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Is it the same? Curriculum Pack 5MIDDLE YEARS

Equation creationEquation manipulationEquation solutions

Remote Schools Curriculum and Assessment Materials

Starting Smart Is it the same?

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MATHEMATICS RESOURCE FOR TEACHERS

Starting Smart ~ Term at a Glance 10 weeks At a Glance to help teacher’s develop and extend student’s understanding of

chance and data and number. Incorporates learning journey, assessment contexts, strategies, resources and connections

Starting Smart ~ What knowledge of data collection and analysis do my student’s already have and can demonstrate?

Includes 10 complete lesson plans with resources and assessment tools to help teachers get started.


Starting Smart succinctly and clearly describes opportunities for seeking and interpreting evidence of learning that teachers and students can use to determine:

Where students are in their learningWhere they need and want to beHow best to get there

Starting Smart provides a model of best practice for teachers to:

Plan for assessmentObserve learningAnalyse and interpret evidence of learning to determine a pathway that connects what learners already know, understand and can doProvide feedback opportunities and assist them to set personal learning goals


Background informationAlgebra is the area of mathematics that uses letters and symbols to represent numbers, points and other objects, as well as the relationships between them. Its value is that it enables general statements to be made simply, it enables a wide range of problems to be solved efficiently and it even enables some problems to be solved that can’t yet be solved in any other way.

Developmental stagesAccording to Wright1 there are five stages in the development of Algebraic thinking and processing.

Stage 1 Stage 2 Stage 3 Stage 4 Stage 5Copy a pattern and

create the next elementPredict relationship values by continuing

the pattern with systematic counting

Predict relationship values using recursive methods e.g. table of

values, numeric expression

Predict relationship values using direct rules

e.g. ? x 3 + 1

Express a relationship using algebraic symbols with

structural understanding e.g. m = 6f + 2

or m = 8 + 6(f – 1)

Level 1: copy a pattern and create the next element.In the initial stages students need to spend a great deal of time exploring the idea of pattern. Students are introduced to the concept of a pattern being something that is repeated. They learn to add the next object or number to a sequence in order to extend the pattern; they learn to see that there is a limitless variety of patterns; they learn to invent their own patterns; and they begin to be able to describe what the next object in the pattern will be. An important idea here is the role of ordinal number in the development of relations with spatial patterns.Level 2: continue a pattern using systematic countingNow number takes a more predominant part in pattern. Patterns have a first term; a fifth term; a tenth term and patterns are made with numbers. Students begin to realise that they can predict what number is going to come next. This may be done by recognising the number pattern or by counting on by the number required by the pattern.Level 3: predict values using relationships between successive termsSequences involving numbers now become more complicated where maybe a table of values becomes necessary. As well as understanding what is happening, students can describe in words the rules for given sequences and can create and use their own recursive rules. Finding the number that comes before a given number in a sequence will lead students to develop the skills needed to solve equations.Level 4: predict values using rulesStudents are now able to predict values of terms in a sequence using the direct rules involved. They can also produce the rules of a given number pattern and use these rules to predict future terms. Students start with linear relationships and develop their skills to work with quadratic and exponential sequences.Level 5: find and use an algebraic expression for a relationThe fifth step in Wright’s developmental sequence shows the beginnings of “real” algebra. This stage involves the generalisation of processes that are now well understood. For instance, in solving 2 n + 7 = 35, they should

know the idea of n as a variable that can take a particular value when constrained understand the equality relation implied by the equals sign ‘undo’ the + 7 by subtracting 7 from both sides ‘undo’ the 2n by dividing both sides by 2

The number foundation for AlgebraStudents moving forward into the study of Algebra need a sound knowledge of the properties of number and of the four basic operations.

Equality: One of the first difficulties that students have with algebra may arise because of an incomplete knowledge of what the equals sign signifies. When they initially meet equations they are learning arithmetic and come across equations like 7 + 9 = 16. Here the equals sign indicates that an answer has to be found, so they come to think of the equals sign as meaning ‘find the answer’, rather than “is equal to”.


In algebra they may see equations like 3(x – 1) = 3x – 3. Here the right hand side doesn’t look like an answer. In fact we have a statement about equality. When we want to, we can replace one side of the equation by the other, just like we can use 7 + 9 instead of 16. Students need to know that equals signs represent equalities.

Operations: The answers to a number of problems can be found by using different operations. Difficulties arise for students who are only able to see ‘multiplication’ in terms of repeated addition. Multiplication needs to be understood as an operation in its own right. At the same time, children need to know the properties of multiplication, especially the fact that the order of multiplication is unimportant.

In the same way, division needs to be understood as an operation in its own right and not just as repeated subtraction. Knowing the properties of division, that is the order of division is important (12 / 3 is not equal to 12 / 3) and division is the inverse of multiplication, is vital as well.

The interaction of operations is also important. Just as there is an inverse relationship between multiplication and division, there is a similar relationship between addition and subtraction. Of more importance however is the hierarchy of the operations in an expression. Children need to know how to calculate 5 + 4 x 7 / 3 and they need to practise numerous examples of this type in order to use algebraic expressions with confidence.

Range of Numbers: Many students go through the early years of school only having experienced relatively small whole numbers. They are not sure whether or not the basic operations apply equally as well to large numbers, fractions and decimals. As part of your planning, ensure that they regularly come across problems that involve numbers in the hundreds and thousands, fractions and decimals.Much of the skills involved in the calculation of unknown quantities in equations are contained in the Number Strand. This information won’t always be explicitly written in this unit of work. Where number concepts are needed, these should be taught explicitly, and the lesson will be flagged for your attention.

Adapted fromNZ Ministry of Education, Algebra information http://www.nzmaths.co.nz/algebra-informationWright V, 1998, The Learning and Teaching of Algebra: Patterns, Problems and Possibilities in Exploring Issues in Mathematics Education, Proceedings of a Research Seminar on Mathematics Education, Wellington: Ministry of Education.

