x20 = 40064515962230001330960196215Step 5: Add these numbers together – 3 + 60 + 20 + 400 = 483.00Step 5: Add these numbers together – 3 + 60 + 20 + 400 = 483.This can be explained using expanded form.21 is 20+1, 23 is 20+3Each element of the expanded forms of the first number is multiplied by each element of the second number. 3746500189599 21x23 can be explained as 20 x 20 = 400+ 1 x 20 = 20+ 20 x 3 = 60+ 1 x 3 = 3Total= 483Some students see this more easily in a table:X203Total2040060460120323483The same result can be achieved by using a zero as a place-holder when multiplying by the number in the tens column.21x2363420483998425100391The first step is to multiply the top number (21) with the number in the bottom row in the ones column (3). 21x3 is 63 (or 3x1=1 and 3x20=60) This is written in the first row under the line. 00The first step is to multiply the top number (21) with the number in the bottom row in the ones column (3). 21x3 is 63 (or 3x1=1 and 3x20=60) This is written in the first row under the line. 100076075564Next, the numbers in the top row are multiplied by the digit in the tens column (2 - which represents 20). In this example, 20x21 is 420. To make this easier, a zero is written in the ones column in the next row. The zero is a place-holder. Next the answers to 2x1 and 2x2 are inserted. The resulting numbers are then added (63+420=483)00Next, the numbers in the top row are multiplied by the digit in the tens column (2 - which represents 20). In this example, 20x21 is 420. To make this easier, a zero is written in the ones column in the next row. The zero is a place-holder. Next the answers to 2x1 and 2x2 are inserted. The resulting numbers are then added (63+420=483)MULTIPLICATION AND DIVISION A DIVISION4.19 Multiply two 2-digit numbers together with regrouping. When doing multiplication in unit columns, regrouping occurs when the product of two numbers being multiplied is more than 9. 15x1-32971-3333503 4750415118603You can quickly calculate 15 x 13 mentally:15 x 10 = 15015 x 3 = 45150 + 45 = 1950You can quickly calculate 15 x 13 mentally:15 x 10 = 15015 x 3 = 45150 + 45 = 1951499235120653x5=15. This means 10 need carrying over to the next column. A 1 is added to the tens column.3x5=15. This means 10 need carrying over to the next column. A 1 is added to the tens column.15x1-32971-3333503 3+1515019578859213436800199540729946A zero is added as a place-holder in the ones column.00A zero is added as a place-holder in the ones column.1210756127010002066455140166The addition is then completed, including the ten that were carried over (regrouped).00The addition is then completed, including the ten that were carried over (regrouped).132257422848000Care should be taken to start with calculations that only require regrouping once. Only after this concept is mastered, students can be introduced to calculations with more than one regrouping. Note that it is not necessary for students to perform complex calculations, like the one below, in standard two.76x353 5+302-818678510910011+180266021985831605055x6=30 so 30 is carried over. This is represented by the small 3 added to the tens column. 3x6 is 18 so a small 1 is added to the hundreds column (This is really 30x6)005x6=30 so 30 is carried over. This is represented by the small 3 added to the tens column. 3x6 is 18 so a small 1 is added to the hundreds column (This is really 30x6) 217415313164Note that the addition here also requires regrouping. In the tens column, 8+5+5=16. This is represented by the green 1 in the example.00Note that the addition here also requires regrouping. In the tens column, 8+5+5=16. This is represented by the green 1 in the example.8474396497900This can be checked using expanded form70+6 x 30 + 5= 70x30 + 70x5 + 6x30 + 6x5= 2100 + 350 + 180 + 30 Note 70x30 can be mentally calculated as 7x3x10x10 or 7x3x100= 266070x5 can be mentally calculated as 7x5x10. Students should practice this.MULTIPLICATION AND DIVISION A4.20 Multiply a whole number by a number with one decimal place. Remind students that multiplication is repeated addition. This applies if a decimal is multiplied by a whole number: 0.4x4 = 0.4+0.4+0.4+0.4 = 1.6. It also applies if a number with a decimal is multiplied by a whole number: 3.2 x 3 = 3.2+3.2+3.2 = 9.6.On paper, the unit columns method can be used.14.2x+1342. 61879310226830Note that if the multiplicand is a whole number, then the position of the decimal point remains unchanged. This is not the case when two numbers with decimals are multiplied together – for example 3.2x3.2 = 10.24. This is introduced in in standard three.00Note that if the multiplicand is a whole number, then the position of the decimal point remains unchanged. This is not the case when two numbers with decimals are multiplied together – for example 3.2x3.2 = 10.24. This is introduced in in standard three.MULTIPLICATION AND DIVISION A4.21 Explain why the commutative property applies to multiplication. The commutative property of multiplication states that you can multiply numbers in any order. In math, the commutative property of multiplication allows us to change the places of factors in a product. Multiplying numbers in a different order