Mental Math Yearly Plan Grade 7 - Nova Scotia Department .

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Mental MathYearly PlanGrade 7Draft — September 2006

MENTAL MATHAcknowledgementsThe Department of Education gratefully acknowledges the contributions of the following individuals to thepreparation of the Mental Math booklets:Sharon Boudreau—Cape Breton-Victoria Regional School BoardAnne Boyd—Strait Regional School BoardEstella Clayton—Halifax Regional School Board (Retired)Jane Chisholm—Tri-County Regional School BoardPaul Dennis—Chignecto-Central Regional School BoardRobin Harris—Halifax Regional School BoardKeith Jordan—Strait Regional School BoardDonna Karsten—Nova Scotia Department of EducationKen MacInnis—Halifax Regional School Board (Retired)Ron MacLean—Cape Breton-Victoria Regional School BoardSharon McCready—Nova Scotia Department of EducationDavid McKillop—Chignecto-Central Regional School BoardMary Osborne—Halifax Regional School Board (Retired)Sherene Sharpe—South Shore Regional School BoardMartha Stewart—Annapolis Valley Regional School BoardSusan Wilkie—Halifax Regional School BoardMENTAL COMPUTATION GRADE 7 — DRAFT SEPTEMBER 2006i

MENTAL MATHContentsIntroduction . 1Definitions . 1Rationale . 1The Implementation of Mental Computational Strategies . 2General Approach. 2Introducing a Strategy . 2Reinforcement . 2Assessment. 2Response Time . 3Mental Math: Yearly Plan — Grade 7 . 4Number Sense . 4Fractions and Decimals. 6Decimals and Percent . 9Probability. 12Integers. 13Addition . 13Subtraction. 14Multiplication . 16Division. 17Geometry . 18Data Management . 18Patterns . 19Linear Equations and Relations . 20MENTAL COMPUTATION GRADE 7 — DRAFT SEPTEMBER 2006iii

MENTAL MATHIntroductionDefinitionsIt is important to clarify the definitions used around mental math. Mental math in Nova Scotia refersto the entire program of mental math and estimation across all strands. It is important to incorporatesome aspect of mental math into your mathematics planning everyday, although the time spent eachday may vary. While the Time to Learn document requires 5 minutes per day, there will be days,especially when introducing strategies, when more time will be needed. Other times, such as whenreinforcing a strategy, it may not take 5 minutes to do the practice exercises and discuss the strategiesand answers.For the purpose of this booklet, fact learning will refer to the acquisition of the 100 number factsrelating the single digits 0 to 9 for each of the four operations. When students know these facts, theycan quickly retrieve them from memory (usually in 3 seconds or less). Ideally, through practice overtime, students will achieve automaticity; that is, they will abandon the use of strategies and giveinstant recall. Computational estimation refers to using strategies to get approximate answers bydoing calculations in one’s head, while mental calculations refer to using strategies to get exactanswers by doing all the calculations in one’s head.While we have defined each term separately, this does not suggest that the three terms are totallyseparable. Initially, students develop and use strategies to get quick recall of the facts. These strategiesand the facts themselves are the foundations for the development of other mental calculationstrategies. When the facts are automatic, students are no longer employing strategies to retrieve themfrom memory. In turn, the facts and mental calculation strategies are the foundations for estimation.Attempts at computational estimation are often thwarted by the lack of knowledge of the related factsand mental calculation strategies.RationaleIn modern society, the development of mental computation skills needs to be a major goal of anymathematical program for two major reasons. First of all, in their day-to-day activities, most people’scalculation needs can be met by having well developed mental computational processes. Secondly,while technology has replaced paper-and-pencil as the major tool for complex computations, peopleneed to have well developed mental strategies to be alert to the reasonableness of answers generated bytechnology.Besides being the foundation of the development of number and operation sense, fact learning itself iscritical to the overall development of mathematics. Mathematics is about patterns and relationshipsand many of these patterns and relationships are numerical. Without a command of the basicrelationships among numbers (facts), it is very difficult to detect these patterns and relationships. Aswell, nothing empowers students with confidence and flexibility of thinking more than a commandof the number facts.It is important to establish a rational for mental math. While it is true that many computations thatrequire exact answers are now done on calculators, it is important that students have the necessaryskills to judge the reasonableness of those answers. This is also true for computations students will dousing pencil-and-paper strategies. Furthermore, many computations in their daily lives will notrequire exact answers. (e.g., If three pens each cost 1.90, can I buy them if I have 5.00?) Studentswill also encounter computations in their daily lives for which they can get exact answers quickly intheir heads. (e.g., What is the cost of three pens that each cost 3.00?)MENTAL COMPUTATION GRADE 7— DRAFT SEPTEMBER 20061

