FIRST GRADE NUMBER SENSE

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1 NUMBER SENSE Number sense is a way of thinking about number and quantity that is flexible, intuitive, and very individualistic. It grows as students are exposed to activities that cause them to think about number in many ways and in different contexts. Number sense includes the ability to compute accurately, to self correct by detecting errors, and to recognize results as reasonable. According to the California Framework, a person has Number Sense if he or she has an intuitive feel for number size and combinations as well as the ability and facility to work with numbers in problem situations in order to make sound decisions and reasonable judgments. The mathematics curriculum enables students to work with numbers to develop number sense traits than include a thorough understanding of number meanings, abilities to represent quantities in multiple ways, recognize the magnitude of number, to know the relative effects of operating on numbers, and to estimate and judge the reasonableness of quantitative results. Numbers enable students to count, to measure, to compare, and to make predictions. Helping students to develop number sense requires appropriate modeling, posing process questions, encouraging thinking about numbers, and in general creating a classroom environment that nurtures number sense. By the end of 1st grade, students understand and use the concept of ones and tens in the place value number system. They add and subtract small numbers with ease. They measure with simple units and locate objects in space. It is important that students gain a conceptual understanding of numbers and counting and not simply learn to count to 100 by rote. They need to understand, for instance, that counting can occur in any order and in any direction, not just the standard left-to-right counting, as long as each item is tagged once and only once. Students must understand that numbers represent sets of specific quantities of items. Of particular importance is learning and understanding the counting sequence for teen values and multiples of 10. For the former, it should be emphasized that teen values represent a 10 value and a certain number of unit values 12 does not simply represent a set of 12 items, it also represents one 10 and two 1s. Students should understand that the equation 15 8 = 7 is the same as 15 = Particular attention should be paid to the assessment of these competencies, as students who fail to learn these topics well will be very likely to have serious difficulties in later grades. The achievement of these standards will require that students are exposed to and asked to solve simple addition and subtraction problems throughout the school. 1

2 KEY STANDARDS Count read and write whole numbers to 100. Compare and order whole numbers to 100 using the symbols for less than, equal to, or greater than. Know the addition facts and corresponding subtraction facts (sums to 20) and commit them to memory. Use the inverse relationship between addition and subtraction to solve problems. Identify one more than one less than, ten more than, ten less than a given number. Count by 2 s 5 s and 10 s with numbers to 100. Show the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, finding the difference). Elaboration California Mathematics Framework Elaboration: Teaching students to effectively solve basic addition and subtraction problems and to commit the answers to memory will require considerable practice at solving these problems. As described in chapter 5, the associated practice should be in small doses each day or at the very least several times a week. At the beginning of the school year, practice should focus on smaller problems (with sums less than or equal to 10) and then large-valued problems should be emphasized in practice once students are skilled at solving the easier problems. Frequent assessment should be provided to determine if students are mastering new facts and retaining prior facts. Students have mastered basic facts when they can solve problems involving them quickly and accurately; accurate but slow problem solving indicates that students are still using counting or other procedures to solve simple problems and have not yet committed the basic facts to memory. Committing the basic addition and subtraction facts to memory is a major objective in 1st and 2nd grade. Students who do not commit the basic facts to memory will be at a disadvantage in further work with numbers and arithmetic. Understanding the symmetric relationship between sets of simple addition problems, such as and 2 + 7, can be used to reduce the memorization load in learning facts. The teaching of these relationships is to be incorporated into the sequence for teaching students simple addition and in their practice. For example, after students learn 7 + 2, they can be shown that the same answer applies to Moreover, by placing problems such as and in sequence in practice sheets, students will have the opportunity to "discover" and reinforce this relationship as well. Later, they might learn that the combination of 7, 2 and 9 can be used to create subtraction facts as well as addition facts. While the standard calls for counting by 1 to 100 in 1st grade, counting into the 100s can be begun in the latter part of 1st grade if students have mastered counting to 100. Counting backward for numbers up to 100 should also be done in 1st grade. 2

