This task gives children an opportunity to subtract a three-digit number including a zero that requires regrouping. The solutions show how students can solve this problem before they have learned the traditional algorithm.

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

In this problem, the variables a,b,c, and d are introduced to represent important quantities for this esimate: students should all understand where the formula in the solution for the number of leaves comes from. Estimating the values of these variables is much trickier and the teacher should expect and allow a wide range of variation here.

This task requires students to find linear equations with different numbers of solutions.

This task a could be used as an introduction to writing and graphing linear inequalities. Part (a) includes significant scaffolding to support the introduction of the ideas. Part (b) demonstrates that, in some situations, writing down all possible combinations is not feasible.

The purpose of this task is to help students see the connection between aÖb and ab in a particular concrete example. The relationship between the division problem 3Ö8 and the fraction 3/8 is actually very subtle.

The price of copper fluctuates. Between 2002 and 2011, there were times when its price was lower than $1.00 per pound and other times when its priace was higher than $4.00 per pound. Copper pennies minted between 1962 and 1982 are 95% copper and 5% zinc by weight, and each penny weighs 3.11 grams. At what price per pound of copper does such a penny contain exactly one cent worth of copper?

Pennies have a monetary face value of one cent, but they are made of material that has a market value that is usually different. It is the value of the materials that requires attention in this problem. While it is interesting to compare the face value with the value of the materials, this does not have any bearing on the calculations. Interference between these two notions of value is a possible area of difficulty for some students.

As written, this problem gives students all of the information they need to estimate the thickness of a soda can.

his is a version of ''How thick is a soda can I'' which allows students to work independently and think about how they can determine how thick a soda can is. The teacher should explain clearly that the goal of this task is to come up with an ''indirect'' means of assessing how thick the can is, that is directly measuring its thickness is not allowed.

The equations in this task are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level (e.g, 6.EE.5). By asking students to reason about solutions without explicily solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

The purpose of this modeling task is to have students use mathematics to answer a question in a real-world context using mathematical tools that should be very familiar to them. The task gets at particular aspects of the modeling process, namely, it requires them to make reasonable assumptions and find information that is not provided in the task statement.

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. The task could also be used to generate a group discussion on interpreting functions given by graphs.

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match.

These problems are meant to be a progression which require more sophisticated understandings of the meaning of fractions as students progress through them.

The purpose of this task is for students to apply the concepts of mass, volume, and density in a real-world context.

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

Teachers who use this version of the task will need to bring tree leaves (or prepare a good sketch of a tree leaf) to class so that they can work on and discuss how to approximate the area of an irregular shape like a leaf.