MENTAL MATHThe Implementation of Mental ComputationalStrategiesGeneral ApproachIn general, a strategy should be introduced in isolation from other strategies, a variety of differentreinforcement activities should be provided until it is mastered, the strategy should be assessed in avariety of ways, and then it should be combined with other previously learned strategies.Introducing a StrategyThe approach to highlighting a mental computational strategy is to give the students an example of acomputation for which the strategy would be useful to see if any of the students already can apply thestrategy. If so, the student(s) can explain the strategy to the class with your help. If not, you couldshare the strategy yourself. The explanation of a strategy should include anything that will helpstudents see the pattern and logic of the strategy, be that concrete materials, visuals, and/or contexts.The introduction should also include explicit modeling of the mental processes used to carry out thestrategy, and explicit discussion of the situations for which the strategy is most appropriate andefficient. The logic of the strategy should be well understood before it is reinforced. (Often it wouldalso be appropriate to show when the strategy would not be appropriate as well as when it would beappropriate.)ReinforcementEach strategy for building mental computational skills should be practised in isolation until studentscan give correct solutions in a reasonable time frame. Students must understand the logic of thestrategy, recognize when it is appropriate, and explain the strategy. The amount of time spent on eachstrategy should be determined by the students’ abilities and previous experiences.The reinforcement activities for a strategy should be varied in type and should focus as much on thediscussion of how students obtained their answers as on the answers themselves. The reinforcementactivities should be structured to insure maximum participation. Time frames should be generous atfirst and be narrowed as students internalize the strategy. Student participation should be monitoredand their progress assessed in a variety of ways to help determine how long should be spent on astrategy.After you are confident that most of the students have internalized the strategy, you need to helpthem integrate it with other strategies they have developed. You can do this by providing activitiesthat includes a mix of number expressions, for which this strategy and others would apply. Youshould have the students complete the activities and discuss the strategy/strategies that could be used;or you should have students match the number expressions included in the activity to a list ofstrategies, and discuss the attributes of the number expressions that prompted them to make thematches.AssessmentYour assessments of mental math and estimation strategies should take a variety of forms. In additionto the traditional quizzes that involve students recording answers to questions that you give one-at-atime in a certain time frame, you should also record any observations you make during thereinforcements, ask the students for oral responses and explanations, and have them explain strategiesin writing. Individual interviews can provide you with many insights into a student’s thinking,especially in situations where pencil-and-paper responses are weak.2MENTAL COMPUTATION GRADE 7 — DRAFT SEPTEMBER 2006

MENTAL MATHAssessments, regardless of their form, should shed light on students’ abilities to compute efficientlyand accurately, to select appropriate strategies, and to explain their thinking.Response TimeResponse time is an effective way for teachers to see if students can use the mental math andestimation strategies efficiently and to determine if students have automaticity of their facts.For the facts, your goal is to get a response in 3-seconds or less. You would give students more timethan this in the initial strategy reinforcement activities, and reduce the time as the students becomemore proficient applying the strategy until the 3-second goal is reached. In subsequent grades whenthe facts are extended to 10s, 100s and 1000s, a 3-second response should also be the expectation.In early grades, the 3-second response goal is a guideline for the teacher and does not need to beshared with the students if it will cause undue anxiety.With other mental computational strategies, you should allow 5 to 10 seconds, depending upon thecomplexity of the mental activity required. Again, in the initial application of the strategies, youwould allow as much time as needed to insure success, and gradually decrease the wait time untilstudents attain solutions in a reasonable time frame.MENTAL COMPUTATION GRADE 7— DRAFT SEPTEMBER 20063

