Interpreting Numerical Expressions
Lesson 4 of 5
Objective: SWBAT interpret & compare numerical expressions without evaluating them.
Students need to have mastered writing & interpreting numerical expressions, & they have to be familiar with phrases like: four more, & twice as many. To interpret expressions independently, like they will in the I.P., student have to feel comfortable using mathematical language like: groups of, and the value of. Therefore, as the warm up, we have five practice problems with typical mathematical language.
These are words/phrases that I add to my word wall. Word walls are typically only used in primary grades, but I feel like it has really helped my students. I don't keep a running list, but instead use a weekly list on the board.
I use my document camera to show these, and then use cold calling afterward to go over the answers to them. To help students even more, you can have them read aloud the expressions; this strategy will support auditory, visual, and kinesthetic learners.
Complete the numerical expression for each phrase.
1) seventeen added to two groups of five
2) nine less than the result of dividing twelve by four
3) twice the sum of three and nineteen
(For #3, students frequently forget to use parentheses when writing numerical expressions that are grouped. They need to practice looking for the phrases: the sum of, the product of, the result of, groups of, to help them remember to use parentheses.)
Complete the phrase for each numerical expression.
1) (6 + 4 3/4) divided by 2
2) (4 x 3/4) - 5
It's important for students to understand that not all expressions can be compared without evaluating. Students can look for parts of the expressions that are the same or equivalent. It may be helpful to remind students who are having trouble of the Commutative and Associative Properties of Addition.
Jennifer and Kristen bring water bottles to the tennis match for the team. Jennifer brings 3 cases of twenty-four bottles, plus four extra bottles. Kristen brings 3 cases of twenty-four bottles, but she and her sister drank two of the bottles on the way to the tennis match. Write numerical expressions to represent how many water bottles Jennifer and Kristen each bring to the tennis match. How does the number of water bottles that Jennifer brought compare?
I prompt my students to consider a case of water bottles as one group of 24. So, the number of water bottles in three cases is three groups of twenty-four, or 3 x 24.
STEP 1: We write words numerical expressions to represent how many water bottles were brought.
Jennifer: 3 x 24 + 4 (three groups of twenty-four plus four more)
Kristen: 3 x 24 -2 (three groups of twenty-four minus two)
I take time to point out the key words: plus four extra, and drank two, and cold call upon students to show how these were represented in the numerical expressions.
STEP 2: We now interpret the expressions using MP7. (We look at the structure of the expressions, identifying parts that are equivalent.) I point out that the product of 3 x 24 has the same value in both expressions. The value of Jennifer's expression is 4 more than that value. The value of Kristen's expression is 2 less than that value. The value of Jennifer's expression is the greatest.
STEP 3: We then compare the values of the expressions. The value of 3 x 24 + 4 is greater than the value of 3 x 24 - 2. Both expressions contain 3 x 24, but the value of Jennifer's expression is greater than the value of Kristen's expression because it adds 4 to 3 x 24 instead of subtracting 2. (So, Jennifer brings 6 more water bottles to the tennis match.)
Think Pair Share- Using their table partner, students work on MP 7, looking at the structure of the expressions, identifying parts that are equivalent. I ask each group to discuss the statements that the two students, Tanya & Ana, made. The students need to clearly explain how they've reached their own answer to show whether each student is correct. I set up the following questions on the board:
- What is the same about these two numerical expressions? What is different?
- What is the result when you multiply any number by 2?
I model using questions with student input here, and make sure that students do not just copy the questions, but rather are prepared to use them in the answer when answering questions in a few minutes.
12 divided by 3 - 2/3 and (12 +3 - 2/3) x 2
Tanya says (12 + 3 - 2/3) is half as much as (12 + 3 - 2/3) x 2
Ana says (12 + 3 - 2/3) x 2 is twice as much as (12 + 3 - 2/3)
Explain why both students are correct.
In retrospect, I would probably have my students with exceptionalities make a box for each girl and put each respective answer in there to organize their thinking.
As time allows, you could have students tackle another problem as well. Using MP 2, students reason abstractly and quantitatively to identify quantities and their relationships to compare numerical expressions.
T.J. has (17 + 2) x 100 baseball cards. Izzy has 17 + 2 baseball cards. Without evaluating, explain how the number of T.J.'s baseball cards compares to the number of Izzy's baseball cards.
I use cold calling to call upon students to facilitate discussion about how students solved the IP practice.
Using the following vocabulary words, I have students write a quick paragraph about how the vocabulary connects to the I Can Statement: numbers, parentheses, expressions, equivalent, compare, value, evaluate.