may make the problem easier to solve, but the end result will still be the same. For example, the problem 3 x 11 x 2 can be rearranged to multiply the smaller number first followed by the big number 3 x 2 x 11. The answer is still 66 either way. 2854718269788Students should already know that addition is commutative. Multiplication is repeated addition, so multiplication is also commutative.0Students should already know that addition is commutative. Multiplication is repeated addition, so multiplication is also commutative.For example: 3 x 4 = 12 4 x 3 = 12 4.22 Find multiples of a whole number. Multiples are the product of a whole number multiplied by a counting number. To find multiples, we can multiply a whole number by counting numbers. For example, the multiples of 3 are 3, 6, 9, 12, 15. They can also be found through repeated addition.Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32 and so on4 (4x1)4+4+4+4+4=20 (4x5) 4+4+4+4+4+4+4+4+4=36 (4x9)4+4=8 (4x2) 4+4+4+4+4+4=24 (4x6) 4+4+4+4+4+4+4+4+4+4=40 (4x10)4+4+4=12 (4x3) 4+4+4+4+4+4+4=28 (4x7) 4+4+4+4+4+4+4+4+4+4+4=44 (4x11)4+4+4+4=16 (4x4) 4+4+4+4+4+4+4+4=32 (4x8) 4+4+4+4+4+4+4+4+4+4+4+4=48 (4x12)Fractions and Decimals AFRACTIONS AND DECIMALS A 5.13 Identify equivalent fractions using pictures, number line, fraction strips or other manipulatives. Equivalent fractions often challenge students, especially when they are first introduced. Manipulatives give students a concrete way to understand this unfamiliar, abstract mathematical concept. Regular practice with manipulatives -- from student-made paper items to objects you have at home or in the classroom -- gives students a hands-on approach to understanding fractions.Manipulatives such as fraction tiles and pattern blocks may be very useful in helping students to understand equivalent fractions. 9004921905000 To show equivalent fractions on a number line, first divide the number line into equal parts as is indicated by the denominator. The numerator tells the parts represented on the number line. 2/3 is equivalent to 4/6 as shown when both number lines are divided into equal parts. FRACTIONS AND DECIMALS5.14 Find a fraction that is equivalent to another by multiplying both the numerator and the denominator by the same number.We can find equivalent fractions for any fraction simply by multiplying both the numerator and the denominator by the same number. For example: To find an equivalent fraction to ? , we can multiply by any number, as long as we multiply both top and bottom by the same number .3421621318755.15 Compare and sequence groups of proper fractions with unlike denominators. This learning outcome can be achieved by making visual comparisons using fraction strips and pictures. 397573570792Look at the fraction strips. Use <,>or = to compare the two fractions in each example.(a) 2/6 and 3/8(b) 2/6 and 1/4 (c) 4/8 and 1/2(d) 2/3 and 3/4 (d) 6/8 and 3/4 0Look at the fraction strips. Use <,>or = to compare the two fractions in each example.(a) 2/6 and 3/8(b) 2/6 and 1/4 (c) 4/8 and 1/2(d) 2/3 and 3/4 (d) 6/8 and 3/4 5.16 Add and subtract proper fractions with like denominators.Adding fractions together when they have a common denominator is easy to do, because you can just add all the numerators together! The new fraction will use the same original denominator, so all you have to worry about is adding the numbers above the line. The same is true for subtracting fractions that have common denominators.The fractions 3/5 + 2/5 = 5/5The fractions 3/8 + 4/8 = 7/8PatternsPATTERNS2.14 Create a two dimensional pattern design using only pictures.This learning outcome requires students to make designs from pictures. For guidance they might study floor coverings, curtains and clothing. The idea is to use a small number of pictures repeatedly to make a design.21668251457272.15 Explain the difference between odd and even numbers.An even number is a whole number that is a multiple of 2. An even number can be divided exactly by 2 that is without remainder. An even number can be divided exactly in half.An odd number is a whole number that cannot be divided exactly by 2. If an odd number is divided by 2, there will always be a remainder of 1.All even numbers end in one of the following: 2, 4, 6, 8, 0. All odd numbers end in one of the following: 1,3,5, 7, 9. Zero is considered to be an even number because it is a multiple of 2 (0x2=0)This concept does not apply to fractions or decimals. Strategies to teach Odd and even NumbersPlay a heads-down game of identifying odd and even. Have the students put their heads down on their desk. Give them a number, starting with something small. Have students raise their hands for odd numbers or put their hands on their heads if they think it's even. Count even objects as a class to establish a pattern. Start with even numbers. Count the items together by having students pair up the items and count them off by 2s. Ask students if they notice a pattern with the numbers. If they're having trouble understanding, you can take each pair and divide it, so you have 2 groups of items. Count each group aloud. That way, you show each even number can be divided into 2 equal groups. Ask