Readiness Check

Students find the distance between two points that share the same $x$- or $y$-coordinate in preparation for their work with the Pythagorean Theorem.

If most students struggle with this item, plan to spend more than the suggested time on the Warm-Up, and include grid paper or a coordinate plane for support. Ensure that students make the generalization that when the points share either the same $x$-coordinate or the same $y$-coordinate, we can subtract the coordinate that is not the same in both points to find the distance.

In this unit, students investigate whether or not $\sqrt 2$ is a rational number. To do this, they square various fractions in an attempt to get an answer of 2.

If most students struggle with this item, plan to revisit what it means to evaluate an algebraic expression given a value for the variable. Students may need to see an example prior to the Lesson 4 Warm-up of how we can substitute a given value into the expression. If students still struggle, show them how to successfully square fractions and decimals using a calculator, and provide four-function calculators for Lesson 4.

This problem involves practice squaring rational numbers in preparation for studying the square root of 2. An important idea here is that positive numbers less than 1 become smaller after being squared, while positive numbers greater than 1 become larger.

If most students struggle with this item, plan to spend time on the placement of rational numbers on the number line in the Activity Synthesis for Activity 3. Consider using this item as part of the Cool-down, replacing the $0.5^3$ with $0.5^2$.

In this unit, students convert decimal expansions that repeat into rational numbers. They may not have much experience doing the same with terminating decimals. If students struggle here, give them a little more practice in ramping up to the relevant lessons in this unit. Emphasize that fractions like $\frac 4 {10}$ are correct and are generally much simpler to find than an equivalent fraction in lowest terms.

If most students struggle with this item, plan to take opportunities starting in Lesson 2 to practice writing numbers in different forms. There are several opportunities to discuss which form of numbers is easier to deal with in given situations.

This unit includes work on decimals, including infinitely repeating decimals. Students may have difficulty with $\frac 1 9$ because it is a repeating decimal. Students using long division to tackle this problem should be successful, while students who look for benchmark comparisons or look for a power of 10 in the denominator need to try a different method.

If most students struggle with this item, plan to take opportunities starting in Lesson 2 to practice writing numbers in different forms. There are several opportunities to discuss which form of numbers is easier to deal with in given situations.

Students are asked to find the area of triangles and quadrilaterals on a grid in this unit using any strategy. Examples of strategies are drawing a surrounding rectangle and decomposing and rearranging. Students will decompose, rearrange, and calculate areas of squares and triangles when proving the Pythagorean Theorem.

If most students do well with this item, it may be possible to skip or move faster through the Warm-Up and Activity 3.

This problem anticipates students’ work with square and cube roots. The square root of a positive number $a$ is defined as the positive number whose square is $a$. Likewise, the cube root of a number $a$ is a solution to the equation $x^3 = a$. 

If most students struggle with this item, plan to begin the Warm-Up with an equation such as $x \boldcdot x=36$ and connect it to the equation $x^2=36$.

Note that students are not expected to "take the square root of each side" to solve equations like this. If needed, provide a list of perfect squares and cubes, or other opportunities to practice recognizing perfect squares and cubes, throughout the unit.