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TRANSCRIPT
Australian Curriculum Year 5 ACMNA098 Iden%fy and describe factors and mul%ples of whole numbers and use them to solve problems
Key Idea • Each coun%ng number is divisible by one and
itself, and some coun%ng numbers are also divisible by other numbers
• Mul%plica%on which has an inverse rela%onship with with division, is helpful in finding factors
Resources FISH Problem Solving kit Closing the Gap – Ladybirds Cubes / MAB units/Games h#p://nzmaths.co.nz/resource/mul4ples-‐and-‐factors) cardboard squares mul%plica%on table (for differen%ated learners) poster size s%cky notes (for living charts) or alterna%vely use cardboard IWB -‐ Promethean Planet -‐ sample flipchart images displayed below for further clarifica%on. Google images (some aRached below) -‐ this is and individual learning.
Vocabulary factors, mul%ples, prime number, divide, mul%ply, rectangle, length, width, product, %mes, lowest common, highest common Ac>vity Process-‐ Introductory Ac4vity Process The purpose of this lesson is to introduce a revision finding a factor and using mul%ples. (This will demonstrate what the students know and will determine how to proceed for the lessons ahead). Discuss the defini%ons of mul%ples and factors before your students begin this ac%vity. Make sure that they understand that every number is a factor of itself, because if they divide a number by itself, there is no remainder. For example, 12 ÷ 12 = 1 without a remainder, so 12 is a factor of 12. A prime number is a number that has only two factors, itself and 1, for example: 5, 7, 13, and 29. (Note that 1 itself is not considered to be a prime number.) Before the students play the game, ask the following ques%ons: • Imagine you threw a 4 and a 6. Which squares could you choose to cover with your counter?(a number with more than two factors, a factor of 24, a mul%ple of 2, a mul%ple of 3, a mul%ple of 4, a mul%ple of 8, an even number, or a mul%ple of 6) • Imagine you need a mul%ple of 5 to get four counters in a row. Which throws of the dice would give you a mul%ple
of 5? (1 and 5, 2 and 5, 3 and 5, 4 and 5, 5 and 5, 6 and 5, 7 and 5, 8 and 5, or 9 and 5)
(Adapted from h#p://nzmaths.co.nz /resource/mul4ples-‐and-‐factors)
Teacher background informa6on to be shared and discussed or for students to record into a maths journal/dic6onary: The factors of a number are any numbers that divide into it exactly. This includes 1 and the number itself. For example, the factors of 6 are 1, 2, 3 and 6. The factors of 8 are 1, 2, 4 and 8. For larger numbers it is some%mes easier to 'pair' the factors by wri%ng them as mul%plica%ons. For example, 24 = 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6 So the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Learners complete 3 factor and mul%ple examples-‐eg 2 is a factor of 8, 8 is a mul%ple of 2 Ac4vity Process-‐Game On This game could be extended by asking: • What are all the different products you could throw with the two game dice, one labelled 1–6 and the other 4–9? (4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 54) • There are two different ways of gegng a product of 12: throwing a 3 and a 4 or a 2 and a 6. Which other products can you throw more than one way using the game dice? (8: 1 x 8 or 2 x 4; 16: 2 x 8 or 4 x 4; 18: 2 x
their own choosing. List the factors of each number explored into their maths books. (if you have access to Signpost DVD there are some good examples that can be accessed for whole class, small group or individual work). Op6on 2: this is another lesson example which could be used to further explore factors and factor trees or used in conjunc%on/ajer op%on 1. (there some images on Google images that can help with those visual students in the class) you need to show examples Ac>vity Process-‐Using Factor Trees. A factor tree is a diagram used to break down a number by dividing it by its factors un%l all the numbers lej are prime. A prime number is a number that is only divisible by itself and the number 1. The number 2 is the only even prime number. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 … Problem -‐ Cindy wants to find the factors of 36. She knows the following %mes tables -‐ 36 = 2 x 18 36 = 4 x9 36 = 6 x6. Draw a factor tree for 36 = 2 x 18 to show the students what the representa%on will look like. Then draw factor trees that start with 36 = 4 x 9 and 36 = 6 x6 . Discuss why the ends of the tree all show 36 = 2 x 3 x 3 x 2. Now have the students repeat the process by drawing factor trees using 24 and explain that it ends with 2 x 2 x 2 x 3. Repeat the process with 30