students about odd numbers, and then prove the concept together. That is, ask students if they can guess what makes a number odd. As students get closer, count items together again, this time in groups of odd numbers. Ask students to pair the items up. Students will see that there's 1 left over each time with odd numbers. Reinforce this information by saying it aloud for them. Give students a chart of 100 numbers to color in. The chart should have a box for each number from 1 to 100. Have students color the even numbers 1 color and the odd numbers another color.PATTERNS2.16 Explain the difference between prime and composite numbers.A Prime Number is a whole number that cannot be made by multiplying other whole numbers apart from the number itself and 1.For example: 5 is a prime number. It can be made by multiplying 5 by 1 but not by any other combination of whole numbers. Prime numbers can only be exactly divided (without remainder) by one and themselves. They cannot be exactly divided by any other number.We cannot multiply other whole numbers like 2, 3 or 4 together to make 5. A composite number can be made from groups of other numbers., in other words they have more than two factors. For example, 6 can be made by 2×3 so is NOT a prime number, it is a composite number. 6 is a composite number because it has 4 factors including 1 and itself, 6. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17,19, 23, 29, 31.NumberCan be exactly divided by Can be made from Prime or Composite?1(1 is not prime or composite)21, 3Prime31, 3Prime41, 2, 42x2Composite51, 5Prime61, 2, 3, 62x3Composite71, 7Prime81, 2, 4, 82x4Composite91, 3, 93x3Composite101, 2, 5, 102x5Composite111, 11Prime121, 2, 3, 4, 6, 122x6, 3x4Composite131, 13Prime141, 2, 7, 142x7Composite151,3,5,15….3x5CompositePATTERNS2.17 Generate a series of items based on a pattern rule.Patterns are one of the most important ways to develop student's mathematical thinking. When they begin seeing patterns, sequences and order in equations, they will be able to answer problems more fluently.Students need practice with patterns involving numbers or symbols which either repeat or grow. Students need multiple opportunities creating and extending number and shape patterns. Numerical patterns allow students to reinforce facts and develop fluency with operations.A pattern rule defines a pattern. Pattern rules can be numerical or word-based. Given a particular pattern rule, students should be able to create a pattern. Examples of pattern rules are:Alternate blocks of green and blue. This generates Every third letter of the alphabet beginning with the letter h. The generates h j, m, p, s Starting from 12, each item in the series is two more than the previous one. This generates 12,14,16,18. This type of pattern rule can be expressed as a formula (x=x+2). However, since algebra is not covered in standard 2, this learning outcome can be achieved using only word based patternsSome examples of pattern rules that reinforce other knowledge in this unit are:Prime numbers in ascending order.Odd numbers in descending order from 19.A famous pattern is the Fibbonacci Sequence (pronounced Fibbonarchey). The rule is. After 3 and 5 all the rest of numbers are the sum of the two numbers before, that is 3 + 5 = 8, 5 + 8 = 13 etc… Therefore the pattern sequence will be: 3, 5, 8, 13, 21, 34, 55, 89, ...Note that this learning outcome requires students to generate the pattern from the rule – not to identify the rule from the pattern. That is the next learning outcome.2.18 Identify the pattern rule for a given pattern. This learning outcome reverses the process of the previous one. In this learning outcome, students are given the pattern and have to identify the rule.For example, what is the rule for:(a) 2,4,6,8,10 (even numbers or each number is two more than the one before it)(b) 3,6,9,12,15 (multiples of 3)(c) 1,2,4,8,16,32 (each number is double the one before it)(d) 100, 107, 114, 121, 128 (starting at 100, each number is 7 more than the one before it)PATTERNS2.19 Explore patterns for triangular numbers, multiples and factors.5842042608500A triangular number is the number of circles in an equilateral triangle evenly filled with circles. They’re called triangular numbers because they can be made into triangular shapes. Triangular numbers can be calculated as follows:11+2=31+2+3=61+2+3+4=101+2+3+4+5=15 1+2+3+4+5+6=21 and so on,Exploring Multiplication PatternsStudents can explore how multiples form patterns by shading them on a number chart.-254027178000Multiples of 3Students can be asked to describe the pattern in words. On a number chart, multiples of three make diagonal lines.They can also be asked to explain why this pattern occurs and use it to predict other multiples.Multiples of 4All multiples of 4 are even.This pattern shows that if the number of tens is even, then the multiples of 4 will end in a 0, 4, 8 (20, 24, 28, 40, 44, 48, 1020, 1024, 1028 etc). If the number of tens is odd, then the multiples end in 2 or 6 (12, 16, 92, 96). This allows us to predict whether a large number can be divided by 4. 