NTCF connections Mathematics:

Algebra – Relationships – (Band 2- Band 5) Number – Number and Number Systems/ Calculating (Band 2-Band 5)

ESL: Speaking – Communication (Secondary: Level 2-Level 5) Writing – Communication (Secondary: Level 2 -Level 5)

Program area connections Physical fitness –How can physical fitness improvements be evaluated? PE program –How can I improve my athletic ability?

Learning technology: Operating computer components 2

EsseNTial Learnings: Creative learner 3 Constructive learner 2

What evidence do I gather? What the students know, do and understand about; real world problems that involve finding an

unknown expressed as an equation, recognising equivalences, balancing and solving equations and using number, number relationships and calculation strategies.

http://www.nzmaths.co.nz/algebra-information

A rich ongoing mathematics program

Teachers consolidate student’s learning by providing opportunities to practise and revise concepts, knowledge, skills and understandings identified during the explicit teaching of the previous curriculum pack.

Teachers plan informal learning experiences through daily lesson routines to provide a context for developing the concepts, skills and language required in the following term.

Teachers plan, teach, monitor, assess and report on the new learning described in the focus curriculum pack, through targeted strategies and learning design made explicit in daily lesson routines.To fully realise a numeracy perspective in the teaching and learning of maths, it is necessary to provide opportunities for students to engage in mathematics in ways that enable them to recognise when maths might assist to interpret information and then apply their knowledge to solve practical problems.

Goal setting is an essential component as it helps to visualize and plan actions, serve as a guide in making decisions and enables students to evaluate their progress more clearly. Reflecting on goals helps students to be aware of their learning and reveals students' thoughts on their progress. Reflection also gives the teacher feedback about how students learn.


Specific English language vocabulary

algebra inequation expression less than decimals constant

algebraic equality inverse more than percentages symbolformula equation backtracking operations brackets pronumeral

formulae variable substitute whole numbers parentheses unknowninequality linear substitution rational numbers simplify coefficient

A rich mathematics programThe following model outlines a process for teachers to plan a rich ongoing program. This program should consist of strategies and activities that consolidate and build upon students’ existing knowledge, skills and understandings while explicitly teaching the required new learning and preparing students for the learning focus of the following term’s curriculum. It incorporates a learning design approach of daily lesson routines and evidence based instructional strategies and practices.

A rigorous and rich mathematics routine should include a range of components and instructional procedures at the whole class, small group and individual level to move students from what they already know, understand and can do to where they need and want to be. These include the following scaffolded, modelled, shared, guided and independent learning experiences, processes and strategies:

Goal setting and reflection – students set goals for their learning and monitor their progress through reflection

Hands-on problem solving – students learn to think and work like a mathematician Skill development – students explore, develop, select and apply a range of skills in a range of contexts to

solve problems Strategy development – students develop a bank of strategies to use when required to solve problems Focus on Process – students discover, practise and refine processes that will enable them to solve problems Concept development – students have many opportunities to explore, practice and revisit concepts over

time

Mathematics Lesson Sequence

E S C


Adapted from the Australian Curriculum Unit/ Sequence of Work Mathematics template

Mathematics Observation Profile

The profile (Observation_profile.xlsx) is designed as a recording sheet for monitoring an individual student’s progress throughout the school year. NTCF Band level Declarative and Procedural Knowledge requirements suggest tasks and questions that can be used for on-going and summative assessment.

Directions for use: The five strands for Mathematics with their specific Indicators, (Declarative knowledge) are listed in this profile booklet. Three columns, (for each year of the teaching cycle) are provided to record a student’s performance level, as a sliding scale (Emerging – Solid – Comprehensive).

Observations for achievement in the strands of Algebra, Space, Measurement, Chance and Data will be recorded over the term of their focus, whereas achievement in Number will be

recorded over the year. As the students progress to the next phase of learning, their overall achievement level is carried forward.

For example: It is suggested that teachers record an evaluation (achievement level) for the observed Indicator during each term. It is not necessary to record an achievement level for Indicators that have not been addressed. Student work, conversations with the students and observations provide evidence for assessment. Assessment is based on the student’s ability to explain, model and apply learning. Student portfolios will support the assessment.

Tuni

ng in/

Daily rituals

Whole class, 5-10 minutesShort, energetic activities (usually completed mentally/ orally) which consolidate/ review knowledge previously introduced in the mathematics teaching and learning program and precede the explicit teaching component of the main planned learning experience.

Explicit teachi

ng

Whole class 10 – 15 minutesTeacher directed introduction to a particular concept with an explicit statement about what knowledge and skills are to be developed is provided. This element might include a; short task, problem to solve, discussion, explanation or demonstration of possible strategies, approaches or processes. Performance standards and expectations are made explicit to students.

Develop

mental Activities

Group/ partner/ individual 30 – 40 minutesGroups of students are told of the learning activities which will cater for different learning styles. Students may be allocated to groups and concrete materials or visual models could be provided. Students are encouraged to follow the Do-Talk-Record-Reflect learning model where activities have a problem solving focus, correct use of mathematical language is emphasised. One group may receive further explicit teaching while other groups may work independently or supported by the teaching team.

Reflection

Whole class 5 - 10 minutesStudents or groups might provide feedback about the learning activity; discuss strategies and solutions using correct terminology. The reflection time also offers opportunities for teachers to clarify student learning and progress towards short and long term goals, and highlights any misconceptions.

Is it the same? Curriculum Pack 5MIDDLE YEARSMathematics ~ Is it the same?

Big Idea – Real world problems that involve finding an unknown (such as purchasing, finance, painting, fencing, packaging, gardening, building and design) can be expressed as an equation. Equations provide a shorthand way of showing the information with equivalent expressions on both sides of an equal sign. Recognising equivalences helps us to balance an equation and keep it true by applying different methods to find the unknown value. Balancing and solving equations allows us to apply and use our understandings of number,

number relationships and systems; calculation strategies and methods as well as objects and shapes; physical attributes and graduated scales.

