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Number 1 Arithmetic Mathematical Ideas: Number 1 provides a revision and extension of arithmetic techniques from the primary school, and an introduction to some of the basic concepts of number theory. Because students/classes vary in their background, the initial content may have to be varied by the teacher, and teachers should use their own judgement about how much addition/subtraction/multiplication facts need to be covered. They should be mindful though, that many students leave primary school proficient in arithmetic, and not all students need to recover the basics. The scheme emphasises multiplication as an appropriate starting point, because of 1) the difficulty that many students face with multiplication techniques and 2) its applications in high school algebra (expanding brackets etc). Emphasis is also placed on building up mental pictures rather than on the use of a particular algorithm. Students should have at hand a variety of techniques for solving arithmetic problems, and be encouraged ALWAYS to check the reasonableness of their answers. The Number Theory covered (factors, multiples and primes) relies on students both understanding concepts AND learning vocabulary. For less able students, it provides many opportunities to practice the basic number links (eg multiplication/division) needed to perform arithmetic. Many students arrive in High School with a collection of misconceptions concerning place value. Some believe that the longer the decimal, the bigger it is, some the contrary (Stacey, K. & Steinle, V. (1998)). The initial lessons on decimals use the conflicts between these various misconceptions to allow students to build a more accurate visualization of decimal numbers, which is enhanced by the use of (virtual) manipulatives. Decimal arithmetic (particularly addition) is used to build this mental picture. As with decimals, it is important that students have a clear mental picture of directed numbers. Two models can be suggested; one. The one used in this scheme uses the number line and builds on students’ familiar experiences of ordering directed numbers on a line (eg a thermometer). The second model relies on the idea of cancelling +/quantities (the “IOU” model using money), and can be introduced if students have problems with the first model. Multiplication is initially seen as repeated addition; most students seem to accept multiplying two negatives as a way of completing the pattern in the times-table. Ideas covered: Arithmetic. Teachers may need to supplement the basic course which emphasises multiplication Number Theory (factors/multiples/primes) Order of operations (BEDMAS) Powers and roots Decimal Numbers and Place Value Decimal arithmetic Directed Numbers (including arithmetic with directed numbers) Curriculum Coverage: NC Arithmetic Number Theory (factors/multiples/primes) 2 Demonstrate the ability to use the multiplication facts; write and solve story problems which involve whole numbers, using addition, subtraction, multiplication, or division; write and solve story problems which require a choice of any combination of the four arithmetic operations. Recall the basic addition and subtraction facts; mentally perform calculations involving addition and subtraction; demonstrate the ability to use the multiplication facts; write and solve story problems which involve whole numbers, using addition, subtraction, multiplication, or division; write and solve story problems which require a choice of any combination of the four arithmetic operations. [Suggested Learning Experience: exploring number patterns showing multiples.] 3 Order of operations (BEDMAS) Powers and roots Decimal Numbers and Place Value. Decimal Arithmetic. Read any 3-digit whole number; explain the meaning of the digits in 2or 3-digit whole numbers; order any set of three or more whole numbers (up to 99). Demonstrate knowledge of the conventions for order of operations. Explain the meaning of the digits in any whole number; explain the meaning of the digits in decimal numbers with up to 3 decimal places; order decimals with up to 3 decimal places. Directed Numbers 4 5 6 Make sensible estimates and check the reasonableness of answers; explain satisfactory algorithms for addition, subtraction, and multiplication [Suggested Learning Experience: exploring factors of numbers by investigating rectangular (composite) numbers and line (prime) numbers] explain the meaning and evaluate powers of whole numbers; [Suggested Learning Experience: Students investigate the representation of any integer as a product of primes, and show how to use this property in determining greatest common divisors, and lowest common multiples]. express the values of square roots in approximate and exact forms. [Suggested Learning Experience: Investigate surds and other irrational numbers] Explain the meaning of negative numbers;. negative numbers, using practical activities or models if needed. Round numbers sensibly. Solve problems involving positive and Pedagogy: Because of the place of this topic at the start of Year 9, it is especially important for the teacher to build a learning community in their classroom, giving opportunities for pair and group work. Groups should be structured