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Expressions and Equations: Grade 8

Hook 1 8.EE.A.1

Prompt: Rewrite each expression without using the operation given in the expression and then evaluate. 

 

Many students need some scaffolding before jumping into the properties of powers. So here's an incredibly quick and simple hook with the potential for a review of not only exponential form, its vocabulary and the nature of exponential growth but also the difference between multiplication (repeated addition) and exponential form (repeated multiplication). Evaluating exponential expressions using addition rather than multiplication may be one of the most common errors we see in 8th grade math. By allowing students to think on their own about the difference between the two rewritten expressions, this hook serves as a pre-emptive strike and will hopefully help to minimize such errors before they take root.

Video: Pyramid and Ponzi Schemes (1:29)

Hook 2 8.EE.A.1

Level 1 Prompt: Write down three things these three problems have in common and then create your own equation following the patterns you observed. 

Level 2 Prompt: Write in words, being as specific as possible, the pattern you see.

Level 3 Prompt: Explain why mathematically the pattern you see works.

Level 4 Prompt: Write an algebraic rule to represent the pattern you see.

This hook gives students the opportunity to define the property of multiplying like bases themselves. At the basic level, students identify patterns; the need for common bases, the addition of the exponents and the fact that the base remains the same in the product. In both speaking and writing, ask for specificity and proper vocabulary where appropriate to emphasizes the need in mathematics to "attend to precision". If we want our students to think of themselves as mathematicians we must ask them to behave like mathematicians. In the level 3 prompt, students may prove the rule mathematically by introducing expanded form into the conversation and clarifying why the pattern exists. For students with a solid grasp of the property, the level 4 hook offers the opportunity to write an algebraic representation, which requires a solid understanding of both exponents and the use of variables to express properties of numbers. No matter which level you assign to your groups, you want to give them the time to struggle, share ideas and questions. Answer their questions with questions of your own. "What do you see?", "Can you explain out loud what your are thinking before you write it in down?", "How would you explain exponential form to a 5th grader?", "How do we use variables to represent any number in algebra?". This hook should force students to think deeply about their understanding of exponential form and how that knowledge is related to the property of multiplication of expressions in exponential form.

Hook 3 8.EE.A.1

Level 1 Prompt: Write down two things these three problems have in common then create your own equation following the patterns you observed. 

Level 2 Prompt: Write in words, being as specific as possible, the pattern you see.

Level 3 Prompt: Explain why mathematically the pattern you see works.

Level 4 Prompt: Write an algebraic rule to represent the pattern you see.

Obviously the goals here are identical to those of hook 2. If students completed hook 2, this hook should give them the opportunity to improve in the specificity of their discussions and hopefully move up a level (in terms of prompts) in their understanding of properties and their representations. 

Hook 4 8.EE.A.1

Prompt 1: Explain how an equation with only positive exponents could result in a quotient with negative exponents.

Prompt 2: Explain the meaning of negative exponents.

Prompt 2: What does it mean if an exponential expression has an exponent of zero?

We're revving up the rigor for this prompt. If you feel it's too much for your students, I'd simply go back to the prompts from Hooks 2 and 3. However, if your kids are seeing the patterns, identifying the rules and representing them algebraically, these hooks challenge them to explore new ideas; negative exponents and a number taken to the power of zero. Students may not get to these properties on their own, which is okay. Exploring their own ideas, they may deduce some essential ideas about zero and negative exponents. These ideas may help them to internalize these properties quicker and with a deeper understanding once you unveil the rules either through discussion or whole class instruction.

Hook 5 8.EE.A.1

Prompt 1: Explain how each expression proves the property of zero as an exponent.

Prompt 2: Create an algebraic equation to model a rule that would apply to each expression.

By applying the previous properties of exponents, students have the opportunity to prove to themselves that any base raised to the zero power must equal 1. The aim of the exercise is to move students past the memorization of the properties of algorithms and into a deeper understanding of the rules. However, there's no doubt students may get stuck. Remind them that mathematicians look for patterns and ask them to evaluate each expression both in standard and exponential form. Ask them what they see in their answers. Give them time to think, discuss and question. In short, don't give them the answer. Let them work for it. Once they've gone as far as they can go, you can have a class discussion that generates buzz from their productive struggle. 

Video: History of Zero (3:52)

Hook 1 8.EE.A.2

Prompt: Write three questions you have about the expressions above. 

Students struggle with the concept of squares and cubes so the key to this hook is to give them the chance to sit with their ideas, reflect and write questions before they dive into the nature of squares and cubes. Of course this is also an excellent opportunity to formatively assess students' depth of knowledge about roots (and/or irrational numbers).  For beginners, it'll be more helpful to place perfect squares and cubes in the problem to simplify things. For intermediate and advanced students who may be familiar with the symbols, placing non-perfect squares and cubes in each expression should raise the level of challenge and rigor of discussion. Also look at Hook 1 8.NS.A.2 for help with introducing the square root function.

