Integer Product Signs - Using Counters to Discover Signs of Products
Lesson 12 of 27
Objective: SWBAT determine if the product of two integers is positive or negative using counters to represent equal groups
In the introduction, I ask the essential question: How can you use multiplication as repeated addition or equal groups to determine the sign for the product of two integers? We review this concept using a simple multiplication problem - 3 * 2 as 3 groups of 2 or 2 + 2 + 2.
I will give each student 10-15 counters of each type. Next I show how the model for multiplying integers works (MP5). In short, when the first factor is a positive value, students use this factor to determine the number of equal groups of the second factor. Then they evaluate the sum.
If the first factor is a negative, students start with a set of zero pairs and remove groups of the second factor.
Here is a good explanation of the model.
I will guide students through problem a then check for understanding with b. I will follow in the same manner using d&e, g&h, and j&k. Then students will solve c, f, i, and l on their own. I will make sure everyone is able to model the problems. If necessary, I can easily provide other problems that follow the pattern of each column. I will just make sure that the problems require a reasonable amount of counters.
Now that students understand the model for integer products, students work with their partner(s) to evaluate 8 products. They fill in a table to say if the sign of the factors are the same or different, the product itself, and the sign of the product.
The questions that follow are designed to get students to see that products of same signs are positive and different signs are negative. This turns students attention to the structure of a product based on the signs of the factors (MP7). Then, using repeated reasoning students conclude when a sign is positive or negative (MP8).
While students are working, I will make sure that values are being filled in correctly. It may be necessary to ask student pairs to model a problem step by step to help them find any errors being made.
By the end of this section, there will inevitably be students who say "Wait, you said two negatives makes a negative!" Or, they may assume now that the sum of two negatives is positive. When this happens, I will ask them to model -2 * -4 using counters and then -2 + -4 using counters. In fact, whenever this happens I will ask students to recall the models.
The first 3 problems in independent practice just ask student to apply the conclusions from the previous lesson section.
Problems 4-11 ask students to evaluate different products.
Problems 12-15 introduce the notation of exponents with integers and grouping symbols. This may be the first time students see the distinction between (-2)^3 compared to -2^3. Students may need a little guidance with this issue.
Problems 16-18 require student to evaluate algebraic expressions.
Problem 19 is a word problem about the "money" students at our school earn for attendance (or lose for lack of attendance).
Problem 20, requires students to consider all possible integer products that equal +24 and then find the two factors whose have the least sum. This is a good exercise in MP1.
The extension presents a problem about a plane's descent using the expression 22,000 + (-480t). The numbers involved are relatively big, but shouldn't be too difficult for students to calculate without calculators. I may allow calculator use though, so that the focus of students attention can be on completing a table and making a good graph.
The exit ticket assesses whether students have learned the signs of products in question 1. This may be the most important question on the exit ticket since it is the main point of the lesson. Question 2 can be solved using the commutative property (MP7). I inserted this question into the exit ticket because it is deceptively easy, but often students forget to apply what they already know about multiplication. This will be a reminder! Question 3 asks students to use values of +1 and -1 in a string of five factors to make a product of +1. Question 3 may be the second most important question of the exit ticket as it most directly relates to the main point of the lesson.