This collaborative activity is designed to help students understand why finding common denominators is a necessary step when adding or subtracting fractions. Students will collaborate with their peers to identify errors in common mathematical thinking. Error analysis such as this is not only engaging for students, but also provides them a safe environment to explore misconceptions and strengthen their own foundational understanding. This lesson also includes three leveled, independent assignments that can be used as extension opportunities, additional practice, or differentiated assessments of students’ ability to find common denominators to add and subtract fractions.
CCSS.Math.Practice.MP1 | Make sense of problems and persevere in solving them. |
CCSS.Math.Practice.MP2 | Reason abstractly and quantitatively. |
CCSS.Math.Practice.MP3 | Construct viable arguments and critique the reasoning of others. |
CCSS.Math.Practice.MP4 | Model with mathematics. |
CCSS.Math.Practice.MP6 | Attend to precision. |
CCSS.Math.Content.4.NF.A.1 – Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. |
CCSS.Math.Content.4.NF.B.3 – Understand a fraction a/b with a > 1 as a sum of fractions 1/b. CCSS.Math.Content.4.NF.B.3.A – Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. CCSS.Math.Content.4.NF.B.3.C – Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. CCSS.Math.Content.4.NF.B.3.D – Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent them. |
CCSS.Math.Content.5.NF.A.1 - Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) |
CCSS.Math.Content.5.NF.A.2 - Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. |
Students will be able to explain why we find common denominators when adding or subtracting fractions.
Students will be able to add and subtract fractions by finding common denominators and generating equivalent fractions.
• Building Blocks of Math: Moving Beyond Foundations series, specifically Fractions
• Help ‘Em Out Partner Activity (1 per pair of students)
• Independent Assignments (1 per student)
o Option A – High Speeds and Tough Turns
o Option B – Just Hanging Around
o Option C – Tricks Galore
• Fraction Manipulatives (1 per student)
• Optional: Printable Fraction Strips
• Pencils
• Optional: multiplication charts
• Optional: scratch paper
Consider using strategic grouping during this lesson. Heterogenous pairs can be used to help engage and benefit all learners during this activity.
Consider allowing students access to a multiplication chart for support in determining common denominators. It may be helpful to review how to determine common multiples and therefore common denominators.
Finally, consider using leveled assignments in the independent work setting. Option A requires the least cognitive demand whereas Option C requires the highest cognitive demand.
o Option A – High Speeds and Tough Turns
o Option B – Just Hanging Around
o Option C – Tricks Galore
These procedures are general and can be applied to each strategy spotlighted in this activity.
Return to the whole group setting and discuss each situation, error in mathematical thinking, advice, and correct solution. Focus your discussion on why common denominators are useful when adding and subtracting fractions. Consider using the following to guide your conversation:
a. Brielle says 3/5 + 2/3 = 5/8. Brielle is not thinking about the size of the pieces. We cannot add fifths and thirds because they do not represent the same portion of a whole, or the same unit. When we use fraction manipulatives, we can see that 5/8 is actually smaller than 2/3 , so we know it is an unreasonable answer. In order to find the correct answer, we can use a common denominator of 15 and rewrite the problem as 9/15 + 10/15 = 19/15 , or 1 4/15.
b. Thao does not know how to solve the problem 1/2 - 3/8. He says he is confused because when he tries to subtract the numerators from one another, he runs into an issue, that 3 is less than 1. This happens with the denominators, too. He says 8 is less than 2 so he does not know how to subtract them. Thao is applying what he knows about subtracting whole numbers. Although he is right about struggling to subtract a larger number from a smaller one (this is possible with negative numbers, which likely have not yet been introduced to students), this is not the root of his problem. Thao could have contextualized the problem. Thinking about this fraction problem as a real-world situation would help him visualize the situation. He could imagine the problem as if it were pizza: we started with 1/2 of a pizza but someone ate three of the eight slices, or 3/8 of the whole pizza. We could easily visualize this, noticing that half of the pizza is equivalent to four out of eight slices, or 4/8. Now we can subtract 4/8 - 3/8 = 1/8.
c. Jess has been paying attention in math class and knows that finding equivalent fractions can make adding and subtracting easier. When Jess saw the problem 3/12 + 3/6 on their homework, they simplified the fractions and wrote the equivalent expression 1/4 + 1/2. Although Jess is on the right track with finding equivalent fractions, they did not consider the importance of finding common denominators and keeping the size of the pieces consistent. There are two common ways to solve this problem: first, we could rewrite it with a common denominator of 12: 3/12 + 6/12 = 9/12. Alternatively, we could use the simplified expression 1/4 + 1/2 to find a common denominator of 4: 1/4 + 2/4 = 3/4.
Transition to independent work time. Consider differentiating students’ assignments based on their ability to add and subtract fractions with uncommon denominators. Option A requires the least amount of cognitive demand where Option C requires the greatest amount of cognitive demand.
a. Option A – High Speeds and Tough Turns
b. Option B – Just Hanging Around
c. Option C – Tricks Galore