Wk LEARNING JOURNEY ASSESSMENT CONTEXT

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What do the learners already know, can do and understand about number, manipulation of unknown quantities and algebraic thinking?

Activate prior knowledge – lessons 2-8What do we know about equals, equations and algebra? What number relationships do I know? What can I do with numbers and grouping symbols?What can I do with numbers and symbols?

Assessment for Learning Activities – lessons 1 and 9Tables and chairsBroken calculator

Student reflection on learning Journal writing

Use interactive websites, and familiar one or two step problems to revise order of operations, equivalence, equations and recall mathematical words and phrases.

Use regular discussions, journals and (Pack link) to reflect and communicate mathematical thinking and learning.

To cater for all learners;Consider

Explicit teaching Modelling Cooperative and

independent learning

Links to home/ community

Links to learner values

Reports/ presentations

Short, rich tasks

Assessment for LearningConsider

Diagnostic tests Open-ended tasks Problem solving Concept maps Graphic organisers Journals Rubrics Self-assessment

sheets

NOW WHAT? Teachers use band/level information collated from the assessment of learner’s knowledge and skills to determine future planning.In each learning experience planned, use the process of finding missing values, equivalence and solving equations to demonstrate the interconnected understandings of number, space and measurement.

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10

How can I evaluate algebraic expressions and use the order of operations?

How do I compare integers using greater than or less than?

Select a range of equations and inequations (such as 14-6x212+2x8) to apply order of operations calculations and to determine the missing symbol (=,<,>).

How can I evaluate expressions using parentheses and write equivalent expressions using the properties of algebra?

How can I write algebraic expressions involving one operation?

How can I translate phrases to algebraic expressions and solve problems by writing and solving equations?

Given two variables, r and y, with values of 6 and 4 respectively, pupils should be asked to make up algebraic instructions with a value of 12. For example: 6(r - y), 2r, 3y, 4r – 3y

Magic Tricks: Think of a number. Add 4; now add the number you were first thinking of; Divide by 2; subtract 2. Add 5 etc. Relate this to an algebraic expression

2 cans of coke and 3 mars bars cost $2.15, 1 can of coke and 5 mars bars costs $2.30. Calculate the cost of a can of coke and the cost of a mars bar.

s=number of sweets in a boxo If I add 2 how many sweets will I have?o If I add 4, how many sweets will I have?o How many sweets in 2 boxes?o How many sweets in 3 boxes, then I eat 7?

Use familiar real life problems in finance, packaging, gardening etc to construct algebraic equations with an unknown value that is represented by a pro-numeral.

How can I use the distributive property to write equivalent expressions, factor, and collect like terms?

How can I solve for a given variable in a formula?

Add and subtract like terms, expand or factorise expressions to simplify and solve a range of equations with one unknown value.

How can I use the addition property of equality to solve equations/ inequations?

How can I solve equations/ inequations using the multiplication property of equality?

How can I apply both addition and multiplication properties together to solve equations/ inequations?

How can I solve equations/ inequations by moving all variables to the same side of the equals sign?

How can I solve equations/ inequations that contain parentheses?

Solve equations through substitution: If 3X + 4 = 2

If X = 3 13 = 25 If X = 4 16 = 25 Getting closer If X = 5 19 = 25 If X = 6 22 = 25

If X = 7 25 = 25 Which is correct

Use a variety of strategies (such as guess & check, backtracking or balancing) to solve simple one and two step equations

(such as x−3

2=4 , becomes

x−3=4×2 ,then x=8+3 , so x=11)

How can I determine whether a number is a solution of an equation/ inequation?

If X = 4 then what are the values of: 3X 10 – 3X 2(X – 1) 4 + 2(X + 3) 3X + 4 – X 4X + 2

Work backwards to check solutions by substituting back into the original

equation (in the example 11−3

2=4).


Lesson Learning Goal Mathematics Indicators

Lesson 1

What do we know about equals, equations and

algebra?

Band 3 complete equations involving all four operations, fractions and decimals

where one elements is missingBand 4 create and manipulate algebraic expressions and equations involving

the four operations in order to determine unknown values and solve problems

Lesson 2What number

relationships do I know?

Band 3 use a range of mental or written strategies to add and subtract

numbers of any size including decimal fractionsBand 4 use efficient mental, written and technological strategies to calculate

using all 4 operations with integers, fractions, decimals and mixed numbers

Lesson 3This time should be spent teaching an explicit lesson on mental calculation strategies, number facts, operations with whole numbers, fractions and decimals expanding on the students knowledge of number laws and relationships.

Lesson 4

What can I do with numbers and grouping

symbols?

Band 3 use a range of mental or written strategies to add and subtract

numbers of any size including decimal fractionsBand 4 use efficient mental, written and technological strategies to calculate

using all 4 operations with integers, fractions, decimals and mixed numbers

Lesson 5

This time should be spent teaching an explicit lesson on the order of operations. Practise should include operations with whole numbers, fractions and decimals. Introduce the use of calculators and investigate how they operate with the order of operation rules. Help students to become proficient with the memory function and the grouping symbol keys (i.e. (), {}, []) on the scientific calculators.

Lesson 6What can I do with

numbers and symbols?




Lesson 7This time should be spent teaching an explicit lesson on the writing and solving of simple equations by inspection. Practise should include operations with whole numbers, fractions and decimals.

Activities could include, magic squares, arithmagons and guess my number puzzles.

Lesson 8

What can I do with numbers and pronumerals?




Lesson 9 Assessment task – Broken calculator

Lesson 10 Let’s review


Starting Smart Lesson 1: What do we know about equals, equations and algebra?

