Students learn that a fraction $\frac{a}{b}$ is a product of a whole number $a$ and a unit fraction $\frac{1}{b}$, or $\frac{a}{b} = a \times \frac{1}{b}$, and that $n \times \frac{a}{b} = \frac{(n \space \times \space a)}{b}$. Students learn to add and subtract fractions with like denominators, and to add and subtract tenths and hundredths.
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Problem 10
Each bead weighs \(\frac{5}{8}\) gram. How much do 7 beads weigh? Explain or show your reasoning.
Solution
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Problem 11
Exploration
Measure how thick your workbook is to the nearest \(\frac{1}{8}\) inch.
If all of your classmates stacked their workbooks together, how tall would the stack be? Explain or show your reasoning.
Check your answer by measuring, if possible.
Solution
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Problem 12
Exploration
Diego walked the same number of miles to school each day. He says that he walked \(\frac{48}{5}\) miles in total, but does not say how many days that distance includes.
What are some possible number of days Diego counted and the distance he walked each of those days?
Solution
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Section B: Addition and Subtraction of Fractions
Problem 1
Write \(\frac{4}{3}\) in as many ways as you can as a sum of fractions.
Write \(\frac{9}{8}\) in at least 3 different ways as a sum of fractions.
Solution
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Problem 2
Draw “jumps” on the number lines to show two ways to use fourths to make a sum of \(\frac{7}{4}\).
Represent each combination of jumps as an equation.
Solution
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Problem 3
Explain how the diagram represents\(\frac{13}{5} - \frac{4}{5}\).
Use the diagram to findthe value of \(\frac{13}{5}- \frac{4}{5}\).
Use a number line to represent and find the difference \(\frac{9}{4} - \frac{3}{4}\).
Solution
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Problem 4
Show two different ways to find the difference: \(2 - \frac{3}{4}\)
Solution
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Problem 5
Elena is making friendship necklaces and wants the chain and clasp to be a total of \(18\frac{1}{4}\) inches long. She is going to use a clasp that is \(2\frac{3}{4}\) inches long. How long does her chain need to be? Explain or show your reasoning.
Solution
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Problem 6
For each of the expressions, explain whether you think it would be helpful to decompose one or more numbers to find the value of the expression.
\(\frac{4}{3} + \frac{5}{3}\)
\(5\frac{1}{5} - 2\frac{2}{5}\)
\(9\frac{5}{6} - 6\frac{1}{6}\)
Solution
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Problem 7
The lengths of the shoes of a dad and his two daughters are shown.
How much longer is the older daughter’s shoes than her sister’s?
Which is longer, the dad’s shoes or the combined lengths of his daughters’ shoes?
Solution
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Problem 8
Exploration
A chocolate chip cookie recipe calls for \(2\frac{3}{4}\) cups of flour. You only have a \(\frac{1}{4}\)-cup measuring cup and a\(\frac{3}{4}\)-cup measuring cup that you can use.
What are different combinations of the measuring cups that you can use to get a total of \(2\frac{3}{4}\) cups of flour?
Write each of the combinations as an addition equation.
Solution
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Problem 9
Exploration
The table shows some lengths of different shoe sizes in inches.
U.S. shoe size
insole length
1
\(7\frac{6}{8}\)
1.5
8
2
\(8\frac{1}{8}\)
2.5
\(8\frac{2}{8}\)
3
\(8\frac{4}{8}\)
3.5
\(8\frac{5}{8}\)
4
\(8\frac{6}{8}\)
4.5
9
5
\(9\frac{1}{8}\)
5.5
\(9\frac{2}{8}\)
6
\(9\frac{4}{8}\)
6.5
\(9\frac{5}{8}\)
7
\(9\frac{6}{8}\)
What do you notice about the insole lengths as the size increases?
What will the insole length increase be from size 7 to 7.5? What is the insole length of a size 7.5 shoe?
Predict the insole length for sizes 9, 10, and 12. Explain your prediction. Then solve to find out if your prediction is true.
Solution
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Section C: Addition of Tenths and Hundredths
Problem 1
Andre is building a tower out of different foam blocks. These blocks come in three different thicknesses: \(\frac{1}{2}\)-foot, \(\frac{1}{4}\)-foot, and \(\frac{1}{8}\)-foot.
Andre stacks two\(\frac{1}{2}\)-foot blocks, two\(\frac{1}{4}\)-foot blocks, and two\(\frac{1}{8}\)-foot blocks to create a tower. What will the height of the tower be in feet? Explain or show how you know.
Solution
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Problem 2
Find the value of each of the following sums. Show your reasoning. Use number lines if you find them helpful.
\(\frac{1}{10} + \frac{3}{100}\)
\(\frac{24}{100} + \frac{4}{10}\)
\(\frac{7}{10} + \frac{13}{100}\)
Solution
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Problem 3
Is the value of each expression greater than, less than or equal to 1? Explain how you know.
\(\frac{3}{10} + \frac{7}{100}\)
\(\frac{13}{10} + \frac{7}{100}\)
\(\frac{30}{100} + \frac{7}{10}\)
Solution
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Problem 4
Diego and Lin continued to play with their coins.
Diego said that he has exactly 3 coins whose thickness adds up to \(\frac{50}{100}\) cm. What coins does Diego have? Explain or show your reasoning.
coin
thickness in cm
1 centavo
\(\frac{12}{100}\)
10 centavos
\(\frac{22}{100}\)
1 peso
\(\frac{16}{100}\)
2 pesos
\(\frac{14}{100}\)
5 pesos
\(\frac{2}{10}\)
20 pesos
\(\frac{25}{100}\)
Solution
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Problem 5
Exploration
A chocolate cake recipe calls for 2 cups of flour. You gather your measuring cups and notice you have these sizes:\(\frac{1}{2}\) cup, \(\frac{1}{3}\) cup, \(\frac{1}{4}\) cup, and \(\frac{1}{6}\) cup.
What are the different ways you could use all 4 measuring cups to measure 2 cups of flour?
What are other ways you could use just someof the 4 measuring cups to measure exactly 2 cups of flour?
Solution
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Problem 6
Exploration
A dime is worth \(\frac{1}{10}\) of a dollar and a penny is worth \(\frac{1}{100}\) of a dollar.
If I have \(\frac{89}{100}\) of a dollar, how many different combinations of dimes and pennies could I have? Use equations to show your reasoning.
A nickel is worth \(\frac{5}{100}\) of a dollar. How many different combinations of dimes, nickels and pennies could I have if I stillhave\(\frac{89}{100}\) of a dollar? Use equations to show your reasoning.
Solution
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