3 Grade Level Readiness Considerations for Grade 1 Reading and writing numbers. Many students demonstrate lack of understanding of place value when they encounter numbers, such as 16 and 61. Instruction should be very carefully sequenced to show that 16 is 1 ten and 6 ones, while 61 is 6 tens and 1 one. Students need to know prerequisite skills underlying place value, such as 6 tens equals 60 and its corollary, 60 equals 6 tens, as well as addition facts in which a single-digit number is added to the tens number, , , These facts can be taught verbally before students read and write the numbers. Learning the number that represents a group of tens is important for place value and reading numbers. More naive students are more likely to have difficulty with groups of tens in which the tens number does not say the name of the first digit. For example, we say twenty not "twoty", than with tens numbers in which we say the name of the first digit in saying the tens number, sixty, forty, seventy, eighty, ninety. Provide more practice on these more difficult items. Skip counting. In addition to enhancing children's number sense, skip counting is important for facilitating the learning of multiplication and division. Counting by tens should be introduced when students can count by ones to about 20 or 30. Counting by tens helps students learn to count by ones to 100. Skip counting is taught just like counting by ones. The teacher models the first part of the sequence, the students practice the first part. The modeling and practicing continues on new parts of the sequence until students can say the whole sequence. Skip counting requires systematic teaching using a procedure similar to that discussed for counting by ones. Regularly scheduled practice will help students master counting a sequence. Previously introduced sequences should be reviewed as students learn new ones. Addition and subtraction facts. Teaching these basic facts should be very systematic, as previously discussed, as should assessments. Symmetric relationships. Understanding the symmetric relationship of facts can reduce the memorization load in learning facts. This too has been stressed above. Adding and subtracting two multi-digit numbers. The sequence in which basic facts are introduced should be coordinated with the facts selected for column problems and story problems. The facts in those problems should have been sufficiently practiced in prior factlearning activities. 3

4 ALGEBRA AND FUNCTIONS Algebra Learning algebra is important in a student s mathematical development. It opens the door to organized abstract thinking and supplies a tool for logical reasoning. Algebra embodies the construction and representation of patterns and generalization, and active exploration and conjecture. By itself algebra is the language of variables, operations, and symbol manipulation. Every mathematical strand uses algebra to symbolize, clarify, and communicate. According to the California Framework, algebra is the fundamental language of mathematics. It enables students to create a mathematical model of a situation, provides the mathematical structure necessary to use the model to solve problems, and links numerical and graphical representatives of data. Algebra is the vehicle for condensing large amounts of data into efficient algebraic statements. In first grade children learn to recognize patterns and make comparisons. Observations and discussion of how things change will eventually lead to an understanding of functions and relationships. Students begin to represent a mathematical situation using concrete objects and pictures. They begin to work with classification, patterns, and number. The major goal is for students to be able to sort and classify, analyze simple patterns, and make predictions about them. No key standards at this grade level. 4

5 MEASUREMENT AND GEOMETRY Measurement Students begin to make decisions about size of objects by touching and comparing objects directly. The comparing and ordering of objects provides students the skills to see relationships. For example, students can find an object as tall as they are, or as long as their shoe. At this age, students use nonstandard units of measurements which is important for understanding the need for standard units in later grades. Geometry In first grade the children develop spatial sense or an intuitive feel for one s environment and the objects in it. It is important to have students experiment with patterns, symmetry, and balance. Children begin to explore concepts of time and develop ways to measure it. No Key Standards at this grade level. 5

6 STATISTICS, DATA ANALYSIS, AND PROBABILITY In First Grade the goal of statistics is to help children make sense of data. The purpose is to help students organize their thinking, find patterns, and see relationships. Experiences in collecting, organizing, and displaying objects and information help students reach the goal. It is important for children to count and compare, question and discuss, and organize and make decisions. The graphing of data helps children to see the one-to-one correspondence. Describe, extend, and explain ways to get a next element in simple repeating patterns is also very important, but it has to be handled carefully. Students should never get the idea that the next term automatically repeats (unless they are told explicitly that it does); however, it is legitimate to ask what is the most likely next term. In this way students begin to learn the usefulness of patterns in sorting and understanding data, but they also learn careful and precise patterns of thought. Examples are sequences of colors, such as "red, blue, red, blue, " or numbers, "1, 2, 3, 1, 2, 3, 1, 2, 3, ". But more complex series might also be used, such as "1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, ". KEY STANDARDS Describe, extend, and explain ways to get a next element in simple repeating patterns. 6

7 MATHEMATICAL REASONING Making conjectures, gathering evidence, and building an argument to support ideas are fundamental to doing mathematics. Mathematical reasoning is synonymous with sense making. It is how we discern truth. This is generally done through the application of deductive, inductive, spatial, or algebraic reasoning. According to the California Framework, mathematics provides an opportunity to encounter reasoning in one of its purest forms and to establish mathematical truths with a certainty that is rare in other disciplines. The importance of reasoning to mathematics cannot be overstated. Mathematics makes unique and indispensable contributions to the development of the students ability to think and communicate in a logical manner, a major goal of mathematical study. Children s early experiences with mathematical reasoning are important to later mathematical development. Children want to make sense of the world and the experiences provided at kindergarten help develop the reasoning skills that make this possible. Pattern recognition and classification skills are two important elements of mathematical reasoning. 7