Readiness Check

Students find an equation to represent a proportional relationship. Students may or may not use the equation in order to answer the question in Part C. A table would also be effective, but because the numbers are large, the equation may be more efficient. Because the unit rate is given, students might also answer Part C by doing the relevant arithmetic without referring to the equation.

If most students do well with this item, plan to emphasize, in Lesson 3, the value of knowing that the circumference and diameter of a circle have a proportional relationship. Ask students, "If I know that the circumference is always about three times the diameter, what else can I figure out? What if the diameter is 10 units? 100 units? 1 unit? How do you know?" "What if I know the circumference is 60 units?" Make sure that students understand the relationship between the patterns they noticed and the equation $C = \pi{d}$. The Warm-up in Lesson 4 gives another opportunity to assess students' understanding of the value of knowing the relationship. Optional Lesson 5 gives students more opportunities to practice using the proportional relationship.

The formulas for the circumference and area of a circle build naturally on simpler formulas for perimeters and areas of polygons. This item recalls what students have done with perimeter and area of a square. Students who select choice A or choice C have doubled only one side length of the rectangle to find the perimeter. Students who select choice B may have been thinking of perimeter when computing area, using the incorrect formula $A=2 \ell w$.

If most students struggle with this item, plan to incorporate these expressions into Activities 2 and 3 to connect how students used the squares' side lengths to find the perimeters and areas.

In this unit, students will measure circles in order to discover a relationship between the circumference and the diameter. As a result, they will need to deal with measurement error. This problem asks students to explain a discrepancy in measurement.

If most students struggle with this item, plan to incorporate reporting the measures in Activity 2 as millimeters as well as to the nearest centimeter. Ask how using different levels of precision changes how we report measurements. This will be important as students measure parts of a circle. A lack of precision may hide the relationships that students need to notice and use, and not understanding that some error is inherent in human measuring may mean that students discount the overall patterns.

In the second section of this unit, students decompose, rearrange, and enclose shapes while exploring different methods to find the area of a circle.  If most students struggle with this item, plan to use it as the Warm-up for Lesson 6. In grade 6, students learned a variety of strategies for finding areas in Unit 1 of Grade 6, and revisiting those strategies may be enough support. Be sure to call on students who used different strategies.

Students who select choice A found the area of the larger rectangle around the state plus the area of the tip of the decomposed state. Students who select choice B calculated the area of the larger rectangle containing the state: $563 \boldcdot 435$. Students who select choice D calculated the perimeter instead of the area.

If most students struggle with this item, plan to use it as part of the Warm-up for Lesson 6 as well. Ask about this item before asking about Item 4. In the other activities in the lesson, students practice decomposing figures to find areas. If students need more support, consider using activities from Unit 1 in Grade 6.

Students estimate the area of an irregular shape by comparing it with shapes composed of rectangles and triangles. Some students may have already encountered the formula for the area of a circle before and may use that as a response in Part C. If not, students might cut up unit squares to see how many fit in the circle—but it will still be close! Encourage students who want to spend a long time on Part C to do the rest of the assessment first, then come back to it. The idea of this problem is to get a sense of what strategies students use to compare areas.

If most students struggle with this item, it's not necessary to take action, but note their strategies as you prepare to teach lesson 7.