MENTAL MATHMental Math: Grade 7 Yearly PlanIn this yearly plan for mental math in grade 7, an attempt has been made to align specific activitieswith the topic in grade 7. In some areas, the mental math content is too broad to be covered in thetime frame allotted for a single chapter. While it is desirable to match this content to the unit beingtaught, it is quite acceptable to complete some mental math topics when doing subsequent chaptersthat do not have obvious mental math connections. For example practice with integer operationscould continue into the data management and geometry chapters. Integers are so important in grade7 that they should be interjected into the mental math component over the entire year once they havebeen taught.SkillNumberSenseExampleReview multiplication and divisionfacts througha) rearrangement, or commutativeproperty/decompositiona) 8 7 5 8 5 7(rearrangement or commutativeproperty)16 25 4 4 25 4 100b) multiplying by multiples of 10b) 70 80 7 8 10 104 200 6 (7 600 6)c) multiplication strategies such asdoubles, double/double, doubleplus one, halve/double etc.(Intent is to practice facts throughpreviously learned strategies)c) 12.5 4 12.5 2 2(Double 12.5 and then double again)16 x 258 x 50 (half 16 then double 25)3 15 (2 15) (1 15) 30 15Link exponents to fact strategies andproperties for whole numbers. Usepreviously learned strategies such as- distributive strategy2a) 7 7 7 49Use the above fact along with thedistributive property to calculate:3b) 7 7 x 7 x 7 49 x 7 50 x 7 - 7 350 - 73- associative property4c) for 6 , use distributive property6 6 6 36 6 (30 6) (6 6)MENTAL COMPUTATION GRADE 7 — DRAFT SEPTEMBER 2006

MENTAL MATHSkillExample4d) 3 3 3 3 3 9 9 81- working by partse) 2456 8 2400 8 56 8 300 7 307Scientific notation:a) multiplying and dividing bypowers of 10a) 24 000 1024 000 10024 000 10002.4 100.24 1000.024 1000b) dividing by 0.1 and multiplyingby 10 give same result etc.b) 0.024 0.01 0.024 100 2.44.30 0.001 4.30 1000 4 300c) practice converting betweenscientific and standard notationsc) Which exponent would you use to writethese numbers in scientific notation? 87 000 8.7 x 10 310 3.1 x 10Write these numbers in standard form:34 x 1025.03 x 1019.7 x 10The correct scientific notation for thenumber 30100 is:330.1 x 1043.1x 1033.1 x 1043.01 x 10d) comparison of numbers inscientific notation or with powers of10 computationd) Which is larger:48i) 5.07 10 or 2.4 1052ii) 2.3 10 or 234.7 10iii) 670 100 or 6.7 100iv)Apply the divisibility rules toworking with factors and multiplesMENTAL COMPUTATION GRADE 7— DRAFT SEPTEMBER 200688.98.89or100.01a) Which of these are prime?2006, 2003, 2001, 1999b) Is 1998 divisible by 4? 6? 9?c) Quick calculation -find the factors of 48d) Quick calculation -find the first 55

MENTAL MATHSkillExamplemultiples of 26(in c and d use pencils to record answers)Fill in the missing digit(s) so that thenumber isApply the divisibility rules to helpcreate multiples of numbersFractionsandDecimalsa) divisible by 9: 3419b) divisible by 6: 7 158c) divisible by 6 and 9: 5601d) divisible by 5 and 6: 70 81The mental math materialconnected to this topic is extensiveand consideration needs to be givenas to what should be addressedduring the fraction unit and whatcan be done at a later time.Review mentally converting betweenimproper fractions and mixednumbersTeach benchmarks for fractions(0,1 1 3, , , 1) and decimals4 2 4(0, 0.25, 0.50, 0.75, 1)-treat fractions and decimalstogethera) where they are located on anumber linea) Place these fractions in the appropriateplace on the number line1 44 7,,17 51 161 1 1,,, 0.001ii.8 20 9933 6 17iii.,,, 2/2020 20 2071 13iv., 0.42, 1 ,, 0.1583 12i.b) compare and order numberswith the benchmarks6b)63,Is2 3 ?5 4MENTAL COMPUTATION GRADE 7 — DRAFT SEPTEMBER 2006

