Operational Sense and Quantity - Teaching Strategies PDF

**OPERATIONAL SENSE AND QUANTITY** **Now here are some of the strategies you can use in teaching operational sense:** +-----------------------+-----------------------+-----------------------+ | **Grade 1** | **Grade 2** | **Grade 3** | +=======================+=======...

**OPERATIONAL SENSE AND QUANTITY** **Now here are some of the strategies you can use in teaching operational sense:** +-----------------------+-----------------------+-----------------------+ | **Grade 1** | **Grade 2** | **Grade 3** | +=======================+=======================+=======================+ | **providing | **providing | **promoting the | | experiences with | meaningful | development of a | | part-part-whole | experiences with | conceptual | | relationships (e.g., | number lines and | understanding of | | using counters, | hundreds charts --** | multiplication** | | blocks, number | | | | lines);** | **experiences in | **and division by | | | which they use | using story problems | | | movement and patterns | and models with | | | on the lines and** | manipulative | | | | materials and** | | | **charts to represent | | | | addition and | **pictures for all of | | | subtraction | the operations;** | | | questions;** | | +-----------------------+-----------------------+-----------------------+ | **providing | **providing | **providing | | experiences with | opportunities to | experiences using 0 | | number lines and | identify subtraction | and 1 in both | | hundreds charts, and | as both a counting-up | addition (0 plus any | | experiences with | procedure** | number** | | movement on the | | | | number lines and | **(e.g., solving 15 | **equals that number; | | charts to represent | -- 11 by counting up | 1 plus any number | | addition and | from 11) and a | equals the next | | subtraction | counting-down | number in the number | | questions;** | procedure.** | sequence) and | | | | subtraction (any | | | **Counting up is | number minus 0 equals | | | easier for some | that number; any | | | students;** | number minus 1 equals | | | | the previous number | | | | in the number | | | | sequence);** | +-----------------------+-----------------------+-----------------------+ | **providing | | **providing | | opportunities to use | | experiences with | | concrete materials to | | multiplication and | | represent problems | | division using arrays | | that** | | and repeated addition | | | | or subtraction;** | | **involve addition | | | | and subtraction;** | | | +-----------------------+-----------------------+-----------------------+ | **providing addition | | **providing | | and subtraction tasks | | opportunities to | | involving screened | | develop their own | | (hidden) groups;** | | algorithms, using | | | | both written and | | | | mental methods to | | | | find the answers to | | | | computation | | | | questions;** | +-----------------------+-----------------------+-----------------------+ | **providing | | **providing | | opportunities to | | opportunities to | | identify subtraction | | discuss and explain | | as both a counting-up | | their self-generated | | procedure (counting | | algorithms in social | | up is easier for some | | contexts and to hear | | students) and a | | the explanations of | | counting-down | | strategies that | | procedure;** | | other** | | | | | | | | **students use;** | +-----------------------+-----------------------+-----------------------+ | **providing | **guiding them in | | | opportunities to use | probing their own | | | both vertical and | learning, to help | | | horizontal formats | them identify and | | | for addition and | communicate their own | | | subtraction, so that | strategies for | | | students rely on | solving problems;** | | | their own mental | | | | strategies and not | | | | just the formal | | | | algorithms;** | | | +-----------------------+-----------------------+-----------------------+ | **having them use the | **teaching | | | calculator to make | traditional | | | predictions and to | algorithms through | | | self-correct as they | guided mathematics, | | | work with the | including a** | | | operations;** | | | | | **focus on the | | | | meaning behind the | | | | algorithms and the | | | | use of models to | | | | demonstrate the | | | | algorithms, with | | | | reference back to | | | | self-generated | | | | algorithms;** | | +-----------------------+-----------------------+-----------------------+ | **providing | | **encouraging the use | | opportunities to | | of estimating | | create their own | | strategies (e.g., | | strategies for adding | | ask, "Do you think | | and taking away | | this** | | numbers -- strategies | | | | that will often | | **cookie jar would | | involve using what | | hold about 20 cookies | | they know to** | | or about 100 | | | | cookies?");** | | **find out what they | | | | do not know (e.g., | | | | they may extrapolate | | | | from knowing** | | | | | | | | **that 8+2 is 10 to | | | | knowing that 8+3 is | | | | the same as 8+2+1 | | | | more=11);** | | | +-----------------------+-----------------------+-----------------------+ | **supporting them in | **using everyday | | | identifying some of | situations as | | | the useful strategies | contexts for problems | | | for solving addition | or using real-life | | | and subtraction | contexts** | | | problems -- for | | | | example: using known | **for problems;** | | | facts, using doubles | | | | (which are often | | | | readily remembered), | | | | making tens, using | | | | compensation, | | | | counting up, counting | | | | down, using a number | | | | line or hundreds | | | | chart, using the | | | | commutative property | | | | of addition, using | | | | the inverse | | | | relationship of | | | | addition and | | | | subtraction, using 0 | | | | and 1 in both | | | | addition (0 plus any | | | | number equals that | | | | number; 1 plus any | | | | number equals the | | | | next number in the | | | | number sequence) and | | | | subtraction (any | | | | number minus 0 equals | | | | that number; any** | | | | | | | | **number minus 1 | | | | equals the previous | | | | number in the number | | | | sequence);** | | | +-----------------------+-----------------------+-----------------------+ | **using everyday | | **having them use | | situations as | | self-initiated and | | contexts for problems | | teacher-suggested | | (calculating milk | | drawings and | | money,** | | representations of | | | | the operations;** | | **taking attendance) | | | | and/or using | | | | real-life contexts | | | | for problems (e.g., | | | | "How** | | | | | | | | **many players are on | | | | your soccer team? How | | | | many would there be | | | | if 5 players** | | | | | | | | **quit?");** | | | +-----------------------+-----------------------+-----------------------+ | **providing | | **using known facts | | opportunities to | | to derive new facts. | | discuss their | | For example, use 5 | | solutions in social | | times 6 is 30 to help | | contexts with their** | | calculate 6 times 6 | | | | (just add one more 6 | | **classmates and with | | to the 30);** | | the teacher;** | | | +-----------------------+-----------------------+-----------------------+ | **providing | | **frequently using | | opportunities to | | strategies that | | write about the | | involve partitioning | | problems and to | | (e.g., calculate 8 x | | connect solutions** | | 9 by multiplying 8 x | | | | 10 and then | | **with the | | subtracting the extra | | appropriate | | 8 from the product | | algorithms;** | | to** | | | | | | | | **make 72);** | +-----------------------+-----------------------+-----------------------+ | **presenting | | **having them link | | traditional | | concrete and | | algorithms through | | pictorial | | guided mathematics, | | representation with | | including a** | | the written form of | | | | an operation or | | **focus on the | | equation;** | | meaning behind the | | | | algorithms, the use | | | | of models to | | | | demonstrate** | | | | | | | | **the algorithm, and | | | | instruction that | | | | addresses students' | | | | misunderstandings | | | | of** | | | | | | | | **the equals symbol | | | | (e.g., in the | | | | question 2+\_\_=5, | | | | the students may | | | | supply 7** | | | | | | | | **as the missing | | | | addend);** | | | +-----------------------+-----------------------+-----------------------+ | **encouraging them to | | **providing | | use estimating | | experiences with | | strategies (e.g., "Do | | division that involve | | you think this | | either fair sharing | | cookie** | | (e.g., dividing 24 | | | | candies among 4 | | **jar would hold | | friends) or repeated | | about 20 cookies or | | subtraction (e.g., 24 | | about 100 | | candies are to be | | cookies?");** | | eaten in equal | | | | numbers over 6 days. | | | | How many will be | | | | eaten per day?);** | +-----------------------+-----------------------+-----------------------+ | **encouraging them to | | **using "chunking" | | use self-initiated | | strategies, such as | | drawings and | | partitioning out 25's | | representations of | | and 50's (e.g., to | | the** | | calculate 332+227, | | | | they pull out 325 and | | **operations.