Preparation Setup Interactive Whiteboard and open Equations_1.notebook Setup butchers’ paper (textas & blue tac) to record concept map for patterns and algebra Copy the assessment task – Table and Chairs for each student Math exercise book for student glossary and possibly calculator for warm-up activity

Approximate Times Lesson Outline

Preview/Mental and oral warm-ups

Goal Setting5-10 minutes

Display and discuss the learning goal (IWB page 2)Display Mental Telepathy (IWB page 3) and explain to the students what Mental telepathy means. Ask the students to think of a number, (maybe start with a number between 1 and 10) and to mentally calculate each step as you read out the instructions. Before they tell you the answer, announce to the class that their answer is 25. Repeat the process 2 or 3 more times with the students choosing 2 or 3 digit numbers. Explain that by the end of this unit, they will be able to understand the process and dazzle others with their mental telepathy.

Explicit Teaching10-15 minutes

Brainstorm Inform students that the key word you will use for the word web is: Algebra, then

record the key word on the butcher’s paper.Ask students to call out other words or come to the butcher’s paper and draw a picture that shows their understanding of the concept.Keep the completed word webs for review and additions each lesson.

Student Activity20-30 minutes

Tables and Chairs Introduce the task Tables and Chairs (IWB page 4). Ensure students understand what

‘end-to-end’ means (demonstrate using square tiles if needed). Read through each question clarifying the meaning or explaining any term that the students do not know.

Encourage students to provide as much evidence of their mathematical thinking as possible

Provide small square tiles for students’ use if required. Teachers can support students by answering questions without telling them what to

do. The object of the exercise is not that students get the right answer, but that they are given an opportunity to demonstrate what they actually do know and can do largely on their own.

To encourage students to provide as much evidence of their mathematical thinking as possible, display the Assessment Table (IWB page 5) show explain how the tasks will be assessed.

Reflection5-10 minutes

Allow students time to complete their reflection in their maths journal. Sentence starters or questions could include:

o What was your favourite part of the lesson?o What questions do you still have?o Write down 3 things you did well in the lesson

If time permits students may wish to share their response with the class.


Starting Smart Lesson 2: What number relationships do I know?

Preparation Setup Interactive Whiteboard and open Equations_1.notebook Truth tiles game board and cards and scissors


Preview/mental and Oral warm-ups


Display and discuss the learning goal (IWB page 6), then review and revise the class brainstorm from lesson 3 to add new vocabulary or concepts as required.Display Today’s target number is… (IWB page 7), decide what number you would like to use, (consider using fractional or decimal numbers) write this number into the box and pose some or all of the following challenges

o Find 3 numbers whose sum is the target number, Find 2 numbers whose difference is the target number, Find 3 numbers whose product is the target number, Find all the factors of the target number, Describe the target number and have the learners record their answers after each question.


Student Activity20 - 30 minutes

Truth tilesGroup students into pairs, display Truth tiles (IWB page 8) and distribute game boards, cards and scissors.

The puzzle involves placing the nine digits (1 to 9) into a frame so that the three statements are all true simultaneously, (explain True, False and simultaneous).

Using the IWB, demonstrate how to begin the problem by creating true statements, invite students to come to the board to create true statements.

Allow time to experiment then record the first two solutions on the board for example;1 + 7 = 89 – 5 = 42 x 3 = 6

5 + 4 = 98 – 7 = 13 x 2 = 6

When there are two or more solutions found, ask ‘How will we know when we have found all the solutions?’ Continue with the investigation prompting students to make observations like: 2 x 3 is the same as 3 x 2. Raise the question of whether that 'statement' also works for the other operations. For example: Is 2 - 3 the same as 3 - 2? Also ask ‘Can we use the number 1 in the multiplication?’ to make obvious that multiplication by 1 results in the answer being the same as the question. The conversations about number relationships represent fundamental arithmetic laws such as the commutative property and the inverse relationship between addition and subtraction.

The question of when a solution is really different, e.g. 7+1=8, 1+7=8, 3x2=6, 2x3=6, invariably arises and needs to be dealt with. As a class, decide on your protocol as this will change the number of possible solutions


Allow students time to complete a reflection in their maths journal. Sentence starters or questions could include:

o What was your favourite part of the lesson?o What questions do you still have?o Write down 3 things you did well in the lesson

If time permits share responses with the class or revisit class brainstorm


Starting Smart Lesson 4: What can I do with numbers and grouping symbols?

Preparation Setup Interactive Whiteboard and open Equations_1.notebook Four large sized dice Four dice per group of four Butchers’ paper and textas



Goal Setting5– 10 minutes

Display and discuss the learning goal (IWB page 9), then review and revise the class brainstorm from lesson 3 to add new vocabulary or concepts as required.Display Today’s target number is… (IWB page 10), decide what number you would like to use, (consider using fractional or decimal numbers) write this number into the box and pose some or all of the following challenges

o Find 3 numbers whose sum is the target number, Find 2 numbers whose difference is the target number, Find 3 numbers whose product is the target number, Find all the factors of the target number, Describe the target number and have the learners record their answers after each question.



Display the page ‘Target Number 12’ (IWB page 11), choose 4 students who will roll the large dice and then divide the rest of the class into two teams.

Have the four students roll their die and write the numbers on the board. Ask the class,‘Can anyone make 12 by combining some, or all, of these numbers? Each number can only be used once’, (if no one can make 12, roll the dice again).

Have students that can make the target number explain their methods and you write it in words on the board.

If a student suggests a correct way of making 12, then their team wins a point. The other team can challenge and if the challenge is upheld then the challengers take the point instead. Roll the dice again.

Play continues until one team has reached 12 points. Using some of the examples from the game, discuss how to write the ways of making 12

in mathematical notation. Stress the use of brackets for making calculations exact. For example, “add 5 and 1 then multiply by 2” has to be written as (5 + 1) x 2 and not as 5 + 1 x 2 which could be misinterpreted as 7.