101,354 cannot be divided by 4 but 101,356 can.Multiples of 5-46890192334Multiples of 5 make two vertical columns on the number chart. This shows us that all numbers that end in either 5 or 0 are multiples of 5.Factors can also have patterns but these may be harder to spot. One exercise is to list numbers that have half their factors as even numbers and half as odd numbers.For example, odd numbers only ever have odd number factors. The factors of 27 are 1,3,9 and 27. The factors of 51 are 1, 3, 17 and 51.2, 4, 8, 16, 32, 64 (and so on) only have even number factors if you do not include 1.Other numbers have both odd and even numbers as factors. A few of these have exactly the same number of odd and even factors. For example the factors of 6 are 1, 2, 3, 6 (1 & 3 are odd, 2 & 6 are even). The factors of 10 are 1 & 5 (odd) and 2 & 10 (even). The factors of 14 are 1 & 7 (odd) and 2 & 14 (even). The factors of 18 and 1,3 & 9 (odd) and 2, 6 and 18 (even). So the numbers that have exactly the same number of odd and even factors are 2, 6, 10, 14, 18 and so on.PATTERNS2.20 Identify missing elements of a pattern using a pattern rule.If students have understood the content of the previous learning outcomes in this unit, then they should find this one straight forward. The first step in finding a missing number in a series is to work out the pattern ruleFor example: 1, 3, 5, 7, 9, 11, 13….. each number is two more than the one before it – so to continue the pattern just add 2 each time – 15 . . .17 . . . 19.This can also apply to visual patterns. What colour will be next two circles be?-287-512Name the missing shape.-47195196768If the pattern repeats, what will be the next two vehicles to join the line?Multiplication and Division B4.23 Explore divisibility rules for division by 2, 5, and 10.Dividing single and 2-digit numbers by 2,3,4,5 and 10 was introduced in standard one. However, the basic concept of division as sharing items in equal sized groups should be reviewed. 8 ÷ 2 means splitting 8 into two equal groups of 4.For example, imagine there are 8 green-billed toucans flying over a forest.4771267441038-287-1045They divide into two equal groups, one flying south, the other flying north. How many toucans will there be in each group?305641938735Once division has been reviewed, the focus can shift to divisibility rules. These rules can tell us at a glance if one number can be divided by another. The rules for this learning outcome are:A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. These are called even numbers.A number is divisible by 5 if it ends in 0 or 5. A number is divisible by 10 if it ends in 0. For example, 56,930 is divisible by 10. Numbers that can be divided by 10 can also be divided by 2 and 5.939803873500Students can create a colour-coded number chart to show the patterns. Green boundary – divisible by 2Red number – divisible by 5Yellow shading – divisible by 10/Rice, Beans and Salad game.In this game, students count in ascending order from 1. However, if a number is divisible by 2 they say rice, if it is divisible by 5 they say beans and if it is divisible by 10 they say salad. Some numbers will be all three of rice, beans and salad. As followsNumber12345678910What is saidOneRiceThreeRiceBeansRiceSevenRiceNineRice, beans and salad4.24 Divide, mentally and with automaticity, 1-digit and 2-digit numbers by 2, 3, 4, 5, & 10, without remainders.The aim of this learning outcome is to develop students’ fluency in dividing numbers mentally.This skill can be first developed by practicing dividing real objects into groups of equal size. It can be further developed using games, flashcards and drills. Daily practice is key.Having the knowledge of the divisibility for 2, 3, 4, 5, & 10 is also key for students when dividing mentally and with automaticity. 4.25 Identify the remainder after objects are divided into groups.Review the division process through naturally occurring contexts such as a child seeing how many bananas a parent can share among the children in the family. “A parent has 12 bananas to give to her three children. If they are divided equally, how many bananas will each child get? Next use the same or a similar example to introduce the concept of remainders. If a parent has 14 bananas to share among her three children . . . The main idea is to assist students to build meaningful connections between what they know about sharing real objects to the division process. For example: 4 girls are playing marbles. They have 15 marbles in their jar. They decide to share the 15 marbles equally and to leave any remainders in the jar. How many marbles will each girl get? How many will be left in the jar?.-2540190500459168536195003167196734761762883-467000Anna1580228189230242748347973This concept can be reinforced in a PE lesson. Imagine there are 25 students in the class. Play a game that requires students to get into groups with an exact given number. Students who are unable to get in a group are the remainders.349715733937Any sets of real objects can be used to develop this concept0Any sets of real objects can be used to develop this conceptGet in groups of 3 (8 groups, 1 remainder)Get in groups of 4 (6 groups, 1 remainder) Get in groups of 5 (5 groups, No remainder)Get in groups of 6 (4 groups, 1 remainder)Get in groups of 7 (3 groups, 4 remainder)4.26 Divide 1-digit and 2-digit numbers by 2, 3, 4, 5, & 10, with remainders.This learning outcome is designed to be performed mentally. This should be done before introducing the paper-based method in the next learning outcome.4.27 Divide 2-digit numbers by 1-digit numbers, using the short form of division without carrying over within the calculation.Short division is a paper-based procedure for dividing any number by a single digit number. 