In this lesson, students will apply what they already know about adding with positive and negative integers to understand multiplication of integers. From there, students will build on their understanding of operations with integers. Once students understand the rules with multiplication, they will move on to division and predicting signs when there are more than two given numbers. You can expect this lesson with additional practice to take two `45`-minute class periods.
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Students will be able to multiply and divide integers.
Give students a copy of the student sheet to work on throughout the lesson.
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Students are likely to recall that multiplication is repeated addition to help them justify and answer the warm up. Students might say that `4 \times (-3)` can be written as `-3 + (-3) + (-3) + (-3)` which is `-12`. Have a conversation around if we can apply this thinking to all positive `x` negative integers. Try it with two more problems. Students will eventually conclude that a positive integer `x` negative will result in a negative integer.
To expand on the question, ask students if the answer to `-3 \times 4` is the same as `4 \times (-3)`. Through student discussion, they should recall that the commutative property applies to multiplication. Then, ask students if the answer will always be negative if there is a positive and negative number being multiplied.
Allow students a minute to look at the given information and consider what they notice and wonder with the given information. This will help connect students' current understanding of multiplying integers to what happens when both values are negative.
Students will likely notice that the first number is always being multiplied by `-2` and that it goes down by one. They may also notice that the last number is increasing by `2`; however, some students may say it’s going down by `2` because they are ignoring the negative sign in front. Make sure students understand that the last number is adding `2` every time.
To expand on the pattern in the slide, ask students to continue the pattern on their student sheet. When students complete the pattern, they will see that two negative numbers multiply to a positive number:
It may help to use vocabulary like “opposite” to help students better understand the concept. For example, `-(-2)` would be read as “the opposite of `-2`”.
Give students some time to fill in the multiplication table on their student sheet. Once students have filled in the values, consider having them check their work with you before having them use red and green colored pencils or markers to identify the positive and negative values.
When reviewing with students, highlight the areas students mention according to their signs, such as green for positive, and red for negative. Students may start with the products of two positive numbers and recognize the answer is always positive; however, allow students to guide the conversation as much as possible. With the visual cues, students should try to come up with the general rules for multiplying with integers.
Depending on how quickly students have worked through the examples may change what you choose to start the class with. Consider using students’ multiplication tables to write informal rules for multiplying integers. Make sure you thoroughly review what students covered during the first day!
Give students a minute to write the division equations with the given values. They can check with a partner if needed.
Ask students what conclusions they can reach about division based on these examples. For example, for the equation on the left, students can write `12 \div 4 = 3` and `12 \div 3 = 4`. On the right, students should write `-12 \div -4 = 3` and `-12 \div 3 = -4`. Tying the division to what they already know about multiplication will help students recognize that the rules are the same. To further student discussion, ask students to write the division equations related to `-3 \times (-4) = 12`. By this point, ask students to share the patterns they have noticed when multiplying and dividing integers to help support the rules.
Although students should have a concrete understanding of why these rules work, it can be helpful to provide them a reference.
Ask students to give an example to help justify each example for multiplying and dividing integers. Remind students that these rules should be applied to decimals and fractions.
Use the next slide to help students focus on the signs of their answers rather than the actual answers. It may help to encourage students to use their hands as visuals. For example, “thumbs up” can represent positive and “thumbs down” can represent negative. Give students an opportunity to predict on their own. When reviewing each expression, ask students for their predictions ahead of time. This can help you gauge how well students were able to predict the signs on their own.
Go through each example one number at a time. Using visual cues, like thumbs up or down, can help students keep track of their signs. For example, the first example starts with a thumbs down, switches to the opposite because of the negative, stays the same because the next number is positive, etc. Students should be able to come to the conclusion that the first expression is positive, and the next two are negative. Some students might also notice that when the number of negative numbers is even the sign is positive but when it is odd, the sign if negative.
Now that students have had a chance to predict the signs, they should review multiplying and dividing with decimals and fractions. Give students time to work on the problems individually and check with a partner if needed.
These problems can help you identify misconceptions students may have surrounding the signs of the answers. Students may have the following misconceptions with calculations:
After you’ve completed the examples with the whole class, it’s time for some independent practice! ByteLearn gives you access to tons of practice problems for multiplying and dividing integers. Check out the online practice and assign to your students for classwork and/or homework!
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