** | | 225 to make 550, and | | | | then add on the | | | | remaining 7 and 2 to | | | | make a total of | | | | 559);** | +-----------------------+-----------------------+-----------------------+ | | | **providing | | | | opportunities to use | | | | calculators to | | | | explore the effects | | | | of changing** | | | | | | | | **numerals in an | | | | operation and to | | | | identify the patterns | | | | that occur as the | | | | numerals are changed | | | | (e.g., add 10, 100, | | | | 1000,... to a | | | | number and identify | | | | the pattern in the | | | | answers: 10+3=13, | | | | 100+3=103, | | | | 1000+3=1003,** | | | | | | | | **10 000+3=10 003,. | | | |.. );** | +-----------------------+-----------------------+-----------------------+ | | | **providing | | | | opportunities to | | | | build and use | | | | multiplication charts | | | | and to identify the | | | | patterns that occur | | | | in the charts;** | +-----------------------+-----------------------+-----------------------+ | | | **providing | | | | opportunities to | | | | identify | | | | multiplication fact | | | | patterns in | | | | hundreds** | | | | | | | | **charts;** | +-----------------------+-----------------------+-----------------------+ | | | **providing | | | | sufficient | | | | opportunities so | | | | that, with | | | | experience, they are | | | | able to** | | | | | | | | **construct and | | | | understand their own | | | | algorithms for | | | | solving two- and | | | | three digit word | | | | problems and to | | | | justify and explain | | | | their methods. For | | | | example,** | | | | | | | | **in response to the | | | | word problem "Jane | | | | collected 203 pop can | | | | tabs and Julie** | | | | | | | | **collected 318. How | | | | many did they have | | | | altogether?", | | | | students can use | | | | their** | | | | | | | | **own flexible | | | | algorithms to find | | | | and share a solution. | | | | One student may | | | | give** | | | | | | | | **this method: "I | | | | took 18 from the 318 | | | | and 3 from the 200; | | | | then I added the** | | | | | | | | **200 and the 300 to | | | | get 500; then I added | | | | the 18 and 2 (of the | | | | 3 ones) to get** | | | | | | | | **20, so my answer is | | | | 520 plus the extra 1 | | | | (from the original 3 | | | | ones) or 521."** | | | | | | | | **Another child may | | | | respond by adding 18 | | | | and 3 to make 21 and | | | | then adding** | | | | | | | | **the 200+300=500 to | | | | the 21 to make 521. | | | | As long as students | | | | can justify** | | | | | | | | **and explain their | | | | methods, they should | | | | be allowed to use | | | | them. By developing | | | | and understanding | | | | their own algorithms | | | | first, students are | | | | much more** | | | | | | | | **likely to make | | | | sense of more formal | | | | algorithms, and are | | | | able to compare** | | | | | | | | **various methods and | | | | see which method is | | | | more efficient.** | +-----------------------+-----------------------+-----------------------+ ***Quantity.* Quantity represents the "howmuchness" of a number and is a crucial concept in developing number sense. Understanding the concept of amount is necessary before learning about place value, operations, and fractions. Students can estimate and reason with numbers more easily if they have a basic concept of quantity. It\'s crucial for comprehending relative size (e.g., few, many) and proportional reasoning (e.g., bigger, smaller, twice as big, half as big).** **Here are some strategies on how you will teach quantity:** +-----------------------+-----------------------+-----------------------+ | **Grade 1** | **Grade 2** | **Grade 3** | +=======================+=======================+=======================+ | **continuing to | **continuing to | | | provide opportunities | provide opportunities | | | to use their fingers | to estimate, using | | | and hands to build** | concrete materials | | | | and pictures in | | | **up the concepts of | problem-solving | | | 5 and 10, | situations;** | | | particularly in | | | | finger plays or when | | | | doing problem-solving | | | | tasks and singing | | | | songs. Students | | | | benefit from | | | | explorations in which | | | | they make comparisons | | | | that help them to | | | | think about how many | | | | more fingers they | | | | might need to show | | | | other amounts, such | | | | as 13;** | | | +-----------------------+-----------------------+-----------------------+ | **providing | **providing | | | experiences with | experiences with | | | quantities of up to | estimation strategies | | | 50 using concrete | using grouping of | | | materials and | tens and hundreds. | | | pictures in real-life | For example, in | | | or socio-dramatic | response to a | | | situations:** | question about how | | | | many students are in | | | **-- Set up a store | the primary grades, | | | in a socio-dramatic | they use the | | | area of the | knowledge that there | | | classroom. "Sell" | are about 2 groups of | | | objects in** | 10 people in each | | | | classroom to estimate | | | **set quantities, | how many students** | | | assigning prices such | | | | as \$1, \$5, and \$10 | **there are in 4 | | | for sets of 1, 5,** | classrooms;** | | | | | | | **and 10.