4141955617006dividend - number that is being divided.divisor – number doing the dividing.In this example, 4 is the divisor, 48 is the dividend.00dividend - number that is being divided.divisor – number doing the dividing.In this example, 4 is the divisor, 48 is the dividend.Note this learning outcome is an introduction only, which is why carrying over is excluded. This means all digits of the dividend must be exactly divisible (no remainder) by the divisor. To calculate 48 ÷4 using short division, lay out the problem like this:30949556508Working from left to right, each digit in the dividend is divided by the divisor. The answer is written above the line.129605252302Step 1: 4 ÷ 4 1886778248071Divided By & Divided IntoStudents often make mistakes when reading or stating division problems. They might read 8 ÷ 4 as 4 divided by 8 – instead of 8 divided by 4 or 4 divided into 8.This tends to happen when the short division layout is used. can easily by misread as 4 divided by 48.00Divided By & Divided IntoStudents often make mistakes when reading or stating division problems. They might read 8 ÷ 4 as 4 divided by 8 – instead of 8 divided by 4 or 4 divided into 8.This tends to happen when the short division layout is used. can easily by misread as 4 divided by 48.Step 2: 8 ÷ 4 The answer is 12.Short division with carrying over and long division are covered in standard 3.Fractions and Decimals B÷5.17 Add proper fractions with unlike denominators using the area model.Note – the learning outcomes in this unit do not require students to perform paper-based calculations such as finding the lowest common denominator or cross multiplying. These techniques are covered in later grade levels. The aim of this unit is to provide students with a deep understanding of calculations with fractions using visual models. In particular, they use area models that represent fractions as parts of a rectangle divided into equal parts. If available, squared or graph paper should be used.If students are trying to add fractions with unlike denominators, start by drawing an area model for each fraction.6604014033500Step 1: Draw an area model for ? and an area model for ?.Note that both rectangles cover the same number of squares (12). The teacher will have to direct the students to draw the area models with 12 squares. 43947325528600Step 2: Divide the ? area model into thirds. The area model is nowThere are 12 squares, so the denominator will be 12.Students should know that ? is equivalent to 3/12Step 3, Add the squares from the area model for ?. Four squares are shaded in the area model for ? so four squares are added to the combined diagram.22239312642600This shows that the ? + ? = 7/12Here is another example. Note that both rectangles cover the same number of squares (15). The teacher will have to carefully select examples where this is possible.467550597155006604017843500The fifths area model is divided into thirds.This gives a total of fifteen squares – so the denominator will be 15. When the squares are added together we can see that 2/5 +1/3 =11/155.18 Subtract proper fractions with unlike denominators using area model.Using an area model is helpful in subtracting fractions with unlike denominators. To start, children need each fraction to have the same number of parts (again, a “like denominator”).They can set up area models for each fraction just as we did before to compare. Then overlay the models to find a new, common “part” and rewrite each fraction. Once they’ve done this, they can subtract the fractions easily, because they now have a common denominator. Example:What is the difference in the amount of sugar between Finn and Izzy’s drink?Finn energy drink – 1/3 Izzy small soda – 1/4422084521971000Area models:Draw a rectangle and divide it into three equal parts. Shade one partto represent one-third.Draw another rectangle and divide it into 4 equal parts. Shade one part to represent one fourth. 367538022860000 Subdivide each part of the one-third model into four equal parts.Subdivide each part of the one-fourth model into three equal parts.Each model is divided into 12 parts. Both models now have 12 equal parts (12 is the common denominator). The model representing energy drink has 4 parts shaded out of 12. The model representing small soda has 3 parts shaded out of 12. The difference between the small soda and the energy drink was 1/12 of a cup of sugar. 4/12 – 3/12 = 1/12 Find a common denominator:The difference between the small soda and the energy drink was 1/12 of a cup of sugar. 