MENTAL MATHSkillExample7?841 1 ?320.51 Which benchmark is each of thefollowing numbers closest to?0.26, 0.81, 0.95, 0.00099Which benchmark is each of thefollowing numbers closest to?0.51, 0.501,c) Create fractions close to thebench marksPractice until students haveautomaticity of equivalence betweencertain fractions and decimals(halves, fourths, eights, tenths,fifths, thirds, ninths,) This can berevisited during the probability unitso the equivalencies are rememberedand practiced.Review mentally converting betweenimproper fractions and mixednumbers-practice estimation usingMENTAL COMPUTATION GRADE 7— DRAFT SEPTEMBER 20061 4, , 0.95 5c)complete the fraction so that it isclose to the benchmark given:1:2 11i.close toii.close to 1:iii.close to 0.5:iv.close to 0:13259Can use flashcards with fractions on one sideand decimals on the other.If you have a class that works well togetheryou can put fractions and decimals onseparate cards and hand a ‘class set’ out. Godown the rows and as one student calls outtheir fraction or decimal the others have tolisten and call out the equivalent if they havethe card.Estimate:21 3 107

MENTAL MATHSkillExample5 10 8 395 44–6 5benchmarks to add and subtractfractions and mixed numbers-is the sum or difference greaterthan, less than or equal to theclosest benchmark1 1 1 ?6 8 223Is 1.5?34IsWhich sum or difference is larger? Estimateonly?3 43 1 or 4 78 105 25 3b) 4 or 4 6 56 4a)a) Link multiplying a wholenumber by a fraction todivision.a) i.31 20, thinkof 20 is the44Forsame as 20 4 which is 5 so3 20 is 3 sets of 5 which is 154ii. Write a fraction sentence for thispicture:b) Link multiplying a fraction bya whole number to visuallyaccumulating setsb) i.Write a fraction sentence for thispicture:ii. 6 c) When the 2 separate visualpictures are firmly established,8c) i.1 6 thirds 2 wholes316 18MENTAL COMPUTATION GRADE 7 — DRAFT SEPTEMBER 2006

MENTAL MATHSkillpractice should consist ofproblems using both types(The intent here is that students keepa firm connection between numbersentences and visuals at this time.)Revisit the 4 properties associative,commutative, distributive, andidentitya) mentally we want students to(1) recognize when a problemcan be done mentally and (2)do the mental calculationb) create problems using wholenumbers, decimalsc) create problems that involvecombinations of propertiesusing rearrangement etc.Since students need much practicein this area, it is advisable to revisitthis topic several times during theyear, where appropriate.DecimalsandPercentExample4 1053iii. 14 72iv. 93ii.Calculate:a) 1.33 8.25 6.75b) 6 98 6 (100 - 2) (6 100) – (6 2)c) 4 2.25c) 7 2.50 6d) 25 2.08 4e) 46 23 0 55Judgment questions are found in theresource Number Sense: Grades 6-8 (DaleSeymour Publications)pages 18 – 24Incorporate the “Make 1, Make 10,etc” strategy for decimals as well asproperties stated above do the 4 operations andincorporate other strategiesPractice can start with simple wholenumbers, order of operations, and extend todecimals and use multiple strategies:a) 38 14 could be 38 2 12b) 4 7 – 3 7 could be ( 4-3) x 7c) 6 42 72d) 17 – 4e) 1.25 3.81 1.25 3.75 0.06f) 4 0.26 4 0.25 4 0.01g) 4- 1.98 could be 4.02- 2.00or 4 – 2 0.02h) 42 0.07 4 200 7a) This is an extension of thepercents work done earlier.a) Use flashcards with fractions on one sideand percents on the other. A suggestedprogression is to work with halves,fourths, tenths, and fifths on the firstday and eighths, thirds and ninths onthe next day. Mixed practice continuesuntil automaticity is achieved.b) State the % benchmark closest to eachof these:b) Establish benchmarks forpercents: 0%, 25%, 50%, 75%, and 100 %.MENTAL COMPUTATION GRADE 7— DRAFT SEPTEMBER 20069

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Mental Math: Grade 7 Yearly Plan In this yearly plan for mental math in grade 7, an attempt has been made to align specific activities with the topic in grade 7. In some areas, the mental math content is too broad to be covered in the time frame allotted for a single chapter. While it is desirable to match this content to the unit being

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