9 or 3 x 6; 20: 4 x 5 or 5 x 4; 24: 3 x 8 or 4 x 6 or 6 x 4; 30: 5 x 6 or 6 x 5; 36: 4 x 9 or 6 x 6) • What’s the probability of throwing a double? (There are 36 possible combina%ons that can be thrown with these dice, and only 3 of these are doubles: double 4, 5, or 6. So the probability of throwing a double is or.) • Which squares in the game are easier/harder to cover? Can you use the informa%on you have about the possible products that can be thrown to explain why? (Easier to cover: a number with more than two factors [34 out of 36 possible combina%ons have more than 2 factors; only 5 and 7 don’t], an even number, a mul%ple of 2 [27 out of 36 possible combina%ons are even and are therefore also mul%ples of 2], and a mul%ple of 3 [20 out of 36 possible combina%ons]. Harder to cover: a prime number [only 2 out of 36 combina%ons] and a mul%ple of 7 [only 6 chances out of 36].) (adapted from h#p://nzmaths.co.nz/resource/mul4ples-‐and-‐factors) Ac>vity Process-‐ Choose from the 2 op%ons to introduce the concept of factors during the first couple of days of the week and then proceed onto the op%ons for ac%vi%es involving mul%ples. Op6on 1: this lesson will help the students iden%fy and describe the factors of whole numbers. Have each student use 24 cubes (or MAB units) to make a rectangular paRern. Record the number of cubes used on each side of the rectangle, eg: 4 and 6. Have students make different rectangles and discuss the results. Note that 1 x 24 = 24, 2 x 12 = 24, 3 x 8 = 24, Ask students to use the cubes to explore numbers of
Ac>vity Process-‐Introducing Mul6ples. Op6on 1 Give prac%ce at coun%ng by twos, fives, tens and fours. As we count by fours, we say (use the language) mul%ples of fours. Make mul%ples using rows of place-‐value (MAB) ones. Explain that, if 4 is a factor of 28, then 28 is a mul%ple of 4. Discuss now with the students that a mul%ple is the product of any two or more of its factors. Explain and demonstrate on the board that some numbers are mul%ples of several other numbers. For example -‐ 24 is a mul%ple of 3, 4, 6, 12 and 24. An extension ac%vity to this is located on the Signpost Maths DVD 5. Extension -‐ use a calculator to list the first ten mul%ples of 14 and 21. Encourage the students to use the constant mul%plier. i.e: 14 X X 1 = 2 = etc. answer: The first three common mul%ples of 14 and 21 are 42, 84 and 126. Discuss and review the term ‘mul%ple.’ 24 is a mul%ple of 8 because 3 x 8 = 24. Have the students present some other example they have discovered on the IWB
Op6on 2: Review the terms ‘factors’ and ‘mul%ples’ and elicit from the students what knowledge they have to these two terms. Explain that the lesson will be focussing on mul%ples of whole numbers and give some examples using some mul%ples of 5 -‐ 5, 10, 15, 25, 50, etc to start the students thinking paRern. Now move on and draw on the board a Concept box (arrays) -‐ explain that this is another way of represen%ng and loca%ng factors and mul%ples of numbers. Use the example of 3 x 8 = 24 so three rows of eight = 24 and draw the box to show the paRern (especially for visual learners) and keep repea%ng a line of eight to introduce the next mul%ple. You might like to start with a smaller number depending on whiteboard size or alterna%vely give the students some grid paper and allow them to copy and keep mul%plying. This is beneficial for differen%ated learners. For these differen%ated learners you may like to give them a copy of a mul%plica%on table to help them complete the work. Prac%ce with lots of examples (consolida%on) by breaking students up into small groups and have them write onto large s%cky notes (poster size) all the mul%ples for a given number up to 100/150. For example -‐ one group might be doing mul%ples of 3’s, another mul%ples of 8, etc. Spend some %me at the end of the lesson sharing these and display as living charts around the classroom. Op6on 3: Extend the previous ac%vi%es by introducing the terms of lowest and highest common factors and mul%ples. Revise with students what is a factor and what is a mul%ple. Place some examples on the board and look at what is the lowest common factor -‐ explain that all whole numbers have 1 and itself as factors but you want to find the lowest common factor other than 1. For example -‐ 12 and 15 the lowest common factor would be 3. Examples of board work below. Also instruct and look at finding the lowest common mul%ple of two numbers eg: 12 and 24 would be 48 and the highest common mul%ple between 100 and 200.
