Grade 8

Readiness Check

Check Your Readiness

Select all the triangles that can be rotated to match up with Triangle 1.

Triangle A

Triangle B

Triangle C

Triangle D

Answer: A, C

Teaching Notes

Some students already have an intuitive (if not rigorous) understanding of what rotations are. This item probes that understanding by having students identify rotated images of a given triangle. If students can answer this question correctly, then they already have a good intuition for rigid motions of the plane.

Triangle A is a rotation of Triangle 1. Triangle B is a reflection and translation of Triangle 1. Triangle C is a rotation of Triangle 1. Triangle D is a translation of Triangle 1. Some students may note that it's possible to combine a slide with a 360-degree rotation to match up Triangle 1 and Triangle D. This argument shows a decent understanding of both reflections and rotations, but is not technically correct since Triangle D cannot land exactly on Triangle 1 with a single rotation.

This language will be formalized and the concept of rotations will be developed over the span of several lessons. If most students do well with this item, it may be possible to move more quickly through the first two lessons in the unit.

Identify all pairs of lines which appear to be perpendicular in the diagram.
Identify all pairs of lines which appear to be parallel in the diagram.

<p>A diagram. Line r. Line t. Line s. Line m. Line p.</p>

Answer:

$s$ and $p$ ; $t$ and $p$
$s$ and $t$

Teaching Notes

Students identify parallel and perpendicular lines.

If most students struggle with this item, plan to use Lesson 3 Activity 1 to review the term “parallel” using the isometric grid paper. Lesson 5 Activity 3 provides an opportunity to review the term “perpendicular.”

On the coordinate plane, plot and label these points:
Point $A: (2,2)$ , Point $B: (2,6)$ , Point $C: (4,7)$ , Point $D: (4,3)$ .
What is the length of $AB$ ?
What kind of quadrilateral is $ABCD$ ?

Answer:

See plotted points
4 units
Parallelogram

Teaching Notes

Students plot points on a coordinate grid and find distances between points sharing the same $x$ -coordinate or the same $y$ -coordinate. The last part of the problem assesses whether students can identify a parallelogram.

If most students struggle with this item, plan to use Lesson 5 Activity 1 to review the coordinate plane and to consider how to describe translations. For extra practice, do the optional activity in Grade 7 Unit 5 Lesson 7. Grade 6 Unit 7 Lesson 11 has additional activities for more practice.

Lines $AC$ and $BD$ intersect at $E$ .

What is the measure of angle $BEC$ ? Explain how you know.
What is the measure of angle $AEB$ ? Explain how you know.

Answer:

$120^\circ$ . Angles $AED$ and $BEC$ are vertical angles, so they have the same measure.
$60^\circ$ . Angles $AED$ and $AEB$ are supplementary, so their measures add up to $180^\circ$ .

Teaching Notes

Students identify and use facts about adjacent and vertical angles to calculate angles. It is possible that students will use the fact that angle $DEB$ is adjacent to angle $AED$ to answer the second question. Check to see if students remember the vocabulary “vertical” and “supplementary,” since students may remember the properties without remembering those names.

If most students struggle with this item, plan to use Lesson 14 Activity 1 to review supplementary and vertical angles, making those terms explicit during discussion.

For each set of measurements, decide whether or not it is possible to draw a triangle with those measurements. If it is possible, draw the triangle.

Side lengths of 3 cm, 5 cm, and 9 cm
Side lengths of 3 cm, 5 cm, and 6 cm
Angles measures $90^\circ$ , $45^\circ$ , and $75^\circ$
Angles measures $90^\circ$ , $30^\circ$ , and $60^\circ$

Answer:

No
Yes; Answers vary
No.
Yes; Answers vary

Teaching Notes

In grade 7, students investigated whether it was possible to draw a triangle given a set of 3 conditions. When given three sides, they discovered that the sum of the lengths of the two shorter sides of a triangle must be greater than the length of the longest side. They also saw that only certain angle combinations were possible, but did not learn the result that the sum of the interior angles of a triangle is 180.

If most students do well with this item, it may be possible to skip the optional activity in Lesson 15. This activity asks students to collect data on different types of triangles and to conclude that the angles for each add to 180 degrees.

Provide access to measuring tools for this problem.

Find the area of each parallelogram.

Answer:

Parallelogram A: 12 square units.

Sample reasoning:

The parallelogram is 4 units wide and 3 units high, so its area is 12 square units.
The shape can be decomposed into a 3-by-3 square and 2 triangles that fit together to make a 3-by-1 rectangle. So the total area is $9+3=12$ square units.

Parallelogram B: 8 square units.

Sample reasoning:

The parallelogram can be surrounded by a 6-by-3 rectangle, and the areas of the missing triangles, which are $1+1+4+4$ , can be subtracted.

Teaching Notes

Students can use a variety of strategies to solve these problems. Some students may remember the formula for the area of a parallelogram, but it is not necessary that they do. Because of its positioning on the grid, Parallelogram B will require a more creative strategy than using the area formula. In the unit, students will use area as one way to reason about whether two figures might be congruent.

If most students do well on this item, plan to encourage students to explore the Are You Ready for More in Lesson 10. If most students struggle with the parallelogram on the left, plan to use Lesson 10 Activity 2 to explore and review the relationship between strategies for finding the area of a triangle and the area of a parallelogram. If most students struggle with the parallelogram on the right, briefly review strategies for finding the area of figures that aren't oriented with the grid. However, this skill will be more important in a future unit.

Here are two triangles. Describe a way to move triangle $ABC$ so that it matches up perfectly with triangle $DEF$ .

<p>A figure. Triangle ABC. Triangle DEF.</p>

Answer:

Sample response: Triangle $ABC$ can be moved right 8 units and then turned clockwise 90 degrees around point $F$ .

Teaching Notes

The formal definition of congruence is new in this unit, but the idea is intuitive. This item examines the student’s ability to visualize and verbalize the steps that take one figure to another.

If most students do well with this item, it may be possible to move fairly quickly through the first two lessons in the unit.