Math

Students identify, represent, and interpret multiplicative comparisons in patterns, tape diagrams, multiplication equations, measurements, and units of money.  They describe relationship between quantities as times as much as or use other language as applicable to given context (e.g., as many as, times as long as, and times as heavy as ).  Students use multiplication or division to find an unknown quantity in a comparison.

Students name the place value units of ten thousand, hundred thousand, and million.  They recognize the multiplicative relationship between place value units-the value of a digit in one place is ten times as much as the value of the same digit in the place to its right.  Students write and compare numbers with up to 6 digits in standard, expanded, word, and unit forms.

Students name multi-digit numbers in unit form in different ways by using smaller units, and they find 1 more or 1 lesson of a given unit in preparation for round on a vertical number line.  Students round four-digit, five-digit, and six-digit numbers to the nearest thousand, ten thousand, and hundred thousand.  They determine an appropriate rounding strategy to make useful  estimates for a given context.

Students build fluency with addition and subtraction of numbers of up to 6 digits by using the standard algorithm.  They add and subtract to solve two-step and multi-step word problems.  The Read-Draw-Write process is used to help students make sense of the problem and find a solution path.  Throughout the topic, students round to estimate the sum or difference and check the reasonableness of their answers.

Students use multiplicative comparisons to describe the relative sizes of metric units of length, mass, and liquid volume.  They express larger units in terms of smaller units and complete conversion tables.  Students add and subtraction mixed unit measurements.

Students begin multi-digit multiplication and division by multiplying and dividing multiples of 10 by one-digit numbers. They develop conceptual understanding by representing the multiplication and division with place value disks or on a place value chart and by naming multiples of 10 in unit form. Students multiply and divide by using equations. They apply the associative property of multiplication, which allows them to make use of familiar facts to multiply (e.g., 5 × 60 = 5 × 6 × 10 = (5 × 6) × 10).  To divide, students rename two-digit and three-digit multiples of 10 as tens to make use of familiar division facts, and they relate division to an unknown factor problem. Students multiply and divide to find the area or unknown side length of a rectangle with the newly formalized area formula A = l × w.

Students multiply two-digit numbers by one-digit numbers by using the distributive property. They decompose the two-digit numbers into tens and ones and then multiply each part by the one-digit number. Students use a place value chart, an area model, and equations to represent the multiplication. Equations are written in both unit form and standard form to record the multiplication represented in the models. When multiplying by using equations, they write or think about numbers in unit form and recognize that, although the units change, the multiplication facts are familiar. Students apply their learning to solve one-step word problems. An optional lesson at the end of the topic provides an opportunity for students to use simplifying strategies, such as compensation and decomposition, to multiply.

Students divide two-digit and three-digit numbers by one-digit numbers by using the break apart and distribute strategy. They decompose the two-digit or three-digit total into tens and ones and then divide each part by the one-digit number. Students use a place value chart, an area model, and equations to represent the division. Equations are written in both unit form and standard form to record the division represented in the models. When dividing by using equations, students write or think about the numbers in unit form. Students recognize that, although the units change, the division facts are familiar. They apply their learning to solve one-step word problems. Three-digit division in this module is limited to expressions such as 126 ÷ 3 where the number of hundreds is less than the divisor.

Students express larger customary units of length (i.e., yards and feet) in terms of smaller units (i.e., feet and inches) by using tape diagrams, number lines, and conversion tables. They focus on the relationships between the units and use repeated addition, skip-counting, and multiplication to complete the conversions. Students also convert mixed units to a smaller unit (e.g., yards and feet to feet) and add and subtract with mixed units by using number lines, number bonds, the arrow way, and equations. The measurement units provide a context for investigating the perimeter of rectangles. The relationships between the side lengths and perimeter of a rectangle are formalized. Students apply their understanding of customary units of length and the formulas for area and perimeter to solve problems including those that have both additive and multiplicative comparisons. 

Students use what they know about multiplication and division to help them identify factors, multiples, prime numbers, and composite numbers within 100. They list factor pairs by using arrays, division, and the associative property. Number bonds and equations help students to organize their thinking. They skip-count to find multiples and recognize relationships between factors and multiples—a number  is a multiple of each of its factors and factors can be used to find other factors. Students explore properties of prime and composite numbers within 100. They apply their understanding of factors and multiples to determine whether a number is divisible by another number and to find an unknown term in shape or number patterns.