Hook 2 8.EE.A.2

Prompt: Use a sign you know or make one up and explain what it would do to a number.

As I mentioned in the previous hook (Hook 1 8.EE.A.2) , students struggle with the meaning of square and cube roots. By thinking about them as inverse relationships, students can contextualize them into their prior knowledge of squares and cubes. This hook allows some creativity in making up a sign, if needed, while simultaneously giving students the opportunity to think about what function the inverse of a square would perform. If students think deeply about what we are doing in math, rather than simply the rules of math, they will deepen their understanding of concepts and strengthen their skills.

Hook 1 8.EE.A.3

Prompt: Use a calculator to calculate each product. Then write a rule to describe the pattern you see.

 

Here again is an opportunity to scaffold. Every year I've done this exercise with 8th graders and every year I'm disappointed to see how many students find their calculations to be a revelation. Scientific notation will never make sense to students who do not have a solid understanding of multiplication by multiples of ten. A second important element to this hook is that students observe that the decimal place moves to the right each time, which is why the hook intentionally includes a decimal to the thousandth rather than a whole number. It acts as a preemptive strike against the misconception that the exponent in scientific notation indicates how many zeroes to add to the leading number. 

Video: Babylonian Base 60 Number System (3:51)

Hook 2 8.EE.A.3

Prompt: Write three things that numbers written in scientific notation have in common.

Students will jump all over the use of multiplication and the base 10. The challenge will be to identify the fact that the leading numbers are all less than ten and greater than or equal to 1. Demand specificity and mathematical vocabulary (their response should be a compound inequality) in your students' responses and don't allow them to settle for generalities like "there's only one number in front of the decimal". Mathematics is an exact science and students should practice being exact in their use of language. Let them work for the answer and they will build a strong conceptual foundation of scientific notation.

 

Video: History of Numbers (5:07) 

Hook 1 8.EE.A.4

Prompt 1: Write a rule for multiplying numbers written in scientific notation.

Prompt 2: Use the properties of multiplication to prove that each statement is true.

To begin, adjust the given scientific notation equations to match your students' level of skill. Ideally, you want the task to be just above their skill level so that they are appropriately challenged. The third problem throws a curve ball at students because the answer is written in scientific notation but it does not show the conversion that took place to write this answer. By reviewing the associative property, (either by their own discovery, in discussion or whole class instruction) students will move beyond memorization and have an opportunity to gain a conceptual understanding of the algorithm we use to multiply values written in scientific notation.

Hook 2 8.EE.A.4

Prompt: Create an algorithm to divide numbers written in scientific notation.

Building on students' understanding of multiplying numbers written in scientific notation, this hook challenges students to discover an algorithm for division of numbers written in scientific notation. The rigor has been raised, compared to the previous hook, by posing the questions in non-scientific notation form. Thus students aren't simply looking for patterns, they're being asked to create the equations and quotients themselves. Again, we're expecting them to be mathematicians and to perform the rigorous tasks associated with mathematics. In this case, to convert the expression and quotient to scientific notation and then analyze their work for patterns in order to identify the algorithm.

Hook 1 8.EE.B.5

Prompt: A pilot's dashboard has a malfunctioning instrument. Which point of data above is not working correctly?

This hook opens up discussion of how a unit rate presents itself in the different representations of a proportional relationship. Students' discussions and sticking points provide powerful formative assessment in terms of their understanding of each representation and their readiness to make comparisons of different proportional relationships. For those students who struggle to get started, you want to guide them through inquiry. Ask them what they know and, if needed, focus their attention on the speedometer. Once they recognize the speed of the plane, they may be able to make sense of the other representations. Of course, different students need different levels of support so heterogenous grouping is always a good idea to build in some of that support. 

Video: What is a Data Scientist? (1:48)

Hook 1 8.EE.B.6

Prompt: See GRAPHIC ORGANIZER below.

This standard asks students to use similar triangles to observe the nature of slope and the equations of linear relationships. Therefore, this hook asks your students to draw similar triangles and then study their features. How much guidance you give your students will depend on their need. With my students, who were on average right at proficiency or slightly below, I would not offer any verbal guidance but I would pass out graph paper and a ruler. There's no doubt struggling students will need more but push them to go as far as they can on their own before suggesting ideas such as graphing the coordinates, comparing the side lengths of the triangles or analyzing the slopes of each hypotenuse. The graphic organizer alludes to the fact that time needs to be taken for this problem. However, inserting specific time frames (1 minute, 5 minutes, 10 minutes?) may be helpful for some classes. 

Finally, it's worth restating, especially with such a challenging hook, that the goal of productive struggle is not to have 100% of of your class come to a correct conclusion revealing the standard. What is important is that your students think about the work so that when you do reveal this standard. they already have an investment and interest in the answer.

Hook 1 8.EE.C.7

Prompt Level 1: For each illustrated equation, the amount of money in each piggy bank is the same. Find the amount of money in a single piggy bank for each equation.

