Math Grade Level Summary
Grade: 3Year: 2016-2017
Summary: In 3rd grade, students need to not only understand how to work out different math concepts, but to ask “why” and “how” and “How else can this be done?” All students need to have the idea of number sense: What makes this number? What makes this problem true? Talking about numbers allows students to apply mathematical concepts to all aspects of life. 3rd grade will focus more on the exploration of multiplication, division, measurement, fractions and not as much focus should be on addition and subtraction facts and memorization. To develop rigor and perseverance as a math student, the Standards for Mathematical Practice will be interlaced and addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among mathematical topics. In grade 3, instructional time should focus on six critical areas: Numbers and operations in base ten, the relationship between multiplication and division, patterns in addition and multiplication, geometry, fractions, and measurement. First, students build on what they have learned in 2nd grade studying numbers and operations in base ten. They will use place value to round whole numbers and fluently add and subtract within 100. Students will then begin developing an understanding of the relationship between multiplication and division. They will interpret products and quotients by describing real life applications of basic facts and understand and use the properties of multiplication, including the distributive property. Students will develop strategies for multiplication and division within 100. They will find patterns in addition and multiplication to deepen the understanding of what multiplication is conceptually, relating the operations to the concept of finding area. In geometry, students will develop an understanding of the structure of rectangular arrays and of area and describe and analyze two-dimensional shapes. 3rd graders will also develop an understanding of fractions, representing them as a whole split into equal parts on a number line, recognizing equivalent fractions, and comparing fractions with like denominators. Finally, as students prepare to move into 4th grade they solve problems involving measurement and estimation of intervals of time, money, liquid volumes, masses, and lengths of objects. Students also represent and interpret data and apply their understanding of fractions to read a ruler to the nearest ½ and ¼ inch.
Key Concepts: multiplication, division, fractions, measurement and patterns and relationships. Secondary concepts: Numeration (adding, subtracting, rounding, place value), Perimeter area volume, solids and shapes, money, decimals, congruent and symmetry, data, graphs and probability
Practice Standards at this grade level:
8 Standards of Mathematical Practice:
1. Make sense of problems and persevere in solving them: In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.
2. Reason abstractly and quantitatively: Third graders should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities.
3. Construct viable arguments and critique the reasoning: In third grade, students may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking.
4. Model with mathematics: Students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Third graders should evaluate their results in the context of the situation and reflect on whether the results make sense.
5. Use appropriate tools strategically: Third graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles.
6. Attend to precision: As third graders develop their mathematical communication skills, they try to use clear and precise language in their discussions with others and in their own reasoning. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the area of a rectangle they record their answers in square units.
7. Look for and make use of structure: In third grade, students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to multiply and divide (commutative and distributive properties).
8. Look for and express regularity in repeated reasoning: Students in third grade should notice repetitive actions in computation and look for more shortcut methods. For example, students may use the distributive property as a strategy for using products they know to solve products that they don’t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually evaluate their work by asking themselves, “Does this make sense?”
Timeline:
Unit 1: Numbers and Operations in Base 10 (4-5 of weeks)
· Use place value understanding to round two and three-digit whole numbers to the nearest 10 or 100
o Try this to practice this concept! https://www.georgiastandards.org/Georgia-Standards/Frameworks/3rd-Math-Unit-1.pdf
o Task 4: The Great Round Up! (p.36)
· Standard and expanded form of numbers within place value (up to 10,000)
· Mental strategies for addition and subtraction (number sense!)
o Number talks
o Sums of 100
o Combinations of 100
· Add two and three-digit whole numbers (limit sums from 100 to 1000) and/or subtract two and three-digit whole numbers from three-digit whole numbers
· Build on conceptual knowledge of the relationship between addition and subtraction
o Properties (commutative, identity, and associative)
· Order a set of whole numbers from least to greatest or greatest to least (up through 9,999 and with up to four numbers
· Problem solving within addition and subtraction
o Problems are placed within four basic categories: Joining, Separating, Part-Part Whole, and Comparing where either the result is unknown (2 + 3 = ?), change is unknown (2 + ? = 5), or start is unknown (? + 3 = 5)
o Limit to problems with whole numbers and having whole-number answers
· Assess the reasonableness of answers
· Patterns within the addition tables as well as multiplication tables (leading to multiplication)
· Compare total values of combinations of coins (penny, nickel, dime, and quarter) and/or dollar bills less than $5.00.
· Make change for an amount up to $5.00 with no more than $2.00 change given (penny, nickel, dime, quarter, and dollar).
· Round amounts of money to the nearest dollar.
