Illustrate and explain the product of two factors using equations, rectangular arrays, and area models.
Use strategies based on place value, properties of operations, and/or the relationship between multiplication and division to find whole-number quotients and remainders with one-digit divisors and up to four-digit dividends.
Illustrate and/or explain quotients using equations, rectangular arrays, and/or area models.
Using area and length fraction models, explain why one fraction is equivalent to another, taking into account that the number and size of the parts differ even though the two fractions themselves are the same size.
Apply principles of fraction equivalence to recognize and generate equivalent fractions.
Example: $\frac{a}{b}$ is equivalent to $\frac{n \times a}{n \times b}$.
Compare two fractions with different numerators and different denominators using concrete models, benchmarks (0, $\frac{1}{2}$, 1), common denominators, and/or common numerators, recording the comparisons with symbols >, =, or <, and justifying the conclusions.
Explain that comparison of two fractions is valid only when the two fractions refer to the same whole.
Model and justify decompositions of fractions and explain addition and subtraction of fractions as joining or separating parts referring to the same whole.
Decompose a fraction as a sum of unit fractions and as a sum of fractions with the same denominator in more than one way using area models, length models, and equations.
Add and subtract fractions and mixed numbers with like denominators using fraction equivalence, properties of operations, and the relationship between addition and subtraction.
Solve word problems involving addition and subtraction of fractions and mixed numbers having like denominators, using drawings, visual fraction models, and equations to represent the problem.
Apply and extend previous understandings of multiplication to multiply a whole number times a fraction.
Model and explain how a non-unit fraction can be represented by a whole number times the unit fraction.
Example: $\frac{9}{8} = 9 \times \frac{1}{8}$
Extend previous understanding of multiplication to multiply a whole number times any fraction less than one.
Example: $4 \times \frac{2}{3} = \frac{4 \times 2}{3} = \frac{8}{3}$
Solve word problems involving multiplying a whole number times a fraction using visual fraction models and equations to represent the problem.
Examples: $3 \times \frac{1}{2}, 6 \times \frac{1}{8}$
Express, model, and explain the equivalence between fractions with denominators of 10 and 100.
Use fraction equivalency to add two fractions with denominators of 10 and 100.
Use models and decimal notation to represent fractions with denominators of 10 and 100.
Use visual models and reasoning to compare two decimals to hundredths (referring to the same whole), recording comparisons using symbols >, =, or <, and justifying the conclusions.
Interpret data in graphs (picture, bar, and line plots) to solve problems using numbers and operations.
Create a line plot to display a data set of measurements in fractions of a unit ($\frac{1}{2}, \frac{1}{4}, \frac{1}{8}$).
Solve problems involving addition and subtraction of fractions using information presented in line plots.
Select and use an appropriate unit of measurement for a given attribute (length, mass, liquid volume, time) within one system of units: metric - km, m, cm; kg, g, l, ml; customary - lb, oz; time - hr, min, sec.
Within one system of units, express measurements of a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.
Use the four operations to solve measurement word problems with distance, intervals of time, liquid volume, mass of objects, and money.
Solve measurement problems involving simple fractions or decimals.
Solve measurement problems that require expressing measurements given in a larger unit in terms of a smaller unit.
Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
Apply area and perimeter formulas for rectangles in real-world and mathematical situations.
Identify an angle as a geometric shape formed wherever two rays share a common endpoint.
Use a protractor to measure angles in whole-number degrees and sketch angles of specified measure.
Decompose an angle into non-overlapping parts to demonstrate that the angle measure of the whole is the sum of the angle measures of the parts.
Solve addition and subtraction problems on a diagram to find unknown angles in real-world or mathematical problems.
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines, and identify these in two-dimensional figures.
Identify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size.