4/12 – 3/12 = 1/12 310515159672005.19 Multiply proper fractions with unlike denominators using area model.Before beginning to multiply fractions, begin by reviewing whole numbers multiplied by fractions.For example, 8 x 1/2 is asking, “What is half of 8?” In other words, the answer will only be part of the whole (8), not more. In this case, the answer is 4.The same is true when you multiply fractions by fractions. If problem says, 1/3 x 1/2 it means, “What is half of a third.548132059309000451825167035500Step 1:Work out what the length and width of the rectangle should be. The length (or height) should be the denominator of the first fraction. The width should be the denominator of the second fraction. In the problem 1/3 x 1/2 the rectangle should be 3 squares by 2 squares.Step 2: Draw an area model of 1/3 Note that diagonal lines are used for the shading. This will make the next step easier to understand.Step 3: Draw and area model of 1/2 This can be shaded in full colour in a different direction than the first diagram.Step 4: To multiply, 1/3 x 1/2 combine the two diagrams.2133602032000The shading for 1/2 is done on top of the shading for 1/3. This is a visual representation of 1/3 x 1/2. The numerator of the answer is the number of squares shaded both ways. One square is shaded with both red lines and yellow colour so the numerator will be 1.There are six squares, so the denominator will be 6.The answer is therefore 1/3 x 1/2 = 1/6. 5.20 Divide proper fractions with unlike denominators using area model.429577556642000What does it mean to divide by a fraction? This question can be investigated using whole numbers. If we ask what is 4 divided by ? we are asking how many halves are there in 4 wholes. If I have 4 oranges and divide them each in half, how many half oranges will there be? The answer is 8.1/2 ÷ 1/4 means how many quarter are there is a half. Half an orange can be divided into two quarter oranges.Note that students have not yet been introduced to mixed numbers or improper fractions in this curriculum. It is therefore better to only use examples that have an answer which is a proper fraction. In a division calculation, this means the second fraction should be larger than the first one.1/3 ÷ 1/2 = 2/3 (a proper fraction) but . 1/2 ÷ 1/3 = 3/2 (an improper fraction)Dividing Fractions Using Area Models – Method 1Step 1:Draw a rectangle with the number of rows and columns equaling the denominators of the two fractions.21971025527000The denominator of 1/4 is 4, so there are 4 columns.The denominator of 1/3 is 3, so there are 3 rows.Step 2:Shade the fraction of the rectangle that represents the dividend using diagonal lines.Shade the fraction of the rectangle that represents the divisor using full colour.127025273000Step 3: Write the number of shaded squares representing the dividend as the numerator. Write the number of shaded squares representing the divisor as the denominator. 1/4 ÷ 1/3 = 3/4This means you can fit three quarters of a third into a quarter as demonstrated if the squares are rearranged so that as of the coloured squares as possible overlay the diagonally shaded squares.122535118745Three out of the four squares overlap.This works because 1/3 and 4/12 are equivalent fractions and ? and 3/12 are equivalent fractions. The diagram is showing how much of 4/12 can fit into 3/12. Three out of the four parts can!472808745392 Ask students to find other ways of visually representing the same relationship.3683026606500Here is another example.Step 1: Draw a rectangle with 3 columns and 4 rows.Step 2:Shade 2 out of the 3 columns in diagonal linesStep 3:Shade 3 out of the 4 rows in full colourStep 4:The numerator is the number of diagonally shaded squares (8)Step 5:The denominator is the number of fully shaded squares (9)2/3 ÷ 3/4 = 8/9 Eight-ninths of three quarters fits into two-thirds.This can be demonstrated by rearranging the squares so they lie over each other.112722482408 of the 9 squares representing 3/4 fit over the 8 squares representing 2/3This works because 3/4 and 9/12 are equivalent fractions and 2/3 and 8/12 are equivalent fractions. The diagram is showing how much of 9/12 can fit into 8/12. Eight out of the nine parts can!This technique is sometimes explained with an analogy to rooms in an apartment block.Area models can also be used when a smaller fraction is divided into a larger one. Note, however, that the answer will be a whole number, an improper fraction or a mixed number. These concepts have not yet been introduced.1270-127000Step 1: Draw a rectangle with 6 columns and 3 rows.Step 2: Shade in 4 of the 6 columns. 