In this lesson plan, your learner will practice applying number facts to two-digit numbers through engaging addition and subtraction stories. We'll use base-ten blocks and part-part-whole diagrams to illustrate these concepts, helping your learner understand and apply number facts efficiently.
Key Concepts for Applying Number Facts to Two-Digit Numbers
Performing calculations with two-digit numbers can be a challenging transition for learners. To help them move forward confidently, it is important to build connections with concepts and visual aids that they are familiar with.
Relating Single-Digit Facts to Two-Digit Numbers:
Your learner's understanding of addition and subtraction facts with single-digit numbers forms the foundation for working with two-digit numbers.
For example, if your learner knows that 7 - 2 = 5, they can apply this fact to solve 27 - 2 = 25. Similarly, knowing that 4 + 3 = 7 helps them understand that 14 + 3 = 17. This knowledge enables them to tackle more complex problems with confidence.
Using Base-Ten Blocks (Dienes) and Number Lines:
Base-ten blocks (Dienes) and number lines are used in this lesson to bridge the gap between single-digit and two-digit numbers.
Dienes blocks help visualize numbers physically, making it easier for your learner to see how tens and ones work together. For instance, representing 23 with two ten blocks and three one blocks, then adding six one blocks, helps your learner understand why 23 + 6 equals 29.
Similarly, number lines help your learner visualize the step-by-step process of addition and subtraction, reinforcing their understanding of number relationships.
Encouraging Pattern Recognition and Use of Number Facts:
It's important for your learner to look for patterns and use their knowledge of number facts rather than relying on counting on or back strategies.
Recognizing that the ones digits remain consistent across similar addition or subtraction problems helps your learner see the underlying structure of numbers.
For instance, noticing that in the sequence 4 + 3, 14 + 3, 24 + 3, the ones digit is always 7, regardless of the tens digit, reinforces their understanding of how addition works.
Encourage your learner to use these patterns and their number facts to solve problems more efficiently.
Teaching Plan
The following activities will help your learner develop confidence in applying number facts to two-digit number problems. Be sure to work at a pace that is comfortable for your learner.
Examples and visuals to support the lesson:
1. Revisiting Addition within Ten
Begin with the story: “Diego walked for three minutes to get to his friend’s house, and then walked for another six minutes to get to school. His journey took nine minutes altogether.”
Ask your learner to represent this story on a part-part-whole diagram using base-ten cubes, then write the corresponding equation (3 + 6 = 9). Have your learner describe, in full sentences, what each number in the equation represents.
In this example, the total journey time is given to ensure the focus remains on understanding the structure rather than on calculation. But if they did need to calculate the total time, remind your learner that they wouldn't need to count the total number of ones to find the answer. They may just know that 3 + 6 = 9. If not, prompt them to solve 3 + 6 by relating to their knowledge that 4 + 6 = 10.
Skill Check
I can use different strategies to quickly add and subtract within 10.
2. Two-Digit and Single-Digit Addition
Present a similar story: “Dana walked for twenty-three minutes to get to her friend’s house, and then walked for another six minutes to get to school. Her journey took twenty-nine minutes altogether.”
As before, ask your learner to represent the story on a part-part-whole diagram, now using base-ten reds for tens and cubes for ones, and write the corresponding equation (23 + 6 = 29).
Compare the two stories by asking: “What’s the same? What’s different?” Draw attention to the fact that the only difference is the two additional tens in the second calculation.
It is also useful to compare the calculations on a number line representation.
Skill Check
I can see how adding two-digit numbers is similar to adding single-digit numbers.
3. Applying Addition Facts
Once your learner understands these examples, start demonstrating that we can apply our addition facts within ten to any two-digit numbers.
Begin by exploring a single-digit fact and the set of corresponding two-digit plus single-digit calculations to reveal the underlying pattern. For instance, in the expressions 4 + 3, 14 + 3, and 24 + 3, the sums will have the same ones digit, regardless of the value of the tens digit.
Your learner can continue to represent the calculations using base-ten blocks, as in the previous step. Also, show the related calculations on a number line, ensuring that the representation is used to expose the structure and generalize the pattern rather than as a tool for calculation.
Continue to encourage your learner to use known facts rather than counting on in ones. As you model the calculations, record the corresponding equations.
In each case, ask your learner: “What do you notice about the ones digits? Can you see any patterns?” “What is changing? What is staying the same?” “Which other equations would belong in this pattern? Why?”
Skill Check
I can use what I know about adding within 10 to add with two-digit numbers.
4. Using Various Representations
The bead bar is a useful representation to show alongside the number line. For example, the four ones and three ones can be made from what is now an incomplete block of ten (with three beads ‘discarded’ to the right). This allows your learner to pull blocks of ten beads from the left to explore the related calculations, rather than needing to make the four and three each time. This draws attention to what is staying the same (4 + 3).
Choose carefully whether/when to introduce this extra representation. Modeling a given concept/calculation in different ways can improve conceptual understanding by revealing underlying mathematical structures; however, using too many representations at the same time can be confusing.