Group students into four and give each group 4 dice, butcher’s paper and a texta. The groups will now play a similar game with TARGET 24. Students write down the numbers thrown each time, along with the methods of getting

24 in words and then with the correct grouping symbols. Observes their work and collect good examples for discussion later.



o What was challenging part of today’s lesson?o What questions do you still have?o Write down 3 things you did well in the lesson

If time permits learners may wish to share response with the class.


Starting Smart Lesson 6: What can I do with numbers and symbols?

Preparation Setup Interactive Whiteboard and open Equations_1.notebook Butchers’ paper with the heading ‘The Four Fours Challenge’ and textas




Display and discuss the learning goal (IWB page 12), then review and revise the class brainstorm from lesson 3 to add new vocabulary or concepts as required.

Display the butchers’ paper and explain to the students that the challenge will be to make as many numbers as possible using any mathematical sign (i.e. +, -, x, /, (), {}, []) and only 4 fours for example; 4x4+4+4=24 or, 44+4/4=45...

Have each student create a number sentence and write it on the butchers’ paper. The rest of the class need to check for accuracy.

This activity can be repeated each day to complete number sentences for all numbers up to 100 (or an agreed end point)



Display the page Guess my rule (IWB page 13). Tell the students that you have a mystery rule (like "take a number, double it, and add 2

to the answer") and that they have to guess the rule after they hear some clues. Ask a student for a starting number, e.g. 9 then you tell them the answer is 20

(9x2+2=20). Other students suggest inputs and you supply the outputs while keeping a record of the attempts on the board.

Keep taking input numbers until a student is able to tell you the rule, in words. Play another round, or two, recording the algorithm in words each time. Divide the class into small groups. Students take turns to be the person thinking of the

rule. Each person should keep a record of the pairs of numbers to assist in checking that no mistakes are made.

Bring the class back together and focus on the rules, written in words, from the first few games.

Discuss with students the need to write mathematical statements in a more shorthand method

Ask for suggestions from the class as to how to write the first rule, e.g. let be the input number and the rule would be written as x 2 +2

To consolidate understanding, construct rules for the remaining games Have each pair of students write their rules as number sentences, each checking the

others work





Starting Smart Lesson 8: What can I do with numbers and symbols?


Preparation Setup Interactive Whiteboard and open Equations_1.notebook Butchers’ paper with the heading ‘The Four Fours Challenge’ and textas




Display and discuss the learning goal (IWB page 14), then review and revise the class brainstorm from lesson 3 to add new vocabulary or concepts as required.

Display the ‘Four Fours Challenge’ poster and remind the students of the challenge i.e. to make as many numbers as possible using any mathematical sign (i.e. +, -, x, /, (), {}, []) and only 4 fours for example; 4x4+4+4=24 or, 44+4/4=45...




Display the page Mental Telepathy (IWB page 15). Play a round of the game, (encouraging students to use numbers other that whole

numbers) to remind them of the process Instruct the students to focus on the list of instructions and ask ‘Why does this work?’ Set the challenge (and a time limit) for people to work n groups to try to establish why

everyone got the same answer. Ask for students to share their thinking or strategies they were working on. Depending on the student’s responses, introduce them to the idea of letting a letter, say

x, to stand for a general number and have them try the challenge again. If the students are still having difficulty, take them through the algebraic reasoning (IWB

page 16)

There may need to be discussion around the idea of how to double x + 4, (manipulation of algebraic expressions will be consolidated later in the term)

Show them one more ‘trick’ (IWB page ?) and ask the students to investigate the logic behind the solution

Once understood, set the students to develop their own mental telepathy ‘trick’ for others to analyse.





Starting Smart Lesson 9: Assessment task – Broken calculator

Preparation


Setup Interactive Whiteboard and open Equations_1.notebook Copy the assessment sheet for students Computer access for each student Assessment rubric




Display and discuss the learning goal (IWB page 17), explain that the goal for the lesson will be an assessment task.





Display the page ‘Broken Calculator’ (IWB page 18) and explain the taskYou sit down to take a test when you suddenly realise that your calculator is broken. Some keys will not work at certain times. What are you going to do? This experience sounds like quite a nightmare! Fortunately, numbers are very powerful tools that can be used in a variety of ways.

Click the image on the IWB page and demonstrate how to ‘break’ the keys, "break" the 1, 8, 5, 2, 0, 3 and - (minus) keys (only keys that work are: 7, divide, 4, 6, multiply, open and close parenthesis, add, equal, and cancel.)

Demonstrate how to use the calculator and while the students go to the website, http://seeingmath.concord.org/broken_calculator/ , hand out the student response sheet.

List all the different ways you can make 25 on this calculator with keys that are NOT broken, (the first one has been done).

If needed, use the extra challenges for the students to do. Mark and collate student results on the ‘Broken calculator – Analysis Tool. Look for

trends in your class data to inform the teaching and learning sequence for the rest of the term. This could be recorded on the Whole class analysis.





http://seeingmath.concord.org/broken_calculator/


Starting Smart Lesson 10: Let’s review

Preparation Setup Interactive Whiteboard and open Equations_1.notebook Six Puzzle pages




Display the page ‘Let’s review’ (IWB page 19) and explain that the goal for the lesson will be to review what has been learnt.





Decide on which puzzles you would like to work on and display each page ‘as you go. Students could work individually or in groups. Distribute the pages and use the following

questions/ talk points to make the learning richer; Puzzle 1

o Using the digits of the year of your birth, in order, what other numbers can you make?

Puzzle 2o The solution is pairs of numbers between 10 and 20 which add to 42 when

used with 9. There are 4 pairs. Suppose you still used the numbers 10 to 20 and a total of 42, but they were used with 8 in the centre?

Puzzle 3o Can you rewrite the puzzle so that the numbers in the rhyme change, but the

answer is still the same? Which numbers can’t be used? How many possible ways are there of rewriting the rhyme?

Puzzle 4o How do you think the inverter of the puzzle thought it up? (work backwards)o Can you think up a puzzle like this that uses halves/ quarters/ two sorts of

fractions? Puzzle 5

o Has mum ever been five times my age before? Will she ever be five times my age again?

