Mathematics — Problem Solving and Number Sense

Bar Model Practice: Part-Whole Reasoning and Problem Solving

Grades K-8 | Aurora Curriculum Pack

Aurora Mathematics standard: Draw the bar model BEFORE writing any equation. The model shows your thinking. The equation records it. A scholar who draws the model correctly will write the equation correctly every time. A scholar who writes the equation without the model is guessing at structure.

How to use this sheet: This worksheet is organized into three grade bands. Use the section that matches your scholar's current level. All problems use real contexts drawn from community, cooperative economics, and African and African American history.

Grade Band: K-2 — Counting, Comparing, and Basic Part-Whole

Grades K-2

Worked example (read together before starting):
Amara has 5 books. Her brother gives her 3 more. How many books does she have now?

Step 1 — Draw it: Draw a long bar for the total. Split it into two parts: one part for 5, one part for 3.
Step 2 — Write it: 5 + 3 = 8
Step 3 — Say it: "Amara has 8 books in all."

Problem 1 — Adding (Part + Part = Whole)

The co-op garden has 6 rows of sunflowers and 4 rows of tomatoes. How many rows are in the garden altogether?

Draw your bar model in the box. Show the two parts and the whole.

Draw your bar model here — show two parts and the total

Equation: _______ + _______ = _______

Answer in a complete sentence: _______________________________________________

Problem 2 — Subtracting (Whole - Part = Part)

There were 12 scholars at the community school. 5 scholars went home early. How many scholars stayed?

Draw your bar model — show the whole and what was taken away

Equation: _______ - _______ = _______

Answer: _______________________________________________

Problem 3 — Comparing (How Many More?)

Zara counted 9 birds at the feeder. Marcus counted 6 birds. How many more birds did Zara count?

Draw two bars — one for Zara, one for Marcus

Equation: _______ - _______ = _______

Answer: _______________________________________________

Who counted more? _______________

By how many? _______________

Problem 4 — Number Sense: True or False?

Circle TRUE or FALSE. Explain your thinking by drawing a bar or using objects:

4 + 7 is the same as 7 + 4. TRUE FALSE Why? _______________________________________________

10 is more than 8 + 3. TRUE FALSE Why? _______________________________________________

Problem 5 — Word Problem: Scholar Creates One

Write your own word problem that uses the numbers 7 and 4. Draw the bar model and solve it:

Write your word problem here:

Draw the bar model for your problem

Answer: _______________________________________________

Aurora Pause — K-2

Mansa Musa of the Mali Empire (1280-1337 CE) gave away so much gold on his pilgrimage to Mecca that he lowered gold prices across North Africa for 10 years. If he gave 3 bags of gold in Cairo, 4 bags in Medina, and 2 bags in Mecca, how many bags did he give in all? Draw the bar model and solve.

Draw your bar model here

Answer: _______________________

Grade Band: 3-5 — Multiplication, Division, and Multi-Step Problems

Grades 3-5

Worked example — Multi-step problem:
A community garden has 6 plots. Each plot has 8 plants. The gardeners harvested 23 plants last week. How many plants are left?

Step 1 — Total plants: Bar model with 6 equal parts of 8 = 6 × 8 = 48 plants total
Step 2 — Remaining plants: 48 - 23 = 25 plants remaining
Always draw TWO bar models for a two-step problem — one for each step.

Problem 1 — Multiplication as Equal Groups

The Frederick Douglass Community Library has 7 shelves. Each shelf holds 24 books. How many books are in the library altogether?

Draw a bar model with 7 equal parts, each showing 24

Equation: _______ × _______ = _______

Answer: _______________________________________________

Problem 2 — Division: Sharing Equally

A co-op received 156 pounds of sweet potatoes. They will share equally among 12 families. How many pounds does each family receive?

Draw a bar showing 156 split into 12 equal parts

Equation: _______ ÷ _______ = _______

Show your work: _______________________________________________

Answer: _______________________________________________

Problem 3 — Multi-Step: Part-Whole with Extra Information

Harriet Tubman made 13 trips on the Underground Railroad and guided approximately 70 freedom seekers to liberty. If she brought 4 people on each of the first 8 trips, how many did she bring on the remaining 5 trips combined?

This problem requires two bar models and two equations. Show all steps.

Step 1 — First 8 trips:

Bar model for Step 1

Equation 1: _______________

Step 2 — Remaining trips:

Bar model for Step 2

Equation 2: _______________

Final answer: _______________________________________________

Problem 4 — Fractions with Bar Models

A community savings fund has $240. Three-fourths of it is for housing repairs. One-fourth is for education supplies. How much money is for housing? How much for education?

Draw a bar divided into 4 equal parts. Label each part. Show the amounts.