4/6 Count the squares. There are 12.Step 3: Shade in 1 of the 3 rows. 1/3. Count the squares. There are 6.Step 4: Divide the totals 12÷6=2 27167814605000 This can also be demonstrated by rearranging the squares so the coloured ones overlay the diagonally shaded ones.Sometimes this method will lead to remainders.1333511493500 After the area model has been drawn and the squares have been rearranged, we can see that six of the squares representing a quarter fit into the squares representing four-sixths but there are two left over. Two out of the eight squares are not filled. The answer, therefore is one and two eighths.13654521704700 Fractions and Decimals CFRACTIONS AND DECIMALS C5.21 Identify the value of any digit in a number that has up to 2 decimal places using the decimal place value chart. Begin this unit by reviewing place value.Then introduce the place value chart with decimal places.0444500 The concept of place value, in base ten, is that each digit has a value ten times greater than the one to its right. This is true of numbers on both sides of the decimal point. Tenths have a value ten times greater than hundredths. Another way of looking at it is that after the decimal point, each digit has a value ten times smaller than the one before it. Hundredths are ten times smaller than tenths. Thousandths are ten time smaller than hundredths.12.57, is 1 ten, 2 ones, five tenths and seven hundredths.2.04 is 2 ones, 0 tenths and four hundredths.Note the number 12.57 is pronounced “twelve point five seven” or twelve and fifty-seven hundredths. Students should be told “twelve point fifty seven” is incorrect.2.04 can be said as “two point oh four” or “two point zero four” or two and four hundredths.5.22 State, read and write decimal numbers up to 2 decimal places in expanded form.The expanded form of a decimal number is the number written as the sum of its whole number and decimal place values. For example,: 23.76 is 20 + 3 + 0.7 + 0.06 in expanded form.100 + 3 + 0.06 is the expanded form of the number 103.06.23.76 can also be expanded as:20x10 + 6x1 + 7x0.1 + 6 x 0.01 5.23 Round-off numbers with 2 decimal places to the nearest tenth.Review the rules for rounding introduced in learning outcome 1.35.Underline the rounding place value.Look to the right of the rounding place.If the number is 5 or greater, ROUND UP! By adding 1 to the underline digit.If the number is less than 5, it stays the same. Numbers to the right of the rounding place turn to zeroes. The same steps can be followed in rounding to any decimal place.For example, to round 273.48 to the nearest tenth.The number in the tenth place is underlined. 273.48The number to the right of the tenths place is identified 273.48Since 8 is 5 or greater then the number in the tenths place is rounded up.The answer is 273.5To round 12,983.82 The number in the tenth place is underlined. 12,983.82The number to the right of the tenths place is identified 12,983.82Since 2 is less then 5, then the number is the tenths place is not changed.The answer is 12,983.85.24 Compare and sequence groups of decimal numbers up to 2 decimal places using pictures, number line, a place value chart or other manipulatives.When comparing decimals, it is helpful to write one below the other. This is shown in the examples below:Which is the greatest out of 0.57, 0.54 and 0.05Let's compare these decimals using a place-value chart.OnesDecimal pointtenthshundredths0.570.060.54If the numbers to the left of the decimal point are the same, then the next step is to compare the digits in the tens column. In this example, two numbers have a 5 in the tens column and one has a 0. The one with the zero is the smallest. It doesn’t matter what is in the hundredths column.Two numbers have the same value in the tenths column (5). However, when we look at the digits hundredths column we see one is higher than the other. 0.57 is greater than 0.54. 0.57 is fifty-seven hundredths. 0.54 is fifty-four hundredths.This can also be demonstrated with a number line.Example IIWhich is greater, 3.78 or 3.61?OnesDecimal pointtenthshundredths3.783.61First, compare the whole-number parts. Both are the same, so proceed to the next digit. Compare the tenths: 7 tenths is larger than 6 tenths. So 3.78 is larger than 3.paring decimals on number line shows the intervals between two numbers which will help students to increase the basic concept on formation of decimal numbers. On a number line 0.8 and 0.34 are both between 0 and 1. Because 0.8 is to the right of 0.34 we know that 0.8 is greater than 0.34.Pictures and fractions can also be used to compare decimals. Polygon Pizza Place caters children's parties with square-shaped pizza. Each pizza is exactly the same size and is divided into equal parts called slices.3810-381000 At Sam's party, each child had 2 out of 10 slices from a single pizza. At Elena's party, each child had 15 out of 100 slices from a single pizza. At which party did each child have more pizza?We can write a fraction to represent each party.PartyFractionDecimalSam’s 2/100.2Elena’s15/1000.15Each child got more pizza at Sam's Party.In the example, we compared two decimal numbers and found that 0.2 is greater than 0.15. Some students would argue that 0.15 is a longer decimal with more digits, and is therefore greater than 0.2. However, if we think about money, we know that 20 cents is greater than 15 cents. Thus, 0.2 > 0.15As a note, the more digits in a decimal number the smaller that number is!5.25 Identify the equivalent decimal forms of 1/4, 1/2 and 3/4.Both fractions and decimals tell us about part of a whole. There are two different ways of saying the same thing. This can be shown by comparing a fraction strip with a decimal strip.-6355067300635The shaded area in each case is the same. Five tenths are shaded in the top diagram. 