** | | | | | | | | **-- Have children | | | | calculate the cost of | | | | items for the | | | | classroom, such as | | | | pencils,** | | | | | | | | **erasers, or | | | | stickers.** | | | | | | | | **-- Play barrier | | | | games using different | | | | quantities of | | | | materials and have | | | | students make a | | | | conjecture about the | | | | number of items their | | | | partner is holding: | | | | "Do you have 10 | | | | objects? Do you have | | | | more than 10? Less | | | | than 50?", and so | | | | on;** | | | +-----------------------+-----------------------+-----------------------+ | **providing | **providing | | | experiences with | opportunities to use | | | estimating, using | estimation in | | | concrete materials | mental-math | | | and pictures, and | situations (e.g., | | | opportunities to use | estimating how many | | | benchmarks to help | apples the class | | | determine a range of | might have in their | | | numbers (or values). | lunch bags and | | | Teacher- or | recognizing that the | | | student-initiated | number would probably | | | questions that limit | be the same as or | | | the range of | less than the | | | possibilities for | quantity of | | | determining an amount | students);** | | | are helpful with this | | | | (e.g., "Is it less | | | | than or more than | | | | 10?"; "Is it closer | | | | to 5 or closer to | | | | 20?");** | | | +-----------------------+-----------------------+-----------------------+ | **providing | **providing | | | experiences that | experiences with | | | repeat the same types | teacher- or | | | of estimation | student-initiated | | | activities, so that | questions that limit | | | students can build up | the range of | | | their conceptual | possibilities for | | | understanding of the | determining an amount | | | amount of something | (e.g., "Does the | | | familiar. For | amount seem closer to | | | example, ask them to | 5 or to 50? Could you | | | estimate the number | hold this many blocks | | | of cans in the | in your hands?** | | | recycling bin. Give | | | | them a benchmark | **In a cup?");** | | | (e.g., "Remember that | | | | there were 10 | | | | yesterday and it was | | | | full on the | | | | bottom");** | | | +-----------------------+-----------------------+-----------------------+ | **providing | **using "nice | | | experiences with the | numbers" such as 5, | | | numbers 5 and 10 to | 10, 25, and 50 to | | | consolidate an | make estimates (e.g., | | | understanding of | ask, "If this pile | | | those numbers as | has 5 pennies, how | | | anchors for the | many do you think are | | | numbers below and | in that pile? Is it | | | above them. For | less than or more | | | example, use ten | than 20? Is it closer | | | frames to show 30:** | to 25 or closer to | | | | 50?");** | | +-----------------------+-----------------------+-----------------------+ | **linking instruction | **using the "nice" | **providing | | related to the big | fraction of 1⁄2 to | experiences with | | idea of counting with | help reason through | estimation strategies | | the concept of | problems. For | (e.g., grouping in | | quantity, so that the | example,** | tens and/or hundreds, | | two ideas are | | rounding to tens and | | developed | **knowing that if | to hundreds);** | | simultaneously;** | half the box of | | | | baseballs contains 5 | | | | balls, then 1⁄3, 1⁄4, | | | | 1⁄5, and so on, of | | | | the box would hold | | | | fewer than 5 balls;** | | +-----------------------+-----------------------+-----------------------+ | **providing | **providing | | | experiences of | experiences with the | | | "divvying up" various | numbers 10 and 100 to | | | amounts, to give | consolidate an | | | students informal | understanding of the | | | experiences with | importance of 10 and | | | fractions of sets | 100 in our | | | before they have to | place-value system;** | | | attach such | | | | experiences to formal | | | | notation.