Puzzle 6o Can you use matches to build other sequences of shapes? What number

patterns develop? How can you use them to work out the number of matches in the 10th or 100th shape of your sequence?



What was your favourite part of the lesson? What was challenging part of today’s lesson? Write down 3 things you did well in the lesson


Assessment tool 1: Suggested rubric format

Alge

bra

Num

ber

Mat

hem

atica

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Use the Whole Class Analysis table (Assessment tool 2), to determine the next steps in advancing student learning as it highlights what learners know and can do and what confusions, preconceptions, or gaps they may have.

Use the Whole Class Analysis table (Assessment tool 2), in conjunction with the Targeted Group Learning organiser (Assessment tool 3), the Scope and sequence for Mathematics (Assessment tool 4) and the Scope and sequence for ESL (Assessment tool 5) to identify the targeted outcomes for the learning program.

Suggested resources, concepts and assessment opportunities are detailed in Term at a Glance.

Ideas and activities are also listed in the Strategies and Resource Bank which may be used in the next teaching and learning cycle.

Band 2 Band 3 Band 4 Band 5

Complete equations involving simple addition or subtraction where one of the elements (addend, minuend or subtrahend) is missing

Complete equations involving all four operations, fractions and decimals where one element is missing

Create & manipulate algebraic expressions & equations involving the four operations in order to determine unknown values & solve problems

Use efficient mental, written and technological strategies to calculate using all 4 operations with integers and decimals.

Use a range of mental strategies to add & take numbers up to 2-digits & use the relationship between multiplication and division to solve simple problems

Use a range of mental or written strategies to add and subtract numbers of any size including decimal fractions

Use mathematical terms to draw conclusions from the different calculations.

Use algebraic terms when interpreting or drawing conclusion from the written problems and calculations.

Use algebraic language to critically interpret written problems and discuss any implications for calculating.

Interpret and create written algebraic related problems and discuss the strategies used in the calculation using algebraic terminology.

Apply algebraic techniques & rules, including index laws & fractions, to simplify algebraic expressions, resolve equations and solve real-world problems


Assessment tool 2: Whole Class Analysis

Legend√ Broad understanding+ Sound understanding• Basic understanding^ Limited understanding× No understanding

Student

Understanding of Common

Measurement Units

Estimating Accurately

Select Appropriate Measuring Tool

Measure Accurately

e.g. Jo Blog + • + ^

Class: _____________________

Date: _____________________


Assessment tool 3: Targeted Group Learning

Create focussed class groupings based on Assessment Tool 2: Whole Class Analysisinformation. Each group will have different learning needs and require different teaching emphasis.

Teaching Emphasis Targeted Groups√ +

• × or ^

√ +

• × or ^

√ +

• × or ^

√ +

• × or ^

To start, select a teaching emphasis and work out some a small group activity that you can focus on. The other members of the class are working

independently on work without teacher/teacher aide facilitating. These activities can focus on concepts that extend learners knowledge of time. This

way you can give support to individual learners’ needs.


Assessment tool 4: Scope and sequence for Mathematics


Alg

ebra

A 2 Algebra recognise and continue physical patterns formed by

repeatedly adding or subtracting a predictably increasing or decreasing number of elements

express patterns as a number sequence generate patterns and number sequences given a

description or set of instructions continue and complete number sequence patterns

involving repeated addition or subtraction complete equations involving simple addition or

subtraction where one of the elements (addend, minuend or subtrahend) is missing

A 3 Algebra recognise, complete or continue number sequence

involving repeated multiplication or division by a constant

convert a pattern to a number sequence and use this to construct a table of values

complete missing numbers in a given table of values precisely describe a number pattern in a way that it

could be exactly reproduced complete equations involving all four operations,

fractions and decimals where one elements is missing(note this indicator is missing from the NTCF)

A 4 Algebra recognise, complete or continue a number sequence

based on repeatedly applying one or two operations convert a number sequence to a table of values and

determine a rule linking the value of any term to its position in the sequence

create and manipulate algebraic expressions and equations involving the four operations in order to determine unknown values and solve problems

A 5 Algebra apply quadratic formulae to generate paired number

sequences and represent them graphically investigate and interpret linear and quadratic

functions and their graphs apply algebraic techniques and rules, including index

laws and rules involving simultaneous equations and fractions, to simplify algebraic expressions, resolve equations and solve real-world problems


Num

ber

N 2.1 Numbers and number systems count forwards and backwards by 10s and 100s on

and off the decade estimate, compare, order, read and represent

numbers up to five-digits round to the nearest 10, 100 and 1 000 in

estimation compare, order, and represent commonly used

fractions separate and rearrange collections and objects in a

variety of ways to show equal parts determine equivalence between commonly used

fractions

N 2.2 Calculating recall basic addition and subtraction facts to 20 use mental strategies to calculate multiplication

facts to 10 x 10 use a range of mental strategies to add and subtract

numbers up to 2-digits add and subtract whole numbers to thousands and

decimals with money add and subtract simple fractions expressed in

words use informal written strategies for multiplying and

dividing a two and three digit by a one digit number use the relationship between multiplication and

division to solve simple problems estimate both sums and products by rounding to a

single digit or multiples of ten

N 3.1 Numbers and number systems order, read and represent whole numbers in the

millions round decimals to the nearest whole number to

assist with estimation order, represent and manipulate decimals, simple

fractions and common percentages find equivalence between fractions including

expressing a mixed number as an improper fraction equate common percentages to fractions and

decimals round decimal numbers to a given place when

estimating use ratio to describe relationships between

quantities identify prime numbers

N 3.2 Calculating mentally calculate addition and subtraction facts to

100, recall multiplication facts to 10 x 10 and derive related division facts

use a range of mental or written strategies to add and subtract numbers of any size including decimal fractions

mentally multiply two digit numbers by single digit numbers and divide by two digit numbers by factors and multiples of 10

