This problem provides an opportunity for students to reason about ratio and proportion in the realistic context of mixing a fruit drink from concentrate. The Teachers' Notes page offers suggestions for implementation, discussion questions, ideas for extension and support, and links to related problems (Blackcurrantiest is cataloged separately).

Patterns in Rational and Irrational Numbers: Standard 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
This formative assessment exemplar was created by a team of Utah educators to be used as a resource in the classroom. It was reviewed for appropriateness by a Bias and Sensitivity/Special Education team and by state mathematics leaders. While no assessment is perfect, it is intended to be used as a formative tool that enables teachers to obtain evidence of student learning, identify assets and gaps in that learning, and adjust instruction for the two dimensions that are important for mathematical learning experiences (i.e., Standards for Mathematical Practice, Major Work of the Grade).

In this problem students apply basic proportional reasoning in the context of a pie recipe. Given a recipe for 80 pies, Peter needs to determine whether the ingredients he has on hand are enough to make 2 pies. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, ideas for extension, and printable lists of ingredients (doc).

The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation. A formal explanation requires the idea of a limit to be made precise, but 7th graders can start to wrestle with the ideas and get a sense of what we mean by an "infinite decimal."

This three-act math task utilizes videos and questioning to help students explore multiplying decimals (scale).

This three-act math task utilizes videos and questioning to help students explore dividing decimals with whole number quotients.

This three-act math task utilizes videos and questioning to help students explore multiplying and dividing decimals.

In this video segment from Cyberchase, Harry must adjust his dinner budget after his cousin, Harry, unexpectedly decides to join him on his date

This task highlights a slightly different aspect of place value as it relates to decimal notation. More than simply being comfortable with decimal notation, the point is for students to be able to move fluidly between and among the different ways that a single value can be represented and to understand the relative size of the numbers in each place.

Why Be Rational?: Standard 8.NS.2 - Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
This formative assessment exemplar was created by a team of Utah educators to be used as a resource in the classroom. It was reviewed for appropriateness by a Bias and Sensitivity/Special Education team and by state mathematics leaders. While no assessment is perfect, it is intended to be used as a formative tool that enables teachers to obtain evidence of student learning, identify assets and gaps in that learning, and adjust instruction for the two dimensions that are important for mathematical learning experiences (i.e., Standards for Mathematical Practice, Major Work of the Grade).

This problem provides an opportunity to introduce a visual way of representing operations on unknown numbers to help lead students to using a symbolic representation. Learners are asked to think of a number and then through an interactivity are given a sequence of operational instructions to follow which leads all students to the same final number. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support.