0.5 is shaded in the one beneath it because 0.1+0.1+0.1+0.1+0.1 = 0.5. This shows that 5/10 is equivalent to 0.5Money can be used to teach the equivalence between ? and 0.25. There are four shillings on one dollar so a shilling is ? of a dollar. Each shilling is worth 25 cents. This is written as $0.25. So, ? of a dollar is 0.25 of a dollar. The same example works for ?. Some fractions are so common it’s worth memorizing their decimal format. The equivalent decimal for ? to fraction out of 100 is done by multiplying the numerator and the denominator by 25 to form an equivalent fraction. =1 x 25 4 x 25=25 100= 0.25-3175-635-3104-2258? is equivalent to 0.25 because they are in the same location on the number line. 5.26 Add and subtract decimal numbers, up to 2 decimal places.This learning outcome can be introduced by reviewing adding mixed numbers in which the fraction is expressed as a tenth. For example, 1 5/10 + 2 2/10 = 3 7/10. Examples that require carrying over from can also be practiced. 3 5/10 + 2 7/10 = 6 2/10. These calculations can be compared to ones using decimals.1 5/10 + 2 2/10 = 3 7/10. can be compared to 1.5 + 2.2 = 3.7.The same idea can be used with two decimal places. 3 53/100 + 2 21/100 = 5 74/100 can be compared to 3.53+2.21=5.74When asked to add or subtract decimals using the column method on paper, the most important step is to line up the decimal points.TesOnesDecimal pointtenthshundredths14.86+22.90=37.76Step 1: Line up the decimal points so that similar place values are lined up. In other words, the tens place in both numbers should be lined up, the ones place in both numbers should be lined up, etc. The decimal points will line up vertically, as will the numbers after the decimal point: the tenths and the hundredths.Step 2: Add the digits together that are lined up. If the numbers do not have the same number of digits after the decimal point, you can use "filler zeros" to help you line up the numbers. In the example: 45 + 6.98, a common mistake that is often made is that the 45 is added to the 98. However, here the 45 represents a whole number. It is 4 tens and 5 ones. The 98 represents 9 tenths and 8 hundredths. These numbers do not hold the same place values.522821051200So instead, we can add a decimal point at the end of the 45 and then add two filler zeros so that the numbers can be lined up. -53975287655 The steps for subtracting numbers with decimals are exactly the same as adding decimals. Let's take a look at a couple of examples.First we will line up the numbers by making sure the decimals are lined up. Then we just subtract the numbers that are in common place value positions.In the subtraction examples, the filler zeros are even more helpful.With the zero here, we can see that we need to borrow in order to subtract.-2540190500 So the steps to adding and subtracting are this:1.) Line up the place values of the numbers by lining up the decimals.2.) Add in filler zeros if needed.3.) Add or Subtract the numbers in the same place value positions.Geometry BGEOMETRY B7.25 Construct 3-D figures from given nets.3D Nets are used to show the correlation between 2D plane shapes and three dimensional shapes.A geometry net is a 2-dimensional shape that can be folded to form a 3-dimensional shape or a solid. Or a net is a pattern made when the surface of a three-dimensional figure is laid out flat showing each face of the figure. A solid may have different nets. Here are some steps to determine whether a net forms a solid: 1. Make sure that the solid and the net have the same number of faces and that the shapes of the faces of the solid match the shapes of the corresponding faces in the net. 2. Visualize how the net is to be folded to form the solid and make sure that all the sides fit together properly. Nets are helpful when we need to find the surface area of the solids. Here are some examples of nets of solids: Prism, Pyramid, Cylinder and Cone. 15748000035674304953000 7.26 Identify the figure of a given net.A net is a pattern made when the surface of a three-dimensional figure is laid out flat showing each face of the figure. A solid may have different nets.33664237465000 A rectangular prism or cuboid is formed by folding a net as shown: 16700566738500A cube is a three-dimensional figure with six equal square faces. There are altogether 11 possible nets for a cube as shown in the following figures. 7.27 Describe turns using quarter, half, three-quarter and full turn and 0°, 90°, 180°, 270°, 360°.21652146977900Turns and Right AnglesA quarter turn = 1 right angle? of a complete turn = 90° 20112577979900A half turn = 2 right angles? of a complete turn = 180° 311151905 The arrow points to the north (N) of the compass. 22148806096000A three-quarter turn = 3 right angles? of a complete turn = 270° 19005553746500A complete turn = 4 right anglesA complete turn = 360° 7.28 Classify an angle as acute, right, obtuse or straight based on the approximate size of the angle.An acute angle is defined as an angle that is more than 0° but less than 90° in measure. A right angle is defined as an angle that is 90° in measure. An obtuse angle is defined as an angle that is between 90° and 180° in measure. A Straight angle is 180°. ................
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