use a range of mental and written strategies to multiply and divide whole numbers by one and two digit whole numbers and decimal numbers by one digit whole numbers and by 10, 100 and 1 000

add and subtract fractions and multiply familiar fractions by a whole number

calculate common percentages of a quantity

N 4.1 Numbers and number systems demonstrate understanding of system place value order or determine the equivalence of fractions,

decimals and percentages interpret, represent and calculate using ratios and

rates determine squares and cubes and their related roots

and represent these using index notation

N 4.2 Calculating use efficient mental, written and technological

strategies to calculate using all 4 operations with integers, fractions, decimals and mixed numbers

express a number as a product of its prime factors apply and manipulate ratios and rates to solve

problems use effective estimation strategies including rounding

in context and solve problems where exact calculations are not required

N 5.1 Numbers and number systems order, read, interpret and represent very large and

small numbers using scientific notation express recurring decimals as common fractions apply index laws to simplify and evaluate arithmetic

expressions, interpret and manipulate negative indices

round numbers to a specified number of significant figures

convert rates from one set of units to another

N 5.2 Calculating use technology to undertake tasks involving iterative

processes and basic statistics explain the effect of truncating and rounding on

accuracy in calculations; determine appropriate levels of accuracy for the context and take account of truncation variations when using calculators

solve problems involving rates and ratios requiring conversion of units

Enlarge this page to A3 for easy reading.


Assessment Tool 5: ESL Scope and Sequence

Level 2 Level 3 Level 4 Level 5

Spea

king

- Co

mm

unic

ation

S L2.1 CommunicationCommunicate verbally and non-verbally in social, expressive and learning situations,make requests and sometimes initiate action

Communication respond to simple questions and requests suggest, request, initiate and direct action with

simple two word commands retell story or event in sequence using known

sentence patterns and visual support give reason for action, with contextual support participate in face to face conversation, with

support.

S L3.1 CommunicationCommunicate and learn through SAE in predictable situations and construct sequenced oral texts using limited SAE

Communication use SAE to negotiate simple transactions, recount main ideas and recall details in sequence

from oral texts give a series of short directions in known context

with support recount events/actions/stories in sequence, using

speech and non-verbal language describe and identify people, places and things

using simple vocabulary for colour, size, place, location and time

participate with peers in supported small group tasks

participate in interaction with structured support contribute to short dialogue on known topic express ideas, sometimes fragmented chunks in

learning areas express humour, opinions and describe feelings.

S L4.1 CommunicationExpress the main point and some detail of ideas and opinions in supportive classroomsituations, using a range of familiar spoken texts types

Communication communicate in a range of situations and give

messages in connected speech across curriculum contexts, with some support

contribute ideas in group and class tasks give reasons and express opinions in SAE give short sequences of instructions attempt to express complex thoughts and

feelings, humour and opinions.

S L5.1 CommunicationParticipate actively in social, expressive and informational contexts and elaborate ideas,with support

Communication use SAE in a range of contexts across the

curriculum interact and negotiate with peers in planning

and presenting a project or special event contribute to new topic discussion with

contextual support give a short prepared formal report and

answer some questions recount main ideas/details and connect ideas

from oral texts conduct an interview from a prepared format use SAE to make a simple hypothesis or

generalisation

Writi

ng -

Com

mun

icati

on

W L2.1 CommunicationWrite short, simple, coherent texts containing a few ideas related to task/topic and showing organisation of subject matter with modelled support

Communication write short, simple texts on familiar topics for

the maths learning areas incorporating familiar language with some

specialised terms, with support, e.g. simple reports

write brief texts which show simple logical sequencing of ideas

contribute to group construction of texts/writing activities

present information in a variety of forms, e.g. simple tables, charts, graphs

W L3.1 CommunicationWrite simple informational texts using familiar spoken and written language from modelled texts

Communication write information texts based on modelled

language for general school use, e.g. reports take part in shared writing activities, e.g.

suggest words, phrases or sentences present information in a variety of forms, e.g.

simple tables, charts, graphs initiate own writing, e.g. labels, short notes,

texts. write a variety of texts with some elaboration,

integrated ideas and information

W L4.1 CommunicationUse a basic structure of the information text type in the maths learning area for a given audience

Communication write extended texts in the maths learning area

to convey integrated ideas and information to given audience, with some support

incorporate information from other sources in their writing, e.g. report, simple explanation, notes, summaries, discussion

plan and sequence information for specific text type, e.g. report, explanation

write according to structure of text genre make summaries by writing sentences expanded

from key words.

W L5.1 CommunicationWrite informational texts in the maths learning area incorporating language and ideas from different sources, with support for the more complex texts

Communication write extended texts for a range of purposes

across the curriculum, with support, e.g. report, explanation, personal reflection

write reports incorporating information from two or three sources

incorporate language and ideas from selected sources drawn from classroom activities

use a range of formats, e.g. graphs, diagrams, to convey information

Enlarge this page to A3 for easy reading.


Resources Bank

Strand Resource

AlgebraMostly algebra These teaching and learning resources, based on algebra, include lesson plans, activities and computer programs focusing on: interpreting algebraic expressions creating and solving equations evaluating algebraic expressions performing number magic http://nationalstrategies.standards.dcsf.gov.uk/node/367445?uc%20=%20force_uj

Computer interactive materialsBalance 1http://212.219.3.14/lrc/resources/standards_unit/maths_for_tlc/software/six_applets_FI_May21/balance.htmlBalance 2http://212.219.3.14/lrc/resources/standards_unit/maths_for_tlc/software/six_applets_FI_May21/balance_game.html

Mr Barton maths.comA collection of resources targeted at enhancing the teaching of maths in the Middle Yearshttp://www.mrbartonmaths.com/algebra.htm