Using one of the representations as a scaffold, provide your learner with practice writing linked facts. For example, begin with your learner representing 6 + 2 with base-ten blocks, then ask them to make and write the related calculations, working forwards through the sequence (16 + 2, 26 + 2, etc.).
Use the stem sentence: “I know that ___ plus ___ is equal to ___ (single-digit fact), so ___ plus ___ is equal to ___ (related two-digit plus single-digit calculation).”
To encourage your learner to think more deeply about the patterns, ask questions such as: “True or false - when we add five ones to a number that ends in three, the total always ends in eight?” Ask your learner to explain their answers using concrete or pictorial representations to support their descriptions.
Skill Check
I can use equations and math tools to find patterns for adding two-digit numbers.
5. Single-Digit and Two-Digit Subtraction
Begin with a single-digit number fact and then explore related two-digit minus single-digit calculations. For example: “I know that 7 - 2 = 5, so 27 - 2 = 25.”
Have your learner practice writing linked facts and answering questions such as: “True or false — when we subtract four ones from a number that ends in nine, the total always ends in five?”
When your learner has mastered working with related facts and can generate their own examples, introduce part-part-whole models with numerals. Begin with number sequences that you explored in detail for addition and subtraction, so that only the representation is being varied.
Present the part-part-whole models alongside the familiar base-ten block representations and ask your learner to identify similarities and differences between the representations.
The part-part-whole representations are particularly useful for linking addition and subtraction as inverse operations. For each diagram, ask your learner to identify the addition and subtraction equations.
Note that, at this stage, we are looking at calculations involving the addition or subtraction of a single-digit number, so the equations 37 - 34 = 3 (and 3 = 37 - 34) have not been included in the example set; however, these are the final two which make up the set of eight equations supported by the part-part-whole diagram, so your learner may mention equations such as these.
Skill Check
I can see how subtracting from two-digit numbers is similar to subtracting from single-digit numbers.
6. Practicing with Equations
When they are ready, your learner should also practice working solely with equations, although part-part-whole models can be used initially for support.
As usual, present missing number problems with variation in terms of the location of the missing number (sum, addend, difference, subtrahend, and minuend) and the position of the equals sign (to emphasize equivalence).
Encourage your learner to reason about their answers. For example, when solving 57 = ? + 3, your learner may say, “I know that fifty-seven is equal to fifty-four plus three because to make a seven digit in the ones, I need to add four to the three.”
To provide further challenge, use a problem of the form: “Chang is sorting expressions into sets, and he puts 81 + 8 and 85 + 4 together.” “Why did Chang put these together?” “Can you write more expressions that are part of this set?”
Skill Check
I can solve missing number problems with two-digit numbers.
7. Word Problem Practice
When your learner is confident working in the abstract, provide them with practice using real-life problems, such as money contexts. By choosing examples carefully, varying the structures and stories, you can draw attention to the underlying mathematical structures.
Ask your learner to identify the similarities and differences between the questions, and for each question ask: “Which addition/subtraction fact would help you solve this problem?”
Part-Whole Examples:
“Omar bought a chocolate bar for thirty-two cents, and a lollipop for seven cents. How much did Omar spend in total?”
“Omar bought a chocolate bar for thirty-two cents, and a lollipop; he spent thirty-nine cents in total. How much did the lollipop cost?”
“Omar bought a chocolate bar, and a seven-cent lollipop; he spent thirty-nine cents in total. How much did the chocolate bar cost?”
Augmentation (Joining) and Reduction (Take Away) Examples:
“At first, Faris had twenty-one dollars in his piggy bank; then he added his three dollars of pocket money. How much does Faris have now?”
“At first, Faris had twenty-one dollars in his piggy bank; then he added his pocket money. Now he has twenty-four dollars. How much pocket money did Faris add?”
“At first, Faris had some money in his piggy bank; then he added his three dollars of pocket money. Now he has twenty-four dollars. How much money did Faris have at first?”
Difference Examples:
“Aliona spent forty-two dollars, and James spent more than Aliona. The difference between the amounts they spent is five dollars. How much did James spend?”
“Aliona spent less than James, and James spent forty-seven dollars. The difference between the amounts they spent is five dollars. How much did Aliona spend?”
“Aliona spent forty-two dollars, and James spent forty-seven dollars. What is the difference between the amounts they spent?”
Skill Check
I can solve different types of word problems with two-digit numbers.
Summary
By the end of this lesson, your learner will be able to confidently apply their knowledge of single-digit addition and subtraction facts to solve two-digit problems. They'll understand the connections between single-digit and two-digit numbers, use Dienes blocks and number lines effectively, and recognize patterns to solve problems without relying on counting strategies.
With over 12 years of experience as a classroom teacher, tutor, and homeschool parent, my specialty is easing math anxiety for students of all ages. I'm committed to empowering parents to confidently support their children in math!
Skill Check