Grade 8

Readiness Check

Check Your Readiness
1.

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

&lt;p&gt;Five congruent triangles in a grid.&lt;/p&gt;<br>  
<p>Five congruent triangles in a grid. Triangle 1 is located with a vertex at four units right and 7 units up. Another vertex at 3 units right and 10 units up. Another vertex at 2 units right and 9 units up. A triangle labeled A is flipped and moved to the right 2 from Triangle 1. A triangle labeled B is rotated down and shares a vertex with Triangle 1 at four units right and 7 units up. A triangle labeled C is moved down 6 units. A triangle labeled D is flipped and moved down on the grid.</p>  

A.

A

B.

B

C.

C

D.

D

Answer: B, D

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 reflection of Triangle 1. Triangle B is a rotation of Triangle 1. Triangle C 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 the triangles. This argument shows a decent understanding of both reflections and rotations, but is not technically correct since Triangle C cannot land exactly on Triangle 1 with a single rotation. Triangle D is a rotation of Triangle 1.

This language will become 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.

2.
  1. Identify all pairs of lines which appear to be parallel in the diagram.
  2. Identify all pairs of lines which appear to be perpendicular in the diagram.

&lt;p&gt;Five lines.&lt;/p&gt;<br>  
<p>Five lines. Line I is a horizontal line. Line J slants downward and to the right. Line F slants downward and to the right. Line G slants downward and to the right. Line H slants upward and to the right. Lines I, J, and H intersect at a single point. Lines H and F intersect at a single point. Lines F and I intersect at a single point. Lines H and G intersect at a single point.</p>  

Answer:

  1. gg and ff
  2. hh and gg; hh and ff

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.”

3.

  1. On the coordinate plane, plot and label these points:

    A:(2,1)A: (2,1),

    B:(5,1)B: (5,1),

    C:(7,2)C: (7,2),

    D:(4,2)D: (4,2)

    &lt;p&gt;X y plane, origin O. Horizontal axis, labelled x, scale from negative 1 to 10 by 1’s. Vertical axis, labelled y, scale from negative 1 to 10, by 1’s.&lt;/p&gt;<br>  

  2. What is the length of CDCD?
  3. What kind of quadrilateral is ABCDABCD?

Answer:

<p>Parallelogram ABCD.</p>

  1. See plotted points
  2. 3 units
  3. Parallelogram

Teaching Notes

Students plot points on a coordinate grid and find distances between points sharing the same xx-coordinate or the same yy-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.

4.

Lines ABAB and CDCD intersect at EE.

&lt;p&gt;Lines A B and C D intersect at point E. Angle A E C is 50 degrees.&lt;/p&gt;<br>  

  1. What is the measure of angle AEDAED? Explain how you know.

  2. What is the measure of angle DEBDEB? Explain how you know.

Answer:

  1. 130∘130^\circ. Angles AECAEC and AEDAED are supplementary, so their measures add up to 180∘180^\circ.
  2. 50∘50^\circ. Angles AECAEC and DEBDEB are vertical angles, so they have the same measure.

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 DEBDEB is adjacent to angle AEDAED 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.

5.

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.

  1. Side lengths of 2 cm, 3 cm, and 4 cm
  2. Side lengths of 2 cm, 3 cm, and 6 cm
  3. Angles 90∘90^\circ, 45∘45^\circ, and 45∘45^\circ
  4. Angles 90∘90^\circ, 60∘60^\circ, and 60∘60^\circ

Answer:

  1. Yes, it is possible. Answers vary.
  2. No, it is not possible.
  3. Yes, it is possible. Answers vary.
  4. No, it is not possible.

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 degrees.

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.

6.

Find the area of each parallelogram.

Figure A
&lt;p&gt;A quadrilateral on a grid.&lt;/p&gt;
<p>A quadrilateral on a grid. The figure has a vertex at two units right, two units up. Another vertex at 5 units right and 2 units up. Another vertex at 7 units right and 4 units up. Another vertex at 4 units right and 4 units up.</p>  

Figure B
&lt;p&gt;A quadrilateral on a grid.&lt;/p&gt;
<p>A quadrilateral on a grid. The figure has a vertex at 10 units right and 3 units up. Another vertex at 11 units right and 4 units up. Another vertex at 15 units right and 2 units up. Another vertex at 14 units right and 1 unit up.</p>  

Answer:

Parallelogram A: 6 square units. Sample reasoning:

  • The parallelogram is 3 units wide and 2 units high, so its area is 6 square units.
  • The shape can be decomposed into a 2-by-1 rectangle and 2 triangles that fit together to make a 2-by-2 square. So the total area is 2+4=62 + 4 = 6 square units.

Parallelogram B: 6 square units. Sample reasoning: The parallelogram can be surrounded by a 3-by-5 rectangle, and the areas of the missing triangles, which are 12+12+4+4\frac{1}{2} + \frac{1}{2} + 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 area of figures that aren't oriented with the grid. However, this skill will be more important in a future unit.

7.

Here are two triangles:

&lt;p&gt;Two triangles A B C and D E F on a grid. Let (0 comma 0) be the bottom left corner. Then the coordinates are: A(2 comma 5), B(4 comma 6), C(7 comma 4) and D(10 comma 2), E(13 comma 4), F(15 comma 3).&lt;br&gt;<br>  &lt;/p&gt;<br>  

Describe a way to move triangle ABCABC so that it matches up perfectly with triangle FEDFED.

Answer:

Answers vary. Sample response: Triangle ABCABC can be moved down 2 units and then flipped over a vertical line that lies halfway between the 2 triangles.

Teaching Notes

The formal definition of congruence is new in this unit, but the idea is intuitive. This item examines students' 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.