Exploring Algebra This interactive mathematics resource allows the user to explore and interpret word problems by evaluating expressions and solving equations in a variety of different contexts. This resource includes print activities, solutions, learning strategies, and a board game.http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.PATT&ID2=AB.MATH.JR.PATT.ALG&lesson=html/object_interactives/algebra/use_it.html

Connect FourStudents play a generalized version of connect four, gaining the chance to place a piece on the board by solving an algebraic equation. Parameters: Level of difficulty of equations to solve and type of problem. Algebra Four is one of the Interactive assessment games.http://www.shodor.org/interactivate/activities/AlgebraFour/

NumberMr Barton maths.comA collection of resources targeted at enhancing the teaching of maths in the Middle Yearshttp://www.mrbartonmaths.com/number.htm

Engagement Tasks (commonly used as lesson starters) and Discovery Cards (for use as individual or small group problem-solving) have been prepared as ready-to-use classroom resources. http://www.learningplace.com.au/deliver/content.asp?pid=28567

Math magicThis site has been designed to help those who would like a good site to use as a reference to all of their number sense questions. This site has many number sense tricks and methods for middle school. http://math-magic.com/

http://math-magic.com/

http://www.learningplace.com.au/deliver/content.asp?pid=28567

http://www.mrbartonmaths.com/number.htm

http://www.shodor.org/interactivate/activities/AlgebraFour/

http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.PATT&ID2=AB.MATH.JR.PATT.ALG&lesson=html/object_interactives/algebra/use_it.html

http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.PATT&ID2=AB.MATH.JR.PATT.ALG&lesson=html/object_interactives/algebra/use_it.html

http://www.mrbartonmaths.com/algebra.htm

http://212.219.3.14/lrc/resources/standards_unit/maths_for_tlc/software/six_applets_FI_May21/balance_game.html

http://212.219.3.14/lrc/resources/standards_unit/maths_for_tlc/software/six_applets_FI_May21/balance.html

http://nationalstrategies.standards.dcsf.gov.uk/node/367445?uc%20=%20force_uj


Tables and Chairs …The community is planning to have a party on the Council lawn. They have lots of small square tables.

Each table seats 4 people like this:

The community decides to put the tables in an end-to-end line along the lawn to make one big table.

a) Make or draw a line with 2 tables. How many people will be able to sit at it?

b) Make or draw a line of 4 tables. How many people will be able to sit at it?

c) Make or draw a line of tables that would seat 8 people. How many tables are needed?

d) Make or draw a line of tables that would seat 12 people. How many tables are needed?

e) Make or draw a line of tables that would seat 20 people. How many tables are needed?

f) Fill in the shaded boxes to show your results so far.

Number of tables 1 2 4Number of people 4 8 12 20

g) Can you find another way to describe your results so far? Show this in the space below.

Adapted from Street Party, Assessment Research Centre, University of Melbourne (2001) for the Scaffolding Numeracy in the Middle Years-Linkage Project 2003-2006 Assessment Materials Option 2


h) The community can borrow 99 tables. How many people could they seat using 99 tables placed end-to-end? Show your working and explain your answer in as much detail as possible.

i) The community can borrow rectangular tables that seat 6 people. Draw one of these tables showing the people sitting around it.

j) Draw a line of 5 of these rectangular tables placed end-to-end. How many people will be able to sit at it?

k) Explain what happens to the number of people as more rectangular tables are placed end-to end. Describe or show your findings in at least two ways.

l) How many people could be seated if 46 of these rectangular tables were placed end-to end? Show your working and explain your answer in as much detail as possible.

m) How many of these rectangular tables would you need to place end-to-end to seat 342 people? Show your working and explain your answer in as much detail as possible.

Adapted from Street Party, Assessment Research Centre, University of Melbourne (2001) for the Scaffolding Numeracy in the Middle Years-Linkage Project 2003-2006 Assessment Materials Option 2

=

=

=

+

-

X


Truth tiles

Cut out these tiles and use them on the board above to make a true addition.


Reproducible Page ... © Maths300, Curriculum Corporation

Possible solutions – Truth Tiles

1 + 7 = 89 – 4 = 52 x 3 = 6

1 + 7 = 89 – 5 = 42 x 3 = 6

1 + 7 = 89 – 4 = 53 x 2 = 6

1 + 7 = 89 – 5 = 43 x 2 = 6

7 + 1 = 89 – 4 = 52 x 3 = 6

7 + 1 = 89 – 5 = 42 x 3 = 6

7 + 1 = 89 – 4 = 53 x 2 = 6

7 + 1 = 89 – 4 = 53 x 2 = 6

4 + 5 = 98 – 1 = 72 x 3 = 6

4 + 5 = 98 – 7 = 12 x 3 = 6

4 + 5 = 98 – 1 = 73 x 2 = 6

4 + 5 = 98 – 7 = 13 x 2 = 6

5 + 4 = 98 – 1 = 72 x 3 = 6

5 + 4 = 98 – 7 = 12 x 3 = 6

5 + 4 = 98 – 1 = 73 x 2 = 6

4 + 5 = 98 – 7 = 13 x 2 = 6


Broken calculator

You sit down to take a test when you suddenly realise that your calculator is broken. Some keys will not work at certain times. What are you going to do?Go to: http://seeingmath.concord.org/broken_calculator/ and "break" the 1, 8, 5, 2, 0, 3 and - (minus) keys.

List all the different ways you can make 25 on this calculator with keys that are NOT broken. The first one is done for you.

1 7 x 3 + 4 = 25

2

3

4

5

6

7

8

Challenge questions: Without using the "0" key, how would you calculate 22 x 40 on your calculator? Without using the "2" key, how would you calculate 18 x 25 on your calculator? Without using the "5" key, how would you calculate 18 x 25 on your calculator?

http://seeingmath.concord.org/broken_calculator/


Puzzle 1:

Adapted from Puzzle 24, Professor Morris Puzzles, Black Douglas Professional Education Services 1997


Puzzle 2:



Puzzle 3:



Puzzle 4:



Puzzle 5:



Puzzle 6:


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