# Unit 3

Math Expressions Unit 3 Background Sheet

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Spotlight on: Whole Numbers/Decimals

Compared to our first unit, this unit will feel like a breath of fresh air. We will be working hard on developing a deeper understanding of decimals and making sure we all have memorized the place value names after the decimal (tenths, hundredths, thousandths).

Lessons on the commutative, associative, and distributive properties (now for ADDITION) will give us an opportunity to review those properties and ensure that students are able to name the difference between the commutative and associative properties.

5th Grade Objectives targeted in this unit include:

· Understanding place value from billionths to billions (NC Standard Course of Study says 5th graders need to master thousandths to 999,999); this means that student worksheets will contain larger numbers than they need to master;
· Writing decimals and whole numbers in standard, word, and expanded forms
· Comparing and Ordering whole numbers and decimals (to thousandths)
· Rounding whole numbers and decimals in order to estimate for addition/ subtraction
· Use Commutative, Associative, and Distributive Properties to solve simple equations
· Write equations to solve problems with whole numbers and decimals
· Solve “change,” “collection,” and “comparison” problems (see explanations)
· In this unit, students will also see line graphs, bar graphs, and pictographs as a way of learning about ROUNDING

Math Expressions Vocabulary: How do we talk about Decimals?

Decimal: a special kind of number that tells the number of equal parts a number is divided into. While fractions can show a number divided into any number of equal parts (1/3, 1/4, 1/5, 1/92), a decimal shows part of the whole using tenths, hundredths, or thousandths (.1, .01, .001). Timed races, money, percentages, batting averages, and scores at competitive events like gymnastics and ice skating are examples of ways we see decimals used in the real world.

Math Expressions Vocabulary: How do we talk about

Many addition/subtraction problems involve CHANGES—the starting number, the change to the starting number, or the result will be an unknown. When word problems are made up of larger numbers, students need to understand how to set up situation and solution equations to represent the problems.

Situation Equation: represents the situation
Solution Equation: represents the steps you go through to solve the problem

 Change Addition (plus) Change Subtraction (minus) Unknown Total (sum) 6 children were playing tag in the yard. 3 more children came to play. How many children are in the yard now? Situation/Solution Equation: 6 + 3 = n Unknown Starting # Some children were playing tag in the yard. 3 more children came to play. Now there are 9 children in the yard. How many children were in the yard to begin with? Situation Equation: n + 3 = 9 Solution Equation: 9 - 3= n Unknown Change 6 children were playing in the yard. Some more came to play. Now there are 9 children in the yard. How many children came to play? Situation Equation: 6 + n = 9 Solution Equation: 9 – 6 = n Unknown Result (difference) Jake has 10 trading cards. He gave 3 to his brother. How many trading cards does he have left? Situation/Solution Equation: 10 – 3 =n Unknown Starting # Jake has some trading cards. He gave 3 to his brother. Now he has 7 cards. How many cards did he give to his brother? Situation Equation: n – 3 = 7 Solution Equation: 7 + 3 = n Unknown Change Jake has 10 trading cards. He gave some to his brother. Now he has 7 trading cards left. How many cards did he give to his brother? Situation Equation: 10 – n = 7 Solution Equation: 10 – 7 = n

Comparison Models are used in this unit for Comparison Problems.

Example Problems:

The nursery has 83 rose bushes and 66 lilac bushes. How many fewer lilac bushes are there than rose bushes?

The problem already tells us that there are more rose bushes. We draw this as a rectangle because it is the larger amount. We draw a smaller rectangle above or below to represent the smaller amount (lilacs) and the oval represents
the difference between the two (d).
This visual representation makes it clear that we need to subtract 66 from 83 to find the difference. Equation: 83 – d = 66, or 66 + d = 83; Solution Equation: 83 – 66 = d

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There are 10 dogs at the kennel. There are 6 fewer dogs than cats. How many cats are at the kennel?

[[image:file/view/cat_prob.png align="left"]]

The problem tells us there are fewer dogs than cats. We draw a larger rectangle to represent cats because we know it is the larger amount. We draw a smaller rectangle above or below to represent 10 dogs (the smaller amount). We place 6 in the oval because the problem tells us 6 is the difference between cats and dogs. The visual representation makes it clear that we need to add 10 and 6 to get the total number of cats. Solution Equation: 10 + 6 = c

In this unit, students will also review algebraic properties of addition. Students need to be able to use these properties to INFER the value of a variable.

Commutative Property of Addition: changing the order of the addends does not change the total—Addition is commutative!

Example:
3 + 5 + 7 = 7 + 3 + 5
8 + 7 = (12) + 3
15 = 15

In the problem above, we might use the commutative property to add the numbers in an order that makes it simpler for us. We could do 7+3 first to make a “10,” making our adding of 5 more easier.

Associative Property of Addition: changing the way addends are grouped does not change the total—Addition is associative!

Example:
3 + (5 + 7) = (3 + 5) + 7
3 + (12) = (8) + 7
15 = 15

So, if I see the problem 9 + x + 34 = 9 + 3 + 34, I can infer that x = 3.
If I see 9 + (37 + 82) = (9 + x) + 82, I can infer that x = 37.

If I encounter a problem like: 1.02 + (3.44 + .98) = n, although 3.44 and .98 are grouped in parenthesis, I know that I can combine .98 and 1.02 first (to add easier numbers and get 2) because the associative property says I can group addends in any order and still get the same result. Now I can easily add 2 + 3.44 to get 5.44.