Extensions and Varia>ons Ac>vi>es • Ask students use a calculator to find the product of any two numbers less than 20. They can ask a partner to use the calculator to find the factors of a product?
This could also be repeated for mul%ples.
• Ask students solve problems using whole numbers. They will need to iden%fy and describe the factors and mul%ples of whole numbers to solve these problems.
• Ask your high level learners make A4 size (living charts if needed) charts dealing with mul%ples above 150 etc. This can also be done for factors.
• Have students work with calculators to mul%ply larger numbers eg: 24 x 25 using factors and mul%ples -‐ 24 is 4 x 6 x 25 or 2 x 12 x 25, or (2 x 3) x ( 4 x 25 ) etc.
• Factor crossword
Assessment-‐ By the end of Year 5, students solve simple problems involving the four opera%ons using a range of strategies. They check the reasonableness of answers using es%ma%on and rounding. Students iden%fy and describe factors and mul%ples. They explain plans for simple budgets. Students connect three-‐dimensional objects with their two-‐dimensional representa%ons. They describe transforma%ons of two-‐dimensional shapes and iden%fy line and rota%onal symmetry. Students compare and interpret different data sets. Students order decimals and unit frac%ons and locate them on number lines. They add and subtract frac%ons with the same denominator. Students con%nue paRerns by adding and subtrac%ng frac%ons and decimals. They find unknown quan%%es in number sentences. They use appropriate units of measurement for length, area, volume, capacity and mass, and calculate perimeter and area of rectangles. They convert between 12 and 24 hour %me. Students use a grid reference system to locate landmarks. They measure and construct different angles. Students list outcomes of chance experiments with equally likely outcomes and assign probabili%es between 0 and 1. Students pose ques%ons to gather data, and construct data displays appropriate for the data. Independent Assessment-‐Students choose one op6on to complete Op%on 1 A furniture shop sells chairs in sets of four. Which of the following numbers of chairs would make complete sets. 10, 12, 15, 20, 22, 28, 35? Jus%fy
Complete a Factor Crossword Across 1. Mul%ples of 7 3. Mul%ples of 5 5. Factor of 24 6. Has factors of 2 and 5 9. Factor of 42 Down 1. Number %mes 10 divides by 5 2. Mul%ple of 10 4. Factor of 27 6. Factor of 27 8. Mul%ple of 2 Digital Learning – The following digital resources can be used for reinforcement or consolida%on ac%vi%es at teachers discre%on Promethean Planet has a number of various flipcharts available for the IWB and below are some screenshots with the names of some of the flipcharts available.
Inves>ga>on: You have been asked to create a display of field trip photos for a class webpage. Create a diagram of your various layout op%ons for arranging them on the screen. Select one and jus%fy why it would best fit suit the screen?
your answer using a green fish example Op%on 2 A gardener has 12 flowering plants that he wants to plant in equal rows at the front entrance to the garden. What are all the ways he can arrange the flowers in equal rows? Jus%fy your answer using a green fish example Op%on 3 A concert organiser has 200 seats he needs to arrange in rows with access aisles on the sides and one in the middle. What are his possible op%ons for moving concert aRendees quickly and safely in and out of their seats. Jus%fy your choice of most efficient sea%ng arrangement you would recommend he use. Jus%fy your answer using a green fish example
Background-‐Learners need to be able to break down a number into all its factors.
Students need to be able to solve number problems that they will come across in everyday life. In order to achieve this they need to have a knowledge bank of number facts which will allow them to develop effec%ve methods of calcula%on. To do this they need to create, use and explore number paRerns in a wide range of ac%vi%es which include prac%cal, mental and formal work in different contexts. For example purchasing mul%ple items when shopping or planning a celebra%on etc.
Was the
learne
r able to Above Expected Below
Iden%fy appropriate mathema%cal concepts, skills and strategies necessary to prove a solu%on
Use skills and strategies that lead to a simple solu%on
Use strategies that lead to an unsuccessful solu%on
Demonstrate the difference between factors and mul%ples
Use simple factors and mul%ples Unsucessfully use factors and mul%ples
Able to use calculator func%ons effec%vely to check answers
Able to use calculator independently to check answers
Unable to use calculator independently to check answers