** | | | +-----------------------+-----------------------+-----------------------+ | | **providing | **providing | | | experiences with | experiences with | | | manipulatives and ten | numerals and amounts | | | frames to build up an | of 10, 100, and 1000, | | | understanding of 10 | using manipulatives | | | as an anchor for all | and place-value | | | other numbers in our | charts to build up an | | | place-value system, | understanding of 10, | | | and linking such | 100, and 1000 as | | | grouped quantities | anchors for all other | | | with two-digit | numbers in our | | | numbers** | place-value system;** | +-----------------------+-----------------------+-----------------------+ | | **using | | | | "fair-sharing" | | | | problems that relate | | | | to students' prior | | | | personal knowledge -- | | | | for example, how to | | | | share a chocolate bar | | | | among 3 people. Have | | | | the students try to | | | | think about what they | | | | will call each of the | | | | pieces, so that the | | | | pieces can be | | | | distinguished from | | | | the whole chocolate | | | | bar. Link their | | | | suggestions with the | | | | traditional symbols | | | | for fractions;** | | +-----------------------+-----------------------+-----------------------+ | | **providing | **providing | | | experiences with | opportunities to use | | | games using pattern | fraction | | | blocks, fraction | manipulatives -- | | | blocks, and/ or | pattern blocks, | | | Cuisenaire rods to | Cuisenaire rods, and | | | model fractions and | fraction blocks -- to | | | to compare fractions | explore fractions and | | | as parts of a** | mixed numbers. The | | | | overuse of any one | | | **whole;** | representation (e.g., | | | | pizza, pie) may | | | | create difficulties** | | | | | | | | **in understanding | | | | other types of models | | | | (e.g., number-line | | | | representations,** | | | | | | | | **rectangles);** | +-----------------------+-----------------------+-----------------------+ | | **providing | **using fraction | | | experiences in | problems. For | | | informal | example, give this | | | problem-solving | problem: Ellen has 2 | | | situations using | cupcakes that she | | | fractions. Ask | wants to share with 3 | | | students how they | friends. Then probe | | | would divide 8 pizzas | with questions (e.g., | | | among 6 people. Use | "Will each of the | | | manipulatives such as | friends be able to | | | toy people or drawn | get at least one | | | people and 8 paper | whole cupcake? Why or | | | plates as pizzas and | why not? Will they be | | | let** | able to get 1⁄2 or | | | | more of a cupcake? | | | **students come up | Why or why not?");** | | | with their own | | | | solutions;** | | +-----------------------+-----------------------+-----------------------+ | | **providing | **using prompts to | | | experiences with the | remind students | | | number 10 and the | that:** | | | bundling of tens into | | | | hundreds to | **-- fraction | | | consolidate an | portions must be of | | | understanding of the | equal size;** | | | importance of 10 in | | | | our place value | **-- a fraction | | | system;** | represents a | | | | relationship, not a | | | | particular amount. It | | | | is important for | | | | students to know that | | | | 1⁄2 of a small amount | | | | may be much smaller** | | | | | | | | **than 1⁄3 of a large | | | | amount;** | | | | | | | | **-- a fraction | | | | represents part of | | | | the whole. Students | | | | often make the | | | | mistake of comparing | | | | a part with the | | | | remaining parts. For | | | | example, when asked, | | | | "What fraction of the | | | | grid is chequered?", | | | | a student might reply | | | | 2⁄4** | | | | | | | | **instead of 2⁄6;** | | | | | | | |  | +-----------------------+-----------------------+-----------------------+ | | **providing | **using labelled | | | experiences with the | fractions in the | | | incidental labelling | classroom (e.g., | | | of fractional | labelling one of 6 | | | amounts,** | windows as l⁄6 of the | | | | windows in the | | | **especially common | room);** | | | fractions that | | | | students are less | | | | likely to hear in | | | | their** | | | | | | | | **everyday activity, | | | | such as 1⁄3 or 1⁄4 | | | | (e.g., say, "I am | | | | going to give each | | | | of** | | | | | | | | **these children 1⁄4 | | | | of these balls").** | | +-----------------------+-----------------------+-----------------------+ | | | **encouraging | | | | students to develop a | | | | mental image of | | | | fractional amounts, | | | | to help** | | | | | | | | **with reasoning in | | | | problem-solving | | | | activities that | | | | involve fractions | | | | (e.g., ask,** | | | | | | | | **"What does this | | | | sheet of paper look | | | | like when it is | | | | divided in halves? In | | | | fourths? In | | | | eighths?").** | +-----------------------+-----------------------+-----------------------+