Students multiply and divide multiples of 10, 100, and 1000 by focusing on place value units. They use place value disks and write equations in unit form to help them recognize that they can use familiar multiplication and division facts to find products and quotients. Application of the associative property helps students to rewrite two-factor multiplication expressions as three-factor expressions so, again, they can multiply by using familiar facts. To help prepare for multiplication of two-digit numbers by two-digit numbers, an area model is used to show that multiplying a multiple of 10 by a multiple of 10 results in a number with the unit of hundreds.

Students divide numbers of up to four digits by one-digit numbers. They draw an area model, represent the divisor as one side length, and compose the unknown side length by building up to the total. Students also represent the division on a place value chart. They decompose the totals into place value units, divide each unit, and record long division in vertical form alongside the place value chart to reinforce conceptual understanding. They recognize that, although the value of the unit is different, the process of dividing each unit remains the same.

Students apply the distributive property to multiply numbers of up to four digits by one-digit numbers. They break apart the larger factor by place value and multiply the number of each unit by the one-digit factor. They represent the multiplication by using place value charts, area models, and vertical form. Students record partial products in vertical form by recording each partial product separately and by recording them together on one line.

Students apply the associative and distributive properties to multiply a two-digit number by a multiple of 10 and then progress to multiplying two-digit numbers by two-digit numbers. Area models are used to represent the multiplication and to help students recognize how each factor is broken apart and multiplied. Students see that each part of one factor is multiplied by each part of the other factor. They record four partial products in the area model and in vertical form alongside the area model and then transition to recording two partial products in the same way. Students add the partial products to find the product.

Students use multiplicative relationships to convert units of time and customary units of weight and liquid volume to smaller units. They use conversion tables and number lines to express larger measurement units in terms of smaller units and recognize that the smaller units are all multiples of the same number. Students notice relationships in the conversion tables and use the tables to convert other amounts. Throughout the topic, students add and subtract mixed units by using different methods including the method of expressing larger units in terms of smaller units before adding or subtracting and the method of adding or subtracting like units, renaming as necessary.

Students divide with numbers that result in whole-number quotients and remainders. They recognize the remainder as the amount remaining after finding a whole-number quotient, and they solve word problems that require interpretation of the whole-number quotient and remainder. Students estimate quotients by finding a multiple of the divisor that is close to the total and then dividing. They reason about the relationship between their estimate and the actual quotient and apply their thinking to assess the reasonableness of their answers to division word problems. Students use the four operations to solve multi-step word problems. They draw tape diagrams that represent the known and unknown information in the problem to help them find a solution path. After solving, students assess the reasonableness of their answers.

Students decompose fractions into a sum of unit fractions and into a sum of non-unit fractions. They use familiar models such as number bonds, tape diagrams, and number lines to represent fractions. They recognize that the area model may be a useful model to represent fractions. Students decompose fractions greater than 1 into a sum of a whole number and a fraction  less than 1. This decomposition helps students rename fractions greater than 1 as equivalent mixed numbers. Students express mixed numbers as a sum of a whole number and a fraction less than 1. Then they rename the whole number as an equivalent fraction that they then add to the fraction. This helps students rename mixed numbers as equivalent fractions greater than 1.

Students generate equivalent fractions and equivalent mixed numbers. They decompose fractional units to find an equivalent fraction with smaller units and record their work with multiplication. They compose fractional units to find an equivalent fraction with larger units and record their work with division. Students use area models, as well as tape diagrams and number lines, to represent fractions and compose or decompose fractional units to generate equivalent fractions.

Students use various methods to compare fractions less than 1, fractions greater than 1, and mixed numbers. They consider the relationship between the numbers and use what they know about unit fractions to compare fractions to benchmark numbers such as 0, 1, and 1. When the fractions have related numerators or denominators, students use what they know about generating equivalent fractions to rename one fraction to create a common numerator or common denominator. They rename both fractions as equivalent fractions to compare any two fractions. They use similar methods to compare fractions greater than 1 and mixed numbers.

