Check Your Readiness

1.

How many inches are there in one foot?
How many centimeters are there in one meter?
How many feet are there in one yard?
How many meters are there in one kilometer?

Answer:

12
100
3
1,000

Teaching Notes

Throughout Unit 7.1, students work with length and area in a variety of contexts. They need to know how to convert units fluently and efficiently.

If most students struggle with this item, plan to use this problem to create an anchor chart early in the unit for students to use as they think about appropriate scales to use throughout this unit.

2.

There are 12 inches in 1 foot and 5,280 feet in 1 mile. Jada ran $3\frac14$ miles.

How many feet did Jada run?
How many inches did Jada run?

Answer:

17,160 ( $3\frac14 \boldcdot 5,280=17,160$ )
205,920 ( $12\boldcdot17,160=205,920$ )

Teaching Notes

Students perform multiple unit conversions, working with fractions and large numbers.

If most students struggle with this item, plan to practice unit conversions before doing Lesson 11. In Grade 6 Unit 3 Lessons 3 and 4, you will find activities to support conversions with fractions and large numbers.

3.

Jada drank 2 liters of water yesterday. Andre drank $\frac14$ times as much water as Jada drank. Lin drank three times as much water as Andre drank.

Did Andre drink more or less water than Jada drank? Explain how you know.
Did Lin drink more or less water than Jada drank? Explain how you know.

Answer:

Less. $\frac14<1$ , so multiplying by $\frac14$ makes a positive number smaller.
Less. Three times $\frac14$ of a quantity is the same as multiplying the quantity by $\frac34$ , and $\frac34<1$ , so the result is smaller.

Teaching Notes

The language of scaling appears in grade 5. While the scaling in this unit is geometric in nature, the language and thought processes associated with this grade 5 standard are very helpful for students.

If most students struggle with this item, plan to emphasize scaling language when launching and synthesizing Lesson 2 Activity 3, "Scaled Triangles." As students are examining the triangles, they will encounter the fractional multipliers 2, $\frac{1}{2}$ , and $\frac{2}{3}$ . MLR2: Collect and Display can be used to highlight student language such as "twice as big" and "half the size" and connect that language to scale factors.

4.

Each small square in the graph paper represents 1 square unit. Find the area of each figure. Explain your reasoning.

<p>Graph paper. Figure A.</p> — Figure A

<p>Graph paper. Figure B.</p> — Figure B

Answer:

Figure A has an area of $10\frac12$ square units. It can be divided into a 3-unit-by-3-unit rectangle (with an area of 9 square units) and a 1-unit-by-3-unit triangle (with an area of $1\frac12$ square units). Figure B has an area of $18\frac12$ square units. It can be surrounded by a 5-unit-by-5-unit square with two triangles removed. Those triangles have areas of 4 and $2\frac12$ square units. The area of Figure B is $18\frac12$ square units because $25−4−2\frac12=18\frac12$ .

Teaching Notes

One important use of scale models is to help students make calculations about the object represented. Part of these calculations use the scale, and other parts involve finding the lengths or areas of the model. This task makes sure students can effectively calculate the areas of complex shapes.

If most students struggle with this item, plan to do Lesson 6 Activity 3, "Area of Scaled Parallelograms and Triangles," focusing students on attending to the base and height of their polygon. During the activity launch, include a discussion on how students can determine the area of figures if they aren't sure of a formula. During the launch, review the pre-unit diagnostic item featuring students who decomposed and rearranged figures as well as students who found parallelograms and triangles and can speak to how they used the base and height. If students need more support with calculating the area of complex shapes, refer to Grade 6 Unit 1.

5.

This rectangle has side lengths $a$ and $b$ .

For each expression, say whether it gives the perimeter of the rectangle, the area of the rectangle, or neither.

$a \boldcdot b$
$a+b$
$a^2+b^2$
$2a+2b$

Answer:

Area
Neither (It’s half the perimeter)
Neither
Perimeter

Teaching Notes

Students have been calculating perimeter and area of figures since elementary school. In grade 6, they began to use variables and exponential notation in these formulas. This exercise checks students’ understanding of perimeter and area using variables.

If most students struggle with this item, plan to use this problem to review area and perimeter. Grade 6 Unit 1 Lessons 5 and 6 could also be used as review.

6.

A recipe for 1 pizza crust calls for 2 cups of flour, 9 tablespoons of water, and 2 teaspoons of olive oil. The recipe can be scaled up to make multiple pizza crusts. Complete the table to show the quantities to use for multiple pizza crusts.

number of pizza crusts	cups of flour	tablespoons of water	teaspoons of olive oil
1	2	9	2
2	4
5		45
	8

Answer:

number of pizza crusts	cups of flour	tablespoons of water	teaspoons of olive oil
1	2	9	2
2	4	18	4
5	10	45	10
4	8	36	8

Teaching Notes

Students scale figures up and down, which is similar to scaling recipes up and down. This item checks that students are comfortable representing scaling with a table.

If most students struggle with this item, plan to do Lesson 2 Activity 3, Scaled Triangles, with extra attention to Question 3 and the activity synthesis. The synthesis uses the table of scaled copies to help students begin thinking about and articulating scale factor.

7.

Here is a polygon. Draw a scaled copy of the polygon with scale factor 4.

Answer:

Answers vary. Sample response:

Teaching Notes

Some students may have seen or worked with scale drawings before. This item checks whether or not they can reproduce a drawing at a given scale on a grid.

If most students do well with this item, it may be possible to skip Lesson 3 Optional Activity 2.