Domain: Operations and Algebraic Thinking

Strand: Write and interpret numerical expressions.

Standard:

• • • (5.OA.1) Use parentheses to construct numerical expressions, and evaluate numerical expressions with these symbols.

    • (5.OA.2) Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

Examples & Resources:

• (5.OA.2) Express the calculation “add 8 and 7, then multiply by 2” as 2 x (8 + 7). Recognizing that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Strand: Analyze patterns and relationships.

Standard:

• (5.OA.3) Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

Examples & Resources:

• (5.OA.3) Given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Domain: Number and Operations - Fractions

Strand: Use equivalent fractions as a strategy to add and subtract fractions.

Standard:

• • • (5.NF.1) Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

    • (5.NF.2) Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem). Use benchmark fractions and number sense of fractions to estimate mentally and check the reasonableness of answers.

Examples & Resources:

• • • (5.NF.1) 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

    • (5.NF.2) Recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Literature Connections:

• Spaghetti and Meatballs for All! by Marilyn Burns

• Speed Mathematics: Secret Skills for Quick Calculation by Bill Handley

Strand: Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Standard:

• • • (5.NF.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.

    • (5.NF.4) Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

• (5.NF.5) Interpret multiplication as scaling (resizing), by:

a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a 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 (division of a fraction by a fraction is not a requirement at this grade).

• (5.NF.6) Solve real world problems involving multiplication of fractions and mixed numbers (e.g., Use visual fraction models or equations to represent the problem).

• (5.NF.7) Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

b. Interpret division of a whole number by a unit fraction, and compute such quotients.

• Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions (e.g., by using visual fraction models and equations to represent the problem).

Examples & Resources:

• • • (5.NF.3) 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.NF.4a) Use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b ) x (c/d) = ac/bd).

    • (5.NF.7a) Create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationsip between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.

    • (5.NF.7b) Create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

    • (5.NF.7c) How much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Literature Connections:

• • • Anno's Mysterious Multiplying Jar by Masaichiro & Mitsumasa Anno

    • Eating Fractions by Bruce McMillan

Domain: Geometry

Strand: Graph points on the coordinate plane to solve real-world and mathematical problems.

Standard:

• • • (5.G.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 plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y coordinate).

    • (5.G.2) Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Examples & Resources:

Literature Connections:

• Grandfather Tang's Story by Ann Tompert

• The King's Chessboard by David Birch

• The Librarian Who Measured the Earth by Kathryn Lasky

• The Boy Who Reversed Himself by William Sleator

• Sea Clocks: The Story of Longitude by Louise Borden

Strand: Classify two-dimensional (plane) figures into categories based on their properties.

Standard:

• (5.G.3) Understand that attributes belonging to a category of two-dimensional (plane) figures also belong to all subcategories of that category.

• (5.G.4) Classify two-dimensional (plane) figures in a hierarchy based on attributes and properties.

Examples & Resources:

• (5.G.3) All rectangles have four right angles and squares are rectangles, so all squares have four right angles.

Use:

• • • geo solids

    • geo boards

    • paper models