Students estimate to assess reasonableness when solving word problems to establish an underlying theme for the topic. They add and subtract fractions with like units and subtract a fraction from a whole number. Using unit form and a number line helps students relate their previous work with adding and subtracting whole numbers to adding and subtracting fractions. They see similar part-total relationships and the importance of adding and subtracting like units. Students may apply their part-total understanding to think of a subtraction problem as an unknown addend problem. In an optional lesson, students add fractions with related units by generating equivalent fractions.

Students add a fraction to a mixed number and add two mixed numbers. They also subtract a fraction from a mixed number and subtract two mixed numbers. Students apply previously learned strategies for adding and subtracting whole numbers to add and subtract mixed numbers. They use number bonds, the arrow way, and an open number line to represent and record the addition and subtraction. Understanding a mixed number as a sum of a whole number and a fraction helps students as they compose and decompose mixed numbers to add and subtract. Students create and interpret line plots, including solving addition and subtraction problems with fractional data.

Students use what they know about multiplying whole numbers to multiply fractions and mixed numbers by whole numbers. They use unit form and the associative property to multiply a fraction by a  whole number ( e.g., 3 × _5= (3 × 5) sixths 6). To multiply a mixed number by a whole number, students express the mixed number as a sum and then apply the distributive property.

Within the familiar context of money, students use decimal points to record amounts of money as decimal numbers for the first time. Students see that numbers can be represented in different ways. They use tape diagrams, number lines, and area models to represent the fractional unit of tenths. They write tenths in unit form and fraction form and then see that decimal form is another way to write the numbers. Students decompose 1 one into 10 tenths and compose 10 tenths into 1 one by using familiar representations, including place value disks, and recognize tenths as both a fractional unit and a place value unit. They record mixed numbers of ones and tenths in unit form, fraction form, and decimal form.

Students decompose tenths into hundredths by using tape diagrams, number lines, and area models. They recognize hundredths as a fractional unit and write hundredths in fraction form and decimal form. Students see that the decomposition of 1 tenth as 10 hundredths and the composition of 10 hundredths as 1 tenth follows the same pattern as other place value units, and they recognize that hundredths is also a place value unit. 

Students compare decimal numbers by applying their prior understanding of whole number and fraction comparison and by using strategies of their choice. They justify their comparisons and see how different strategies can be used. Students then use an area model, number line, and place value disks to represent decimal numbers. Students express decimal numbers in decimal form, fraction form, and unit form. They compare numbers by using different strategies such as making like units, comparing the value of each digit starting with the largest unit, and using mental math strategies. Students apply their knowledge to compare mixed numbers and to order decimal numbers.

Students extend their understanding of fraction equivalence and fraction addition with like units to add fractions and mixed numbers with the unlike units of tenths and hundredths. They rename tenths as hundredths to create like units and then use familiar strategies to add. For example, they might combine like units or break apart an addend to make an easier problem. Students also solve word problems that require the addition of metric measurements or amounts of money expressed as decimal numbers. They express the decimal numbers in fraction form, add the numbers in fraction form, and then use decimal form within the solution statement.

Students define, name, and draw points, lines, line segments, rays, angles, parallel lines, perpendicular lines, and intersecting lines. They identify different types of angles and describe them in relationship to each other (e.g., an obtuse angle is larger than a right angle and smaller than a straight angle). Students use their understanding of the words parallel, perpendicular, and intersecting to identify and name relationships between lines, line segments, rays, and sides in polygons.

Students apply fractional understanding to see an angle as a fractional turn through a circle that is measured in degrees. They describe turns in real-world situations, and they refine their definitions of angle types to include degree measures. Students use protractors to measure and draw angles with accuracy and use benchmark angles to estimate the measures of angles.

Students recognize and apply the additive nature of angle measure to find the unknown measures of angles within figures without using a protractor. They use what is known and the part–total relationship to determine an unknown angle measure when right angles, straight angles, and angles of known measures are decomposed. Students extend the strategy to find the measures of multiple unknown angle measures around a point.

Students recognize, identify, and draw lines of symmetry. They identify attributes of polygons including side length and the presence or absence of pairs of parallel sides, pairs of perpendicular sides, and angle types to sort and classify them. Special emphasis is placed on triangles. Students classify triangles based on side lengths and angle measures and draw triangles based on given attributes.