by the teacher for various purposes, rather than relying on friendship groupings. The department website gives some suggestions for grouping. The first lesson introduces the students to each other, and gives the teacher a chance to get to know them, and something of their mathematical experience. Alongside of the lesson activities are both an enhancement and an extension course. Both should be used by class teachers. It is suggested that teachers put aside one lesson a week to work on enhancement activities, that free students from the demands of assessment, allow them to “take risks”, and introduce them to a variety of “maths rich” environments. Focus is put on developing a “can do” attitude (Managing Self), and students are given strategies for dealing with situations when they are stuck. A variety of problem situations are introduced, and students are encouraged to believe that when they are stuck they can, maybe with scaffolding, at least make progress, even if they can’t go onto solve the problem. The scheme provides two such activities (“Sports Hall” and “Towers”). The department are beginning to integrate the use of virtual manipulatives into their teaching, as far as the available technology allows. These “objects” allow students to build up concrete mental pictures of some of the basic objects and concepts of mathematics, before they begin to abstract them into algebra later in the year. Reading: The incidence of misconceptions of decimal notation amongst students in Grades 5 to 10 Stacey, K. & Steinle, V. (1998) NUMBER 1 LESSONS Lesson 1 Support Main Lesson2 Lesson 3-4 1. 2. Go through Course Booklet, equipment, HW expectations etc. What is Maths? MindMap. Whole Class. Extract: Number, Statistics, Geometry, Algebra, Measurement. 3. The answer is 10. Put the title “The answer is 10” in the centre of the page. Divide the page (roughly) into 5 sectors (Number, Statistics etc), and in each one give examples of questions with answer of 10. Think (silence->Pair (random)->Share (2 pairs). Produce poster as a group, taking a response in turn from each person in the group. Each person may pass. LO: Expectations for the year Learning names Participation Modelling group work for future use Assessing prior knowledge For Support students, this should be seen as times table practice, or multiplication of powers of ten. LO: Estimate the answer to a multiplication problem 2. 3. SC: Every person in the room contributed an idea in class, pair and group situations. SC: Can estimate 23x47 to 20x50=1000 1. Two digit by two digit multiplication. Introduce concept of estimating by rough (informal) rounding, eg 13x47 is approx. 10x50=500. Insist that approx. is used over coming lessons. Ask for a range of methods of multiplying (ad hoc and algorithm). See attachment. Spend 2 lessons on algorithms – emphasise “box method”. “Closest Product Game”: approx. and checking skills (“last digit” check). LO: Perform multiplication using a range of different algorithms Estimate the answer to a multiplication problem SC: Can estimate, and calculate, without a calculator 23x47 As for Main, but extend with Near Misses 1, Near Misses 2 and Near Misses 3. Extension Effective teachers provide students with opportunities to work both independently and collaboratively to make sense of ideas. Support Main Extension Lesson 5-6 Lesson 7 Lesson 8-9 “Sports Hall” worksheet “I’m Stuck” Cards and poster Factor trees. Introduce “prime” as the end points. Express a number as a product of prime factors. Factor trees continued. LO: 1) uses multiplication to solve a problem 2) uses “resilience” in solving a problem (managing self) Pairs (random) Hand out sheet. Read (individually). Brainstorm “what to do when stuck”. Hand out cards. Separate into two piles – useful in this problem. Not useful in this problem. Order the useful ones. LO: Understands “factor”/”factor pair”/”prime”/”product of primes factors”. LO: Understands “factor”/”factor pair”. Can draw a factor tree. Using factor trees to derive highest common factors. “Highest Common Factor” worksheet. Definition: A prime is a number with exactly two factors. LO: Can use a factor tree to derive the highest common factor of two (or more) numbers. Lesson ender: “Fizz-Buzz” Game Lesson ender: “Fizz-Buzz” Game Start problem as whole class. As for main, but product of prime factors in index form. Managing Self Support Main Lesson 10 Lesson 11 Lesson 12 Primes. Understand the concept of a prime number. Recognise primes up to 100. Primes are learnt by experience of “playing” with primes. Discuss primes as the “ends” of factor trees. Primes are numbers with two factors. Few examples and then sieve. Then Pirate Maze (using sieve). Factors Multiples and Primes. Factors Multiples and Primes. Review vocab.: factor, factor pair, multiple, prime, prime factor, common factor, HCF, common multiple, LCM. Introduce square number. Revise vocab.: factor, factor pair, multiple, prime, prime factor, common factor, HCF, common multiple, LCM, square number. LO: Understand “prime number” Eratosthenes’s Sieve (needs 100 grid). Pirate Prime Maze from 10Ticks. SC: Can explain why 17 (for example) is prime. LO: Learns the meaning of, and distinguishes between “factor”, “multiple” and “prime” “Factors and Multiples game’. Talk about setting targets; if we can’t solve the problem, at least we can improve our score. “Starting from Scratch” is another possible problem solving Strategy. LO: Learns the meaning of, and distinguishes between “factor”, “multiple” and “prime” Complete a poster showing the vocab, with examples. Run “prime factorization” animation. Worksheet: “poster” Extension Lesson ender: “Fizz-Buzz” Game As above, plus: Is 221 prime? Explain. What is the highest prime you can find? (use prime tester.lgo to test). SC: Can name 3 factors of 12, and 3 multiples of 12. Can identify primes up to at least 20. Extension: Effective teachers are able to facilitate classroom dialogue that is focused on mathematical argumentation. Managing Self. Effective teachers shape mathematical language by modelling appropriate terms and communicating their meaning in ways that students understand. SC: can give an example of each of the words in the above list. Support Main Lesson 13-14 Lesson 15 Lesson 16 “Towers” problem Powers and Roots BEDMAS LO: Understands “powers of 2”, “powers of 3 ” etc and associated notation. Understands “power”, “exponent” and “base”. Can calculate a power on a calculator. LO: Can calculate any number to a power, using a calculator. LO: Can use BEDMAS Alpha text book Chapter 5, Powers and roots. Learn Alberta BEDMAS manipulative. Alpha text book chapter 6 Four Fours Problem (Also Decimals Diagnostic as a pretest for next lesson) SC: Can calculate 25 and (-3)4 on a calculator. Extension group can calculate √(5041) on a calculator, and (√2)2 without a calculator. SC: Can calculate (without a calculator) 2+3x5 using the BEDMAS convention. “The Towers” worksheet. Use “what to do when stuck” cards. Summarize with vocabulary and the use of a calculator. SC: can read 24, and calculate it on a calculator. Understands that it means 2x2x2x2 Roots (section 5.8) is extension only. Extension Managing Self Effective teachers carefully select tools and representations to provide support for students' thinking. Support Main Lesson 17-18 Lesson 19 Lesson 20-21 Ordering Decimal Numbers. Decimals/Place Value Adding and subtracting decimals. LO: Can order ANY group of decimal numbers. LO: Can order ANY group of decimal numbers. LO: Can add or subtract two decimal numbers. Group the class by misconception. Attempt to have a “Longer-Larger”, “Shorter-Larger” and “Task-Expert” in each group, if possible. Allow each group 5 minutes to discuss and then mark each of the homeworks, and then discuss as a class. Use decimals grid to discuss the values of a decimal number. (unit square, tenths, hundredths etc). Use decimals grid and Decimals Blocks from the National Library of Virtual Manipulatives. 10Ticks L4P4 p.29-33 Ordering decimal numbers. Repeat diagnostic Make sure that students have a clear picture before allowing the algorithm. SC: Each group can nominate a person who can explain why eg 0.56 is bigger than 0.496 and 0.437 is smaller than 0.6 SC: Whole class “task expert” Then some text book exercises on adding and subtracting decimals. SC: class can add eg 1.45 and 1.36 Multiplying and dividing decimals (emphasise the use of estimation to check reasonableness). Extension Effective teachers are able to facilitate classroom dialogue that is focused on mathematical argumentation. Effective teachers carefully select tools and representations to provide support for students' thinking. Support Main Lesson 22-24 Lesson 25 Lesson 26-27 Directed Numbers Arithmetic with directed numbers Revision LO: Can order a list of directed numbers Can add and subtract negative numbers LO: Can multiply and divide using directed numbers Can use a calculator to perform arithmetic with directed numbers LO: Can develop a plan to revise for an assessment. Introduction of negative numbers, using a temperature/thermometer model (half a lesson). Learn Alberta has a short introductory video. ‘Thermometer” is also useful. “Walk the Line”. The physical/concrete aspect of this lesson is enormously important, and should be referred to often. Use the “Multiplication table”, and complete the three quadrants (+/+, +/- and -/+) using repeated addition. Ask students to “guess’ the -/- quadrant, or (ultimate proof!) try it on their calculator. Discuss division also. Discuss (using Year 9 Revision section on website) how to revise, and sources of help and questions. Very few students will have revised for anything at primary school. Emphasise the need to do lots of questions, and allow access to lots of questions. SC: By the end of lesson 26, ALL students can name the topic/topics they intend to revise in lesson 27, and how they plan to revise. p.58-60 in Alpha text book When students are fairly confident with “walking the number line”, then text book type exercises can be introduced. Eg 10Ticks L5P4 p3-6 SC: class can calculate (both mentally and on calculator) eg 3x-4, -4x-5, 50/-5. SC: class can calculate eg 4-7, 4+-6, -3+-4, 2--8 Extension Students who have a good grasp of the concepts of directed numbers should still complete some text book exercises, but can then try the more “puzzle like” activities in 10Ticks L5P4 p7-10 Effective teachers carefully select tools and representations to provide support for students' thinking. Managing Self