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Operations and Algebraic Thinking Write and interpret numerical expressions. Number and Operations in Base Ten Understand the place value system. Use equivalent fractions as a strategy 5.1. Recognize that in a multi-digit to add and subtract fractions. number, a digit in one place represents 10 times as much as it represents in the place 5.1. Add and subtract fractions with 5.1. Use parentheses, brackets, or to its right and 1/10 of what it represents unlike denominators (including mixed braces in numerical expressions, in the place to its left. numbers) by replacing given fractions and evaluate expressions with with equivalent fractions in such a way these symbols. 5.2. Explain patterns in the number of as to produce an equivalent sum or zeros of the product when multiplying a difference of fractions with like 5.2. Write simple expressions number by powers of 10, and explain denominators. For example, 2/3 + 5/4 that record calculations with patterns in the placement of the decimal = 8/12 + 15/12 = 23/12. (In general, numbers, and interpret numerical point when a decimal is multiplied or a/b + c/d = (ad + bc)/bd.) expressions without evaluating divided by a power of 10. Use wholethem. For example, express the number exponents to denote powers of 10. 5.2. Solve word problems involving calculation “add 8 and 7, then addition and subtraction of fractions multiply by 2” as 2 × (8 + 7). referring to the same whole, including Recognize that 3 × (18932 + 5.3. Read, write, and compare decimals to cases of unlike denominators, e.g., by 921) is three times as large as thousandths. using visual fraction models or 18932 + 921, without having to a. Read and write decimals to equations to represent the problem. Use calculate the indicated sum or thousandths using base-ten benchmark fractions and number sense product. numerals,number names, and expanded of fractions to estimate mentally and form, e.g., 347.392 = 3 × 100 + 4 ×10 + assess the reasonableness of answers. Analyze patterns and 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × For example, recognize an incorrect relationships. (1/1000). result 2/5 + 1/2 = 3/7, by observing that b. Compare two decimals to 3/7 < 1/2. thousandths based on meanings of the 5.3. Generate two numerical digits in each place, using >, =, and < patterns using two given rules. symbols to record the results of Identify apparent relationships comparisons. between corresponding terms. Form ordered pairs consisting of 5.4. Use place value understanding to corresponding terms from the round decimals to any place. two patterns, and graph the ordered pairs on a coordinate Perform operations with multi-digit whole plane. For example, given the numbers and withdecimals to hundredths. rule “Add 3” and the starting number 0, and given the rule 5.5. Fluently multiply multi-digit whole “Add 6” and the starting number numbers using the standard algorithm. 0, generate terms in the resulting sequences, and observe that the 5.6. Find whole-number quotients of terms in one sequence are twice whole numbers with up to four-digit the corresponding terms in the dividends and two-digit divisors, using other sequence. Explain strategies based on place value, the informally why this is so. properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular 5.7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Number and Operations—Fractions Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? 5.6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Measurement and Data Convert like measurement units within a given measurement system. 5.1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. Geometry Graph points on the coordinate plane to solve real-world and mathematical problems. 5.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the Represent and interpret data. plane located by using an ordered 5.7. Apply and extend previous pair of numbers, called its understandings of division to divide 5.2. Make a line plot to display a data set coordinates. Understand that the unit fractions by whole numbers and of measurements in fractions of first number indicates how far to whole numbers by unit fractions. a unit (1/2, 1/4, 1/8). Use operations on travel from the origin in the a. Interpret division of a unit fraction fractions for this grade to solve problems direction of one axis, and the by a non-zero whole number,and involving information presented in line second number indicates how far to compute such quotients. For plots. For example, travel in the direction of the second 5.4. Apply and extend previous understandings of example, create a story context for given different measurements of liquid in axis, with the convention that the multiplication to multiply a fraction or whole (1/3) ÷ 4, and use a visual fraction identical beakers, find the amount of names of the two axes and the number by a fraction. model to show the quotient. Use the liquid each beaker would contain if the coordinates correspond (e.g., x-axis a. Interpret the product (a/b) × q as a parts of a relationship between multiplication total amount in all the beakers were and x-coordinate, y-axis and ypartition of q into b equal parts; equivalently, as and division to explain that (1/3) ÷ 4 redistributed equally. coordinate). the result of a sequence of operations a × = 1/12 because (1/12) × 4 = 1/3. q ÷ b. For example, use a visual fraction model to Geometric measurement: understand show (2/3) × 4 = 8/3, and create a story context concepts of volume and relate volume to 5.2. Represent real world and for this equation. Do the same with (2/3) × (4/5) b. Interpret division of a whole multiplication and to addition. mathematical problems by graphing = 8/15. (In general, (a/b) × number by a unit fraction, and points in the first quadrant of the (c/d) = ac/bd.) compute such quotients. For 5.3. Recognize volume as an attribute of coordinate plane, and interpret b. Find the area of a rectangle with fractional side example, create a story context for solid figures and understand concepts of coordinate values of points in the lengths by tiling it with unit squares of the 4 ÷ (1/5), and use a visual fraction volume measurement. context of the situation. appropriate unit fraction side lengths, and model to show the quotient. Use the a. A cube with side length 1 unit, called show that the area is the same as would be found relationship between multiplication a “unit cube,” is said to have Classify two-dimensional figures by multiplying the side lengths. Multiply and division to explain that 4 ÷ (1/5) “one cubic unit” of volume, and can be into categories based on their fractional side lengths to find areas of rectangles, = 20 because 20 × (1/5) = 4. used to measure volume. properties. and represent fraction products as rectangular b. A solid figure which can be packed areas. without gaps or overlaps using n unit 5.3. Understand that attributes cubes is said to have a volume of n belonging to a category of 5.5. Interpret multiplication as scaling (resizing), c. Solve real world problems cubic units. twodimensional figures also belong by: involving division of unit fractions to all subcategories of that category. a. Comparing the size of a product to the size of by non-zero whole numbers and 5.4. Measure volumes by counting unit For example, all rectangles have one factor on the basis of the size of the other division of whole numbers by unit cubes, using cubic cm, cubic in, cubic ft, four right angles and squares are factor, without performing the fractions, e.g., by using visual and improvised units. rectangles, so all squares have four indicated multiplication. fraction models and equations to right angles. b. Explaining why multiplying a given number by represent the problem. For example, 5.5. Relate volume to the operations of a fraction greater than 1 results in a product how much chocolate will each multiplication and addition and solve real 5.4. Classify two-dimensional greater than the given number (recognizing person get if 3 people share 1/2 lb of world and mathematical problems figures in a hierarchy based on multiplication by whole numbers greater than 1 chocolate equally? How many 1/3- involving volume. properties. as a familiar case); explaining why multiplying a cup servings are in 2 cups of raisins? given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.