Grade 6

Readiness Check

Check Your Readiness

Here are three fractions: $\dfrac{3}{4}$ , $\dfrac{5}{6}$ , $\dfrac{6}{8}$ . Two of these fractions are equivalent to each other. Which two? Explain or show your reasoning.

Answer:

$\frac{3}{4}$ and $\frac{6}{8}$ are equivalent to each other. Sample reasoning:

3 and 4 can each be multiplied by 2 to get 6 and 8.
Here is a diagram:

Teaching Notes

Students identify equivalent fractions and explain what characteristics make the fractions equivalent. This concept ties in with nearly all the work in this unit and the next—equivalent ratios, ratio tables, scaling, unit rates, and constants of proportionality.

If most students struggle with this item, plan to use this item for some error analysis before beginning Lesson 5.

42 is 6 times what number?
700 is 7 times what number?
4 is 28 times what number?
Choose one of these questions and explain how you know that your answer is correct.

Answer:

7
100
$\frac{1}{7}$
Answers vary.

Teaching Notes

In their work with ratio and proportion, students will often need to consider questions of the form, “B is A times what number?” The second part requires reasoning with place value. The third part of this question involves reasoning with fractions.

If most students struggle with this item, plan to highlight the relationship between multiplication and division and support students in reasoning about an unknown factor. For instance, when discussing the question in Activity 3, “How many batches can you make with 15 cups of flour if one batch uses 5 cups of flour?”, consider rewording the question as “15 is 5 times what number?” or “5 times what number gives 15?” Consider asking students to write equations that can represent the situation, such as $5 \, \boldcdot \,?=15$ or $15 \div 5 = \,?$

Label each tick mark with its location on the number line:

Answer:

Each tick mark should be labeled with a multiple of 3 (3, 6, 9, 12, . . .).

Teaching Notes

Difficulty with placement of numbers and tick marks may be an indication that students need work with measurement conventions as discussed in the geometric measurement progression. This issue may need more instructional attention than what is currently given. In this case, students need to use proportional reasoning to figure out how to mark the three equally-spaced tick marks between 0 and 12. Students will use this skill when they study double number lines in the upcoming unit.

If most students struggle with this item, plan to expand Activity 1 by including a few examples of partitioning a number line.

Here is a number line:

Write the number at $A$ as a fraction.
Write the number at $A$ as an equivalent fraction.

Answer:

$\frac{1}{3}$ (or equivalent)
$\frac{2}{6}$ (or equivalent, but different from previous answer)

Teaching Notes

This question is intended to assess whether the student understands that the interval from 0 to 1 must be partitioned into $n$ parts of equal length in order for each subinterval to have length $\frac{1}{n}$ .

If most students struggle with this item, plan to expand Activity 1 by including a few examples of equivalent fractions.

To make one batch of cookies, these ingredients are needed:

cookie mix	eggs	water	oil
1 box	2 eggs	$\frac18$ cup	$\frac13$ cup

What amounts of eggs, water, and oil are needed to make 2 batches of cookies?

Answer:

4 eggs, $\frac{1}{4}$ cup water, $\frac{2}{3}$ cup oil (or equivalent)

Teaching Notes

To answer the question, students will need to keep track of this information as they scale up the recipe from one batch to two batches. In addition to the conceptual work of scaling, this problem assesses students’ comfort with multiplying a fraction by a whole number.

If most students struggle with this item, observe how students think about scaling in Activity 3 and make note of areas of confusion. Then, in Lesson 5 Activity 2, students will practice multiplying a fraction times a whole number.

Which skateboarder is faster: Jada, who travels 2 miles in 12 minutes at a constant speed, or Clare, who travels 2 miles in 10 minutes at a constant speed? Explain how you know.

Answer:

Clare is faster because she covers the same distance that Jada covers but she covers it in less time.

Teaching Notes

This problem assesses students’ conceptual understanding of speed. Jada and Clare both skateboard the same distance, but Jada takes longer. The fact that a longer time results from a slower speed may seem counterintuitive to students who do not have a solid understanding of the relationship between time, rate, and distance. Students who have difficulty may need more real-life examples to help make this idea plausible to them.

If most students struggle with this item, plan to share a few examples similar to the one in the assessment before launching Activity 2.