ALEX Classroom Resource

Students:

• Model and explain a fraction as a multiple of a unit fraction.

Example: 5/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 or 5 x 1/3 or (5 x 1)/3.
• Multiply a whole number times any fraction less than 1 and justify the product.

Example: 5 x 2/3 is 5 sets of two-thirds, which is ten-thirds. or 5 x 2/3 = 5 x (2 x 1/3) = (5 x 2) x 1/3 = 10 x 1/3 or 10/3.
• Solve word problems involving multiplying a whole number times a fraction using a visual model and equation to represent the problem.

Students know:

• Models or equations to represent multiplication situations.
• The fraction a/b is equivalent to the unit fraction 1/b being iterated or "copied" the number of times indicated by the numerator, a.

Students are able to:

• Model and explain how a non-unit fraction can be expressed as multiplication.
• Multiply a whole number times any fraction less than one.
• Solve word problems involving a whole number times a fraction using a visual fraction model and equation to represent the problem.

Students understand that:

• Previous work involving multiplication with whole numbers can be extended to fractions in showing multiplication as putting together equal-sized fractional groups.
• Problem solving situations involving multiplication of a whole number times a fraction can be solved using a variety of strategies, models, and representations.

Learning Objectives:
M.4.16.1: Recognize fractions in their simplest forms.
M.4.16.2: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
M.4.16.3: Demonstrate an understanding of fractional parts.
M.4.16.4: Apply properties of operations as strategies to multiply and divide.
M.4.16.5: Recall basic multiplication facts.
M.4.16.6: Define multiple.
M.4.16.7: Compare two fractions with the same numerator or the same denominator by reasoning about their size.
M.4.16.8: Recognize that comparisons are valid only when the two fractions refer to the same whole.
M.4.16.9: Record results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
M.4.16.10: Name the first ten multiples of each one-digit natural number.
M.4.16.11: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
M.4.16.12: Solve simple fractions using multiplication strategies.
M.4.16.13: Recognize equivalent forms of fractions.
M.4.16.14: Multiply proper fractions with common denominators 2-10.
M.4.16.15: Solve word problems using whole numbers.
M.4.16.16: Write number sentences for word problems.
M.4.16.17: Identify key terms in word problems.
M.4.16.18: Multiply and divide within 100.
M.4.16.19: Recall basic multiplication facts.

Students:
Given a fraction times a whole number,

• use visual models to illustrate the product to develop the procedure (a/b) × q.
• Create a story context for the equation (a/b) × q.

Given a fraction times a fraction,

• Use visual models to illustrate the product to develop the procedure (a/b) × (c/d).
• Create a story context for the equation (a/b) × (c/d).

Given a rectangle with two fractional side lengths,

• Use an area model to illustrate and find the rectangular area.
• Find the area by tiling it with unit squares of the appropriate unit fraction.

Given a rectangle with fractional side lengths including mixed numbers,

• Use an area model to illustrate and find the rectangular area to lead to answers in the form of whole numbers or mixed numbers.

Example: Using an area model, a rectangle with dimensions of 1 1/2 x 1 2/3 will have partial products of 1, 1/2, 1/3, and 1/6 and the sum of the partial products will give an area of 2 sq units.
• Use an area model to find the area of a rectangle by tiling the rectangle with unit squares.

Example: Using an area model, a rectangle with dimensions 1 1/2 x 1 1/3 will be tiled with unit squares of 1/6 size showing the tiled partial products as 6/6, 3/6, 2/6, and 1/6 for a total area of 12/6 sq units, so it would take 12 tiles of size 1/6 units to cover the area of the rectangle.

Students know:

• How to write an equation involving repeated addition with fractions as a multiplication equation of a whole number times the fraction.
  Example: 2/9 + 2/9 + 2/9 + 2/9 = 4 x 2/9 = 8/9.
• The relationship of partial products to an area model when multiplying by two whole numbers.
• Area of a rectangle is determined by multiplying side lengths and is found in square units.

Students are able to:

• Use previous understandings of multiplication to
• Find products of a fraction times a whole number and products of a fraction times a fraction.
• Use area models, linear models or set models to represent products.
• Create a story context to represent equations (a/b) × q and (a/b) × (c/d) to interpret products.
• Find area of rectangles with fractional side lengths and represent products as rectangular areas.
• Find the area of a rectangle by tiling the area of a rectangle with unit squares.

Students understand that:

• Any whole number can be written as a fraction.
• The general rule for multiplication involving fractions can be justified through visual models.
• A variety of contextual situations can be represented by multiplication involving fractions.
• Tiling with unit squares can be used to find the area of a rectangle with fractional side lengths.

Learning Objectives:
M.5.12.1: Define proper fraction.
M.5.12.2: Multiply fractions using denominators between 2 and 5.
M.5.12.3: Identify proper and improper fractions.
M.5.12.4: Recall basic multiplication facts.
M.5.12.5: Model changing a whole number to a fraction.
M.5.12.6: Partition a rectangle into rows and columns of same-size squares, and count to find the total number of them.
M.5.12.7: Label the numerator and denominator of a fraction.
M.5.12.8: Count the area squares for the length and width.
M.5.12.9: Identify the width and length of a rectangle.