Housing (3/4): $ _______

Education (1/4): $ _______

Check: Do both amounts add to $240? _____ Show: _______ + _______ = _______

Problem 5 — Comparison with Difference

In 1865, the Freedmen's Bureau helped establish 4,329 schools for formerly enslaved people. By 1870, there were 9,503 schools. How many more schools existed in 1870 than 1865?

Draw two bars and mark the difference

Equation: _______________________________________________

Answer: _______________________________________________

Problem 6 — Scholar-Created Problem

Create a two-step word problem that involves multiplication and subtraction. Your problem must use a real context (money, food, books, distance, people). Draw both bar models and solve.

Write your problem here:

Step 1 model

Equation 1: _______________

Step 2 model

Equation 2: _______________

Answer: _______________________________________________

Aurora Pause — Grades 3-5

Madam C.J. Walker (1867-1919) was the first self-made female millionaire in American history. She employed 40,000 people across her company. If she paid each employee an average of $3 per week and there are 52 weeks in a year, what was her approximate annual payroll? Draw the bar model and show your work.

Bar model

Work: _______________________________________________

Annual payroll: $ _______________

Grade Band: 6-8 — Ratios, Proportions, Percents, and Algebraic Thinking

Grades 6-8

Worked example — Percent bar model:
A community store sold $800 in goods. 35% of sales were produce. How much money came from produce sales?

Draw a bar representing $800 (the whole = 100%). Mark 35% of the bar.
35% = 35/100 = 0.35
0.35 × $800 = $280 came from produce sales.

Problem 1 — Ratios and Proportional Reasoning

A recipe for jollof rice serves 4 people and requires 2 cups of rice, 3 cups of tomato, and 1 cup of onion. Complete the ratio table for different serving sizes:

Servings	Cups of Rice	Cups of Tomato	Cups of Onion
4	2	3	1
8
12
20
2

The ratio of rice to tomato is always _______ : _______. Explain why this ratio must stay the same: _______________________________________________

Problem 2 — Percent: Finding the Part

The Howard University Class of 1956 had 340 graduates. 60% became doctors, lawyers, or teachers. 25% went into business. The remaining percent entered other fields. How many graduates went into each category?

Draw a percent bar model showing the 340 graduates divided into the three groups

Doctors/lawyers/teachers (60%): _______

Business (25%): _______

Other (___%): _______ Show how you found the percent for "other": _______________________________________________

Problem 3 — Percent: Finding the Whole

A co-op store collected $630 in dues, which represents 70% of its annual dues goal. What is the full annual dues goal?

Draw a bar where $630 = 70% of the whole. Find the whole.

Setup: 70% of ? = $630

Work: _______________________________________________

Annual goal: $ _______________

Problem 4 — Algebraic Thinking: Writing and Solving Equations

A scholar earns the same hourly wage tutoring younger students. Last week she tutored for 6 hours and earned $45. This week she tutored for 9 hours. How much did she earn this week?

Let w = hourly wage. Write the equation from last week's information: _______________________________________________

Solve for w: w = $ _______

Write the equation for this week: _______________________________________________

This week's earnings: $ _______

Problem 5 — Multi-Step with Percents and Operations

Documented fact: The GI Bill (1944) gave veterans access to college loans and home loans. However, banks in the South routinely denied these loans to Black veterans. Of the 3,229 VA home loans approved in Mississippi and Georgia in 1947, only 2 went to Black veterans. What percentage of loans went to Black veterans (round to the nearest hundredth of a percent)? What does this number document?

Calculation: _______________________________________________

Percent: _______% Show your division: _______________________________________________

What this number documents: _______________________________________________

Problem 6 — Unit Rates and Proportional Comparison

Compare two community stores: Store A sells 5 pounds of rice for $3.75. Store B sells 8 pounds of rice for $5.60. Which store has the better unit price per pound? Show the calculation for both and write a recommendation.

Store A unit price: $_______ per pound Store B unit price: $_______ per pound

Better deal: _______ Recommendation: _______________________________________________

Aurora Pause — Grades 6-8

The Tuskegee Airmen flew 1,578 missions during World War II and had a documented record of never losing a bomber they were assigned to escort — the only fighter group of the war to hold this record. If they flew an average of 4.7 missions per day, approximately how many days were they actively flying missions? Round to the nearest whole number. Show your work using a bar model where appropriate.

Bar model or calculation space

Work: _______________________________________________

Approximate days: _______

Self-Assessment — All Grade Levels

After completing your grade band's problems, answer honestly:

Which problem was most difficult? _______ What made it difficult? _______________________________________________

Did you draw the bar model BEFORE writing the equation every time? YES MOST TIMES NO

If no: on which problem did you skip the model? _______ What happened when you tried to solve without it? _______________________________________________