## Mathematical practices

Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

## Related Lessons

Lesson 24

Use the number line to place, compare, and order decimals to the hundredths. Use estimation and an algorithm based on place value to add decimals to hundredths. Use the number line to place, compare, order, and add decimals to the hundredths.

Lesson 23

Match visual models to the numbers they represent in fraction and decimal format. Solve problems with decimal fractions. Match visual models to the numbers they represent in fraction and decimal format.

Lesson 22

Use visual models and equations to solve problems involving multiplying and dividing fractions. Use models & equations to solve problems involving multiplying and dividing fractions.

Lesson 21

Use visual models, repeated addition, and multiplication to solve problems with equal sized groups. Compare the strategies to see which one is most efficient. Use visual models, addition, & multiplication to solve problems with equal sized groups.

Lesson 20

Use estimation and visual models to solve problems involving adding fractions with unlike denominators. Solve problems involving adding fractions with unlike denominators.

Lesson 19

Put together the fraction puzzle pieces by matching the visual model, expression, and fraction representations. Match visual models, expressions, and fractions to explore adding fractions.

Lesson 18

Use visual models and benchmark fractions to help compare and order fractions with unlike numerators and denominators. Use a number line and benchmark fractions to compare and order fractions.

Lesson 14

Use estimation and visual models to solve problems involving adding fractions with unlike denominators. Use visual models to solve problems involving adding fractions with unlike denominators.

Lesson 13

Divide two- and three-digit numbers by a single-digit number using strategies based on place value (partial quotients). Divide multi-digit numbers by a single-digit number using strategies based on place value.

Lesson 10

Explore side length, area, and volume and apply the formulas V = l × w × h and V = B × h for rectangular prisms to solve problems. Explore side length, area, and volume and apply the formulas for volume to solve problems.

Lesson 08

Use visual models, decompose shapes, and apply the area formula for rectangles to solve problems in real world contexts. Use visual models & the area formula for rectangles to solve problems.

Lesson 07

Decompose shapes into rectangles to determine the area of the composite shape using the area formula for rectangles. Decompose shapes into rectangles to determine the area of the composite shape.

Lesson 6

Decompose shapes into rectangles to determine the perimeter of the composite shape using the perimeter formula for rectangles. Decompose shapes into rectangles to determine the perimeter of the composite shape.

Lesson 5

Use attributes of quadrilaterals (number of sides and type of angles) to sort and classify shapes. Use attributes of quadrilaterals to sort and classify shapes.

Lesson 4

Use your body and form several yoga poses to help create, describe, and name different types of triangles. Name their angles. Move into yoga poses to help create, describe, and name different types of triangles.

Lesson 3

Move on a physical place value chart to help understand that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Move on a physical place value chart to understand the "times 10" pattern.

Lesson 2

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right. Hunt for patterns in the number of zeros of the product when multiplying a number by powers of 10. Hunt for patterns in the product when multiplying a number by 10.

Lesson 01

Multiply a two-digit by a one-digit whole number using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Multiply whole numbers using place value strategies & the properties of operations.

Lesson 303

Compare fractions with the same numerator and different denominators. Compare and order fractions using bar models. Represent one whole using a fraction with equal numerator and denominator.