Prompt Level 2: For each illustrated equation, the amount of money in each piggy bank is the same. Rewrite each equation algebraically and solve.

This pictorial approach is similar to one taken by the Connected Math Project (CMP) curriculum and one that, I believe, is a very powerful tool to helping students articulate and understand the processes of solving equations. Instead of introducing variables, which may confuse some, the pictorial approach offers students pictures to represent the unknown amount. Of course, all students will have to grasp the idea of a variable representing this unknown value at some point, but the piggy banks are one way to ease students into the concept. In this case, the idea of not having a solution or having infinite solutions will be new to many students, who will come to these conclusions for the second and third equations and assume that they've done something wrong. One word of caution: Some students, perhaps because of the pictorial representations, assume that fraction values are impossible and, when they solve equations and arrive at a fraction for a variable, conclude "no solution" is possible. Therefore, it may be helpful to dissuade this idea directly during discussion.

Video: Solving an Impossible Problem (2:47)

Hook 1 8.EE.C.7.B

Prompt Level 1: Each piggy bank holds the same amount of money. Fill in the missing values.

Prompt Level 2: Each piggy bank holds the same amount of money. Rewrite as algebraic equations and fill in the missing values.

The key here is to look for understanding of the distributing property and combining like terms. Sticking to pictorial representations is helpful for struggling students when discussing these properties and can be elaborated by drawing in steps in the spaces between the equations given. Manipulatives, of course, are always helpful as well. In any case, you should allow students to use trial and error to try to solve the problem but then force them to think about the process being used. One way to sharpen this distinction is to write it directly into your lesson objective. Let kids know that the goal of the lesson is not to find the answer to an equation, but rather to be able to articulate the process. This objective needs to be reinforced by giving students opportunities to speak aloud and write about the steps involved in solving equations. If we expect our students to be mathematicians we must expect them to speak like mathematicians.

Hook 1 8.EE.C.8.A

Prompt: Tell a story to fit the given graph.

The standard centers on the point of intersection as a solution for both linear relationships but this hook will give you much more formative assessment in terms of students' understanding of linear relationships and their graphs while also allowing your kids to be creative. The depth of your discussion will depend on the skills of your students to some degree, but it's important that all students have a conceptual understanding of the fact that graphs are visual representations of the relationships between the given variables. By asking students to create their own stories, this hook allows us to evaluate that understanding and to create a class discussions that strengthens students' grasp of this concept. To heighten the challenge, you can create a similar graph without labels and/or scales. After your discussion, have fun and show them this video of The Freeze.

Hook 1 8.EE.C.8.B

Prompt: What is Marcy thinking and is she correct?

This hook asks students to study the combination or elimination method. Some may see the pattern in Marcy's thinking, adding the equations together, but it's important for students to discuss and articulate why this works as well. Students have the habit of taking the equal sign for granted and not thinking deeply about what the sign tells us. When we ask students to deconstruct this problem, we're asking them to think about equality and what aspects of equality allow us to manipulate the equation in a way that allows us to solve it.  

Hook 2 8.EE.C.8.B

Prompt: What is Peter thinking and is he correct?

In order to create some challenge, this hook displays the lazy habit many students have of skipping steps when solving equations. I'd suggest using the opportunity to note that writing out the process of the distributive property adds clarity to the work, prevents easy mistakes and is the regular practice of mathematicians who are concerned with precision. However, this hook's main goal, similar to the previous hook, is to allow students to deconstruct a system-solving strategy, in this case, substitution. Once again, students have the opportunity to think about the nature of equality. As always, allow students time to struggle, share ideas and think deeply about the equations given before opening the class to discussion or direct instruction.

Hook 1 8.EE.C.8.C

Prompt: Will he make it?

Kids will immediately ask "Well, how far is the house?". That, of course, is the real question of this hook. Whether or not our guy survives depends on a single measurement much as every airplane lift off, building construction and computer program depends on precise measurements. So ask your students to figure out the window of safety putting an emphasis on precision and proof. We know he'll be fine if he only needs to run 1 meter or two or even ten but what about 20? Writing and solving linear systems is a difficult task for first year algebra students so it's important we break down the process into all of its parts. For those who struggle, begin by prodding them to identify and label the variables. Think aloud with students (with small groups or whole class discussion) by simply asking "What do we want to know?" and "What do we know?". Ask them about the relationship between variables. "How are seconds and meters related for the runner? The bear?". Get them to define each component of their equation so that equations are packed with useful data and not just a collection of numbers and variables. Then look for students' ideas of how to represent this data. A picture is a good place to start but what happens when we translate this information to a graph or table or equation? Let the students see the utility and efficiency of each model. In short, break it all down into small pieces and use those pieces to build up to a solution. Math should not be a set of rules for them to follow. Math is logic and when we understand the underlying logic in a math problem everything we do and write has meaning.