Unit 2: The Relationship Between Multiplication and Division (6-7 of weeks)
· Determine factors of a product by creating equal groups of counters/tiles
· Use arrays to solve multiplication problems
· Interpret/describe whole number products of whole numbers (up to and including 10 x 10) by creating equal groups with manipulatives
o i.e.: Interpret 35 as the total number of objects in 5 groups, each containing 7 objects
o i.e.: Describe a context in which a total number of objects can be expressed as 5 x 7
· Interpret/describe whole number quotients of whole numbers (limit dividends through 50 and divisors to 10) by creating equal groups with manipulatives
o i.e.: Interpret 48 ÷ 8 as the numbers of objects in each share when 48 objects are partitioned equally into 8 shares
o i.e.: Describe a context in which a number of shares or a number of groups can be expressed as 48 ÷ 8
· Use multiplication and/or division to solve word problems in situations involving equal groups, arrays, and/or measurement quantities
· Assess the reasonableness of answers
· Determine the unknown whole number in a multiplication (up to and including 10 X 10) or division (limit dividends through 50), and limit divisors and quotients through 10) equation relating three whole numbers
· Interpret and/or model division as a multiplication equation with an unknown factor
o i.e.: Find 32 ÷ 8 by solving 8 x ? = 32
· Identify the missing symbol that makes a number sentence true
· Identify arithmetic patterns (including patterns in the addition table or multiplication table) and/or explain those using properties of operations
· Associative and commutative property of multiplication (not identification by name necessarily, by an understanding of the concept and ability to apply it)
Unit 3: Patterns in Addition and Multiplication (3-4 of weeks)
· Use multiplication and/or division to solve word problems in situations involving equal groups, arrays, and/or measurement quantities
· Assess the reasonableness of answers
· Solve two-step equations using order of operations (equation is explicitly stated with no grouping symbols)
· Create or match a story to a given combination of symbols and numbers
· Identify arithmetic patterns (including patterns in the addition table or multiplication table) and/or explain those using properties of operations
o i.e.: observe that 4 times a number is always even; explain why 4 times a number can be decomposed into two equal addends
· Associative and commutative property of multiplication (not identification by name necessarily, by an understanding of the concept and ability to apply it)
· Number sense, number sense, number sense!
Unit 4: Geometry (6-7 of weeks)
· Explain that shapes in different categories may share attributes and that the shared attributes can define a larger category.
o i.e.: A rhombus and a rectangle are both quadrilaterals since they both have exactly four sides.
· Recognize rhombi, rectangles, and squares as examples of quadrilaterals and/or draw examples of quadrilaterals that do not belong to any of these subcategories.
· Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
o i.e.: Partition a shape into 4 parts with equal areas or describe the area of each of 8 equal parts as 1/8 of the area of the shape.
· Perimeter of polygons
o Find the perimeter given the side lengths, finding an unknown side length, exhibiting rectangles with the same perimeter and different areas, and exhibiting rectangles with the same area and different perimeters. Use the same units throughout the problem.
· Measure area by counting unit squares
o Differentiate between area and perimeter (a common misconception or struggle area)
· Discovery through inquiry of the area formula
· Multiply side lengths the find areas of rectangles with whole-number side lengths in the context of real-world and mathematical problems
· Relate area to the operations of multiplication and division
Unit 5: Representing and Comparing Fractions (5-6 of weeks)
· Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
o Partition a shape into 4 parts with equal areas.
o Describe the area of each of 8 equal parts as 1/8 of the area of the shape.
· Conceptual understanding of fractions as numbers
o Demonstrate that when a whole or set is partitioned into y equal parts, the fraction 1/y represents 1 part of the whole
· Represent fractions on a number line diagram.
· Explain equivalence of fractions through reasoning with visual fraction models.
· Compare fractions by reasoning about their size
· Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line
· Recognize and generate simple equivalent fractions with denominators 2, 3, 4, 6, and 8
· Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers (i.e.: 3 = 6/2)
· Compare two fractions with the same numerator or the same denominator using reasoning about their size
· Recognize that comparisons are valid only when the two fractions refer to the same whole (i.e.: Is ½ ALWAYS larger than ¼?)
Unit 6: Measurement and Data (5-6 weeks)
· Tell and write time to the nearest minute
· Measure elapsed time intervals in minutes.
· Solve word problems involving addition and subtraction of time intervals in minutes
o Represent the problem on a number line diagram, drawing a pictorial representation on a clock face, etc.
· Measure and estimate liquid volumes and masses of objects using standard units (cups (c, pints (pt), quarts (qt), gallons (gal), ounces (oz) and pounds (lb)) and metric units (grams (g), kilograms (kg), and liters (l)).
· Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units (i.e.: utilize drawings (such as a beaker with a measurement scale) to represent the problem.
· Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories.
· Solve one and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs (i.e.: draw a bar graph in which each square in the bar graph might represent 5 pets
· Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch.
o Use a ruler to measure lengths to the nearest quarter inch or centimeter.
· Show data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters